Simulations of Interfacial Dynamics of Complex Fluids Using Diﬀuse Interface Method with Adaptive Meshing by Chunfeng Zhou B.Sc., Peking University, 2001 M.Sc., Peking University, 2003 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Chemical and Biological Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2008 c Chunfeng Zhou 2008 Abstract A diﬀuse-interface ﬁnite-element method has been applied to simulate the ﬂow of twocomponent rheologically complex ﬂuids. It treats the interfaces as having a ﬁnite thickness with a phase-ﬁeld parameter varying continuously from one phase to the other. Adaptive meshing is applied to produce ﬁne grid near the interface and coarse mesh in the bulk. It leads to accurate resolution of the interface at modest computational costs. An advantage of this method is that topological changes such as interfacial rupture and coalescence happen naturally under a short-range force resembling the van der Waals force. There is no need for manual intervention as in sharp-interface model to eﬀect such event. Moreover, this energy-based formulation easily incorporates complex rheology as long as the free energy of the microstructures is known. The complex ﬂuids considered in this thesis include viscoelastic ﬂuids and nematic liquid crystals. Viscoelasticity is represented by the Oldroyd-B model, derived for a dilute polymer solution as linear elastic dumbbells suspended in a Newtonian solvent. The Leslie-Ericksen model is used for nematic liquid crystalswhich features distortional elasticity and viscous anisotropy. The interfacial dynamics of such complex ﬂuids are of both scientiﬁc and practical signiﬁcance. The thesis describes seven computational studies of physically interesting problems. The numerical simulations of monodisperse drop formation in microﬂuidic devices have reproduced scenarios of jet breakup and drop formation observed in experiments. Parametric studies have shown dripping and jetting regimes for increasing ﬂow rates, and elucidated the eﬀects of ﬂow and rheological parameters on the drop formation process and the ﬁnal drop size. A simple liquid drop model is used to study the neutrophil, the most common type of white blood cell, transit in pulmonary capillaries. The cell size, viscosity and rheological properties are found to determine the transit time. A compound drop model is also employed to account for the cell nucleus. The other four cases concern drop and bubble dynamics in nematic liquid crystals, as determined by the coupling among interfacial anchoring, bulk elasticity and anisotropic viscosity. In particular, the simulations reproduce unusual bubble shapes seen in experiments, and predict self-assembly of microdroplets in nematic media. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . xii 1 Introduction . . . . . . . . . . . . 1.1 Interfacial dynamics . . . . . . 1.2 Complex ﬂuids . . . . . . . . . 1.3 Methodology . . . . . . . . . . 1.3.1 Constitutive equations . 1.3.2 Diﬀuse interface method 1.3.3 Finite element method . 1.3.4 Adaptive meshing . . . 1.4 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 3 3 4 7 8 11 2 Formation of simple and compound drops in microﬂuidic devices 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Theory and numerical methods . . . . . . . . . . . . . . . . . . . . . 2.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Drop formation at an oriﬁce . . . . . . . . . . . . . . . . . . . 2.3.2 Formation of simple drops in a ﬂow-focusing device . . . . . . 2.3.3 Formation of compound drops in a ﬂow-focusing device . . . 2.3.4 Eﬀects of viscoelasticity . . . . . . . . . . . . . . . . . . . . . 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 15 17 18 18 21 32 37 40 42 3 Simulation of neutrophil deformation and transport in 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Theory and numerical methods . . . . . . . . . . . . . . 3.3 Simulations using the simple Newtonian drop model . . 3.3.1 The process of cell deformation and entrance . . 3.3.2 The entrance time . . . . . . . . . . . . . . . . . 3.3.3 Eﬀect of cytoplasmic viscosity . . . . . . . . . . 3.3.4 Eﬀects of capillary diameter and geometry . . . . . . . . . . . . . . . . . . . . 47 47 49 51 53 55 58 60 iii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . capillaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 64 67 69 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 74 75 76 81 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 86 87 88 91 92 rise of bubbles and drops in a nematic liquid crystal Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . Theory and numerical method . . . . . . . . . . . . . . . . Results and discussion . . . . . . . . . . . . . . . . . . . . . 6.3.1 Simple shear ﬂow as validation . . . . . . . . . . . . 6.3.2 Static orientational defects . . . . . . . . . . . . . . 6.3.3 Flow-induced transformation of defect conﬁguration 6.3.4 Rising velocity, drag force and the ﬂow ﬁeld . . . . . 6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 95 97 100 100 101 102 108 114 115 nematic liquid crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 119 121 124 127 135 141 142 . . . . . . . . . . 145 145 147 148 150 3.4 3.5 3.6 3.3.5 Comparison with experiment Viscoelastic eﬀects . . . . . . . . . . Summary . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . 4 Deformation of a compound drop through 4.1 Introduction . . . . . . . . . . . . . . . . 4.2 Theory and numerical methods . . . . . . 4.3 Numerical results . . . . . . . . . . . . . . 4.4 Summary . . . . . . . . . . . . . . . . . . 4.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Droplet interaction and self-assembly 7.1 Introduction . . . . . . . . . . . . . 7.2 Theory and numerical methods . . . 7.3 Results and discussion . . . . . . . . 7.3.1 Pairwise interactions . . . . . 7.3.2 Multi-drop interactions . . . 7.4 Summary . . . . . . . . . . . . . . . 7.5 Bibliography . . . . . . . . . . . . . in a . . . . . . . . . . . . . . . . . . . . . 8 Dynamic simulation of capillary breakup 8.1 Introduction . . . . . . . . . . . . . . . 8.2 Theory and numerical methods . . . . . 8.3 Results and discussion . . . . . . . . . . 8.3.1 Bulk elasticity . . . . . . . . . . iv . . . . . . . . . . . . a contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Heart-shaped bubbles rising in anisotropic liquids 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . 5.2 Theory and numerical methods . . . . . . . . . . . 5.3 Results and discussion . . . . . . . . . . . . . . . . 5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . 5.5 Bibliography . . . . . . . . . . . . . . . . . . . . . 6 The 6.1 6.2 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . of nematic ﬁbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table of Contents 8.3.2 Interface anchoring . 8.3.3 Anisotropic viscosity 8.4 Summary . . . . . . . . . . 8.5 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusions and recommendation . . . . . 9.1 Theoretical model and numerical algorithm 9.2 Physical insights . . . . . . . . . . . . . . 9.3 Theoretical and numerical limitations . . 9.4 Recommendations for future work . . . . 9.5 Bibliography . . . . . . . . . . . . . . . . v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 156 159 161 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 164 165 165 166 168 List of Tables 8.1 Wavelength and pinchoﬀ time at various surface anchoring energies, with AK = 0.833, Ca = 1 and Re = 150. Time is made dimensionless by ηa/σ. The last column, for an isotropic Newtonian ﬂuid with a viscosity equal to the average viscosity η of the LC’s, will be cited in the next subsection. 155 vi List of Figures 1.1 1.2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 Molecular origin of interfacial tension. . . . . . . . . . . . . . . . . . . . . An unstructured triangular mesh generated by GRUMMP with interfacial reﬁnement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Formation and detachment of a drop at the tip of a capillary. The drop and matrix ﬂuids are both Newtonian, with density ratio α = 2 and viscosity ratio β = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the drop size between our simulation and Wilkes et al. (1999). α = 2, β = 1. Bo is varied by using diﬀerent ﬂuid densities and interfacial tension; W e varies between 1.90 × 10−4 and 2.68 × 10−4 , and Ca between 2.87 × 10−5 and 3.55 × 10−3 . . . . . . . . . . . . . . . . . . . Variation of the minimum neck radius Rn on the thread with time t for the simulation in Fig. 2.2. . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric setup for simulating drop formation in a ﬂow-focusing device. Shown is the meridian plane of the axisymmetric device. . . . . . . . . . Snapshots of drop formation in the dripping regime at Ca = 1.70 × 10−3 , W e = 5.75 × 10−4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The radius of the neck, measured at its thinnest part, oscillates in time before dropping to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of the capillary wave inside a long nozzle. All other parameters are the same as Fig. 2.5. (a) Geometry of the domain; (b) radius of the thread as a function of the axial distance from the inlet of the nozzle to the outlet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dependence of the drop radius rd on the ﬂow-rate ratio Γ. (a) Eﬀect of varying one ﬂow rate while keeping the other ﬁxed. (b) Comparison between two ﬁxed Qi values while Qo varies. . . . . . . . . . . . . . . . . The drop radius rd increases with the radius of the downstream collection tube re . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The velocity ﬁeld near the drop at (a) tVi /a = 6.59 and (b) tVi /a = 6.67 shortly before pinch-oﬀ. . . . . . . . . . . . . . . . . . . . . . . . . . . . A cycle of drop formation in the jetting regime at Ca = 1.06 × 10−2 , W e = 2.25 × 10−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tipstreaming at Ca = 5.36 × 10−4 , W e = 5.75 × 10−5 , Γ = 1.89 × 10−2 , α = 1.11, β = 0.167. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometric setup for simulating the formation of compound drops. Shown is the meridian plane of an axisymmetric device. . . . . . . . . . . . . . . Snapshots showing the diﬀerent stages of a successful encapsulation process. Ca = 4.50 × 10−3 , W e = 3.24 × 10−4 . . . . . . . . . . . . . . . . . . vii 2 10 20 21 22 23 24 24 25 26 27 28 28 31 32 34 List of Figures 2.15 Compound drop formation is sensitive to the ﬂow rates. The ﬂow-rate ratios are kept ﬁxed at 3 : 6 : 40, and the three streams have equal density and viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Phase diagram for compound drop formation. The inner and the outer ﬂuids are identical, and the viscosity ratio β is between the middle and the outer ﬂuid. Ca is deﬁned for the ﬂow in the inner tube. The innermiddle-outer ﬂow-rate ratios are ﬁxed at 3 : 6 : 40. . . . . . . . . . . . . . 2.17 Simple drop formation in the dripping regime when the inner ﬂuid is (a) viscoelastic, and (b) Newtonian. The snapshots are taken shortly before the drop detaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.18 Simple drop formation in the jetting regime when the inner ﬂuid is (a) viscoelastic and (b) Newtonian. . . . . . . . . . . . . . . . . . . . . . . . 2.19 Viscoelastic eﬀects on compound drop formation. All 3 ﬂuids are Newtonian in (a). The middle ﬂuid is viscoelastic in (b) and (c), two snapshots following pinch-oﬀ of the inner jet. . . . . . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 The geometric setup for simulating a neutrophil’s entrance into a capillary. Two cylindrical tubes are connected by an arc of 90◦ . . . . . . . . . . . . (a) The microchannel of Yap & Kamm (2005b). The scale bar is 100 μm, and the two arrows indicate the microchannel and the reservoir. (b) Schematic showing the dimensions of the microchannel. Its cross-section is rectangular with a width of 5 μm and a depth of 2.5 μm. After Yap & Kamm (2005b); c 2005 the American Physiological Society. . . . . . . . . Snapshots of the neutrophil during its entrance into the capillary. Ca = 0.0893, β = 3 and ζ = 1.4. . . . . . . . . . . . . . . . . . . . . . . . . . . Variations of the ﬂow rate Q and cell length l during the entrance process. Ca = 0.0893, β = 3 and ζ = 1.4. . . . . . . . . . . . . . . . . . . . . . . . Log-log plot of the dimensionless entrance time τent as a function of the capillary number Ca. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The cell length le at the end of the entry process as a function of the capillary number Ca for ζ = 1.4 and β = 3. . . . . . . . . . . . . . . . . . The ﬂow rate (a) and cell length (b) during cell entry for three viscosity ratios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The entrance time as a function of the viscosity ratio. The matrix viscosity is kept ﬁxed while the cell viscosity is varied. ζ = 1.4. . . . . . . . . . . . Eﬀects of the capillary size on (a) the entrance time τent and (b) the ﬂow rate Qe at the end of the cell entry. Ca = 0.0893 and β = 3. . . . . . . . A snapshot of a neutrophil entering the capillary with an elliptic obstacle after the entrance. Ca = 0.0893, β = 3 and ζ = 1.4. . . . . . . . . . . . . The entrance time as a function of the obstacle height h. The correlation of Bathe et al. (2002) is also shown for comparison. . . . . . . . . . . . . Comparison of the dimensional entrance time with experimental data. A pressure drop ΔP = 100 Pa corresponds to Ca = 0.0893. . . . . . . . . . viii 35 36 38 39 40 51 52 54 54 55 57 58 59 60 61 62 63 List of Figures 3.13 The entrance time τent decreases with the Deborah number De when the relaxation time λH increases. . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 The streamline pattern and contours of the shear stress τpxy inside the cell during its entrance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Eﬀects of the Deborah number on the temporal evolution of the ﬂow rate. Ca = 0.0893, β = 3 and ζ = 1.4. . . . . . . . . . . . . . . . . . . . . . . . 3.16 The ratio of entrance times between the viscoelastic cell and the Newtonian one increases with the ﬂow rate, indicated by Ca. . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5 Geometric setup for simulating the deformation of a compound drop through a 2:1 contraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Snapshots of the transit of a compound drop into the capillary. Ca = 0.179, Re = 1.56 × 10−2 and ζc = 0.72. . . . . . . . . . . . . . . . . . . . Temporal variations of the instantaneous ﬂow rate Q and the length of the compound drop l for the process of Fig. 4.2. . . . . . . . . . . . . . . . . (a) Eﬀect of the inner drop on the transit time with changing pressure drop with ζc = 0.72. The Reynolds number varies in the range 7.81 × 10−3 ≤ Re ≤ 7.81 × 10−2 . (b) Transit time as a function of the core radius, represented by ζc , at Ca = 0.179 and Re = 1.56 × 10−2 . . . . . . . . . . . Flow ﬁelds inside the compound drop toward the end of the entry process for two core radii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 66 66 67 76 78 79 80 80 5.1 An air bubble rising in the wake of a falling steel ball in a micellar solution. The bubble volume is roughly 2 cm3 and the images are separated by 33 ms. 87 5.2 Equilibrium shapes of stationary bubbles in a nematic with a vertical farﬁeld orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 Snapshots of the rising bubble in a nematic with planar anchoring. The far-ﬁeld molecular orientation is vertical. . . . . . . . . . . . . . . . . . . 90 6.1 6.2 6.3 6.4 6.5 Simple shear ﬂow of a nematic with homeotropic anchoring on the walls. The director orientation is indicated by θ(y), and the velocity v(y) deviates from a linear proﬁle because of the anisotropic viscosity. . . . . . . . . . Comparison between our results and the 1D exact solution of Carlsson (1984). (a) The director orientation proﬁle; (b) the velocity proﬁle. . . . Defect conﬁgurations near a drop with homeotropic anchoring. (a) The satellite point defect, indicated by the black dot, within the n ﬁeld. Aσ = 0.05, AK = 100. (b) The ﬁnite-element mesh for (a) is reﬁned around the interface and the satellite defect. (c) Drawing of the director ﬁeld for a Saturn-ring defect. (d ) The surface-ring defect for Aσ = 0.05, AK = 100, indicated by black dots on the equator of the drop. . . . . . . . . . . . . Position of the satellite point defect near a stationary drop with homeotropic anchoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational domain for a Newtonian drop rising in a nematic whose far-ﬁeld orientation is vertical. . . . . . . . . . . . . . . . . . . . . . . . . ix 100 101 103 104 104 List of Figures 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 A “phase diagram” of steady-state defect conﬁgurations. AK and Er are varied by tuning K and g, respectively. . . . . . . . . . . . . . . . . . . . Steady-state position of the defect near a rising drop as a function of the Ericksen number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Director orientation around a steadily rising oblate drop in zone VI, with a small surface ring indicated by two black dots in the rear of the drop. . Transient rising velocity of a drop in a nematic liquid crystal with (a) planar anchoring and (b) homeotropic anchoring. . . . . . . . . . . . . . Director and ﬂow ﬁelds around the drop with homeotropic anchoring at three stages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Terminal velocity U of the drops in Fig. 6.9 as aﬀected by viscous anisotropy. The range of U corresponds to 4.93 < Re < 10.9 and 12.1 < Er < 26.8 for planar anchoring, and 4.29 < Re < 6.64 and 10.5 < Er < 16.3 for homeotropic anchoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . The drag coeﬃcient for drops rising in a nematic as a function of Re or Er. For homeotropic anchoring, the data correspond to the satellite conﬁguration. η1 /η2 = 3, Aσ = 0.2, AK = 40. . . . . . . . . . . . . . . . Non-spherical drop shapes produced by the nearby defects. Eo = 0.3, M o = 5.56 × 10−5 , Aσ = 0.5, AK = 15, η1 /η2 = 4.7. . . . . . . . . . . . . The geometry of the computational domain. The radius of the drops is a. With axisymmetry around the z axis, the domain is half of the meridian plane. For 2D computations, the domain is the entire rectangular box. . Representation of the director ﬁeld. (a) Birefringent pattern for an axisymmetric n ﬁeld through crossed polarizers; (b) Grayscale representation of n in 2D geometries with contours of (n2x −1/2)2 . White indicates a vertical or horizontal n, while black means a 45◦ tilt. . . . . . . . . . . . . . . . Attraction between a longitudinal pair of drops in the parallel conﬁguration. (a) The center-to-center distance R as a function of time. (b) The attraction force F (R), computed by integrating the elastic stress around the drop, compared with prior experimental data and theoretical results. Interaction between a parallel longitudinal pair of droplets with an initial separation smaller than the equilibrium value. . . . . . . . . . . . . . . . The repulsion between an anti-parallel longitudinal pair. (a) The separation R(t) increases from an initial value of 2.5a. (b) The repulsion force F has a long range and decays slowly with R. . . . . . . . . . . . . . . . Experimental observation of the repulsion between an anti-parallel pair of droplets. The insets are images through crossed polarizers. . . . . . . . . The interaction between a lateral pair of droplets in the parallel arrangement. (a) The center-to-center distance R(t) increases as the droplets repel each other. (b) The repulsion force F as a function of R in the dynamic simulation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The interaction between a lateral pair of droplets in the anti-parallel arrangement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x 106 107 107 109 110 111 112 113 125 127 128 130 130 131 133 134 List of Figures 7.9 Four droplets assemble into a vertical chain in a vertically aligned nematic. The height of the domain is 12.5a with periodic conditions in the vertical direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.10 Interaction between two chains in the parallel arrangement. (a) Separation between the second drops from the top. (b) Repulsion force on the second drop from the top of the right chain. . . . . . . . . . . . . . . . . . . . . 7.11 Interaction between two chains of droplets in the anti-parallel arrangement. (a) The separation between the chains R(t) decreases from an initial value of 2.9a to an equilibrium one of 2.44a. (b) The attractive force F as a function of R in the dynamic simulation. . . . . . . . . . . . . . . . . . 7.12 Self-assembly of 8 drops in a doubly periodic domain. Note that the time is made dimensionless by ηa2 /K. . . . . . . . . . . . . . . . . . . . . . . 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Schematic of the computational domain, which is half of the meridian plane of the axisymmetric geometry. . . . . . . . . . . . . . . . . . . . . Eﬀect of bulk elasticity on capillary waves on a nematic ﬁber. The domain is 25a in length and 6.66a in width. The viscosity coeﬃcients are those of MBBA, and surface anchoring is ﬁxed at AW = 1. . . . . . . . . . . . . . Evolution of the interface and the director ﬁeld during breakup. AW = 1.0, AK = 0.0167. The snapshots are at diﬀerent times: (a) t = 476, (b) 502, (c) 507, (d ) 512, (e) 566, (f ) 658. . . . . . . . . . . . . . . . . . . . . . . Thinning of the minimum neck radius at diﬀerent levels of bulk elasticity. Eﬀect of AW on the director ﬁeld (a) inside the ﬁber during breakup and (b) inside a daughter drop. . . . . . . . . . . . . . . . . . . . . . . . . . . Eﬀect of viscous anisotropy on the thinning of the neck during the breakup of nematic ﬁbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axial velocity proﬁles vz (z) at two radial positions r = 0.2a and 0.65a at dimensionless time t = 499. AW = 1, AK = 0.833, and the viscosities correspond to ν = 4.34 in Fig. 8.6(a). . . . . . . . . . . . . . . . . . . . . xi 136 136 138 139 149 151 153 154 156 157 158 Statement of Co-Authorship This thesis contains seven co-authored and self-contained manuscripts that are either published in or submitted to peer-reviewed journals. Following is a list of contributions to the following areas: • Identiﬁcation and design of research program The identiﬁcation and the design of the research program for this thesis has been decided by the author in collaboration with and supervised by Professor James J. Feng. • Performing the research The research for this thesis has been performed entirely by the author. Discussion with group members and the supervisor took place at regular group meetings. • Data analyses The data analyses for the current research has been performed by the author under the direct supervision of the thesis advisor, Professor James J. Feng. • Manuscript preparation This thesis has been prepared and organized by the author under the direct supervision of Professor James J. Feng. The ﬁnal version has incorporated comments and suggestions by the supervisory committee and the examining committee. xii Chapter 1 Introduction 1.1 Interfacial dynamics The interface separating two immiscible ﬂuids such as water and oil plays a central role in the ﬂuid dynamics of the mixture. This may be demonstrated by stirring an oil drop in a cup of water. With gentle stirring, the oil drop moves about on the water surface. Stronger stirring causes the oil drop to break up into many smaller droplets. After the stirring stops, the droplets gradually coalesce and in time reform a single oil drop. This simple process reveals three key interfacial phenomena: interfacial movement, breakup and coalescence. Interfacial dynamics refers to the motion and topological change of the interface subject to ﬂows in the bulk phases or external forces due to electric and magnetic ﬁelds. In equilibrium, the oil drop assumes a round shape because of surface tension or interfacial tension, which is a result of the cohesive forces between similar molecules in each component. This is illustrated by Fig. 1.1. The interior molecules have all the same kind of molecules around them. However, those on the surface also have the dissimilar molecules as neighbors, and this reduces the attractive force on them and puts those surface molecules in an unfavorable energy state. In other words, it takes work to move a molecule from the interior onto the interface because of the cohesive force. This interfacial energy can also be seen as a tension against the creation of interfacial area, and is known as the interfacial tension. If the interface is between a liquid and air, the tension is also called a surface tension. This interfacial force adds another dimension to the ﬂuid dynamics and plays a signiﬁcant role in many interfacial phenomena. Many industrial process involve the control and manipulation of the interface. Familiar examples include coating, polymer coextrusion and foaming. Interfacial ﬂows in microﬂuidic devices have recently gained much attention (Stone et al., 2004), with intriguing applications such as the generation of monodisperse single and double emulsions (Anna et al., 2003; Nisisako et al., 2003; Utada et al., 2005). The signiﬁcance of the interfacial dynamics in microﬂuidics is easy to understand from a scaling standpoint. The small length-scale gives rise to large ratios of surface area to the bulk volume, and therefore highlights the role of interfacial eﬀects. The next chapter contains further details on simple and compound drop formation in microﬂuidic devices. To predict the behavior of the interface, one needs to solve the ﬂuid dynamics of both ﬂuids in the bulk, as well as the evolution of the interface. The interfacial tension gives rise to a normal force proportional to the curvature of the interface, which exerts its eﬀects on the ﬂow ﬁeld. This makes interfacial ﬂows a diﬃcult task since the interfacial forces are nonlinear, even in the case of inertialess creeping ﬂows. Besides the nonlinearity of the interfacial dynamics, the two ﬂuids typically have mismatched densities, viscosities and other physical properties. These also contribute to the diﬃculty of analyzing interfacial 1 Chapter 1. Introduction Figure 1.1: Molecular origin of interfacial tension. ﬂows. Therefore, only a few theoretical solutions exist for interfacial ﬂows; they are the linear theory of the breakup of inﬁnite cylindrical ﬂuid threads (Rayleigh, 1879; Tomotika, 1935; Mikami & Mason, 1975), and a self-similarity theory (Eggers, 1997) that yields scaling arguments. In recent years, computer simulation has emerged as an important tool for studying interfacial dynamics. On the one hand, numerical simulations can reveal fundamental physics of the process that is hard to obtain experimentally. On the other hand, they may also provide guidelines for design and fabrication in both experiments and industrial applications. 1.2 Complex ﬂuids Complex ﬂuids are microstructured ﬂuids. The evolution of the internal microstructures aﬀects the macrodynamics of the ﬂuids and vice versa. Examples include polymeric solutions and melts, liquid crystals, gels, suspensions, emulsions and micellar solutions (Larson, 1999). These materials often have unique industrial applications as the microstructure can be manipulated by means of ﬂow or other methods to achieve desired mechanical, optical or other physical properties. One such application is polymer blends (Utracki, 1990) made of incompatible components. Under certain ﬂow conditions, the dispersed phase is stretched into ﬁlaments. After solidiﬁcation, these long polymeric ﬁlaments turn into strong ﬁbers and give the materials great strength. Another example is polymer-dispersed liquid crystals (PDLC), in which liquid crystalline droplets are dispersed in a polymer matrix before they are solidiﬁed. It shows great potential in electro-optical applications (West, 1990); for example, one can control the opacity of a PDLC glass by manipulating the electric ﬁeld across it (Mucha, 2003). Complex ﬂuids often exhibit unique interfacial behavior. For example, a thread of Newtonian liquid breaks up into drops via the well-known Rayleigh instability. A thread of non-Newtonian polymer solution, on the other hand, develops the so-called bead-on2 Chapter 1. Introduction a-string morphology (Oliveira et al., 2006), with a chain of droplets connected by a thin liquid string. The string persists for long times without breaking because the polymer lends the liquid a great resistance against breakup. In fact, this strong polymer stress plays key roles in many other situations, such as in partial coalescence (Yue et al., 2006a) and drop formation in microﬂuidics (Zhou et al., 2006). Liquid crystals are another class of complex ﬂuids that show unusual interfacial dynamics. An anisotropic interfacial tension gives rise to non-spherical shapes for bubbles and drops dispersed in the liquid crystal (Zhou et al., 2007). These phenomena have motivated the research described in the following chapters. 1.3 Methodology Recently, we have developed a ﬁnite-element based algorithm with adaptive meshing that is capable of simulating the ﬂow of two-component systems of rheologically complex ﬂuids (Yue et al., 2006b). The interfaces are treated as having a ﬁnite thickness with a phase-ﬁeld parameter varying continuously from one phase to the other. Fluid properties such as density and viscosity, and ﬂow quantities such as pressure may change steeply yet continuously across the interfaces. Therefore no discontinuity appears in the system. The phase-ﬁeld parameter, which deﬁnes the position of the interfaces, is described by a mixing energy consisting of two components, one “hydro”-philic and the other “hydro”phobic. An advantage of this method is that topological changes such as interfacial rupture and coalescence happen naturally under a short-range force resembling the van der Waals force. There is no need for manual intervention as in sharp-interface models to eﬀect such events. Moreover, this energy-based formulation easily incorporates complex rheology as long as the free energy of the microstructures is known. Our theoretical model has two major features: constitutive equations for complex rheology and the diﬀuse interface method for the interface. The numerical algorithm has two main components as well: a ﬁnite element formulation and an adaptive mesh generator. In the following subsections, we brieﬂy describe these four items. 1.3.1 Constitutive equations Newtonian ﬂuids have a stress tensor that is proportional to the instantaneous strain rate tensor. Microstructured ﬂuids have a much more complex behavior since the microstructural conformation evolves in the ﬂow and the stress tensor generally depends on the strain history. Generally, the stress and strain-rate tensors are related by constitutive equations instead of an explicit functional form. This dissertation focuses on polymeric ﬂuids and liquid crystals as two important classes of complex ﬂuids. In the following, we brieﬂy describe the Oldroyd-B model for polymer solutions and the Leslie-Ericksen model for nematic liquid crystals. More detailed derivations are available in the reference books cited. The Oldroyd-B ﬂuid model describes a dilute polymer solution as linear elastic dumbbells suspended in a Newtonian solvent. The elastic stress τd is given by the Maxwell 3 Chapter 1. Introduction equation (Bird et al., 1987a) τ d + λH τ d(1) = 2μp D, (1.1) where λH is the relaxation time that may range from microseconds to seconds, μp is the polymer viscosity and D = [∇v + (∇v)T ]/2 is the strain-rate tensor. μp /λH gives the elastic modulus for the dumbbells, and the subscript (1) denotes the upper convected derivative (Bird et al., 1987a): A(1) = dA − (∇v)T · A − A · ∇v, dt (1.2) for a second-rank tensor A, ∇v being the velocity gradient tensor. The total stress τ is the sum of the polymer elastic stress and the solvent viscous stress. In terms of τ , the Oldroyd-B constitutive equation reads: τ + λH τ (1) = 2μ(D + λ2 D (1) ), (1.3) where λH is the total viscosity, and λ2 is the retardation time. The Leslie-Ericksen theory treats the liquid crystal molecules as rigid rods whose average orientation may be spatially distorted in three basic ways: splay, bend and twist. If we take equal contributions from these three elastic modes, an elastic stress can be written out (de Gennes & Prost, 1993): σ e = K∇n · (∇n)T (1.4) where K is an elastic constant and n is the molecular orientation. In addition to the elastic stress, the liquid crystal also has an anisotropic viscous stress σ = α1 D : nnnn + α2 nN + α3 N n + α4 D + α5 nn · D + α6 D · nn (1.5) where α1−6 are the Leslie viscous coeﬃcients, and N = ddtn − Ω · n is the rotation of liquid crystal molecules with respect to the background ﬂow ﬁeld with vorticity tensor Ω. In general, the molecules perpendicular to the ﬂow direction cause greater viscous stress than do those parallel to it. The orientation ﬁeld n evolves in the ﬂow ﬁeld according to a balance between elastic and viscous torques: K∇2 n = γ1 N + γ2 D · n. (1.6) Where γ1 = α3 − α2 and γ2 = α3 + α2 . Thus, the ﬂow and stress are coupled through the orientation ﬁeld n(x) and have to be solved together. 1.3.2 Diﬀuse interface method A straightforward way of handling the moving interfaces is to have a mesh that has grid points on the interfaces, and deforms according to the ﬂows on both sides of the boundary. This has been implemented in boundary integral and boundary element methods (Cristini 4 Chapter 1. Introduction et al., 1998; Toose et al., 1995; Khayat, 2000), ﬁnite-element methods (Hu et al., 2001; Ambravaneswaran et al., 2002; Hooper et al., 2001a,b; Kim & Han, 2001) and a ﬁnitediﬀerence method (Ramaswamy & Leal, 1999b,a). Keeping track of the moving mesh entails a computational overhead. Furthermore, large displacement of internal domains causes mesh entanglement as happens, for example, when one drop overtakes another. Most importantly, the moving-mesh methods cannot handle topological changes such as breakup and coalescence; the sharp interface formulation breaks down in such cases. As an alternative, ﬁxed-grid methods that regularize the interface have been quite successful in treating deforming interfaces. These include the volume-of-ﬂuid (VOF) method (Li & Renardy, 2000), the front-tracking method (Unverdi & Tryggvason, 1992) and the levelset method (Chang et al., 1996). Instead of formulating the ﬂow of two domains separated by an interface, these methods represent the interface by a scalar “order parameter” that obeys a convection equation. Then a single set of governing equations can be written over the entire domain, and solved on a ﬁxed grid in a purely Eulerian framework. The diﬀuse-interface model can be viewed as a physically motivated level-set method. Instead of choosing an artiﬁcial smoothing function for the interface, the diﬀuse-interface model describes the interface by a mixing energy. Thus, the structure of the interface is determined by molecular forces; the tendencies for mixing and separation are balanced through the nonlocal mixing energy. When the interfacial thickness approaches zero, the diﬀuse-interface model becomes identical to a sharp-interface level-set formulation. It also reduces properly to the classical sharp-interface model (Elder et al., 2001). Besides its handling of the interface, the method has another advantage in its easy incorporation of the complex rheology of microstructured ﬂuids. As long as such microstructures are described by an energy, it can be added into the mixing energy to form the total free energy of the multi-phase system. Then the formal variational procedure applied on the total free energy will lead to the proper constitutive equation for the microstructured ﬂuids. Thus, interfacial dynamics and complex rheology are handled in a uniﬁed theoretical framework. The free energy description The diﬀuse-interface method is based on an energy description. Introducing a phaseﬁeld variable φ which takes on values of ±1 in the two bulk components, the interface is indicated by the level-set of φ = 0. The interface stores a mixing energy. fmix (φ, ∇φ) = λ (φ2 − 1)2 , |∇φ|2 + 2 22 (1.7) where λ is the mixing energy density and is a capillary width indicative of the interfacial thickness. The mixing energy ensures that the interface neither spreads wide nor collapses into a singular surface. In the limit of → 0, λ/ gives the interfacial tension. By following the evolution of φ by a convection-diﬀusion equation (see Eq. 1.14 below), the interfacial motion and deformation are tracked. If one or both components are complex ﬂuids, additional terms have to be included in the total free energy. Take for example the blend of a Newtonian ﬂuid and a polymer so5 Chapter 1. Introduction lution described by the Oldroyd-B model. There is an elastic energy due to the dumbbell elasticity. If the dumbbells have an elastic constant H and a conﬁguration distribution Ψ(Q) about the connector vector Q, the average energy is (Bird et al., 1987b) fd = R3 1 kT ln Ψ + HQ · Q ΨdQ, 2 (1.8) where k is the Boltzmann constant, T is the temperature, and the integration is over all possible conﬁgurations of Q. Now the total free energy density of the two-phase system is: 1+φ f = fmix + nfd , (1.9) 2 where n is the number density of the dumbbells. Note that φ = 1 in the polymer solution and φ = −1 in the Newtonian ﬂuid. A similar free energy can be written out for a Newtonian-liquid crystal system. For brevity, however, we only write the equations based on Eq. 1.9 in the following. Details on the liquid crystalline mixture can be found in Yue et al. (2004) and chapter 5 and 6. Governing equations The general procedure for deriving the governing equations have been outlined in our recent work (Yue et al., 2004, 2006b). In the following, therefore, we will simply list the equations for our particular system of a blend of a Newtonian ﬂuid and a polymer solution. The ﬁeld variables are velocity v, pressure p, phase function φ and polymer stress τd. We write the continuity and momentum equations in the usual form: ∇ · v = 0, ρ ∂v + v · ∇v ∂t = −∇p + ∇ · σ, (1.10) (1.11) where σ is the stress tensor consisting the viscous stress, interfacial stress σ i and the viscoelastic stress denoted as τ d due to the dumbbell energy fd . The interfacial stress tensor is derived from the mixing energy following a variational procedure(Yue et al., 2004): (1.12) σ i = −λ(∇φ ⊗ ∇φ). τ d obeys the Maxwell equation: τ d + λH τ d(1) = μp [∇v + (∇v)T ], (1.13) which has been given earlier as Eq. 1.1. The phase-ﬁeld variable φ obeys the Cahn-Hilliard equation: ∂φ + v · ∇φ = γ∇2 G, ∂t 6 (1.14) Chapter 1. Introduction where G = λ[−∇2 φ + φ(φ2 − 1)/ 2 ] is often called the chemical potential, and γ is the interface mobility. The boundary conditions for the velocity and stress components depend on the geometric and physical setup of the problem, as is typical in the literature of non-Newtonian ﬂuid mechanics. For the phase-ﬁeld φ, the following conditions are used on the outer boundary of the domain: ∂φ = 0, ∂m ∂3φ = 0. ∂m3 (1.15) (1.16) Note that in the diﬀuse-interface framework, the phase boundaries are seen as part of the interior and require no boundary conditions. 1.3.3 Finite element method The ﬁnite element method is attractive since our applications involve complex geometries. Furthermore, the unstructured grid allows convenient adaptive meshing to reﬁne the interfacial region. The discretization of the governing equations follows the standard Galerkin formalism (Hu et al., 2001). However, the Cahn-Hilliard equation requires special attention. With C 0 elements, which are smooth within each element and continuous across their boundaries, one cannot represent spatial derivatives of higher order than 2. Thus the fourth-order Cahn-Hilliard equation has been decomposed into two second-order equations: γλ ∂φ + v · ∇φ = 2 Δ(ψ + sφ) (1.17) ∂t ψ = − 2 Δφ + (φ2 − 1 − s)φ, (1.18) where s is a positive number to enhance stability of the numerical method (Shen, 1995). For all the calculations reported hereafter, s = 0.5 is used. The chemical potential G in Eq. (1.14) is now simply G = λ2 (ψ + sφ). We seek the following weak solution: (v, p, τ d , φ, ψ) ∈ U × P × T × S × S. For 2D or 3D axisymmetric ﬂows, the solution spaces satisfy U ∈ H 1 (Ω)2 , P ∈ L2 (Ω), T ∈ L2 (Ω)3 ˜ ψ), ˜ we (L2 (Ω)4 for axisymmetric ﬂow) and S ∈ H 1 (Ω). Using basis functions (˜ v , p˜, τ˜, φ, write the following weak forms of the governing equations: ρ Ω ∂v ˜ + (−pI + σ) : ∇˜ + v · ∇v − g − G∇φ · v v dΩ = 0, ∂t (1.19) (∇ · v)˜ pdΩ = 0, (1.20) Ω Ω τ d + λH τ d(1) − μp ∇v + (∇v)T 7 : (˜ τ + αv · ∇˜ τ )dΩ = 0, (1.21) Chapter 1. Introduction γλ ∂φ + v · ∇φ φ˜ + 2 ∇(ψ + sφ) · ∇φ˜ dΩ = 0, ∂t Ω Ω ψ − (φ2 − 1 − s)φ ψ˜ − 2 ∇φ · ∇ψ˜ dΩ = 0, (1.22) (1.23) where σ consists of viscous stress and polymer stress. In Eq. (1.19), the interfacial stress tensor is replaced by an equivalent form G∇φ that can be seen as a body force. We have adopted the streamline upwind Petrov-Galerkin scheme for the constitutive equations to improve stability (Brooks & Hughes, 1982). In the derivation of the weak formulation, all surface integration has been neglected for simplicity, which implies that natural boundary conditions are satisﬁed. The boundary conditions can be thus summarized as: v (−pI + τ ) · m τd ∇φ · m = ∇ψ · m = = = = vg , 0, τ in , 0, on (∂Ω)u on (∂Ω)τ on (∂Ω)in on ∂Ω (1.24) (1.25) (1.26) (1.27) where ∂Ω = (∂Ω)u (∂Ω)τ and (∂Ω)u (∂Ω)τ = ∅, (∂Ω)in is the inﬂow boundary, and m is the unit normal to the boundary. The zero-ﬂux conditions in Eq. (1.27) help maintain volume conservation of the phases. On an unstructured triangular mesh (see next subsection for adaptive meshing), we have used piecewise quadratic (P2) elements for v, φ and ψ , and piecewise linear (P1) elements for p and τd. Second-order implicit methods are used for time discretization. Details of the discretization scheme can be found in Yue et al. (2006b). The non-linear algebraic system generated by Eqs. (1.19) to (1.23) is solved using an Inexact Newton’s method with backtracking (Eisenstat & Walker, 1996). Within each Newton iteration, the sparse linear system is solved by preconditioned Krylov methods (van der Vorst, 2000; Saad, 1996) such as the Generalized Minimum Residual (GMRES) method and the Biconjugate Gradient Stabilized (BCGSTAB) method, both of which turn out to have similar performance, although GMRES is expected to be more stable(Saad, 1996). ILU(0) and ILU(t) preconditioners (Saad, 1996; Benzi, 2002) are found to be robust for our simulations (Yue et al., 2006b; Zhou et al., 2006). By design, our grid size varies greatly from the interface to the bulk. This gives rise to a highly ill-conditioned sparse matrix and numerical diﬃculties. This is resolved by a scaling procedure whereby a diagonal scaling matrix is left-multiplied to the mass matrix prior to applying the aforementioned preconditioners. Thus, each row of the matrix is scaled by the inverse of the sum of the absolute values of the entries in that row before the linear system is sent to the preconditioned Krylov solvers. 1.3.4 Adaptive meshing As mentioned in the above subsection, the phase-ﬁeld model allows one to use a ﬁxed Eulerian mesh to capture moving internal boundaries. To achieve high accuracy at 8 Chapter 1. Introduction modest computational cost, we need a mesh with dense grids covering the interfacial region and coarser grids in the bulk. As the interface moves out of the ﬁne mesh, the mesh in front needs to be reﬁned while that left behind needs to be coarsened. Such adaptive meshing is achieved by using a general-purpose mesh generator GRUMMP, which stands for Generation and Reﬁnement of Unstructured Mixed-Element Meshes in Parallel (Freitag & Ollivier-Gooch, 1997). GRUMMP generates a mesh by using Delaunay reﬁnement, and allows control of the internal grid size by using a scalar ﬁeld. GRUMMP produces triangular elements in 2D and tetrahedral elements in 3D. GRUMMP controls the spatial variation of grid size using a length scale LS , which speciﬁes the intended grid size at each location in the domain. One determines the local grid size by: LS (p) = min |q − p| ls (p) LS (q i ) + i , min R Neighbors q i G (1.28) where the inner minimum is taken among all pre-existing neighboring points q i , and R and G are user-supplied constant resolution and grading parameters. In our framework, the grid size distribution is dictated by the need to resolve thin interfaces. Since the phase-ﬁeld variable φ is constant (±1) in the bulk but varies steeply across the interface, we can impose a prescribed small grid size h1 on the interface by making LS depend on |∇φ| on every node: 1 √ LS (x, y) = (1.29) |∇φ| C2 + h1∞ where h∞ is the mesh size in the bulk, and the constant C controls the mesh size in the interfacial region: h1 = LS |φ=0 ≈ C · , being the capillary width. In previous work, we have used C values between 0.5 and 1; h1 ≤ ensures that the thickness of the interface typically contains on the order of 10 grid points (Yue et al., 2004). Figure 1.2 shows an example of the mesh inside a square containing an ellipse. Note that the interface is enveloped in a ribbon of ﬁne grids. Depending on the normal velocity of the interface, it will approach the boundary of the “ﬁne-mesh ribbon” in a certain number of time steps, at which time GRUMMP is called to reﬁne and coarsen the neighboring regions in the front and rear, respectively. This ribbon is taken to comprise 3 layers of the ﬁnest triangles. When the interface goes beyond the second triangle from the middle, remeshing is invoked. The solution of the last time step is then projected onto the new grid in a scheme described by Hu et al. (2001) The speed at which the interface traverses the ﬁne-mesh ribbon also constrains the time step Δt. We require that the interface not advance more than a whole cell at one time step: Δt ≤ Δtint = h , All interfacial cells |v · n| min (1.30) where v is the interfacial velocity, n is the unit normal to the interface and h is the cell size along n. In the above, we have outlined the theoretical model and numerical methods common to all the applications that will be described in detail in the rest of the thesis. Each of the 9 Chapter 1. Introduction (a) (b) Figure 1.2: (a) An unstructured triangular mesh generated by GRUMMP with interfacial reﬁnement. The parameters are G = 3, outer boundary mesh size h2 = 0.5, interior mesh size h3 = h2 /2, and interfacial mesh size h1 = h2 /64. (b) Magniﬁed view of a portion of the interfacial region. The bold curve indicates the interface φ = 0, which is centered in a belt-like region of reﬁned triangles. following chapters is a self-contained manuscript that is either published in or submitted to peer-reviewed journals. Chapter two is a study on the simple and compound drop formation in microﬂuidic ﬂow-focusing devices. Chapter three models the neutrophil transit and transport in pulmonary capillaries. Chapter four is on compound drop deformation through a contraction. Chapter ﬁve to chapter six concern the rise of a bubble or drop in a liquid crystal. For Chapter ﬁve, we studied a unique “inverted-heart” shape of a bubble when it rises in the liquid crystal. Chapter six focuses on the transformation of the topological defects near the rising bubble or drop. Chapter seven simulates the self-assembly of many droplets dispersed in a liquid crystal matrix. Finally, chapter eight investigates capillary breakup of nematic ﬁbers, a problem of direct applications in self-reinforcing polymer composites. 10 Bibliography Ambravaneswaran, B., Wilkes, E. D. & Basaran, O. A. 2002 Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops. 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A. 2005 Monodisperse double emulsions generated from a microcapillary device. Science 308, 537–541. Utracki, L. A. 1990 Polymer Alloys and Blends. Munich: Hanser. van der Vorst, H. A. 2000 Krylov subspace iteration. Computing in Science and Engineering 2, 32–37. West, J. L. 1990 Polymer-dispersed liquid crystals. In Liquid-Crystalline Polymers (ed. R. A. Weiss & C. K. Ober), ACS Symp. Ser., vol. 435, chap. 32, pp. 475–495. Washington, D.C.: ACS. Yue, P., Feng, J. J., Liu, C. & Shen, J. 2004 A diﬀuse-interface method for simulating two-phase ﬂows of complex ﬂuids. J. Fluid Mech. 515, 293–317. Yue, P., Zhou, C. & Feng, J. J. 2006a A computational study of the coalescence between a drop and an interface in Newtonian and viscoelastic ﬂuids. Phys. Fluids 18, 102102. Yue, P., Zhou, C., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2006b Phase-ﬁeld simulations of interfacial dynamics in viscoelastic ﬂuids using ﬁnite elements with adaptive meshing. J. Comput. Phys. 219, 47–67. Zhou, C., Yue, P. & Feng, J. J. 2006 Formation of simple and compound drops in microﬂuidic devices. Phys. Fluids 18, 092105. 13 Bibliography Zhou, C., Yue, P., Feng, J. J., Liu, C. & Shen, J. 2007 Heart-shaped bubbles rising in anisotropic liquids. Phys. Fluids 19, 041703. 14 Chapter 2 Formation of simple and compound drops in microﬂuidic devices ∗ 2.1 Introduction The recent upsurge in microﬂuidic research has arisen from the conﬂuence of maturing microfabrication techniques and the increasing need to manipulate small amounts of ﬂuids, often for the purpose of fast and inexpensive analysis of biological and chemical samples (Stone & Kim, 2001; Stone et al., 2004; Thilmay, 2005; Squires & Quake, 2005). An important issue in microﬂuidics is the control of ﬂuid interfaces, in particular the generation and manipulation of droplets (Atencia & Beebe, 2005; Joanicot & Ajdari, 2005). From a fundamental standpoint, the signiﬁcance of interfacial dynamics in microﬂuidics is easy to understand. The miniaturization of ﬂow geometries greatly increases the ratio of surface area to volume, therefore accentuating the role of interfacial forces. In practical applications, droplets in microchannels form the centerpiece of many lab-on-a-chip devices. For instance, a variety of mechanisms can be employed to move a droplet as a means of pumping ﬂuids (Stone et al., 2004; Laser & Santiago, 2004). Another oft-cited application is the use of droplets as chemical reactors where the kinetics can be monitored and controlled very precisely (Bringer et al., 2004). Recently, droplets have also been used as templates for producing microparticles with specialized biological, chemical and optical properties (Lu et al., 2004; Dendukuri et al., 2005; Fernandez-Nieves et al., 2005). The application of the greatest relevance to this paper is the use of microﬂuidic channels for producing highly monodisperse emulsions. Traditionally, emulsions are made by shearing immiscible ﬂuids in macroscopic mixers (Becher, 2001). The droplets thus produced have a wide range of sizes, and the size distribution is often poorly controlled. Recently, two types of microﬂuidic devices have been developed for producing droplets with precisely controlled sizes: the T-junction and the ﬂow-focusing device (Stone et al., 2004). Integration of multiple units allows the eﬃcient production of monodisperse emulsions consisting of a great number of droplets (Link et al., 2004; Fernandez-Nieves et al., 2005). Moreover, compound drops may be produced either using a two-step procedure of two T-junction units in tandem, with alternating surface wetting properties (Okushima et al., 2004), or by clever use of multiple injection ports in a one-step procedure (Utada et al., 2005). These allow one to engineer double emulsions with much more precisely controlled dimensions and properties than those produced using conventional methods (Grossiord & Seiller, 1998; Goubault et al., 2001), with potential impacts in drug delivery, controlled release and other applications (Olivieri et al., 2003). ∗ A version of this chapter has been published. Zhou, C., Yue, P. & Feng, J. J., Formation of simple and compound drops in microﬂuidic devices, Phys. Fluids 18, 092105 (2006) 15 Chapter 2. Formation of simple and compound drops in microﬂuidic devices In comparison with the rapid experimental advances, very little theoretical or numerical work has been done on interfacial dynamics in microﬂuidic channels. Aside from scaling arguments on the drop size and conditions for drop formation (Utada et al., 2005; Umbanhowar et al., 2000), we know of only one numerical simulation of drop formation in the ﬂow-focusing device (Davidson et al., 2005). Our current knowledge about the working of various devices is mostly empirical. Many fundamental questions on interfacial behavior in small dimensions and close proximity of bounding walls remain to be answered. For example, what physical mechanisms govern the process? Do drops form through the same capillary instability as operates in macroscopic scales? How does the ﬁnal drop size depend on various geometric, material and ﬂow parameters? To answer such questions, systematic theoretical and numerical studies need to be carried out. Moreover, a coherent understanding of micro-scale interfacial ﬂows is essential for a rational design and manufacturing strategy. Thus, this computational study was motivated not only by the technological potential of microﬂuidics, but also by the fundamental ﬂuid dynamics behind the devices. By exploring the process over ranges of the material and operating parameters, we seek to clarify the underlying physics, and to provide guidelines for further technological developments. Furthermore, we introduce non-Newtonian rheology into the picture. This is an important addition because the sample ﬂuids in the targeted applications are often biological complex ﬂuids that contain macromolecules. Despite the typically slow ﬂow rates, the small dimension of micro-channels implies large strain rates capable of distorting the microstructure of the ﬂuids and thus producing viscoelasticity (Stone et al., 2004). Besides, the elasticity of polymer molecules may be exploited to produce ﬂuidic control and memory elements (Groisman et al., 2003). So far, little experimental attention has been given to non-Newtonian ﬂuids in microﬂuidics. Indeed, Stone et al. (2004) have identiﬁed this as one of the major thrust areas in microﬂuidic research in the near future. Our numerical work will provide the ﬁrst indications of the eﬀects of non-Newtonian rheology on drop formation and behavior in microﬂuidic ﬂows. In the context of computational ﬂuid dynamics, the process of drop formation in the ﬂow-focusing devices is a formidable task. Besides the complex geometry, the moving and deforming internal boundary is a well-known numerical diﬃculty. Moving-grid methods use grid points to track the interfaces, and tend to break down when the interfaces undergo topological changes such as breakup and coalescence (Cristini & Tan, 2004). Fixed-grid methods avoid this problem by regularizing the interface and converting the Lagrangian description of moving boundaries into an Eulerian description. The price is the need for an additional scalar ﬁeld and adequate resolution of the interfacial region (Feng et al., 2005). Incorporating non-Newtonian rheology is a further challenge. Recently, we have developed a ﬁnite-element algorithm with adaptive meshing that possesses the strengths of both ﬁxed and moving grids (Yue et al., 2006). Couched in a diﬀuse-interface framework, the method is particularly suited for simulating interfacial dynamics in complex ﬂuids. In this model, the two components are assumed to mix in a narrow interfacial layer, across which physical properties change steeply but continuously. The interfacial position and thickness are determined by a phase-ﬁeld variable whose 16 Chapter 2. Formation of simple and compound drops in microﬂuidic devices evolution is governed by a mixing energy. This way, the structure of the interface is rooted in molecular forces and calculated from a convection-diﬀusion equation; there is no longer a need for tracking the interface. Moreover, the model uses an energy-based formulation that incorporates the rheology of microstructured ﬂuids with ease. We have set three objectives for this study: (a) to demonstrate that our diﬀuseinterface method is capable of reproducing the experimentally observed processes of simple- and compound-drop formation; (b) to map out the ranges of operating and material parameters so as to provide guidelines for future design and development; and (c) to explore the eﬀects of viscoelastic rheology on the process. 2.2 Theory and numerical methods Yue et al. (2004) have described a theoretical model that incorporates non-Newtonian rheology into a diﬀuse-interface framework, presented a numerical algorithm using spectral discretization, and conducted numerical experiments to validate the code. Recently, Yue et al. (2006) have developed a more versatile and powerful version of the numerical toolkit, dubbed AMPHI, using ﬁnite elements with Adaptive Meshing using a phase ﬁeld (φ). Since the numerical schemes in AMPHI and their validation have been discussed in detail, we will only summarize the main ideas and give the governing equations here. Consider a pair of “immiscible” ﬂuids in contact. To be speciﬁc, we take one ﬂuid to be Newtonian and the other a viscoelastic Oldroyd-B ﬂuid. The diﬀuse interface has a small but non-zero thickness, inside which the two components are mixed and store a mixing energy. We deﬁne a phase-ﬁeld variable φ such that the concentrations of the non-Newtonian and Newtonian components are (1 + φ)/2 and (1 − φ)/2, respectively. Then φ takes on a value of 1 or −1 in the two bulk phases, and the interface is simply the level set φ = 0. Starting with the system’s free energy, comprising the mixing energy of the interface and the bulk elastic energy in the Oldroyd-B ﬂuid, we can derive the following set of governing equations (Yue et al., 2005b, 2006): ∂φ + v · ∇φ = γ∇2 G, ∂t (2.1) G = λ −∇2 φ + ρ φ(φ2 − 1) 2 , τ p + λH τ p(1) = μp [∇v + (∇v)T ], 1+φ 1+φ 1−φ μn + μs )[∇v + (∇v)T ] + τ p, τ = ( 2 2 2 ∂v + v · ∇v = ∇ · (−pI + τ ) + G∇φ + ρg, ∂t ∇ · v = 0, (2.2) (2.3) (2.4) (2.5) (2.6) where G is the chemical potential and γ is the mobility parameter; λ is the interfacial energy density and the capillary width represents the interfacial thickness. With decreasing , the diﬀuse-interface model approaches the classical sharp-interface model and 17 Chapter 2. Formation of simple and compound drops in microﬂuidic devices λ/ gives the interfacial tension (Yue et al., 2004). The polymer stress τ p obeys the Maxwell equation, with the subscript (1) denoting the upper convected derivative and λH being the polymer relaxation time. μp and μs are the polymer and solvent contributions to the shear viscosity of the Oldroyd-B ﬂuid, and μn is the viscosity of the Newtonian ρ1 + 1−φ ρ2 , ρ1 and ρ2 being the densities for the phase. ρ is a mixture density: ρ = 1+φ 2 2 Oldroyd-B and Newtonian components, and g is the gravitational acceleration. These equations are discretized on a ﬁnite-element grid using the Galerkin formulation with streamline upwinding for the constitutive equation. We will concern ourselves only with axisymmetric geometry in this study, and the 2D computational domain is covered by an unstructured grid of triangular elements. Prior experience has shown that the key to accurate phase-ﬁeld simulations is suﬃcient resolution of the interfacial region (Feng et al., 2005). Thus, we have built an adaptive meshing scheme using the method contained in the public-domain package GRUMMP (Freitag & Ollivier-Gooch, 1997). The scheme allows one to control the spatial gradient of grid size using a scalar ﬁeld. In our application, the phase-ﬁeld variable φ is a natural choice for this function. Thus, we have a belt of reﬁned triangles covering the interfacial region. As the interface approaches the edge of the belt, remeshing is performed with the mesh upstream of the interface being reﬁned by edge bisection and/or node insertion while that left behind being coarsened. Typically the interfacial layer requires roughly 10 grid points to resolve, and remeshing happens over tens of time steps. We use implicit time-stepping, with Newton iteration at every step to handle the nonlinearity in the equations. The time step is automatically adjusted according to a set of criteria based on the normal velocity of the interface and the bulk velocity. Numerical experiments with grid reﬁnement and time-step reﬁnement have been carried out (Yue et al., 2006), and adequate resolution is ensured for the simulations presented in the following. 2.3 2.3.1 Results and discussion Drop formation at an oriﬁce We ﬁrst consider the growth and detachment of drops from a tip of a capillary tube that discharges into an ambient ﬂuid at a small constant ﬂow rate. This relatively simple process has been studied in the past by several groups, theoretically by lubrication analysis (Eggers, 1997), experimentally (Wilkes et al., 1999; Notz et al., 2001; Christanti & Walker, 2001; Shore & Harrison, 2005) and computationally (Wilkes et al., 1999; Notz et al., 2001; Ambravaneswaran et al., 2002). A summary of the main results can be found in a recent review (Basaran, 2002). Important aspects of the process, e.g., the drop size, the critical length of the thread prior to pinch-oﬀ and satellite drop formation, have been explored as functions of geometry, ﬂow rate and ﬂuid properties, including viscoelasticity. A comparison of our simulation with results in the literature will serve as a validation of our numerical method and code. Although Yue et al. (2006) presented numerical examples for benchmarking the method, they did not examine interface rupture and pinch-oﬀ, phenomena that are key to the drop formation in microchannels. For our 18 Chapter 2. Formation of simple and compound drops in microﬂuidic devices purpose, we will focus on a computational study by Wilkes et al. (1999), who used a moving-grid ﬁnite-element method to simulate the formation of drops of Newtonian liquids issuing from a capillary into air. Let us denote the viscosity and density of the ﬂuid inside the capillary by μi and ρi , and those of the ambient matrix ﬂuid by μo and ρo . Other parameters of the problem are the radius of the nozzle a, the average velocity in the capillary Vi , interfacial tension σ and gravitational acceleration g. The process is controlled by ﬁve dimensionless groups: Bo = Ca = We = α = β = (ρi − ρo )ga2 , σ μi Vi , σ ρi Vi2 a , σ ρi , ρo μi , μo (2.7) (2.8) (2.9) (2.10) (2.11) where the Bond number Bo, the capillary number Ca and the Weber number W e indicate the ratio of gravity, viscous force and inertia to capillarity. The Reynolds and Ohnesorge numbers are also used in the literature: Re = W e/Ca, Oh = Ca/W e1/2 . We start with an initial condition where the interface is ﬂush with the opening of the capillary tube, and a parabolic velocity proﬁle with average Vi is imposed upstream. Figure 2.1 shows two snapshots of a typical simulation, one at an early stage of drop formation and the other immediately after the drop pinches oﬀ. Note that the ﬁniteelement grid is reﬁned near the interface. As the interface deforms, the mesh evolves accordingly. In this example, the mesh is generated by enforcing a grid size h1 = 0.003a on the interface, h2 = 0.15a inside the inner ﬂuid and h3 = 0.2a in the far ﬁeld of the outer ﬂuid. Numerical experiments with mesh reﬁnement have shown such resolutions to be suﬃcient. The small h1 , relative to the capillary width = 0.006a, implies that there are about 15 grid points across the interfacial layer, more than adequate for resolving the interfacial proﬁle and producing an accurate interfacial tension (Yue et al., 2004). The scenario of drop growth and pinch-oﬀ is similar to prior simulations by Wilkes et al. (1999) To be more quantitative, we compare the drop radius r as a function of Bo in Fig. 2.2. Note that in both computations, W e and Ca are very small. Thus, inertial and viscous forces are negligible, and the process is dominated by the balance between capillary force and gravity: σa ∼ (ρi − ρo )gr3 . This leads to the well-known scaling (de Gennes et al., 2004): r ∝ Bo−1/3 . (2.12) a Note ﬁrst that the power law is borne out by both sets of results. In addition, the agreement between the two results is within 5% throughout the Bo range. This is despite the diﬀerences in α and β as inertia and viscosity have negligible eﬀects in this limiting 19 Chapter 2. Formation of simple and compound drops in microﬂuidic devices (a) (b) Figure 2.1: Formation and detachment of a drop at the tip of a capillary. The drop and matrix ﬂuids are both Newtonian, with density ratio α = 2 and viscosity ratio β = 1. Other dimensionless parameters are Bo = 0.465, Ca = 3.05×10−3 and W e = 2.95×10−4 . (a) Early stage of drop growth at dimensionless time tVi /a = 2.71. The left half shows the ﬁnite-element grid and the right half plots the interface and streamlines. (b) Shortly after the pinch-oﬀ, tVi /a = 4.14. regime as long as the density diﬀerence is matched properly through Bo. This validation indicates that our numerical scheme handles interfacial deformation and pinch-oﬀ accurately. In fact, the diﬀuse-interface treatment aﬀords an advantage over sharp-interface methods in simulating breakup and coalescence. These topological changes are no longer mathematical singularities to be circumvented by ad hoc schemes that cut and reconnect the interface. Rather they are controlled by a short-range molecular force akin to the van der Waals force (Yue et al., 2005a). To further illustrate this point, Fig. 2.3 plots the thinning of the neck in time. Except for the ﬁnal moment, Rn (t) approximates a 2/3 power-law which would obtain for an inertio-capillary pinch-oﬀ (Eggers, 1997). In our case, Oh = 0.178, Bo = 0.465, and the small viscous and gravity eﬀects may have caused the deviation from the scaling. However, the most interesting feature is the abrupt dive of Rn once it reaches a threshold value Rn∗ ≈ 0.01a. This is when the short-range force inherent in the Cahn-Hilliard model begins to dominate. Owing to the phenomenological nature of this model, however, the degree to which this short-range force reﬂects reality 20 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.2: Comparison of the drop size between our simulation and Wilkes et al. (1999). In our calculations, α = 2, β = 1. Bo is varied by using diﬀerent ﬂuid densities and interfacial tension; W e varies between 1.90 × 10−4 and 2.68 × 10−4 , and Ca between 2.87 × 10−5 and 3.55 × 10−3 . Wilkes et al. (1999) treated the drop surface as a free surface such that α = ∞, β = ∞. W e and Ca are of the same order of magnitude as ours. in a particular experiment is not known a priori. Thus, we may consider the threshold Rn∗ the smallest length scale that is resolved with conﬁdence in our simulations. Note that Rn∗ is close to 2 and is roughly half of the interfacial thickness. 2.3.2 Formation of simple drops in a ﬂow-focusing device The simulations to be discussed in this subsection are motivated by the experiments of Anna et al. (2003), who produced water drops in oil in a ﬂow-focusing geometry. However, the setup of the numerical problem diﬀers from its experimental counterpart in several ways. First, the experimental channels have rectangular cross-sections, and the ﬂow ﬁeld is 3D. We have adopted a 2D axisymmetric geometry illustrated in Fig. 2.4. The length scales are similar to those in the experiment but a precise match is impossible. Our main motivation for using an axisymmetric geometry is to reduce the computational cost. But axisymmetric devices have practical advantages over the more common rectangular microchannels for drop encapsulation (Takeuchi et al., 2005). Second, the solid surfaces in Anna et al.’s microchannel seem to be hydrophobic since the interface apparently never makes contact with the outer walls in the experiment. In our phase-ﬁeld formulation, a contact angle of 90◦ is the default although it may be modiﬁed through the free energy 21 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.3: Variation of the minimum neck radius Rn on the thread with time t for the simulation in Fig. 2.2. Rn is scaled by the nozzle radius a and t by the ﬂow time a/Vi . (Jacqmin, 2000). As a result, our interface may touch the wall at very low ﬂow rates, which we have avoided in the simulations to reﬂect the experimental hydrophobicity. Finally, surfactants (Span 80) are added to the oil phase in the experiment to prevent coalescence of the drops after they are formed. It is unclear whether the surfactants have aﬀected the drop formation process, and we have made no such provisions in the simulations, except assuming a reduced interfacial tension of σ = 10 mN/m. This is approximately the limiting interfacial tension between water and hexacane with saturating Span 80 surfactants on the interface (Campanelli & Wang, 1999). At the inlet, parabolic velocity proﬁles are imposed for both the inner ﬂuid and the outer ﬂuid, with average velocities Vi and Vo , respectively. At the outlet, the normal stress and the radial velocity vanish, while at the axis of symmetry, the radial velocity and all radial gradients are put to zero. The physical parameters are taken from the experiments (Anna et al., 2003), with the inner ﬂuid being water and the outer silicone oil. The density and viscosity ratios are α = ρi /ρo = 1.11 and β = μi /μo = 0.167. Other dimensionless parameters of the problem are the various length ratios, W e and Ca for the inner ﬂuid: W e = ρi Vi2 a/σ and Ca = μi Vi /σ, and the ratio of ﬂow-rates Γ = Qi /Qo between the inner and outer streams. Except for results in Figs. 2.8 and 2.12, we have ﬁxed Γ = 1/4 in this subsection according to a series of experiments in Anna et al. (2003) The W e and Ca values cited below correspond to ﬂow rates and material constants from the experiments and a characteristic length a = 20 μm, close to half of the actual oriﬁce width of 43.5 μm. Mesh sizes are similar to those used in Fig. 2.1: h1 = 0.005a on the 22 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.4: Geometric setup for simulating drop formation in a ﬂow-focusing device. Shown is the meridian plane of the axisymmetric device, and the computational domain is the top half. The characteristic length is the radius of the nozzle a. interfaces, h2 = 0.08a in the bulk of the inner ﬂuid and h3 = 0.1a for the bulk of the outer ﬂuid. The capillary width is = 0.01a. Two regimes, dripping and jetting, have been identiﬁed in previous experiments on drop formation at the tip of a capillary discharging into stationary air (Ambravaneswaran et al., 2004) or a co-ﬂowing stream (Umbanhowar et al., 2000; Cramer et al., 2004), and most recently in ﬂow-focusing devices (Utada et al., 2005). Similar regimes have been observed in our simulations. The dripping regime is characterized by periodic formation of highly uniform spherical drops outside the oriﬁce, and prevails at low ﬂow rates. With increasing ﬂow rates, there is a transition to the jetting regime, where a jet extends downstream from the oriﬁce, with drop formation at the tip of the jet. In the following, we will analyze each regime in turn and discuss the transition. 1. Dripping regime The ﬂow focusing occurs through the strong contraction upstream of the nozzle, which stretches the inner ﬂuid into a ﬁlament inside the nozzle. Upon exiting the oriﬁce, the expansion decelerates the ﬂow, and interfacial tension produces a nearly spherical bulb at the end of the ﬁlament (Fig. 2.5a). The bulb is fed by the ﬁlament and grows gradually. Meanwhile a neck forms at the base of the bulb (Fig. 2.5b). Shortly after, oscillations are observed on the neck (Fig. 2.5c,d). The oscillation is ampliﬁed in time, and after several periods the neck pinches oﬀ and the bulb disconnects from the ﬁlament and travels downstream (Fig. 2.5e). Then a new cycle commences. This process is highly periodic and produces drops of very uniform size (Fig. 2.5f). The oscillation of the neck before drop pinch-oﬀ is more clearly shown by the timesequence in Fig. 2.6. During periodic drop formation, the interfacial shape upstream of the nozzle hardly varies in time, and the oscillation is largely localized to the neck region. This oscillation calls to mind the well-known Rayleigh instability. Because the ﬁlament ends in the bulb, however, capillarity works through both the Rayleigh instability and the end-pinching mechanism (Stone & Leal, 1989). To investigate the development of capillary waves on the ﬁlament, we have carried out a simulation where the length of the nozzle is elongated to 3 times of the original size (Fig. 2.7). The growth of capillary waves within the nozzle, apparent in Fig. 2.7(a), is more precisely represented by the ﬁlament 23 Chapter 2. Formation of simple and compound drops in microﬂuidic devices (a) t = 3.84 (b) t = 4.96 (c) t = 6.06 (d) t = 6.59 (e) t = 6.68 (f) t = 12.1 Figure 2.5: Snapshots of drop formation in the dripping regime at Ca = 1.70 × 10−3 , W e = 5.75 × 10−4 . The time is made dimensionless by a/Vi . In dimensional terms, the frequency of drop formation is roughly 189 Hz. Figure 2.6: The radius of the neck, measured at its thinnest part, oscillates in time before dropping to zero. Rn is scaled by a and t by a/Vi . The period of oscillation, converted using dimensional parameters in Anna et al.’s experiment, is about 0.16 ms. radius in Fig. 2.7(b). Clearly, the capillary wave travels downstream with increasing amplitude. The wave-speed vw = 40.6Vi is close to the ﬂuid speed on the interface 24 Chapter 2. Formation of simple and compound drops in microﬂuidic devices (a) (b) Figure 2.7: Development of the capillary wave inside a long nozzle. All other parameters are the same as Fig. 2.5. (a) Geometry of the domain; (b) radius of the thread as a function of the axial distance from the inlet of the nozzle to the outlet. The radius and the position are scaled by the nozzle radius a. vi = 37.1Vi . (vi itself agrees with the theoretical prediction of Eq. 2.16 to within 3%.) Thus, the capillary wave grows while riding on the interface and traveling downstream. A Fourier analysis identiﬁes a dominant wavelength λ = 9.24rf , rf being the average radius of the ﬁlament inside the nozzle. Theoretical prediction by (Mikami & Mason, 1975), who extended Rayleigh’s theory to a viscous cylinder surrounded by another viscous ﬂuid and conﬁned in a tube, gives the fastest-growing wavelength as 9.42rf , in close agreement with our numerical value. Since the bulb and the neck hardly move forward, the traveling wave produces the oscillation in Fig. 2.6 as it arrives at the neck. The period at the neck is roughly twice that of the wave inside the nozzle. Thus we have conﬁrmed the existence and growth of capillary waves. But is capillary instability the direct cause of the breakup of the ﬁlament and pinch-oﬀ of the drop? For capillary breakup, the drop radius rd would be proportional to the radius of the ﬁlament rf . The latter can be easily estimated from the ﬂow-rate ratio Γ and viscosity ratio β by assuming parabolic velocity proﬁles in the nozzle and matching the shear stress at the 25 Chapter 2. Formation of simple and compound drops in microﬂuidic devices (a) (b) Figure 2.8: Dependence of the drop radius rd on the ﬂow-rate ratio Γ. (a) Eﬀect of varying one ﬂow rate while keeping the other ﬁxed. The intersection of the two curves, at Γ = 0.167, Ca = 1.70 × 10−3 and W e = 5.75 × 10−4 for the inner ﬂuid and Cao = 2.49 × 10−2 and W eo = 2.84 × 10−3 for the outer ﬂuid, is near the transition point; decreasing either Qi or Qo causes dripping while increasing either leads to jetting. (b) Comparison between two ﬁxed Qi values while Qo varies: lower Qi with Ca = 1.70×10−3 , W e = 5.75 × 10−4 and higher Qi with Ca = 1.06 × 10−2 , W e = 2.25 × 10−2 . The former experiences the transition at Γ = 0.167 while the latter is entirely in the jetting regime. interface: rf = a 1+Γ+ Γ √ 1 + Γβ . (2.13) For our simulations, β = 0.167 and Γ < 1, and the above is well approximated by a powerlaw: rf /a ∼ Γ1/2 . We have conﬁrmed that the computed rf follows this scaling closely. Then we would expect the radius of the drops rd to scale with Γ1/2 as well. This scaling, as it turns out, fails to represent the numerical results in Fig. 2.8. Moreover, the drop size also depends strongly on the diameter of the downstream collection tube (Fig. 2.9). Therefore, the pinch-oﬀ of the drop is not entirely determined by the capillary instability. The complex geometry—expansion from the nozzle to the collection tube—and the outer ﬂow ﬁeld play important parts as well. For drop formation in a co-ﬂowing ambient ﬂuid without ﬂow-focusing, Umbanhowar et al. (2000) suggested a scenario of drag-induced pinch-oﬀ similar to that of a pendant drop under gravity. This seems to be consistent with the trend in Fig. 2.9, where the slower outer ﬂow for larger expansion ratio re /a results in less drag on the bulb and larger drop size. Balancing a Stokes-like viscous drag and the capillary force on the ﬁlament: μo vo rd ∼ σrf , where vo ∼ Qo /re2 is the velocity of the outer ﬂuid in the collection tube, 26 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.9: The drop radius rd increases with the radius of the downstream collection tube re . Γ = 0.25, α = 1.11, β = 0.167, Ca = 1.70 × 10−3 and W e = 5.75 × 10−4 . A linear ﬁt to the numerical data is rd /a = 0.38re /a + 0.65. one obtains 1 3 rd σΓ 2 re2 σΓ 2 re2 ∼ ∼ . a μo Qo μo Qi 3 (2.14) According to this, rd should scale with Γ 2 for ﬁxed Qi , and with re2 for ﬁxed Γ. However, the former is contradicted by the data in Fig. 2.8 while the latter by Fig. 2.9. We are led to the conclusion that the pinch-oﬀ is the combined eﬀect of both capillary instability and stretching of the neck by viscous drag (Davidson et al., 2005). Imagine setting Qi and Qo suddenly to zero. The ﬁlament hanging outside the nozzle will break into droplets by end-pinching: the end forms a bulb and the neck pinches in with the ﬂuid being squeezed out by the high capillary pressure in the neck. In the actual simulations, the ﬂow in and around the ﬁlament changes this picture. On the one hand, the inner ﬂow supplies ﬂuid to the neck and resists its continuous thinning (Fig. 2.10a). On the other hand, the outer ﬂow exerts a drag on the bulb that tends to stretch and thin the neck. As a result, the neck radius reaches smaller minima in successive cycles of oscillation with the incoming waves (Fig. 2.6). During the shrinking phase of the neck’s oscillation, the forward velocity in the ﬁlament is impeded, and the ﬂuid accumulates upstream of the neck (Fig. 2.10b). Periodically, this ﬂuid mass merges into the end drop as the neck size reaches its maximum. Thus, as the minimum neck radius diminishes in the oscillation, the maximum increases as shown in Fig. 2.6. Eventually, a critical point is reached when the neck size gets so thin that the forward ﬂow in the ﬁlament is completely arrested 27 Chapter 2. Formation of simple and compound drops in microﬂuidic devices (a) (b) Figure 2.10: The velocity ﬁeld near the drop at (a) tVi /a = 6.59 and (b) tVi /a = 6.67 shortly before pinch-oﬀ. The parameters are the same as in Fig. 2.5. The maximum velocity (dark) is nearly 120Vi and the minimum (white) is below 0.6Vi . (a) t = 30.8 (b) t = 30.9 (c) t = 31.4 (d) t = 32.6 (e) t = 34.8 (f) t = 34.9 Figure 2.11: A cycle of drop formation in the jetting regime at Ca = 1.06 × 10−2 , W e = 2.25 × 10−2 . The time is made dimensionless by a/Vi . In dimensional terms, the frequency of drop formation is roughly 1027 Hz. (Fig. 2.10b). The vortex ring before the neck is caused by the high capillary pressure at the neck that drives the ﬂuid into reverse motion. Then end-pinching and stretching by viscous drag conspire to eﬀect the pinch-oﬀ. The interplay of various factors is such that an argument based on capillary instability or viscous drag alone is inadequate. 2. Jetting regime As the ﬂow rates increase, dripping gives way to jetting. While dripping produces drops right after the nozzle, jetting features a long jet that extends several oriﬁce di- 28 Chapter 2. Formation of simple and compound drops in microﬂuidic devices ameters downstream into the collection tube. Drops form at the forefront of the jet periodically. A cycle of drop pinch-oﬀ is shown in Fig. 2.11. The mechanism for drop formation seems to be qualitatively the same as in the dripping regime, i.e., via a combination of viscous drag and capillary instability, the latter comprising Rayleigh waves and end-pinching. Inside the nozzle, viscous shear stretches the interface and produces a thin ﬁlament as before (cf. Eq. 2.13). Upon entering the collection tube, the expansion reduces the outer ﬂow velocity and the viscous shear on the ﬁlament. Capillary retraction then causes the jet to become thicker. Nevertheless, capillary waves are evident on the jet. As in the dripping regime, it does not seem that the drop pinch-oﬀ is due entirely to the growth of the capillary wave. If it is, the period of drop formation should be given by the time required for capillary breakup, which in our low-viscosity case scales as tc ∼ (ρi rj3 /σ)1/2 , rj being the jet radius in the collection tube. Then the drop size rd can be calculated from (Qi tc )1/3 . If we assume rj to follow the same scaling as rf in Eq. (2.13), then the drop size rd scales with Γ1/4 with Qi ﬁxed, and Γ7/12 with Qo ﬁxed. Neither agrees with the trend of the data in Fig. 2.8. Instead, the rd (Γ) relationship seems to be qualitatively the same in the dripping and jetting regimes. Thus, we surmise that the same mechanism underlies the pinch-oﬀ in both regimes. However, there are notable diﬀerences from dripping. The most obvious is the appearance of the long jet in the collection tube. This is related to the faster speed of the ﬁlament at higher Qi and Qo , and will be discussed in detail below as related to the dripping-jetting transition. Second, because the jet gets rather thick, the bulb at the tip of the jet moves downstream with a velocity comparable to the jet velocity. Thus, there is no oscillation at the neck, in contrast to the dripping regime where the waves travel into an essentially stationary drop and get absorbed. The thicker jet also leads to larger drops than in the dripping regime for the same geometry and Γ (cf. Fig. 2.8b). Finally, the jetting regime exhibits somewhat greater irregularity than the dripping regime. The polydispersity, deﬁned by the standard deviation in drop radius divided by the mean, is about 1%. In the dripping regime, no variation of drop size is detectable in the numerical data; the process is perfectly periodic up to the accuracy of our numerical resolution. The transition from dripping to jetting is often seen as a competition between two time scales: a capillary time tc for the growth of interfacial disturbance and a ﬂow time tf for the convection of the ﬂuid (Ambravaneswaran et al., 2004). If tc < tf , pinch-oﬀ occurs shortly after the oriﬁce, giving rise to dripping. Conversely the disturbance is carried downstream before it gets ampliﬁed, and jetting obtains. Depending on the Ohnesorge number of the ﬂow, the capillary disturbance can be dominated by viscosity or by inertia (Lister & Stone, 1998). In our case, Oh ∼ O(10−2 ) and the pertinent capillary time is the inertial time: tc = C(ρi rj3 /σ)1/2 . Numerical tests show C ≈ 26 for the geometry and ﬂuid properties used here. We take tf to be the time needed for one waveform to travel downstream: tf = λw /vw , where λw and vw are the capillary wavelength and wave speed in the collection tube. As noted before, inside the nozzle the wave speed is roughly equal to the interfacial speed. In the collection tube, we assume a similar equality: vw = vj , vj being the interfacial velocity on the jet. Now the critical condition for dripping-jetting 29 Chapter 2. Formation of simple and compound drops in microﬂuidic devices transition can be formulated in terms of a modiﬁed We number: C 2 ρi rj3 vj2 t2c We = 2 = = 1. tf σλ2w (2.15) Thus, we set the dripping-jetting transition at the point where the jet consists of one waveform beyond the oriﬁce. Such a criterion is necessarily arbitrary insomuch as the transition is gradual in the simulations as well as the experiments (Ambravaneswaran et al., 2004). The jet radius rj and jet surface velocity vj can be estimated using the same arguments as led to Eq. (2.13). In particular, vj = 2Qo Γ √ , · 1+ 2 πre 1 + 1 + Γβ (2.16) where re is the radius of the expansion tube. For the dripping simulation depicted in Fig. 2.5, W e = 0.539. For the jetting run in Fig. 2.11, W e = 21.52. These are consistent with the criterion in Eq. (2.15). We carried out a series of simulations to determine the transition point, from the visual criterion of whether the jet carries one or more waveforms between the oriﬁce and the neck behind the bulb. Results put W e between 0.877 and 1.02, as expected. From the dependence of rj and vj on Γ, one may show that for small Γ, W e ∼ Γ−1/2 for ﬁxed Qi and W e ∼ Γ3/2 for ﬁxed Qo . This explains the observations in Fig. 2.8 that dripping gives way to jetting with increasing Γ at a ﬁxed Qo , and with decreasing Γ at a ﬁxed Qi . In other words, this transition takes place when either Qi or Qo increases. 3. Comparison with prior work As mentioned before, our axisymmetric simulations approximate the conditions of the 3D ﬂow in the microﬂuidic device of Anna et al. (2003). Over a range of ﬂow rates, they reported interfacial morphologies resembling our dripping and jetting regimes. The drop size decreases with the outer ﬂow rate and increases with the inner ﬂow rate. This trend has been conﬁrmed by our simulations; see Fig. 2.8. More quantitative comparison is possible if we take the width of the experimental nozzle to be our nozzle diameter 2a, and the thickness of the expansion channel to be our 2re . For instance, we predict a drop radius rd = 36.4 μm at Ca = 1.70 × 10−3 , W e = 5.75 × 10−4 , whereas the drops in the experiment have rd = 36 − 39 μm at roughly matching ﬂow rates; the agreement is within about 10%. Davidson et al. (2005) did similar simulations using a VOF method, with which our results agree well. In both dripping and jetting regimes, the drop diameter agrees within 2%, and the critical jet length at pinch-oﬀ is within less than 5% for jetting and 10% for dripping. This discrepancy may be due to the diﬀerent strategies for handling interface rupture. The VOF method requires external intervention to eﬀect breakup and reconnection, while our phase-ﬁeld theory allows such events to evolve naturally (Yue et al., 2005a). Moreover, the oscillation of the neck is a prominent feature of our simulations that reveals the mechanism for pinch-oﬀ (cf. Fig. 2.6). In their ﬁgures and online movies, on the other hand, no such oscillation can be discerned. An online video 30 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.12: Tipstreaming at Ca = 5.36 × 10−4 , W e = 5.75 × 10−5 , Γ = 1.89 × 10−2 , α = 1.11, β = 0.167. clip for Utada et al.’s experiment (Utada et al., 2005) shows neck oscillation very similar to our simulation. Utada et al. (2005) noted that the jetting regime exhibits greater irregularity than the dripping regime, with a polydispersity of 3% in the former and 1% in the latter. These are greater than predicted here but the trend is the same. The frequency of drop formation ranges from 100 to 5000 Hz in the experiments. Our numerical predictions fall in that range. Another interesting feature of Utada et al.’s experiment is that their dripping-jetting transition is accompanied by a discontinuous jump in the drop size. This has to do with the geometry of their setup, where the outer ﬂuid does not go through a long conduit before exiting into the collection tube. Thus it has a roughly ﬂat velocity proﬁle at the oriﬁce, and develops a parabolic proﬁle only far downstream. The drop formation at the oriﬁce in dripping, consequently, diﬀers qualitatively from the pinchoﬀ far downstream in jetting. This contrasts our geometry where both streams develop parabolic proﬁles inside the relatively long nozzle. Our dripping-jetting transition is gradual and there is no sudden change in the drop size (Fig. 2.8). Finally, Utada et al. considered drop pinch-oﬀ in their jetting regime as due entirely to capillary instability, and this is supported by a scaling relationship for the drop size. As indicated earlier, a similar scaling does not hold in our case. In addition to dripping and jetting, we have also observed a regime for large outer ﬂow rates Qo coupled with small inner ﬂow rates Qi where a very thin jet is drawn into the nozzle and breaks up into small droplets inside the nozzle, with diameter on the order of 1/10 of the nozzle radius a (Fig. 2.12). For its resemblance to the so-called “tipstreaming” phenomenon (Taylor, 1934), we may call this the tipstreaming regime. In the literature, tipstreaming is usually regarded as due to accumulation of surfactants at the downstream stagnation point of a drop (de Bruijn, 1993; Eggleton et al., 2001; Renardy et al., 2002). In appearance, it is also similar to electrospraying where a micro-thread is drawn from the tip of a Taylor cone by electrostatic forces (Cloupeau & Prunetfoch, 1994). But our tipstreaming involves neither surfactants nor electric ﬁelds, and is purely hydrodynamic. Previous theoretical models have shown the possibility of surfactant-free tipstreaming when a thin ﬁlament is drawn from a drop by an extensional ﬂow (Tomotika, 1936; Sherwood, 1984). In our simulation of the ﬂow-focusing devices, the contraction ﬂow into the nozzle creates a similar scenario in the limit of small Γ = Qi /Qo . When the 31 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.13: Geometric setup for simulating the formation of compound drops. Shown is the meridian plane of an axisymmetric device, and the computational domain is the top half. thin thread enters the nozzle, it breaks up into small drops due to Rayleigh instability. Finally, Garstecki et al. (2005) reported a peculiar capillarity-independent “displacement regime” for forming bubbles and drops in a ﬂow-focusing device at exceedingly low ﬂow rates. We did not attempt to reproduce it computationally. 2.3.3 Formation of compound drops in a ﬂow-focusing device A compound drop consists of an inner ﬂuid enclosed by a ﬂuid shell that is suspended in an outer ﬂuid. For the past few decades, double emulsions composed of water-in-oilin-water compound drops have been studied as a means for delivering drugs dissolved in the inner aqueous phase enterally (Engel et al., 1968; Oba et al., 1992). But the conventional method of shearing in mixers produces double emulsions whose size distribution is hard to control. New techniques for making compound drops have appeared, including the breakup of compound jets under electrohydrodynamic forcing (Loscertales et al., 2002) and microencapsulation by a solvent exchange method (Yeo et al., 2004). More recently, several groups demonstrated the possibilities of using microﬂuidic devices for mass-producing highly monodisperse compound drops (Atencia & Beebe, 2005; Joanicot & Ajdari, 2005) and for fabricating micron-sized polymer capsules and vesicles (Utada et al., 2005; Takeuchi et al., 2005). A great advantage is the precise control, through the ﬂow rates, of the inner and outer drop sizes as well as the number of droplets encapsulated in each larger drop. In this subsection, we describe numerical simulations aimed at elucidating the formation of compound drops in a ﬂow-focusing device. The axisymmetric computational domain in Fig. 2.13 is loosely based on the experimental device of Utada et al. (2005), which employs three streams to produce compound drops in a single step. In our setup, the three streams ﬂow in the same direction, while in the experiment, the outermost stream issues from openings located on the shoulder of the contraction. Thus, it comes in the opposite direction and forms a contactline with the intermediate ﬂuid on the outer 32 Chapter 2. Formation of simple and compound drops in microﬂuidic devices wall. In either case, a compound jet forms and is forced through the contraction. Mesh sizes are h1 = 0.01a on the interfaces, h2 = 0.1a in the bulk of the middle ﬂuid and h3 = 0.2a for the bulk of the inner and outer ﬂuid. The capillary width = 0.02a, and the interface is resolved adequately. One may choose the following set of dimensionless parameters for the process: a capillary number Ca and a Weber number W e for the inner ﬂuid (deﬁned using the average velocity in the inner tube, the inner ﬂuid density and viscosity, and the innermiddle interfacial tension), the density, viscosity and ﬂow-rate ratios among the three ﬂuids, the ratio of interfacial tensions among the ﬂuid pairs, and various length ratios. In this work, we did not aim to map out the entire parameter space; indeed that task would be extremely laborious if at all possible. Instead, we have probed a few parameters, including Ca, W e and the viscosity ratios, with the goal of achieving a preliminary understanding of the process. Utada et al. used two grades of silicone oil for the inner and outer ﬂuid and a water-glycerol mixture for the middle ﬂuid, with inner-middleouter viscosity ratios of 1 : 1 : 9.6. In our phase-ﬁeld model, however, it is cumbersome to represent three species, and we have thus made the inner and outer ﬂuids identical. Furthermore, we ﬁx the inner-middle-outer ﬂow-rate ratios at 3 : 6 : 40, and the middleouter density ratio at α = 1. The middle-outer viscosity ratio β = 1 except for the results in Fig. 2.16. As an initial condition, we have the middle ﬂuid forming a hemispherical shell enclosing the inner ﬂuid, and the outer ﬂuid ﬁlling the rest of the domain. Parabolic velocity proﬁles are speciﬁed at the inlets for the three streams with zero stress conditions at the exit and symmetry conditions along the centerline. Most results are presented in dimensionless form, but a few quantities of more direct practical relevance, such as the frequency of drop formation, are given in dimensional form. These are computed using characteristic values based on experiments: a = 20 μm, ρ = 1 g/cm3 and σ = 20 μN/m. Figure 2.14 illustrates the temporal evolution of the process that consists of the formation of a co-axial compound jet and its subsequent breakup into compound drops. The inner jet undergoes what appears to be Rayleigh instability, with capillary waves growing on the interface. Meanwhile, capillary waves develop on the outer interface as well, with a longer wavelength. The shrinking neck on the outer interface pinches in on the inner interface, and helps the inner jet to pinch oﬀ at a point some distance upstream of its naturally occurring neck (Fig. 2.14a). Thus, the waveform behind the front bulb of the inner jet is divided, with the front part shrinking forward and merging into the inner drop (Fig. 2.14b). Then the outer interface pinches oﬀ and encapsulates the inner drop (Fig. 2.14c,d). The process then starts again. One notable diﬀerence from the formation of simple drops is that the process here is much more irregular. The position on the inner jet where the outer jet pinches in varies somewhat from one cycle to the next, resulting in 3.4% and 4.5% variations in the inner and outer drop sizes. The duration of each cycle varies by 14.6%. Averaged over 10 drops in the run depicted in Fig. 2.14, the inner drop radius is 0.97a and the outer drop radius is 1.40a, and the frequency of drop formation is 1672 Hz. These are comparable to experimental values. The loss of near-perfect periodicity is apparently due to the two modes on the two interfaces not being in harmony. 33 Chapter 2. Formation of simple and compound drops in microﬂuidic devices (a) t = 1.30 (b) t = 1.31 (c) t = 1.32 (d) t = 1.34 Figure 2.14: Snapshots showing the diﬀerent stages of a successful encapsulation process. The times are made dimensionless by the ﬂow-time tf = a/Vi . Ca = 4.50 × 10−3 , W e = 3.24 × 10−4 . Compound jets are known to be liable to multiple modes of capillary instability (Radev & Tchavdarov, 1988; Chauhan et al., 2000). In this case, the hydrodynamic mechanisms for breakup of the compound jet are similar to those previously discussed for simple jets. For the parameters used here, the inner and middle ﬂuids have roughly the same velocity in the compound jet. Thus, there is little viscous shear on the inner interface and the breakup is dictated by capillary forces. In contrast, there is considerable shear on the outer interface. Its breakup must have been the combined eﬀect of capillary waves and stretching by a viscous drag. Note, however, that the pinch-oﬀ on the inner and outer interfaces are coupled through capillary pressure in the middle ﬂuid. Indeed, this coupling is essential for achieving successful encapsulation and controlling the morphology of the compound drop. While the process involves a large number of geometric, material and ﬂow parameters, we have only systematically explored the eﬀects of the ﬂow rates and viscosity ratio. The great sensitivity of the encapsulation process to the ﬂow rates is illustrated in Fig. 2.15. Panel (b), a later snapshot from the simulation depicted in Fig. 2.14, shows a proper compound drop with a single inner drop, at Ca = 4.50 × 10−3 and W e = 3.24 × 10−4 . Panel (a) has the three ﬂow rates reduced by 25%, at Ca = 3.38×10−3 and W e = 1.82×10−4 , while (c) has the ﬂow rates increased by 25%, at Ca = 5.63 × 10−3 and W e = 5.06 × 10−4 . In (a), the smaller ﬂow rates imply a thicker jet and breakup nearer to the expansion, similar to observations of simple drop formation. The wavelength on the outer jet is much longer than that on the inner jet so that two inner drops are encapsulated in one outer drop. In (c), on the other hand, the jet is thinner and longer at higher ﬂow rates. The outer jet necks at a position that is roughly half a waveform upstream from the base of the inner bulb. After pinch-oﬀ, the inner drop has a thin thread attached to its base which breaks oﬀ into a satellite drop. Thus, we 34 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.15: Compound drop formation is sensitive to the ﬂow rates. (a) 25% ﬂow rate decreases from (b), (c) 25% of ﬂow rate increases from (b). Conditions for (b) are identical to those in Fig. 2.14. The ﬂow-rate ratios are kept ﬁxed at 3 : 6 : 40, and the three streams have equal density and viscosity. Time is scaled by a/Vi of (b). have qualitatively conﬁrmed observations by Utada et al. (2005) that encapsulation of one or more drops can be controlled by varying the ﬂow rates. In principle, the number of inner drops depends on the ratio of the wavelengths on the two interfaces. In the narrow range of parameters tested, we have successfully encapsulated no more than two inner drops. With the ﬂow-rate ratios ﬁxed at 3 : 6 : 40 for the inner, middle and outer streams, we have constructed a “phase diagram” in Fig. 2.16 for ranges of the viscosity ratio β and inner-ﬂow capillary number Ca. Successful encapsulation of one or two inner drops occurs only within a narrow band on the Ca − β plane. In particular, if β and/or Ca is too low, the middle ﬂuid goes through the contraction much more readily than the more viscous inner ﬂuid. The shell thus ruptures too early and fails to encapsulate the inner ﬂuid (see left panel). On the other hand, if β and/or Ca is too high, the breakup of the outer interface is delayed, partly because of viscous damping within the middle ﬂuid and partly because of the small viscous drag exerted on the outer bulb by the low-viscosity outer ﬂuid. In the meantime, the inner jet advances without breaking up for lack of pinching from the outer interface. Thus, the inner bulb eventually presses and ruptures the outer interface, again causing loss of the inner ﬂuid and failure of encapsulation (right panel). Utada et al. (2005) have observed a dripping regime at low ﬂow rates and a jetting 35 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.16: Phase diagram for compound drop formation. The circle indicate formation of compound drops with one or two inner drops, and the crosses failure of encapsulation. The inner and the outer ﬂuids are identical, and the viscosity ratio β is between the middle and the outer ﬂuid. Ca is deﬁned for the ﬂow in the inner tube. The inner-middle-outer ﬂow-rate ratios are ﬁxed at 3 : 6 : 40. regime at high ﬂow rates. Our simulation in Fig. 2.14 corresponds to the latter. In Fig. 2.15, we see that the position of drop pinch-oﬀ moves upstream as the ﬂow rates decrease. But encapsulation fails when the ﬂow rates become too small, and we have not reproduced the dripping regime for compound drop formation. Comparing our simulations with video footage from Utada et al.’s experiment, we recognize several possible reasons for this discrepancy. First, we have rectangular corners in the channel that produce a sudden contraction and expansion, while the experiment had smooth and gradual transitions. The latter geometry seems to be more favorable to the innermost ﬂuid getting into the nozzle. A common mode of failure in our simulations, when the ﬂow rates are small, is the failure to draw the inner ﬂuid into the nozzle. It is excluded from the contraction for long, and eventually presses and ruptures the middle ﬂuid upstream of the nozzle (Fig. 2.16, left panel). Second, our ﬁxed ﬂow-rate ratios 3 : 6 : 40 may inhibit dripping. The relatively large middle ﬂow rate implies a thick middle shell, which tends to jet forward before the inner ﬂuid makes its way into the nozzle. In contrast, Utada et al.’s inner-to-middle ﬂow-rate ratio is 4 : 1; the inner and middle streams co-ﬂow into the nozzle as a compound jet and pinch-oﬀ together. Finally, Utada et al.’s outer ﬂuid is 10 times as viscous as the other two ﬂuids. This may have played a role in dripping as well. To sum up, many parameters aﬀect the formation of compound drops, of which we have examined only two: Ca and β. We may speculate that for diﬀerent ﬂow-rate ratios, the boundaries in Fig. 2.16 will shift. If a compound drop does form, the inner-to-outer drop radius ratio should be determined by the inner-to-middle ﬂow-rate ratio because of mass conservation. Experiments show that the outer drop size decreases with increasing 36 Chapter 2. Formation of simple and compound drops in microﬂuidic devices outer ﬂow rate (Utada et al., 2005), which is consistent with our simulations for simple drop formation (Fig. 2.8). A more comprehensive parametric study is required to generate a more detailed “operation map” that indicates how drop size and morphology can be controlled via geometry, ﬂow rates and ﬂuid properties. 2.3.4 Eﬀects of viscoelasticity Many microﬂuidic applications involve non-Newtonian ﬂuids. Macromolecules occur naturally in biological ﬂuids, and synthetic polymers may be introduced into the ﬂuid components for fabricating micro-scale polymer capsules (Utada et al., 2005; Takeuchi et al., 2005). When the ﬂuids deform, so do the macromolecules. Then viscoelastic stresses arise that may modify the ﬂow and interfacial behavior in return. Such ﬂow-microstructural coupling is well studied in non-Newtonian ﬂuid mechanics. The simulations reported next represent an exploration of similar coupling in drop formation in ﬂow-focusing devices. The viscoelastic component is modeled by the Oldroyd-B equation (cf. Eqs. 2.3, 2.4), based on a dilute suspension of elastic dumbbells in a Newtonian solvent. The relaxation time of the dumbbells, λH , scaled by the characteristic ﬂow time tf = a/Vi , gives rise to the Deborah number λH Vi . (2.17) De = a Rheological predictions of the Oldroyd-B model can be found in Bird et al. (1987) The following parameters are used in the simulations. In simple drop formation, we ﬁx the density ratio α = 1.11, viscosity ratio β = 0.167 and ﬂow rate ratio Γ = 0.25. We deﬁne β using the total viscosity μt = μp + μs of the Oldroyd-B model, with equal contribution from the polymer and the solvent μp = μs . This β is matched with the Newtonian β when comparing with simulations in preceding subsections. The geometry is the same as in Fig. 2.4 except that the length of the nozzle is shortened to 2a. In compound drop formation, we use the same geometry as in Fig. 2.13 and again take the innermost ﬂuid to be identical to the outermost. Furthermore, the viscosity ratio and density ratio between the inner and middle ﬂuids are set to 1, while the inner-middle-outer ﬂow-rate ratios are ﬁxed at 3 : 6 : 40. We have simulated the formation of simple drops when the inner ﬂuid (drop phase) is viscoelastic and the outer ﬂuid is Newtonian. The viscoelastic eﬀect turns out to be quite diﬀerent for the dripping and jetting regimes. Figure 2.17 compares a viscoelastic simulation with its Newtonian counterpart in the dripping regime, all parameters being the same except for the Deborah number. With the non-Newtonian inner ﬂuid, the drop size is larger and the pinch-oﬀ occurs further downstream. The polymer tensile stress τpyy , in the ﬂow direction, attains its maximum at the entrance into the contraction because of the strong acceleration and elongation of the inner ﬂuid. Scaled by the nominal viscous shear stress μt Vi /a, this maximum value is τpyy = 3.33 × 103 . Once inside the nozzle, the ﬂow becomes mostly shear and τpyy relaxes gradually and vanishes upon exiting the nozzle. Therefore, this stress has no direct bearing on the drop formation. A second local maximum of τpyy occurs at the neck where the drop pinches oﬀ. Its most signiﬁcant eﬀect is to delay the pinch-oﬀ of the drop. This is evident from the oscillation of the neck. 37 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.17: Simple drop formation in the dripping regime when the inner ﬂuid is (a) viscoelastic, and (b) Newtonian. The snapshots are taken shortly before the drop detaches. Ca = 2.26 × 10−3 , W e = 1.02 × 10−3 . In (a), De = 0.113, and the grey-scale contours depict the polymer tensile stress τpyy along the ﬂow direction. The drop size is 7.5% larger than in (b), while the critical jet length, from the oriﬁce to the point of pinch-oﬀ, is 10.1% longer. Before the last cycle, the oscillation is similar to its Newtonian counterpart in Fig. 2.6. The period of oscillation T = 0.20 ms is slightly longer than the Newtonian period, as the wave speed and the interfacial ﬂuid velocity are both smaller in the viscoelastic case. The last cycle of oscillation, however, takes nearly 0.40 ms. The situation is similar to the capillary breakup of a polymeric thread, where the elevated tensile stress (or elongational viscosity) is known to prolong the thinning of the thread and delay the pinch-oﬀ (Li & Fontelos, 2003). As a result, the frequency of drop formation is reduced from 272 Hz in the Newtonian case to 218 Hz. This gives the drop more time to grow from imbibing the ﬂuid through the neck. By the same token, the location of pinch-oﬀ is also further downstream. In the jetting regime, the viscoelastic eﬀect seems to be opposite to that in dripping: the drop size is smaller and the jet length is shorter (Fig. 2.18). The latter can be attributed to the normal stress τpyy in the jet, which, unlike in the dripping regime, extends some way downstream beyond the nozzle. This tensile stress, along with capillarity, resists the stretching of the jet by the faster outer ﬂuid. Thus, the jet is shorter. Besides, the most unstable wavelength is shorter on a viscoelastic jet than on a comparable Newtonian one. This has been shown by analysis (Funada & Joseph, 2003) as well as experimental data (e.g., Fig. 8 of Christanti & Walker (2001)). The shorter wavelength implies smaller drops for viscoelastic ﬂuids, as is indeed the case in our simulations. For the same inner ﬂow rate Qi , the pinch-oﬀ frequency is higher for the Oldroyd-B drops, 681 Hz compared with 602 Hz for Newtonian drops. Similar to the dripping regime, a second maximum of polymer stress occurs at the neck before the bulb. Although this 38 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.18: Simple drop formation in the jetting regime when the inner ﬂuid is (a) viscoelastic and (b) Newtonian. Ca = 7.15×10−3 , W e = 1.02×10−2 . In (a), De = 0.357, and the grey-scale contours depict the polymer tensile stress τpyy along the ﬂow direction, with a maximum τpyy = 1.49×103 , scaled by μt Vi /a. Compared with the Newtonian case, the jet length is 9.0% shorter for the viscoelastic case and the drops are 4.0% smaller in diameter. tends to prolong the pinch-oﬀ process, its eﬀect is apparently overwhelmed by the eﬀects discussed above. In compound drop formation, viscoelasticity in each of the three streams may have diﬀerent eﬀects on the process. Motivated by experiments that employ compound drops for making polymeric hollow spheres (Utada et al., 2005; Takeuchi et al., 2005), we have only simulated the situation where the middle ﬂuid is an Oldroyd-B ﬂuid while the inner and outer streams are identical Newtonian ﬂuids (Fig. 2.19). Results show that the viscoelastic eﬀect on the compound drop is similar to that on the simple drop in the jetting regime (Fig. 2.18). The length of the outer jet (from the oriﬁce to the point of pinch-oﬀ) is shorter by 5.3%, and the outer drop size is smaller by 2% than their Newtonian counterparts. The inner-outer drop size ratio is unaﬀected by viscoelasticity as long as a single inner drop is enclosed in the outer drop, since it is determined by the ﬂow-rate ratio. With the Deborah number De as a new parameter, the phase diagram should be modiﬁed as well, but we have not done extensive parameter sweeps to determine such modiﬁcations. Another interesting and potentially useful eﬀect of viscoelasticity in the middle ﬂuid is in suppressing satellite drops when the inner jet breaks up. In an all-Newtonian system, a satellite drop is sometimes enclosed in the outer drop along with the primary drop; see Fig. 2.19(a) and Fig. 2.15(c). Depending on the applications, this may be undesirable. Figure 2.19(b) and (c) show that in otherwise identical conditions, viscoelastic stress in the middle ﬂuid eliminates the satellite drop. This is similar to the suppression of satellite drops by polymers in a jet (Christanti & Walker, 2001). As the inner jet pinches at the neck, the tensile stress τpyy develops in a thin sheath around the neck. Apparently 39 Chapter 2. Formation of simple and compound drops in microﬂuidic devices Figure 2.19: Viscoelastic eﬀects on compound drop formation. All 3 ﬂuids are Newtonian in (a). The middle ﬂuid is viscoelastic in (b) and (c), two snapshots following pinch-oﬀ of the inner jet. Ca = 4.50 × 10−3 , W e = 3.24 × 10−4 , and De = 0.090 for (b) and (c). The times are made dimensionless by the ﬂow-time tf = a/Vi . The grey-scale contours indicate the level of polymer tensile stress τpyy in the middle ﬂuid, with a maximum on the order of τpyy = 4.20 × 103 , scaled by μt Vi /a. the stretching inside the jet induces similar deformation in its immediate surroundings. Upon pinch-oﬀ, the newly formed ends retract much more slowly than in a Newtonian surrounding ﬂuid. Evidently, the large tensile stress in the middle ﬂuid resists the capillary retraction and suppresses end-pinching (Fig. 2.19b,c). Eventually the thread shrinks into the primary drop without secondary breakup. Recent experiments (Chen et al., 2006) have shown that polymer dissolved in the surrounding ﬂuid has similar eﬀects on capillary breakup to that inside the thread. That is, it tends to produce a thin and long-lasting ﬁlament and delay the breakup, and sometimes lead to the beads-on-a-string morphology. In our case, we do not see beads-on-a-string. This is probably because the time scale of the process is too short. 2.4 Summary The numerical simulations presented in this paper have reproduced scenarios of jet breakup and drop formation previously observed in experimental ﬂow-focusing devices. The diﬀuse-interface model is suitable for such ﬂows involving interface rupture. For 40 Chapter 2. Formation of simple and compound drops in microﬂuidic devices simple drops, parametric studies have shown dripping and jetting regimes for increasing ﬂow rates, and elucidated the eﬀects of ﬂow and rheological parameters on the drop formation process and the ﬁnal drop size. In particular, the pinch-oﬀ is shown to depend on both capillary instability and stretching of the interface by viscous drag. 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Yue, P., Feng, J. J., Liu, C. & Shen, J. 2005b Transient drop deformation upon startup of shear in viscoelastic ﬂuids. Phys. Fluids 17, 123101. Yue, P., Zhou, C., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2006 Phaseﬁeld simulations of interfacial dynamics in viscoelastic ﬂuids using ﬁnite elements with adaptive meshing. J. Comput. Phys. 219, 47–67. 46 Chapter 3 Simulation of neutrophil deformation and transport in capillaries using Newtonian and viscoelastic drop models ∗ 3.1 Introduction Neutrophils often encounter narrow capillary segments during their transit through the pulmonary and systemic microcirculations. Because the neutrophil diameter (6–8 μm) often exceeds the diameter of a pulmonary capillary (2–15 μm) (Doerschuk et al., 1993; Yap & Kamm, 2005b), cell deformation is necessary for passage. It is well known that neutrophils take much longer to traverse the pulmonary capillary bed than erythrocytes (Hogg et al., 1994). As a result, white cells accumulate in the lungs and form a reservoir from which they may be readily recruited when needed (Lien et al., 1987; Doerschuk et al., 1993). As the pulmonary bed consists of 50–100 capillary segments (Huang et al., 2001; Bathe et al., 2002), the longer transit time for neutrophils likely reﬂects their relatively poor deformability (Wiggs et al., 1994; Doerschuk, 1999), which in turn depends on the structure and mechanical properties of the cell: membrane rigidity, cytoplasmic viscosity and viscoelasticity, and the properties of the nucleus. Measuring leukocyte deformation and transit in vivo is a diﬃcult task (Hogg et al., 1994), and so far our understanding of the mechanical behavior of leukocytes has come mostly from micropipette aspirations in vitro. By measuring the time-dependent cell deformation at a controlled pressure, cell viscosity, cortical tension and elastic modulus can be estimated by treating the cell as a homogeneous continuum (Evans & Yeung, 1989; Hochmuth, 2000). As a sort of rheometry, aspiration experiments yield fundamentally important data. But they do not produce direct information on cell transit in capillaries. by particle-tracking while the cell undergoes structural changes and activation. This cannot be reﬂected in our simulations as an advantage. The Cell Transit Analyzer (Fisher et al., 1992) combines the aspiration process with subsequent passage through cylindrical pores, and allows rapid measurements of transit time. However, the cell entry and departure from the micropore is monitored from the attendant pulses in electric conductivity across the ﬁlter. The actual deformation of the cell is unknown and the interpretation of the signal is subject to much uncertainty (Fisher et al., 1992; Drochon, 2005). Recently, Yap & Kamm (2005b) carried out an experiment using microﬂuidic channels ∗ A version of this chapter has been published. Zhou, C., Yue, P. & Feng, J. J., Simulation of neutrophil deformation and transport in capillaries using Newtonian and viscoelastic drop models, Ann. Biomed. Eng. 35, 766-780 (2007) 47 Chapter 3. Simulation of neutrophil deformation and transport in capillaries that seems to have overcome previous diﬃculties. The device allows direct observation of neutrophil deformation and activation upon entering a microﬂuidic channel, as well as measurement of the cell entrance time as a function of the pressure drop imposed across the microﬂuidic channel. The present work is motivated by Yap and Kamm’s experiment, and aims to extract a fundamental understanding of the cell entry process using dynamic simulations. Yap & Kamm (2005b) also reported interesting results on neutrophil activation probed by microrheology and pseudopod formation. These will not be accounted for in our simulations. Analyzing the physical and mechanical processes on the single-cell level remains a challenge, especially when taking the living and dynamic nature of the cell into consideration (Bao & Suresh, 2003). Lim et al. (2006) reviewed the quantitative mechanical models developed so far for the cell. Generally, these adopt either a microstructural approach or a continuum approach. The former is based on the cytoskeleton as the main structural component, and has been used widely to investigate cytoskeletal mechanics in adherent cells (Stamenovic & Ingber, 2002). On the other hand, the continuum approach takes the cell as comprising homogeneous materials with certain eﬀective properties. Although providing less insight into the details of intra-cellular processes, the latter is simpler and has found applications in simulating large-scale transient behavior of suspended cells such as the corpuscles in the blood (Kamm, 2002; Pozrikidis, 2003a, 2005). Continuum cell models come in several ﬂavors. The Newtonian liquid drop model (Yeung & Evans, 1989; Evans & Yeung, 1989) sees the cell as a homogeneous Newtonian drop enclosed by a cortex that has a constant and isotropic tension and negligible viscous dissipation. Non-Newtonian drop models, such as the shear-thinning model (Tsai et al., 1993) and the viscoelastic Maxwell model (Dong et al., 1988), have since been developed to reﬂect the non-Newtonian rheology of the cytoplasm in leukocytes. Furthermore, compound drop models (Dong et al., 1991; Hochmuth et al., 1993; Kan et al., 1998, 1999) explicitly account for the cell nucleus as a drop suspended in the cytoplasm. The cell and nuclear membranes are taken to be interfacial layers with diﬀering interfacial tensions. So far, micropipette aspiration has been the benchmark problem for testing these models, and the prediction has been remarkably good considering the simplicity of the models (Dong et al., 1988; Yeung & Evans, 1989; Dong & Skalak, 1992; Drury & Dembo, 2001). Moreover, the models have been used to simulate other modes of controlled deformation in relatively simple geometries, e.g., recovery in a quiescent medium (Tran-Son-Tay et al., 1991, 1998; Kan et al., 1999) and deformation in elongational ﬂows (Kan et al., 1998) and shear ﬂows (Pozrikidis, 2003b, 2005). Most recently, the compound drop models have been applied in simulating the adhesion of leukocytes on the endothelium (Li & Wang, 2004; Khismatullin & Truskey, 2005; Jadhav et al., 2005). However, only a few simulations have been carried out on the transit of suspended neutrophils, either through a single capillary (Tran-Son-Tay et al., 1994; Bathe et al., 2002) or an idealized capillary network (Huang et al., 2001). The present study consists of dynamic simulations, based on the Newtonian and viscoelastic drop models, of the deformation and transit of neutrophils through capillaries that roughly correspond to the experiments of Yap & Kamm (2005b). Speciﬁcally, we 48 Chapter 3. Simulation of neutrophil deformation and transport in capillaries study the entry of a neutrophil from a larger vessel into a narrow capillary using geometry and parameter values based on the Yap-Kamm experiments. The entrance time, the cell deformation and the subsequent motion within the capillary will be compared with observations. The focus of the study will be on how cytoplasmic rheology aﬀects the entry of a neutrophil into a thin capillary. The nucleus is not explicitly accounted for, and the cytoplasmic rheology should be understood as an averaged property of the complex mixture of the nucleus, cytosol and suspended organelles. Furthermore, we have neglected two physiologically important factors. First, the cell membrane is treated as thin interfaces having a constant cortical tension, with no elastic resistance against bending and inplane shearing and no viscous dissipation. We recognize that membrane elasticity has an important role in cell deformation that has received a great deal of research (Eggleton & Popel, 1998; Pozrikidis, 2003a). Although membrane elasticity may be incorporated into the phase-ﬁeld theory (Du et al., 2004; Du & Zhu, 2006), we defer the implementation to a future eﬀort. Second, we have not accounted for the glycocalyx on the inner walls of the capillary, which is known to greatly increase resistance on red blood cells (Feng & Weinbaum, 2000; Secomb et al., 2001). This is partly because very little is known of the glycocalyx on the pulmonary endothelium, and partly because incorporating the porous layer model would greatly increase the complexity of the ﬂow simulation. So far, the only multi-dimensional computation that accounts for the glycocalyx used a drastically simpliﬁed model of a frictionless contact surface (Bathe et al., 2002). In the context of computational ﬂuid dynamics, our problem is a complex one because of the moving and deforming interfaces and the non-Newtonian rheology of the ﬂuid components, each being a major computational challenge (Sethian & Smereka, 2003; Owens & Phillips, 2002). Recently, we have developed a diﬀuse-interface method that incorporates the moving interface and non-Newtonian rheology in a uniﬁed variational framework (Yue et al., 2004; Feng et al., 2005). Implemented using spectral methods and ﬁnite elements, the method has been applied successfully to several problems in drop dynamics of complex ﬂuids (Yue et al., 2005a,b,d ,c, 2006b; Zhou et al., 2006; Yue et al., 2006a), and will be adapted here to the task of neutrophil deformation and transport. 3.2 Theory and numerical methods Our diﬀuse-interface model was developed mainly for simulating interfacial dynamics in complex ﬂuids. For any binary blend, the two nominally immiscible components are assumed to mix in a narrow interfacial layer and store a mixing energy. Across the interfacial layer, physical properties such as viscosity and density change steeply but continuously. The interfacial position and thickness are determined by a phase-ﬁeld variable φ whose evolution is governed by a Cahn-Hilliard equation. The interfacial tension is given by the mixing energy. This way, the structure of the interface is rooted in molecular forces and calculated from a convection-diﬀusion equation; there is no longer a need for tracking the interface. Moreover, the model uses an energy-based formulation that incorporates the non-Newtonian rheology of microstructured ﬂuids with ease. This is the main reason for our selecting this methodology. A more in-depth discussion of 49 Chapter 3. Simulation of neutrophil deformation and transport in capillaries the advantages and disadvantages of the diﬀuse-interface model, vis-`a-vis the classical sharp-interface model and other interface regularization methods, can be found in the literature (Lowengrub & Truskinovsky, 1998; Yue et al., 2004; Feng et al., 2005). The above methodology has been implemented in a ﬁnite-element package AMPHI (Adaptive Meshing with phase ﬁeld φ). Yue et al. (2006b) have described the code in detail and presented numerical experiments to establish its validity and accuracy. In this paper, we will only summarize the main ideas and give the governing equations. To be speciﬁc, consider a Newtonian ﬂuid in contact with a viscoelastic Oldroyd-B ﬂuid, the interface between the two being diﬀuse with a small but non-zero thickness. We deﬁne a phase-ﬁeld variable φ such that the concentrations of the non-Newtonian and Newtonian components are (1 + φ)/2 and (1 − φ)/2, respectively. Then φ takes on a value of 1 or −1 in the two bulk phases, and the interface is simply the level set φ = 0. Starting with the system’s free energy, comprising the mixing energy of the interface and the bulk elastic energy in the Oldroyd-B ﬂuid, we can derive the following set of governing equations (Yue et al., 2005a, 2006b): ∂φ + v · ∇φ = γ∇2 G, ∂t (3.1) G = λ −∇2 φ + ρ φ(φ2 − 1) 2 , τ p + λH τ p(1) = μp [∇v + (∇v)T ], 1+φ 1+φ 1−φ μn + μs )[∇v + (∇v)T ] + τ p, τ = ( 2 2 2 ∂v + v · ∇v = ∇ · (−pI + τ ) + G∇φ + ρg, ∂t ∇ · v = 0, (3.2) (3.3) (3.4) (3.5) (3.6) where G is the chemical potential and γ is the mobility parameter; λ and are the interfacial energy density and capillary width, respectively. The polymer stress τ p obeys the Maxwell equation, with the subscript (1) denoting the upper convected derivative and λH being the polymer relaxation time (Bird et al., 1987). μp and μs are the polymer and solvent contributions to the shear viscosity of the Oldroyd-B ﬂuid, and μn is the viscosity ρ1 + 1−φ ρ2 , ρ1 and ρ2 being of the Newtonian phase. ρ is a mixture density: ρ = 1+φ 2 2 the densities of the Oldroyd-B and Newtonian components, and g is the gravitational acceleration. As demonstrated elsewhere (Yue et al., 2005a, 2006b), the diﬀuse interface has two important features: (a) The interface has a thickness on the order of 5 . The Cahn-Hilliard dynamics ensures that it neither collapses into a sharp surface nor diﬀuses into a wide region. (b) In the limit √ of → 0, the above system reduces to the familiar sharp interface formulation, and 2 2λ/3 gives the interfacial tension (Yue et al., 2004). The velocity and shear stress are continuous across the interface and the normal stress has a jump consistent with the interfacial tension. To accurately reproduce the cortical tension on a thin “membrane”, therefore, we must use an that is much smaller than the overall dimension and must resolve the φ proﬁle adequately within the thin interface. This is why adaptive meshing is essential to AMPHI (Yue et al., 2006b). 50 Chapter 3. Simulation of neutrophil deformation and transport in capillaries Figure 3.1: The geometric setup for simulating a neutrophil’s entrance into a capillary. Two cylindrical tubes are connected by an arc of 90◦ . Shown is the meridian plane and the upper half is the computational domain. These equations are discretized on a ﬁnite-element grid using the Petrov-Galerkin formulation with streamline upwinding for the constitutive equation. We will concern ourselves only with axisymmetric geometry in this study, and the 2D computational domain is covered by an unstructured grid of triangular elements. To resolve the thin interfacial region, we have used an adaptive meshing scheme based on the public-domain package GRUMMP (Freitag & Ollivier-Gooch, 1997). The scheme allows one to control the spatial gradient of grid size using a scalar ﬁeld. In our application, the phase-ﬁeld variable φ is a natural choice for this function. Thus, we have a belt of reﬁned triangles covering the interfacial region. As the interface approaches the edge of the belt, remeshing is performed with the mesh upstream of the interface being reﬁned by edge bisection and/or node insertion while that left behind being coarsened. Typically the interfacial layer requires roughly 10 grid points to resolve, and remeshing happens over tens of time steps. We use implicit time-stepping, with Newton iteration at every step to handle the nonlinearity in the equations. The time step is automatically adjusted according to a set of criteria based on the normal velocity of the interface and the bulk velocity. Numerical experiments with grid reﬁnement and time-step reﬁnement have been carried out (Yue et al., 2006b), and adequate resolution is ensured for the simulations presented in the following. 3.3 Simulations using the simple Newtonian drop model As the simplest model for a neutrophil, we have a liquid drop containing a homogeneous viscous Newtonian ﬂuid that represents the cytoplasm and nucleus in an average sense. The interfacial tension σ represents a constant and isotropic tension in the cell membrane. The drop ﬂuid, or “cytoplasm”, has a density ρc and a viscosity μc , and the cell is suspended in a matrix of density ρm and viscosity μm . Before deformation, the spherical cell has a radius rc . To simulate the entry of the neutrophil into a capillary, we use the axisymmetric computational domain illustrated in Fig. 3.1. The narrow capillary downstream of the contraction has a radius a and length L = 10a. A constant pressure drop ΔP is applied over the entire length of the domain 19a. To construct the 51 Chapter 3. Simulation of neutrophil deformation and transport in capillaries (a) (b) Figure 3.2: (a) The microchannel of Yap & Kamm (2005b). The scale bar is 100 μm, and the two arrows indicate the microchannel and the reservoir. (b) Schematic showing the dimensions of the microchannel. Its cross-section is rectangular with a width of 5 μm and a depth of 2.5 μm. After Yap & Kamm (2005b); c 2005 the American Physiological Society. dimensionless groups controlling the process, we use a as the characteristic length and Vf = ΔP a2 /(8μm L) as the characteristic velocity. Note that Vf is the average velocity in a Poiseuille ﬂow through a uniform pipe of radius a with pressure gradient ΔP/L. Then ﬁve dimensionless groups can be constructed: μm Vf , (3.7) σ ρm V f a Re = , (3.8) μm ρc α = , (3.9) ρm μc β = , (3.10) μm rc ζ = , (3.11) a where the capillary number Ca indicates the ratio between viscous and capillary forces, and the Reynolds number Re represents the ratio between inertial and viscous forces. The characteristic ﬂow time is tf = a/Vf , and the ﬂow rate will be scaled by Qf = πa2 Vf . For brevity, we use the same symbols for dimensional and dimensionless variables, but will explicitly indicate which is meant where confusion may arise. The cell radius, eﬀective viscosity and cortical tension of the neutrophil are taken from the literature (Evans & Yeung, 1989; Yap & Kamm, 2005b; Lim et al., 2006): rc = 3.5 μm, μc = 2.2 poise and σ = 0.035 dyn/cm. The plasma density and viscosity are essentially those of water. The neutrophil is nearly neutrally buoyant and we have taken the density ratio α to be unity in all the simulations. The dimensions in Fig. 3.1, with a = 2 to 2.5 μm, approximate these in the experiment of Yap & Kamm (2005b) (Fig. 3.2). But their microchannel has a rectangular cross-section and an exact match is impossible. For the typical ﬂow rates in the experiments (Yap & Kamm, 2005b), the capillary number Ca ranges up to 0.1 and the Reynolds number Re = O(10−3 ). Thus inertia has little Ca = 52 Chapter 3. Simulation of neutrophil deformation and transport in capillaries part in the dynamics to be discussed. The viscosity ratio β is on the order of 200 for an activated neutrophil and may be as large as 104 before activation (Yap & Kamm, 2005b; Evans & Yeung, 1989; Lim et al., 2006). Highly viscous cells deform less and tend to press tightly against the channel walls at the entrance. For lack of a proper treatment of the membrane-wall interaction, such cells often stick to the wall. This drawback limits us to β values on the order of 50. For convenience, therefore, we have used relatively small β values. Varying β between 1 and 16 shows a clear trend in the results and we did not explore higher β values systematically. As mentioned before, we use an adaptive meshing scheme to resolve the thin interfacial region so as to produce an accurate cortical tension. In all the simulations, we have used a capillary width = 0.008a. The ﬁnest grids occur at the interface with grid size h1 = 0.005a, while the bulk mesh size inside and outside the cell are h2 = 0.12a and h3 = 0.2a, respectively. These, along with the time step Δt, have been tested in numerical experiments (Yue et al., 2006b) to ensure adequate resolution. At the beginning of the simulation, the cell is placed on the centerline at x = 5a and the velocity is zero everywhere. Then the pressure drop ΔP , imposed over the whole length of the domain, drives a ﬂow from the left to the right in Fig. 3.1. At the inlet and the outlet, we set the boundary conditions to be v = 0 and ∂u/∂x = 0. On the centerline we use symmetry conditions: v = 0 and ∂u/∂y = 0. 3.3.1 The process of cell deformation and entrance For one set of parameters, the process of cell entrance is illustrated by the snapshots in Fig. 3.3. The variations of the ﬂow rate Q and cell length l are plotted in Fig. 3.4 as functions of time. One may discern four stages in the process. (i ) First, the neutrophil deforms and moves into the contraction while the ﬂow rate drops sharply (t < 5). This is because the front of the cell is sucked into the capillary and plugs most of the ﬂow area. The cell length increases steadily, and this stage continues until roughly half of the cell is within the capillary. (ii ) Once the plugging of the capillary has reached its maximum level, the ﬂow rate more or less keeps constant until the whole cell enters the capillary (5 < t < 9). The elongation of the cell continues in this second stage; its rear is “held” by the contraction while its front is stretched by the ﬂow (Fig. 3.3c). (iii ) The third stage is a transient as the rear of the cell clears the contraction (9 < t < 12). As the contraction loses its “grip” on the cell (Fig. 3.3d ), the high capillary pressure inside its rear produces a sudden forward ﬂow and a retraction of the cell’s back surface. This temporary shortening of the cell (see Fig. 3.4) in turn increases the blockage in the capillary and causes the ﬂow rate to drop. Both l and Q recover in time as the cell attains an equilibrium shape. (iv ) Finally, the cell moves downstream with a constant shape and velocity (t > 12). The scenario described above is observed for most of the simulations but is not universal ; some aspects vary depending on the parameter values. In section 3.3.3, we will see diﬀerent behaviors of the ﬂow rate in stage (ii) for higher and lower cell viscosities (cf. Fig. 3.7). In the more extreme case of very high pressure drops, both the ﬂow rate 53 Chapter 3. Simulation of neutrophil deformation and transport in capillaries (a) t = 2.49 (b) t = 4.41 (c) t = 6.16 (d) t = 8.84 (e) t = 10.96 (f) t = 12.74 Figure 3.3: Snapshots of the neutrophil during its entrance into the capillary. Ca = 0.0893, β = 3 and ζ = 1.4. Time is made dimensionless by tf = a/Vf . Figure 3.4: Variations of the ﬂow rate Q and cell length l during the entrance process. Q is made dimensionless by Qf . The cell length l is the distance between the foremost and rearmost points of the cell and is scaled by a. Ca = 0.0893, β = 3 and ζ = 1.4. and the cell length vary monotonically, and no obvious stages can be discerned. This is because the cell is highly elongated by the contraction ﬂow upstream of the capillary and the entry becomes relatively uneventful. We have observed in the simulations that the ﬂow rate Q and cell length l generally vary in opposite directions (cf. Fig. 3.4). This will be explained in the next subsection in terms of the increased ﬂow resistance due to the cell. 54 Chapter 3. Simulation of neutrophil deformation and transport in capillaries Figure 3.5: Log-log plot of the dimensionless entrance time τent as a function of the capillary number Ca. The data points are numerical results while the solid curve represents Eq. (3.18) derived from scaling arguments. β = 3 and ζ = 1.4. Time is made dimensionless by a/Vf , and Ca is deﬁned using the applied pressure ΔP in Eq. (3.7). 3.3.2 The entrance time As in Yap and Kamm’s experiment (Yap & Kamm, 2005b), we deﬁne the entrance time τent as the time interval between the leading edge of the cell crossing the entry to the capillary (namely, the axial position where the straight portion of the capillary starts) and its trailing edge clearing the entry. Figure 3.5 shows the numerically computed τent for a range of applied pressure drop (or capillary number). The dimensionless entrance time increases with the imposed pressure; the slope suggests a weak power-law with an index of around 0.1. Intuitively, the dimensional τent should decrease with ΔP or the velocity Vf since a higher pressure will induce a faster ﬂow and a more rapid entry of the cell. If the cell followed the surrounding ﬂuid perfectly, its velocity would be proportional to Vf and the dimensionless entrance time, scaled by a/Vf , would be independent of the imposed pressure or the ﬂow rate. In reality, however, the cell’s motion is hindered by the channel walls. The weak increase of τent with Ca in Fig. 3.5 suggests that the cell lags the matrix ﬂuid more at faster ﬂow rates. In the following, we will provide a more quantitative explanation for the eﬀect using scaling arguments. With negligible inertia, the constant pressure drop ΔP is entirely expended on overcoming the viscous friction on the channel walls. The presence of the cell increases the wall friction in its vicinity. To quantitate this eﬀect, we adopt a simpliﬁed geometry of the cell. Figure 3.3 suggests that during much of the entry process, the rear of the cell is constricted by the pinch and hardly moves forward. The front of the cell extends into the capillary in a slug shape. We make three assumptions about the gap δ between the cell and the capillary wall: (a) δ is much smaller than the capillary radius a; (b) δ is constant 55 Chapter 3. Simulation of neutrophil deformation and transport in capillaries along the length of the cell lc that is inside the capillary; (c) δ does not change in time during the entry process. Volume conservation of the cell then allows us to calculate δ from the cell length le at the end of the entry process: δ =a− 4rc3 , 3le (3.12) rc being the radius of the initial spherical cell. Note that we have assumed a cylindrical shape for the part of the cell inside the capillary. Now the pressure drop ΔP can be divided into two parts: ΔPcell to overcome the elevated wall friction over the cell length lc inside the capillary, and ΔPwall for the rest of the channel wall. The former will be calculated from the shear rate inside the gap δ, while the latter is to be estimated from the pressure drop needed to drive the same ﬂow rate in the absence of the cell. This dichotomy is not exact, and further simpliﬁcations will be made in the following. The errors will be lumped in the end into a single adjustable coeﬃcient. At the instantaneous ﬂow rate Q, the velocity within the gap scales with Q/(2πaδ) and the viscous shear stress scales with μm Q/(2πaδ 2 ). Because of the tangential velocity on the cell surface, the actual ﬂow rate through the gap δ is smaller than Q. But this discrepancy will be accounted for by the adjustable coeﬃcient. Balancing ΔPcell against the shear stress in the gap leads to the scaling ΔPcell ∝ μm Qlc . πa2 δ 2 (3.13) For ΔPwall , we modify the Poiseuille formula on account of the wider section and the contraction upstream of the capillary: ΔPwall = 9.9μm Q(L − lc ), πa4 (3.14) where the coeﬃcient 9.9 is determined for our geometry from pressure drop in the absence of the cell. Now we may write the total pressure drop as ΔP = ΔPwall + ΔPcell = 9.9μm μm Q(L − lc ) + c 2 2 Qlc . 4 πa πa δ (3.15) The adjustable parameter c accounts for the geometric simpliﬁcations made above and the slip velocity on the cell surface. Therefore, it should depend on the geometry and the cell-matrix viscosity ratio β. It is to be determined by ﬁtting the entrance time. As the presence of the cell increases the local wall friction, Eq. (3.15) implies that if lc increases, the ﬂow rate Q will decrease and vice versa. This explains the trend in Fig. 3.4. As lc (t) is the cell length within the capillary, it increases in time from 0 at the start of cell try to le at the end. Thus, the entry time τent is largely determined by how fast the cell is elongated by the ﬂow. Since the rear of the cell moves little in this process (cf. Fig. 3.3), cell elongation depends on the motion of its leading edge. We assume that this 56 Chapter 3. Simulation of neutrophil deformation and transport in capillaries Figure 3.6: The cell length le at the end of the entry process as a function of the capillary number Ca for ζ = 1.4 and β = 3. motion is at the instantaneous average velocity within the capillary: dlc /dt = Q/(πa2 ). Then Eq. (3.15) leads to an ordinary diﬀerential equation for the cell length lc (t): ΔP = μm 9.9(L − lc ) clc dlc . + 2 · a2 δ dt (3.16) Now the entrance time τent can be obtained by integrating the above equation: τent = μm ΔP cle2 4.95(2Lle − le2 ) . + a2 2δ 2 (3.17) Noting that L = 10a and scaling τent by tf and the lengths by a, we arrive at the dimensionless entrance time τent = 1 80 99le − 4.95le2 + cle2 2δ 2 . (3.18) The cell length le depends on ΔP or Ca. Figure 3.6 indicates that the cell is longer (and thinner) inside the capillary at higher capillary number. This dependence cannot be easily modeled, however. Generally speaking, le (Ca) is determined by the balance between capillary and viscous forces. But it is also inﬂuenced by the wall conﬁnement and the inner circulation. Thus, we have decided to use the numerical results Fig. 3.6 in Eq. (3.18). Finally the coeﬃcient c can be determined by a least-square ﬁtting of the equation to numerical data in Fig. 3.5. The best ﬁtting is achieved for c = 0.13 in this case. The fact that c < 1 is mainly because the ﬂow rate through the gap δ is typically only a fraction of Q on account of the cell’s motion. Equation (3.18) describes the numerical results well at low Ca, but underestimates τent at high Ca. As the pressure drop ΔP and Ca increase, the cell becomes thinner and 57 Chapter 3. Simulation of neutrophil deformation and transport in capillaries (a) (b) Figure 3.7: The ﬂow rate (a) and cell length (b) during cell entry for three viscosity ratios. Ca = 0.0893, ζ = 1.4. Q, t and l have been made dimensionless by πa2 Vf , a/Vf and a. The inset in (b) compares the blockage of the entrance for two β values. more elongated. Not only does this violate the assumption δ a, but the cell develops a conic nose and the uniform gap assumption becomes less accurate. Hence the failure of the scaling at higher Ca. Finally, the scaling argument indicates that the weak increase of τent with Ca in Fig. 3.5 is due to the weak rise of the cell length le with Ca in Fig. 3.6. Since the latter is plotted in dimensionless parameters, it can be interpreted alternatively as the cell length le decreasing with the interfacial tension σ, which is intuitively obvious. Thus, the entrance time τent is expected to decrease with the cortical tension σ at a constant pressure drop. 3.3.3 Eﬀect of cytoplasmic viscosity In the above simulations, the viscosity ratio between the ﬂuid inside the cell and the suspending medium is set to be β = 3. This is much below the cytoplasm-plasma viscosity ratio in vivo (Yap & Kamm, 2005b; Evans & Yeung, 1989), as well as the experimental value in Yap & Kamm (2005b) who used water as the suspending ﬂuid. As indicated earlier, the use of a modest β value is a numerical expedient. In this subsection, we will vary the viscosity ratio to see how the cytoplasmic viscosity aﬀects the process of neutrophil entry and passage in a capillary. We carried out a series of numerical simulations at capillary numbers Ca = 0.0893. The viscosity ratio β is varied between 1/16 to 16 with all other dimensionless groups unchanged. At higher Ca, we were able to reach larger β values without having the cell stuck on the walls, but the results have the same trend as discussed below. For three β values, Fig. 3.7 illustrates the temporal evolutions of the ﬂow rate Q and total cell length l. In all three cases, the ﬂow rate manifests the stages described in Section 3.3.1. However, the character of the second stage changes with β. For a small β, Q keeps 58 Chapter 3. Simulation of neutrophil deformation and transport in capillaries Figure 3.8: The entrance time as a function of the viscosity ratio. The matrix viscosity is kept ﬁxed while the cell viscosity is varied. ζ = 1.4. constant or even increases in this stage, whereas for a large β, Q continues to decrease, albeit at a milder slope than in the previous stage. Recall that at the second stage, the cell has reached maximum blockage of the capillary, with roughly half of the cell inside the capillary (cf. Fig. 3.4). The diﬀering trends in Q are because the cell approaches the capillary entrance with diﬀering shapes and thus causes diﬀering degrees of blockage. For a lower cell viscosity and smaller β, the cell deforms more quickly in the contraction ﬂow upstream of the capillary, and has already developed the protruding nose by the time the cell starts to enter the capillary. As the front of the cell extends further into the capillary, the rear deﬂates simultaneously, thereby enlarging the gap between the cell surface and the wall at the “shoulder” of the contraction where the blockage is the greatest. Thus, Q increases in time in stage two. A more viscous cell, on the other hand, has a stouter shape when it approaches the capillary and thus plugs the entry more severely. The inset in Fig. 3.7(b) shows a narrower gap at the shoulder for β = 16 than β = 1, and this explains the generally lower Q for higher β. Moreover, as the cell continues to deform and its front protrudes into the capillary, the gap between the cell and the wall is squeezed further at the shoulder, causing the continued decline in Q until the cell is completely inside the capillary. Then it comes as no surprise that the entrance time τent increases with β (Fig. 3.8). The eﬀect is rather weak, and follows a power law with an index of 1/7. Evidently, this is because a more viscous cell deforms more slowly as in Fig. 3.7(b). In fact, the cell has an ¯ is a certain combination of the cell and inherent visco-capillary time scale μ ¯rc /σ, where μ matrix viscosities (Yue et al., 2006a). The fact that τent scales with β 1/7 instead of β is because the external ﬂuid, whose viscosity is kept constant, also aﬀects the deformation process. 59 Chapter 3. Simulation of neutrophil deformation and transport in capillaries (a) (b) Figure 3.9: Eﬀects of the capillary size on (a) the entrance time τent and (b) the ﬂow rate Qe at the end of the cell entry. Ca = 0.0893 and β = 3. 3.3.4 Eﬀects of capillary diameter and geometry While the human neutrophil has a diameter close to 7 μm, the diameter of the pulmonary capillary ranges from 2 μm to 15 μm (Yap & Kamm, 2005b). Naturally, a neutrophil will enter and traverse a larger capillary much more readily than a narrower one. In exploring how the entrance time τent depends on the capillary diameter a, it is more convenient to use the cell radius rc as the characteristic length. Then varying a amounts to varying the size ratio ζ = rc /a without aﬀecting any of the other dimensionless groups. However, to match the data in prior subsections where rc = 1.4a, we have used rc /1.4 instead of rc as the characteristic length. Similarly, the characteristic velocity Vf = ΔP (rc /1.4)2 /(8μm L) is used in scaling τent and Q and in deﬁning the capillary and Reynolds numbers. The radius of the upstream vessel is kept constant at 1.43rc . Not surprisingly, the entrance time increases steeply as the capillary narrows (Fig. 3.9a). The τent (ζ) curve does not follow a power-law, its slope on the log-log plot increasing from 3.2 for the smallest ζ to 5 at the upper bound. Bathe et al. (2002) computed the transit time of a cell in a capillary with a constriction, and correlated the results with the minimum radius at the nip of the constriction. If we equate this minimum radius with our capillary radius a, their correlation is τent = τ0 (ζ 5 − 1) in our notation. Despite the diﬀerent geometry, the 5th power law is comparable to our data in Fig. 3.9(a) for large ζ. As ζ decreases toward unity, Bathe et al.’s empirical equation deviates from a 5th power law, as do our data. Since the pre-factor τ0 cannot be deﬁned unambiguously for our geometry, a more detailed comparison cannot be made. As an indication of the transit time of the cell once it is entirely inside the capillary, Fig. 3.9(b) plots the instantaneous ﬂow rate Qe at the end of the entrance process (when the cell’s trailing edge clears the entry) as a function of the capillary radius. Note that Qe has been scaled by a characteristic value π(rc /1.4)4 ΔP/(8μm L) that is independent 60 Chapter 3. Simulation of neutrophil deformation and transport in capillaries Figure 3.10: A snapshot of a neutrophil entering the capillary with an elliptic obstacle after the entrance. Ca = 0.0893, β = 3 and ζ = 1.4. of a. For the Poiseuille ﬂow in a pipe, the ﬂow rate is expected to scale as a4 or ζ −4 . Over the range of ζ in Fig. 3.9(b), the data fall on a gentle curve whose slope is close to −4. The deviation from the power law is such that toward the upper bound of ζ (i.e., for the smallest capillaries), Qe decreases more than ζ −4 as the capillary radius decreases. This is because of the impenetrable cell surface hindering the ﬂow in its vicinity. The eﬀect becomes stronger for smaller capillaries which are more severely plugged by the cell. A related issue is how a neutrophil traverses a partially blocked capillary. It has long been known that in falciparum malaria, parasitized erythrocytes tend to adhere and block the lumen of brain capillaries (Yoeli & Hargreaves, 1974), with potentially fatal consequences. More recently, direct visualization in a microﬂuidic channel demonstrated that P. falciparum-infected erythrocytes lose their elasticity and deformability and become lodged in the channel (Shelby et al., 2003). Bathe et al.’s simulation (Bathe et al., 2002) used a geometry of a cylindrical capillary with a constriction formed by a smooth protrusion on the inner walls. Although this is intended to mimic the entrance to a segment in the pulmonary capillary network, their result suggests that the passage of a neutrophil will be greatly delayed by blockage of a capillary. Our computational geometry in Fig. 3.10 is based on images of brain capillaries partially obstructed by sequestered erythrocytes (Yoeli & Hargreaves, 1974). The blockage is modeled by an annular pinch in the shape of half an ellipse in the meridian plane. The major axis of the ellipse is ﬁxed and equal to the capillary radius a, and its minor axis is varied to change the degree of constriction. We denote the height of the protrusion (or the minor semi-axis of the ellipse) by h. The constriction is right after the entry; the ellipse starts where the circular arc of the contraction would have connected to the wall of the capillary. In the results to be presented, we have reverted to using a as the characteristic length. Figure 3.11 shows that the entrance time τent increases with h, and the increase becomes steeper when the blockage gets more severe. When the obstacle height is 30% of the capillary radius, τent is almost doubled. Conceivably, for a critical h value, the cell will fail to pass completely, and this critical h should increase with the imposed pressure drop or Ca. Figure 3.11 also plots the aforementioned correlation of Bathe et al. (2002) recast in terms of h. Since our obstacle is elliptic and theirs is part of a circular arc, the pre-factor τ0 is determined by taking the average between the major and minor axes of our ellipse as the diameter of the circular arc. The correlation shows a much stronger eﬀect than our results. This is mostly because in their geometry, the obstacle has a radius 61 Chapter 3. Simulation of neutrophil deformation and transport in capillaries Figure 3.11: The entrance time as a function of the obstacle height h. τent is made dimensionless by a/Vf , and h by a. Ca = 0.0893, β = 3 and ζ = 1.4. The correlation of Bathe et al. (2002) is also shown for comparison. greater than the cell radius rc . Therefore, during much of the passage the cell is entirely within the constricted segment. In our geometry, the extent of the constriction is much smaller (cf. Fig. 3.10), and therefore the correlation does not apply. 3.3.5 Comparison with experiment As mentioned before, this numerical work was motivated by the experiment of Yap & Kamm (2005b), and naturally the numerical results should be compared with their measurements. The experimental device includes a microchannel connecting two water-ﬁlled reservoirs that maintain a constant pressure drop during the transit of a single neutrophil (Fig. 3.2). By adjusting the water level, the pressure drop can be varied systematically in a series of experiments. The microchannel is rectangular in its cross section with an eﬀective radius of 2 μm. The human neutrophils have a diameter close to 7 μm and a very large viscosity, around 2.2 poise in the adherent spread cells and even higher in passive round cells. Yap and Kamm observed that if the pressure drop is below a threshold of 3.92 Pa (0.4 mm H2 O), the cell fails to enter the oriﬁce. We observed a similar stoppage at a pressure of 3.0 Pa, and found the threshold pressure to be sensitive to the geometric parameters. For instance, the threshold is lower for the rounded corner shown in Fig. 3.1, and increases markedly as the entry corner gets sharper. It also increases when the reservoir-to-capillary contraction ratio increases. However, the resemblance between simulation and experiment may belie an important diﬀerence. In micropipette aspiration at a pressure lower than that required to suck the whole cell into the pipette, the cell seems to completely block the ﬂow and make solid contact with the walls (Hochmuth, 2000). It is not clear whether the same happened in Yap and Kamm’s experiment. In 62 Chapter 3. Simulation of neutrophil deformation and transport in capillaries Figure 3.12: Comparison of the dimensional entrance time with experimental data. A pressure drop ΔP = 100 Pa corresponds to Ca = 0.0893. In both simulations and experiments, the cell-to-capillary size ratio is ζ = 1.75. The experimental viscosity ratio β = 220 for the activated cell while the simulation has β = 16. The triangles are data extrapolated from β = 16 to 220 using the 1/7 power law of Fig. 3.8. our simulations, the cell is always separated from the walls by a thin ﬂuid layer, and the stoppage is owing to the cell’s cortical tension resisting deformation. Yap and Kamm reported entrance time data at several driving pressure drops, and these are reproduced in Fig. 3.12. As expected, the dimensional entrance time decreases with increasing pressure drop. We have carried out a series of simulations for comparison with the experiments. The capillary radius a = 2 μm matches that in the experiment. However, it is diﬃcult to reproduce the experimental ﬂow conditions owing to a numerical limitation. When the cell viscosity is high and/or the pressure drop is small (but still above the threshold), the cell deformation is mild and its surface is pressed against the channel walls. In the experiment, Yap and Kamm treated the PDMS walls with a copolymer surfactant solution to passivate the surface and deter cell adhesion. In the simulations, on the other hand, there is no cell membrane. The Cahn-Hilliard energy implies neutral wettability of the cell ﬂuid. So the cell tends to adhere to the walls with its surface at a 90◦ contact angle (Yue et al., 2006b; Zhou et al., 2006). This problem does not arise if the cell deforms readily, say at high pressure or low cell viscosity. Thus, we have a numerical dilemma between probing low pressure and high cell viscosity. Although the diﬃculty can be alleviated by modifying the expression for the surface energy to increases the hydrophobicity of the wall (Jacqmin, 2000), we have not yet implemented this capability in our code. In Fig. 3.12, we present data for β = 16 and ΔP above the experimental range. Despite the non-overlapping pressure ranges, the numerical data exhibit a trend that is consistent with the experimental data. Quantitatively, the numerical τent appears 63 Chapter 3. Simulation of neutrophil deformation and transport in capillaries lower than the experimental value, owing probably to the low β value. Since the eﬀect of cytoplasmic viscosity has been established in subsection 3.3.3, we have extrapolated the τent data for β = 16 to β = 220 by using the 1/7 power law of Fig. 3.8. These are in better agreement with the measured values in Fig. 3.12; the diﬀerence is roughly 18% if extrapolated to lower ΔP . 3.4 Viscoelastic eﬀects The idea of the leukocyte cytoplasm being viscoelastic comes not only from its content— numerous organelles behaving as deformable capsules and various biopolymers—but also from phenomenological observations of transient eﬀects in micropipette aspiration (Dong et al., 1988; Lim et al., 2006). The Maxwell model has been used to simulate cell deformation during aspiration (Dong et al., 1988) and passage through a capillary with a constriction (Bathe et al., 2002). In the latter study, Bathe et al. examined the transit time as dependent on the viscoelastic parameters of the model and the geometry. We have simulated the entrance of a neutrophil when the cytoplasm is modeled as a viscoelastic Oldroyd-B ﬂuid and the outer matrix is Newtonian. The Oldroyd-B model (cf. Eqs. 3.3, 3.4), based on a dilute suspension of elastic dumbbells in a Newtonian solvent (Bird et al., 1987), is essentially the same as the Maxwell model except for an additional viscous stress due to the solvent. This viscous stress has two beneﬁts: it avoids the unphysical situation of a Maxwell cell having zero viscosity at startup of deformation, and it enhances numerical stability (Owens & Phillips, 2002). The viscoelasticity is represented by a new dimensionless group, the Deborah number De = λH Vf , a (3.19) which is the relaxation time of the dumbbells λH scaled by the characteristic ﬂow time tf = a/Vf . The geometry is the same as in Fig. 3.1, and the following parameters are used in the simulations. We ﬁx the density ratio α = 1 and viscosity ratio β = 3. We deﬁne β using the total viscosity μt = μp + μs of the Oldroyd-B model, with equal contribution from the polymer and the solvent: μp = μs . This β is matched with the Newtonian β when comparing with simulations in the preceding section. The radius of the undeformed cell is still rc = 1.4a, and the capillary number will be given later for individual runs. Inertia is negligible. For the viscoelastic relaxation time, Bathe et al. (2002) obtained λH = 0.167 s from cell indentation. Dong et al. (1991) found a somewhat larger value of 0.25 s by ﬁtting micropipette aspiration data. Under our ﬂow conditions, these correspond to De = 416 and 624, respectively. Figure 3.13 illustrates the eﬀect of the Deborah number De on the entrance time τent with all other parameters being ﬁxed. This corresponds, in dimensional terms, to varying the relaxation time of the cytoplasm. The entrance time decreases monotonically when the Deborah number increases. To understand this trend, we show in Fig. 3.14 the ﬂow and stress ﬁelds inside a viscoelastic cell that is roughly 2/3 through the entry. Roughly 64 Chapter 3. Simulation of neutrophil deformation and transport in capillaries Figure 3.13: The entrance time τent decreases with the Deborah number De when the relaxation time λH increases. Ca = 0.0893, β = 3 and ζ = 1.4. The arrow indicates the entrance time for the comparable Newtonian cell. speaking, the ﬂow ﬁeld consists of three zones. The middle of the cell contains mostly rotational ﬂow with closed streamlines. The rear is practically a “dead water” zone with low deformation and stress. Finally, there is a small area in the front where the ﬂow is extensional. Except for the front, the elastic dumbbells within the cytoplasm experience a very low level of strain rate. Note that our De is deﬁned using the nominal shear rate Vf /a in the capillary. The actual strain rate within the cell is much smaller, and the actual De is as low as 0.15. As a result, the dumbbells largely remain in the coiled state and the stress level is low. This is reﬂected by the low shear stress τpxy in Fig. 3.14. Note that τpxy is scaled by τcell = μt um /a, um being the maximum horizontal velocity in the cell. The fact that τpxy < 1 in most of the cell implies that the cytoplasm assumes a lower shear stress than expected for steady shear at the strain rate um /a. This is because the viscoelastic stress takes a ﬁnite time (λH ) to develop, and attains only a fraction of τcell under the transient strain on the recirculating streamlines. For longer relaxation times or higher values of De, even lower levels of stress can be achieved. Therefore, the cytoplasm manifests a lower eﬀective viscosity at a larger De. Thus, the entry time becomes shorter as the relaxation time increases for the same reason as shown in Fig. 3.8. This argument is also borne out by Fig. 3.15, which shows that with increasing De, not only the ﬂow rate increases, but the Q(t) curve develops a more prominent upswing in stage two, resembling the case of a lower cytoplasmic viscosity β = 1 in Fig. 3.7(a). It is interesting to compare the above with Bathe et al.’s results (Bathe et al., 2002). They observed that the transit time increases with the modulus G of the Maxwell model for smaller G and levels oﬀ for large G. They interpreted the saturation as the limit of purely Newtonian rheology because the elastic spring becomes too stiﬀ to stretch. Since the relaxation time λH = μp /G, their increase of τent with G corresponds to our decrease with De. Moreover, their insight on the elastic and viscous responses is entirely 65 Chapter 3. Simulation of neutrophil deformation and transport in capillaries Figure 3.14: The streamline pattern (upper half) and contours of the shear stress τpxy (lower half) inside the cell during its entrance. The nominal Deborah number is De = 25 and the dimensionless time t = 7.66. The streamlines are in a reference frame ﬁxed on the front of the cell. The shear stress is scaled by τcell = μt um /a, where um is the maximum horizontal velocity in the cell. Figure 3.15: Eﬀects of the Deborah number on the temporal evolution of the ﬂow rate. Ca = 0.0893, β = 3 and ζ = 1.4. Note that with increasing De, the Q(t) curve assumes a resemblance to that of a Newtonian cell with a lower cell viscosity (cf. Fig. 3.7a). consistent with our analysis above. Note that Bathe et al. used the linear Maxwell model which would diﬀer from our nonlinear model for large strains. This and the diﬀerence in geometry preclude a quantitative comparison. But the viscoelastic response is qualitatively the same. The dumbbell stretching can be boosted by increasing the ﬂow rate or ΔP while keeping the relaxation time λH ﬁxed. This amounts to increasing the Deborah number De and the capillary number Ca simultaneously. Figure 3.16 shows that the entrance time for the viscoelastic cell increases toward that of the comparable Newtonian cell as Ca increases, and catches up with the latter approximately at Ca = 1.01, estimated 66 Chapter 3. Simulation of neutrophil deformation and transport in capillaries Figure 3.16: The ratio of entrance times between the viscoelastic cell and the Newtonian one increases with the ﬂow rate, indicated by Ca. β = 3 and ζ = 1.4. The Deborah number De increases with Ca in proportion: De = 25 at Ca = 0.0893 and De = 250 at Ca = 0.893. from interpolation. For still higher ΔP , τent exceeds that of the Newtonian cell. This is because at high ﬂow rates, the cytoplasm experiences increasingly severe deformation and the viscoelastic stress can grow beyond that of the Newtonian ﬂuid (Bird et al., 1987; Yue et al., 2005a). 3.5 Summary In this paper, we have examined the entrance time of a neutrophil as aﬀected by the size and geometry of the capillary and the viscosity and viscoelasticity of the cytoplasm. The results are explained by investigating the ﬂuid mechanics of the process. Qualitatively, the results are consistent with prior numerical and experimental data. Within the ranges of parameters covered, the results can be summarized as follows. (a) The entrance time τent decreases when the pressure drop over the capillary is increased, and the numerical results are in semi-quantitative agreement with the measurements of Yap & Kamm (2005b). (b) The entrance time increases sharply with decrease of the capillary diameter, and also when an obstacle inside the entry constricts the capillary. (c) The entrance time increases with the cell viscosity according to a power-law with an index of 1/7. (d ) Viscoelasticity inside the cell tends to facilitate cell deformation and shorten τent at moderate ﬂow rates. With increasing ﬂow rate, this eﬀect is reversed when the cytoplasm develops large viscoelastic stresses. Harking back to the sequestration of leukocytes in the lung, the longer transit time 67 Chapter 3. Simulation of neutrophil deformation and transport in capillaries for leukocytes, as compared with erythrocytes, seems to involve two of the above mechanisms: the white cells are larger in size (b), and they have a more viscous interior than the red cell (c). Besides, the white cell also has a highly viscous nucleus, but that is neglected in the simulations. The simulations predict the correct trend, but a more detailed comparison is hampered by simpliﬁcations in the models and the geometric setup of the simulations. In particular, we should emphasize the limitations in the physical models employed in the simulations. First, the cell membrane is represented by a ﬂuid interface with a constant and isotropic tension. No elastic resistance to in-plane shearing and bending is incorporated. Second, the neutrophil membrane is wrinkled with roughly 100% excess area over that of a smooth sphere enclosing the same volume, whereas in our simulations, the interface in principle has unlimited extensibility. Third, the cell undergoes internal structural adaptation when it deforms and activates under mechanical load (Yap & Kamm, 2005a). Such dynamics are ignored in our model, and indeed little work has been done on the multi-scale coupling between the cytoskeleton conformation and the mechanics of the cell as a whole. 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Previous studies on W/O/W emulsions have been mostly concerned with formulation and stabilization (Jiao & Burgess, 2003; Onuki et al., 2004; Bozkir & Hayta, 2004), and relatively little has been done on the deformation and morphological evolution of compound drops in ﬂow ﬁelds. The latter process is practically important since shear-induced burst of the oil shell is an important mechanism for drug release (Muguet et al., 2001). The hydrodynamics of the multiple interfaces is also central to the preparation of multiple emulsions, either through intense shearing in a mixer (Goubault et al., 2001) or through compound jet breakup in microﬂuidic devices (Utada et al., 2005; Zhou et al., 2006). Finally, compound drop dynamics is relevant to the deformation and migration of eukaryotic cells, with the inner drop representing the cell nucleus suspended in the cytoplasm (Kan et al., 1998; Khismatullin & Truskey, 2005; Jadhav et al., 2005). For the most part, ﬂuid mechanical studies of compound drops have dealt with three types of ﬂow geometries: translation in a quiescent ﬂuid (Johnson & Sadhal, 1985), dynamics in extensional ﬂows (Stone & Leal, 1990; Kan et al., 1998), and dynamics in shear ﬂows (Stroeve & Varanasi, 1984; Smith et al., 2004). In particular, Kan et al. (1998) investigated the deformation, relaxation and breakup in uniaxial elongation, and interpreted the coupling between the inner drop and the outer shell in terms of two time scales. If the relaxation time of the inner drop matches that of the shell, the compound drop will behave like a homogeneous one. Toose et al. (1999) incorporated non-Newtonian rheology into the shell ﬂuid and computed the deformation of the compound drop in elongational ﬂow. More recently, Smith et al. (2004) constructed a phase diagram depicting the morphology of daughter drops after shear-induced breakup at various values of the capillary number and interfacial tension ratio between the inner and outer surfaces. Notably, all prior work has been done in homogeneous far-ﬁeld ﬂows. Little is known about ∗ A version of this chapter has been published. Zhou, C., Yue, P. & Feng, J. J., Deformation of a compound drop through a contraction in a pressure-driven pipe ﬂow, Int. J. Multiphase Flow 34, 102-109 (2007) 74 Chapter 4. Deformation of a compound drop through a contraction compound drop deformation caused by inhomogeneous ﬂows in conﬁned geometries, as may be relevant to transport of cells and vesicles in microcirculation and drug delivery using multiple emulsions. Simulating the deformation of a compound drop is a computational challenge because of the two moving and deforming interfaces. Recently, we have developed a diﬀuseinterface method that accounts for the moving interfaces in a variational framework (Yue et al., 2004; Feng et al., 2005). Implemented using ﬁnite elements with adaptive meshing, the method has been applied successfully to several problems in drop dynamics (Yue et al., 2006a,b). In particular, we simulated the deformation of a simple drop through a contraction in a pressure-driven pipe ﬂow (Zhou et al., 2007). This note represents an application of the same methodology to compound drops deformation. The geometry is a prototype for entry of eukaryotic cells into capillaries or micropipettes (Wiggs et al., 1994; Hochmuth, 2000) and the transport of double emulsions (Garti, 1997). It generates a mixed-type ﬂow having shear and extensional characters in diﬀerent regions, and is thus an extension of prior studies in simple shear and uniform elongational ﬂows. 4.2 Theory and numerical methods We treat the interface between two nominally immiscible ﬂuids as a thin but ﬁnite mixing layer characterized by a capillary width and a Ginzburg-Landau mixing energy in terms of a phase ﬁeld φ (Lowengrub & Truskinovsky, 1998). In such a diﬀuse-interface framework, the scalar ﬁeld φ determines the position of the interface, and the governing equations can be written uniformly throughout the two-phase system. The interfacial tension arises from the mixing energy density, and appears in the momentum equation as a forcing term. A more detailed discussion of the advantages and disadvantages of the diﬀuse-interface model, vis-`a-vis the classical sharp-interface model and other interface regularization methods, can be found in the literature (Lowengrub & Truskinovsky, 1998; Yue et al., 2004; Feng et al., 2005). Since a compound drop consists of three ﬂuid components separated by two interfaces, a general diﬀuse-interface representation requires the introduction of an additional phase ﬁeld and additional interaction energies. The resulting theoretical model is rather complex and cumbersome for numerical computations (Kim et al., 2004). In this initial study, therefore, we have limited ourselves to the W/O/W type of compound drops made of two rather than three ﬂuid components. Then the conventional phase-ﬁeld description is adequate as the innermost and outermost ﬂuids are identical. This simpliﬁed model allows exploration of the fundamental hydrodynamic mechanisms, but precludes a comprehensive parametric study of general three-component compound drops. The system of equations governing the motion of a two-component Newtonian mixture 75 Chapter 4. Deformation of a compound drop through a contraction Figure 4.1: Geometric setup for simulating the deformation of a compound drop through a 2:1 contraction. Two cylindrical tubes are connected by an arc of radius a and central angle 90◦ . Shown is the meridian plane and the upper half is the computational domain. is as follows (Yue et al., 2004): ∇ · v = 0, ρ (4.1) ∂v + v · ∇v = −∇p + ∇ · μ ∇v + (∇v)T ∂t ∂φ + v · ∇φ = γ∇2 G, ∂t φ(φ2 − 1) , G = λ −∇2 φ + 2 + G∇φ, (4.2) (4.3) (4.4) where G is the chemical potential and γ is the mobility parameter; λ and are the interfacial energy density and capillary width, respectively. The phase ﬁeld φ takes on values of ±1 in the two bulk phases, and the average density and viscosity are simply ρ = 1+φ ρ1 + 1−φ ρ2 and μ = 1+φ μ1 + 1−φ μ2 . Note that the G∇φ term in the momentum 2 2 2 2 equation is a diﬀuse-interface representation of the interfacial tension. The interface typically has a thickness ∼ 5 ; the Cahn-Hilliard equation (Eq. 4.3) ensures that it neither collapses into a sharp surface nor diﬀuses into a wide region. In the limit √of → 0, the above system reduces to the familiar sharp-interface formulation, and 2 2λ/(3 ) gives the interfacial tension σ (Yue et al., 2004). To accurately capture the interfacial tension, we must use an that is much smaller than the overall dimension and then resolve the φ proﬁle adequately within the thin interface. For this purpose, we have developed a ﬁnite-element package AMPHI (Adaptive Meshing with phase ﬁeld φ) that has adaptive meshing as an essential ingredient. Yue et al. (2006b) have described the algorithm in detail and validated the numerical toolkit by benchmark problems. In the following, we use unstructured triangular elements in an axisymmetric computational domain, with time steps and grid sizes that are ﬁne enough to ensure accuracy of the numerical results. 4.3 Numerical results The axisymmetric ﬂow geometry is illustrated in Fig. 4.1, consisting of two cylindrical tubes connected by a circular arc. The downstream tube has radius a and length L = 10a, 76 Chapter 4. Deformation of a compound drop through a contraction while the upstream tube is twice as thick with a length of 8a. The compound drop has a core ﬂuid of density ρc and viscosity μc , and a shell ﬂuid of ρs and μs . As mentioned above, the suspending ﬂuid (matrix) is identical to the core ﬂuid. The core-shell and shell-matrix interfaces have the same constant interfacial tension σ. Initially, the two interfaces are concentric and spherical with radii rc and rs , centered at z = 6a, and there is no ﬂow throughout the domain. At t = 0, a constant pressure drop ΔP is applied over the entire length (19a) of the domain. On the upstream and downstream boundaries z = 0. On the centerline (z = 0 and 19a), we set the boundary conditions to vr = 0 and ∂v ∂z ∂vz we use symmetry conditions: vr = 0 and ∂r = 0. Thus, the ﬂow rate Q varies as the drop traverses the conduit. To construct the dimensionless groups controlling the process, we use a as the characteristic length and V = ΔP a2 /(8μc L) as the characteristic velocity. Note that V is the average velocity in a Poiseuille ﬂow through a uniform pipe of radius a with pressure gradient ΔP/L. Then six dimensionless groups can be constructed: Ca = Re = α = β = ζc = ζs = μc V , σ ρc V a , μc ρs , ρc μs , μc rc , a rs , a (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) where the capillary number Ca indicates the ratio between viscous and capillary forces, and the Reynolds number Re represents the ratio between inertial and viscous forces. The characteristic ﬂow time is tf = a/V , and the ﬂow rate will be scaled by Qf = πa2 V . For brevity, we use the same symbols for dimensional and dimensionless variables, but will explicitly indicate which is meant where confusion may arise. In the simulations presented here, we have ﬁxed α = 1, β = 1 and ζs = 1.4. We will explore a range of ζc to examine the core size eﬀect on the transit process. The entry of the compound drop into the contraction consists of three distinct stages, which are illustrated by the snapshots of Fig. 4.2 and the temporal variations of the instantaneous ﬂow rate and drop length in Fig. 4.3. In the ﬁrst stage (0 < t < 4), the compound drop approaches the contraction. The strong elongational ﬂow causes the shell to form a protrusion, while the core also experiences moderate deformation. The shoulder of the drop progressively blocks the ﬂow area at the contraction, thus causing the continual decrease in ﬂow rate Q (Fig. 4.3a). The length of the drop l increases in the mean time (Fig. 4.3b). At the beginning of the second stage (4 < t < 8), maximum blockage at the contraction corresponds to a minimum Q. Afterwards, the core moves forward along with the shell ﬂuid, thereby deﬂating the rear of the drop (Fig. 4.2c). This enlarges the gap between the outer surface and the wall at the contraction and causes 77 Chapter 4. Deformation of a compound drop through a contraction (a) t = 0.266 (b) t = 4.11 (c) t = 5.30 (d) t = 8.06 (e) t = 9.06 (f) t = 9.85 Figure 4.2: Snapshots of the transit of a compound drop into the capillary. Ca = 0.179, Re = 1.56 × 10−2 and ζc = 0.72. Time is scaled by a/V . a recovery of Q in stage two. Note the small dip in the Q(t) curve at t = 5; it is the result of the core passing the constriction. At the end of the second stage, both the ﬂow rate Q and the drop length achieve a local maximum. In stage three (t > 8), the rear of the drop, consisting of only the shell ﬂuid, passes the contraction into the thinner tube. As the contraction loses its “grip” on the drop (Fig. 4.2d ), the capillary pressure due to the high curvature in its rear produces a sudden forward ﬂow and a temporary retraction of the drop’s overall length (Fig. 4.3b). This temporary shortening of the drop in turn increases the blockage in the capillary (Fig. 4.2e) and causes the ﬂow rate to drop sharply (Fig. 4.3a). Then both Q and l recover as the compound drop translates in the downstream tube. Simulations using longer tubes indicate that Q and l approach roughly constant values. But the core continues to move slowly forward relative to the shell ﬂuid. This will be seen (cf. Fig. 4.5) as due to the recirculation in the shell ﬂuid. Eventually the two interfaces are pressed into each other and the shell breaks. Diﬀuse interfaces are known to coalesce prematurely (Yue et al., 2006a), and the rupture of the shell may not reﬂect reality. Qualitatively, the transit process is similar to that of a simple drop, which is also shown in Fig. 4.3 for comparison. But the core tends to resist deformation of the compound drop and this modiﬁes the process quantitatively. Throughout the drop entry, both Q and l are below those for the simple drop. The ﬂuctuation in Q also has larger magnitudes. After the core is inside the capillary, its surface hampers the recirculation in the shell ﬂuid and causes a slight bulge on the outer surface (Fig. 4.2c onward). A more viscous core (β < 1) should amplify these diﬀerences although we have not explored this systematically. Note that in terms of suppressing the ﬂow rate and drop deformation, the presence of the core is tantamount to an elevated viscosity in a simple drop (Zhou et al., 78 Chapter 4. Deformation of a compound drop through a contraction Figure 4.3: Temporal variations of the instantaneous ﬂow rate Q and the length of the compound drop l for the process of Fig. 4.2. For comparison, we have also plotted results for a simple drop of the same size (radius 1.4a) and ﬂuid properties as well as for a compound drop with a more viscous shell (β = 3). 2007). From an energetic viewpoint, the impenetrable inner surface causes more dissipation inside the compound drop, and deformation of the inner drop entails an additional energy penalty in the increased interface area. We have also explored the eﬀect of a more viscous shell ﬂuid as is relevant to typical W/O/W emulsions. The most prominent diﬀerence from the equal-viscosity case occurs in the second stage (Fig. 4.3). Instead of a strong recovery, Q remains more or less constant, or even decrease somewhat for larger β. This is because a more viscous shell reacts more slowly to the ambient ﬂow. As the core enters the downstream tube, the rear of the drop does not deﬂate rapidly enough to boost the total ﬂow rate Q. By the same token, the cell length l is generally smaller for larger β, and the transit time τtrans is longer. The eﬀect of shell viscosity is similar to that of the drop viscosity for a simple drop (Zhou et al., 2007). It is no surprise that the compound drop takes longer time to traverse the passage than a simple drop of the same size. Figure 4.4(a) plots the “transit time” τent , deﬁned as the interval between the moments when the leading and trailing edges of the drop enter the thinner tube, as a function of the capillary number. As τent has been scaled by the ﬂow time tf = a/V , its increase with Ca does not withstand the decrease of the dimensional transit time with the pressure drop or ﬂow rate. Surprisingly, τent shows a non-monotonic dependence on the core size ζc (Fig. 4.4b). Intuitively one expects τent to increase with ζc since the larger the inner drop, the larger the energy penalty in deforming it so that the drop can enter the capillary. This seems to hold for ζc up to 0.6. To understand the anomalous decrease of τent for larger ζc , we compare the ﬂow patterns for ζc = 0.64 and 0.80 in Fig. 4.5. When the drop ﬁrst approaches the entry, the above intuition is indeed borne out and 79 Chapter 4. Deformation of a compound drop through a contraction (a) (b) Figure 4.4: (a) Eﬀect of the inner drop on the transit time with changing pressure drop with ζc = 0.72. The Reynolds number varies in the range 7.81×10−3 ≤ Re ≤ 7.81×10−2 . (b) Transit time as a function of the core radius, represented by ζc , at Ca = 0.179 and Re = 1.56 × 10−2 . In both plots the size of the outer drop is ﬁxed with ζs = 1.4. (a) (b) Figure 4.5: Flow ﬁelds inside the compound drop toward the end of the entry process for two core radii: (a) ζc = 0.64, (b) ζc = 0.80. In both cases, Ca = 0.179, Re = 1.56 × 10−2 . The streamlines are drawn in a reference frame ﬁxed to the leading edge of the compound drop, which has an instant velocity of 0.81V in (a) and 1.21V in (b). The gray-scale contours are for the horizontal velocity u. the drop with the larger core attains a lower speed. After the core completely enters the capillary, however, the larger core, though more elongated, requires no additional energy to maintain its shape. Now the smaller inner drop continues to move forward relative to the outer drop surface (Fig. 4.5a); its instantaneous velocity is 0.11V relative to the front tip of the compound drop. But the larger core has practically stopped moving forward as a whole (Fig. 4.5b), with a relative velocity of 0.012V . Meanwhile, internal eddies develop within the inner drop in Fig. 4.5(b) to accommodate the recirculation in the shell ﬂuid. 80 Chapter 4. Deformation of a compound drop through a contraction As a consequence, the smaller core in Fig. 4.5(a) creates stronger velocity gradients on its ﬂanks than the larger core. This translates to a larger drag on the compound drop in Fig. 4.5(a) and a lower speed. This explains the shorter transit time for the drop with the larger core. The diﬀerence in drop speed is reﬂected by the ﬂow rate. The instantaneous Q at the moment when the drop completely enters the smaller tube is also a non-monotonic function of ζc . Besides, one notes the greater drop length l in Fig. 4.5(b) with the larger core. This contrasts the trend in Fig. 4.3(b) and conﬁrms that the drop length also varies non-monotonically with the core size. With even larger ξc , however, the trend is bound to reverse once more since in the limit of ξc → ξs , the compound drop approaches a simple one with twice the interfacial tension. It is interesting to compare our results with compound drop deformation in unbounded elongational ﬂows. With β = 1, equal interfacial tension on the inner and outer surfaces and rc = 0.5rs , Stone & Leal (1990) and Kan et al. (1998) found that for capillary numbers much below the critical value Cacr for breakup, the deformation of the compound drop in an elongational ﬂow is nearly the same as a simple drop of the same ﬂuid and size. This is because the recirculation within the shell ﬂuid is weak and the core deforms little. Thus, the inner surface hardly hinders the overall deformation of the drop. As Ca approaches Cacr , however, the shell is so stretched that the outer interface presses against the core. Then the core does aﬀect the deformation and breakup of the compound drop (Stone & Leal, 1990; Kan et al., 1998). In our conﬁned ﬂow geometry, on the other hand, the compound drop is subject to a geometric constraint that dictates the deformation of the inner drop as well as the shell, regardless of the capillary number. Hence the compound drop sustains milder deformation than the simple drop in Fig. 4.3, and the diﬀerence increases with ﬂow speed or Ca. Obviously, as ζc becomes suﬃciently small, the diﬀerence between simple and compound drop deformation should vanish. 4.4 Summary This note presents simulations of the morphological evolution of a compound drop as it moves along the centerline of a circular tube with a 2:1 gradual contraction. The ﬂow is driven by a ﬁxed pressure diﬀerence imposed on the matrix ﬂuid. The deformation of the two interfaces is captured by a phase-ﬁeld representation, with an interfacial tension determined by the mixing energy in the thin but diﬀuse interfaces. Results show that the inner core generally hinders deformation of the compound drop and prolongs the transit time. However, the eﬀect is non-monotonic in the core size; it is greatest for an intermediate core radius. The underlying mechanism is the core hampering the inner circulation and subjecting the compound drop to stronger shear inside the shell. The expedient of using a binary phase-ﬁeld model places a limitation on our study: the inner core ﬂuid must be identical to the suspending ﬂuid. This is appropriate for W/O/W compound drops encountered in drug delivery. A more general compound drop involves three diﬀerent ﬂuid components. Such systems call for tertiary phase-ﬁeld models as have recently appeared in the literature (Kim et al., 2004; Burman et al., 2004). For 81 Chapter 4. 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Fluids 18, 092105. 84 Bibliography Zhou, C., Yue, P. & Feng, J. J. 2007 Simulation of neutrophil deformation and transport in capillaries using Newtonian and viscoelastic drop models. Ann. Biomed. Eng. 35, 766–780. 85 Chapter 5 Heart-shaped bubbles rising in anisotropic liquids ∗ 5.1 Introduction To investigate the impact of solids on viscoelastic liquids, Akers & Belmonte (2006) dropped spheres of diameter d ∼ 1 cm into an aqueous solution of the wormlike micellar system cetylpyridinium chloride (CPCl)/sodium salicylate (NaSal). Occasionally air bubbles were entrained into the ﬂuid, and would rise in the wake of the ball (Fig. 5.1). Such a bubble assumes a peculiar shape while in the near wake, resembling an inverted heart or a spade (a). The upper surface has sloped shoulders that join in a point. The bottom is relatively ﬂat with a small conical protrusion in the middle. As it rises, both points on top and bottom quickly retract and the bubble appears roughly spherical (b). Further up, the bubble assumes the familiar shape with a round top and a long pointed tail at the bottom (c). The last image resembles that seen of bubbles in viscoelastic polymer solutions (Liu et al., 1995; Herrera-Velarde et al., 2003), the tail being produced by the tensile stress in the wake of the bubble. The inverted-heart shape in the ﬁrst image, on the other hand, has never been reported before. A possible explanation is that the micellar solution has been temporarily transformed into an anisotropic nematic liquid in the near wake of the falling ball. The ordered micelles have a preferred orientation (the “easy direction”) with respect to the bubble surface, deviation from which is penalized by an anchoring energy (Rapini & Papoular, 1969; de Gennes & Prost, 1993; Rey, 2000). Such surface anchoring may compete with the interfacial tension and the bulk molecular order and force the bubble into the peculiar shape. Farther away from the ball, the micelles relax and lose the nematic order, and the bubble shape reverts to that commonly seen in viscoelastic liquids. Although Akers and Belmonte did not present direct evidence for the orientational order in the near wake, a ﬂow-induced nematic state can be inferred from two facts. First, a falling ball produces strong elongation in its wake that tends to modify the microstructural conformation of the ﬂuid. Both ﬂexible polymers and wormlike micelles have been observed to align into “birefringent strands” in the near wake (Harlen, 1990; Handzy & Belmonte, 2004). Second, semi-dilute and concentrated micellar solutions are known to undergo an isotropic-to-nematic transition under shear, for surfactant concentrations down to 1.09 wt.%(Berret et al., 1994; Kadoma & van Egmond, 1998; Fischer et al., 2002). At higher concentrations, micelles commonly exhibit a nematic phase even in equilibrium (Poulin et al., 1999; Mondain-Monval et al., 1999). The CPCl concentration used by Akers and Belmonte is estimated at 2.87 wt.%, well into the semi-dilute regime. ∗ A version of this chapter has been published. Zhou, C., Yue, P., Feng, J. J., Liu, C. & Shen, J., Heart-shaped bubbles rising in anisotropic liquids, Phys. Fluids 19, 041703 (2007) 86 Chapter 5. Heart-shaped bubbles rising in anisotropic liquids (a) (b) (c) Figure 5.1: An air bubble rising in the wake of a falling steel ball in a micellar solution. The bubble volume is roughly 2 cm3 and the images are separated by 33 ms. From Akers & Belmonte (2006) with permission, c 2006 Elsevier B.V. It is therefore reasonable to assume an anisotropic nematic state in the near wake, with the micelles predominantly oriented vertically. In such an environment, the bubble shape is aﬀected not only by the hydrodynamic forces and interfacial tension, but also by the surface anchoring and bulk molecular orientation. 5.2 Theory and numerical methods To test this hypothesis, we have carried out dynamic simulations of bubbles rising in a nematic ﬂuid having a vertical far-ﬁeld orientation. The rheology and orientation of the ﬂuid are modeled by the Leslie-Ericksen theory for liquid crystals (de Gennes & Prost, 1993), and the moving and deforming bubble surface is captured in a diﬀuse-interface framework (Yue et al., 2004). Details about the theoretical model and numerical method can be found elsewhere (Yue et al., 2005, 2006), and only a brief summary is given here. The free energy of a Newtonian-nematic mixture has three contributions: a mixing energy, a bulk elastic energy and a surface anchoring energy: λ λ |∇φ|2 + 2 (φ2 − 1)2 , 2 4 1 (|n|2 − 1)2 , = K ∇n : (∇n)T + 2 4δ 2 A (n · ∇φ)2 . = 2 fmix = (5.1) fbulk (5.2) fanch 87 (5.3) Chapter 5. Heart-shaped bubbles rising in anisotropic liquids In fmix , φ is the phase-ﬁeld variable with φ = 1 and −1 in the nematic and Newtonian bulk phases and φ = 0 at the interface, λ is the mixing energy density and is the capillary width. fbulk is the Frank energy with a single elastic constant K, n being the director, regularized to permit defects where |n| deviates from unity over a small region of size δ (de Gennes & Prost, 1993; Liu & Walkington, 2000). For fanch , we adapt the Rapini-Popoular form (Rapini & Papoular, 1969) for planar anchoring to our diﬀuseinterface formalism, with A being the anchoring energy density and the easy direction being perpendicular to the interface normal ∇φ. We have tested homeotropic anchoring as well, but it turns out√ to be irrelevant to Fig. 5.1 and will not √ be discussed here. In the sharp-interface limit, 2 3 2λ gives the interfacial tension σ and 2 32A becomes the anchoring strength W (Rapini & Papoular, 1969; Yamamoto, 2001; Yue et al., 2004). A variational procedure on the free energy, supplemented by the appropriate dissipative terms, leads to the governing equations (Yue et al., 2004): ∇ · v = 0, ρ ∂v + v · ∇v = −∇p + ∇ · σ, ∂t ∂φ φ(φ2 − 1) , + v · ∇φ = γλ∇2 −∇2 φ + 2 ∂t h = γ1 N + γ2 D · n, (5.4) (5.5) (5.6) (5.7) where the deviatoric stress tensor 1+φ 1+φ σ = −λ(∇φ∇φ) − K (∇n) · (∇n)T − A(n · ∇φ)n∇φ + σ + (1 − φ)μD, (5.8) 2 2 with the Leslie viscous stress (de Gennes & Prost, 1993) σ = α1 D : nnnn + α2 nN + (|n|2 −1)n ∇n − 1+φ α3 N n+α4 D+α5 nn·D+α6 D·nn, and the molecular ﬁeld h = K ∇ · 1+φ − 2 2 δ2 A(n · ∇φ)∇φ. In the above, γ is the Cahn-Hilliard mobility parameter, D = 12 [∇v + (∇v)T ] and N = ddtn − 12 [(∇v)T − ∇v] · n. The α’s are the Leslie viscous coeﬃcients, γ1 = α3 − α2 , γ2 = α2 + α3 = α6 − α5 , and μ is the viscosity of the Newtonian phase. The governing equations are discretized on a ﬁnite-element grid using the PetrovGalerkin formulation with streamline upwinding for the constitutive equation (Yue et al., 2006). With axisymmetry, the 2D computational domain is covered by an unstructured grid of triangular elements. A key element of the numerical algorithm is an adaptive meshing scheme that deploys the ﬁnest grids around the interface and adaptively coarsens and reﬁnes the grid as the interface moves. Numerical experiments with grid reﬁnement and time-step reﬁnement have been carried out to ensure adequate resolution, and the accuracy and robustness of the code has been established by benchmarking against rising bubbles in Newtonian ﬂuids and other known solutions (Yue et al., 2006). 5.3 Results and discussion To analyze the eﬀects of ﬂow and molecular order separately, we ﬁrst computed the equilibrium shape of a stationary bubble in a nematic (Fig. 5.2). The static shape 88 Chapter 5. Heart-shaped bubbles rising in anisotropic liquids (a) (b) (c) Figure 5.2: Equilibrium shapes of stationary bubbles in a nematic with a vertical far-ﬁeld orientation. The curves depict the director ﬁeld and the bubbles are axisymmetric with planar anchoring. Both σ and K are ﬁxed at σa/K = 3, and the three plots correspond to increasing anchoring strength: W a/K = 0.6, 15 and 150. depends on the competition among the interfacial tension σ, the anchoring strength W and the bulk elastic energy K. Minimizing the total free energy, the bubble typically takes on a lemon shape (Fig. 5.2b), and the degree of elongation is determined by two dimensionless groups: W/σ and W a/K, a being the equivalent radius of the bubble. For weak anchoring (W a/K < 1), n readily deviates from the easy direction and bulk elasticity can exert little inﬂuence on the bubble shape (Fig. 5.2a). For strong anchoring, however, the bubble becomes more elongated to reduce the bulk distortion at the expense of increased interface area (Fig. 5.2c). Note the “boojum” defects at the poles of the bubble, where bulk distortion creates large surface curvatures. Both the lemon shape and the boojum defects have been reported in prior experiments (Nastishin et al., 2005). The shape of a rising bubble is also inﬂuenced by ﬂow eﬀects, expressed by two digη 4 otvos number mensionless groups (Grace, 1983): the Morton number M o = ρσ 3 and the E¨ 2 Eo = ρga , ρ and η being the liquid density and viscosity and g the gravitational accelerσ ation. In Akers and Belmonte’s experiment (Akers & Belmonte, 2006), the micellar solution is strongly shear-thinning, and its surface tension σ also varies depending on the relaxation and redistribution of the surfactants. Based on the data given, the experimental conditions correspond to 8.70 < Eo < 20.3 and 10−2 < M o < 103 . The simulations will use Eo and M o in these ranges, with the characteristic viscosity η = (α4 −α2 +α6 )/2 being the average between two Miesowicz viscosities (de Gennes & Prost, 1993) and the Leslie coeﬃcients scaling as α1 : α2 : α3 : α4 : α5 : α6 = 0 : −0.32 : −0.08 : 1.0 : 0.28 : −0.12. The anchoring strength W and bulk energy K are more diﬃcult to estimate. We have tested a range of W/σ and W a/K values, and their relevance to real materials will be discussed later. For computational conveniences, we have assigned equal density and 89 Chapter 5. Heart-shaped bubbles rising in anisotropic liquids (a) t = 0 (b) t = 1.40 (c) t = 2.64 (d ) t = 5.27 Figure 5.3: Snapshots of the rising bubble in a nematic with planar anchoring. The farﬁeld molecular orientation is vertical. Eo = 11.2, M o = 1.01, W/σ = 5 and W a/K = 15. Time is scaled by 2a/g. Steady state is reached in frame (d ). viscosity to the “bubble” and matrix ﬂuid; the buoyancy force is replaced by an upward body force acting on the bubble. The density and viscosity of the bubble phase aﬀect the internal recirculation but not so much the bubble shape. Figure 5.3 shows snapshots of a bubble during its rise. The initial shape is spherical, with a uniform director ﬁeld (Fig. 5.3a). Within a time scale of ηa/W (=1.41 in this case), the director relaxes toward the easy direction on the bubble surface, deforming it into a lemon shape resembling Fig. 5.2(b). At this point the bubble velocity is about 46% of its terminal value. As the rise velocity increases, so does the hydrodynamic drag due to viscous and inertial forces. As a result, the shoulders are pushed down and ﬂattened, and the bubble loses fore-aft symmetry (Fig. 5.3c). Eventually a steady state is reached in Fig. 5.3(d ), with a bubble shape closely resembling the experimental picture in Fig. 5.1(a). The terminal velocity U corresponds to a Reynolds number Re = ρU a = 3.39, while the experimental value in Akers & Belmonte (2006) is estimated as η a 1.72 < Re < 3.90. The steady-state Ericksen number Er = ηU = 2.05. The ﬂat bottom K is reminiscent of the bubble shape in Newtonian liquids(Grace, 1983) at a comparable Re, except for the protrusion in the middle due to the boojum defect. This simulation 90 Chapter 5. Heart-shaped bubbles rising in anisotropic liquids oﬀers strong support for our hypothesis that it is the nematic order in the wake of the ball that aﬀords the bubble its unusual shape. 5.4 Summary Basic features of the inverted-heart shape—sloped shoulders and a ﬂat bottom with a conical protrusion—are obtained if both W and K are suﬃciently large; a rough guideline is W σ and K 0.1σa. These numerical parameters need to be related to the experiment (Akers & Belmonte, 2006). Since no in-situ characterization was done on the micellar solution, we are limited to circumstantial evidence. For nematic wormlike micelles, anchoring arises from entropic eﬀects such as excluded volume, and a scaling argument on W predicts strong planar anchoring (W a/K 1) (Poulin et al., 1999). This has been conﬁrmed experimentally through quadrupolar interactions among colloidal particles (Poulin et al., 1999; Mondain-Monval et al., 1999). Thus, it is reasonable to assume planar anchoring in our computations, and the condition W a/K 1 is consistent with the numerical parameters in Fig. 5.3. However, the estimated W is smaller than the surface tension of common thermotropic liquid crystals (Sonin, 1995). In the experiment (Akers & Belmonte, 2006), the abundance of surfactants may have reduced σ below W . For lyotropic nematics made of self-assembled molecular aggregates similar to wormlike micelles, W/σ > 1 has been observed for domains in a bi-phasic system (Kaznacheev et al., 2003; Nastishin et al., 2005). The mechanism revealed by the heart-shaped bubble has potential applications in other complex ﬂuids that feature nematic-isotropic interfaces, such as nematic emulsions (Tixier et al., 2006) and polymer-dispersed liquid crystals (PDLC) (Mucha, 2003). In self-assembly of colloids for making photonic crystals (Manoharan et al., 2001), a nematic matrix will aﬀord better control of the spatial periodicity as well as the possibility of nonspherical voids with better performance and tunability (Nazarenko et al., 2001; Velikov et al., 2002). In manufacturing PDLC ﬁlms, planar anchoring inside nematic drops tends to produce a bipolar shape similar to those in Fig. 5.2. The drop shape and orientation can be exploited to maximize the contrast between the on- and oﬀ-states (Drzaic & Muller, 1989; Chan et al., 2001). Finally, the phase morphology of bicontinuous polymerliquid crystal networks (Jeong et al., 1999) and “reversed mode PDLC” (Macchione et al., 2000) depends on the coupling between surface anchoring, bulk elasticity and deformation, and the mechanism discussed in this Letter is expected to play a key role. 91 Bibliography Akers, B. & Belmonte, A. 2006 Impact dynamics of a solid sphere falling into a viscoelastic micellar ﬂuid. J. 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Phys. 219, 47–67. 94 Chapter 6 The rise of bubbles and drops in a nematic liquid crystal ∗ 6.1 Introduction Nematic liquid crystals exhibit special electro-optical properties and ﬁnd applications in numerous modern technologies. As complex ﬂuids, they are distinguished microscopically by molecular alignment and long-range orientation order, and macroscopically by a liquid-solid duality in that they ﬂow as anisotropic viscous ﬂuids but resist orientational distortion as elastic solids (de Gennes & Prost, 1993). In a ﬂuid mechanical context, the motion of a particle or drop in a nematic is of fundamental interest, being the counterpart of the Stokes or Hadamard-Rybczynski problem in viscous Newtonian ﬂuids. Besides, suspensions and emulsions in nematic matrices show intriguing mesoscopic structures and mechanical properties that suggest new applications (Poulin et al., 1997b; Poulin & Weitz, 1998; Loudet et al., 2000; Tixier et al., 2006). Particle motion in nematic liquid crystals is much more complex than the Stokes problem. Even in a static nematic, insertion of a particle or drop normally causes the nucleation of orientational defects (Poulin et al., 1997b; Lavrentovich, 1998; Feng & Zhou, 2004). The liquid crystal molecules prefer a certain orientation on interfaces, the most common being homeotropic (normal) and planar (tangential) anchoring. If the orientation ﬁeld surrounding the drop or particle comes into conﬂict with the far-ﬁeld orientation, defects form. These may be seen as singularities in the director ﬁeld n(r), which represents the average molecular orientation at each spatial point. For a particle with homeotropic anchoring, experiments have recorded two types of defects: a “satellite” point defect (Poulin et al., 1997b; Poulin & Weitz, 1998; Lubensky et al., 1998) and a “Saturn-ring” line defect that encircles the particle on its equator (Mondain-Monval et al., 1999; Gu & Abbott, 2000). With planar anchoring, two surface defects known as “boojums” form at the poles (Poulin & Weitz, 1998). Orientational defects have long been an important subject of liquid crystal physics and indeed condensed matter physics in general (Trebin, 1982; Kl´eman, 1983). For moving particles, the earliest studies were falling-ball experiments to measure the eﬀective viscosity of liquid crystals (White et al., 1977; Kuss, 1978). More recently, Poulin et al. (1997a) used the “Stokes drag” to verify the dipolar attraction force between two droplets in a nematic ﬂuid. In such dynamic situations, the ﬂow modiﬁes the director ﬁeld and defect conﬁguration near the particle. The latter in turn aﬀect the rheology of the liquid crystal and thus the ﬂow ﬁeld. Therefore, the two-way coupling between ﬂow and microstructure is the key physics governing particle motion in nemat∗ A version of this chapter has been published. Zhou, C., Yue, P. & Feng, J. J., The rise of Newtonian drops in a nematic liquid crystal, J. Fluid Mech. 593, 385-404 (2007) 95 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal ics. In general, such coupling has been formulated by constitutive theories for nematic liquid crystals (de Gennes & Prost, 1993; Rey & Tsuji, 1998; Feng et al., 2000). Owing to the rheological complexity, only a handful of theoretical studies have appeared on the moving particle problem, most of which sought to decouple the ﬂow ﬁeld and the director ﬁeld (Stark, 2001). For instance, the director ﬁeld may be ﬁxed at the static solution, and the resulting ﬂow ﬁeld and drag are calculated (Ruhwandl & Terentjev, 1996; Stark & Ventzki, 2001). This corresponds to the low Ericksen number (Er) limit, where the viscous forces are too weak to modify the orientational ﬁeld maintained by elasticity. Conversely, the Newtonian ﬂow ﬁeld may be prescribed, and the director ﬁeld n(r) is calculated as a result (Diogo, 1983; Yoneya et al., 2005). This may be linked to the high-Er limit. Stark & Ventzki (2002) seem to be the ﬁrst to tackle the ﬂow-director two-way coupling at ﬁnite Er. In ﬂow around a sphere with a satellite point defect, they predicted a counter-intuitive ﬂow eﬀect that moves the defect upstream. This was contradicted by Yoneya et al. (2005) who showed that the defect shifts downstream at a similar Ericksen number. But the latter study prescribed the Stokes ﬂow ﬁeld, and it is unclear whether the decoupling accounts for the discrepancy. To our knowledge, the only other coupled study is Fukuda et al. (2004), who showed that the ﬂow tends to convect the n ﬁeld downstream, while n modiﬁes the velocity ﬁeld and makes it fore-aft asymmetric. Unfortunately, Fukuda et al. assumed an isotropic viscosity and thus omitted an important component of the liquid crystal rheology. Therefore, a “complete solution” that fully couples ﬂow and director ﬁelds and incorporates viscous-elastic duality and anisotropy is not yet available. Most prior calculations assumed rigid anchoring on the particle surface. In reality, the anchoring strength is ﬁnite, representable by an anchoring energy, and has a major role in deﬁning the defect conﬁguration (Lubensky et al., 1998; Mondain-Monval et al., 1999). Furthermore, only solid particles have been considered in theoretical studies so far, even though most of the experimental observations have come from emulsions with isotropic droplets suspended in nematics (Poulin & Weitz, 1998). If the interfacial tension is not so strong as to overwhelm the surface anchoring, the interplay between the two is known to lead to unique drop and bubble shapes (Nastishin et al., 2005; Akers & Belmonte, 2006; Zhou et al., 2007). These lacunae in our current understanding have motivated the present simulations based on the Leslie-Ericksen theory. Simulating the rise of Newtonian drops in a nematic liquid crystal is a computational challenge because of the well-known numerical diﬃculties in handling moving and deforming interfaces as well as the complex rheology of the nematic liquid crystal. Not only are the rheological properties and stresses discontinuous across the interface, they are anisotropic and evolving with the microstructure in the nematic component. In principle, the balance between the stresses and the surface tension determines the motion of the interface, which must be tracked dynamically on a moving grid while solving for the ﬂow in each component. As an alternative, we have developed an energy-based diﬀuseinterface method that handles both the interface and the rheology in a uniﬁed framework (Yue et al., 2004; Feng et al., 2005). The interface is now a thin diﬀuse layer deﬁned by a phase-ﬁeld variable. A mixing energy governs the interaction of the two components. 96 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal As long as the microstructure of the complex ﬂuid is describable by a free energy, as is the case for liquid crystals, that energy can be combined with the mixing energy to form the total free energy of the system. A formal variational procedure then leads to the proper governing equations of the two-ﬂuid system. To solve these, Yue et al. (2006b) have developed 2D and axisymmetric ﬁnite-element algorithms based on adaptive mesh generation, which is key to resolving the thin interface. The method has proved accurate and eﬃcient in simulating dynamics of viscoelastic drops and jets (Yue et al., 2005d , 2006a; Zhou et al., 2006), and will be adapted to the problem at hand. This study has three objectives: (1) to demonstrate that the motion of Newtonian drops in a nematic ﬂuid can be successfully simulated by our diﬀuse-interface method, incorporating complex rheology, deformable interfaces and a variable anchoring strength; (2) to investigate how ﬂow modiﬁes the orientational ﬁeld and especially the defect conﬁguration near the drop; and (3) to investigate how the director ﬁeld modiﬁes the ﬂow in return, especially how the rheological anisotropy aﬀects the rising velocity of the drops and the drag force on them. 6.2 Theory and numerical method Yue et al. (2004, 2006b) have described the theoretical model and the numerical method in detail, and validated the methodology by benchmark problems. Planar 2D and axisymmetric applications to drop dynamics have been reported recently (Yue et al., 2005a,b,c,d , 2006a; Zhou et al., 2006). Therefore, we will only summarize the main ideas and give the governing equations for a two-component mixture of a Newtonian ﬂuid and a nematic liquid crystal. The diﬀuse interface has a small but non-zero thickness, inside which the two components are mixed and store a mixing energy. We deﬁne a phase-ﬁeld variable φ such that the concentrations of the nematic and Newtonian components are (1 + φ)/2 and (1 − φ)/2, respectively. Then φ = 1 in the bulk nematic phase, and φ = −1 in the bulk Newtonian phase. The interface is taken to be the level set φ = 0. There are three types of free energies in this system: mixing energy of the interface, bulk distortion energy of the nematic, and the anchoring energy of the liquid crystal molecules on the interface: λ (φ2 − 1)2 , |∇φ|2 + 2 22 K (|n|2 − 1)2 , = ∇n : (∇n)T + 2 2δ 2 A (n · ∇φ)2 (planar anchoring) 2 = A 2 2 2 [|n| |∇φ| − (n · ∇φ) ] (homeotropic anchoring). 2 fmix = (6.1) fbulk (6.2) fanch (6.3) √ In fmix , λ is the mixing energy density, is the capillary width and the ratio 2 2λ/3 produces the interfacial tension σ (Jacqmin, 1999; Liu & Shen, 2003; Yue et al., 2004). fbulk is the Frank energy with a single elastic constant K. Diﬀerent elastic constants may be assigned to diﬀerent modes of distortion (de Gennes & Prost, 1993), but we use the 97 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal one-constant approximation for simplicity. Note that fbulk is regularized to permit defects where |n| deviates from unity over the defect core of size δ. The second term, devised by Liu & Walkington (2000) after the Ginzburg-Landau energy of Eq. (6.1), represents the distortion energy of the defect by an energy penalty against the shortening of |n|, eﬀectively using |n| as an order parameter. For fanch , we adapt the Rapini-Popoular form (Rapini & Papoular, 1969) to√our diﬀuse-interface framework, with A being the anchoring energy density and W = 2 2A/3 giving the surface anchoring strength (Yamamoto, 2001). Now we have the total free energy density for the two-phase material: f (φ, n, ∇φ, ∇n) = fmix + 1+φ fbulk + fanch . 2 (6.4) A variation on the free energy, supplemented by the various dissipative terms, leads to the following governing equations for the conﬁguration variables v, p, φ and n (Yue et al., 2004): ∇ · v = 0, ρ ∂v + v · ∇v = −∇p + ∇ · σ − ρgez , ∂t φ(φ2 − 1) ∂φ , + v · ∇φ = γλ∇2 −∇2 φ + 2 ∂t h = γ1 N + γ2 D · n. (6.5) (6.6) (6.7) (6.8) The density ρ is the average between the nematic density ρ1 and the Newtonian density ρ1 + 1−φ ρ2 , g is the gravitational acceleration and ez is the upward unit ρ2 : ρ = 1+φ 2 2 vector. The phase-ﬁeld variable φ obeys the Cahn-Hilliard equation, γ being the mobility parameter of the diﬀuse interface (Yue et al., 2004, 2007). The deviatoric stress tensor σ = −λ(∇φ ⊗ ∇φ) − K 1+φ 1−φ 1+φ (∇n) · (∇n)T − G + σ + μ[∇v + (∇v)T ], (6.9) 2 2 2 with G = A(n · ∇φ)n ⊗ ∇φ for planar anchoring and G = A[(n · n)∇φ − (n · ∇φ)n] ⊗ ∇φ for homeotropic anchoring, and μ being the viscosity of the Newtonian component. σ is the Leslie viscous stress in the nematic phase (Leslie, 1968) σ = α1 D : nnnn + α2 nN + α3 N n + α4 D + α5 nn · D + α6 D · nn, (6.10) where α1 to α6 are the Leslie viscous coeﬃcients observing the Onsager relationship (de Gennes & Prost, 1993): α2 + α3 = α6 − α5 . D = 12 [∇v + (∇v)T ] is the strain rate tensor, and N = ddtn − 12 [(∇v)T − ∇v] · n is the rotation of the director n with respect to the background ﬂow ﬁeld. The n ﬁeld evolves in the ﬂow according to a balance between elastic and viscous torques as given in Eq. (6.8). The elastic torque is represented by the molecular ﬁeld (de Gennes & Prost, 1993): h=K ∇· 1 + φ (n2 − 1)n 1+φ ∇n − − g, 2 2 δ2 98 (6.11) Chapter 6. The rise of bubbles and drops in a nematic liquid crystal with g = A(n · ∇φ)∇φ for planar anchoring, and g = A[((∇φ · ∇φ)n − (n · ∇φ)∇φ] for homeotropic anchoring. Both g and G derive from the anchoring energy fanch through a variational procedure (Yue et al., 2004). The coeﬃcients γ1 = α3 − α2 and γ2 = α3 + α2 . Obviously, the apparent viscosity of the nematic depends on the orientation of n relative to the ﬂow. This viscous anisotropy is commonly represented by the Miesowicz viscosities in a simple shear ﬂow (de Gennes & Prost, 1993): 1 (−α2 + α4 + α5 ), 2 1 (α3 + α4 + α6 ), = 2 η1 = (6.12) η2 (6.13) which are measured with n held perpendicular and parallel to the ﬂow direction, respectively; η1 > η2 . If n makes an angle θ with the ﬂow, the general formula for the shear viscosity is (Carlsson, 1984): η(θ) = η2 − (α2 + α3 ) sin2 θ. (6.14) A third Miesowicz viscosity may be deﬁned with n along the vorticity axis. This is irrelevant to the present study which constrains n to the meridional plane in axisymmetric geometries. Without external ﬁelds or wall anchoring, n achieves a steady alignment in simple shear if α2 /α3 > 0, but tumbles endlessly if the ratio is negative. Most liquid crystals are of the aligning type (de Gennes & Prost, 1993), and the distinction is insigniﬁcant in complex ﬂow ﬁelds as simulated here. Thus, we have used α values based on the aligning PAA and MBBA in the rest of this paper. To arrive at the Cahn-Hilliard equation (6.7), we have omitted from the right-handside coupling terms between fmix and the nematic energies. These are insigniﬁcant as long as the interface stays narrow. In fact, the Cahn-Hilliard diﬀusive dynamics has a visible eﬀect only during singular events such as ﬁlm rupture (Yue et al., 2005a). In the current context, the diﬀuse interface may be seen as merely a numerical device for treating a moving internal boundary. The governing equations are solved, in axisymmetric geometries, by a numerical scheme AMPHI that employs Galerkin ﬁnite elements with adaptive meshing (Yue et al., 2006b). The latter has proved to be key to accurate phase-ﬁeld simulations, and the interface of thickness O(4 ) requires roughly 10 grid points to resolve (Feng et al., 2005). In addition, the defect regions are covered with ﬁne grids as well (see Fig. 6.3b). Since the size of the defect core is comparable to the interfacial thickness (Stark, 1999), δ is chosen to be 4 . We use implicit time-stepping, with the time step automatically adjusted according to the motion of the interface. Numerical experiments with grid reﬁnement and time-step reﬁnement have been carried out (Yue et al., 2006b), and adequate resolution is ensured for the simulations presented in the following. 99 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal Figure 6.1: Simple shear ﬂow of a nematic with homeotropic anchoring on the walls. The director orientation is indicated by θ(y), and the velocity v(y) deviates from a linear proﬁle because of the anisotropic viscosity. 6.3 6.3.1 Results and discussion Simple shear ﬂow as validation Consider the simple shear ﬂow of a nematic between parallel walls in Fig. 6.1, with rigid homeotropic anchoring on the walls. A 1D analytical solution is available if α1 vanishes, and we will compute the same ﬂow in a 2D domain and use the exact solution to validate our numerical treatment of the Leslie-Ericksen theory. In the 1D solution due to Carlsson (1984), the velocity proﬁle v(y) and orientation proﬁle θ(y) are given by coupled equations: y τ dy − V, 2 0 η2 − (α2 + α3 ) cos θ(y) √ θ 1 K y(θ) = √ dθ, 2τ 0 F (θ) − F (θm ) √ 2 θm 2 K τ = , L2 0 F (θ) − F (θm ) v(y) = F (θ) = α2 + √ η1 η2 η1 η2 tan−1 η2 tan θ − θ, η1 (6.15) (6.16) (6.17) (6.18) where τ is the constant shear stress on the plates determined from θm = θ L2 , and θm , the largest rotation angle at the center between the walls, is in turn determined by the condition V = v(L). This solution assumes α1 = 0, and is written in a slightly simpliﬁed form here because of the one-constant approximation. Our 2D computation uses a domain of length 5L divided into 8235 triangular elements. Figure 6.2 compares our solution with Carlsson’s analytical solution, and the two are in excellent agreement. Note that the maximum director angle θm = 77.5◦ is short of the 100 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal (a) (b) Figure 6.2: Comparison between our results and the 1D exact solution of Carlsson (1984). (a) The director orientation proﬁle; (b) the velocity proﬁle. The Leslie coeﬃcients are α1 = 0, α2 /α4 = −1.78, α3 /α4 = −0.056, α5 /α4 = 1, α6 /α4 = −0.83, and the Leslie angle θ0 = tan−1 α2 /α3 = 80◦ . The ﬂow velocity corresponds to an Ericksen number Er = 35. Leslie angle θ0 = 80◦ because at Er = η¯V L/K = 35, the viscous eﬀect is not strong enough to completely dominate the elastic eﬀect. In Er, the characteristic viscosity of the nematic is taken to be η¯ = (η1 + η2 )/2 = (α3 + α4 + α5 )/2. The rotation of n into the ﬂow direction reduces the local viscosity, and the velocity proﬁle v(y) reacts by diminishing the shear rate at the walls and increasing it in the center to maintain a constant shear stress. In this case, the minimum viscosity at the center ηm is such that η1 /ηm = 13.4 and ηm /η2 = 2.55. 6.3.2 Static orientational defects When a drop has planar anchoring on its surface, boojums are the only possible defects, even when the drop is moving in the liquid crystal. With homeotropic anchoring, on the other hand, multiple defect patterns may appear and interesting transformations take place. Thus we will only consider homeotropic anchoring in this subsection, examining defects surrounding stationary particles as a preface to the ﬂow-induced transformation discussed in the next subsection. A more or less coherent picture has emerged about defects near a stationary particle (Ruhwandl & Terentjev, 1997; Stark, 2001; Feng & Zhou, 2004). The Saturn ring and the satellite point defect are the two possible conﬁgurations (Fig. 6.3a and c), and their stability depends on the relative importance of surface anchoring and bulk elasticity, represented by the dimensionless group AK = Wa , K 101 (6.19) Chapter 6. The rise of bubbles and drops in a nematic liquid crystal a being the eﬀective radius of the drop. A Saturn ring incurs more distortion to the surface anchoring while a satellite costs more bulk energy. Thus, rings are favored at smaller AK . Indeed, the point defect becomes unstable below a critical AK , and spontaneously opens into a Saturn ring. For suﬃciently weak anchoring, the Saturn ring shrinks onto the particle surface or even into the particle as an “imaginary ring” (Kuksenok et al., 1996). For larger AK , both point and ring defects are stable, and either can be realized from proper initial conditions. The point defect becomes energetically more favorable with increasing AK , but it is unclear whether the ring ever becomes unstable (Ruhwandl & Terentjev, 1997; Feng & Zhou, 2004). In regularizing the Leslie-Ericksen theory to allow defects (Eq. 6.2), we treat n(r) as a vector ﬁeld. In reality, the molecular orientation is a pseudo-vector that does not distinguish n and −n. As a consequence, our vector-based theory cannot allow defect lines of half strength; the surrounding n ﬁeld inevitably contains an apparent discontinuity between n and −n and thus incurs an inﬁnite elastic energy. Instead of the Saturn-ring defect with a strength of −1/2, therefore, we predict a surface ring as shown in Fig. 6.3(d ). With this caveat, we reproduce all the features noted above, including the eﬀect of AK on defect stability. In particular, for strong enough anchoring, the satellite defect is produced if the initial n is radial near the drop surface, while the surface ring arises from an initially uniform n ﬁeld. An additional parameter, Aσ = W , σ (6.20) governs the shape of the drop, and a small value is used in most of the simulations to ensure a nearly spherical drop. Figure 6.4 plots the position of the satellite point defect as a function of AK . The increase of anchoring energy moves the point defect farther from the drop, as is noted by Ruhwandl & Terentjev (1997). At the limit of AK → ∞, rd /a → 1.35, √ which agrees well with prior calculations (rd /a = 1.26 by Lubensky et al. 1998 and 2 by Pettey et al. 1998) and measurement (rd /a = 1.4 ± 0.1 by Cluzeau et al. 2001). With decreasing AK , the defect approaches the drop and causes a protrusion on the interface. As AK falls below a threshold value, around 10 in this case, the point defect opens up into a surface ring. This threshold AK is close to the previous Monte Carlo prediction of approximately 7 (Ruhwandl & Terentjev, 1997). The distance rd has some practical implications. Potentially it can be used as a measurement of the anchoring strength W , which is otherwise diﬃcult to determine. Furthermore, rd also determines the particle spacing in self-assembled arrays of droplets in nematic emulsions (Poulin & Weitz, 1998). 6.3.3 Flow-induced transformation of defect conﬁguration Static defects may be driven from one conﬁguration to the other by an external electric or magnetic ﬁeld (Terentjev, 1995; Loudet & Poulin, 2001). It will be interesting to see whether similar transitions can be eﬀected by the ﬂow surrounding a rising drop. Figure 6.5 shows schematically the computational domain for simulating rising drops in a nematic medium. A spherical drop of radius a is initially centered at (0, 4a), with either 102 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal (a) (b) (c) (d ) Figure 6.3: Defect conﬁgurations near a drop with homeotropic anchoring. (a) The satellite point defect, indicated by the black dot, within the n ﬁeld. Aσ = 0.05, AK = 100. (b) The ﬁnite-element mesh for (a) is reﬁned around the interface and the satellite defect. (c) Drawing of the director ﬁeld for a Saturn-ring defect. (d ) The surface-ring defect for Aσ = 0.05, AK = 100, indicated by black dots on the equator of the drop. 103 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal Figure 6.4: Position of the satellite point defect near a stationary drop with homeotropic anchoring. rd is the distance between the defect core and the centroid of the drop. For AK < 10, the point defect loses stability and gives way to a surface ring on the equator. Aσ = 0.05. Figure 6.5: Computational domain for a Newtonian drop rising in a nematic whose farﬁeld orientation is vertical. The geometry is axisymmetric and only half of the meridian plane is used in the computation. The drawing is not to scale; H = 20a and L = 10a. 104 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal homeotropic or planar anchoring, although the latter will not be discussed until the next subsection. The far-ﬁeld director orientation is vertical and parallel to the drop motion. A horizontal far-ﬁeld n would upset axisymmetry and require a fully 3D simulation. We also disallow azimuthal components of n and v. Thus, n = (0, 1) on the bottom, side and top walls. The velocity vanishes on the bottom and side walls, but the top is assigned a stress-free condition. On the axis of symmetry, we require ∂/∂r = 0 for all variables except the radial components of n and v: nr = 0 and vr = 0. For numerical parameters, the capillary width = 0.01a, and a small grid size h1 = 0.006a is used on the interface and near the defect (cf. Fig. 6.3b). Inside the drop and in the matrix, the grid sizes are h2 = 0.08a and h3 = 0.1a, respectively. These prescribed values are guidelines for mesh generation, and the actual mesh is spatially unstructured and varies adaptively during the simulation. The rise of drops or bubbles in a nematic ﬂuid is governed by 10 dimensionless numbers: ρ2 (drop-to-matrix density ratio), ρ1 μ β = (drop-to-matrix viscosity ratio), η¯ Δρga2 Eo = (E¨otvos number), σ Δρg η¯4 Mo = (Morton number), ρ21 σ 3 α = (6.21) (6.22) (6.23) (6.24) plus the 4 ratios of the 5 independent Leslie viscosities and the static parameters Aσ and AK . In Eo and M o, Δρ = ρ1 − ρ2 . Drop and bubble shapes deviate from the spherical at large Eo (Grace et al., 1976), and our code has been shown to accurately capture this eﬀect for Newtonian ﬂuids (Yue et al., 2006b). For all results presented hereafter, α = 0.5 and β = 0.514. A set of Leslie coeﬃcients based on those of PAA (Chandrasekhar, 1992) and MBBA (de Gennes & Prost, 1993) are chosen as the baseline. Values of α1 , α3 , α4 and α5 are ﬁxed at the ratio of α1 : α3 : α4 : α5 = 5 : −1 : 18 : 18. This also ﬁxes the characteristic viscosity η¯ = (α3 + α4 + α5 )/2. Then the viscosity ratio η1 /η2 is varied through α2 (and α6 according to the Onsager relationship) to probe the eﬀect of viscous anisotropy. Based on the terminal velocity U of the rising drop, we determine η and Ericksen number Er = η¯U a/K. the steady-state Reynolds number Re = ρ1 U a/¯ Our results show that the ﬂow shifts the orientation ﬁeld downstream, and modiﬁes the relative stability of the ring and point defects. The “phase diagram” in Fig. 6.6 depicts the stability of each conﬁguration near a steadily rising drop, parametrized by Er and AK that denote respectively the strengths of ﬂow and surface anchoring as compared with the bulk elasticity. Six zones may be identiﬁed with diﬀerent defect conﬁgurations. In zone I, the drop has an imaginary ring inside but no defects outside. In zone II, the surface ring is the sole stable conﬁguration, while in zone IV, only point defects occur. In zone III, V and VI, both the surface ring and the satellite point defect are locally stable and either may appear depending on initial conditions. 105 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal Figure 6.6: A “phase diagram” of steady-state defect conﬁgurations. AK and Er are varied by tuning K and g, respectively. Aσ = 0.01, η1 /η2 = 3. For vanishing Er, the transitions from zone I to II and III with increasing anchoring strength are well known from static studies (Kuksenok et al., 1996; Ruhwandl & Terentjev, 1997). This simple picture holds up to Er ∼ 1. In this weak-ﬂow regime, the only ﬂow eﬀect is to shift the surface ring or satellite defect downstream. The shift is more pronounced for higher AK , since stronger surface anchoring favors a smaller ring. For higher Er, a transition from zone II to V or from zone III to IV takes place depending on AK . In zone V, the point defect becomes locally stable; an initial point defect can now be stabilized in the wake of the drop by a suﬃciently strong ﬂow whereas in zone II, it would have opened up into a ring defect. If the initial condition has a ring defect, it remains stable in zone V but shifts downstream and shrinks in radius with increasing Er. Going from zone III to IV with increasing Er, the ring defect loses stability. On a drop that initially bears a surface ring, the ﬂow sweeps the ring downstream on the drop as it rises. If its terminal velocity puts it in zone IV, the surface ring will be shed into a satellite point defect in the wake. Starting with an initial point defect, we have only simulated the conﬁguration with the defect in the wake of the drop. Having a point defect upstream of the drop appears unlikely in reality and may indeed be unstable to 3D disturbances. Throughout zone III and zone IV, the point defect remains stable and shifts downstream with increasing Er. The steady-state position of the ring and point defects is shown in Fig. 6.7 as a function of Er that crosses from zone III to IV. Increasing Er further from zone IV, there is another transition to zone VI where the surface ring regains stability. At such high Er, the drop assumes an oblate shape, and an initial equatorial ring turns into a small surface ring near the bottom that cannot be shed into the wake as a point defect. Figure 6.8 shows an example of the steady-state director ﬁeld in zone VI. Thanks to the oblate shape, the ﬂow near the rear stagnation 106 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal Figure 6.7: Steady-state position of the defect near a rising drop as a function of the Ericksen number. rd is the distance between the centroid of the drop and the point defect (rd /a > 1) or the center of the surface ring (rd /a < 1). The arrow indicates the transition from zone III to IV at Er = 1.10 when the surface ring defect gives way to a point defect. Aσ = 0.01, AK = 30, η1 /η2 = 3. Figure 6.8: Director orientation around a steadily rising oblate drop in zone VI, with a small surface ring indicated by two black dots in the rear of the drop. η1 /η2 = 3, Aσ = 0.01, AK = 20, Er = 53.6, Eo = 1.20, M o = 1.22 × 10−6 and Re = 21.9. 107 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal point of the drop is much reduced as compared with zone IV. The viscous forces are thus much weaker and can no longer drive the surface ring oﬀ. Similar conﬁgurations occur at high Er in zone V. In fact, zone V and VI are connected at the top, and their division is mostly a result of our describing the phase diagram in terms of increasing Er from equilibrium. The multiplicity of defect conﬁgurations in zone III, V and VI implies hysteresis. For example, raising Er transforms a ring in zone III into a point defect in IV. Upon lowering Er back into zone III, however, the point defect remains (cf. Fig. 6.7). The trend in Fig. 6.7, showing defects being “convected” downstream by ﬂow, agrees with the results of Yoneya et al. (2005) and Fukuda et al. (2004), but contradicts the prediction of Stark & Ventzki (2002) that defects shifts upstream under ﬂow. As defects are orientation patterns rather than material properties, the convective eﬀect is not intuitively obvious. Thus, we designed an experiment using silicone oil drops rising in the nematic 5CB (Khullar et al., 2007). Both the Saturn ring and the point defect shift downstream as the rising velocity increases. At high enough speed, the Saturn ring is shed into the wake as a point defect. This settles the question on the direction of convection, and conﬁrms the ﬂow-induced ring-to-point transformation predicted here. The experimental conditions correspond to AK ≈ 25 and Er ≈ 0.25, comparable to the values in Fig. 6.7. Note that Yoneya et al. (2005) have predicted a similar transformation at Er ∼ 10, but with the ﬂow ﬁeld prescribed as the Stokes solution and with rigid anchoring (AK → ∞). We should mention that the ranges of dimensionless parameters in the preceding discussion correspond to common small-molecule nematics under reasonable ﬂow conditions, and the same is true for the next section. For example, the Leslie viscosity ratios are close to those of ﬂow-aligning nematics such as PAA and MBBA. Take MBBA (de Gennes & Prost, 1993): η¯ = 7.25 × 10−2 Pa·s, K = 5 × 10−12 N (average of the three elastic constants for splay, twist and bend). Then the range of 0.1 Er 100 in Figs. 6.6 and 6.7 corresponds to rising velocities from 0.14 to 140 μm/s for a drop of diameter 100 μm, which are consistent with the experimental values of Khullar et al. (2007). 6.3.4 Rising velocity, drag force and the ﬂow ﬁeld In this subsection, we investigate the eﬀect of the nematic microstructure on the ﬂow ﬁeld, with special attention to the implications of the viscous anisotropy and defect conﬁguration. The geometric setup of the computation is the same as in the last subsection (cf. Fig. 6.5), but some of the parameter values diﬀer. In particular, we will focus on a range of rise velocity that corresponds to zone IV for homeotropic anchoring, with the satellite defect being the sole stable conﬁguration. This range displays the most interesting behavior when a ring defect transforms into a satellite during the rise of the drop. Planar anchoring will be considered as well. Figure 6.9 shows the transient rising velocity of a drop with planar and homeotropic anchoring. To give a sense of the time and velocity scales, a silicone oil drop 100 μm in diameter rising in MBBA would have η = 32.5 μm/s, both within the experimental range of η¯/(Δρga) = 1.54 s and Δρga2 /¯ Khullar et al. (2007). Within each plot, we examine the eﬀects of viscous anisotropy by 108 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal (a) (b) Figure 6.9: Transient rising velocity of a drop in a nematic liquid crystal with (a) planar anchoring and (b) homeotropic anchoring. Eo = 0.3, M o = 5.56 × 10−5 , Aσ = 0.2 and η. AK = 40. Time is made dimensionless by η¯/(Δρga), and velocity by Δρga2 /¯ varying the ratio of two Miesowicz viscosities. As explained before, this is achieved by varying α2 and α6 while keeping the characteristic viscosity η¯ = (η1 + η2 )/2 and the other Leslie coeﬃcients ﬁxed. With planar anchoring, the rising velocity V increases monotonically in time toward the terminal velocity U . For homeotropic anchoring, on the other hand, V experiences an overshoot. This is caused by the transition from a surface ring to a point defect as explained in the last subsection. Figure 6.10 shows the director and ﬂow ﬁelds near the drop with homeotropic anchoring at three times. Initially, n is vertical throughout the domain. A surface ring forms quickly on the equator of the drop and shifts downstream as the drop rises (t = 6.33). With the drop accelerating, the ﬂow sweeps the defect ring towards the rear of the drop (t = 28.4), and eventually transforms it into a point defect as V attains the terminal velocity (t = 52.7). Comparing Fig. 6.10(b) and (c), the n ﬁeld with the point defect has a larger area—including the wake—in which n are nearly orthogonal to the streamlines. According to Eq. (6.14), the nematic exhibits higher viscosity there than in areas where n is aligned to the ﬂow. Thus, the transformation from surface ring to point defect increases the viscous dissipation in the entire domain and thus the drag on the drop. This explains the overshoot in Fig. 6.9(b). Our recent experiment has conﬁrmed such an overshoot during the ring-to-point defect transformation (Khullar et al., 2007). If the drop has a point defect at the start, or if the steady state falls in zone III, V or VI where the surface ring is stable, there will be no overshoot in V . For planar anchoring, two boojums stay at the poles throughout the rise (Fig. 6.10d ), and the rising velocity again shows no overshoot. Roughly speaking, the orientation distortion extends into the nematic bulk for a fraction of the drop diameter; within this region the ﬂow is aﬀected by the anisotropic viscosity. 109 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal (a) (b) (c) (d ) Figure 6.10: Director and ﬂow ﬁelds around the drop with homeotropic anchoring at three times: (a) t = 6.33, (b) t = 28.4, and (c) ﬁnal steady state at t = 52.7, with Er = 15.8 and Re = 6.43. (d ) The steady state for a drop with planar anchoring; Er = 25.2 and Re = 10.3. These correspond to the curves in Fig. 6.9 with η1 /η2 = 34. For the streamlines, the reference frame is aﬃxed to the centroid of the drop. 110 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal Figure 6.11: Terminal velocity U of the drops in Fig. 6.9 as aﬀected by viscous anisotropy. η . The range of U corresponds to 4.93 < Re < 10.9 U is made dimensionless by Δρga2 /¯ and 12.1 < Er < 26.8 for planar anchoring, and 4.29 < Re < 6.64 and 10.5 < Er < 16.3 for homeotropic anchoring. With either type of anchoring, the rise velocity increases with viscous anisotropy as measured by η1 /η2 . Figure 6.11 gives a clearer view of this eﬀect in terms of the terminal velocity U . Furthermore, planar anchoring produces a higher U than homeotropic anchoring under otherwise identical conditions. This diﬀerence can again be explained by the drag as aﬀected by diﬀerent director ﬁelds. With planar anchoring (Fig. 6.10d ), n aligns with the streamlines in most of the domain, apparently minimizing the total dissipation (Jad˙zyn & Czechowski, 2001). This leads to a smaller drag and hence a greater U than the drop with homeotropic anchoring. In both cases, however, alignment between n and v is more prevalent throughout the domain than their being orthogonal, and larger areas experience η2 than η1 . With increasing η1 /η2 , therefore, the overall viscous dissipation diminishes with η2 and the rising velocity U increases as in Fig. 6.11, and more signiﬁcantly for planar anchoring. In fact, for each value of η1 /η2 , the U values for both anchoring types are bounded by the Hadamard-Rybczynski predictions using η1 and η2 . The eddies in the wake of the drop in Fig. 6.10(d ) form a vortex ring. It is not expected for Newtonian ﬂuids at Re = 10.3 and viscosity ratio β = 0.514 (Dandy & Leal, 1989), nor does it appear for homeotropic anchoring. The explanation seems to rest with the anisotropic viscosity. The Reynolds number cited above is deﬁned using the characteristic viscosity η¯. In Fig. 6.10(d ), the streamlines align with the director n to varying degrees in the ﬂow ﬁeld. Thus, the local viscosity may be much below η¯ in some regions. In particular, the streamlines next to the drop surface follow the n ﬁeld precisely. If we take the Miesowicz viscosity η2 to be the local viscosity, the local Reynolds number will be around 180. The high momentum of this layer of ﬂuid then leads to ﬂow separation in the wake. In contrast, homeotropic anchoring causes n to 111 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal Figure 6.12: The drag coeﬃcient for drops rising in a nematic as a function of Re or Er. For homeotropic anchoring, the data correspond to the satellite conﬁguration. η1 /η2 = 3, Aσ = 0.2, AK = 40. be mostly orthogonal to v near the drop surface. The local Reynolds number is much lower and no separation occurs. We have also noticed that for planar anchoring, the recirculating zone shrinks with decreasing η1 /η2 and disappears altogether when η1 = η2 . This is consistent with viscous anisotropy being the cause of the vortex ring. The drag on the drop can be further analyzed in terms of the drag coeﬃcient CD = 4π Δρga3 3 1 ρ U 2 (πa2 ) 2 1 (6.25) deﬁned from the terminal velocity U . The Hadamard-Rybczynski formula gives CD ·Re = 8(1 + 1.5β)/(1 + β) for a spherical Newtonian drop moving in a Newtonian matrix at vanishing Re (Bachelor, 1980). In view of this formula, we plot the product CD Re against Re and Er in Fig. 6.12. U and hence Re and Er are varied through the buoyancy force while keeping the viscosity ratio β ﬁxed. As noted before, homeotropic anchoring gives a higher drag than planar anchoring. If the matrix were a Newtonian ﬂuid, CD Re would be constant for small Re and increase with Re for ﬁnite inertia. That CD Re decreases with increasing Reynolds number reﬂects the enhanced alignment of n by the ﬂow ﬁeld. This is better illustrated by Er marked on the upper abscissa. For small Ericksen numbers, say Er < 1, the director orientation is hardly aﬀected by the surrounding ﬂow. Thus CD Re remains roughly constant. As Er exceeds unity, viscous ﬂow eﬀects become comparable to the elastic eﬀects and the ﬂow-alignment of n reduces CD Re. This decline eventually levels oﬀ as the ﬂow-alignment saturates around Er = 10. All prior results in the literature on the drag force are for rigid spheres with homeotropic anchoring at Re = 0. Nevertheless, a comparison is interesting. In the limit of vanishing Er, Stark & Ventzki (2001) ﬁxed the n(r) ﬁeld to the equilibrium solution with a 112 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal (a) (b) (c) Figure 6.13: Non-spherical drop shapes produced by the nearby defects. Eo = 0.3, M o = 5.56 × 10−5 , Aσ = 0.5, AK = 15, η1 /η2 = 4.7. (a) A drop with homeotropic anchoring and a surface ring rising at Re = 5.54, Er = 2.04. (b) A drop with homeotropic anchoring and a point defect. Re = 3.92, Er = 1.44. (c) A drop with planar anchoring and boojums. Re = 7.16, Er = 2.64. The boojums give the highest rising velocity while the point defect the lowest. point defect, and computed the drag in terms of an eﬀective viscosity ηeﬀ deﬁned from the Stokes formula. In our case, a similar ηeﬀ can be estimated from the HadamardRybczynski formula. At the lowest Er = 0.109 for homeotropic anchoring, our data gives ηeﬀ /η2 = 2.63, which is comparable to the results of Stark & Ventzki (2001): 1.83 for MBBA and 2.32 for 5CB. Our somewhat larger drag may have to do with wall conﬁnement in the geometric setup (Fig. 6.5). The decrease of ηeﬀ with Er is consistent with the ﬁndings of Stark & Ventzki (2002). At the highest Er = 14.1 in Fig. 6.12, ηeﬀ /η2 = 2.22. Although the Hadamard-Rybczynski formula no longer applies exactly at the ﬁnite Re in this case, ηeﬀ asymptoting to a value signiﬁcantly larger than η2 bespeaks the “orientation boundary layer” on the drop surface due to the homeotropic anchoring (cf. Fig. 6.10c), inside which there is still considerable misalignment between n and v. With even higher Er, the experiment of White et al. (1977) suggests that ηeﬀ approaches η2 . For planar anchoring, our data give ηeﬀ /η2 = 2.18 and 1.60 for the low-Er and high-Er limits of Fig. 6.12. So far we have used relatively large values for the interfacial tension σ (or small values of Aσ ) to keep the surface curvature of the drop smooth. At larger Aσ , drops deform in response to the nearby orientational ﬁeld, especially the presence of defects. Figure 6.13 illustrates three typical situations with the surface ring, satellite and boojum defects. The proximity of defects causes large curvature on the drop surface as a result of the competition between interfacial tension and anchoring energy. The cost in anchoring energy due to the defects is reduced at the expense of interfacial area such that the total energy is minimized. The lemon shape in Fig. 6.13(c) is well known in nematic drops with planar anchoring, and has also been reported for isotropic drops in nematic medium (Nastishin et al., 2005; Zhou et al., 2007). The drop with boojums rises the fastest while that with the satellite the slowest. 113 Chapter 6. The rise of bubbles and drops in a nematic liquid crystal 6.4 Summary This paper presents a computational study of the rise of a Newtonian drop in a nematic liquid crystal. The problem is a rough counterpart of the Hadamard-Rybczynski problem in Newtonian ﬂuids, although the Reynolds number ranges up to about 10 and mild drop deformation occurs. The key physics revealed by the simulation is the two-way coupling between the ﬂow ﬁeld and the molecular orientation ﬁeld, and especially the conﬁguration of orientational defects. The results can thus be summarized as follows. (a) Eﬀect of ﬂow on the orientational ﬁeld. With either a satellite point defect or a surface ring defect, the ﬂow sweeps the defect downstream. Thus, the surface ring shrinks and moves toward the rear stagnation point, and at high enough Ericksen number may be transformed into a point defect. An initial point defect moves farther downstream with increasing Ericksen number. The stability of the two defect conﬁgurations is depicted by a phase diagram in terms of the Ericksen number and the ratio between surface anchoring and bulk elastic energies. (b) Eﬀect of orientation on the ﬂow ﬁeld. This is mainly manifested through the viscous anisotropy of the ﬂuid. Drops with planar anchoring rise faster than those with homeotropic anchoring since the director ﬁeld is better aligned with the streamlines. With homeotropic anchoring, a drop experiences an overshoot in the transient rising velocity when a ring defect changes into a detached point defect. With both types of anchoring, the drag coeﬃcient decreases with the Ericksen number because stronger viscous ﬂow aligns the director to the streamline and reduces frictional dissipation. Through a systematic examination of the coupling between ﬂow and molecular orientation, we strive to construct a coherent picture for the ﬂuid mechanics of a particle moving in a nematic liquid. So far, the predicted ﬂow eﬀects on defect convection and transformation have been veriﬁed experimentally, as has the overshoot in rise velocity accompanying the defect transformation (Khullar et al., 2007). Finally, we point out two limitations in our work. The ﬁrst is the vectorial nature of the Leslie-Ericksen theory. The original version cannot handle defects as they would constitute singularities. A relaxation of the unit-length requirement on the director allows integer-strength defects to be simulated, but the Saturn ring has to be represented by a surface ring. The latter has fewer degrees of freedom and possibly diﬀerent stability regimes from an unattached Saturn ring. This restriction can be removed by adopting a tensorial representation of the molecular orientation (Rey & Tsuji, 1998; Feng et al., 2000; Yoneya et al., 2005). 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In their well-known monograph, de Gennes & Prost (1993) described the aggregation of bubbles on the free surface of a cholesteric into strings that delineate the molecular orientation. In nematics, Poulin and coworkers carried out a series of experimental observations on pattern formation of suspended particulates (Poulin et al., 1994, 1997b; Loudet et al., 2000, 2001). Water droplets suspended in the thermotropic liquid crystal pentylcyanobiphenyl (5CB) in the nematic phase form chains with neighboring drops separated by a constant distance, and the chains tend to align with the background nematic orientation (Poulin et al., 1997b). This was later conﬁrmed by experiments using PDMS oil drops in another thermotropic nematic (Loudet et al., 2000, 2001). The formation of parallel chains requires a strong homeotropic anchoring on the drop surfaces, i.e., with the director n normal to the surface. With planar anchoring or weak homeotropic anchoring, the droplets form kinked lines at an angle with the background orientation (Poulin & Weitz, 1998; Mondain-Monval et al., 1999; Poulin et al., 1999; Musevic et al., 2006). Ruhwandl & Terentjev (1996, 1997) obtained analytical solutions in the weak anchoring limit for particles bearing the Saturn ring, and thus provided an explanation for the kinked lines. In the rest of this paper, we limit ourselves to droplets with strong homeotropic anchoring. Poulin et al. (1997b); Poulin & Weitz (1998) proposed a theoretical framework for understanding the self-assembly of droplets into chains. As the water drop possesses homeotropic anchoring on its surface, its inclusion in an otherwise uniformly oriented liquid crystal necessitates the appearance of defects. In this case, a point defect known as a “hyperbolic hedgehog” accompanies each water droplet, and the two form a dipole when viewed from a great distance. Attraction between such dipoles explains the formation of chains of droplets separated by the satellite point defects. Furthermore, the elastic energy incurred by the point defect prevents the neighboring drops from contact and coalescence. Thus, long-range attraction and short-range repulsion both play roles in the formation and stability of the self-assembled pattern. Poulin et al. (1997b); Poulin & Weitz (1998) and Lubensky et al. (1998) used phenomenological ansatz director ﬁelds to ∗ A version of this chapter has been published. Zhou, C., Yue, P. & Feng, J. J., Dynamic simulation of droplet interaction and self-assembly in a nematic liquid crystal, Langmuir 24, 3099-3110 (2008) 119 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal compute the long-range attraction between neighboring particles as due to the interaction among eﬀective dipoles and quadrupoles. Qualitatively, the above explanation is clear and convincing. But the nature of pairwise interaction needs to be clariﬁed in a more rigorous way. For example, the dipolea, dipole interaction accounts for long-range attractions, and predicts F ∼ R−4 for R F being the attraction force, R being the center-to-center separation and a being the particle radius. Both experimental measurements (Poulin et al., 1997a; No¨el et al., 2006) and numerical computations (Fukuda et al., 2004b) have borne out this long-range scaling. What becomes of the attractive force as the particles move close to each other? Lubensky et al. (1998) used the repulsion between quadrupolar moments to predict a decline of the attraction, but could not produce a repulsive force even when R → 2a. Data of Poulin et al. (1997a) show that the attraction force F starts to fall below the scaling at R ≈ 3a. No¨el et al. (2006) measured F for a wider range of R. The decline of the attraction force for short separations is much steeper than predicted by Lubensky et al.’s model. Finally, Fukuda et al. (2004b) computed the repulsion between two “anti-parallel” dipoles, with the two defects on the outer side of the droplets. They found F ∼ r−3 in the whole range of r from slightly above 2a to 9a, in disagreement with the r−4 scaling predicted by dipolar interaction. Thus, there is still much uncertainty about the nature of the pairwise interaction. More recently, two-dimensional (2D) colloidal crystals have been achieved via nematicmediated self-assembly (Nazarenko et al., 2001; Musevic et al., 2006). They are of potential importance as a novel way to control the stability and structures of colloids (Loudet & Poulin, 2002) and as templates for novel optical materials (Rudhardt et al., 2003). In particular, Musevic et al. (2006) identiﬁed lateral interaction between parallel chains as the key mechanism in governing 2D crystallization. As in Poulin et al. (1997b), particles with the hedgehog defect form chains along the direction of the background nematic orientation. If another particle is in the vicinity of the chain with its dipole parallel and in the same direction as those in the chain, it is repelled by the chain. Conversely, if the single particle’s dipole is opposite to that of the chain (anti-parallel conﬁguration), attraction occurs. Hence, anti-parallel chains aggregate and form a regular and robust hexagonal lattice. Using an energy minimization procedure similar to that of Fukuda et al. (2004b), Musevic et al. (2006) computed the equilibrium arrangement of particles, which turns out to be a periodic pattern similar to observations. But the defect separating particles in the chain is a small ring instead of the observed point defect. All the above computations are concerned with the static equilibrium position, which has been sought through a straightforward energy minimization procedure. In reality, the particles move as a result of elastic relaxation, and hence the director ﬁeld is not at equilibrium until the motion ceases in the end. The process cannot be treated as quasistatic in general, and the dynamic pairwise and multi-particle interaction may diﬀer appreciably from predictions based on equilibrium director ﬁelds. The dynamic problem is much more diﬃcult than the static one. The motion of the particles is typically driven by an elastic force due to nematic distortion. The ﬂuid ﬂow and evolution of the director ﬁeld are coupled, and both are in turn dependent on the position and motion of the 120 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal particle surfaces on which velocity and anchoring boundary conditions are enforced. This complex task seems to have been ﬁrst tackled by Stark & Ventzki (2002) using the Leslie-Ericksen theory (de Gennes & Prost, 1993) to model the nematic. Up to now, dynamic simulations have been done on the motion of single particles. Yamamoto (2001) developed a scheme for evolving the position of multiple particles by elastic forces in a quasi-static manner. The director ﬁeld is equilibrated at every step and no ﬂuid motion is involved. But there have been no dynamic simulation on pairwise and multi-particle interactions in a nematic. Previous experiments and computations on a single particle (Fukuda et al., 2004a; Zhou et al., 2007a; Khullar et al., 2007) have shown rich dynamics in the approach to multiple defect conﬁgurations and in the transition among them. For multiple particles, the dynamic evolution will be important to the self-assembly process, and therefore warrants an in-depth study. In this paper, we present what appears to be the ﬁrst dynamical simulation of the self-assembly of multiple droplets in a nematic liquid crystal. The numerical methodology is based on a phase-ﬁeld representation of the interface, and employs ﬁnite elements with adaptive meshing to resolve the interfaces as well as defects in the nematic bulk. After describing the numerical methodology and the algorithm, we will report axisymmetric and 2D planar simulations on pairwise interactions and self-assembly of multiple droplets. In particular, we will explore the nature of longitudinal and lateral pairwise interaction forces. In view of prior experimental observations, we will then investigate the formation of chains, chain-chain attraction and repulsion as well as 2D assembly of a cluster of droplets. Where possible, comparisons will be made to experiments as well as previous static computations. 7.2 Theory and numerical methods Recently, we have developed a general ﬁnite-element algorithm AMPHI for simulating interfacial dynamics in two-component rheologically complex ﬂuids (Yue et al., 2006b). The interfaces are treated as having a small but ﬁnite thickness with a phase-ﬁeld variable changing continuously from one phase to the other. Fluid properties, such as density and viscosity, and ﬂow quantities, such as pressure and velocity, change steeply yet continuously across the interfaces. Hence, no discontinuity appears in the system. The phase-ﬁeld φ, which indicates the position of the interfaces, is governed by a mixing energy consisting of two components, one “hydro”-philic and the other “hydro”-phobic. This energy-based formulation easily incorporates complex rheology as long as the free energy of the microstructures are known. The package has been extensively validated (Yue et al., 2006b), and applied to simulate drop deformation, coalescence and jet breakup in Newtonian and viscoelastic liquids (Yue et al., 2006a; Zhou et al., 2006). The nematic liquid crystal admits a natural energetic description. Bulk distortions may be described by the Frank energy (de Gennes & Prost, 1993), and surface anchoring by the Rapini-Papoular anchoring energy (Yue et al., 2004; Rapini & Papoular, 1969). Thus, the elastic characteristics are easily incorporated into the phase-ﬁeld framework of AMPHI. Anisotropic viscosities may be introduced via Leslie coeﬃcients. Thus, we 121 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal have recently adapted AMPHI to explore the defect dynamics around a single drop rising in a nematic (Zhou et al., 2007b,a). For the multi-droplet simulations to be presented, the code has been generalized to allow periodic boundary conditions in two directions. Details of the theoretical model and computational algorithm can be found in Yue et al. (2004, 2006b) and Zhou et al. (2007a) In the following, we will only summarize the main ideas and give the governing equations. Consider a two-component immiscible blend of a Newtonian liquid and a nematic liquid crystal. For the problem at hand, we may visualize Newtonian drops of arbitrary shape and size suspended in the nematic medium. The Newtonian bulk is represented by φ = −1 and the nematic by φ = 1, and the interfaces are simply the level set of φ = 0. The governing equations are the continuity and momentum equations, supplemented by the Cahn-Hilliard equation for the transport of the phase ﬁeld φ (Yue et al., 2004) and the Leslie-Ericksen equations for the nematic director n (Zhou et al., 2007a): ∇ · v = 0, ρ ∂v + v · ∇v = −∇p + ∇ · σ, ∂t φ(φ2 − 1) ∂φ + v · ∇φ = γλ∇2 −∇2 φ + , 2 ∂t h = γ1 N + γ2 D · n, (7.1) (7.2) (7.3) (7.4) where λ, and γ are the interfacial energy density, capillary thickness and mobility of ρ1 + 1−φ ρ2 is an average between the diﬀuse interface, respectively. The density ρ = 1+φ 2 2 the two components. The stress tensor σ in the momentum equation is: σ = −λ∇φ∇φ − K 1+φ 1+φ 1−φ ∇n · (∇n)T − G + σ + μ[∇v + (∇v)T ], 2 2 2 (7.5) where G is the anchoring stress of the nematic director on the interface, G = A(n · ∇φ)n∇φ for planar anchoring and G = A[(n · n)∇φ − (n · ∇φ)n]∇φ for homeotropic anchoring, A is the surface anchoring energy density, K is the bulk elastic constant of the nematics under the one-constant approximation, and μ is the Newtonian viscosity. σ is the Leslie viscous stress (Leslie, 1966, 1968) in the nematic phase σ = α1 D : nnnn + α2 nN + α3 N n + α4 D + α5 nn · D + α6 D · nn, (7.6) where α1−6 are the Leslie viscous coeﬃcients obeying an Onsager relation α2 +α3 = α6 −α5 so ﬁve of them are independent (de Gennes & Prost, 1993). D = 12 [∇v + (∇v)T ] is the strain rate tensor, Ω = 12 [(∇v)T − ∇v] is the vorticity tensor, and N = ddtn − Ω · n is the rotation of n with respect to the background ﬂow ﬁeld. The director ﬁeld n evolves in the ﬂow ﬁeld according to a balance between elastic and viscous torques as given in Eq. (7.4). The molecular ﬁeld h, denoting elastic torque in the nematic, derives from the free energies of the system (de Gennes & Prost, 1993): h=K ∇· 1 + φ (n2 − 1)n 1+φ ∇n − − g, 2 2 δ2 122 (7.7) Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal with g = A(n · ∇φ)∇φ for planar anchoring and g = A[((∇φ · ∇φ)n − (n · ∇φ)∇φ] for homeotropic anchoring. The term involving δ arises from regularizing the Frank energy to allow defects (Zhou et al., 2007a): freg (n) = K (|n|2 − 1)2 1 . |∇n|2 + 2 4δ 2 (7.8) Thus, a defect is represented by |n| that falls below unity within a small area of size δ. This regularization is inspired by the Landau-Ginzburg equation (Lin & Liu, 2000; Liu & Walkington, 2000), and amounts to a simpliﬁed form of Ericksen’s theory with a variable order parameter (Ericksen, 1991), with |n| acting as the local order parameter. We have used δ = 4 since the defect core size is comparable to the interfacial thickness. Note that in the limit of → 0, the diﬀuse interface model reverts to the classic Navier-Stokes sharp-interface hydrodynamics (Yue et al., 2004). In particular, the interfacial tension σ and Rapini-Papoular anchoring√constant W can be √ recovered from the diﬀuse-interface parameters for small : σ = 2 2λ/3 and W = 2 2A/3 . We have used = 0.01a for most of the calculations, a being the radius of the droplets. Previous computations indicate that such an is small enough to approximate the sharp-interface limit in the present simulations (Yue et al., 2006b,a), especially because they involve no small-scale phenomenon such as surface rupture. In axisymmetric and planar 2D geometries, these equations are discretized on a unstructured grid of triangular elements using the Petrov-Galerkin formulation with streamline upwinding for the constitutive equation (Yue et al., 2006b). A critical ingredient of the algorithm is an adaptive meshing scheme that accurately resolves the drop interfaces as well as the defect cores at a manageable computational cost. As the interface and defects move, the mesh quality is monitored and updated by coarsening and reﬁnement as needed. Time integration is by an implicit second-order scheme with the time step automatically adjusted according to the motion of the interface (Yue et al., 2006b). Typical grid sizes are h1 = 0.006a on the interface and near defects, h2 = 0.2a in the bulk of the drop ﬂuid and h3 = 0.5a in the matrix. The meshing module of the program smooths the transition between diﬀerent regions. Numerical experiments have shown that these grid sizes and the time step used in the simulations are suﬃcient for numerical convergence. As a general numerical algorithm for computing nematic-particle interactions, our package is versatile and powerful. We should perhaps mention some of its features, though not all will be important to the computations reported below. First, our code accounts for several factors that have been largely ignored in the past. These include the drop deformation, which in some cases may interact with the nematic order in the vicinity of the surface (Zhou et al., 2007b), ﬁnite anchoring strength that may conceivably be tuned to manipulate the resultant colloidal structure, and the fully anisotropic rheology of the nematic, which plays a major role in the motion of dispersed particulates in a nematic medium (Zhou et al., 2007a). Second, the phase-ﬁeld formulation has the advantage of simulating topological changes such as interfacial rupture and coalescence naturally under a short-range force resembling the van der Waals force (Yue et al., 2005). There is no need for manual intervention as in sharp-interface models to eﬀect such events. 123 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal Finally, the ﬁnite-element method with adaptive meshing makes possible simulations of multiple interfaces and defects in complex geometries while maintaining accurate spatial resolution of the large-gradient regions. We must note that the code is limited at present to two spatial dimensions. Some of the geometric setups to be simulated are axisymmetric and thus can be readily handled by the code. Others, such as the self-assembly of multiple droplets, occur in 3D and we are forced to compute a planar 2D analogy of them. However, this is not as severe a restriction as it might appear. Considerable theoretical and experimental work has been done on particle interaction in freely suspended smectic C ﬁlms, where the dynamics is essentially 2D. Such results provide direct benchmarking for our computations. Furthermore, their general similarity to observations in 3D nematics suggest that the underlying physics is common, and that our 2D numerical simulations are relevant to 3D reality. More details will be given in the next section. Some remarks on terminology seem necessary to avoid confusion in discussing the results. The term “2D” has two meanings in this paper. One refers to the 2D patterns formed by particles as opposed to 1D chains. The other refers to the spatial dimensions in the computations. Similarly, we sometime use the term “dipole” to refer to the dropdefect ensemble, with the drop-to-defect vector indicating its orientation. This is to be distinguished from the electrostatic analogy that treats the particle-particle interaction as that between “dipolar” and “quadrupolar” moments (Lubensky et al., 1998). In fact, an important conclusion to be drawn from our simulations is that the pairwise interaction is not of the dipolar nature in general. 7.3 Results and discussion This section has two main parts. The ﬁrst deals with pairwise interactions, with each of the droplets bearing a single hyperbolic hedgehog defect. The two droplets are initially arranged so that their line of centers is parallel or perpendicular to the far-ﬁeld nematic orientation. These will be called, respectively, longitudinal and lateral pairs. In each case, the two drop-to-defect dipoles may be in the same direction (parallel) or opposite to each other (anti-parallel). Special attention will be given to the pairwise interaction forces, in connection with the open questions in the literature regarding the dipolar nature of this interaction. The second part explores the interaction and self-assembly of multiple droplets. We will study the formation of linear chains, interaction between neighboring chains in parallel and anti-parallel orientations, and ﬁnally 2D assembly of droplets in a doubly periodic domain. For a longitudinal pair of droplets, the geometry is axisymmetric. For the other conﬁgurations, the real physics is 3D and we are forced to simulate a 2D analogue of it in a planar domain. With axisymmetry, the computational domain is half of the meridian plane, shown in Fig. 7.1. The rectangular domain has a width of L = 15a in the radial direction and length H = 24a along the axis of symmetry, where a is the drop radius. The two droplets are located on the z axis with top-bottom symmetry, at some initial separation. The outer boundary at r = L is a no-slip wall, and periodic 124 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal Figure 7.1: The geometry of the computational domain. The radius of the drops is a. With axisymmetry around the z axis, the domain is half of the meridian plane. For 2D computations, the domain is the entire rectangular box. boundary conditions are imposed on the top and bottom. The z axis has symmetric boundary conditions for all the variables: ∂/∂r = 0. Note that no boundary conditions are needed on the drop surfaces; velocity and shear-stress continuities occur naturally and the interfacial tension is accounted for by the interfacial stress −λ∇φ∇φ in Eq. (7.5). The 2D domains are rectangles with periodic conditions in the vertical or both directions. For the former, the non-periodic boundaries are non-slip walls with rigid anchoring. The initial n(r) ﬁeld is uniformly vertical everywhere except within a small distance from each drop; in this shell n aligns radially. Thus, the initial director ﬁeld has an abrupt jump between the near ﬁeld and far ﬁeld orientation. After the simulation starts, there is a rapid rearrangement in the near ﬁeld, resulting in a point defect near each droplet. For all cases except one (cf. Fig. 7.4), this initial transient is short and insigniﬁcant to the ensuing dynamic evolution of the director ﬁeld. Even though the director does not distinguish between n and −n in reality, we have found the direction of n a convenient means to control the location of the defect. If the far ﬁeld n is upward and the near ﬁeld is radially outward, the point defect will nucleate below the drop. Changing either will put the defect above the drop. This scheme will be used to produce parallel and anti-parallel dipoles in the simulations. 125 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal The complete set of dimensionless groups governing our system is: ρ2 α = (drop-to-matrix density ratio), (7.9) ρ1 μ β = (drop-to-matrix viscosity ratio), (7.10) α4 /2 Wa (anchoring-to-bulk energy ratio), (7.11) AK = K W (anchoring-to-interfacial tension ratio), (7.12) Aσ = σ ρ1 U a Re = (Reynolds number), (7.13) η ηU (capillary number), (7.14) Ca = σ along with the various length ratios of the geometry and ratios between the Leslie viscous K and the coeﬃcients. Re and Ca are deﬁned using a characteristic velocity U = ηa viscosity η = (α3 + α4 + α5 )/2 that is the average between the largest and smallest Miesowicz viscosities (de Gennes & Prost, 1993). The ratio α3 /α2 distinguishes “ﬂowaligning” nematics from “tumbling” ones in simple shear ﬂows. In the complex ﬂows due to droplet self-assembly, this distinction is insigniﬁcant. Experiments have also used a wide range of thermotropic and lyotropic liquid crystals of both aligning and tumbling types. We have adopted the Leslie viscosities of a common nematic MBBA at 25◦ C for all computations (de Gennes & Prost, 1993): α1 = 6.5 centipoise (cp), α2 = −77.5 cp, α3 = −1.2 cp, α4 = 83.2 cp, α5 = 46.3 cp, α6 = −32.4 cp. Furthermore, we have assumed that the isotropic drop phase has the same density as the nematic and the same viscosity as the isotropic part of the nematic viscosity: α = 1, β = 1. For the anchoring energy W , prior experiments cited “strong anchoring” without giving a value (Poulin & Weitz, 1998) while computations typically assumed rigid anchoring (Lubensky et al., 1998). In all the computations, we have used a large AK = 100. In addition, W is typically much smaller than the interfacial tension σ (Sonin, 1995). We have used Aσ = 0.2. This, along with a small Ca = O(10−3 ), ensures that the droplets never deviate visibly from the spherical shape. The Reynolds number Re = O(10−2 ) for all the computations and inertia is negligible. In presenting the results, we have used two diﬀerent methods to visualize the nematic orientation (Fig. 7.2). The ﬁrst is a computed light intensity map through crossed polarizers, which would correspond to birefringent images recorded in the experiments. This will be used for the longitudinal pairs in axisymmetric geometry. The numerical scheme is detailed by Han & Rey (1995). For a satellite point defect near a drop, Fig. 7.2(a) shows the light intensity map along with vector lines for the director ﬁeld n(r). The pattern of two bright lobes separated by a darker line, with the point defect at the tip, closely resembles experimental pictures in the literature (Poulin et al., 1997a; No¨el et al., 2006; Khullar et al., 2007). The second presentation (Fig. 7.2b) uses contours of (n2x − 1/2)2 in a 2D planar domain, where nx is the horizontal component of n (Yamamoto, 2001; Yamamoto et al., 2004). Thus, bright areas indicate a horizontal or vertical n, while dark 126 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal (a) (b) Figure 7.2: Representation of the director ﬁeld. (a) Birefringent pattern for an axisymmetric n ﬁeld through crossed polarizers; (b) Grayscale representation of n in 2D geometries with contours of (n2x − 1/2)2 . White indicates a vertical or horizontal n, while black means a 45◦ tilt. areas have n at a 45◦ angle. The point defect is clearly marked by the focal point from which four dark brushes emanate. The gradient of the grayscale indicates the degree of elastic distortion, which reaches a maximum at the point defect and dies oﬀ in the far ﬁeld. 7.3.1 Pairwise interactions 1. Longitudinal pairs Consider ﬁrst the interaction of two particles in the parallel conﬁguration, with a centerto-center distance of R. The defects are on the line of centers, which is parallel to the far-ﬁeld director orientation. Representing the long-range force by the interaction of dipolar and quadrupolar moments, Lubensky et al. (1998) constructed a phenomenological formula for the attraction force FP between the particles: FP a = −24.97 4πK R 4 + 62.21 a R 6 . (7.15) Note that the dipolar attraction (the ﬁrst term) dominates the quadrapolar repulsion (the second term) at large distances. Mathematically, the repulsive force becomes signiﬁcant for smaller R, but the formula is supposed to be used only for R a. In the rest of the paper, we will use the same sign convention, i.e., repulsion being positive and attraction being negative. 127 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal Figure 7.3: Attraction between a longitudinal pair of drops in the parallel conﬁguration. Time is made dimensionless by the elastic relaxation time τe = ηa2 /K, length by a and force by 4πK. The same non-dimensionalization is used for all later plots. (a) The center-to-center distance R as a function of time. The insets show the defect conﬁguration, visualized as in Fig. 7.2(a), at time t = 56.1 and 145.1. (b) The attraction force F (R), computed by integrating the elastic stress around the drop, compared with prior experimental data (symbols) and theoretical results (lines). The dash line is the drag force estimated from Eq. (7.16), and the dash-dot line is the phenomenological FP of Eq. (7.15). The long-dash line indicates the R−4 scaling. In our dynamic simulation in the axisymmetric computational domain of Fig. 7.1, the two droplets indeed attract each other, and their separation decreases in time from an initial 7a to a ﬁnal equilibrium value Re = 2.45a (Fig. 7.3a). The speed of approach increases initially as R shrinks, and reaches a maximum around R = 3.5a. Afterwards, the speed quick drops to zero. In the equilibrium state, one point defect is roughly midway between the two drops, a distance of 1.22a from either drop center. Compared with the position of the point defect near a single drop (Zhou et al., 2007a), the defect between the droplets is “compressed”. The other defect is at a greater distance of 1.32a, close to that near a single drop. Note that the equilibrium separation of 2.45a agrees with prior experimental (No¨el et al., 2006) and numerical results (Fukuda et al., 2004a) to within 3%. The R(t) curve closely resembles that of Poulin et al. (1997a) In the experiment, the particles’ approach takes about 5 seconds, which corresponds to a dimensionless time of 136, comparable to that in Fig. 7.3(a). We have also computed the instantaneous attractive force F between the droplets, by integrating the elastic stress components over the drop surface, and this is plotted in Fig. 7.3(b). To extract an attractive force from the particle trajectory, Poulin et al. (1997a) calculated a Stokes drag with an “eﬀective viscosity” μe measured from a capillary tube: (7.16) FD = 6πμe aV 128 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal where V is the instantaneous velocity of the particles. Then F is equated to FD by assuming zero inertia for the particles. In our computations, Re ∼ 10−2 and inertia is negligible. But strictly speaking, the idea of a constant eﬀective viscosity is suspect since the anisotropic viscosity may vary signiﬁcantly depending on the ﬂow ﬁeld (Zhou et al., 2007a). As a test of Eq. (7.16), we have computed FD by taking μe to be η, the average between the largest and smallest of the Miesowicz viscosities. This turns out to be a reasonable approximation of F ; it is some 12% smaller in magnitude but follows the same trend. The discrepancy comes from the choice of μe . Since the nematic director is mostly normal to the drop surface, it is no surprise that μe underestimates the local viscosity and hence the drag force FD . For large separations (R > 5a), all computations and measurements agree that the attraction force F obeys the R−4 scaling. This is consistent with the dipolar attraction in Eq. (7.15). The magnitude of our F is slightly below Lubensky et al.’s formula and No¨el et al.’s data. Poulin et al.’s data, however, are lower by a factor of about 4. This may be due to their using a μe in Eq. (7.16) that is too low. Prior studies (Poulin et al., 1997a; Fukuda et al., 2004a; No¨el et al., 2006) put the lower bound of R for the R−4 scaling between 3a and 4a, whereas our F starts to fall below the power-law at R ≈ 5a. The reason for this diﬀerence is unclear at present. Our F reaches a maximum near R = 3.6a and then declines with decreasing R. The data of No¨el et al. show a much stronger attraction for smaller R, followed by a precipitous drop within R ≈ 2.5a. Thus, their interaction is much more “hard-sphere-like”, with a shorter range than in our case. Equation (7.15) is a poor approximation for short-range interaction, as expected. The quadrupolar repulsion is far too weak to represent the decline of attractive force with decreasing R. In fact, the formula never predicts much reduction in the attraction before the drops touch. If the drops are initially closer than the equilibrium separation Re , they are expected to repel each other. To explore this scenario, we simulated the separation of two droplets from an initial separation of 2.37a. Upon start of the simulation, the n ﬁeld relaxes into one with two point defects, similar to the insets of Fig. 7.3(a). This elastic relaxation, an artifact due to the initial condition, produces the anomalous behavior in Fig. 7.4 for the initial period of the simulation (t 2). At the beginning, R decreases momentarily (t 0.5) before increasing with time. Then the droplets separate from each other with increasing velocity until t ≈ 2. With negligible inertia, the acceleration reﬂects the changing forces on the droplets. Indeed, the repulsive force F also increases with R in this period, which is counterintuitive. Since time t is scaled by the elastic relaxation time τe = ηa2 /K, the duration of this initial transient (t ≈ 2) being of order one is reasonable. Only afterwards does F decrease with increasing R as expected. The motion ceases at an equilibrium separation of Re = 2.45a, the same value as reached in the pairwise attraction simulation of Fig. 7.3. Therefore, only the latter part of the simulation, with t > 2 and R > 2.4, can be meaningfully compared with static measurements of F (R) (Fig. 7.4b). The numerical result parallels the trend of the experimental data (No¨el et al., 2006), although shifted to larger R, again indicating longer-range interaction in our computation than in the experiment. Incidentally, No¨el et al. (2006) reported that 129 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal (a) (b) Figure 7.4: Interaction between a parallel longitudinal pair of droplets with an initial separation smaller than the equilibrium value. (a) Temporal evolution of the centerto-center distance R(t) and elastic force F (t). F is computed by integrating the elastic stress, and positive values denote repulsion. (b) Comparison between the computed F (R) and measurements by No¨el et al. (2006) (a) (b) Figure 7.5: The repulsion between an anti-parallel longitudinal pair. (a) The separation R(t) increases from an initial value of 2.5a. (b) The repulsion force F has a long range and decays slowly with R. pushing the two particles too close will result in the point defect opening into a ring. Our simulations show the same transition for initial separations below 2.37a. We turn our attention now to two droplets in the anti-parallel conﬁguration, with the two point defects lying outside of the pair, one on the top of the upper drop and the other below the lower drop. The initial center-to-center separation between the drops 130 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal Figure 7.6: Experimental observation of the repulsion between an anti-parallel pair of droplets. The insets are images through crossed polarizers. The drop on top is at the end of a vertical chain (not shown) with upward dipolar orientation. R and t are made dimensionless by the drop radius a and τe = ηa2 /K, with η = 0.0765 Pa·s, a = 50 μm and K = 10−11 N. is 2.5a. There is a rapid elastic relaxation at the beginning of the run, but unlike in Fig. 7.4, this transient is insigniﬁcant compared with the length of the simulation. The two droplets separate in time with a slowly decreasing velocity (Fig. 7.5a). The simulation is terminated as R approached 10a and the droplets get close to the top and bottom of the computational domain (cf. Fig. 7.1). At that time the two are still moving apart very slowly. As before, we plot the repulsion force F as a function of R in Fig. 7.4(b). For R up to 6a, F decays as R−1/3 . For larger separations, the decline of F becomes even milder. Fukuda et al. (2004a) computed the static repulsion between two anti-parallel dipoles ﬁxed in space. Their results show the scaling F ∼ R−3 , in disagreement with the R−4 scaling expected, at least for large R, from dipolar interactions. Why does our F exhibit an even slower decay? For the last part of the simulation, say R > 6a, one can imagine that the periodic boundary conditions on the top and bottom of the domain have introduced interference. A plausible explanation for the mild R−1/3 initial decay is the dynamic nature of our simulation. As the droplets move apart, the surrounding director ﬁeld is continuously evolving. This will produce a diﬀerent elastic force on the droplets than if the latter are ﬁxed in space. Even though the total simulation lasts more than 100τe , the drops move an appreciable 0.1a apart within τe . Therefore, the velocity of separation is too high for the process to be considered quasi-static. To probe the anti-parallel repulsion further, we have conducted an experiment on the interaction among droplets of silicone oil in the nematic 5CB. Occasionally the antiparallel conﬁguration appears and repulsion is observed. We present one such scenario in Fig. 7.6, where a single droplet (at bottom) is repelled by another (at top) with an 131 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal opposite dipolar direction, the latter being the end of a chain of particles with upward dipoles. Over the entire period of observation, the separation R increases almost linearly in time. If we ignore the later portion of the data (say R > 3.5a) because nearby droplets may have interfered, there does appear to be a weak power-law decay of the velocity v ∼ R−0.5 for the range of 2.6a < R < 3.5a. This implies that the repulsion force also decays weakly according to F ∼ R−0.5 . That this power law is much closer to our dynamic computation than prior static computations lends support to our argument above. 2. Lateral pairs When a pair of droplets are placed side by side, with their line-of-centers orthogonal to the far-ﬁeld nematic orientation, the geometry is no longer axisymmetric. Therefore, we resort to 2D planar simulations for lateral pairwise interactions. The same is true for the multiple particle interactions in the next section. Fortunately, essentially 2D dynamics can be realized experimentally by incorporating relatively large particles or droplets into freely suspended smectic-C and smectic-C* ﬁlms (Cluzeau et al., 2001). In fact, considerable research has been devoted to this special setup as a rare opportunity to study “anisotropic, two-dimensional emulsions” (V¨oltz & Stannarius, 2004). The 2D character greatly simpliﬁes theoretical analysis (Pettey et al., 1998; Cluzeau et al., 2004) and facilitates experimental observations (Dolganov et al., 2003). For example, in a smectic-C* ﬁlm of thickness smaller than the helical pitch, the droplets are observed to interact through the accompanying point defect and form chains, in qualitatively the same fashion as in 3D nematics (Cluzeau et al., 2001). Therefore, our 2D simulations enjoy greater relevance to reality than can be generally expected. Consider ﬁrst the parallel conﬁguration for a lateral pair, with their dipole direction initially in parallel (Fig. 7.7). The rectangular computational domain has a horizontal width of 17a and a height of 10a, and the two drops are initially placed side by side in the middle with a separation of 3.33a. Upon start of the simulation, there is again a very brief transient caused by the initial n(r) ﬁeld relaxing to lower the elastic distortional energy. As the two hedgehog defects take shape, they move inward toward each other. There is no rotation of the drops in this process. Then the two droplets repel each other and move apart laterally (Fig. 7.7a). This motion slows down in time and eventually stops at a dimensionless time t ≈ 80. The apparent steady-state has a drop separation Re = 4.02a and an angle between the two dipoles of 46.9◦ . Musevic et al. (2006) measured the interaction potential for a single particle when it is placed beside a chain of particles in 5CB, with the dipole of the single particle parallel to that of the chain. Although their result may have involved contributions from multiple particles in the chain, we have estimated their repulsion force F and compared it with our numerical result in Fig. 7.7(b). Two diﬀerences stand out. Our F is several times smaller, and it drops to zero rather abruptly at a relatively short separation. The larger F in Musevic et al.’s measurement is probably due to repulsion from neighboring particles in the chain. The short range of our F may have to do with the side walls on which ﬁxed vertical anchoring is imposed. Besides, the drop-to-defect vectors tilt toward each 132 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal (a) (b) Figure 7.7: The interaction between a lateral pair of droplets in the parallel arrangement. (a) The center-to-center distance R(t) increases as the droplets repel each other. The n ﬁeld is visualized as in Fig. 7.2(b) in the insets. (b) The repulsion force F as a function of R in the dynamic simulation. The data of Musevic et al. (2006) are also shown for comparison. other as the drops separate. This conﬁguration promotes attraction between drops (cf. Fig. 7.10), which may have shortened the range of repulsion relative to that for parallel dipoles. Note also that the simulation is dynamic in a 2D geometry, while the experiment is static in 3D next to a substrate. It is diﬃcult to speculate on the implications of such factors for F . Additional measurements of Musevic et al. (2006) suggested that a lateral pair with their dipoles in anti-parallel arrangement will attract each other. We have simulated such a scenario in the same computational domain as above, with the droplets initially side by side at a separation of 2.62a (Fig. 7.8). The droplets not only approach each other laterally, but also shift vertically so as to move the two point defects closer (see insets). As the line of centers makes an angle of roughly 23◦ with the vertical far-ﬁeld nematic orientation (t = 125), the two point defects sit on opposite sides of the line of centers instead of moving into the gap between the droplets. Soon afterwards, the two drops coalesce at t = 136. In this process, the attractive force F varies non-monotonically with the separate R, with a peak at R = 2.3a (Fig. 7.8b). With decreasing R, the attraction initially increases as one would expect. Then the relative rotation between the droplets changes the defect conﬁguration and the nature of the attraction. This causes the decline of the attraction force for R < 2.3a. In the static measurements of Musevic et al. (2006), a single particle is placed beside a chain of particles, with its dipole anti-parallel to that of the chain. The single particle is not abreast with one in the chain, but rather is staggered between two neighboring ones (cf. Fig. 7.11a). Thus, the measured F comes from two or more particles in the chain, and there is no relative rotation among the particles as exhibited by our doublet. Subject 133 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal (a) (b) Figure 7.8: The interaction between a lateral pair of droplets in the anti-parallel arrangement. (a) The center-to-center distance R(t) decreases as the drops approach each other. Their line of centers also rotates clockwise. In the end, the two droplets coalesce. (b) The attractive force F as a function of R in the dynamic simulation. The data of Musevic et al. (2006) are also shown for comparison. to these complications, perhaps only the initial portion of our result, say R > 2.4 in Fig. 7.8(b), are comparable with the experimental data. Similar to the repulsion between a parallel pair (Fig. 7.7b), our force is smaller in magnitude, and also occurs in a much shorter range than in the experiment. The apparent agreement in the humped shape of the curves is probably fortuitous. In the experiment, the attraction dies down when the single particle is pushed too close to the chain. In our simulation, on the other hand, the two drops maneuver in two dimensions. The downturn in the attraction force with shrinking R is due to the evolving defect pattern toward the end. To summarize the results and discussion in this subsection, our simulations reproduce the characteristics of pairwise interactions as measured in experiments and predicted by ad hoc models. For longitudinal doublets, a parallel pair tend to attract each other while an anti-parallel pair repel, unless their initial separation is very small. For lateral pairs, parallel pairs repel while anti-parallel ones attract. Besides, computations and measurements exhibit common trends in the magnitude of the forces. For example, longitudinal interactions are stronger than lateral ones. For longitudinal pairs, the parallel attraction is greater than the anti-parallel repulsion, whereas for lateral pairs, the parallel repulsion and the anti-parallel attraction have comparable magnitudes. These can be rationalized by the relative position of the point defects as it determines the elastic distortion surrounding the droplets. Quantitatively, the long-range longitudinal attraction between parallel pairs is the only situation with a well-established scaling (F ∼ R−4 ), which is consistent with the simple picture of dipole-dipole attraction. For closer ranges, the inter-particle force shows considerable divergence among diﬀerent computational and experimental studies. To a 134 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal large extent, this discrepancy reﬂects the dynamic nature of the interaction when the particles are allowed to move. However, a clearer understanding requires more careful computations as well as measurements that account for complicating factors such as diﬀerences in spatial dimensionality and the presence of substrates. 7.3.2 Multi-drop interactions 1. Chain formation Among experimental observations, the most prominent feature of self-assembly is the formation of chains parallel to the far-ﬁeld director orientation (Poulin & Weitz, 1998). We simulate this by arranging 4 droplets in a regular zig-zag pattern in a rectangular domain (Fig. 7.9 inset), with initial center-to-center separation of 3a and the line of centers making an angle of 39◦ with the vertical. The width of the domain is 20a, and its height is 12.5a. The director is ﬁxed in the vertical direction on the side walls, where the velocity vanishes. Periodic boundary conditions are used for the upper and lower boundary. As before, the initial director ﬁeld has a thin ribbon of radially outward orientation around each drop, outside of which n points uniformly upward. Upon start of the simulation, the director ﬁeld near each droplet quickly rearranges itself and produces a satellite point defect. These defects are not directly above or below the drop, as one would expect for a single droplet in an otherwise vertically oriented nematic. Rather, the droplet-defect dipole points toward the neighboring drop below it (Fig. 7.9). Then each drop experiences attraction from both neighbors, according to the pairwise attraction force of Fig. 7.3. The net eﬀect is to pull the drops into a straight line. Since this horizontal force is proportional to sin θ, where θ is the angle between the line-of-centers and the vertical, the motion is fast initially when θ is large, and slows down toward the end as θ → 0. The process of chain formation is qualitatively the same as observed in 3D nematics (Poulin et al., 1997b; Poulin & Weitz, 1998) and 2D smectic C* ﬁlms (Cluzeau et al., 2001). However, the periodicity of the computational domain in the vertical direction introduces an artifact. As the four-particle array is repeated above and below, the equlibrium inter-particle separation Re is predetermined as 1/4 of the height of the domain, independent of the physical parameters of the system. In this case, Re = 3.125a is somewhat higher than the 2D observation of 2.6a (Cluzeau et al., 2001). Note also that we have used the regular zig-zag initial conﬁguration for ease of analyzing the forces in the droplets. Similar chains should form from a more random initial conﬁguration. We will explore such a scenario toward the end. 2. Chain-chain interactions Experiments indicate that chain-chain interaction is central to the formation of regular 2D arrays (Musevic et al., 2006). From a fundamental viewpoint, chain-chain interaction is more complex than pairwise interactions since it involves the collective motion of multiple particles and defects. Motivated by the experiment of Musevic et al. (2006), we 135 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal Figure 7.9: Four droplets assemble into a vertical chain in a vertically aligned nematic. The height of the domain is 12.5a with periodic conditions in the vertical direction. The side walls are 20a apart and are not shown in the plots. The angle θ, between the neighboring drops and the vertical axis, decreases from 39◦ toward zero. (a) (b) Figure 7.10: Interaction between two chains in the parallel arrangement. (a) Separation between the second drops from the top. (b) Repulsion force on the second drop from the top of the right chain. have simulated the interaction between two chains, each consisting of four droplets, in the parallel and anti-parallel conﬁguration. The computation domain has a width of 18a and a height of 13.3a, and the boundary conditions are the same as in Fig. 7.9. The parallel chains are initially separated by a center-to-center distance of R = 3.2a (Fig. 7.10a). The defects are directly below the 136 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal droplet and the chains are parallel to the far-ﬁeld director orientation. Upon start of the simulation, the two defects at the bottom of the chains quickly move inward toward each other (Fig. 7.10a). This is reminiscent of the defect motion in the lateral pair of Fig. 7.7. Meanwhile, the other three pairs of defects are conﬁned between the neighboring droplets and are not free to move. In view of the pairwise repulsion, one expects the two chains to move laterally away from each other in time. This is largely true, except for the two droplets at the bottom. Breaking the mirror symmetry, the left droplet moves downward while the right one upward, the two forming an attractive doublet similar to that in Fig. 7.3. At this point, the motion of the droplets has practically stopped, and a steady-state conﬁguration emerges. The symmetry breaking may have been triggered by numerical noise such as asymmetry in the unstructured grid, but is indicative of the instability of the symmetry pattern. The branched conformation closely resembles experimentally observed patterns for water droplets in a 3D nematic, e.g., Fig. 11 of Poulin & Weitz (1998). We have also computed the repulsive force F on one of the upper particles and plotted F (R) in Fig. 7.10(b). Based on the pairwise repulsion of Fig. 7.7, we expect F to decline monotonically with R. The curious rise for 3.2a < R < 3.5a is not due to the initial elastic relaxation, since its duration is roughly 20 times the elastic time scale τe . Rather it is the result of the complex dynamic interaction between the chains. As the two bottom drops attract each other, their attraction force “propagates” up the chains as if along a string. This amounts to an additional attraction on the upper drops, which weakens the repulsion force for the initial period. As the two bottom particles approach, their attraction dies out and so does this eﬀect. Then the repulsion force F assumes its normal decay with R. The magnitude of F is between that of our pairwise repulsion and Musevic et al.’s measurement in Fig. 7.7. Therefore, interaction with multiple particles is responsible, partly at least, for the larger repulsion force here and in Musevic et al.’s experiment than our pairwise repulsion. Note also that F decays over a longer range than the pairwise repulsion, and the steady separation is larger. These are consistent with the idea that for the pairwise interaction in Fig. 7.7, the tilting of the drop-defect vectors toward each other promotes attraction and suppresses repulsion. The attraction between anti-parallel chains turns out to be simpler as it does not greatly distort the conformation of each chain (Fig. 7.11). The initial condition is similar to that of Fig. 7.10(a), but the right chain is ﬂipped vertically so that the point defect is on top of each droplet. The two chains are also oﬀset vertically by a distance of a. The initial separation between the lines of centers is 2.9a. This setup approximates what Musevic et al. (2006) used experimentally. The two chains approach each other with a velocity that is roughly constant up to t = 900. Afterwards, the motion slows down and at t ≈ 1300, the chains approach an equilibrium state with a separation of 2.44a between the centerlines (Fig. 7.11a). This is reasonably close to the experimental value of 2.31a (Musevic et al., 2006). The chain-chain attraction F , plotted in Fig. 7.11(b), diﬀers markedly from the pairwise attraction in Fig. 7.8(b). First, F (R) does not show the humped shape. The decline with shrinking R should be compared with similar decline in Musevic et al.’s data (Mu- 137 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal (a) (b) Figure 7.11: Interaction between two chains of droplets in the anti-parallel arrangement. (a) The separation between the chains R(t) decreases from an initial value of 2.9a to an equilibrium one of 2.44a. (b) The attractive force F , averaged among the droplets, as a function of R in the dynamic simulation. sevic et al., 2006) in Fig. 7.8(b) rather than our computed pairwise attraction, as the latter is due to the rotation of the doublet that is absent here. Perhaps because our initial separation is too small, F in Fig. 7.11(b) does not exhibit the experimentally recorded long-range regime where the attraction decays with the chain-chain separation. Starting from an initial separation of 4a, the chains barely move in thousands of time units, indicating F ≈ 0 for R ≥ 4a. Furthermore, F is some two orders of magnitude smaller than the numerical and experimental data in Fig. 7.8(b). This is because two anti-parallel chains vertically oﬀset by a induce an n ﬁeld that is nearly left-right symmetric about the line of centers for each chain. This is apparent when contrasting the insets in Fig. 7.11(a) with those in Fig. 7.8(a). In the former, the dark brushes are conﬁned within the gap between neighboring particles in the chain, while in the latter they extend to the other particle. Thus, the anti-parallel chain-chain interaction is much weaker than that between two individual particles. Such symmetry does not exist for the parallel chains in Fig. 7.10, and thus the chain-chain repulsion is comparable to the pairwsie repulsion. 3. Multidrop self-assembly We simulate the self-organization of eight droplets in a square domain of dimensions 14a × 14a. Periodic boundary conditions are imposed in both directions; the lack of a Dirichlet condition for n implies that the director ﬁeld will evolve under the inﬂuence of the anchoring on the droplets rather than a far-ﬁeld alignment. Initially, the droplets are randomly positioned in the periodic domain (Fig. 7.12a). Similar to the preceding simulations, the initial director ﬁeld is such that n is radial within a thin ribbon surrounding 138 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal (a) t= 6.38 (b) t= 17.2 (c) t= 33.1 (d ) t= 547.8 Figure 7.12: Self-assembly of 8 drops in a doubly periodic domain. Note that the time is made dimensionless by ηa2 /K. each droplet and uniformly vertical outside. After the simulation begins, a point defect immediately nucleates near each particle (Fig. 7.12a), visualized by the intersection of four dark brushes. The ensuing self-assembly seems to be driven by the pairwise interactions discussed in Section 7.3.1. The point defects quickly move toward the nearest neighbor so the drop-defect dipole points to the latter, similar to Fig. 7.9. This is seen in Fig. 7.12(a) for drop 8 (toward drop 6) and drop 1 (toward drop 4). This is followed, in Fig. 7.12(b), by attraction between longitudinal pairs (drops 1 and 4, 5 and 6, and 6 and 8) as well as repulsion between lateral pairs (drops 3 and 5, 5 and 7). The repulsion between 3 and 139 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal 5 is short-lived, however, as the drop-to-defect dipole for drop 5 rotates clockwise until it points toward drop 3. Afterwards the two attract in a similar way to the two bottom droplets of Fig 7.10(a). Toward the end of the simulation (t = 547.8), a predominant chain has taken shape, consisting of drops 8, 6, 5, 3, and possibly 1 (Fig. 7.12d ), oriented diagonally. Drops 1, 4 and 2 are forming a minor chain along the vertical direction. Drop 7 is the only “free” droplet at this moment, but may eventually be absorbed into the diagonal chain by attractions from drop 1 and 3. Because of the double-periodicity of the computational domain, the interaction among the droplets are not guided by a ﬁxed farﬁeld orientation, and the assembly proceeds slowly, especially in the late stage. Because of the modest number of droplets, chain-chain interaction is absent and the formation of 2D arrays cannot be simulated. Qualitatively, the self-assembly of droplets into chains is in agreement with experimental observations in 3D nematics (Poulin et al., 1997b; Loudet et al., 2000; Musevic et al., 2006) and 2D smectic-C ﬁlms (Cluzeau et al., 2001, 2004). Quantitatively, we can compare the time required for the self-assembly and the equilibrium drop spacing in the chain. Cluzeau et al. (2001) reported chain formation in 2.8 seconds in their experiment, which translates to a dimensionless time of roughly 110. This is comparable to the time scale of chain formation in Fig. 7.12. In the diagonal chain of Fig. 7.12(d ), the point defect is more or less midway between the neighboring particles, with a distance of 1.37a to the drop center. Thus, the drop spacing is Re = 2.74a. This is in reasonable agreement with experimental measurements for chains both in 3D nematic (Re = 2.6a) (Poulin et al., 1997b) and in 2D smectic-C ﬁlms (Re = 2.6a±0.2a) (Cluzeau et al., 2001). The most comparable computational study seems to be that of Yamamoto and coworkers (Yamamoto, 2001; Yamamoto et al., 2004). As mentioned in the Introduction, their scheme is quasi-static and ignores hydrodynamic eﬀects. It amounts to assuming that the ﬂow is much slower than elastic relaxation, and thus decouples the particle motion from ﬂow in the limit of a vanishing Ericksen number. This is perhaps justiﬁable in the ﬁnal stage of self-assembly, but the dynamics can be important early on as shown by our simulation of pairwise interactions. There is a second and perhaps more signiﬁcant diﬀerence. The theoretical model of Yamamoto (2001) is such that the satellite point defect is unstable in 2D, and the pairwise interaction is of the “quadrupolar” type rather than the “dipolar” type seen here and in most experiments. Consequently, the particles aggregate into clusters that tend to assume an angle with the far-ﬁeld orientation, which is ﬁxed in his computation. These clusters should be compared with the “kinked chains” formed of droplets bearing Saturn rings (Musevic et al., 2006) or particles with planar anchoring (Poulin & Weitz, 1998). The straight chains linked by point defects, prevalent in experiments with 3D nematics as well as 2D smectic-C ﬁlms, cannot be realized using his theoretical model. In more recent work, Yamamoto et al. (2004) presented results for a smectic-C ﬁlm. The free energy appears slightly diﬀerent and a stronger anchoring parameter is used. This produces satellite point defects near individual particles, which assemble into chains that are similar to ours in Fig. 7.12. The ﬁnal center-to-center separation Re = 2.88a is somewhat larger than ours and prior experimental values. The time scale of self-assembly cannot be compared because of the quasi-static nature of their 140 Chapter 7. Droplet interaction and self-assembly in a nematic liquid crystal computation. 7.4 Summary Through dynamic simulations, we have explored the interaction of particles in a nematic using a more or less rigorous theory of nematohydrodynamics. The goal is to gain a more rational understanding of the self-assembly process than previously achieved through analogies and ad hoc models. The results on pairwise and multi-particle interactions can be summarized as follows. (a) The long-range attraction force between pairs of droplets, each having a point defect that is on the line-of-centers of the droplets and facing the same direction, obeys a scaling F ∼ R−4 with the drop separation R. This is the most well-established fact about pairwise interaction, and is consistent with the idea of attraction between dipoles. (b) Pairwise interaction in shorter range is poorly documented and there is considerable discrepancy among the few theoretical and experimental studies. We have examined several conﬁgurations for the drop-defect ensemble in which the droplets attract or repel each other. The dynamic nature of the interaction is important, especially in the early stage, and the force between droplets cannot be represented by that between dipolar and quadrupolar moments. (c) Multiple droplets form linear chains via pairwise attraction and repulsion between neighboring droplets. Preformed parallel chains repel each other if their drop-to-defect vectors are in the same direction. They attract if their orientations are reversed. We have compared our simulations with prior static calculations and experiments to the greatest extent possible. There is qualitative, and sometimes semi-quantitative, agreement. More detailed comparison is hampered by experimental complications such as the presence of substrates and diﬃculties in quantifying anchoring strength, as well as two limitations in our simulations: two-dimensionality and the small number of particles. 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J., Liu, C. & Shen, J. 2007b Heart-shaped bubbles rising in anisotropic liquids. Phys. Fluids 19, 041703. 144 Chapter 8 Dynamic simulation of capillary breakup of nematic ﬁbers: molecular orientation and interfacial rupture ∗ 8.1 Introduction Immiscible blends of nematic liquid crystals (LCs) and isotropic liquids occur in several contexts. In optical applications such as polymer-dispersed liquid crystals (Mucha, 2003), the desirable phase morphology is liquid crystal droplets suspended in a polymer matrix. In liquid-crystalline polymer (LCP) composites, on the other hand, it is essential to have the minor LCP phase stretched into thin ﬁbers with strong molecular alignment in the axial direction (Dutta et al., 1990; Kernick & Wagner, 1999). If the composite is rapidly frozen to retain the ﬁbrous morphology in the solid state, the LCP ﬁbers act as ultrastrong in-situ reinforcement (Acierno & Collyer, 1996). In addition, LCP nanoﬁbers (Srinivasan & Reneker, 1995), with diameters on the order of tens of nanometers, form an essential building block in many areas of nanotechnology. Although these are typically electrospun in a gaseous medium, the ﬁber surface morphology is a central concern as well (Reneker & Chun, 1996). More recently, LC ﬁlament breakup has been used in microﬂuidic devices for making monodisperse nematic droplets (Hamlington et al., 2007). The nematic order is a signiﬁcant determinant of the speed of drop pinchoﬀ and the drop size. Therefore, it is important to understand the capillary stability of the LC ﬁber and its breakup process. From a fundamental viewpoint, the breakup of a nematic LC ﬁber is an interesting process. As the physical dimension of the ﬁber narrows down to the micro- or nanometer scale, interfacial eﬀects become increasingly dominant (Stone et al., 2004). Aside from the conventional isotropic interfacial tension, the anchoring of LC molecules on the interface contributes in eﬀect an anisotropic part to the interface tension, which may play a signiﬁcant role in the phase morphology and ﬂuid dynamics of nematic-isotropic two-phase systems (Yue et al., 2005b; Nastishin et al., 2005). This has been illustrated by recent work on bubble and drop behavior in a nematic matrix (Zhou et al., 2007b,a; Khullar et al., 2007). Typically the coupling between surface anchoring and ﬂuid ﬂow is mediated by bulk elasticity of the LC. For example, enforcing the anchoring condition leads to bulk distortion that modiﬁes the anisotropic rheology of the LC bulk. Evidently, surface anchoring and bulk elasticity are the two major factors governing the energetics of LC-isotropic interfaces. When considering the dissipative dynamics of interfacial ∗ A version of this chapter has been submitted for publication. Zhou, C., Yue, P. & Feng, J. J., Dynamic simulation of capillary breakup of nematic ﬁbers: molecular orientation and interfacial rupture (submitted 2008) 145 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers deformation and ﬂow, a third factor—anisotropic viscosity—must be considered as well. Previous experimental work on nematic ﬁber breakup seems to consist of two qualitative observations of the evolving interface and the birefringent pattern inside the ﬁber. For a lyotropic liquid-crystalline polymer ﬁber about 50 μm in initial diameter, Tsakalos et al. (1996) reported Rayleigh instability that proceeds much as for Newtonian ﬂuids. Birefringent patterns do reﬂect eﬀects of the ﬂow ﬁeld, however, with the uniaxial elongation at the neck producing strong orientational order. Eventually, ﬁber breakup gives rise to bipolar droplets with a range of sizes. Machiels et al. (1997) observed similar breakup of thermotropic liquid-crystalline polymer ﬁbers. Neither study was able to shed much light on the coupling between capillary breakup and molecular orientation, and polydomains may have complicated the microstructural order. Theoretical studies are limited to linear instability analysis based on simpliﬁed models. Rey (1997) did the ﬁrst linear analysis on an inﬁnite nematic ﬁber. The nematic director ﬁeld n(r) is uniform and ﬁxed along the axis of the ﬁber, unperturbed by the capillary waves. But undulation of the interface forces n to deviate from the planar easy direction and is penalized by a Rapini-Papoular anchoring energy (Rapini & Papoular, 1969). Thus, surface anchoring is accounted for in the weak anchoring limit, since it does not modify the bulk orientation. But in the bulk, Ericksen’s transversely isotropic ﬂuid (TIF) model is used that does not allow distortional elasticity. Results show that the anchoring tends to stabilize the ﬁber against capillary waves; the threshold for unstable wavelengths is raised and the growth rate of the fastest growing mode is damped when compared with Newtonian ﬁbers of the same viscosity. Similar conclusions were reached by Wang (2001) using the Doi theory. Bulk elasticity is omitted, and anchoring is accounted for by an anisotropic surface energy. More recently, Cheong & Rey (2001, 2002, 2004) have extended linear analysis to “onion” and radial director ﬁelds and non-axisymmetric instability modes. In spite of the progress made, our theoretical understanding of capillary breakup of nematic ﬁbers suﬀers from several limitations. First, only linear instability modes have been analyzed, and we have no knowledge of nonlinear growth of capillary waves and the eventual breakup. Second, theoretical analysis has necessitated the use of drastically simpliﬁed models. These may capture one or two of the key factors: surface anchoring, bulk elasticity or anisotropic viscosity, but not all three. Finally, the ﬂow ﬁeld and director ﬁeld are almost always decoupled to simplify analysis. The motivation for our work is to carry out a fully coupled ﬂuid-dynamic simulation of nematic ﬁber breakup using the Leslie-Ericksen theory of nemato-hydrodynamics. The complex rheology of LCs and the need to capture an evolving interface make this a challenging computation. We overcome these diﬃculties using a ﬁnite-element algorithm based on a diﬀuse-interface model (Yue et al., 2006b). The methodology was developed for simulating interfacial ﬂows of complex ﬂuids in general. In the current application, we will analyze the eﬀects of surface anchoring, bulk elasticity and anisotropic viscosity separately, with an emphasis on the coupling between nano-scale molecular order and micro-scale ﬁber morphology. 146 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers 8.2 Theory and numerical methods We treat the nematic-isotropic interface as a thin diﬀuse layer across which physical properties change rapidly but continuously. The diﬀuse-interface theory uses a variational formulation based on the free energy of the two-phase system. This is most convenient for the current application where bulk distortion of LCs can be represented by the Frank energy (de Gennes & Prost, 1993), and surface anchoring by the Rapini-Papoular anchoring energy (Rapini & Papoular, 1969). Finally, anisotropic viscosity is incorporated by Leslie’s viscous stress tensor. Thus, our algorithm integrates the Leslie-Ericksen theory naturally into an interface-capturing ﬂow solver, bridging the gap between molecular orientation and macroscopic hydrodynamics. Another advantage of the diﬀuse-interface model, essential to the current simulation, is that it handles topological changes such as interfacial rupture and coalescence rationally via a short-range force resembling the van der Waals force. There is no need for manual intervention as in sharp-interface models to eﬀect such events. The numerical package has been extensively validated (Yue et al., 2006b), and applied to simulate drop deformation, breakup and coalescence in polymeric and nematic liquids (Yue et al., 2004, 2005a,b, 2006a; Zhou et al., 2006, 2007b,a, 2008). Thus, we will brieﬂy summarize the main ideas in the following, and refer the reader to prior publications for details of the theoretical model and computational algorithm. We employ a scalar phase ﬁeld φ(r) to represent the composition of a Newtoniannematic two-component system. The Newtonian bulk is represented by φ = −1 and the nematic by φ = 1, and the interfaces are simply the level set of φ = 0. The governing equations are the continuity and momentum equations, supplemented by the CahnHilliard equation for the transport of the phase ﬁeld φ and the Leslie-Ericksen equations of nematohydrodynamics (Yue et al., 2004; Zhou et al., 2007a): ∇ · v = 0, ρ ∂v + v · ∇v = −∇p + ∇ · σ, ∂t ∂φ φ(φ2 − 1) , + v · ∇φ = γλ∇2 −∇2 φ + 2 ∂t h = γ1 N + γ2 D · n, (8.1) (8.2) (8.3) (8.4) where λ, and γ are the interfacial energy density, capillary thickness and mobility of ρ1 + 1−φ ρ2 is an average between the diﬀuse interface, respectively. The density ρ = 1+φ 2 2 the two components. The stress tensor σ in the momentum equation is: σ = −λ∇φ∇φ − K 1+φ 1−φ 1+φ ∇n · (∇n)T − G + σ + μ[∇v + (∇v)T ], 2 2 2 (8.5) where K is the Frank elastic constant of the bulk nematic under the one-constant approximation, and μ is the Newtonian viscosity. G is the anchoring stress of the nematic director on the interface, G = A(n · ∇φ)n∇φ for planar anchoring, A being the surface anchoring energy density. Homeotropic anchoring can be readily modeled but is not as 147 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers relevant here since stretched LC ﬁbers tend to have n aligned axially (Tsakalos et al., 1996; Rey, 1997). σ is the Leslie viscous stress (Leslie, 1966) in the nematic phase σ = α1 D : nnnn + α2 nN + α3 N n + α4 D + α5 nn · D + α6 D · nn, (8.6) where α1−6 are the Leslie viscous coeﬃcients obeying an Onsager relation α2 +α3 = α6 −α5 so ﬁve of them are independent (de Gennes & Prost, 1993). D = 12 [∇v + (∇v)T ] is the strain rate tensor, Ω = 12 [(∇v)T − ∇v] is the vorticity tensor, and N = ddtn − Ω · n is the rotation of n with respect to the background ﬂow ﬁeld. The director ﬁeld n evolves in the ﬂow ﬁeld according to a balance between elastic and viscous torques as given in Eq. (8.4). The molecular ﬁeld h, denoting elastic torque in the nematic, derives from the free energies of the system (de Gennes & Prost, 1993): h=K ∇· 1 + φ (n2 − 1)n 1+φ − g, ∇n − 2 2 δ2 (8.7) with g = A(n · ∇φ)∇φ for planar anchoring. The term involving δ arises from an energy penalty added to the Frank energy to allow defects to be represented by reduced |n| values within a small area of size δ (Liu & Walkington, 2000). Thus, |n| acts like a local order parameter, and the model closely resembles Ericksen’s generalization of the Leslie-Ericksen theory by a variable order parameter (Ericksen, 1991). We have used δ = 4 since the defect core size is comparable to the interfacial thickness. Note that in the limit of → 0, the diﬀuse interface model reduces to the classic sharp-interface hydrodynamics (Yue et al., 2004). In particular, the interfacial tension σ and RapiniPapoular anchoring√constant W can be √ recovered from the diﬀuse-interface parameters for small : σ = 2 2λ/3 and W = 2 2A/3 . To faithfully approximate the sharpinterface limit, needs to be O(10−2 a), a being the macroscopic length scale of typical problems. Although non-axisymmetric instability modes have been considered before (Cheong & Rey, 2002), experiments have shown only axisymmetric capillary waves and drop pinchoﬀ (Tsakalos et al., 1996; Machiels et al., 1997; Hamlington et al., 2007). Thus, we will assume axisymmetry throughout this study. The governing equations are discretized on a unstructured grid of triangular elements using the Petrov-Galerkin formulation (Yue et al., 2006b). A critical ingredient of the algorithm is an adaptive meshing scheme that accurately resolves the evolving interface and any orientational defects. Typical grid sizes are h1 = 0.006a at the interface and near defects, h2 = 0.2a inside the ﬁber and h3 = 0.5a in the matrix, with smooth transitions between diﬀerent regions. Time integration is by an implicit second-order scheme with the time step automatically adjusted according to the motion of the interface. Numerical experiments have shown that the grid sizes and the time step used in the simulations are suﬃcient for numerical convergence. 8.3 Results and discussion The geometry of the axisymmetric computational domain is shown in Fig. 8.1, with the nematic LC ﬁber surrounded by a quiescent Newtonian ﬂuid. We apply periodic 148 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers Figure 8.1: Schematic of the computational domain, which is half of the meridian plane of the axisymmetric geometry. boundary conditions along the z direction, and no slip boundary conditions on the outer boundary (r = R). On the axis of symmetry r = 0, n is in the z direction and the radial velocity vanishes. Note that the velocity, stress and anchoring conditions on the nematic-isotropic interface have been embedded into the diﬀuse-interface formulation and do not constitute boundary conditions. Initially both phases are at rest, the interface is a perfect cylinder and the molecular orientation is uniform and axial inside the ﬁber. Surface disturbances arise spontaneously from numerical noise. Most results presented are for a domain length H = 25a and width R = 3.33a. Since the dominant capillary wavelength is not known a priori, and in any event varies with the physical parameters, imposing periodicity over a ﬁnite H necessarily introduces errors to the result. Comparison with simulations in longer domains, with H up to 60a, shows that H aﬀects the results quantitatively but does not modify the qualitative trend. Thus, H = 25a represents a tradeoﬀ between computational cost and accuracy. For Newtonian ﬂuids, the conﬁnement eﬀect of the outer boundary on capillary instability of a ﬁlament has been studied by Mikami & Mason (1975). For R = 3.33a, the growth rate of the dominant mode should decrease by approximately 10%. For our nematic ﬁber, comparing the result with that in a wider domain with R = 10a shows that the conﬁnement reduces the growth rate by 4.6%. Therefore, the ﬁnite size of the computational domain, while exerting a quantitative inﬂuence, does not hinder the main purpose of the simulations. 149 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers The complete set of dimensionless groups governing our system is: α = β = AK = AW = Ca = Re = ρ1 (nematic-to-matrix density ratio), ρ2 α4 /2 (nematic-to-matrix viscosity ratio), μ K (bulk elasticity-to-interfacial tension ratio), σa W (anchoring-to-interfacial tension ratio), σ ηU (capillary number), σ ρ1 U a (Reynolds number), η (8.8) (8.9) (8.10) (8.11) (8.12) (8.13) along with the various length ratios of the geometry and ratios between the Leslie viscous coeﬃcients. Re and Ca are deﬁned using the visco-capillary velocity U = σ/η. Therefore the capillary number is 1, and the Reynolds number is kept at 150 throughout this study. Note that the typical velocity during the ﬁber breakup is roughly 1% of U , and the actual Ca and Re are much smaller. The viscosity η = (α3 + α4 + α5 )/2 is the average between the largest and smallest Miesowicz viscosities (de Gennes & Prost, 1993). The ratio α3 /α2 determines whether the nematic “tumbles” or “ﬂow-aligns” in simple shear ﬂows. But the distinction is unimportant here as ﬁber breakup engenders predominantly elongational ﬂows. Thus we adopt the Leslie viscosities of a common nematic MBBA at 25◦ C as the basis for the computations (de Gennes & Prost, 1993): α1 = 6.5 centipoise (cp), α2 = −77.5 cp, α3 = −1.2 cp, α4 = 83.2 cp, α5 = 46.3 cp, α6 = −32.4 cp. Furthermore, we match the density and isotropic viscosity of the nematic with the surrounding ﬂuid: α = 1, β = 1. In the following subsections, we study the eﬀects of the bulk elasticity, interface anchoring and anisotropic viscosity in turn, by varying AK , AW and the viscosity ratios, respectively. 8.3.1 Bulk elasticity With the anchoring energy ﬁxed at AW = 1, Fig. 8.2 compares the development of capillary waves at diﬀerent strengths of bulk elasticity. Note that the baseline case (Fig. 8.2a) is not for a Newtonian ﬂuid; the ﬁber still retains the same anisotropic viscosity and surface anchoring. In our periodic domain of length 25a, the dominant mode has three wave forms, with a wavelength of 8.33a. In comparison, an inﬁnitely long Newtonian ﬁber of the same viscosity would have a fastest growing wavelength of 9.66a according to Mikami and Mason’s model (Mikami & Mason, 1975). For a nematic ﬁber with a weak bulk elasticity, AK = 0.0167, the modiﬁcation to the dominant wavelength is too small to be manifested, and the three wave forms persist (Fig. 8.2b). At AK = 0.833, however, the dominant wavelength has lengthened to 12.5a with two wave forms (Fig. 8.2c). Further doubling the bulk elasticity to AK = 1.67 does not change the wavelength in Fig. 8.2(d ). 150 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers Figure 8.2: Eﬀect of bulk elasticity on capillary waves on a nematic ﬁber. The domain is 25a in length and 6.66a in width. The viscosity coeﬃcients are those of MBBA, and surface anchoring is ﬁxed at AW = 1. The bulk elasticity K increases from left to right: (a) AK = 0, t = 491; (b) AK = 0.0167, t = 497; (c) AK = 0.833, t = 547; (d ) AK = 1.67, t = 575. Time is made dimensionless by ηa/σ. One observation is that bulk elasticity tends to increase the wavelength of the capillary waves. This is in qualitative agreement with the predictions of linear stability analysis. Cheong & Rey (2004) showed that the fastest growing wavelength on an inﬁnitely long inviscid nematic ﬁber is √ (8.14) λmax = 2 2πa 1 + 2AK , which reduces to Rayleigh’s classical result for an inviscid ﬁber at K = 0. This formula predicts that the bulk elasticity would increase λmax by 1.7%, 63% and 108% for the three cases in Fig. 8.2(b–d ), which is consistent with the numerical results considering the constraint of the forced periodicity over H = 25a. However, further increasing AK up to 10 does not produce a single wave form in our domain, as expected from the linear formula above. We will return to this discrepancy shortly. A second observation is that the bulk elasticity tends to dampen the growth of the capillary waves. The four panels in Fig. 8.2 correspond to roughly the same wave amplitude. The time needed for reaching this amplitude increases with increasing AK . Again, this may be compared with the linear growth rate on an inviscid nematic ﬁber (Cheong & Rey, 2004): αmax = √ σ 8ρa3 1 + 2AK . (8.15) For our three cases (Fig. 8.2b–d ), this predicts a reduction in αmax of 2%, 39% and 52%, respectively. The actual damping of the growth rate in the simulations is smaller in magnitude: 1.2%, 10% and 15% for the three cases. It is perhaps unreasonable to expect a closer correspondence between our numerical results and Eqs. (8.14) and (8.15). Besides the aforementioned constraints of H = 25a, 151 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers R = 3.33a and the nonlinear nature of our results, the physical models diﬀer in that Cheong & Rey (2004) assumed inviscid ﬁbers and rigid anchoring on the interface. Rigid anchoring tends to amplify the eﬀects of bulk elasticity since it couples the interfacial deformation to bulk distortion more directly, without the “buﬀering” eﬀect of the anchoring energy. Therefore, it is not surprising that in our simulations using a ﬁnite W = σ, the dominant wavelength does not increase as much as predicted by Eq. (8.14), and the growth rate does not decrease as much as predicted by Eq. (8.15). A nonlinear feature of Fig. 8.2 is that the waves are not precisely periodic along the axial direction. The thinning of the ﬁber proceeds more rapidly at the upper “neck” than the lower. This can be easily understood from the capillary pressure in the ﬁber. Let us assume that two neighboring wave forms are initially identical. The high capillary pressure at the neck drives the ﬂuid toward the crest of the wave. If some small disturbance should slightly delay the thinning of one neck relative to the next, the thinner neck experiences a greater capillary pressure that more eﬀectively pumps ﬂuid away, thereby further widening the diﬀerence between the two necks. Thus, the uneven growth among the waveforms is a natural outcome of capillary instability. This behavior has been observed in Newtonian (Kowalewski, 1996) and nematic ﬁber breakup experiments (Tsakalos et al., 1996; Machiels et al., 1997), and is related to “volume scavenging” between coupled spherical-cap droplets (Theisen et al., 2007). As a result, polydisperse drops are produced. To produce monodisperse droplets, one can resort to strongly elongational ﬂows, as have been used in microﬂuidic devices (Zhou et al., 2006; Hamlington et al., 2007). Figure 8.3 depicts the late stage of ﬁber breakup for a nematic ﬁber with AK = 1, one expects the anchoring eﬀect to 0.0167 and AW = 1. Since AW /AK = W a/K dominate the bulk elasticity in determining the director ﬁeld (Zhou et al., 2007b). Indeed, throughout the breakup process, n follows the undulation of the interface except near the centerline, where the elongational ﬂow aligns n axially. The same elongational ﬂow stretches the neck into a thread (Fig. 8.3b), which then pinches oﬀ at both ends to form a satellite drop between the two daughter drops (Fig. 8.3c-d ). The pinchoﬀ produces pointed tips where the director ﬁeld converges. The high curvature there induces a large capillary force that pulls the tips back sharply, giving rise to ﬂat ends (Fig. 8.3d ) or even ﬂattened drops (Fig. 8.3e). In the meantime, the converging director ﬁeld develops “boojums” defects at the ends. Finally, the thread breaks up into three primary droplets and three satellite droplets. The drops display a bipolar conﬁguration (Fig. 8.3f ) with two boojums at the poles. The shape is nearly spherical in this case, but becomes more prolate with increasing AW and AK , similar to previous observations (Yue et al., 2005b). As anticipated earlier, the primary drops are not monodisperse; the bottom drop is some 5.7% smaller than the other two. Note also that the satellite drops shrink in time and eventually disappear owing to the Cahn-Hilliard diﬀusion. The implications of this diﬀuse-interface phenomenon has been examined at length (Yue et al., 2007). The nematic ﬁber breakup process, as simulated and discussed above, may be compared with experimental observations (Tsakalos et al., 1996; Machiels et al., 1997). First, the simulation and experiments agree in that the breakup of a nematic ﬁber does not 152 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers (a) (b) (c) (d ) (e) (f ) Figure 8.3: Evolution of the interface and the director ﬁeld during breakup. AW = 1.0, AK = 0.0167. The snapshots are at diﬀerent times: (a) t = 476, (b) 502, (c) 507, (d ) 512, (e) 566, (f ) 658. Note that n is shown only on a small number of interpolation points; the ﬁnite-element mesh is much denser. diﬀer markedly from that of a Newtonian ﬁber. The uneven wave growth, the pinchoﬀ at the neck and even the formation of satellite drops are qualitatively the same as in Newtonian ﬂuids (Notz et al., 2001). There is nothing as spectacular as, say, the beadon-string morphology for highly viscoelastic polymeric threads (Christanti & Walker, 2001). Quantitatively, the nematic order makes the breakup proceed more slowly, and we will amplify this point shortly in connection to the thinning of the neck. Second, the general features of the experiments are captured by the simulations, including the highly aligned n ﬁeld at the necks, the formation of satellite drops, the bipolar conﬁguration and the polydispersity of the primary drops (Tsakalos et al., 1996). Finally, there are a few observations that the computation fails to reproduce. For instance, a “banded structure”, visible through crossed polarizers, sometimes emerges prior to capillary instability (Tsakalos et al., 1996). This is probably due to the relaxation of the molecular order that has been elevated during the formation of the ﬁbers by stretching. Our Leslie-Ericksen theory does not account for such molecular relaxation. Furthermore, thermotropic LCP ﬁbers often break up into spherical drops containing polydomains whose disordered orientation renders the drop essentially isotropic on the whole (Machiels et al., 1997; Yu et al., 2004). The origin of defects and polydomains is a long-standing problem in LCP dynamics, and requires more sophisticated models than that used here. To examine more quantitatively the eﬀect of bulk elasticity on the breakup process, we plot in Fig. 8.4 the minimum neck radius Rn for several values of AK , which decreases in time until pinchoﬀ. Interestingly, the initial growth of the capillary waves (t < 300) is little inﬂuenced by the diﬀering bulk elasticity. At the beginning of the simulations, 153 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers Figure 8.4: Thinning of the minimum neck radius at diﬀerent levels of bulk elasticity. the nematic director n is aligned axially in the bulk and tangentially on the interface, which induces neither bulk elastic energy nor surface anchoring energy. As the capillary wave develops, surface undulation causes both surface and bulk distortion and the energy penalties amount to an elastic force that resists the growth of the capillary wave. This is the explanation for the stabilizing eﬀects of the nematic order. As a reaction to interfacial deformation, however, the eﬀect only becomes signiﬁcant as the capillary wave reaches a certain amplitude. As measured in Fig. 8.4, the amplitude is only about 0.05a at t = 300, and thus the director ﬁeld has yet to exert a signiﬁcant eﬀect on the capillary wave development. Later, with growing capillary waves, the interfacial and bulk distortion continue to absorb some of the energy released from interfacial area reduction. As a result, less is available to drive capillary breakup against viscous dissipation and inertia. This explains the longer pinchoﬀ time for larger AK values in Fig. 8.4. Finally, it is important to note that real liquid crystals typically have a weak K ∼ −11 N (de Gennes & Prost, 1993). With a surface tension σ ∼ 10−3 N/m (Kim et al., 10 2004; Wu & Mather, 2005), for example, AK = 0.01 for a ﬁber 1 μm in radius. Thus, LC bulk elasticity plays a signiﬁcant role only for nanoﬁbers, such as produced by electrospining (Srinivasan & Reneker, 1995; Reneker & Chun, 1996). However, certain lyotropic systems possess exceedingly low interfacial tensions (Kaznacheev et al., 2003), for which bulk elasticity eﬀect may be manifested at larger length scales. 8.3.2 Interface anchoring We have chosen to discuss bulk elasticity and surface anchoring separately, but obviously the two must cooperate for either to have an eﬀect. It is perhaps appropriate to say that the surface anchoring communicates interfacial deformation to the director ﬁeld in the bulk. In previous linear instability analyses, Rey (1997) isolated the eﬀect of surface anchoring by making the bulk elasticity inﬁnitely strong, and Cheong & Rey 154 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers AW Wavelength pinchoﬀ time 0 0.1 0.5 1 2 5 10 Newtonian 8.33a 8.33a 8.33a 12.5a 12.5a 12.5a 12.5a 8.33a 520 524 536 560 614 638 704 686 Table 8.1: Wavelength and pinchoﬀ time at various surface anchoring energies, with AK = 0.833, Ca = 1 and Re = 150. Time is made dimensionless by ηa/σ. The last column, for an isotropic Newtonian ﬂuid with a viscosity equal to the average viscosity η of the LC’s, will be cited in the next subsection. (2004) isolated the eﬀect of bulk elasticity by making surface anchoring rigid. In this subsection, we ﬁx bulk elasticity at AK = K/σa = 0.833, and vary the anchoring strength through AW = W/σ. The eﬀect of AW is illustrated by Table 8.1 that compares the wavelength and pinchoﬀ time for AW values ranging from 0 to 10. Thus, the surface anchoring tends to raise the threshold wavelength for unstable modes, and reduce their growth rates. In terms of hindering the growth of capillary instability, AW is similar to AK . This is no surprise because, as we alluded to above, the surface anchoring and bulk distortion are allied in bringing about the stabilizing eﬀect on capillary waves. For an inﬁnite nematic ﬁber with AK = ∞ but a ﬁnite AW , Rey’s analysis (Rey, 1997) gives the fastest growth wavelength √ λ = 2 2πa 1 + AW + 2 Ca Re 1 + AW (8.16) . (8.17) and the fastest growth rate αmax = σ 1 √ 3 8ρa 1 + A + W Ca Re For Ca = 1, Re = 150 and AW = 1, for example, the above formulae predict a 49% increase of the fastest-growing wavelength due to the interface anchoring, and a 34% decrease of its growth rate. Table 1 gives, for the corresponding conditions, a 50% lengthening of the wavelength and 7.1% decrease in the growth rate (estimated from the inverse of the total pinchoﬀ time). Considering the diﬀerences in the physical models, parameters and geometric setup, the qualitative agreement is reasonable. Wang (2001) carried out a similar normal mode analysis using the Doi theory for liquid-crystalline polymers in the limit of vanishing bulk elasticity. At a capillary number of unity, with AW increasing from 0 to 10, the most dangerous wavelength is roughly doubled, and its grow rate decreases by 40%. These numbers are again consistent with our results in Table 1. The later stage of the ﬁber breakup process does not vary qualitatively for the AW range simulated. Necking, drop pinchoﬀ and satellite drop formation are similar to those depicted in Fig. 8.3. In fact, these features are basically the same as in Newtonian 155 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers (a) (b) Figure 8.5: Eﬀect of AW on the director ﬁeld (a) inside the ﬁber during breakup and (b) inside a daughter drop. The radial dimension in (a) is ampliﬁed by a factor of 3 for a clearer view. In each plot, the left image corresponds to weak anchoring at AW = 0.1, with time t = 113 in (a) and t = 694 in (b), while the right to AW = 6 with t = 170 in (a) and t = 780 in (b). All other parameters are the same as in Table 1. ﬁber breakup, as noted in previous experimental observations (Tsakalos et al., 1996; Machiels et al., 1997). But the molecular orientation inside the ﬁber and later inside the drops does reﬂect the anchoring strength as shown in Fig. 8.5. For weak anchoring, n readily deviates from the easy direction on the interface so as to avoid comparatively expensive bulk distortions. During the growth of capillary waves (Fig. 8.5a), therefore, the interfacial undulation only aﬀects the outer layer of the nematic. In the daughter drops that result from the breakup (Fig. 8.5b), n does not nucleate boojum defects on the surface but maintains a relatively uniform orientation. For strong anchoring, the interfacial contour has a much greater impact on the bulk n ﬁeld, both in the ﬁber and the ﬁnal bipolar daughter drops. 8.3.3 Anisotropic viscosity The foregoing discussion on anchoring and bulk elasticity concern energetic interactions. In the later stage of breakup, ﬂuid ﬂow introduces considerable dissipation into the system. Thus, anisotropic viscosity, a key rheological feature of nematic LCs, becomes a factor in the development of ﬁnite-amplitude capillary waves and ﬁnal breakup. In fact, that is why in the preceding subsections, we compared the nematic ﬁbers not against an isotropic Newtonian baseline, but one with nil bulk or anchoring energy and the same anisotropic viscosity. The latter being kept the same, the eﬀects of AK and AW were thus isolated. For a truly Newtonian ﬁber with an isotropic viscosity matching the LC average viscosity η, the pinchoﬀ time is 686 in Table 1. Comparing this with the 156 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers (a) (b) Figure 8.6: Eﬀect of viscous anisotropy on the thinning of the neck during the breakup of nematic ﬁbers. The average LC viscosity η is ﬁxed such that Ca = 1, Re = 150. In addition, AW = 1, AK = 0.833. (a) Varying α2 ; (b) varying α3 . nematic ﬁbers having AW = 0 and AW = 10 in the same table, it is apparent that the viscous anisotropy may have as great an eﬀect on LC ﬁber breakup as AW (and AK ). To quantify this systematically, we ﬁx the surface anchoring and bulk elasticity at Aσ = 1 and AK = 0.833 and vary the degree of viscous anisotropy through the Leslie coeﬃcients. Note that this is essentially a nonlinear eﬀect in that it does not aﬀect the initial stages of linear instability. Given the ﬁve independent Leslie coeﬃcients, it is not obvious how to quantify viscous anisotropy. In simple shear ﬂows, a convenient gauge is the Miesowicz viscosities measured with the director n uniformly ﬁxed perpendicular or parallel to the ﬂow direction (de Gennes & Prost, 1993; Zhou et al., 2007a): −α2 + α4 + α5 , 2 α3 + α4 + α6 . = 2 η⊥ = (8.18) η (8.19) Borrowing the same idea to the ﬁber breakup problem, we have kept the average LC viscosity η = (η⊥ + η )/2 constant, and varied the viscous anisotropy via the ratio ν = η⊥ /η . Using the Onsager relationship α2 + α3 = α6 − α5 , we rewrite ν as ν= 2η − α2 − α3 4η = − 1, 2η + α2 + α3 2η + α2 + α3 (8.20) which shows that ν can be manipulated via either α2 or α3 . To keep η constant, α6 or α5 have to be adjusted accordingly. In the following, we vary α2 or α3 on the basis of the Leslie coeﬃcients of MBBA, which has η = 64.15 cp and ν = 4.34. 157 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers Figure 8.7: Axial velocity proﬁles vz (z) at two radial positions r = 0.2a and 0.65a at dimensionless time t = 499. AW = 1, AK = 0.833, and the viscosities correspond to ν = 4.34 in Fig. 8.6(a). The velocity is made dimensionless by σ/η, and the outline of the ﬁber at this instant is also shown. Figure 8.6 plots thinning of the neck radius for various degrees of viscous anisotropy. First, note that in the parameter ranges explored, the eﬀect of ν on the pinchoﬀ time is comparable to that of AK (Fig. 8.4) and AW (Table 1). With varying α2 , Fig. 8.6(a) shows a monotonic trend, with the ﬁber breaking up faster for larger ν (or smaller α2 ). Naively, one might rationalize this by the fact that in the neck region, the LC molecules are predominantly aligned to the ﬁber axis. Thus, the viscosity η should matter much more than η⊥ . Increasing the viscosity ratio ν then amounts to reducing η and consequently the “eﬀective viscosity” of the ordered LC. The same argument fails for Fig. 8.6(b), however, where the pinchoﬀ time does not depend on ν or α3 monotonically. The solution to this puzzle is that the ﬂow within the ﬁber has both elongation and shear components. While the former dominates at the thinning neck, the higher capillary pressure there drives the ﬂuid toward the wave crest, creating a shear ﬂow that may be likened to the Poiseuille ﬂow. This is illustrated by the axial velocity proﬁles in Fig. 8.7. The radial variation ∂vz /∂r gives the degree of shear while the axial one ∂vz /∂z indicates the stretching or compression. Therefore, it is necessary to consider the elongational viscosity of the nematic as well. Assuming a uniform director ﬁeld perfectly aligned with the stretching direction, n = (0, 0, 1) in cylindrical coordinates, the stress tensor in the nematic undergoing uniaxial elongation along z can be calculated from Eq. (8.6): α4 ˙ α4 ˙ ,− , (α1 + α4 + α5 + α6 ) ˙ , (8.21) σ = diag − 2 2 where ˙ is the strain rate. Note that α2 and α3 represent rotational friction and do not appear here. Thus, an elongational viscosity can be deﬁned from the normal stress 158 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers diﬀerence 3 α4 (8.22) η e = α1 + α4 + α5 + α6 = 4η + α1 − + α2 − α3 , 2 2 where we have invoked the average shear viscosity η and the Onsager relation. When we increase ν by decreasing α2 in Fig. 8.6(a), the elongational viscosity η e decreases together with the “eﬀective shear viscosity” mentioned above. The consequence is unequivocal: the ﬁber breaks up more rapidly. In Fig. 8.6(b), on the other hand, increasing ν by decreasing α3 tends to reduce the eﬀective shear viscosity, but in the meantime raises the elongational viscosity. The non-monotonic eﬀect on the breakup of the nematic ﬁber, therefore, can be interpreted as the outcome of the competition between these two mechanisms. 8.4 Summary In this paper we have investigated the breakup of nematic ﬁbers with planar anchoring on the surface and initially axial molecular orientation in the bulk. The process is simulated from the onset of linear disturbances to nonlinear growth and ﬁnally to formation of drops. The main ﬁndings can be summarized as follows: (a) Interface anchoring and bulk elasticity conspire to dampen the growth of capillary waves and the breakup process. In particular, the threshold wavelength for instability is raised and the growth rate of unstable modes is suppressed. (b) Anisotropic viscosity plays a signiﬁcant role in the growth of the capillary wave. The eﬀect of individual Leslie coeﬃcents depends on how it changes the elongational and shear viscosities of the nematic liquid crystal separately since both types of ﬂow are involved in the breakup. (c) The nonlinear growth of the capillary waves leads to the loss of axial periodicity and eventually the formation of polydisperse daughter drops. The nematic order within the ﬁber does not change the main features of the breakup, besides the quantitative eﬀect of slowing down the process. On the other hand, the interfacial deformation and ﬂuid ﬂow do have a direct eﬀect on the molecular orientation: the nematic is highly aligned in the neck region, and typically displays the bipolar conﬁguration in the daughter drops. (d ) The numerical results are in reasonable agreement with prior work in the literature where comparisons can be made. Speciﬁcally, the eﬀect of bulk elasticity and anchoring in suppressing capillary instability is in qualitative agreement with linear analysis. The numerically predicted breakup process captures the main features of experimental observations, and agrees with the latter in that nematic ﬁbers break up in basically the same way as Newtonian ones. In comparison with linear instability analysis, numerical simulations have the advantage of accessing the later stage of the breakup process. A disadvantage, however, is that the ﬁnite domain size tends to inﬂuence the wavelength that emerges. Comparison with linear analysis shows quantitative diﬀerences because of this restriction. Besides, this work leaves out several physical factors, including homeotropic or more general anchoring directions, bulk textures (radial or onion), non-axisymmetric modes of instability and 159 Chapter 8. Dynamic simulation of capillary breakup of nematic ﬁbers ﬁnally the role of molecular order parameter. The ﬁrst 3 have been analyzed in the linear limit (Cheong & Rey, 2004). The last is known to be relevant to the banded textures observed in nematic ﬁber breakup (Tsakalos et al., 1996). These are open issues that future work should explore. Nevertheless, this study appears to be the ﬁrst to explore the nonlinear stage of the capillary breakup and drop pinchoﬀ, and to include all the three factors—anchoring, bulk elasticity and viscous anisotropy—in a self-contained nemato-hydrodynamic theoretical framework. The results will be particularly relevant to the processing and manipulation of nano-scale nematic ﬁbers whose small dimension accentuates distortional elasticity relative to interfacial tension. 160 Bibliography Acierno, D. & Collyer, A. A. 1996 Rheology and Processing of Liquid Crystal Polymers. New York: Chapman and Hall. Cheong, A.-G. & Rey, A. D. 2001 Capillary instabilities in thin nematic liquid crystalline ﬁbers. Phys. Rev. E 64, 041701. Cheong, A.-G. & Rey, A. D. 2002 Cahn-hoﬀman capillarity vector thermodynamics for liquid crystal interfaces. Phys. Rev. E 66, 021704. Cheong, A.-G. & Rey, A. D. 2004 Texture dependence of capillary instabilities in nematic liquid crystalline ﬁbres. Liq. Cryst. 31, 1271–1284. Christanti, Y. & Walker, L. M. 2001 Surface tension driven jet break up of strainhardening polymer solutions. J. Non-Newtonian Fluid Mech. 100, 9–26. Dutta, D., Fruitwala, H., Kohli, A. & Weiss, R. A. 1990 Polymer blends containing liquid crystals: A review. Polym. Eng. Sci. 30, 1005–1018. Ericksen, J. L. 1991 Liquid crystals with variable degree of orientation. Arch. Rat. Mech. Anal. 113, 97–120. de Gennes, P. G. & Prost, J. 1993 The Physics of Liquid Crystals. New York: Oxford. Hamlington, B. D., Steinhaus, B., Feng, J. J., Link, D., Shelley, M. J. & Shen, A. Q. 2007 Liquid crystal droplet production in a microﬂuidic device. Liq. Cryst. 34, 861–870. Kaznacheev, A. V., Bogdanov, M. M. & Sonin, A. S. 2003 The inﬂuence of anchoring energy on the prolate shape of tactoids in lyotropic inorganic liquid crystals. J. Exp. Theor. Phys. 97, 1159–1167. Kernick, W. A. & Wagner, N. J. 1999 The role of liquid-crystalline polymer rheology on the evolving morphology of immiscible blends containing liquid-crystalline polymers. J. Rheol. 43, 521–549. Khullar, S., Zhou, C. & Feng, J. J. 2007 Dynamic evolution of topological defects around drops and bubbles rising in a nematic liquid crystal. Phys. Rev. Lett. 99, 237802. Kim, J.-W., Kim, H., Lee, M. & Magda, J. J. 2004 Interfacial tension of a nematic liquid crystal/water interface with homeotropic surface alignment. Langmuir 20, 8110– 8113. 161 Bibliography Kowalewski, T. A. 1996 On the separation of droplets from a liquid jet. Fluid Dyn. Res. 17, 121–145. Leslie, F. M. 1966 Some constitutive equations for anisotropic ﬂuids. Quart. J. Mech. Appl. Math. 19, 357–370. Liu, C. & Walkington, N. J. 2000 Approximation of liquid crystal ﬂows. SIAM J. Numer. Anal. 37, 725–741. Machiels, A. G. C., Dam, J. V., de Boer, A. P. & Norder, B. 1997 Stability of blends of thermotropic liquid crystalline polymers with thermoplastic polymers. Polym. Eng. Sci. 37, 1512–1525. Mikami, T. & Mason, S. G. 1975 The capillary break-up of a binary liquid column inside a tube. Can. J. Chem. Eng. 53, 372–377. Mucha, M. 2003 Polymer as an important component of blends and composites with liquid crystals. Prog. Polym. Sci. 28, 837–873. Nastishin, Y. A., Liu, H., Schneider, T., Nazarenko, V., Vasyuta, R., Shiyanovskii, S. V. & Lavrentovich, O. D. 2005 Optical characterization of the nematic lyotropic chromonic liquid crystals: Light absorption, birefringence, and scalar order parameter. Phys. Rev. E 72, 041711. Notz, P. K., Chen, A. U. & Basaran, O. A. 2001 Satellite drops: Unexpected dynamics and change of scaling during pinch-oﬀ. Phys. Fluids 13, 549–552. Rapini, A. & Papoular, M. 1969 Distortion d’une lamelle nematique sous champ magnetique conditions d’ancrage aux parois. J. Phys. (Paris) C 30, 54–56. Reneker, D. H. & Chun, I. 1996 Nanometre diameter ﬁbres of polymer, produced by electrospinning. Nanotech. 7, 216–223. Rey, A. D. 1997 Linear stability theory of break-up dynamics of nematic liquid crystalline ﬁbers. J. Phys. II France 7, 1001–1011. Srinivasan, G. & Reneker, D. H. 1995 Structure and morphology of small diameter electrospun aramid ﬁbers. Polym. Int. 36, 195–201. Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering ﬂows in small devices: Microﬂuidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411. Theisen, E. A., Vogel, M. J., Lopez, C. A., Hirsa, A. H. & Steen, P. H. 2007 Capillary dynamics of coupled spherical-cap droplets. J. Fluid Mech. 580, 495–505. Tsakalos, V. T., Navard, P. & Peuvrel-Disdier, E. 1996 Observations of the break-up of liquid crystalline polymer threads imbedded in an isotropic ﬂuid. Liq. Cryst. 21, 663–667. 162 Bibliography Wang, Q. 2001 Role of surface elasticity in capillary instability of cylindrical jets of nematic liquid crystalline polymers. J. Non-Newtonian Fluid Mech. 100, 97–114. Wu, J. & Mather, P. T. 2005 Interfacial tension of a liquid crystalline polymer in an isotropic polymer matrix. Macromol. 38, 7343–7351. Yu, R., Yu, W., Zhou, C. & Feng, J. J. 2004 Dynamic interfacial properties between a ﬂexible isotropic polymer and a TLCP investigated by an ellipsoidal drop retraction method. J. Appl. Polym. Sci. 94, 1404–1410. Yue, P., Feng, J. J., Liu, C. & Shen, J. 2004 A diﬀuse-interface method for simulating two-phase ﬂows of complex ﬂuids. J. Fluid Mech. 515, 293–317. Yue, P., Feng, J. J., Liu, C. & Shen, J. 2005a Diﬀuse-interface simulations of drop coalescence and retraction in viscoelastic ﬂuids. J. Non-Newtonian Fluid Mech. 129, 163–176. Yue, P., Feng, J. J., Liu, C. & Shen, J. 2005b Interfacial force and Marangoni ﬂow on a nematic drop retracting in an isotropic ﬂuid. J. Colloid Interface Sci. 290, 281–288. Yue, P., Zhou, C. & Feng, J. J. 2006a A computational study of the coalescence between a drop and an interface in Newtonian and viscoelastic ﬂuids. Phys. Fluids 18, 102102. Yue, P., Zhou, C. & Feng, J. J. 2007 Spontaneous shrinkage of drops and mass conservation in phase-ﬁeld simulations. J. Comput. Phys. 223, 1–9. Yue, P., Zhou, C., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2006b Phase-ﬁeld simulations of interfacial dynamics in viscoelastic ﬂuids using ﬁnite elements with adaptive meshing. J. Comput. Phys. 219, 47–67. Zhou, C., Yue, P. & Feng, J. J. 2006 Formation of simple and compound drops in microﬂuidic devices. Phys. Fluids 18, 092105. Zhou, C., Yue, P. & Feng, J. J. 2007a The rise of Newtonian drops in a nematic liquid crystal. J. Fluid Mech. 593, 385–404. Zhou, C., Yue, P. & Feng, J. J. 2008 Dynamic simulation of droplet interaction and self-assembly in a nematic liquid crystal. Langmuir 24, 3099–3110. Zhou, C., Yue, P., Feng, J. J., Liu, C. & Shen, J. 2007b Heart-shaped bubbles rising in anisotropic liquids. Phys. Fluids 19, 041703. 163 Chapter 9 Conclusions and recommendation We developed the diﬀuse interface based numerical toolkit, AMPHI, to simulate interfacial dynamics in complex ﬂuids, and applied it to a number of physical problems. The numerical algorithm has two major elements: an eﬃcient ﬁnite-element ﬂow solver and an adaptive meshing scheme. The ﬂow solver was originally developed for solving NavierStokes systems, and the adaptive meshing deploys a ﬁne grid near the moving interface and a coarse grid away from the interface to achieve high numerical accuracy at a reasonable computational cost. The phase-ﬁeld parameter φ obviates the need to track the moving and deforming interface, and allows discretization and numerical solution within an Eulerian framework. Moreover, its gradient ∇φ supplies a convenient criterion for the meshing scheme to determine the local mesh size. In this thesis, two types of complex ﬂuids have been investigated: viscoelastic ﬂuid and nematic liquid crystal. The viscoelastic ﬂuid is described by the Oldroyd-B model, and the liquid crystal is modeled by the Leslie-Ericksen theory. For each simulation, the numerical results have been summarized at the end the preceding chapters. In this chapter, we conclude the thesis by oﬀering a more general view of the theoretical model, the numerical method and physical insights gained from this work, and by suggesting further research to be undertaken in the area of interfacial dynamics in complex ﬂuids. 9.1 Theoretical model and numerical algorithm For the problems of interest here, the diﬀuse interface model proves to be a convenient and uniquely revealing theoretical framework. In terms of interface capturing on a Eulerian grid, the method is comparable to level set and volume-of-ﬂuid methods. However, it is unique in that the theoretical formulation is rooted in a physically meaningful idea: the free energy of the mixing between the species. This endows the resulting model several desirable features. (a) Complex rheology can be included along with interfacial dynamics in a uniﬁed variational procedure. (b) The model incorporates coarse-grained molecular interactions in the form of a disjoining potential comparable to the van der Waals potential. (c) The variational formulation ensures an energy law for the two-phase system and facilitates analysis of the convergence and regularity of the numerical solution (Lin & Liu, 2000; Liu et al., 2005; Feng et al., 2005). In terms of numerical eﬃciency, AMPHI represents a major improvement over a previous spectral algorithms (Yue et al., 2004). Adaptive meshing results in considerable savings in grid numbers, thus making it possible to simulate thinner interfaces with better resolution. This also opens up opportunities for tackling larger-scale ﬂow problems, especially in 3D. Moreover, the ﬁnite-element method easily accommodates complex ﬂow geometry and various boundary conditions. The theoretical model and numerical schemes have been validated extensively in various ﬂow conditions and the numerical accuracy 164 Chapter 9. Conclusions and recommendation has been demonstrated (Yue et al., 2004, 2006b,a). The implicit scheme fully couples the ﬂow equations and interface Cahn-Hilliard equations, and computes ﬂows dominated by large interfacial tension accurately with robust numerical stability. The convergence with the interfacial thickness and mesh size has been established. Note also that the adaptive meshing can be viewed as a compromise between ﬁxed Eulerian grid and moving Lagrangian grid; our grid remains ﬁxed for a certain number of time steps and then adapts to the moving interface. 9.2 Physical insights One may summarize the ﬁndings of the whole thesis by one insight: interfaces of complex ﬂuids behave diﬀerently than those of Newtonian ﬂuids. The ultimate cause of the “anomalous” behavior is the coupling among several factors: the microstructural conformation in the bulk ﬂuids, the interfacial morphology, and the hydrodynamics of the ﬂow ﬁeld. These can be viewed as pertaining, respectively, to the micro-, meso- and macroscopic length scales of the problem, and all the analysis in the preceding chapters was devoted to unraveling the interplay among these length scales. The insights gained in this exercise are not only of scientiﬁc interest, but also important to engineering applications. For example, multi-layer coextrusion of polymer ﬁlms depends critically on maintaining the interfacial shape and position, and these in turn depends on the rheology of each component on the one hand, and the overall ﬂow ﬁeld on the other. Another example is polymer dispersed liquid crystals, where the drop shape depends on the microstructure of the mesogenic phase and the bulk deformation that the mixture undergoes during processing. 9.3 Theoretical and numerical limitations We must point out the limitations of our theoretical model and numerical scheme. First, resolving the interfacial layer constitutes a formidable numerical challenge. Physical systems typically have an interfacial thickness of nanometers and a macroscopic length scale ranging from micrometers to millimeters. This disparity in length scales is a generic conundrum for numerical resolution. Thus, one has to use as thin an interface as computationally aﬀordable while still deploying enough grid points within this layer to ensure accurate evaluation of the interfacial tension. This is the main motivation for our adaptive meshing scheme. Even with it, interfacial resolution remains the bottleneck for our scheme. Thus, the advantage of having a physically meaningful treatment of the interfacial proﬁle comes at the price of computing the 4th-order Cahn-Hilliard equation. There are physical problems, such as the moving contact line (Jacqmin, 2000), where the former is essential. Then 3D simulation will require parallel computation. We will return to this point when discussing future work. Second, the phase-ﬁeld model is not easily generalized to more than two ﬂuid components. A three-component system entails multiple order parameters and interaction energies; such phase-ﬁeld models have appeared in the 165 Chapter 9. Conclusions and recommendation literature but their complexity makes numerical computation diﬃcult (Kim et al., 2004). Finally, the Leslie-Ericksen theory for the nematic liquid crystals is the simplest that couples molecular orientation and hydrodynamics. Its vectorial nature is inadequate for several aspects of the microstructure. The original version (de Gennes & Prost, 1993) cannot handle defects as they would constitute singularities. A relaxation of the unitlength requirement on the director allows integer-strength defects to be simulated, but the Saturn ring has to be represented by a surface ring. The latter has fewer degrees of freedom and possibly diﬀerent stability from an unattached Saturn ring. This restriction can be removed by adopting a tensorial representation of the molecular orientation (Rey & Tsuji, 1998; Feng et al., 2000; Yoneya et al., 2005). 9.4 Recommendations for future work The work described in this thesis may be extended along two directions. One is to upgrade the numerical toolkit to 3D and to incorporate more features, and the other is to apply the tools to other interesting physical problems. Upgrading the numerical code may include the following aspects. (a) Extending the current 2D code (AMPHI-2D) into a full 3D version AMPHI-3D. Both the current adaptive mesh generator and the Navier-Stokes ﬂow solver are available in 3D already. One will need to generalize the diﬀuse-interface solver (for the CahnHilliard equations) to 3D. This will greatly expand the range of problems that can be investigated using AMPHI, a few examples of which are given below. (b) Extending the current two-component diﬀuse-interface formulation to allow three components. This feature will allow us to model compound drops, double emulsions, and eukaryotic cells more realistically. Up to now, we have had to make the innermost ﬂuid identical to the outermost one. This is a poor representation of the cell nucleus, for instance, which is much more viscous and stiﬀer than the cytoplasm. Phase-ﬁeld formulations for three-component systems are available in the literature, and the main challenge will be to solve the more complex equations eﬃciently. (c) Generalizing the wetting conditions on solid walls to allow a wide range of contact angles. This can be achieved by adding a surface energy functional to the current formulation (Jacqmin, 2000). With the freedom to deﬁne varying degrees of wettability on diﬀerent parts of the domain boundary, one will be able to investigate the control of interfacial motion through substrate patterning. The microﬂuidic device with a Tjunction, for one, uses alternating hydrophobicity and philicity in diﬀerent branches to create double emulsions. The role of wettability can then be elucidated systematically by simulations. (d) Incorporating thermocapillary ﬂows. This consists in adding a heat transfer equation into the solver and allowing the interfacial energy to be temperature-dependent. The motivation is to control and manipulate microscopic droplets by temperature gradients. This is a common strategy in thermo-capillary ﬂows that depends on Marangoni forces. Furthermore, thermal gradients may provide a convenient means for moving particles and droplets in liquid crystals by way of modifying the temperature-dependent bulk elasticity 166 Chapter 9. Conclusions and recommendation and surface anchoring. The upgraded numerical code can be applied fruitfully to a wide range of physical problems, three of which are outlined below. (a) Drop formation in microﬂuidic devices. One may extend the studies of drop formation in the microﬂuidic ﬂow-focusing device from axisymmetric geometries into full 3-dimensional ones. Experiments have almost always used rectangular micro-channels since they are easier to fabricate than ones with circular cross sections. In addition, one may investigate asymmetric microﬂuidic devices such as the T-junction (Thorsen et al., 2001). In this geometry, the conﬂuence of two immiscible ﬂows, possibly water and oil, yields micro-droplets. Surface wettability is a key player in the process, especially in compound drop formation. The numerical simulations will quantify a variety of parametric eﬀects to guide the microfabrication of the devices, the selection the ﬂuid properties and the control of ﬂow conditions in the experiments. (b) Drop migration and manipulation. Drop migration in 3D channels and channel bifurcations has potential applications in drop-size-based separation, emulsion densiﬁcations and separation of blood cells (Shevkoplyas et al., 2005) (see item c below). Moreover, the introduction of temperature gradients gives an additional means of manipulating the drops, and possibly separating them according to size. The induced Marangoni ﬂow inside the drop would also enhance the internal mixing and facilitate chemical reactions (Bringer et al., 2004). (c) Biological applications. There is a close resemblance between liquid drops and blood cells, and simple and compound drop models may be used to simulate the motion and deformation of leukocytes in narrow capillaries. The cytoplasm can be either Newtonian or viscoelastic, and the nucleus can be represented by an inner drop of higher viscosity and interfacial tension. Three processes are of particular interest: entry of neutrophils into pulmonary capillaries that have smaller size than the blood cells under a constant pressure drop; migration of cells in blood vessels (the F˚ ahræus and F˚ ahræus-Lindqvist eﬀects (Popel & Johnson, 2005)); and the separation of blood cells at bifurcations (the “bifurcation law” (Pries et al., 1996; Popel & Johnson, 2005)). Interestingly, the last eﬀect has been exploited to extract white cells from blood using a microﬂuidic device (Shevkoplyas et al., 2005). 167 Bibliography Bringer, M. R., Gerdts, C. J., Song, H., Tice, J. D. & Ismagilov, R. F. 2004 Microﬂuidic systems for chemical kinetics that rely on chaotic mixing in droplets. Phil. Trans. R. Soc. Lond. A 362, 1087–1104. Feng, J. J., Liu, C., Shen, J. & Yue, P. 2005 An energetic variational formulation with phase ﬁeld methods for interfacial dynamics of complex ﬂuids: advantages and challenges. In Modeling of Soft Matter (ed. M.-C. T. Calderer & E. Terentjev). New York: Springer. Feng, J. J., Sgalari, G. & Leal, L. G. 2000 A theory for ﬂowing nematic polymers with orientational distortion. J. Rheol. 44, 1085–1101. de Gennes, P. G. & Prost, J. 1993 The Physics of Liquid Crystals. New York: Oxford. Jacqmin, D. 2000 Contact-line dynamics of a diﬀuse ﬂuid interface. J. Fluid Mech. 402, 57–88. Kim, J., Kang, K. & Lowengrub, J. 2004 Conservative multigrid methods for ternary Cahn-Hilliard systems. Comm. Math. Sci. 2, 53–77. Lin, F. H. & Liu, C. 2000 Existence of solutions for the Ericksen-Leslie system. Arch. Rat. Mech. Anal. 154, 135–156. Liu, C., Shen, J., Feng, J. J. & Yue, P. 2005 Variational approach in two-phase ﬂows of complex ﬂuids: transport and induced elastic stress. In Mathematical Models and Methods in Phase Transitions (ed. A. Miranville). Nova Publications. Popel, A. S. & Johnson, P. C. 2005 Microcirculation and hemorheology. Annu. Rev. Fluid Mech. 37, 43–69. Pries, A. R., Secomb, T. W. & Gaehtgens, P. 1996 Biophysical aspects of blood ﬂow in the microvasculature. Cardiovasc. Res. 32, 654–667. Rey, A. D. & Tsuji, T. 1998 Recent advances in theoretical liquid crystal rheology. Macromol. Theory Simul. 7, 623–639. Shevkoplyas, S. S., Yoshida, T., Munn, L. L. & Bitensky, M. W. 2005 Biomimetic autoseparation of leukocytes from whole blood in a microﬂuidic device. Anal. Chem. 77, 933–937. Thorsen, T., Roberts, R. W., Arnold, F. H. & Quake, S. R. 2001 Dynamic pattern formation in a vesicle-generating microﬂuic device. Phys. Rev. Lett. 86, 4163– 4166. 168 Bibliography Yoneya, M., Fukuda, J.-I., Yokoyama, H. & Stark, H. 2005 Eﬀect of a hydrodynamic ﬂow on the orienation proﬁles of a nematic liquid crystal around a spherical particle. Mol. Cryst. Liq. Cryst. 435, 75–85. Yue, P., Feng, J. J., Liu, C. & Shen, J. 2004 A diﬀuse-interface method for simulating two-phase ﬂows of complex ﬂuids. J. Fluid Mech. 515, 293–317. Yue, P., Zhou, C. & Feng, J. J. 2006a A computational study of the coalescence between a drop and an interface in Newtonian and viscoelastic ﬂuids. Phys. Fluids 18, 102102. Yue, P., Zhou, C., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2006b Phase-ﬁeld simulations of interfacial dynamics in viscoelastic ﬂuids using ﬁnite elements with adaptive meshing. J. Comput. Phys. 219, 47–67. 169
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Simulations of interfacial dynamics of complex fluids using diffuse interface method with adaptive meshing Zhou, Chunfeng 2008
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Title | Simulations of interfacial dynamics of complex fluids using diffuse interface method with adaptive meshing |
Creator |
Zhou, Chunfeng |
Publisher | University of British Columbia |
Date Issued | 2008 |
Description | A diffuse-interface finite-element method has been applied to simulate the flow of two-component rheologically complex fluids. It treats the interfaces as having a finite thickness with a phase-field parameter varying continuously from one phase to the other. Adaptive meshing is applied to produce fine grid near the interface and coarse mesh in the bulk. It leads to accurate resolution of the interface at modest computational costs. An advantage of this method is that topological changes such as interfacial rupture and coalescence happen naturally under a short-range force resembling the van der Waals force. There is no need for manual intervention as in sharp-interface model to effect such event. Moreover, this energy-based formulation easily incorporates complex rheology as long as the free energy of the microstructures is known. The complex fluids considered in this thesis include viscoelastic fluids and nematic liquid crystals. Viscoelasticity is represented by the Oldroyd-B model, derived for a dilute polymer solution as linear elastic dumbbells suspended in a Newtonian solvent. The Leslie-Ericksen model is used for nematic liquid crystals，which features distortional elasticity and viscous anisotropy. The interfacial dynamics of such complex fluids are of both scientific and practical significance. The thesis describes seven computational studies of physically interesting problems. The numerical simulations of monodisperse drop formation in microfluidic devices have reproduced scenarios of jet breakup and drop formation observed in experiments. Parametric studies have shown dripping and jetting regimes for increasing flow rates, and elucidated the effects of flow and rheological parameters on the drop formation process and the final drop size. A simple liquid drop model is used to study the neutrophil, the most common type of white blood cell, transit in pulmonary capillaries. The cell size, viscosity and rheological properties are found to determine the transit time. A compound drop model is also employed to account for the cell nucleus. The other four cases concern drop and bubble dynamics in nematic liquid crystals, as determined by the coupling among interfacial anchoring, bulk elasticity and anisotropic viscosity. In particular, the simulations reproduce unusual bubble shapes seen in experiments, and predict self-assembly of microdroplets in nematic media. |
Extent | 2191237 bytes |
Subject |
Interfacial dynamics Complex fluids Diffuse-interface method Adaptive meshing |
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Thesis/Dissertation |
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Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-07-22 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0058512 |
URI | http://hdl.handle.net/2429/1062 |
Degree |
Doctor of Philosophy - PhD |
Program |
Chemical and Biological Engineering |
Affiliation |
Applied Science, Faculty of Chemical and Biological Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 2008-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
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