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Capillary menisci between particles absorbed at a liquid-fluid interface Hou, Linda 1993

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Capillary Menisci between Particles Adsorbedat a Liquid-Fluid InterfacebyLinda HouB.A.Sc. University ofBritish Columbia, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Department of Chemical Engineering)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIA© Linda Hou, November 1993In presenting this thesis in partial fi.ilfihlment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make it freelyavailable for reference and study. I fi.irther agree that permission for extensive copying ofthis thesis for scholarly purposes may be granted by the head of my department or by hisor her representatives. It is understood that copying or publication of this thesis forfinancial gain shall not be allowed without my written permission.Department of Chemical EngineeringUniversity ofBritish Columbia2216 Main MallVancouver, B.C., CanadaV6T 1Z4Date: November 1993.UABSTRACTOne aspect of the stability of solids-stabilized emulsions was indicated by Denkov et al. (8)who reported that the capillary pressure required for an emulsion droplet’s liquid to squeezebetween the interfacial particles and come into contact with another droplet’s free liquid interfacemust be overcome and provides a barrier to the thinning of the film between droplets. In light ofthe work done by Denkov et al., the objective of the thesis has been to determine the effect severalfactors such as particle size, separation distance, wettability, fluid properties, and contact anglehysteresis have on emulsion stability, represented in terms of capillary pressures, using a two-layerparticle model for coalescence.Two general models were developed, one which is based on a uniform layer of sphericalparticles adsorbed on a fluid-fluid interface, and the other, a two dimensional analogue, in whichparallel, horizontal cylinders are situated on the interface. Numerical techniques were applied inthe solution of both models since no simplification was made by the neglect of gravity as is foundin similar models (8, 11, 12). As a consequence, the models can be used to describe macroscopicsystems whose characteristic dimensions are well above the micron scale. A correspondingexperimental system employing parallel cylinders was then constructed and used to generatecapillary pressures and meniscus profiles for comparison with the computed model results.The model results agree with the experimental findings in the literature that the smaller theparticles, the closer the packing of particles, or the rougher the solids are, the more stable theemulsions. Furthermore, a decrease in the Bond number or a decrease in wettability of theparticles to the disperse phase would increase stability based on capillary pressure considerations.Overall trends for the cylinders model and the spheres model were similar. The contact angleswhich yield optimal stability from capillary and thermodynamic considerations lie in the range of900 O 1800.111The study revealed that good agreement between the model and the experimentalmeasurements was obtained when apparent contact angles, which take into account hysteresis,were used to generate the model profiles for each meniscus.ivTABLE OF CONTENTSAbstract.iiList of Figures .iiList of Tables xiiiAcknowledgements xiv1. Introduction 12. Background 52.1 General Concepts of Solid-Liquid Interfacial Thermodynamics 5a) Interfacial Tension and Free Energy 5b) Contact Angle 6c) Hysteresis 8d) Minimization of Free Energy in Emulsions 82.2 Equilibrium of Particles on Horizontal Interfaces 92.3 Emulsions Stabilized by Finely Divided Solids 102.4 The Young-LaPlace Equation and the Equilibrium Shape of Menisci 123. Mathematical Analysis of the Axisymmetric Interface between A Sphere and itsNeighboring Spheres 173.1 Literature Review of Related Sphere Models 183.2 The Sphere Model Configuration 193.3 Mathematical Formulation of the Differential Equations of the Meniscus Shape 233.3.1 Concave Menisci 243.3.2 Convex Menisci 263.4 System of Equations in Alternate Coordinates 283.5 Boundary Conditions 303.6 Effect of Contact Angle Hysteresis 31V3.7 Determination of Film Thickness .343.8 Numerical Solution 354. Mathematical Analysis of the Capillary Shapes between Parallel, Horizontal Cylinders 424.1 Literature Review 434.2 System Configuration 434.3 Mathematical Formulation 454.3.1 Convex Meniscus 464.3.2 Concave Meniscus 504.4 Determination of Film Rupture Threshold 534.5 Numerical Computation 545. Theoretical Model Results 615.1 Relationship to Emulsion Droplet Coalescence 615.2 Characterization of Profile Curves 635.3 Cylinders Model 645.3.1 Effect of the Contact Angle 665.3.2 Effect of the Bond Number 665.3.3 Effect of the Separation Distance 735.4 Spheres Model 775.4.1 Effect of the Contact Angle 775.4.2 Effect of the Bond Number 805.4.3 Effect of the Separation Distance 855.5 Effect of Hysteresis 855.6 Comparison of Cylinders and Spheres 905.7 Relationship to Emulsion Stabilization with Finely Divided Solids 916. The Experimental Program 986.1 Meniscus Profile Image Recording - Background Trials 986.2 Final Image Definition Technique 101vi6.3 Experimental Equipment and Setup .1016.3.1 The Experimental Cell 1036.3.2 Micromanometer 1046.3.3 Lighting 1066.3.4 Photographic Equipment 1066.4 Experimental Preparation of Rods 1066.5 Experimental Procedure 1077. Results and Discussion 1127.1 Comparison with Theoretical Curves 1127.2 Separation Distance Between Rods 1167.3 Effect of Contact Angle 1197.4 Effect of Hysteresis 1217.5 Apparent Contact Angles 1227.6 Hysteresis and Kinetic Forces 1327.7 Sources of Experimental Error 1378. Conclusions 139Nomenclature 143References 146Appendix A - Experimental Data 152Appendix B - Sample Calculations and Derivations 164B. 1 Mathematical Formulation for the Negligible Gravity Case for Cylinders 165B.2 Jacobian Matrix for the Solution of the Spheres Model 169B.3 Capillary Pressure Measurements for Experiments and Sample Calculations 170B.4 Digitizing Program 173Appendix C - Computer Programs 181CALP 182CBO 184viiCD! .187CHYS .189SPALP .192SBO 195SD! 198SHYS 202Digitizing Program 206vu’LIST OF FIGURESFigure 1.1 Denkov et. al.’s one layer-particle model for coalescence 2Figure 1.2 Two-layer particle model of present work 3Figure 2.1 Intermolecular forces at an interface 5Figure 2.2 Illustration of the three interfacial tensions at the s/o/w interface 7Figure 2.3 Emulsion droplet with spherical particle on interface 10Figure 2.4 Radii of curvature of a curved surface 12Figure 3.1 Emulsion droplet with adsorbed layer of particles 18Figure 3.2 Cylindrical cell model for spheres 20Figure 3.3 Coordinate system for sphere cell model 21Figure 3.4 Radii of curvature of surface of revolution described by the curve 21Figure 3.5 Capillary Rise 22Figure 3.6 Capillary Depression 22Figure 3.7 Concave meniscus profile 25Figure 3.8 Convex meniscus profile 26Figure 3.9 The surface of an idealized rough solid with a sine-wave corrugation 32Figure 3.10 Point of rupture between two emulsion droplets 34Figure 3.11 Program flowsheet for meniscus between spheres 40Figure 4.1 Experimental cell (simplified) 42Figure 4.2 Coordinate system for parallel cylinders 45Figure 4.3 Angles for convex (a) and concave (b) menisci 52Figure 4.4 Cylinder model flowsheet of program CALP 57Figure 4.5 Cylinder model flowsheet of program CHYS 59Figure 5.1 Meniscus profiles between a pair of cylinders (Bo3.247, 0=90°, b/a = 1.5) 65Figure 5.2 Meniscus between parallel horizontal cylinders, Bo=0. 130, 0=30° 67Figure 5.3 , Bo0.130, 0=60° 67ixFigure 5.4 “, Bo=0.130, 0=900.67Figure 5 5 “, Bo=0 130, 0=120° 67Figure 5.6 “, Bo=0.130, 0=150° 68Figure 5.7 I’ “,Bo=0.130, 0=180° 68Figure 5.8 Threshold rupture pressure vs. contact angle (cylinders), b/a= 1.50 69Figure 5.9 Meniscus profiles between parallel horizontal cylinders, Bo=1.30x1050=90° 70Figure 5.10 ‘I Bo=1.30x103,0=90° 70Figure 5.11 H “, Bo3.248, 0=90° 70Figure 5.12 H u I’ , Bo=1.30x10’, 0=90° 70Figure 5.13 Rupture capillary pressure vs. bond number (cylinders), b/a 1.5 71Figure 5.14 Rupture capillary pressure vs. bond number (cylinders), b/a3 .0 71Figure 5.15 Meniscus profiles between a pair of cylinders, Bo0.130, 0 =90°, b/a = 1.05 74Figure 5 16 “ H H ‘,Bo=O 130 0=90° b/a = 1 50 74Figure 5 17 “ I’ Bo0 130 0=90°, b/a = 3 00 74Figure 5.18 H “, Bo0.130, 0=90°, b/a = 6.00 74Figure 5.19 Rupture capillary pressure vs. separation distance (cylinders), Bo=0. 130 75Figure 5.20 Rupture capillary pressure vs. separation distance (cylinders), Bo=3 .247 75Figure 5.21 Meniscus profiles between spheres, Bo=0.130, 0=30° 78Figure 5.22 H , Bo=0.130, 060° 78Figure 5.23 “ “,Bo=O. 130, 0=90° 78Figure 5.24 “ H , Bo=0.130, 0120° 78Figure 5.25 “ “,Bo0.130, 0=150° 79Figure 5.26 “ ‘I H Bo=0.130, 0=180° 79Figure 5.27 Threshold rupture pressure vs. contact angle (spheres), b/a 1.50 81Figure 5.28 Meniscus profiles between spheres, Bo=1.30x1050=90° 82Figure 5.29 H H , Bo1.30x1030=90° 82Figure 5.30 I’ , Bo=3.247, 0 =90° 82xFigures.31 ‘I “,Bo=l.30x10’, 0=900.82Figure 5.32 Rupture capillary pressure vs. bond number (spheres) , b/a=1.50 83Figure 5.33 , b/a3.0 83Figure 5.34 Meniscus profiles between spheres, b/a= 1.05, 0=90° 86Figure 5.35 “ I H b/a1.50, 0 =90° 86Figure 5.36 “ ‘I “,b/a=3.00, 0 =90° 86Figure 5.37 ‘I b/a=6.00, 0 =90° 86Figure 5.38 Rupture capillary pressure vs. separation distance (spheres) , Bo0. 130 87Figure 5.39 H ‘I , Bo=3.247 87Figure 5.40 Rupture capillary pressure vs. film thickness (cylinders) , Bo=0. 130, b/a= 1.50 88Figure 5.41 “ “ “, Bo=3.247, b/a1.50 88Figure 5.42 Rupture capillary pressure vs. film thickness (spheres) , Bo=0.130, b/a=1.50 88Figure 5.43 “ ‘I , Bo=3.247, b/a=1.50 88Figure 5.44 Particle size effect on cap (cylinders and spheres) 93Figure 5.45 Cylinders-separation distance effects on 95Figure 5.46 Spheres-separation distance effects on 95Figure 5.47 Hysteresis Plots:Rupture capillary pressure vs. Film Thickness (cylinders),Bo1.30x107,b/a1.05 96Figure 5.48 Hysteresis Plots (cylinders), Bo=1.30x107b/a=3.00 96Figure 5.49 Hysteresis Plots (spheres), Bo=1.30x107b/a1.05 96Figure 5.50 Hysteresis Plots (spheres), Bo=1.30x107b/a=3.00 96Figure 6.1 Original test setup 99Figure 6.2 Negatives produced by laser light technique 99Figure 6.3 Experimental equipment setup 102Figure 6.4 Experimental cell 103Figure 6.5 Plan view of the bottom cell; front view of the test cell 105Figure 6.6 Coating of rods with polymeric resin 108xiFigure 6.7 Calibration sphere on microscope slide 109Figure 6.8 Experimental program flowsheet 111Figure 7.1 Experiment B meniscus profiles 113Figure 7.2 Experiment C meniscus profiles 113Figure 7.3 Experiment D meniscus profiles 114Figure 7.4 Experiment E meniscus profiles 114Figure 7.5 Experiment I meniscus profiles 115Figure 7.6 Experiment J meniscus profiles 115Figure 7.7 Model relationship between Pcap’ and q& for various b/a 117Figure 7.8a Comparison of set 1 data- Pcap’ V5. 95c 118Figure 7.8b Comparison of set 2 data-‘3cap’ VS. c 118Figure 7.9 Model relationship between capillary pressure and cS for various 0 120Figure 7.10 Set 1 - Apparent contact angles vs. a position 123Figure 7.11 Set 2 - Apparent contact angles vs. a position 123Figure 7.12 Experiment B comparison with model-V5. c 124Figure 7.13 Experiment C comparison with model-1cap’ VS. 124Figure 7.14 Experiment D comparison with model-VS. c 125Figure 7.15 Experiment E comparison with model-VS. øc 125Figure 7.16 Experiment I comparison with model-VS. øc 126Figure 7.17 Experiment J comparison with model-VS. Ø 126Figure 7.18 Experiment B curve (B-CVXI6) fit with model curves for 0=78° 128Figure 7.19 Experiment C curve (C-CVX16) fit with model curves for 0=88.5° 128Figure 7.20 Experiment D curve (D-CVXI7) fit with model curves for 0=84.5° 129Figure 7.21 Experiment E curve (CVX28) fit with model curves for 0=75.1° 129Figure 7.22 Experiment I curve (CVX7) fit with model curves for 0=70.5° 130Figure 7.23 Experiment J curve (CVX6) fit with model curves for 0=67.2° 130Figure 7.24 Experiment B comparison with model - a position vs. c 131xiiFigure 7.25 Wilhelmy plate method 132Figure 7.26a Force vs. depth curves for the Wilhelmy plate apparatus with hysteresis 133Figure 7.26b Force vs. depth curves for the Wilhelmy plate apparatus without hysteresis 133Figure 7.27 A hysteresis ioop for a system showing solid-liquid interaction 133Figure 7.28 Hysteresis loop for experiment J 135Figure 7.29 Typical experimental results for the dependence of the dynamic contact angle onspeed of the contact line 137xliiLIST OF TABLESTable 5.1 Properties of selected fluid-fluid interfaces (20°C) 63Table 5.2 Cylinders: Capillary Pressure Dependence on Particle Size (b/a= 1.05, c= 12.99) 91Table 5.3 Cylinders: Capillary Pressure Dependence on Separation Distance (a= 1 tm) 92Table 5.4 Spheres: Capillary Pressure Dependence on Particle Size (b/a=1 .05, c=12.99) 92Table 5.5 Spheres: Capillary Pressure Dependence on Separation Distance (a1 .tm) 92Table 5.6 Comparison of Denkov et al.’s results with the present work 97Table 7.1 Contact Angle Hysteresis 121Table 7.2 Reproducibility of the Level Meniscus for Set 2 136Table A. 1 Micromanometer Readings 153Table A.2 Set 1 Compiled Experimental Data 154Table A.3 Set 2 Compiled Experimental Data 156Table A.4 Experiment B - Comparison of baSed on a and c5 158Table A.5 Experiment C - Comparison ofPcap’ based on a and 159Table A.6 Experiment D- Comparison of based on a and c 160Table A. 7 Experiment E - Comparison of based on a and c 161Table A.8 Experiment I - Comparison of based on a and cb 162Table A.9 Experiment J - Comparison of based on a and c5 163xivACKNOWLEDGEMENTSI wish to thank the following people who have helped in making this thesis possible, mysupervisors Dr. B. Bowen and Dr. S. Levine for their expertise, direction and proofreading; theworkshop crew notably John Baranowski and Chris Castles for construction of the variousexperimental cells, Dr. 3. Yeung for supplying the dental rubber; Horace Lam for advice onphotography and darkroom film processing; various students and friends for their encouragementand help, E. Becher, B. Richardson, M. Choi, C. Brereton, M. Labecki, S. Sharareh, P. Yue, 3.Simrose, D. Baird, D. Marr, W. Yee; my parents; and God for helping me to persevere.This thesis is dedicated to the memory of my nephew, Winston Lee.1Chapter 1INTRODUCTIONSolids-stabilized emulsions are encountered in the mining of tar sands and the recovery ofpetroleum. Their presence on the most part is troublesome since the separation and removal ofthese emulsions is essential for the purification of the petroleum product and for the prevention offouling of process equipment in the refining process (1, 2). The properties of these three-phaseemulsions, known as Pickering emulsions, can however be desirable, as in pharmaceutical and foodmanufcturing for the stabilization of creams, lotions, or edibles like mayonnaise (3). It is notsurprising therefore that the inordinate resilience of these emulsions formed between twoimmiscible fluids, and finely divided solids has long been a subject of much study over the years.One of the earliest studies of the factors affecting the formation and stability of solids-stabilized emulsions was conducted at the turn of the century by S.U. Pickering, to whom theseemulsions owe their name (4). He studied the effectiveness of a number of different metal sulfatesas emulsion-stabilizing agents. Since then, investigations have covered the interfacial rheology, theinterfacial structure, the kinetics of coalescence and aggregation, the formation mechanism, solid-liquid interactions, the free energy and force analyses of the stabilized interface at equilibrium, andalso the methods of destabilizing these emulsions. Several comprehensive reviews have beenwritten on this topic (5, 6, 7). Tn summary, the literature shows that the main factors which areinvolved in the stabilization of oillwater emulsions by finely divided particles are (7, 8, 9):i) the particles must be much smaller than the droplet size;ii) the contact angle must be close to 900, i.e., the solids must be partially wetted by bothliquids;iii) the particles must be in a state of incipient flocculation ; andiv) the increased roughness of the particles produces a higher stabilizing efficiency.2The physico-chemical mechanisms by which the particles stabilize emulsions were dealtwith by Tadros and Vincent (10) in a simplified model, and Levine et al. (11) in a more refmedtheoretical investigation. Both papers deal with the energy of adsorption of solid particles at aliquid-liquid interface. Levine et al. (11) hypothesized by thermodynamic arguments that theparticles partitioned on the interface of these emulsion droplets are caught in a deep free energywell (approximately 106 times the typical Brownian thermal energy) and thus they remain in thatstate unless sufficient energy is applied to the system for them to desorb. Another factor for thestabilizing effect of the adsorbed particles was pointed out by Denkov et al. (8) who reported thatthe capillary pressure required for the liquid between the particles to squeeze through and comeinto contact with another such droplet would be a barrier to the thinning of the liquid film betweenthese droplets. For example, for a densely packed monolayer of particles of 1 tm radius adsorbedwith a contact angle of 60° (measured through the film), on an interface having a tension y of 30mN/rn, their model (see Figure 1.1) predicts that the minimum capillary pressure required for filmrupture to occur would be very high, approximately Pa.p2Figure 1.1 Denkov et al.’s one-layer particle model for coalescence (8)In this thesis, the objective is to investigate several factors which determine the stability ofsolids-stabilized emulsions in light of the paper written by Denkov et al. (8). Their model for3coalescence involved the thinning of a liquid film between two similar emulsion droplets with onelayer of spherical particles wetted above and below by the two droplets (Figure 1.1). Theyproceeded to prove that particle hysteresis should produce a further stabilizing effect since it wouldcause the required capillary pressure at the threshold of film rupture to occur at higher values.They also determined that at larger interparticle distances the threshold capillary pressure for filmrupture was much lower. Furthermore, they showed that adsorption energy considerations alonepredict a maximum emulsion stability at an equilibrium contact angle of 90° and a minimumstability at 0e = 00 or 1800 when the particle is fully wetted by one phase or the other. However,in terms of liquid capillary menisci, maximum instability occurs when 0e 90°, while 0e = 1800(assuming the contact angle is measured through the dispersed phase) would yield the most stableemulsions. These two opposing effects would lead to the conclusion that the most stable filmsoccur when the equilibrium contact angle is neither too close to 90° or 180°. The capillarypressure effect was found to be highest when the hysteresis is larger and when the particle size issmaller which is in agreement with experimental results.To test the idea that a principal factor for coalescence is the contact of particle-freesurfaces extruded by the capillary pressure from two emulsion droplets, we investigate thenecessary conditions for this to occur, in the case of a film formed by a two-monolayer particlemodel (Figure 1.2).Figure 1.2 Two-layer particle model of present workSince both colliding droplets each have an adsorbed layer of particles, the geometrydepicted in Figure 1.2 is a more realistic representation than the single monolayer configuration4shown in Figure 1.1. The developed pressures for the two-particle layer geometry considered hereare expected to be larger than those of the one-layer model of Denkov et al. The effects of varyingthe wettability of the solids, the interparticle distance, the particle size, and the extent of contactangle hysteresis will be investigated.Two general models are developed, one which is based on a uniform adsorbed layer ofsolid spherical particles at the interface of a spherical droplet dispersed in another fluid (similar tothe Denkov article), and then as an extension, we look at a two-dimensional analogue, in which weconsider the meniscus interactions due to our parameters on a cylindrical interface betweenparallel, horizontal stationary cylinders. Both models were solved by numerical techniques sincesimplifications generally found in other models (8, 11, 12), such as the neglect of gravity andconstancy in interface curvature, were not used here. The main impetus for tackling this morecomplex problem is that the solutions obtained can be applied to the more macroscopicexperimental systems needed to verif the results.In the accompanying experimental study, the capillary pressures were measured using amicromanometer while the equilibrium shapes of the menisci formed between two parallel cylinderswere photographed and later digitized for comparison with the model-predicted profiles. Thevariables investigated in this experimental study were the separation distance between the rods, thevariation in the contact angle obtained by changing the adsorbed surface film on the solids, and theeffect of hysteresis. The predictions obtained from the cylindrical interface model were thencompared with experimental results. The results of the two theoretical models are also discussedwith respect to the findings by Denkov et al. (8). Then, in light of the larger picture of the stabilityof emulsions, our observations about capillary effects will be discussed.5Chapter 2BACKGROUND THEORY2.1 General Concepts ofSolid-Liquid Interfacial ThermodynamicsCapillary phenomena are responsible for such fundamental occurrences as tears on a wineglass, the wetting and spreading of liquids on textiles, dewdrops on grass blades, the wailcing ofinsects on water, and of course, the rise of liquids in capillary tubes. These capillary effects occuras a consequence of intermolecular forces of attraction which are unbalanced at the interfacebetween two immiscible fluids (Figure 2.1).Figure 2.1 Intermolecular forces at an interfacea) Interfacial Tension and Free EnergyA measure of this excess force for any pair of immiscible fluids is known as the interfacialtension, y or as surface tension if one of the phases is air. Surface tension is often equated withTTphase 26the surface stress or the surface free energy which is true only in certain circumstances. It ishowever, more precisely defined not as a “tension”, but as the reversible work required to create aunit area of new surface. For solid-liquid systems, the surface tension is equal to the surface stressand surface free energy only in the case of a solid in equilibrium with a pure liquid and its vapor.Otherwise, surface tension is related to the specific surface free energy, w, by the equation:w=y+>p1 [2.1]where p, is the chemical potential and f is the surface excess per unit area of component i.The surface free energy is responsible for determining the shapes of interfaces. For liquiddrops in air or drops on solids, they will take the shape which minimizes the free energy of thesystem. Gibbs (1878), whose thennodynamic model of surfaces (13) is the fundamental treatisewhich underpins the field of wettability, demonstrated that the minimization of free energy for asolid-fluid-fluid system (e.g. a solid/oil/water system) corresponds to the minimization of thefollowing terms:y0A +y0A + YH’S A3 [2.2]where A is an area, and the subscripts ow, os, and ws denote the oil-water, oil-solid, and water-solid interfaces, respectively.b) Contact AngleThis minimization leads to the following relationship, known as Young’s equation (1805),involving the three surface tensions and the three-phase contact angle 0 (as illustrated in Figure2.2):7row cos(6) = — [2.3]wsFigure 2.2 Illustration of the three interfacial tensions at the solidloillwater interfaceThis equation is applicable for surfaces which are ideally smooth, homogeneous, planar, andnondeformable and when all phases are in mutual equilibrium. Young’s equation, though generallyaccepted and deceivingly simple, has never been experimentally verified due to the difficulty ofmeasuring the solid-fluid interfacial tensions (14, 15).The equilibrium contact angle that the intersection of the two fluid phases makes with thesolid is a useful measure of the wettability of the solid. Gibbs’ analysis defines the existence of onestable thermodynamic contact angle. However, in practical systems, a range of different stableangles is often measured. The literature is replete with examples of uncertain, conflicting andconfusing results (16, 17, 18, 19). Thus, disagreement and doubt about Gibbs’ analysis has arisen.However, in Gibbs’ classical treatment, there are underlying hypotheses which are satisfied undervery stringent circumstances, usually requiring purity of liquids, cleanliness of all materials,smoothness, homogeneity, and nondeformability of solid surfaces.The variation in the contact angle has been termed “hysteresis”. The classical model ofGibbs’ was later extended by Johnson and Dettre (19), who include hysteresis by modeling asurface with concentric sinusoidal grooves. Other qualitative treatments have been given byBikerman (20) and Neumann and Good (21).oilwater8c) HysteresisThere are two types of hysteresis:i) true or thermodynamic hysteresis, andii) kinetic or time-variant hysteresis.The interpretation of thennodynamic hysteresis is generally based on the idea ofmetastable states caused by a number of sources such as contamination, roughness, surfaceheterogeneity, and “surface immobility on a macromolecular scale” (14, 22). Kinetic hysteresis isusually associated with slow equilibrium times. Furthermore, it should be noted that there is someevidence that the contact angle may also be dependent on the speed with which the contact line isadvanced or receded (14, 15).d) Minimization of Free Energy in EmulsionsNow how do these concepts relate to solids-stabilized emulsions? In fact, all thesethermodynamic properties in some way affect how the solids remain on the interface and mayexplain the mechanism of stabilization. With the concept of surface free energy in mind, it is clearthat in the absence of gravity, a free liquid drop will try to take on a spherical shape to minimize itsfree energy and thus its free surface area. Similarly, a dispersed droplet within another continuousphase will create a spherical interface, but a swarm of such droplets will be thermodynamicallyunstable because of its high free surface area and high free energy compared to that of coalesceddroplets. The ultimate stability of such a dispersion depends on the presence of stabilizing agentssuch as surfactants or particles adsorbed on the interfaces. When a third phase such as a solid issituated at the fluid interface, the thennodynamic equilibrium involves many new factors.92.2 Equilibrium ofParticles on Horizontal InterfacesUnder gravity, a solid particle may move through one fluid and enter the fluid/fluidinterface or pass through it into the second fluid depending on the densities of the particle and thefluids, the size of the particle, the interfacial tension, and the contact angle at the three-phaseboundary, 0(23, 24). If the particle density is intermediate between the two fluid densities, thenthe particle will always find an equilibrium position at the interface no matter what the interfacialtension, contact angle, or the size and shape of the particle (23). However, if the particle density isgreater or less than both fluid densities, its final position will be determined by other factors. In thefirst case (i.e., particle with greater density), for a given interfacial tension, contact angle and set ofdensities, there exists a critical particle size above which the interface cannot support the particleand it therefore passes through the interface and continues to travel through the lower fluid.Similarly, for the second case, there is a critical size below which the particle can assume anequilibrium position at the interface but, in this case, the motion of the particle is in the oppositedirection.In terms of the surface energetics, those solid particles which meet the size criterion willremain at equilibrium on the planar interface only if they are partially wetted by both liquids. Inother words, the following condition, based on Young’s equation, must also be met:rlsr2s< [2.4]where subscripts 1, 2 and s represent the upper fluid, the lower fluid, and the solid, respectively.Otherwise the particles will remain in the vicinity of the interface but entirely in one of the fluids.Di Meglio and Raphael (24) have shown on the basis of total free energy considerationsthat condition [2.4] also applies to curved interfaces as in the case of a particle adsorbed on adroplet (subscript 2) in a continuous phase (subscript 1) (Figure 2.3).101Figure 2.3 Emulsion droplet with spherical particle on interfaceOn a horizontal interface, the presence of any solid particles will cause distortions to theinterface which can either be convex or concave relative to fluid 1. However, in the case ofspherical droplets in a continuous phase, the meniscus will always be convex relative to fluid 1,because the pressure inside the dispersed phase must be greater than in the continuous phase forthe droplet to exist (7).2.3 Emulsions Stabilized by Finely Divided Solidslii order to be effective for emulsion stabilization, the adsorbed particles must beconsiderably smaller than the droplets, be partially wettable by both fluids, and be in a state ofincipient flocculation (6, 9). In a gravitational field, these small adsorbed particles in closeproximity to each other will experience a mutual attraction and will tend to aggregate together onthe interface (25). A cohesive film of particles thus forms on the surface of the droplets. However,if there is an insufficient quantity of particles to cover the droplet, it will not be fully protectedfrom coalescence with other droplets. A complete, close-packed layer of adsorbed particlesprovides steric stabilization and, if charged, also provides repulsive charge stabilization preventingcoalescence of the emulsion droplets. In the absence of gravity, this capillary interaction does notexist for a perfectly planar interface (26), but becomes important again for highly curved interfacesbecause of the significant pressure difference which exists between the droplet and the continuousmedium (11). For very fine particles, there are also other, relatively short-range forces such as van11der Waals attraction as well as double layer and Born repulsion, which may affect the stabilizingability of adsorbed particle films (11).The type of emulsion formed depends very much on the solid/liquid/liquid contact angle.For solid/oil/water systems, it is widely accepted that, if the contact angle is less than 900(measured through the water phase), then oil-in-water (01W) emulsions are formed. For contactangles above 90°, water-in-oil (W/0) emulsions are formed (9, 27).The mechanism by which these Pickering emulsions are stabilized is still not well known,and continues to be a subject under much study (28, 29, 30). Levine et al. (7, 9, 11) analyzed thethermodynamics of particles partitioned between the 01W interface of a droplet and the continuousphase. They showed that an isolated spherical particle with an appropriate contact angle is trappedin a deep energy well at the interface of the emulsion droplet. Pieranski (31) came to a similarconclusion for polystyrene beads at a water/air interface. As a result of these large adsorptionenergies, in order to maintain an equilibrium partition between particles on the interface and thoseremaining in suspension, these adsorbed particles must experience a counterbalancing repulsiveforce, which Levine et a!. believe to most likely be the short-range Born repulsion forces whichbecome important at close-packing conditions (11). The electric double layer, short-rangesolvation, and solid elastic forces, as well as the van der Waals and capillary forces between theparticles were either found to be of the wrong sign or of insufficient magnitude to account for it(32). It was also determined that the most stable position (minimum free energy) occurred if theparticles had a contact angle close to 90°.Although this large adsorption energy is important in the stabilization of individualemulsion droplets, it does not, on its own, explain how the coalescence of such droplets can beinduced by applied forces such as electrostatic, acoustic and shear forces. Denkov et a!. (8)hypothesize that these forces must be sufficient to overcome the adsorption energy such that theparticles are completely immersed in the non-wetting (disperse) phase. In this manner, liquid fromthe droplet is squeezed between the particles in the film thereby creating a free surface which couldcome into contact with a corresponding free surface on a second droplet. The strength of the12applied force required to achieve coalescence by this mechanism is related to the capillary pressureengendered in the droplet due to the required distortion of the free (i.e., liquid-liquid) interface. Themagnitude of this pressure can be determined using the Young-LaPlace equation which relates thecapillary pressure to the equilibrium shapes of the menisci formed.2.4 The Young-LaPlace Equation and the Equilibrium Shape ofMenisciAs a consequence of surface tensions, when two phases are separated by a curved interfacethere is a pressure difference (zIP) across it which balances these forces. The higher pressureexists on the concave side. The relationship between the two principle radii of curvature of thesurface (Rj and R2), the interfacial tension (fl, and the pressure difference can be derived fromthermodynamic concepts (14, 33).Figure 2.4 shows a segment of an arbitrarily curved surface. The surface is bound bytwo planes which are perpendicular to each other and normal to the surface. On each plane, thesurface describes an arc whose length will be denoted by x andy on the respective planes. The tworadii of curvature which describe the surface are designated Rj and R2 and lie on either plane. IfFigure 2.4 Radii of curvature of a curved surface13the surface is now displaced a small distance outward, the arclengths on each plane would beincreased to x+dx and y+dy, respectively. This increase in surface area isdA =(x +cLv)(y +)—xy [2.5]which is approximatelydA=xdy+ydx. [2.6]The free energy involved in the increase of area isdG =ydA =y(xdy -i-ydx) [2.7]and the expansion of volume due to the pressure work across the interface acting on the area xythrough a distance of dz, can be writtendW=APxydz. [2.8]Equating the free energy and work equations yields:y(xdy +yd) =APxydz. [2.9]In terms of the principle radii of curvature, Rj and R2, we know that from similar triangles theequality of the following ratios holds:x+cfr [2.10]R1+dz R14which yields[2.11]Furthermore,y+dy [2.12]R2+dzRwhich yields[2.13]Using these equations we can eliminate the differential elements in equation [2.9] which results inthe following final form of the relationship between the pressure drop, the radii of curvature, andthe interfacial tension:[2.14]This equation is the fundamental equation of capillarity known as the Young-LaPlace equation(1805). Its use has been invaluable in describing the equilibrium shapes of interfaces ranging fromsessile and pendant drops to open network soap films. The equation can be used for any curvedinterface in a gravitational, centrifugal, or other force field.15Generally, the Young-LaPlace equation yields a second-order differential equation whichcan only be solved analytically in very few cases. Usually, this occurs if the radii of curvature areequal, as in the case of spherical menisci, or if one radius is infinite as in the case of cylindricalinterfaces. Analytical solutions are also possible for general radii of curvature if gravitationaleffects are either nonexistent or negligible, as in the case of very small sessile drops in a liquid ofdifferent density or menisci of microscopic dimensions (23) . For a planar surface, both radii ofcurvature are infinite. Therefore, there is no pressure drop across the interface, i.e. zlP=O. Butplanar interfaces are not the only ones which have zero pressure drop. For example, curved soapfilms over an open framework have the shape of a catenoid for which1 1 [2.15]Substituting equation [2.15] into [2.14] again produces the result that J.P=O. In this case, theopposite signs mean that the radii of curvature lie on either side of the interface thus producing asaddle-shaped interface.With emulsion interfaces adsorbed by finely divided solids, it is often reasonable to assumeconstant curvatures and thus constant zIP. For macroscopic systems, the radii of curvature areoften not constant but vary spatially (33). For surfaces with an axis of symmetry in Cartesiancoordinates, analytical geometry can be used to derive the following relationships for Rj and R2 asfunctions of (xz):d2z[2163/2 116anddz_1=:j: [217]F (dz1”2xI 1+1Likewise, the pressure difference will vary along the meniscus since it is a function of capillaryheight, i.e.,zSP(z)= 1 + 1 [2.18]LR1(x,z) R2(x,z)]The convention of signs adopted here for curvature will be as follows. If the centre of curvature ofRj lies in the upper phase, phase 1, then R1 is positive. Similarly, for R2. If both Rj and R2 arepositive, then Pj is larger than P2. If the interface is saddle-shaped as it is around a sphere, thenR1 and R2 must be opposite in signs, and the pressure difference can be either positive or negative.In terms of emulsion stability, we will consider two different two-dimensional modelswhich will characterize the solids-stabilized interface of a dispersed droplet in a continuous phase.The first model involves an aqueous emulsion droplet covered with fine spherical particlesdispersed in a continuous oil phase. The capillarity will be investigated around a single spherebounded on all sides by other spheres whose saddle-shaped meniscus has circular symmetry aboutthe vertical axis. The second model is a simplification for experimental purposes, and involvesconsidering a cylindrical interface between two horizontal parallel cylinders of infinite length.17Chapter 3MATHEMATICAL ANALYSIS OF THE AXISYMMETRIC INTERFACE BETWEEN ASPHERE AND ITS NEIGHBORING SPHERESExcept in a few cases, the meniscus shapes for which there are explicit numerical oranalytical descriptions in the literature are usually two-dimensional (e.g., cylindrical interfaces) oraxisymmetric (e.g., sessile drops). Furthermore, only when gravity and other body forces areneglected, are there convenient analytical solutions since, under these circumstances, the surfaceshave uniform mean curvature (e.g., cylindrical, spherical, nodoid, catenoid, and unduloid memsci)(26). In the stabilization of emulsion droplets by fine particles, since the size of the particles is inthe microscopic regime, it is often accepted that gravity has a negligible effect on the system and isignored (8, 11).In contrast, we develop two general mathematical models for macroscopic systems inwhich gravity plays an important role, as well as for microscopic systems where the gravitationaleffects are comparable or larger than the capillary effects. The reason for doing so is to extend themodels into the regime where comparisons of their predictions with our macroscopic laboratoryexperiments become possible.The first model is an idealization of a three-dimensional case where spherical particles areadsorbed in a well-ordered array on a spherical oil-water interface. The second model is asimplified two-dimensional problem in which parallel cylinders of infinite length lie horizontally ona fluid interface. The latter case was selected because it is more amenable to laboratoryinvestigation.In both models, we assume that the diameter of the emulsion droplet is significantly largerthan the diameter of the particles sitting on its boundary such that, if we consider a small section ofthe droplet interface, we would see the particles adsorbed at an essentially planar surface (7, 11)with curvature of the interface arising only from the capillary interactions between the solids18(Figure 3.1).ae>> aFigure 3.1 Emulsion droplet with adsorbed layer of particlesThe requirement that a >> a is one of the conditions which is necessary for solids tostabilize emulsions as discussed earlier. The other conditions are that the solids must be partiallywettable by both fluids which means the contact angle can be neither 00 or 1800 (6, 7) and that theparticles adsorbed at the interface must form a close-packed monolayer (9).3.1 Literature Review ofRelated Sphere ModelsThe profile produced around an isolated sphere, a vertical rod, fluid drops or lenses, oroutside a Du Nouy ring is axisymmetric and, unlike the menisci for sessile drops, does not intersectits axis of symmetry. It is considered a bounded menisci having the contact line with the solid atone end and the free level interface at the other. Several different numerical approaches for thesolution of such axisymmetric menisci profiles were treated by Padday and Pitt (34), Huh andScriven (35), and Princen et at. (36, 37). Padday and Pitt developed equations for both boundedmenisci (e.g., sessile drops, vertical rod in a free surface) and unbounded menisci (e.g., liquidbridges between spheres) following the Bashforth and Adams’ approach for sessile and pendantdrops except with a redefinition of the meniscus shape factor. Hartland and Hartley (38) compiled19numerical solutions and tabulated the results for profiles of the rod in a free surface (completelywet) case using an approach similar to that of the Bashforth and Adams’ tables. However, the caseof the rod in a free surface was limited by assuming the rod to be either completely wetted ornonwetted by the lower fluid. In the literature, treatments of meniscus profiles in vertical tubes,cylinders, plates, or rings in contact with an unlimited fluid interface, often assume completewetting (i.e. 0 = 00) (39, 40). This, however, avoids the effect played by contact angle, especiallyas it relates to the equilibrium position of a solid such as a sphere on the interface. Huh andScriven formulated solutions which were general for axisymmetric interfaces of “unboundedextent” (their definition being the opposite to that of Padday and Pitt which we adopt here) whichextended far from a circular contact line to a free flat interfuce. Their analysis can be applied to afloating sphere as well as spheres submerged to different extents, and includes variations in contactangle. Princen et al. determined profiles for inuniscible liquid lenses situated at the interfacebetween a second liquid and air, but only for a limited range of conditions. Unlike the abovemodels, we consider the presence of neighboring solid spheres such that the interface becomes flatat a finite symmetry boundary, and determine the full gamut of possible menisci as one raises orlowers the liquid volume in the space between the spheres.3.2 The Sphere Model ConfigurationThe emulsions encountered in petroleum field operations are often stabilized by finemineral and clay particles of various shapes and sizes with adsorbed asphaltenes (5). As in mostinstances where real systems are characterized in mathematical terms, one often finds the need toincorporate simplifications and assumptions into the model in order to make its solutionmanageable. Thus, for the simulation of this three-phase system, we employ idealized solids whichare uniformly spherical and which have identical, homogeneous wetting characteristics.We consider a single sphere amidst a hexagonal array of identical spheres adsorbed on a20macroscopically planar interface between an upper fluid phase 1, and a lower fluid phase 2. Onefurther simplification made in the model is that there is cylindrical symmetry around the spheresuch that the meniscus is axisymmetric about the sphere’s center. The meniscus profile is thenessentially equivalent to that found for a sphere of radius a bounded by a coaxial circular cross-section ring having an inner radius a0 (Figure 3.2). Such a cylindrical “cell model” has been usedpreviously by Levine et al. (11) to determine the capillary interaction between neighbouringspherical particles adsorbed at an oil-water interface.Figure 3.2 Cylindrical cell model representation of a sphere surrounded by a uniformly-spaced ring ofneighbouring spheresEach spherical particle is situated in the middle of a cell having a symmetry radius b, i.e.,the point, midway between the centre sphere and particles in the ring, where the meniscus exhibitsa maximum or minimum. However, the dimensionless separation distance b/a or B can be nosmaller than t/3, which is the case for hexagonal close-packing of spheres.For the case of a sphere semi-submerged in the interface, the three-phase contact Line ishorizontal and circular when viewed from above. The fluid interface has an axis of revolutionaround the centre of the sphere which coincides with the vertical z-axis. The interfacial profile isthen a function only of the radial coordinate r measured outward from the z-axis. The horizontal21z=O plane of our coordinate system is to coincide with the free level interface at which the pressuredifference ziP=0, presumably at r —* where the solids no longer influence the interface shape.Figure 3.3 illustrates the configuration of the system, a sphere sitting on the interface between twoimmiscible fluids whose densities are P1 and P2 upper and lower, respectively.p1zAhFigure 3.3 Coordinate system for sphere cell modelThe pressure difference across the interface arises from the curvature of the meniscusdescribed by the two radii of curvature Rj and R2. Figure 3.4 shows a surface of revolutionaround the z-axis described by a curve in the r-z plane. Rj is the radius of curvature of the point(rz) and R2 is the length of the normal to the curve at (r,z) from the z-axis. The angle betweenR2 and the symmetry axis is qi. Rj and R2 are perpendicular to each other.zplane)Figure 3.4 Radii of curvature of surface of revolution described by the curve22At the contact position (r,z) on the sphere, qS becomes q5,. fi is the supplementary angleof çb. The smaller of these two angles is also equal to the angle the memscus slope makes with thehorizontal axis, otherwise, referred to as the “meniscus slope angle”. The contact angIe 0 is theangle between the tangent to the solid and the tangent to the liquid interface at the contact line andis measured through the lower phase. And finally, a is the angle which indicates the displacementof the sphere relative to the liquid interface. It is measured from the negative z-axis to the line ofcontact inscribed by the interface on the sphere.Depending upon the nature of the fluids and their contact angle with the solid, a capillaryrise or a capillary depression may result between the particles. For example, if a glass capillarywere placed in a tray of water, the water would rise in the capillary, yielding a concave meniscus(Figure 3.5); whereas in a tray of mercury, the liquid level inside the capillary would fall giving thecharacteristic convex menisci found in many thermometers and barometers (Figure 3.6).Figure 3.5 Capillary Rise Figure 3.6 Capillary DepressionIn the former case, the water wets the glass yielding a very low contact angle while, in thelatter case, mercury exhibits a very high, non-wetting contact angle with glass. Theminimum/maximum (concave/convex) capillary height relative to the zero datum level (i.e., the z=Oaxis) at the centre of the tube is denoted by z0, the latter being negative in the case of convexmenisci. Although, the pressure difference across the interface or capillary pressure varies alongthe meniscus radially (due to nonconstant curvature), we will concern ourselves only with thecapillary pressure at z0 as a point of comparison for different cases.233.3 Mathematical Formulation ofthe Differential Equations ofthe Meniscus ShapeTo describe the meniscus in terms of our chosen coordinates, the Young-LaPlace equationcan be transformed as follows. At equilibrium, the memscus capillary pressure must equal thehydrostatic head of the meniscus (23), i.e.,zP (p2 —p1)gz= [3.2]where c is the capillary constant which is given by(p2-1)gc= [3.3]rand which has units of inverse length squared.Unlike the Bashforth and Adams (41) form for sessile and pendant drops, equation [3.2] ishomogeneous with respect to z because it is measured from the level surface rather than a point onthe meniscus. In their analysis, Bashforth and Adams chose the apex of the sessile/pendant drop asthe origin because the radii of curvature are equal here. They then used the curvature value b’ tonon-dimensionalize the meniscus equation and obtained the well-known shape factor:[3.4]7and the following dimensionless form of the axisymmetric, Cartesian Young-LaPlace equation:241 + sinçb=—+2. [3.5]R1b’ x/b’ b’Since there is no corresponding b’ for unbounded axisymmetric menisci, one often findsthat the Young-LaPlace equation is non-climensionalized using the radius of a solid, or some othersignificant length term such as the capillary constant which has units of inverse length squared.For our purposes, we have chosen to use the radius of the sphere as a reference value.We introduce the dimensionless group, the Bond number Bo:[3.8]rIt represents the ratio of effects due to gravity, which tend to distort the meniscus, to the capillaryforces, which act to minimize the surface area and distortions. As can be seen, this number issimilar to the shape factor used by Bashforth and Adams (41) who tabulated extensive data forsessile and pendant drop shapes for 0.125 B’lOO and 00 < <1800.For the profile around a spherical solid, we derive the differential form of the YoungLaPlace equation from our knowledge of the radii of curvature expressed in our coordinate system,considering both the concave and convex forms.3.3.1 Concave MenisciFor the concave meniscus cross-section (Figure 3.7), by our conventions, the principleradius of curvature Rj is positive while R2 is negative.25Figure 3.7 Concave meniscus profileThe curvature K of a point on a curve is the rate of turning of the tangent line with respect toarclength (42). Thus, the curvature K is defined mathematically as:K=— [3.9]dswhere s is the arclength of the curve and w is the angle between the tangent to the curve and thehorizontal plane. In the limit of vanishingly small arclength,a’V .A’P . z’P .___1urn = lirn = hm = —. [3.101ds ‘-° Ss ‘-‘° R1 sin(’1’) ‘-‘°R1(A’P) R1In terms of radial coordinates,1 d’Vdr dP,—=—————=—————cosT) . [3.11]R drds drSince ‘P= r—qS,[3.12]R dr dr26Sinced Siflis negative, then Rj is positive. Furthermore,drr [3.13]sin (qS)is negative. The Young-LaPlace equation therefore becomesdsinq5sinçS [3.14]dr rwhere qS decreases from q5 at contact to 0 at r=b.3.3.2 Convex MenisciFigure 3.8 Convex memscus profileSimilarly, for the convex meniscus (Figure 3.8), Rj must be negative and= (cos’P)= dsin ‘P = dsin [3.15]R dr dr drwhile R2 is positive and27—. [3.16]sinThus, the corresponding differential Young-LaPlace equation is:clsinçS sinç5+—=cz [3.17]dr rwhich is applicable for 00 < < 1800 as before, but in this case qS increases from q5 to180° at r=b.In terms of Cartesian coordinates, the Young-LaPlace equation takes the form (fromequations [2.16] and [2.17]):d2z dzdr2 + dr=cz [3.18]F (dzl2T5 F (dz21°511+1—I I rIl+I—[ ‘dr)] L drwhere the upper sign corresponds to the convex case, and the lower to the concave.Chan et a!. (43) solved a form of equation [3.18] obtained by neglecting the first derivativein the denominators of both quotients. In this case, the Young-LaPlace equation reduces toBessel’s equation:d2’ d’ [3.19]dr2 rdr28which can be solved using modified Bessel functions.3.4 System ofEquations in Alternate CoordinatesThe forms of the Young-LaPlace equations derived thus far, i.e., equations [3.14], [3.17]and the Cartesian coordinates version [3.18] produce unbounded solutions in certain regions of thedomain of interest. For example, at çit/2, the solutions are discontinuous and undefined. Analternative approach is to make ç the independent variable which leads to the following first-orderparametric equations for the convex case:dX XcosqS [3.20]dqS XZ—sin.6anddZ Xsinq [321]db XZ—sinq5where X = and Z = (upper-case variables will denote the dimensionless forms). However,this approach merely shifts the discontinuity to b=0. Huh and Scriven (35), who similarly derivedprofiles for axisymmetric interfaces around isolated objects, used this approach but sidestepped theproblem by employing Nicolson’s (44) approximation near contact angles of 90°.Yet another method which circumvents the blowup of the solution is to make thecontinuous, single-valued arclength function s the independent variable. This transformationproduces the following numerically stable form of the Young-LaPlace equation with çS, z, and r asdependent variables (45):29dg sinØ [3.22]ds rdr —-=+cosq5 [3.23]and=±sinç [3.24]where the upper sign corresponds to the convex menisci, and the lower sign, to the concave. Thislast approach was taken by Princen and Mason (36, 37) who solved the meniscus profiles for fluidlenses on a two-fluid interface, by Princen and Aronson (46) for rotating menisci in vertical tubes,and by Rotenberg et al. (47) for sessile and pendant shapes.In dimensionless terms, the system of nonlinear equations [3 .22]-[3 .24] becomes:[3.25]dS X}4L=FcosqS [3.26]dSand4=±sinçb [3.27]where S=s/a.30The radial location of the meniscus at the point where it contacts the sphere, X, can berelated to the immersion angle a by:X = sin(a) . [3.28]The arclength is measured from this reference point, i.e., S=O at X=sin(a). At the outersymmetry boundary of the cell model, the final arclength is not known a priori, but weoverestimate by equating it to twice the radius of the cell. How this endpoint is dealt with will bediscussed later in the numerical procedures.3.5 Boundary ConditionsTo determine the shape of the axisymmetric meniscus, the Young-LaPlace equation mustbe solved from the three-phase contact line on the sphere to the cell model symmetry boundary atr=b. For finite contact slopes, the boundary condition isdZ dZ—=±tanq5 or --=±sin(q) [3.29]where the upper sign is for the convex case, and the lower one, for the concave. The contactmeniscus angle can be determined from its geometric relationship with the immersion angle andthe contact angle at the three-phase contact line. For the concave meniscus,= r— 9— a [3.30]whereas for the convex meniscus,31[3.31]or likewise, in terms of the meniscus slope angle fi,/J=a—2r+8. [3.32]At the other end, the meniscus lies flat, with O at r=b where the height is Z0. Therefore,dZ dZ—=0 or —=0. [3.33]dX dSUp to this point, we have only considered that the solids are uniform, ideal spheres whosewettability is characterized by one equilibrium contact angle specific to the solid-fluid-fluid system.However, in practical situations, one often finds that the contact angle the solid makes with thefluids may vary over several degrees. This phenomenon, known as contact angle hysteresis, mustalso be included in the analysis.3.6 Effect of Contact Angle HysteresisThe most reproducible contact angles measured for a given system are its advancing andreceding angles, O and °r’ respectively (14). These angles are so named because in the sessiledrop method of measuring contact angles, the advancing angle, the maximum bound for the contactangle, is measured by increasing the volume of a drop until its contact line with a planar solid justbegins to move; whereas the receding angle, which is the smallest possible contact angle, ismeasured by causing the drop periphery to retreat over the solid. Figure 3.9 shows two liquiddrops on a solid showing very different apparent contact angles, but in fact, have the same actual32contact angle as seen in the magnified views (Figure 3.9). The solid is an idealized surface havingconcentric sine-wave corrugations. This model is often used to explain the discrepancy betweentheory and observation as it shows hysteresis can be caused by microscopic roughness. It can alsoexplain why the contact line does not appear to move as the drop is advanced while achieving themacroscopic advancing angle since, from a microscopic point of view, the liquid is moving downinto the valleys between the ridges whilst maintaining a constant contact angle. Other sources ofhysteresis are surface heterogeneity, liquid impurities, or noncleanliness of surfaces.solidBFigure 3.9 The surface of the idealized rough solid has a sine-wave corrugation, as seen in the twomagnified views, and has rotational symmetry about the z axis. Two static configurations are indicated byA and B with very different apparent contact angles, but in the microscopic view, the actual contact angleis the same.The advancing and receding angles can also be determined from other contact angle measurementmethods (14,19). The difference between the two angles is referred to as the contact anglehysteresis, Oj, i.e.,6h = 8a — [3.34]Hysteresis can be quite large depending on the solid-fluid-fluid system. For example, water onmineral surfaces can have an advancing angle as much as 500 larger than the receding one (14).liquid33For our experimental system, contact angle hysteresis will be measured by starting off witha level interface between a pair of interfacial particles and then increasing the meniscus pressureuntil the contact line on the solid just begins to move. At the level interface obtained bywithdrawing the lower fluid from the system, the contact angle is at its smallest value, 0,.. Thelocation of the three-phase solid-fluid-fluid boundary relative to the sphere is expressed in terms ofthe positional angle a, which is the complement to °r’ as follows:a=lr—Or. [3.35]We will consider the advancement of the meniscus position due to increased fluid volumeto occur in two stages (8). The first stage is characterized by the angle a remaining stationarywhile the rest of the meniscus shape bulges upward as the contact angle increases. At some point,the restraining forces at the contact line will be overcome and the periphery of the meniscus willbegin to move. This second stage movement of the meniscus upward along the solid ischaracterized by the contact angle remaining constant at its final value 61aTo include the effect of hysteresis in our previous analysis, which describes only thesecond stage when the contact line is moving around the sphere, we will include an initial stage inwhich the contact angle is increased from the receding angle to its final value while the contact lineremains stationary. In other words, the contact angle hysteresis does not affect the shape of themeniscus directly; however, it does affect the position of the meniscus on the solid. Thus,hysteresis will influence the value of the capillary pressure required to rupture the film between twoemulsion droplets.343.7 Determination ofFilm ThicknessThe coalescence of two oil-in-water or water-in-oil droplets, stabilized by micron-sizedparticles, is assumed to occur when the film of continuous liquid between the two droplets thinsand the meniscus protruding between any pair of adjacent particles on one droplet touches thecorresponding meniscus on the second droplet. It is assumed for convenience that opposingparticles on the two approaching droplet surfaces are aligned. Thus, rupture occurs when theprotruding menisci from the two droplets meet the line joining the points of contact betweenopposing particles (Figure 3.10). At any equilibrium state of the system prior to rupture, h is takento be the distance from this line to the closest distance to each meniscus, i.e., the meniscus height atthe cell model boundary, r=b. Hence, rupture takes place when h=0.Figure 3.10 Point of rupture between two emulsion dropletsThis process neglects the effects of “dimpling” (48, 49, 50) or the deformation of the approachingmenisci as the continuous phase film is squeezed out of the space between them.Since the coordinates of the meniscus are known in terms of the distance from the free flatinterface, but the line at which rupture takes place is unknown relative to the datum level, the filmthickness can be determined by using the contact line as the reference height. Thus, the onset offilm rupture can be followed by comparing the existing position of the meniscus relative to itscontact line to that where contact is made with the corresponding meniscus above. Thedimensionless distance from the point of rupture H is determined by comparing the actual distanceDroplet 1h Droplet 235of the meniscus from the contact line Hactto the required distance Hr (Figure 3.10):H=!i=Hact_Hr. [3.36]Both Hact and Hr are measured from the vertical coordinate of the contact line. Hact is thedifference between the meniscus height Z0, and the contact line location Z of the actual meniscus:= Z0 —Z [3.37]where the vertical coordinates Z0 and Z are determined from the simultaneous solution of theequations [3.25-3.27].The required distance for rupture, for a level meniscus having the same contact line, isdetermined from simple geometry:Hr 1+cos(a) [3.38]The difference between Hact and Hr is to be minimized to determine the meniscus shapeand capillary pressure at the threshold for film rupture.3.8 Numerical SolutionGiven the size of the solid spheres, their distance apart, the fluid densities and theirinterfacial tension, the contact angle, and the angle a which defines the contact line on the sphere,we can determine the resulting meniscus shape. The unknown parameters in the set of governingequations are the vertical coordinates of the meniscus at the contact line and at the cell model36boundary relative to the absolute datum line (taken to be the position of the free interface far fromthe influence of the particles).This is a boundary value problem which we chose to solve by the shooting method appliedto the simultaneous solution of three ordinary differential equations. We begin at the knowncontact line on the sphere and work outward to the symmetry condition at the cell model boundary.Our variables at contact, X is obtained from equation [3.28], and qS, from [3.30] or [3.31]. Weset Z, the vertical coordinate at the contact end, initially to be zero (as for a level meniscus), thenproceed to solve the equations at incremental arclength positions until the radial co-ordinatereaches the outer boundary, at a distance b/a from the centre of the sphere. In this fashion, theboundary value problem is transformed into a simpler initial value problem. If the meniscus slopehappens to be zero at the outer boundary, we have achieved our desired goal. Using the initialestimate of Z, this condition will generally not be met. The determination of the true value of Zis accomplished iteratively by means of a numerical routine employing Muller’s method (DRZFUNin the UBC NLE nonlinear equation solving subroutines package (51)). Muller’s method is aniterative procedure capable of finding both complex and real roots of a single nonlinear equationefficiently and reliably. It has the advantage of not requiring a good initial estimate(s) of theroot(s) for finding single or multiple roots of a nonlinear equation.The desired condition to be met is that q must be zero at the outer boundary, i.e., themeniscus slope must be horizontal. Thus, the function to be minimized,/ is given byf = (z, X = B) —0 [3.39]1ff is positive, then qS, was too small (i.e., Z is too small), and a larger radial distance is necessarybefore the meniscus reaches its maximum (convex), or minimum (concave) height. Likewise, if f isnegative, the selected c5 chosen was too large (i.e., Z is too large) and the apex was encounteredbefore the cell symmetry boundary was reached. If first one condition and then the other areobtained for two consecutive guesses of Z, the true value lies between them.37Initially, the ordinary differential equations (ODEs) describing the meniscus profile weresolved using a Runge-Kutta or an Adams-Moulton predictor-corrector technique, but overflowconditions occurred in certain regions of the solution domain which were later attributed to thepartial stiffness of the system. Stiffness behaviour apparently arises in many physical situationssuch as in reaction kinetics (52, 53). It occurs when the mathematical solution to the first-ordersystem contains terms that decay at different rates and although certain terms may disappearrapidly with negligible effect on the long-term solution, they can cause numerical difficulties byrequiring excessively small stepsizes in the computation of the solution (54).In order to avoid inaccuracies, we require a solution routine which has the characteristicof absolute stability in the face of relatively large stepsizes, so that the solution progresses at anacceptable rate (52). The most widely used stiff equation solvers are based on multi-step implicitbackward differentiation formulas first implemented by C.W. Gear (54). The stiff ODE equationsolving routine, LSODAR, used here, is also based on GEAWs method, and was obtained fromNETLIB, a collection of software from Argonne National Laboratory, Argonne, Illinois (52).The LSODAR (Mar 30, 1987 version) routine solves problems of the form= f(S,Y) [3.40]where 7 is a vector of n dependent variables. The n equations of the form of equation [3.40] canbe either stiff or nonstiff or both. The routine automatically determines whether the system is stiffor not by repeatedly monitoring the data and then implementing the nonstiff (Adams) or stiff(Backward Difference) methods. The user has the option of supplying a full or banded Jacobian(preferable if the system is stiff) or to allow the internal computation of the Jacobian by numericalmeans.LSODAR has the added capability of finding the roots of any given function while thesolution of the n differential equations continues. This feature is especially useful for stopping the38solution at certain boundaries. For example, in tracking the path of a particle in an enclosed boxwe might wish to stop the routine exactly when the particle reaches a wall. Similarly, in our case,this feature is put to good use. Since the limit of integration with respect to the arclength of themeniscus is unknown a priori, we tenninate the simultaneous ODE solver by continuouslysearching for the root of the function g, defined byg=X—B [3.41]where B is the location of the outer boundary of the cell model. The stability of the resultantmeniscus is not considered here, but references (55, 56) outline a methodology for testing thisaspect of the equilibrium menisci. Once the meniscus shape is known, the capillary pressure canbe detennined. This pressure in terms of the capillary rise is‘capYo [3.42]or, in dimensionless form, obtained by dividing both sides of equation [3.41] through by 2y/a is“Z”2 B’•ZP — ° — 0 [343]cap 2 — 2The resultant capillary pressure at rupture is denoted byBoZP —P — 0 F344cap . LThe numerical solution of this problem is outlined in the program flowsheet shown inFigure 3.11. The rupture capillary pressure is plotted as a function of several variables. A similaranalysis for the cylindrical interface was perfonned.3940Figure 3.11 Program flowsheet for meniscus between spheresnstant XcDetermine Zc, Zo, H, PStore H, and P in arrays Hi (i,j), P1 (i,j)Determine minimum and maximum H,Calculate increment in +cConstant 0—-CaIculato=and XcDetermine Zc, Zo, H, PStore in arrays H2(i,j), P2(i,j)Determine minimum and maximum H, Ii=j±lf Increment initialO =0 + dO I[sTAj1. Radius, distance a B // DATA2. Physical data p,y,’O /Xc, and hysteresis DhCall HVSP (6..Call HVSP Xc,)Graph all curves of film thickness vs. pressure Call MYPLOT (Hi ,Pl ,H2,P2,M,N)Figure 3.11 (continued)Subroutine JACPD(1,2)=q*cos(Y(2))P0(2,2) =-cos(Y(2))IY(3)PD(2,1)=q*BoP0(2,3) = sin(Y(2))/(Y(3) *(PD(3,2)-sin(Y(2))41Subroutine HVSP ( Xc, a, H, Pcap)Determine Zo, ZcHc=abs(Zo-Zc)Hr=(1-cosl))H=Hr-HcPcap=BoZoI2eJFwwt1irFCN(Z)UBC DRZFUNsolves for rootsFCNInitialize Y(1)=øc, Y(2)ZArclength limits T, TaDetermine o, Zo at X = Bøo=Yn(1)Zo = Yn(2)FCN=0-0Return]Call LSODAR (Y,T,To,FEX,JAC,GEX)UBC LSODAR (Y,T,To,FEX,JAC,GEX)Partial differential equation solver’usingJacobian matrix JAC, equation derivatives in FEX,and GEX for program halt at the specifiedboundary.Subroutine FEXDY(1)=q*sin(Y(2))DY(2) q*Dy(1)/\f(3)+q*y(1)*89DY(3)=cos(Y(2))Subroutine GEXGo(1)=Y(3)-B[ Return142Chapter 4MATHEMATICAL ANALYSIS OF THE CAPILLARY SHAPES BETWEENPARALLEL, HORIZONTAL CYLINDERSThe capillary interactions between two approaching emulsion droplets each with a layer offine particles on their respective interfaces has thus far been described by considering a smallsection of the interface near a representative spherical particle.Unfortunately, even on a macroscopic scale, an experimental investigation of this three-dimensional scenario is problematic because of the idealizations associated with the cell model andthe difficulties with visualizing the interface. Thus it was decided a simpler two-dimensional modelof two macroscopic parallel cylinders lying horizontally on a liquid-liquid interface would bestudied. Because of the large scale needed to allow observation of the interface, we cannot neglectthe effects of gravity, and the theoretical basis takes this into consideration. Nevertheless, a systemwith the same geometry but on a smaller scale can be extrapolated back from our experiments.In the experimental apparatus, the cylinders are attached to plates and the cylinder/plateassemblies are sealed into a rectangular chamber such that the lower liquid enclosed in the chambercan come into contact with the upper fluid only via the slit between the two parallel cylinders(Figure 4.1). Increasing the volume of liquid into the lower chamber causes it to be displacedupward forming the meniscus shapes whose mathematical treatment is given in this chapter.Fig 4.1 Experimental cell (simplified)434.1 Literature ReviewThe meniscus profile between floating horizontal, parallel cylinders was considered byAllain and Cloitre (57), Gifford and Scriven (26), and Chan et al. (43) who were concerned withthe capillary forces of attraction that arise when gravity acts on such particles, and with the forcesnecessary for equilibrium and stability on an interface. However, the equilibrium shape betweencylinders found by Allain and Cloitre, and Chan et al., was found only approximately bysuperimposing the results obtained for two individual cylinders, each on an infinitely largeinterface. This superposition approximation is referred to Nicolson’s method (44) and is applicablefor low Bond numbers or when the cylinders are not too close together such that their profile canmerely be summed. Only Gifford and Scriven supplied an exact analysis for the equilibrium shapebetween a pair of floating cylinders. Another fundamental analysis of this configuration is givenby Princen (12) who studied the liquid bridges formed between assemblages of two or morehorizontal cylinders, but in the absence of gravity which is a reasonable approximation for verythin cylinders or for two fluids of similar density.4.2 System ConfigurationAs we increase the volume of liquid entering the lower chamber of the cell, the meniscuswill protrude further through the slit, and its shape will change accordingly. This capillarypressure zIP will be proportional to the hydrostatic head measured from the line joining the centersof the two cylinders (z’) plus a contribution from the applied pressure (b’):AP ApgZ’,,I [4.1]rA more comprehensive derivation of this approach can be found in (58).44Alternatively, we can apply equation [3.2] to this system as well, if we change ourreference line for measuring z back to the horizontal level interface whose position is determinedsimultaneously with the solution of the problem. This is equivalent to pushing an isolated pair ofcylinders down into the interface until the capillary pressure generated by the meniscus between thecylinders is matched by the hydrostatic head measured from the level interface above the cylinders(Figure 4.2). The desired shape of the meniscus occurs only until the cylinders are completelysubmerged below the level surface. For very narrow gaps between the cylinders, the high values ofrequired may be physically unrealistic since the meniscus extending from the outer side of thecylinders would collapse without any vertical support. Nevertheless, the analogy still holds andcan also be likened to the cylindrical interface developed between two vertical flat plates which areinclined to the interface with mirror symmetry about a vertical axis between them.Consider two parallel cylinders of infinite length and a distance d apart sitting horizontally,on an oil/water interface, phase 1 and 2, respectively. The meniscus between the cylindrical rodscan be described by one radius of curvature; since there is no axis of revolution, the other radius ofcurvature is infinite. The Young-LaPlace equation for cylindrical interfaces is, therefore given by:[4.2]rR1We need only look at a cross-sectional plane of the cylinders since the interface varies inonly two dimensions. We introduce Cartesian coordinates such that z=O coincides with thehorizontal plane at which the interface is not distorted by the cylinders (i.e., far from the cylindersat x=x’, where ziP=O) and the x-axis originates at the midpoint between the two cylinders (i.e., atthe axis of symmetry). We define qS as the angle that the normal to the interface makes with thepositive z-axis (Figure 4.2). The distance d is the separation distance between the two cylindersmeasured from their closest points, and d is the distance between the contact lines of the meniscuson each cylinder. The previous nomenclature for other variables holds.45Figure 4.2 Coordinate system for parallel cylindersSince the meniscus is symmetric about the z-axis, we need only consider the half-plane x>0. For the convex meniscus, qS increases from 0 to in the positive x direction with 0 < <900whereas for the concave meniscus, 0 decreases from 180° to in the same direction with 90° <ç< 180°. z0 is the height (or depression) of the meniscus at x=O. We will consider the convexmeniscus first.4.3 Mathematical FormulationThe following derivation describing the cylindrical interface follows a similarapproach taken by Princen (23) in his development of the meniscus between two verticalflat plates and by Levine et al. (58) for menisci between clamped cylinders.zx464.3.1 Convex MeniscusFrom simple geometry, the radius of curvature isldcos [43]dzIntegration of the latter equality with qS = 0 at z z0 yields:z-4[ = —[cz02+ 2(1— cos [44]The slope of the meniscus is[4.5]where 0 < çf’ < it/2. The expression for the derivative of the horizontal dimension with respect tois[4.6]dqS dbdzBy differentiating equation [4.4] with respect to qi one obtainsd(zJ)________________<0 [4.7]dq5 [cz02 +2(1_cosØ)]1’247which is negative for the convex meniscus, and positive for the concave meniscus. Substitutingequations [4.5] and [4.7] into [4.6] then yieldsd(xJ) cosç48dçS— [cz +2(1—cosqS)]” [ . Iwhich supports the choice of negative sign for equation [4.4]. This also agrees with the hydrostaticmodel, since convex menisci occur in capillary depressions with z below the datum line.Equation [4.8] can be written in a more convenient form, if we transform the equation interms of half-angles by making the following trigonometric substitutioncz +2(1—cosç5)=cz +4sin2(-) [4.9]=cz +4[1_cos2)] [4.10]= (cz +4)_[4cos2()] [4.11]and introducing the parameter k,k= I [4.12]+4)equation [4.8] becomes:48kd(x1Jj= 1_2sin2()JdqS 1/2 [4.1312L1_k2(cos2())]Integration of equation [4.13] and allowing[4.14]22such thatsin2=cos- [4.15]2yields1_2sin2)) k(25jfl22_1)/2Jr 2_ d2. [4.16]X%J I2 [i — k2 hi2 2j1/2o 2[1_k2cos2(’l2)]The numerator can be rewritten ask(2sin22—1) = kL_(k2 sin2 2— i)+—-— 11 [4.17]Lk2 k2 ]and introducing the elliptic integrals of the first and second kind,49F(k,qS)=rd21/2 [4.18]° [i_i sin2 2]E(k, ) = — k2 sin22]1”d2 [4.19]the transcendental equation [4.16] finally becomes:= k[(_ 1){F(k.) _F(k. g; ø)}_..{E(k.)_E(k.‘j}][4.20]Nondimensionalizing the equations [4.20] and [4.4] leads to= [(b— 1){F(k)_.F(k ‘n 0)}_ {E(k.)_ E(k, ‘z— j}][4.21]and1/2=_[Z02+_(1_cos(c))] [4.22]where capitals sigrnfy dimensionless terms.504.3.2 Concave meniscusNow for the concave meniscus, Rj is positive such that1 = dcosqSdcos/3 [4.23]R1 dz dzwhere- 8 r. Integration of equation [4.23] leads to:z-..J = [cz + 2(1 + cos [4.24]with the slope of the meniscus being defined as[4.25]db dzdbwhere=tan.. [4.26]The differential equation governing the horizontal dimension is obtained by substituting equation[4.26] and the derivative of equation [4.24] into [4.25] to obtaind(x./)—— cos [4 27]db— [cz +2(1+cos)]”51Finally, using the approach in the previous section, equation [4.27] can be integrated to yieldxiJ = — 1){F(k!j — F(k)} — {E(k.)— E(k .)}][4.28]To simplify equations in the program, for the concave case we chose to write the equationsin terms of/) the supplementary angle to q. Thus, equation [4.24] becomes1/2z-4J = [cz +2(1_cosfi)j [4.29]and equation [4.28] becomes= k[( — x){F(k) — F(k. fl)} — {p.f,.) — E(k 1)}][4.30]which is the same form as equation [4.20] obtained for the convex case.The relationship between the meniscus slope angle and the contact angle and positionalangle on the cylinder is different however for the two orientations. For the convex memscus (Figure4.3(a)),[4.31]where 0 8 r, 0 a r, and 0 Ø, - , whereas for the concave meniscus (Figure4.3(b)), itis52andfi = r— 0— a= 9+ awhere-qS r and PC = r— bC.tangent(b)[4.32][4.331Figure 4.3 Angles for convex (a) and concave (b) menisciTo determine the correct values for the unknowns we need to satisfy the meniscusdimensions within the boundaries of our physical setup. From the physical geometry, the meniscushorizontal length must bedr =d+2a(1—sin a)and from the Young-LaPlace equation, the horizontal distance between the meniscus ends isdC=— 1){F(k f)— F(k, lr—c—— E(k Jr—c )}]•[4.34]tangent to meniscus(a)tangent to meniscustangent to cylinder[4.35]53Equations [4.34] and [4.35] must yield the same result. The dimensionless function dL,which needs to be minimized, is the difference between the two lengths:dL=(dc_dr)i_. [4.36]The solution of dL yields the coordinates for any meniscus generated between a pair of cylinders.4.4 Determination of Film Rupture ThresholdIn order to determine the shape of the memscus at the film rupture threshold, the verticaldimensions need to be fitted also. The dimensionless meniscus height (measured between contactwith the solid and the topmost point), H, is calculated asH=Z0—Z. [4.37]H is then compared with the dimensionless required height Hr whereHr=1±cos(a) [4.38]where the upper sign is to determine the convex case, and the lower one, the concave case.The function dH is the difference between the two equations:dH=HcHr . [4.39]54Once the two functions, dL and dH, are minimized to within the error tolerance of 10, the rupturecapillary pressure can be calculated as1cap = P,,,, = y•cz0 [4.40]where the subscript “rup” signifies the threshold rupture point, or in dimensionless form asBo•Zap= 2[4.41]where the apostrophe signifies a dimensionless value.4.5 Numerical ComputationA series of profiles were generated for the meniscus between the cylinders as its contactpoint was moved along their perimeter. To determine the shape of the memscus for a given set ofconditions, the solution of the Young-LaPlace equation has to be forced to correspond to the actualphysical dimensions of the system according to equation [4.36]. As in the previous model, weapply a root-finding method to this problem since none of the dimensions of the interface areknown at the outset. Given the Bond number, the cylinder radius, the contact angle, and theseparation distance between cylinders, our unknown variables at the outset of the solution are thecapillary rise z0 and the positional angle a at which rupture would occur. Since equations [4.34],[4.35], and [4.36] constitute an algebraic equation for the single unknown parameter k they can besolved for this variable thereby allowing subsequent solutions for a, and z0.55The concavity or convexity of a profile can be easily determined by its contact position onthe cylinders and its contact angle. The turning point coincides with the location of the levelmeniscus which is calculated from equation [3.35]:=Above this level, menisci will take on a convex shape, and below this level, menisci are concavewith a general increase in curvature as a is increased above or reduced below acr.For incremental steps of ct along the inner perimeter of the cylinders, we determine the rootk of equation [4.36] by means of the numerical algorithm UBC DZERO (51) which uses acombination of linear interpolation, rational interpolation, and bisection. The values of k liebetween 0 and 1. The capillary rise or depression can then be calculated from the dimensionlessversion of equation [4.12], i.e.,= [4.42]and Z subsequently from equation [4.22]. The fiowsheet for this algorithm is shown in Figure4.4. The effect of contact angle hysteresis is treated in the same manner described in section 3.6.The algorithm for the inclusion of hysteresis in the determination of profiles is shown in Figure 4.5.In the case in which the rupture point threshold is our main objective, two functions needto be minimized, dL and dH. To reduce the number of calculations and avoid a cumbersomesolution of three coupled transcendental equations, we can manipulate the equations to yield asimpler algorithm in which is the optimizing variable, and each unknown can be successivelyobtained.56By substitution of equations [4.29], [4.37], and [4.38] into [4.39], we obtain the followingrelationship between Z0 and alone:= J[1:Fcos(o c)] — [i_coqsj] 1 [443]2 Bo[lFcos(9FØj]JThe initial step is to test qS, between 00 and 900, and obtain Z0 from equation [4.43]. Once Z0 andare known, the parameter k is determined from equation [4.12] and a is determined from [4.311or [4.32]. Then dL [4.36] is calculated. If dL is not zero, the initial root-finding routine selectsanother value for and the calculations are repeated. Z at contact then is calculated fromequation [4.29]. The programs utilizing this algorithm are used to plot rupture threshold capillarypressures as a relationship with Bond nwnber, separation distance, or contact angle (see AppendixC).57Figure 4.4 Cylinder model fiowsheet of program CALP - generates meniscus profiles and determines rupturethresholdsHo = Abs(Zo-Zo)H = Hr - HoYPoap = WZo/2Zo = -QoSqr(Zo2+2(1-oos( O))fW)Hr = Abs(1+Qotoos( a))58Figure 4.4 (continued)IR=OH=OFRI’s found40STOP59Figure 4.5 Cylinder model flowsheet of program CHYS - generates hysteresis plots (follows the progressof the capillary pressures as the menisci move further up the solid.Part IPart II60Figure 4.5 (continued)STOP61Chapter 5THEORETICAL MODEL RESULTSThe meniscus shapes detennined in the model do not necessarily correspond to that formedbetween stationary objects floating at an interface under equilibrium conditions. Under suchconditions, the position on the interface is governed by a balance of forces which includes the effectof the relative densities of the three phases, the buoyancy forces, the weight of the solid particles,and the interfacial forces. The model instead determines meniscus shapes developed when anapplied external pressure causes the liquid to enter the capillary space between particles fixed inspace. Changing the applied pressure therefore corresponds to changing the extent of immersion inthe lower fluid, of which the free-floating equilibrium meniscus would be one of these profiles. Noconsideration is made about the critical radius of a particle above which it will not float (refer toPrincen (23) for discussion of the critical radii for floating cylinders and spheres).5.1 Relationship to Emulsion Droplet CoalescenceA spherical droplet dispersed in another immiscible fluid has a pressure of[5.1]aewhere ae is the radius of the emulsion droplet, and e must be larger than the pressure that existsin the continuous fluid phase at that point, Thus, only convex (relative to the droplet fluid)menisci can occur (7). If a layer of solids is adsorbed onto the interface of the droplet, the small-62scale curvature at the interface would change and thus a new pressure drop cap would exist suchthat[5.2]i.e., the pressure across the interface is higher in the presence of particles than in their absence.Due to the effects of gravity, the required capillary pressure at rupture should be less forthe upper droplet than that for the lower droplet. The model solves for both the upper and loweremulsion droplets, where the upper profile is taken to be concave and the lower convex. (Also, forthe same wetting characteristics, one would use r— 8 as the contact angle for the upper droplet,since it is measured through the lower liquid). For colloidal systems where gravity has a negligibleinfluence, the bulk film pressure and the emulsion droplet pressure would be uniform, with nodifference between the upper and the lower droplets. However, even for the macroscopic systemsused in our experimental investigations, we have found that the difference in the respectivecapillary pressures is insignificant. The memsci developed for the lower droplet and upper droplet(contact angle of ir— 8) were symmetric about the horizontal axis. The dimensionless capillarypressures calculated for corresponding curves for the upper and lower droplet were identical tothree significant figures. However, a slight difference was noted when six or more significantfigures were compared. Nevertheless, gravity cannot be totally neglected since it does have aneffect on the curvature of the meniscus radially. The theory for the corresponding case wheregravity is zero for the cylinders is developed in Appendix B.Coalescence is assumed to occur when two droplets having stabilizing particle filmsapproach each other, and the liquid memsci protruding between the particles come into contact (weneglect the existence of a critical nonzero film thickness at which spontaneous rupture takes place(48, 59)). The protruding menisci shapes necessary for rupture may be produced when two suchdroplets collide forcing the particles in contact to enter fl.irther into their respective droplet.63Rupture might also take place if the droplets are distorted due to shear or turbulence where theexposed menisci are momentarily in contact. For these collision scenarios, coalescence alsorequires favourable kinetics, i.e., the time for the film to rupture must be less than the time ofcontact between the two droplets (60).5.2 Characterization ofProfile CurvesFour parameters need to be defined to fully describe a particular meniscus between twoparallel cylinders or two adjacent spheres. These are the Bond number, the distance of separationbetween solids, the contact angle 0 of the solid, and the interface position a. The effect of eachparameter on the shape of menisci will be investigated. We consider the simpler geometry of theparallel infinite cylinders first and then go on to discuss menisci between spheres. The exampleinterface selected is that between carbon tetrachioride and water. Table 5.1 shows values of thephysical properties, as well as the capillary constant for various fluid-fluid interfaces.Table 5.1 Properties of Selected Fluid-Fluid Interfaces (20°C)Fluid-fluid Density Capillary Bond BondInterface zip (dynes/cm) Constant c Number Number(glcm3) (cm2) a = 0.1 cm a = 0.5 cmCC14/air 1.594 26.95 58.01 0.580 14.5n-Hexane/air 0.66 18.4 35.18 0.352 8.80H20/air 0.9982 72.75 13.46 0.135 3.37CC14IH9O 0.596 45.0 12.99 0.130 3.25n-Hexane/H20 0.34 51.1 6.53 0.065 1.63* The Handbook of Chemistry and Physics (61)645.3 Cylinders ModelTo illustrate the effect of increasing liquid volume between a pair of parallel, horizontalcylinders on the meniscus shape, a composite plot of several menisci are shown as an example inFigure 5.1 for a system with a Bond number of 3.247, contact angle of 90°, and a dimensionlessseparation distance of 1.5. The profiles are shown for immersion angles starting at 10° up to 1700in 20° increments. As with the experimental runs, which were also started with a concavemeniscus, when fluid is introduced into the cell, the meniscus moves upward becoming less andless concave until it reaches a point where it becomes a level interface and the capillary pressure iszero (represented by a dashed line in the figure). When more fluid is introduced, the meniscuscontinues upward but now with a convex profile. The magnitude of the capillary pressuresgenerally increases as the convex profile creeps upward, but in some cases, depending on the three-phase contact angle and the distance between the cylindrical rods, the pressure peaks at amaximum value and declines thereafter. The dimensionless capillary pressures are shown abovethe corresponding curves. The negative signs of the pressures for the convex menisci serve only todifferentiate them from concave menisci and do not correspond to lower pressures than the latter(the signs are a consequence of the coordinate system chosen in the analysis); thus, capillarypressures will be referred to by their absolute values hereafter unless otherwise specified. For boththe concave and convex portions of the composite plot, the rupture threshold capillary menisci weredetermined. These memsci are shown as dotted lines in the plots. The convex rupture meniscuscorresponds to that of the lower emulsion droplet in our coalescence scenario whereas the concaverupture memscus corresponds to that of the upper emulsion droplet, but for the supplementarycontact angle u=-e In this case, it can be seen that they are essentially mirror images with thesame capillary pressures (to three significant figures).The influence of several factors which define a particular set of profiles between a pair ofcylinders will be discussed and illustrated with reference to similar plots. The factors which will be65x/aFigure 5.1 Meniscus profiles between a pair of cylinders (Bo = 3.247, 0 = 900, b/a = 1.5)66discussed are the contact angle the Bond number Bo, and the separation distance b/a.5.3.1 Effect of the Contact AngleFigures 5.2-5.7 show a series of meniscus profiles for contact angles between 300 to 180°.The plots are for a Bond number of 0.130 and a dimensionless separation distance of 1.50. Someof the figures represent the cylinders as noncircular profiles. This is a consequence of the size ofthe plot and the need to show more pronounced menisci. As the solids are made more wettable bythe lower fluid, the fluid creeps up on the lower part of the cylinders requiring less pressure to doso. However, on the upper portion of the cylinders, the increase in contact angle causesprogressively higher capillary pressures to be developed. This trend is expected since the lowerconcave portion is equivalent to the supplementary contact angle. For this reason we need onlyconsider the convex case in our comparisons. The increase in the contact angle has the effect ofincreasing capillary pressures for curves of a specific immersion position, and of increasing thefilm threshold requirement as shown in Figure 5.8 for b/a = 1.50.5.3.2 Effect of the Bond NumberThe effect that the Bond number has on the meniscus profiles can be seen in Figures 5.9-5.12 for a separation distance b/a of 1.5 and for a contact angle of 90°. Generally, for curveshaving corresponding a values, the curvature of the menisci increases as the Bond numberdecreases. However, the dimensionless capillary pressures first increase as Bond number increasesas seen in Figures 5.9-5.11 for Bond numbers 1 ,3x 10 to 3.25, and then decrease as seen inFigure 5.12 for the Bond number of 13.0. Comparing only the rupture threshold menisci at a fixeddimensionless separation distance b/a, Figures 5.13 and 5.14 show the relationship between6780 = 1.29910 Ba = 1299102.0200 = 30.0 • 0 = 60.0 —0.184b/a1.50 1.5 b/a=1.500 —0.1651,.70 —0 1370.15.. —0.0820.0 0.3 0.107F 0.4 F—0.8 F 0.302—10..—0.50.380 0—‘.5—.o0.30.38—2.5—S.C—3.0 0.2- 0.297—2.5—3.5—4.0 0.175 —3.00.2214.5 I I —3.5 I—3.0 —2.0 —1.5 0.0 1.0 2.0—0.0—1.5 1.0 0.5 0.0 0.5 1.0 1.5 0.0x/a x/aFi re 5 2 Bo=0.130 8=30° Figure 5.3 Bo0.130, 8=600-—Level Meniscus .... Film Rupture Threshold Numbers on profiles are dimensionlesss pressuresMeniscus Profiles between Parallel Horizontal Cylinders3.0 4.0Ba = 1.299101 30 = 1.29910’0 = 90.0 0 = 120.0 •—0.221a.o b/a =1.50 3.0 b/a =1.50 -—0.2055.51.5—0.207/ BE1.0.1.5 —0.363—0.2000.5-. L 0a 0.00.000—0.302—0.50.00 20 .-“ —0.107—1.0--0.0800.5370.26 —1.00.16—0.0—0.50.180.220—2.5— I I I I I —2.0 p—2.5 2,0 1.5 —10 0.5 0.0 0.5 1.0 1.5 2.0 0.5 3.0 2.5 0.0 1.5 —1.0—0.5 0.0 0.5 1.0 .o .o 0x/a x/’aFigure 5.4 Bo0.130, 8=90° Figure 5.5 Bo=0.l30, 8=120°Level Meniscus .... Film Rupture Threshold Numbers on profiles are dimensionlesss pressuresMeniscus Profiles between Parallel Horizontal Cylinders685.5Bo = 1.299I0 Bo = L299104.5 5.0—O 1220 = 150.0 • S = 180.0 _—° b/a =1.50 b/a =1.50 /3.0 4.0 /“ —0.197—0.2683.0—0.317—0.37610-—0.458 —0.4441.0—0.4540.5 1.000—0.330 05-0.44—0.156-0.5 0.0—10.,:.:0n35—05 —0.1650.066—0.044—5.5 —2.0 —L5x/a—5.0xIs10 20 30Figure 5.6 8o0.llO, 8=1500 Figure 5.7 Bo=0.130, 8=180°—- Level Meniscus .... Film Rupture Threshold Numbers on profiles are dimensionlesss pressuresMeniscus Profiles between Parallel Horizontal Cylinders‘1)‘-40:69——Bo = 0.0010.5- --.----Bo=0.130103—— Bo2.018Bo = 4.035—. v— Bo = 6.053b/a 1.500.4 -,A-w//,V0.3 -0.2-0.1 —0.0 -20 40 60 80 100Contact angle 0(0)120 140 160Figure 5.8. Rupture capillary pressure vs. contact angle (cylinders)7005.0 0.0Do = 129910’ Bo = 12991O0 = 90.0 0 = 90.0b/a =1.50 4.0 b/a =1.507 5 —0.039 3.0 —0.120—0.121— 04010 —0.112.5.—o 0.0 - . . 0.00.S —1.05 0 2.00 01—5.0 -2.0—7.5. .—3.0 0 010.0 —4.00..036—5.0—02.5—19.0 —7.5 —5.0 —2.5 0.0 2.5 5.0 7.5 09.0 02.5 —4.0 —3.0 —2.0 1.0 0.0 0.0 2.0 3.0 4.0x/a x/aFigure 5.9 Bo=1.3x10,0=900 Figure 5.10 Bo=l.30xl03,0=900— Level Meniscus .... Film Rupture Threshold Numbers on profiles are dimensionlesss pressuresMeniscus Profiles between Parallel Horizontal Cylinders- 2.9Do 3.24810 Do = L2991000 = 90.02.0 0 = 90.0b/a =1.50-0.179, b/a =1.50—0.2251.0—003 1.5—0,057:__—20—1.00—0.75—050—0.25 0.00 0.25 0.50 0.75 I 00 izs 1.50 l’r’r——--—,—-—-----———-1—-—-———-———r--————-’—_._.x/a x/aFigure 5.11 Bo3.247, 0=90° Figure 5.12 Bo=l.30x10,0=90°Level Meniscus.... Film Rupture Threshold Numbers on profiles are dimensionlesss pressuresMeniscus Profiles between Parallel Horizontal Cylinders71a).z1.a)C,a)Ca).Ca)a).a)C,C)0a)a)a).cC.a)C.C,a)I-.C.a)a).a)C,a)a)0a)a)Rupture Capillaiy Pressure vs. Bond NumberCylindersBond Number BoFigure 5.13 b/a=1.50Bond Number BoFigure 5.14 bla=3.OO72dimensionless rupture capillary pressures and the Bond number. For a given contact angle andseparation distance, by increasing the Bond numbers, the dimensionless rupture pressures reach amaximum and thereafter decrease in magnitude. For small particles such that the Bond numbersare less than the maximum, oil-water interfaces produce lower rupture capillary pressures than forfor a similar system at its corresponding oil-air interface. Above the maximum, the opposite wouldbe true. The Bond number expresses the relative importance of the gravity forces and thecapillary forces operating on the three-phase system. For a single object floating at equilibrium onan infinite interface, the gravitational forces tend to distort the meniscus, while the capillary forcestend to flatten it (57). When Bond numbers are sufficiently small (Bo —* 0), the free equilibriumsituation leads to q&, —* 0, or a level interface (43).In the case of multiple objects in close proximity, the above trend is not observed, as thecapillary forces are accentuated by the occurrence of capillary rise. It can be shown that when theBond numbers are very small (applicable to colloidal particles) that the changes in the capillaryforces are greater than the gravitational forces for the cylinders (57). Thus, the effect of gravitybecomes negligible. For the series shown in Figures 5.9-5.12, the first plots of the menisci shapesresemble vertical ellipses which become more like horizontal ellipses and more oblate in the laterplots as the Bond number is increased. The transition point between the change may be thetransition between a system predominantly capillary force-controlled to gravity-controlled. Ananalysis similar to Denkov et al.’s (8) assuming gravity is absent was performed for the cylinderssuch that the principal radius of curvature is constant:1 — AP — sin—(P2 io1 )gz0 [5 3]R1— — + a(1 — sin a) —(the mathematical formulation of this case is shown in Appendix B). However, from our plots(Figures 5.9-5.12) as the Bond number approaches 1x105 for b/a=1.5, the menisci do not73resemble circular profiles as noted above. Thus, the assumption of constant mean curvature formemsci profiles for micron-sized particle-systems of cylinders would be in error.5.3.3 Effect of the Separation DistanceHolding the Bond number and the contact angle constant at some reasonable value, weconsider the effect of varying the distance of separation between the cylinders. Profiles are shownin Figures 5.15-5.18 for a Bond number of 0.130 and a contact angle of 900 as b/a increases from1.05 to 6.00. It can be seen that the curvatures of all the menisci decrease as the separationdistance increases. In terms of the rupture capillary pressure, their values decreasecorrespondingly as one increases the distance between the cylinders, as shown in Figures 5.19 and5.20 for two values of Bond numbers, 0.130 (a = 0.1 cm), and 3.25 (a 0.5), respectively.One can see that in the series of figures, 5.15-5.18, at small separation distances, theoverall shapes of menisci resemble vertical ellipses. As the gap width between cylinders isincreased, the menisci become similar to horizontal ellipses.When the slit width is sufficiently large, each cylinder would deform the liquid interfacewith very little interference from its neighboring cylinder. Thus, with increasing distance apart, themeniscus flattens out (see Figures 5.15-5.18), approaching the case of isolated cylinders. In termsof the governing equations (Equations[4.7]):dZ sin(qS) [5.4]dq5 [Bo . + 2(1— cos(q5)]2when —-—>, then —---* 0, anda a743.21.29910 Bo = 1.29I02.0= 90.02.0e = 90.0b/a=1.05 20 b/a=1.501.5 —0.425. —0.2657 5.51.0—/ .-o:os.L.a .:0.660.5 —0.2000.0 -.66 0.00.200—0.5—1.0—L.a0.420—‘.5—1.50.26—2.0 0.327—2.00.229—2.5 I F F 2.5 r I—5.2 —1.5 —1.0 —0.5 0.0 2.5 1.0 L.5 2.0 —2.5 2.0 —1.5 —1.0 —0.5 0.0 0.5 1.5 1.5 2.0 2.5x/a x/aFigure 5.15 b/a1.05, 0=90° Figure 5.16 bIa’I.50, 0=90°-- Level Meniscus .... Film Rupture Threshold Numbers onproffles are dimensionlesss pressuresMeniscus Profiles between Parallel Horizontal Cylinders4.0 9.0Bo = 1.299101 Bo 1.2991Ot8.00 9O.O 0 = 90.07.0b/a =3.00—0.103 b/a =6.002.0—0.090 5.0—0.065 4.0-.... —0.03030 1000102030x/a x/aFigure 5.17 bla=3.00, 0=90° Figure 5.18 b/a6.00, 0=90°Level Meniscus .... Film Rupture Threshold Numbers on profiles are dimensionlesss pressuresMeniscus Profiles between Parallel Horizontal Cylinders75a)a)0-.aa(-)aa)Ca)0-.a)a)aC.a)C-)0,0a)b/aFigure 5.19 Bo=0A30‘.5b/aFigure 5.20 Bo=3.247Rupture Capillary Pressure vs. Separation DistanceCylinders76dZ sin(b)/2 [5.5]dqS [2(1—cos(qS)]Likewise for the horizontal coordinate, equation [4.8] becomesdX cos(ç)1/2 [5.6]dqS [2(1—cos())]Equations [5.5] and [5.6] are identical to the equations for the meniscus around a single plane wall(23) or a single cylinder of the same contact angle. In terms of the elliptic equations describing thegap between the cylinders, the parameter k approaches its maximum value of 1 ( this is true forZ0 = 0). Thus, the menisci between pairs of particles far enough apart, can be estimated bylinearly superimposing their respective menisci. This is applicable when the Bond number is small(43,44,57) in order that first derivatives in the Young-LaPlace equation can be ignored. Thissuperposition principle is known as the Nicolson approximation mentioned earlier. Chan et al. (43)also proceeded to show that the superposition principle holds even for cylinders having dissimilarBond numbers but where one of them had to be relatively small. The exact solutions for theequilibrium position of cylinders at an interface found by Gifford and Scriven (26) for this caseshowed that this approximation was valid for Bond numbers <0.5. Similarly, Allain & Cloitre(57) obtained excellent results with this approximation and found it held for horizontal cylinders atall separation distances as long as the Bond number is small. They also found that thissuperposition principle of adding the fields of deformation at any point between particles isdependent on the geometry and the shapes of particles, and fails in cases where capillary riseoccurs as it does between parallel plates at small distances.775.4 Spheres ModelAnalysis of the spherical particles case shows that such factors as the contact angle, theBond number, and the separation distance influence the shapes of the interfaces between theparticles in a similar manner as for the cylinders.5.4.1 Effect of the Contact AngleIt has been determined experimentally that particles whose contact angle is less than 900measured through the water phase would form oil-in-water emulsions whereas those with contactangles greater than 90° (likewise measured through the water phase) would form water-in-oilemulsions (9). However, for the spherical particles and cylinders under consideration here, thecontact angle can hypothetically be above or below 90° to form both 0/W and W/0 emulsionsdepending on the extent of immersion of the particles and the capillary pressure between them sinceconvex profiles can occur for contact angles between the two extremes, 00 < 0< 180°. However,keep in mind, these are not the static equilibrium positions under buoyant conditions. If the contactangle is either 00 (i.e. complete wetting), or 180° (no wetting), the particles will exist entirely inone or the other phase. When the contact angle is considerably less than 90° (measured throughthe lower or dispersed phase), the opportunity for the meniscus reaching the film rupture point ismore likely since more than half of the spherical particle is immersed in the droplet phase and onlya small capillary pressure is required to meet the criteria for rupture. Thus, a more stable positionoccurs if the contact angle is greater than 90°.Figures 5.21-5.26 show a series of profiles as the contact angle is varied from 30° to 180°for the same conditions used in the previous section on cylindrical particles (i.e., Bo = 0.130, andb/a = 1.50). Increasing the contact angle Oof the solids, decreases their ability to be wetted by thelower (or emulsion droplet) fluid and thus, increases the difficulty for the liquid meniscus to reach78r/a r/oFigure 5.21 Bo=0.130, 0=300 Figure 5.22 Bo=0.130, 6=60°— Level Meniscus .... Film Rupture Threshold Numbers on profiles are ditnensionlesss pressuresMeniscus Profiles between Spheresi.0• r/a r/aFigure 5.23 Bo0.130, 8=90° Figure 5.24 Bo0.130, 0=120°—- Level Meniscus .... Film Rupture Threshold Numbers on profiles are dimensionlesss pressuresMeniscus Profiles between Spheres790.0r/aFigure 5.25 Bo0.130, 9=1500—- Level Meniscusr/sFigure 5.26 Bo=0.l30, 0=180°Film Rupture Threshold Numbers on profiles are dimensionlesss pressuresMeniscus Profiles between Spheres80the outer perimeter of the adsorbed particles; it requires higher capillary pressures to do so. Figure5.27 shows the relationship between and contact angle for a separation distance of 1.50.5.4.2 Effect of the Bond NumberFigures 5.28-5.31 show menisci profiles become flatter (capillary pressures lower) as theBond number increases. The relationship between the rupture capillary pressures and the Bondnumber of 0.13 and 3.25 at a constant separation distance of 1.50 are shown in Figure 5.32 and5.33, respectively. If the Bond number increases, the capillary pressure monotomcally decreases.In the limit, if the radius of the sphere or the Bond number is very large, the situationapproaches that of a meniscus between two flat plates (23) of similar contact angles, i.e.,—>>— [5.8]R1 J?2The curvature in the plane of the paper, Rj, is much smaller than the principle radius of curvature,R2, perpendicular to the paper, such that equations [3.14] and [3.17] become[5.9]drwhich is the same equation as [4.3] and [4.23] for parallel vertical plates or parallel horizontalcylinders. Similarly, Padday and Pitt (34) found that for Bo > when c=10 cm2 and theradius of curvature of the rod is 100 cm, there is good agreement between a single rod-in-freesurface profile with experimental data for the meniscus around a Willielmy flat plate.810.7——Bo = 0.130—.— Bo1170.6 -—-v-- Bo3.5/b/a1.5 /4E..0.4- • ... —/,i .0.3-,0.2 - /1.:: II I20 40 60 80 100 120 140 160 180 200Contact angle 0 (°)Figure 5.27. Rupture capillary pressure vs. contact angle (spheres)82r/a r/aFigure 5.28 Bo=l.30x105,9=9O Figure 5.29 Bo1.30x103,=9O°— Level Meniscus .... Film Rupture Threshold Numbers on profiles are dimensionlesss pressuresMeniscus Profiles between SpheresIC 20r/a 2.0r/aFigure 5.30 Bo3.247, 090° Figure 5.31 Bo=1.30x10’, =9o°— Level Meniscus.... Film Rupture Threshold Numbers on profiles are dimensionlesss pressuresMeniscus Profiles between Spheres83O7b/a=1.50o.e O(deBond Number BoFigure 5.32 b/a=1.500.14-b/a = 3.000.12x=1200=1500.10 v=18001::: r::5.5 2.0 25 3.0 3.5 4.0Bond Iumber BaFigure 5.33 b!a3.OORupture Capillary Pressure vs. Bond NumberSpheres84If the Bond number was minimal, i.e., Bo — 0, two possible outcomes would occurdepending on the separation distance. If b/a was very large (approaching infinity), then equation[3.17] becomesdsin(gS)0 [5.10]dXfor all radial values far from the sphere, where qS = 0, and Z = 0. However, if the radialcoordinate b was of the same order of magnitude as a and the dimensionless X value is finite, thenequation [3.17] would reduce tocos(c)d = sin() [5.11]dX XBy separation of variables, this equation becomescos(9S)ddX [5.12]sin(b) Xand integration yieldslnIsin(b)I =ln X+lnC1 [5.13]or C1X = sin(q5) [5.14]and similarly for the vertical coordinate Z,85C1Z=cos(q) [5.15]which shows that in the limit of small separation distances, and small Bond numbers, the meniscusis described by a segment of a circle of radius 1/C. Thus, the negligible gravity assumption canbe used for closely-packed micron-sized spheres stabilizing emulsion droplets.5.4.3 Effect of the Separation DistanceWhen the distance between the spheres increases sufficiently, the capillary pressuredecreases in magnitude and the profile around each sphere becomes less curved (Figures 5.34-5.37). Figures 5.38 and 5.39 summarize the convex rupture capillary pressures at variousseparation distances for Bo0. 130 and Bo=3 .25.In the limiting case where the gap width between spheres becomes very small compared toa, the profiles become more circular. This result is similar to the case for the axisymmetricmeniscus profile in the annulus of two coaxial cylindrical tubes described in the article by Princen(23). The principle radii of curvature Rj would be constant, thus equation [3.14 or 3.17] wouldtransform into=cz [5.16](b—a)where the top sign corresponds to the convex case, and the bottom, the concave case. This is mostclosely shown in Figure 5.34 for the smallest two menisci.5.5 Effect ofHysteresisThe effect of hysteresis can be illustrated in graphical form (Figures 5.40-5.43). If westart out with the contact angle at 0a for the level meniscus, the position of the meniscus and its1.00.0—1.0-1 0Figure 5.34 b/a°°l.05, 0=90°— Level MeniscusFigure 5.35 b/a=l.50, 0=90°Film Rupture Threshold Numbers on profiles are dimensionlesss poessuresMeniscus Profiles between Spheres860o 1.29990”9= 90.0b/= I OS00 10r/a r/ar/aFigure 5.36 b/a°°3.00, 0=90° Figure 5.37 b/a6.00, 0=90°Level Meniscus .... Film Rupture Threshold Numbers on profiles are dhnensionlesss pressuresMeniscus Profiles between Spheres87Do = 0.1301.4O(de0. 0= .312 a= 60• +=90I) x=1201.. o=1500=1801.0__________I1.aa 0.8aLIa0)0 0.60a06)B0.40.20.0 I1.0 1.5 2.0 2.1 3.0 3.8 4.0 4.5 8.0 5Separation Distance b/aFigure 5.38 Bo=’O.130t.3Do = 3.2471.21.1LS (dee)0= 301.0= 60+= 90I-. x=1200.9 0=1500=1800.82)I-3.. 0.72,a0. 0.6aLI0I0.50021 0.40aB0.30.20.1..0.0LO Ls .0 .5 3.0 3.5 .5 .5Separation Distance b/aFigure 5.39 Bo=3.247Rupture Capillary Pressure vs. Separation DistanceSpheres880.aa.I-)CC.0.aUCCCFilm Thickness h/a Film Thickness h/aFigure 5.40 Bo=0.130, b/a1.50 Figure 5.41 Bo=3.247, ba=1.50Rupture Capillary Pressure vs. Film ThicluiessCylindess00 1.0Film Thickness h/a Film Thickness h/aFigure 5.42 Bo=0.130, bIal.50 Figure 5.43 Bo=3.247, bla=l.50Rupture Capillary Pressure vs. Film ThicknessSpheres89corresponding pressure would be determined on the solid line for 0a emanating from the x-axis inthese hysteresis plots. The “ 1800 contact angle “ curve is the limiting envelope for the “constant a”curves. Each “contact angle” curve covers the range for ç’.’ = 00 to 0 which marks the boundary fora equal to 1800. If the hysteresis were, for example, 300, with increased pressure, the meniscusshape would move along the dotted line with the meniscus contact line remaining stationary untilthe hysteresis reached its limit. The final contact angle would then be 30° greater, and themeniscus’ shape would be shifted up to a new curve which one can see requires a higher capillarypressure at film rupture. Notice, that the maximum possible capillary pressure does notnecessarily occur at this threshold, but can occur before, or after h = 0. If it occurs before itcrosses this line, the maximum capillary pressure would still need to be overcome and would be thelimiting pressure for rupture of the film.In the hysteresis plots, the dotted lines show increasing hysteresis to the extent of completenonwetting of the solid (contact angle of 180°). One can see, that this can occur before or afterh = 0. Thus, the location of the dotted line to this point will determine the maximum possiblehysteresis for a particular solid’s wettability characteristics.If the hysteresis of the solid increases, the maximum capillary pressure required forrupture also increases as Denkov et al.(8) also reported. Figures 5.40-5.43 illustrate this effect.Increasing the extent of hysteresis in effect causes the contact angle of the solid to increase. Forboth the cylinders and spheres, pushing the memscus up towards the rupture threshold usuallyincreases the capillary pressure until a maximum pressure is attained. From then on, furthermovement of the meniscus results in lower and declining capillary pressures. The figures alsoshow that as the Bond number increases, the maxima in the curves tend to move further towardsthe right of the plots (i.e. before the rupture threshold is reached). This effect is more evident in thecase for the cylinders than in the case for the spheres.905.6 Comparison of Cylinders and SpheresFrom these graphs, it can be seen that the effect of changing the three parameters mayhave very different results for the case of the cylinders compared to that of the spheres, due to theform of the equations for each case. The series of meniscus profiles shown for various contactangles (Figures 5.2-5.7 and 5.21-5.26) show that the cylinders generally produce more pronouncedand concentric curves than those of the spheres whose curves at the immersion angle extremitiestend to overlap adjacent menisci profiles. Increasing the contact angles has the general effect ofincreasing the rupture capillary pressures for both cases (Figures 5.8 and 5.27).The most dramatic difference is seen in the relationship between the capillary pressure andthe Bond nwnber. For the cylinders case (Figure 5.13 and 5.14), the rupture capillary pressurecurves show a maximum value as the Bond number is increased, whereas for the spheres (Figures5.32 and 5.33) the capillary pressure curves show no such behaviour but decrease monotonicallyas the Bond number increases. Generally, for Bond numbers above the maximum point, thecylinders produce higher values than that for the spheres. However, Bond numbers belowthe maximum, result in the opposite effect. Thus, in practical emulsions, spherical particles wouldproduce more stable systems than cylindrical particles.Increasing the distance between particles, produces the same effect for cylinders andspheres. Once again though, for Bond numbers above the maximum point, the cylinders tend toproduce higher values than the spheres (see data in Appendix A). Nevertheless, the overalltrends are similar. The decrease of hysteresis, the increase in separation distance, and the increasein Bond number, or the increase in wettability lowers the required film rupture threshold makingdestabilization of emulsions efficacious.915.7Relationship to Emulsion Stabilization with Finely Divided SolidsUp to this point, we have considered general trends of the effect of particular variables onthe capillary pressures between particles. When we extend the model down to those sizes typicallyinvolved in emulsion stabilization, we can detennine the magnitude of the capillary pressures whichmust be developed to cause coalescence. Good oil and water emulsions are usually formed withparticles not greater than 1 pm, and even smaller for solids of high densities (27). In contrast, formineral flotation, good results are possible with larger particles greater than 5 pm.Using the carbon tetrachioride/water system as an example, we consider the effect ofvarying particle sizes and separation distances in the micron range. For the cylindrical model, for aseparation distance of 1.05, we vary the particle sizes from one mm to one tenth of one J.Lm for thecontact angle cases of 60° and 120°. These results are given in Table 5.2. Table 5.3 shows, forthe same contact angles and 1 pm size particles, the variation of separation distances from 1.05(hexagonal close-packing for the spheres) to four radii lengths apart. Similarly, Table 5.4 andTable 5.5 show data for the spherical particle case. The maximum pressures shown areapproximate.Table 5.2 Cylinders: Capillary pressure dependence on particle size (b/a 1.05, c = 12.99 cm2)Particle Size P’ P m’ max(cm) (Pa) (°) (Pa) (°)0 = 60°0.1 2.76E-01 2.48E+01 145.62 2.80E-01 2.52E+01 157.50.005 8.28E-02 1.49E+02 137.12 8.89E-02 1.60E+02 157.50.0001 9.34E-03 8.40E+02 126.99 1.28E-02 1.15E+03 157.50.00001 2.30E-03 2.07E+04 123.67 4.06E-03 3.65E+04 157.50 = 120°0.1 7.40E-01 6.66E+02 126.74 1.30E+00 1.17E+03 900.005 2.55E-01 4.58E+03 108.98 3.OOE-01 5.39E+03 900.0001 3.48E-02 3.13E+04 82.81 4.25E-02 3.82E+04 900.00001 5.15E-03 4.63E+04 69.56 1.14E-02 1.03E+05 9092Table 5.3 Cylinders: Capillary pressure dependence on separation distance (a = 1 m)b/a P’ Pmp CL “max’ “max CL(Pa) (0) (Pa) (0)0 = 6001.05 9.34E-03 8.40E+02 126.99 1.28E-02 1.15E+03 157.52 3.42E-03 3.08E+02 124.42 8.83E-03 7.94E+02 1803 2.27E-03 2.04E+02 123.65 7.20E-03 6.48E+02 1804 1.76E-03 1.58E+02 123.23 6.24E-03 5.61E+02 1800 = 120°1.05 3.48E-02 3.13E+04 82.81 4.25E-02 3.82E+04 902 5.34E-03 4.81E+03 69.73 1.17E-02 1.06E+04 1203 3.43E-03 3.09E+03 67.88 8.71E-03 7.84E+03 1354 2.64E-03 2.37E+03 66.94 7.26E-03 6.54E+03 135Table 5.4 Spheres: Capillary dependence on particle size (b/a 1.05, c = 12.99 cm2)Particle Size rup’ rup CL “max’ max CL(cm) (Pa) (°) (Pa) (°)0 = 60°0.1 2.87E-04 2.58E-01 150.84 3.18E-04 2.86E-01 1410.005 5.78E-03 1.04E+02 150.81 6.39E-03 1.15E+02 1410.0001 2.89E-01 2.60E+05 150.81 3.19E-01 2.87E+05 1410.00001 2.89E+00 2.60E+07 150.81 3.19E+00 2.87E+07 1410 = 120°0.1 1.O1E-03 9.13E-01 137.58 5.14E-03 4.63E+00 960.005 2.04E-02 3.67E+02 137.54 1.03E-01 1.85E+03 960.0001 1.02E+00 9.17E+05 137.54 5.14E+00 4,63E+06 96Table 5.5 Spheres: Capillary pressure dependence on separation distance (a 1 .tm)b/a mp’ Ppjp CL “max’ ‘max CL(Pa) (0) (Pa) (°)0 = 60°1.05 2.89E-01 2.60E+05 150.81 3.19E.-01 2.87E+05 1412 6.48E-02 5.83E+04 143.13 6.54E-02 5.89E+04 1443 2.51E-02 2.26E+04 139.29 2.86E-02 2.57E+04 1504 1.29E-02 1.16E+04 137.06 1.59E-02 1.43E+04 1500 = 120°1.05 1.02E+00 9.17E+05 137.54 5.14E+00 4.63E+06 962 2.31E-01 2.08E+05 119.88 2.34E-01 2.1OE+05 1143 8.87E-02 7.99E+04 110.1 9.IOE-02 8.19E+04 1204 4.49E-02 4.04E+04 104.23 4.92E-02 4.43E+04 12093Examination of the results shows that the pressures are usually much higher in the case of thespheres than the cylinders for the given separation distance. For the sizes chosen, the cylinders donot have to exceed the maximum capillary pressure before rupture takes place whereas for thespheres, the maximum occurs before rupture takes place. Thus, the maximum point furtherstabilizes against rupture if it occurs before threshold rupture. The capillary pressures increase asthe particle size decreases for both cylinders and spheres (see Figure 5.44). However, for smallerparticle sizes (below 1 O cm) the spheres produce higher capillary pressures than the cylinders,and vice versa for larger particle sizes.1010.10’- —°—Prup-Cy1inds(0=60°)Pmax- C’Iind (0=600)Pmp - Cy1ind (0 = 120°)8 --o--Pnx-c’Iind(0120°)10- ——Pnp-ph(e’60°)Pmax- Sph (0=60°)—o—Pnip-Sph(e=120°)- -—e-—Pniax-Sph(0=120°)106-,—. 10- O \\ \\\\\\\.6 ‘‘4IIParticle Size (cm)Figure 5.44. Particle size effect on Pcap (Cylinders and Spheres)94As separation distances increase, the pressure falls (Figures 5.45 and 5.46), and thelocation of the maxima moves forward, occurnng much further up the particle (Table 5.3, and5.5). In the case of 1 pm size cylinders, for dimensionless separation distances from 1.05 to 3.0,the maximum pressure occurs after the threshold of rupture, as shown in the hysteresis plots(Figures 5.47-5.48). Thus, it does not affect the stabilization process in any way. For a system ofclose-packed, 1 pm spheres, the maximum capillary pressure occurs well before rupture isencountered, but as the separation distance increases to 3.0, the maximum capillary pressuremoves forward, even beyond rupture and thus no longer provides an energy barrier for stabilization(Figure 5.49-5.50).It can be seen that if the maximum pressure is made to be the limiting pressure, thestability is increased. From this perspective, the most stable position that a particle can take on aninterface is the one furthest from this maximum energy level and from the threshold of film rupture.This occurs if the particle were situated such that the capillary meniscus was level which is themost thermodynamically stable position as determined by Levine et al. (11). Specifically, if theparticle is immersed into the lower fluid to a depth a(1— cos( r— 6)), the free energy is at aminimum, and any other immersion depth is associated with an increase in the free energy of thesystem. The effect of hysteresis on the particles also varies with size and distances, as themaximum extent of hysteresis varies with these changes. With increased sizes and decreasedseparation distances, the maximum possible hysteresis observable increases.In relation to the results of Denkov et al. (8), for the same system we obtained highermaximum capillary pressure results which is as expected due to the differences in the two models(see Table 5.6).950C)Prup (0 600)Prup(0=120°)- -°—- Pmax (0 = 600)---—-Pmax(0=120°)4.Oxt3.Sxl3.Oxl2.Sxl 4.2.Oxl 0I .SxIi.0xi04S.0x103-0.03 4Separation Distance b/aFigure 5.45. Cylinders- Separation distance effects on Pcap4.OxI 03.0xlO-2.0xI0-I .OxI 0’0.0—-----Pup(e=6O)• Prup(0120°)--——Pmax(0=60°)--.—-Pmax(0= 120°)2 3 4Separation Distance b/aFigure 5.46. Spheres - Separation distance effects on PcapFigure 5.49 Bol.30xl07,b/a=l.05 Figure 5.50 Bo=1.3x107,bia=3.00Hysteresis Plots: Rupture Capillary Pressure vs. Film ThicknessSpheres960.2.20-0.0CCC.C2CC.0CCSFigure 5.47 Bo=1.30x10,b/a=l.05 Figure 5.48 Bo=1.3x1(r7 b?a=3.00Hysteresis Plots: Rupture Capillary Pressure vs. Film ThicknessCylindersFilm Thickness h/aC.C-2C-CUCCS0-2CC.UCCm97Table 5.6 Comparison of Denkov et al’s results with the present workb/a Maximum capillary pressure Pmax (Pa)Denkov et al. Present work1.05 2.7x105 3.1x1062.00 7.8x103 1.4x1053.00 2.4x103 5.5x104Thus, for a system having a r=30 mN/rn, (assuming zlp=1.594 as for CC14/air), 0= 120w, and fora 1 pm sphere at a separation distance b/a of 1.05, results in a maximum pressure of 3. lxi 06 Pafor our model, whereas Denkov et al. obtained 2.7x105 Pa. Furthermore, for their one-layer modelwhen the meniscus was advanced towards the outer perimeter of the emulsion droplet, there was nochange in the direction of change in the capillary pressure, i.e., no maxima were experienced as forour model. Their model led to the conclusion that the stablest position based on capillaryconsiderations was when the contact angle is 1800 which opposes the effect due to thermodynamicswhere the stablest position is at 90°. Similarly, we found that the most stable position occurs for alevel meniscus furthest from the point of rupture, or preferably when the contact angle is 180°.The actual position would most likely be determined by the consideration of both thethermodynamic and the capillary forces.The pressures developed for the spherical particles at micron-sized dimensions showextremely high pressures (up to 5 MPa for the cases chosen) which are not easily overcome, andthus show that capillary pressures constitute an important aspect of the stability of emulsiondroplets covered with finely divided solids.98Chapter 6.THE EXPERIMENTAL PROGRAMThe objective of the experimental program is to test some of the findings obtained from thetheoretical model, namely the effect of the solid separation distance, the wettability of the solidsand the injected fluid volume on the meniscus shape and the capillary pressure. The sections whichfollow outline the experimental background for the work, the equipment setup, the experimentalprocedures, and the analytical methods used for interpreting the results.6.1 Meniscus Profile Image Recording - Background TrialsThe parallel-horizontal-cylinder test cell illustrated in Figure 4.1 is a simplified version ofthe one actually used. Two parallel cylindrical rods connected to an enclosed glass cell with only anopening between the rods posed problems in viewing the meniscus profile, because the meniscusadjacent to the glass end wall is distorted. Depending upon the wettability of the wall material, themeniscus either crept up or receded downwards along the glass surface, thereby interfering with theend-on view of the cylinders. Ideally, the meniscus should be measured far from the effects of thecontainer walls. To overcome this difficulty, a technique was developed in which the meniscusmidway down the length of the cell was highlighted by a planar vertical beam of light and thenrecorded photographically (Figure 6.1). A high intensity He/Ne red laser beam (Spectra Physics)was available for this purpose. The initially circular beam was spread out into a sheet by acylindrical lens and then was shone upward through the glass bottom of the cell.This lighting configuration posed another problem because very little incident light entereddirectly into our photomicrographic equipment which consisted of a horizontally mounted Nikon99JEEmIfListereomicroscopeplatform jackFigure 6.1 Original test setupcamera— sheet of lightI glass cellI Içaserlaser housingI I I I I I I I I I I I I I I I I I I I I I Ti . i i . . . .liii i__IIR••*I liii.. iii iiFigure 6.2 Negatives produced by laser light technique100SMZ stereomicroscope equipped with 5x objective lens and a Nikon manual FE2 camera body.The highlighted section was not visible through the viewfinder, so several methods to intensify thelight beam or to increase the exposure density were tested. Exposure time tests showed that imagesappeared on the photographic film only after 2 minutes. However, the outlines were not sharp andhence were unsatisfactory.With increased exposure and/or development times, there was the hazard of fogging up thephotograph. Since the signal was quite weak, any background signals could easily mask ourresults at such extended processing times (62, 63). All experiments were thus done in a darkroomto exclude any extraneous light interference. Other methods of compensation and improving theresolution of the image involved the testing of higher speed films such as Kodak® TMAX 400,3200, and Ilford® HP5+, colored films, varying exposure times, push-processing films, decreasingthe distance between the sample and the camera lens, and using complementary color filters toenhance the image (64-70). A further improvement was also observed when the interface waslightly dusted with a thin layer of light reflecting particles such as pliolite or talc powder (Figure6.2).However, the image quality continued to be specular and poorly discretized. Moreover,the resolution of the images at times was not reproducible. The main reason for these difficultieswas the mode in which the image is formed. Since the light did not directly shine into themicroscope, the image recorded was only due to light refracted and diffracted as it passed throughthe liquid interface. In the formation of highly magnified images, these deletrious diffractioneffects tend to be accentuated by the use of lasers (71). Thus, the quality of the memscus imageson the film, especially after enlargement, was sufficiently poor that the method had to beabandoned. An alternative approach was then devised.1016.2 Final Image Definition TechniqueHigher quality images were obtained by using background illumination. Under theseconditions, the meniscus shape appeared as a dark silhouette against a bright background. Toovercome problems with end effects, the meniscus was measured away from the walls and wasvisibly unobstructed because the parallel rods were slightly curved, being slightly higher in themidsection than at the ends. The meniscus profile at the highest point was then photographed.The recording of liquid-liquid profiles is prone to errors due to optical distortions createdby the difference in refractive indices of the two liquids. In our experimental cell, it was necessaryto photograph the cylindrical solids profile simultaneously with the liquid meniscus. If themeniscus were convex, the solids and meniscus profiles would have to be viewed through the topliquid whereas, if the meniscus were concave, observation would have to be through the lowerliquid. Thus, the magnification would be different for the top and bottom portions of the image(72). Optical distortions can be avoided if the fluids have very similar densities and refractiveindices (72, 73). However, under these circumstances, the interface would be more difficult to see,and thus, the photography would be correspondingly more difficult. Thus, it was decided to keepthe problem as simple as possible by studying an air-water system where the image is measuredthrough the air phase in which refraction of light is minimal.6.3 Experimental Equipment and SetupThe overall setup is shown in Figure 6.3. Its main components consist of the samephotomicrographic equipment described earlier, the new test cell, a light source, and amicromanometer for measurement of capillary pressures.colorfilter/diffuserFigure6.3Experimental equipment setupvalveshutterreleaseclampstandcameramanometerfluidmicromanometermercuryvaporcapillarytubeandorgoose-necksightglass\lampcameraconnector tubewithinteriorlensplatformjack andlevelingplatformlevellingscrewsandplatfonuC1036.3.1 The Experimental CellThe cell, constructed of stainless steel, is made of five separate sections which whenassembled looks like Figure 6.4. The base is the main reservoir for holding the fluid, the secondlayer is the rubber sheet with the two rods glued to it, the third is the upper frame which fits overtop of the rubber sheet, then these pieces are clamped into place, and the caliper mechanism put inplace to straddle the two rods.Figure 6.4 Experimental cellThe rods are clamped at the ends onto a metal frame with built-in slots. The top of theliquid reservoir is covered with dental rubber stretched over the perimeter of the metal cell andclamped into place. The dental rubber sheet is slit in the middle between the two cylindrical rodsclamp; distancecontrolling screwcaliper compassbase cellpipe fittingplatformNleveling screw104which are glued to it. The distance between the rods can be varied by adjusting the screw on thecaliper compass mechanism which in turn pushes on either side of the rods. At each end of themetal base of the cell there is a plastic pipe thread fitted with a nut, either connected with clearTygon tubing to measure the meniscus pressures on one side, or fitted with a septum on the otherside for introducing fluid with a syringe (Figure 6.5).6.3.2 MicromanometerWater was used as the test liquid in order to avoid the use of solvents. To measure thesmall pressure differences needed to detennine the capillary pressure a micromanometer (FlowCorporation Model MM3) was used. The manometer oil employed was a water immiscible bluemanometer oil having a specific gavity of 1.75 (Meriam 175 Blue Fluid). All air pockets andbubbles were meticulously removed from the tubing lines by means of suction using a syringe fittedat a Tee-valve connection (see Figure 6.3). The manometer oil proved to be an aggressive solventsuch that plastic tubing such as Tygon®, CFlex®, or polypropylene were inappropriate for use.Teflon tubing could not be used due to its rigidity and difficulty in fitting over glass capillaries toform a tight seal. Fortunately, Viton® tubing which is a thick-walled flexible tubing used forvacuum pump applications, proved to be resilient and inert to the effects of this oil.The principle of this micromanometer is based on the fact that different pressures on eitherend of the tubing would cause a differential change in the height of the liquid relative to somedatum line. To accentuate the meniscus between the oil and water, the glass capillary is inclined at10° from the horizontal. Measurements are made by maintaining a point on the capillary meniscustangent to the crosshair in the eyepiece of the micromanometer by adjusting the micrometervertically upwards or downwards. The micrometer attachment allows meniscus height readings toan accuracy of 0.0002 inch. The large cross-section reservoir of manometer fluid cancels theeffect of any volume expansion in the tubing (74).septum hollow base cellFigure 6.5. Plan view of the bottom cell; front view of the test cellsteel rod105tubingTeflon nut and boltdental rubberbase frameEZ nut andbolt7.5cm12.5 cmI7.5 cmupper frame\11 dental rubber1066.3.3 LightingTo enhance the sharpness of the image, a 160 W mercury vapor lamp (short wavelengthsclose to ultraviolet wavelengths, 70) was used as the light source. The lamp was placed inside abox with an opening to allow the illumination of only the central area of the test cell. For the firstset of experiments, the rods were made hydrophobic with a Teflon coating (Crown 6065 PermanentTFE Coating Spray) which was green in color. Therefore, a complementary red filter wassometimes employed to enhance the contrast between the solids and the surroundings (68, 70). Thefilter was placed just in front of the lamp. Adjustments were made with the height of the lamp andthe size of the box opening, to control the amount of light and improve the sithouetted images of therods and meniscus against the brighter background.6.3.4. Photographic EquipmentThe stereomicroscope allowed us to magrn1’ our subject to fill the frame of the picture. Itconsisted of a condenser lens (5x magnification) within the microscope tube. A manual cameraequipped with a shutter release was attached to the microscope tube. The height of the microscopewas adjusted to approximately the same height as the meniscus in order to photograph a full frontalbacklit image. The photographic film used typically was Kodak Technical Pan for its fine grainand high contrast.6.4 Experimental Preparation ofRodsTwo sets of rods were prepared. It was necessary to render them hydrophobic to developthe required menisci visible to the camera. For the first set, the stainless steel 5 mm diameter rods107were coated by spraying several times with a Teflon spray until fully covered. For the second set,similar rods were coated with the anionic polymer polyvinyl formaldehyde using the methodoutlined by Bowen (75). The polymer used in our experiments is the commercially available formknown as Formvar® 15/95 E (Canadian Resins and Chemicals Limited). Two wt% of the polymerresin was prepared by dissolving the Formvar powder in reagent grade chloroform (BDH’). Thisconcentration was found by Bowen to produce a reasonably uniform layer which had an averagethickness of 200 A. The coating arrangement, shown in Figure 6.6, consisted of several largesyringes minus their plungers, a four-foot length of 1 mm i.d. glass capillary tube, several tee-valves, and connecting pieces of polyvinyl tubing. The dissolved resin was poured into theuppermost syringe which contained the stainless steel rods to be coated. The solution then drainedslowly through the long glass capillary and collected in the lower syringe flask such that the levelof liquid in the upper syringe fell at a slow, steady rate.6.5 Experimental ProcedureThe test cell was filled with the test liquid, in our case, water, and all tubing was thenconnected and checked for leaks. With the use of the tee-valve, all noticeable air bubbles present inthe lines were meticulously withdrawn using mild suction from the syringe. The lines weresubsequently refilled with water using the same syringe. The test cell and the manometer were thenlevelled carefully. Once the lines were filled with water including the tubing line connected to theglass capillary of the micromanometer, the tee-valve was adjusted to connect the manometer withonly the test cell. To begin the experiment, the microscope was pre-focused by focusing on a wiresuspended over the centre cross-section of the cell. The water level in the cell was adjusted untilthe meniscus was exactly horizontal. The pressure measured at this point corresponds to a zerocapillary pressure. Any pressure differences recorded as water is added or withdrawn from thechamber were then a measure of the capillary pressure at the meniscus maximum or minimum (i.e.,10860 ml syringestopcocksFigure 6.6 Coating of Rods with Polymeric Resinat the midplane between the two cylinders). Additional fluid was added or withdrawn using asyringe. The meniscus cross-section was then photographed using a shutter release to the camerawhich was supported on a platform jack. Pressure measurements were made by controlledadjustments to the micromanometer until the manometer oil/water meniscus in the glass capillarywas observed to be tangent to the crosshair in the eyepiece. Calibration of the objects in thephotographs was made by positioning a small 3 mm ball bearing glued to a glass microscope slidewith a handle clamped on a fine-control micromanipulator supported on a stand over the cross-section of interest (Fig. 6.7). The calibration sphere was then photographed under the sameconditions as the meniscus. Once a set is finished which includes the photographing of a levelmeniscus as well as concave and convex shapes, the roll of film was developed accordingly andthen enlarged for further analysis.Glass capillary1 mmi.d.Rod120 cm1093.155 mmFigure 6.7 Calibration sphere on microscope slideThe shape of the memscus on the enlarged (13 cm x 20 cm) photographs was discretizedusing a digitizing tablet (Kurta Model 1000). The software program written for this application isincluded in Appendix C. The magnification factor for a set of photographs was detennined fromthe ratio of the diameter of the calibration sphere calculated by the digitizing subroutine and theactual micrometer measurement. The sphere’s diameter was determined by calculation from threepoints on the perimeter (Appendix B). In the photographs, the horizontal reference level wasdetermined as the line that connects the centers of the two rods. This line was determined bydigitizing the visible portions of each rod separately, detennining its radius, and hence, the locationof its center. The rods’ center coordinates were initially estimated analytically using a methodbased on three points taken from the uppermost part of the profile, and then more accuratelycalculated using a least squares fitting of nine digitized points. The full circular rod profile couldthen be reconstructed from a knowledge of the center position and the radius. The horizontal linebetween rod centers was used to calculate the baseline orientation and thus reorient the x-ycoordinates of the digitized profiles. The contact points and then the rest of the meniscus profile110were then digitized. Estimates were then obtained for the immersion angle and contact angle asfollows.The digitized data for each meniscus was used to obtain information about its slope at theparticle surface by regressing straight lines to the first and last three points at both ends of thedigitized profile. The immersion angle for both sides of each meniscus was also computed duringthe digitizing process. Furthermore, the immersion angle was measured by determining the centerof the rods independently by finding the intersection of lines bisecting at least two chords in eachcircular profile of the solids and measuring with a protractor. The contact angle was detenninedfrom the supplementary angle to x for the level meniscus. Since these measurements may be proneto error, the contact angle value was further checked by calculation using the depth of immersionof the cylinder measured from the level meniscus, t, and the cylinder radius (a) (76) fromco8)=--—1. [6.1]The calculated values agreed reasonably with the measured values. The reported data are basicallythe measured values.The profiles were then plotted using spreadsheet and graphics software (Lotus 123 andMicroCal Origin® 2.1, respectively). Figure 6.8 shows the flowsheet describing the overall imageprocessing from the experimental stage to the analysis of the data.The meniscus pressures cap’ were determined from fluid hydrostatic considerations basedon the micromanometer readings. Detailed sample calculations are shown in Appendix B. Theresults of the experiments are discussed in the next chapter.111__I—Experiment‘1Measured Capillary Pressuresin Inches of Manometer OilFilm Processing and ImageEnlargement[Digitization and ComputationCalculate Pcap and compare with theoretical valuesPlotting of digitized valuesDetermination ofb/a,4c,Incorporation of theoretical profiles7Images on NegativesFigure 6.8 Experimental program flowsheet112Chapter 7RESULTS AND DISCUSSIONThe experimental data and the calculated values obtained from the experiments areincluded in Appendix A. Some of these numbers have been summarized and tabulated in thischapter. Experiments were performed on two sets of differently treated stainless steel rods. Onepair of cylindrical rods was treated with several layers of a Teflon spray (experimental runs B, C,D, and E), and the other pair was coated with the polymeric resin Formvar© as describedpreviously (experiments I and J). The Teflon spray produced a visibly rough surface which wasexpected to result in high contact angle hysteresis. The Formvar® coating yielded a differentcontact angle and a smoother finish, and thus, should correspond to a low hysteresis system.The independent variables which were investigated experimentally are the distance ofseparation between the rods, the applied fluid pressure in the cell, and the surface wettability of thesolids. For each run, the photographed and digitized images have been superimposed to show howthe meniscus shape changes from concave to convex as more and more fluid is injected into the cell(Figures 7.1-7.6). The composite plots also include the expected theoretical menisci shapes shownat contact positions a from 900 and upwards in 100 increments. The theoretical menisci aregenerated based on the average values for the rod radius a, the distance of separation b/a, and thecontact angle measured from the location of the level meniscus.Z 1 Comparison with Theoretical CurvesA study of the profiles shows that for the chosen contact angle there is some disparitybetween the experimental curves and the theoretically derived curves. This is not surprising as thex/aFigure 7.2. Experiment C meniscus profiles1132—1-2-3 -2 .1 0x/aFigure 7.1. Experiment B meniscus profiles2I0—1-2-3 -2 -1 0 1 2 3114-2-3 -2 -1 0 1 2 3x/aFigure 7.4. Experiment E meniscus profiles210—1-2-3 -2 -1 0 1 2 3xlaFigure 7.3. Experiment D meniscus profiles21I I I • ICalculated Cylinders• Digitized DataTheoretical Profiles—1I I I0x/aFigure 7.6. Experiment 3 meniscus profilesI I115Calculated CylinderDigitized DataTheoretical Profiles210—1-2210—1-2I I I I-3 -2-1 0 1 2 3xlaFigure 7.5. Experiment I meniscus profilesI I I I I-3 -2 -1 1 2116model profiles here do not include any hysteresis effects. The observed menisci have morepronounced curvature than expected at a given a position, which may be evidence of someresistance of the liquid periphery to movement. The nature of this resistance will be consideredlater. Better agreement of the profiles, however, is obtained when higher contact angles are usedfor the model (which will also be discussed later).Similarly, when capillary pressures are based on immersion angles, agreement is poor.However, comparison of the film rupture threshold capillary pressures for the first set (experimentsB-E) shows that the dimensionless capillary pressures are of the same order of magnitude as thetheoretical results for the same system. For example, for Experiment B with a separation distanceof 1.5, the experimental rupture capillary pressure cap’ is -0.1674 whereas the theoretical valueis -0.139 for a contact angle of 60°, and if a higher contact angle is assumed, the theoretical valuewould be higher.Nevertheless, when one solely considers the shape of the meniscus, regardless of itsimmersion angle, by comparing the contact slope angle with that of the theoretical values thereis good agreement between their respective capillary shapes and pressures (this is similar tochanging the contact angle). The contact slope angle can then be used as a basis of comparisonas we look at the effect of changing our variables of interest.7.2 Separation Distance Between RodsA sample calculation for the dimensionless experimental capillary pressures used in thecomparisons is shown in Appendix B. From our model results, when we compared the film rupturecurves for cases where the separation distance is lengthened, with all other variables remainingconstant, we see that the capillary pressure tends to decrease in magnitude (see Figures 5.19 and5.20). In the experimental runs, it was not possible to produce the film rupture point in all casesbecause excessive bulging of the memsci at the centre of the cell sometimes led to drainage of fluid117from near the ends of the rods which were at a lower height due to their curvature. Nevertheless,the film rupture curves which were obtained for some of the experimental runs are approximate asthey involved visual estimation. These curves, as well as curves of similar qS,’s, are compared witheach other. The expected trend for comparison of capillary pressures based on as separationdistance changes is shown in Figure 7.7 (for Bo = 0.328, 0= 900). The trend is similar to thatobtained when only rupture capillary pressures were compared; the absolute capillary pressuresdecreased as separation distance was increased.Comparison of the experimental curves having similar Ø’s show that increasing separationdistance between the rods generally produces a lower capillary pressure for concave menisci butisnt as evident for convex menisci as seen in Figure 7.8a and 7.8b for Set 1 and 2. This result maybe due to interference caused by increasing hysteresis for more convex menisci.1.00.5 -0.0 -—WaL05Wa3.0Ba = 0.3280 = 90’II.0.5 -I I-10 —-100 -50 0 50 100# (‘)Fiwe 7.7. Mc d.Iatistüp beti capillaiy pressiwe aid fIx variws b/a118-0.4 —-40 -20 0 20 40(°)Figure 7.8b. Set 2 - Effect of separation distance on Pcap’ vs.IUU- -—- Experiment B (bla1 .507)Experiment C (bla=1.453)ExperimentD (bla=1.556)Experiment E (b/a1.339)0.4 —0.30.2 -0.10.0 --0.1 --0.2 --0.3 -•-.‘%. •,• (°)Figure 7.8a. Set I - Effect of separation distance on Peap’ vs.0.50.4 -0.3 -0.2 -0.10.0Experiment I (b/a =1.36)-- Experiment 3 (b/a1 .272)-0.1 --0.2 --0.3-50 0 50119The higher hysteresis is tantamount to increasing the apparent contact angle of each meniscus andas Figure 7.9 shows increasing contact angle would increase capillary pressures.Z3 Effect of Contact AngleThe two sets of rods may have unique contact angles since their adsorbed surface films arenot the same. The measurement of the level menisci for the first set places the contact angle of thesystem between 60-70°, whereas for the second set, the contact angle is smaller, at500. Thus, the Formvar© surface is more hydrophilic than the Teflon surface (which most likelycontains additives, binders, etc.).From the model, a comparison based on as the contact angle is varied produces a morecomplicated effect than for a comparison of the rupture curves. Figure 7.9 shows that dependingon the cS value, for a particular contact angle its pressure may be higher or lower than anothercontact angle. But if the experimental point in question is located to the right of the maximumpressure on the figure, the trend is similar to that found for the rupture curves, i.e., capillarypressures increase as the contact angle increases.Experiments E and I are essentially of the same separation distance and are compared witheach other. From a plot of the theoretical Peap’ vs. cc for each run, we know that the experimentaldata points occur to the right of the maximum pressure and thus, for higher contact angles, higherabsolute capillary pressures are expected. If we were to superimpose the respective curves forexperiment E and I from Figures 7.8a and 7.Xb together, we see that for the concave curves, theexpected trend is observed. However, for the more convex curves the trend is not as apparent. Forthe discordant points one can see that they lie just at the bottom of the curves which show thehighest deviation from theory.0.60.5 0.4-,0.3/Bo0.3280.2/-_--..b/a=1.50.1C)0.0o-0.1III-150-100-50050100th(0Figure7.9.Modelrelationshipbetweencapillarypressureand•.for various0C121Z 4 Effect ofHysteresisThere are several measures of hysteresis apparent in our study, one which is the morefmiliar, the determination of the receding and advancing values of the contact angle by slightmovements of the three-phase line. This measurement was not obtainable for all runs becauseslight changes in position were difficult to determine objectively. However, estimates of theapproximate measures can be obtained from the location of the curves in close proximity to thelevel line. These curves were advanced unidirectionally from concave to convex shapes, unlessotherwise specified.Near the level meniscus, it was apparent that the contact lines did not move very much asthe meniscus was advanced such that concave and convex menisci had the same end points. Fromthese observations we are able to measure the contact slope angles for curves having similarcontact lines both above (convex) and below (concave) the level meniscus. The differencesobserved with respect to the level meniscus are a measure of the contact angle hysteresis and aresummarized as follows (these are approximate):Table 7.1 Contact Angle HysteresisExperimental Run Above Level Below LevelB 15.5° 15 5°C <21° 129°D 19.2° <26.2°B 18.2° 13°I 7° 15°J 6.8° 5.8°122As one can see, the second set of experiments, shows low hysteresis compared to the firstset. The first set showing contact angle hysteresis of 15 to 20% whereas the second set showscontact angle hysteresis between 6 to 15%.Z 5 Apparent Contact AnglesAnother manifestation of hysteresis is seen in the calculated apparent contact angles foreach meniscus shown in the data in Appendix A. We expected there to be an initial change in thecontact angle from the level meniscus up to the advancing angle °a and thereafter to remainconstant as the depth of immersion increased. To some degree this is seen in some of theexperiments. But the effect is masked by the presence of this insidious hysteresis. Theexperimental runs show that the effective contact angle increases initially and eventually levels offbut at a much later stage of development for both Set 1 and 2 (Figures 7.10 and 7.11).Generally, the level of hysteresis varies over the circumference of the rods, when oneexpects it to remain constant as proposed theoretically. This change in contact angle varies from 5to 30 degrees. These apparent contact angles were calculated based on the previous equations[2.75] or [2.76].Taking into account the presence of hysteresis which varied from 10 to 20 degrees, thedimensionless capillary pressures for each memsci were compared with those of correspondingfrom the theoretical predictions taken at the corrected contact angles. Tables A.4-A.9 show thatimproved agreement can be obtained over comparisons based on a positions. However, the contactangle which best fits the measured data differs with each run. From Figures 7.12-7.17 which plotPcapvs.bc we can see that the experimental points lie within the linear portion of the curves butdeviate noticeably near the higher c3 values, being higher than expected in all runs. This serves toillustrate that hysteresis is not constant throughout each run. And although the data do not12390— —..-.0-• ——A—°——ExptB60 -- Expt c—-°“-ExptD---A---ExptE5O20 I100 120 140aFigure 7.10. Set 1 - Apparent contact angles o vs. a position50. // // /40::10- I I-• I I •122 124 126 128 130 132 134a (degrees)Figure 7.11. Set 2 - Apparent contact angles 0 vs. a position11240.40.2 -A0.0 --60 -40 -20 0 20 40 60‘I’ (°)Figure 7.12. Experiment B comparison with model - Pcap’ vs.-0.2 --go0.60.40.2L)0-0.2-0.4I I— I I I-60 -40 -20 0 20 40,(0)Figure 7.13. Experiment C comparison with model - Peap’ vs.600.50.40.30.2(0.l10.0-0.1-0.2-0.3-0.40.70.60.50.40.30.2o.100.0-0.1-0.2-0.3-60 -40 -20 0 20 40.c (°)Figure 7.14. Experiment D comparison with model - Peap’ vs.-60 -40 -20 0 20 40(°)Figure 7.15. Experiment E comparison with model - Pcap’ vs.125AI . I . I . I608=48°0=58°0=68°—---—-0=78°A DataI I • I I . I60126a0.7 —0.60.50.4 -0.3 -0.2 -0.1 -0.0 --0.1 --0.2 --0.3 —-800.8—0.60.4-IIIC)U)U)2L)U)U)0U)aI . . I . I • IA-60 -40 -20 0 20 40• (°)Figure 7.16. Experiment I comparison with model - Pcap’ vs.600.2 -0.0 --0.2I 1___ I-60 40 -20 0 20 40,c (°)Figure 7.17. Experiment J comparison with model - Pcap vs.60127follow the full theoretical curves, they do show the presence of a minimum point in the Pcap’vs.Øplot before rising again. As in the theoretical results this shows the presence of a maximumcapillary pressure point (which is not necessarily the film rupture point) above which furtherimmersion of the rods translates into a declining capillary pressure.However, the experimental digitized data can show remarkably good agreement with modelcurves when the apparent contact angles are used. To illustrate the improved results one canobtain when this hysteresis is accounted for, a curve from each experiment was overlayed oncapillary menisci calculated from the model for the same conditions but with its apparent contactangle as shown in Figures 7.18-7.23. For C-CVX16, shown in Figure 7.19, the experimental1cap’ is -0.3180 which agrees favorably with the model meniscus at the same immersion anglehaving a cap of -0.2904.Comparisons based solely on a positions at the contact angle from the level meniscus havebeen shown to be inadequate because of the effects of hysteresis apparent in our system. This pooragreement between experiment and the model are shown in Figure 7.24 for experiment B as anexample, where the immersion angle a is plotted against q. The variations encountered lead us toquestion what may be the cause.0.0-0.50.50.0Figure 7.18. Experiment B curve (B-CVXI6) fit with model curves for o78°xfa1281.51.00.5-1.0-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0x/a2.52.01.51.0-0.5-1.0-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0Figure 7.19. Experiment C curve (CVXI6) fit with model curves for 0=88.5°2.01.5 -1.0 -0.5 -0.0 --05 —-1.0129-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.01.51.00.50.0-0.5 --1.0Figure 7.20. Experiment D curve (CVXI7) fit with model curves for 0=84.5°-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5x/a3.0• Meniscus profile (CVX17)Model profiles (10° increments)Bo0.328b/a1 .5580=84.5°I I • I I______I • I__• Ix/a• Meniscus profile (CVX28)Model profiles (10° increments)Bo0.328b/a1 .3390=75.10I . I . I . I . • I . • I . I . I .Figure 7.21. Experiment E curve (CVX28) fit with model curves for 0=75.100.0-0.5Figure 7.22. Experiment I curve (CVX7) fit with model curves for 8=70.5°1301.51.0 -0.5-0.0 --0.5-1.0-2.5 -2.0 -1.5 -1.0 -0.5 0.0x/a0.5 1.0 1.5 2.0 2.51.51.00.5-1.0-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5x/a• Meniscus profile (CVX6)Model profiles (10° increments)Bo0.328bfal .2790=67.2°I . I I . I . IFigure 7.23. Experiment J curve (CVX6) fit with model curves for 8=67.2°///////////////////////I I1311801601401201008060 —-80 -60 -40 -20 0 20 40 60(°)Figure 7.24. Experiment B comparison with model- a position vs.132Z 6 Hysteresis and Kinetic ForcesIt is worthwhile to determine whether the discrepancies are caused by what we considerhysteresis or whether other factors such as surface interactions, or nonequilibrium conditions maybe the cause. Up to this point we have assumed that the solids surfaces are uniform,nondeformable, and noninteractive with the liquids. But if the surfaces show variation over timewhere a previously wetted surface shows a difference in contact angle, or if the menisci is at ametastable condition which can be overcome if the required energy level is introduced by vibrationsto the system (22) we know that kinetic surface effects may be the cause. A better understandingof hysteresis is necessary to determine how we can differentiate the two effects.To illustrate we use the Withelmy plate method. Consider the schematic of the device usedin this technique (Figure 7.25).Figure 7.25 Wilhelmy plate methodA plate vertically mounted over a liquid is attached to one arm of a balance such that the forceexerted on the plate can be measured while it is being immersed. When the plate is lowered so itjust touches the surface, a force is exerted on the plate which is equal to the weight of the liquid inthe meniscus. As the plate is immersed further into the liquid, this force decreases due tobuoyancy. When there is hysteresis, Figure 7.26a will be the result. When there is no hysteresis,Figure 7.26b will be observed. As the plate is immersed further, the contact angle increases up toz 02Cl)WWII- a.Figure 7.26 Force vs. depth curves for the Wilhelmy plate apparatus; (a) with hysteresis;(b) without hysteresis.133IFORCE(a)FORCE(b)Iz0Cl)wIL0IIa.w0FORCEFigure 7.27 A hysteresis loop for a system showing solid-liquid interaction.134the value of the advancing angle, whilst the measured force decreases. When 1a has been reachedit remains constant, and the force vs. immersion depth curve is a straight line with aslope due to buoyancy. When the plate is then withdrawn from the liquid, the contact angle beginsto recede down to °r• Then at this value, further withdrawal results in a straight line with a slopedue to buoyancy once again. This curve is the hysteresis loop and helps us to define more clearlythe contact angle hysteresis using the terms outlined by Everett (76).Everett used two criteria to define the contact angle hysteresis. The first is that thehysteresis loop is repeatable, that is it can be repeated indefinitely when the independent variable(i.e., depth of immersion) is cycled. The second criterion is the existence of “scanning curves”denoted as dashed lines in Figure 7.26a which are formed by switching from immersion toemersion at different depths. The presence of scanning curves means that all points of the loop areattainable from different directions.Variations in contact angle which do not agree with the above criteria are possible and canbe caused if any of the surface energy components at the interface Ysg’ Ysl’ and hg changes duringthe measurement. These time-dependent effects can be isolated by testing for repeatability of thecontact angle. A typical curve in which surface interactions such as desorption or adsorptionoccurs, causing the loop to shift to different angles is shown in Figure 7.27. These effects can beserious if their time constants are of the same order of magnitude as those of the measuring system.If they are very different, they should not affect the contact angle measured (19).For our experiments, when the meniscus was receded and advanced, there was a definitevariability in the results in terms of the immersion angles and to a lesser degree with the capillarypressures. Figure 7.28 shows a sample hysteresis loop which was performed for Set 2, based onexperiment 3 in which the meniscus level was deliberately raised and lowered for this purpose.However, in this case, it is conceivable that experimental error may mask the true behaviour of theloop. Since experiment 3 proved to be more repeatable than experiment I, it is possible that theproblem may be due to a change in surface energetics caused by wetting and may be avoided bypre-wetting the rods before taking measurements.135uj8A —A— Measured Data0.210.1 2079 A.4 o.o 34A100-5A12A6AJJ3 All‘ I I I-20 -10 0 10 20 30Figure 7.28. Hysteresis ioop for experiment 3136A comparison of the replicates of the level meniscus for I and J (see Table 7.2) show thatthere is fair agreement with the measured capillary pressure, but larger variations with respect tothe position of the menisci on the rods is apparent (about 10 and 30 difference for J and I,respectively). The larger hysteresis evident in run I may be due to the fact it was performed first,without prior wetting of the solids (larger disparity with first point I-LEV5).Table 7.2 Reproducibility of the Level MeniscusCurve a Type P’1-LEV5 126.5 adv. 0I-LEV1O 133.0 rec. 0.009I-LEV12 133.5 rec. 0.01404I-LEV14 133.0 adv. 0.0227J-LEV3 130.3 adv. 0J-LEV7 131.6 rec. 0.01745J-LEV9 130.5 adv. 0.00864AverageI 131.5±2.9 0.01144±0.008w/o I-LEVS 133.3±0.243 130.8±0.6 0.008696±0.007adv.= advancingrec. = recedingOne other possibility for the variations in contact angle is the speed at which the threephase line is moved (15). Dynamic measurements of contact angle show that there is a relationshipwith the speed of movement of the contact front. A typical experimental plot is shown below.137- O (extrapolated)r (static measurement)‘UcU<o 0 u>oCONTACT LINE SPEEDFigure 7.29 Typical experimental results for the dependence of the dynamic contact angle 9, on the speedof the contact line. When U>O (U<O) the contact line is advancing (receding). U denotes the slowestspeed at which an experimental measurement is made (55).It has been suggested that the advancing and receding angles should be extrapolated from dynamiccontact angle measurements rather than the static contact angle generally reported which is usuallymeasured by starting with a static meniscus and advancing it until the contact line is perceived tohave moved. As one can see from the Figure 7.29, these values from the two methods tend to bedifferent.Although there are a few cases where reproducible contact angles are the same for slowadvancement and recession of the interface (77), they are not the rule. It is well known for boththeoretical and experimental work that the advancement and recession (axisymmetric) of liquid ona solid with fine concentric grooves or sawtooth ridges, will produce a steady contact line motionfollowed by “Haines jumps” (78). This phenomenon was observed in the present experimentalwork.Z 7 Sources ofExperimental ErrorThe use of slightly curved rods rather than horizontal rods may cause some unexpectedvariation in the results since the hydrostatics as well as the capillary shapes may be different alongthe length of the rods. This would translate into measured capillary values which might be lower138than the actual value for the meniscus at the center of the cell. The micromanometer readingscould be erroneous if air bubbles are still present in any of the tubing between the manometer oilreservoir and the cell itself. Although no such trapped bubbles were visible in the transparenttubing and glass fittings leading to the manometer oil tubing which was opaque. All glasscapillaries in the micromanometer were carefully cleaned and no apparent hysteresis could be seen.The manometer oil and water interface at the measurement sight glass was sharp and uniform andshowed no signs of adhering to the glass or resisting movement. Minimal error may be incurredwhen judging the line where the cross-hair coincides with the meniscus. The reproducibility erroris reportedly ± 0.000 1” of manometer fluid with an accuracy of ±0.0002” of manometer fluid(73).The photography of the water capillary meniscus in the cell proved to be challenging. Thelighting of the meniscus as well as the rods needed to be precise and was the most difficult aspectto control. However, as some of the photographs show, the lighting was at times uneven, such thatthe top of the rods had too much illumination making parts of the rods, and sometimes part of themenisci (near the contact line), seem to disappear into the bright background. In these cases, themissing areas were extrapolated in the digitization process.Other sources of error were the manual digitization process of images (79), and theenlargement of images by 25 times led to lines which were fuzzy and broader. Thus, someinterpretation of the lines and extrapolation of unclear areas were made. The actual digitizing errorwas determined by repeatedly digitizing the calibration sphere ten times. The mean value for thediameter (calculated from the three-point method) in terms of digitized units was 3317.266 ± 8.776or ± 2.7% standard error of the mean (SEM).The combined effect of these errors can be seen in the variability of the calculated cylinderpoints and the digitized points on the cylinder in the composite profiles obtained from consecutivephotographs. The deviation is approximately a tenth of a millimeter. The gross error involved inthe experimental reproducibility of the level meniscus as shown previously in Table 7.2 shows thatthe immersion angle mean for I is 131.5±2.9° (2% SEM), and for 3 is 130.8 ± 0.6° (0.5% SEM).139Chapter 8CONCLUSIONSThe capillary interactions of cylindrical and spherical shapes situated on a fluid-fluidinterface were considered with relation to the characterization of the capillary phenomenaresponsible for the stability of the disjoining film between two coalescing solids-stabilized emulsiondroplets. Both a theoretical and experimental approach were taken to determine the effect ofseveral factors which play a part in this process.The models represent idealized systems in which the particles are identical andhomogeneous. Meniscus profiles were generated for various configurations for an ideal systemwithout hysteresis. Since all practical systems do contain hysteresis effects, this aspect wasincorporated into the models, and considered with respect to its effect on the coalescence betweenemulsion droplets. In the modelling of the coalescence process, the existence of a critical filmthickness for rupture, as well as the kinetics involved, were neglected. The models represent thecoalescence process by which a layer of particles on approaching emulsion droplets make contactand the capillary menisci between particles are deformed either by pushing the particles further intotheir respective droplets or by squeezing liquid from within the droplets due to shear ordeformation. By varying several factors in the model such as separation distance, contact angle,particle size, or fluid-fluid properties, their effect could be seen.The results of the models agree with experimental conclusions from the literature thati) the finer the particles, orii) the closer the packing of particles on the interface,the more stable the emulsion. Other model trends we found were that a decrease in the Bondnumber or a decrease in the wettability to the dispersed phase (or increase in contact angle) wouldincrease stability based on capillary pressures. These trends are similar for both the cylinders140model and the spheres model which nevertheless produce very different profiles and capillarypressures. The main difference between the two models is seen in the piots of the threshold rupturepressure vs. the Bond number, for which the curves for the cylinders model had a maximum.Thus, for Bond numbers below this maximum, the spheres tend to produce higher rupturepressures than the cylinders.Increasing the hysteresis of the particles in the models was also found to increase thestability of the disjoining film by increasing the required rupture capillary pressure. Varyinghysteresis in effect changes the contact angle of the solids. The maximum observable extent ofhysteresis for a given starting contact angle is determined by the position of the hysteresis curvewith respect to the location of the threshold rupture line.Furthermore, we determined from the hysteresis plots how the capillary pressures can be astabilizing mechanism to prevent the thinning of the disjoining film layer between coalescingemulsion droplets. It was determined that the limiting pressure preventing rupture was notnecessarily the pressure at the threshold of rupture, but the maximum pressure which precedes it.The maximum provides a supplementary “energy barrier” for the system to overcome. Thelocation of the maximum capillary pressure (as seen in the hysteresis plots - Pcap’ vs. h) can bevaried by changing the size of the solids, or the separation distance. For both the cylinders andspheres, the maximum pressure moves to the right of the film rupture line on the hysteresis plotswhen the Bond number increases or the separation distance decreases. For a micron-sized system,this effect is evident for the spheres, but not so for the cylinders. Thus, spherical particles aremore effective as emulsion stabilizers than cylindrical particles.Based on capillary pressure theory, the most stable position that a particle can take on theinterface is the depth to which the meniscus is level at contact (to an immersion depth ofa(1—cos(r— 9))). This position coincides with the minimum free energy of the system asdetermined by Levine et al.(1 1). Similarly to Denkov et al.’s conclusion, the stablest positionwould be the point furthest from rupture which would occur if the solid is nearly completely wettedby the continuous phase (contact angle of 1800). From a thermodynamic standpoint this position141occurs when the contact angle is 900. The actual most stable position may be determined by acombination of both thermodynamic and capillary factors such that 90° 0 . 1800. This mayexplain the experimental results of Schulman and Leja (27) who found that for contact anglesgreater than 90° (measured through the water phase), W/0 emulsions occur, and for contact anglesless than 90°, 0/W emulsions are observed.Several differences were noted between Denkov et aI.’s model and our spheres model. Fora similar system described in their paper, our model yielded higher maximum capillary pressureresults which is as expected due to the different geometries of the two models. Furthermore, as themenisci were made to advance towards the outer perimeter of the emulsion droplet, their resultsproduced a gradual increase in pressure whereas in our model, a maximum was often evident. Oneother difference between the models is in the handling of gravity. Their model neglects gravitywhereas ours does not. It was determined from a theoretical basis that the constant mean curvatureassumption would be satisfactory for systems of small Bond numbers and with very closely-packedparticles. However, for larger systems, a general approach such as the one taken in this workwould be more applicable.The experimental work revealed that comparisons based on contact angles measured fromthe level meniscus were in poor agreement with model-generated profiles which is not surprising asit neglects any effects of hysteresis. However, when comparisons are made based on the contactmeniscus slope angle, results are better. Experiments based on this criterion showed that theexpected trend for the effect of separation distances and the effect of contact angles was supported(increasing contact angle and decreasing separation distances increase rupture pressures).The presence of hysteresis was evident on both sets of rods as there was visible roughness,and variability in advancing and receding contact angles. Hysteresis was much larger on the solidssprayed with Teflon than with the Formvar resin. Hysteresis had the effect of increasing thecapillary pressures. This was most evident for the more convex curves in each run, as they showedthe highest hysteresis, and produced higher capillary pressures than expected from theoreticalcalculations based on a contact angle measured from the level meniscus. Good agreement between142theory and experiment was obtained when the contribution due to hysteresis is included, by usingapparent contact angles in the model. Thus, the proposed model is appropriate for describing thismacroscopic system and the theory may be used to characterize effects involved in the stabilizationof the disjoining film between emulsion droplets by capillary phenomena.The behaviour of the hysteresis was more complex than expected. The hysteresismeasured at the level meniscus did not remain constant as assumed in the theoretical work, butincreased with the upward movement of the interface and finally levelled off much later. Thevarying hysteresis can be due to a combination of kinetic and thennodynamic hysteresis. Possibly,the surfaces are not of uniform roughness, or require previous wetting before measurements aretaken.The main sources of error occurred in the photographic recording of the images, the lackof prewetting of the solids, and the lack of controlled speed of movement of the interface.Improvements can be made to the experimental work by considering these effects. It is alsorecommended that to investigate the effects of the Bond number, and the wettability of the solids,different size rods and smoother ones be used to minimize interference by hysteresis.143NOMENCLATUREa radius of a solid cylinder or sphereae radius of an emulsion dropletA areab half the distance between the centers of a pair of solidsb* curvature at apex of sessile or pendant dropsB dimensionless separation distance between adjacent cylinders/spheres (half), b/aB’ Bashforth and Adams’ shape factor2Bo Bond number; c a(2p1)gIyd distance between closest points on adjacent cylinders/spheresd horizontal distance between contact lines on each cylinderdr horizontal distance determined from the physical geometry as a function of adH dimensionless measure of distance from rupture; difference between actual anddesired vertical coordinate of meniscusdL dimensionless difference between actual and desired horizontal length of meniscusE, F elliptic integrals of the first and second kind, respectivelyJg functionh distance of centre of meniscus from rupture thresholdH h/aHact dimensionless vertical distance between Z, and Z0Hr required vertical distance of memscus from contact level Z to rupture linek modulus in the elliptic integralK curvature of a point on a curven number of equationsO/JJ W/O oillwater, water/oil144P pressurezIP,“2 pressure difference; pressure in the upper fluid, pressure in the lower fluide’ b pressure inside emulsion droplet; pressure in the bulk phasecap’ P,p capillary pressure; capillary pressure at the rupture threshold1cap’ dimensionless capillary pressure, P divided by 2y/ar radial coordinateRj, R2 principle radii of curvatures arclengthU speedU slowest measurable speedw specific surface free energyW workX r/a (spheres model), na (cylinders model)Y set of vectorsx, y, z Cartesian coordinate axesz vertical coordinate measured from the free level interfacez*“ the line joining the centres of the solidsz level of meniscus at contact line from the free level interfacezt, level of meniscus at its lowest or highest point from the free level interface fora capillary rise or depression, respectively.Z z/aa immersion angle, angle between the line connecting the center of the solid to thecontact line and the negative z-axisfi supplementary angle to qangle between the positive z-axis and the normal to the interfacey interfacial tensionF surface excess per unit area1450’s,, 9’ advancing and receding contact angle from extrapolation of dynamic values,respectivelyp chemical potentialA. complementary angle of q2o contact angle0a’ 0,. advancing contact angle; receding contact angleangle between the tangent to the curve and the horizontal planeSubscripts, Superscripts:1,2 upper fluid, and lower fluid, respectivelyact actualb bulkc variable at the contact line of meniscuscap capillarye emulsioni componentmax maximumo oil; or apex of meniscus as in Z0r requiredrup ruptures solidw waterdimensionless146REFERENCES1. 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Matijevic, ed., Plenum Press, New York, 1979, pp. 31-91.77. Everett, D. H., “General Approach to Hysteresis. III. Formal Treatment of the Independent -Domain Model of Hysteresis”, Transactions of the Faraday Society, 50, 1954, pp. 1077-1096 as cited by Johnson and Dettre.78. De Fazio, J.A., and D.C. Dyson, “Stability of Rectilinear Contact Lines I. Single ContactLine Theory”, Journal ofColloid and Interface Science, 135(1), 1990, pp. 45-57.79. Boyce, J.F., S. Schurch, Y. Rotenberg, and A.W. Neumann, “The Measurement of Surfaceand Interfacial Tension by the Axisymmetric Drop Technique”, Colloids and Surfaces, 9,1984, pp. 307-3 17.152APPENDIX A - EXPERIMENTAL DATATable A.1 - Micromanometer ReadingsTable A.2 - Set 1 Compiled Experimental DataTable A.3 - Set 2 Compiled Experimental DataTable A.4 - Experiment B: Comparison of Pcap’ based on a andTable A.5 - Experiment C: Comparison of Pcap’ based on a andTable A.6 - Experiment D: Comparison of Pcap’ based on a andTable A.7 - Experiment E: Comparison of Pcap’ based on a andTable A.8 - Experiment I: Comparison of Pcap’ based on a andTable A.9 - Experiment J: Comparison of Pcap’ based on a and 4TableA.1-MicromanometerReadings(*denotesthelevelmeniscusstandard)ExpiBCCV13CVX16CCV2OLEV21*CVX23CVX24ExptCCCV9LEV11*CVX12CVX15CVX16CVX19CVX23ExptDCCV4CCV6LEV7*cvxlOCVX13CVX17ExptECVX23CCV24LEV26*CVX27CVX280.2903531.664283.1390.5380985.309-170.5060.3080564.080250.7230.4449814.8030.0000.60601109.846-295.0430.56981043.548-228.7450.3802696.309306.5810.54761002.8900.0000.65571200.868-197.9770.70051282.916-280.0250.73821351.961-349.0700.5228957.47145.4190.69601274.675-271.7840.4120754.549247.7920.4605843.373158.9680.54731002.3410.0000.67201230.720-228.3790.77031410.750-408.4090.74001355.257-352.9160.80571475.582-262.0770.4921901.246312.2590.66261213.5050.0000.72281323.757-110.2520.80431473.018-259.5130.3168580.197468.4800.5020.4142758.578290.0980.3110.4386803.265245.4110.2630.4873892.455156.2210.1670.57261048.6760.0000.0000.65791204.897-156.221-0.1670.77971427.965-379.289-0.4070.84071539.682-491.006-0.5260.68231249.584-200.908-0.2150.70911298.666-249.990-0.2680.77971427.965-379.289-0.4070.69451271.927-223.251-0.2390.60921115.706-67.030-0.0720.68231249.584-200.908-0.2150.56971043.365165.9270.1770.62151138.23371.0590.0760.66031209.2930.0000.0000.67331233.101-23.809-0.0250.72511327.969-118.677-0.1260.76391399.029-189.736-0.2020.67331233.101-23.809-0.0250.56971043.365165.9270.1770.66031209.2930.0000.0000.69921280.535-71.243-0.0760.80281470.271-260.979-0.2780.76391399.029-189.736-0.202SET1SET2h(in.)P(dynlcm2)iP(dyn/cm2)m’h(in.)P(dyn/cm2)zP(dyn/cm2)m’0.304-0.1830.2690.000-0.316-0.2450.3290.000-0.2 12-0.300-0.3740.049-0.2910.2660.1700.000-0.245-0.438-0.378-0.2810.3350.000-0.118-0.278ExptIccviCCV2CCV3CCV4LEV5*CVX6CVX7CVXSLEV9cvxi0CVX1 1LEV12CCV13LEV14ExptJccviCCV2LEV3*CVX4cvx5CVX6LEV7ccv8LEV9cvxi0cvx11CVX12153TableA.2-Set1CompiledExperimentalDataApparent0AuPorn’ExptBa=’0.156B1.503CCV13112.5-29.837.70.3010.36170.06080.2402CVX16124.322.578.2-0.181-0.1752-0,0294-0.1516CCV2O123.8-2828.20.2660.17450.02930.2367LEV21123.8056.200.00000.00000.0000CVX23133.535.582-0.313-0.4249-0.0714-0.2416CVX24137.832.574.7-0.243-0.4498-0.0756-0.1674ExptCa0.156B=1.460CCV9118.3-2536.70.32870.16750.02810.3006LEV11119.7060.300.00000.00000.0000CVX1212023.783.7-0.2123-0.1580-0.0265-0.1858CVX1512532.787.7-0.3002-0.3112-0.0523-0.2479CVXI6129.838.388.5-0.3743-0.3351-0.0563-0.3180CVX19130.7352.30.0487-0.14370.00000.0487CVX231344086-0.2914-0.5745-0.0965-0.1949ExptDa0.156B1.558CCV4111.5-31.736.80.2660.22620.03800.2280CCV6108.5-14.5570.17040.11970.02010.1503LEV7111.7068.300.00000.00000.0000CVX1O114.72085.3-0.245-0.1863-0.0313-0.2137CVX13132.33582.7-0.438-0.6121-0.1028-0.3352CVX17-134.53984.5-0.3784-0.7981-0.1341-0.2443154ExptEa=O.156B1.339CVX23142.546.383.8-0.2792-0.6562-0.1103-0.1689CCV24119.513.373.80.33260.10720.01800.3146LEV26121.3058.700.00000.00000.0000CVX27121.518.276.7-0.1175-0.1205-0.0202-0.0973CVX28129.835.385.5-0.2765-0.3214-0.0540-0.2225TableA.2-Set 1CompiledExperimentalData(continued)155ExpiIaBa(°)a(°)(°)0ipp(°)m’cap’(cm)(manual)(measured)CCV#10.161.345123±1.3123.3-38.518.20.46930.25580.04190.4274CCV#20.1591.337124.5±1.9123.5-34.5220.38030.15840.02590.3544CCV#30.1591.340126.9±4.6124.3-22.533.20.23700.13400.02190.2151CCV#40.1641.301129.6±3.3124.3-1540.70.13400.08530.01400.1201LEV#50.1571.364128.4±1126.5053.50.00000.00000.00000.0000CVX#60.1521.404124.6±3127.56.559-0.1427-0.0853-0.0140-0.1287CVX#70.1591.328133.5±1130.52170.5-0.2636-0.2071-0.0339-0.2296CVX#80.151.409132.3±1.2133.52470.5-0.2960-0.2681-0.0439-0.2521LEV#90.1521.403129.4±.55130.5049.50.0053-0.1097-0.01800.0233CVX#100.1531.377131.9±4.2133552-0.0134-0.1365-0.02240.0090CVX#110.1571.342133.52±2.5133.518.364.8-0.1682-0.2071-0.0339-0.1343LEV#120.1541.343134.01±4.3133.5046.5-0.0059-0.1219-0.02000.0141CCV#130.1541.356134.6±3.7133.21157.80.1189-0.0366-0.00600.1249LEV#140.1561.346132.8±81330470.0047-0.1097-0.01800.0227Average0.1561.357Std.Dev.0.0030.027 TableA.3-Set2CompiledExperimental Data156ExptJaBa(°)a(°)(°)0app(°)m’cap’(cm)(manual)(mSUd)CCV#10.1531.304125.9±.7128.8-9.541.70.16970.09060.01480.1548CCV#20.1541.276130±1.3129.5-5.844.70.09700.03880.00640.0906LEV#30.1561.27128.9±2.6130.3049.70.00000.00000.00000.0000CVX#40.1561.254130.9±2.41316.855.8-0.0399-0.0130-0.0021-0.0377CVX#50.1591.216133.±1130.81564.2-0.1500-0.0648-0.0106-0.1394CVX#60.1551.278131±1.5130.81867.2-0.2592-0.1036-0.0170-0.2422LEV#70.1531.307131.6±.1131.64.853.20.0153-0.0130-0.00210.0174CCV#8130.3±.7130.314.864.50.24700.09060.01480.2322LEV#90.1551.298130.5±1130.5049.50.00860.00000.00000.0086CVX#100.1561.268129.±4.6130.86.555.7-0.0503-0.0389-0.0064-0.0439CVX#11131.3±.3131.322.571.2-0.3297-0.1425-0.0233-0.3063CVX#120.1571.262128.8±.2131.51866.5-0.1942-0.1036-0.0170-0.1772Average0.1551.273Std.Dev.0.0010.019TableA.3-Set 2CompiledExperimentalData(continued)157158Table A.4 Experiment B - Comparison of cap’ based on a and cba=0.156 B = 1.503Parameter Meniscus Curve NumberCCV13 CVX16 CCV2O LEV21 CVX23 CVX24Experimentala (°) 112.5 124.3 123.8 123.8 133.5 137.8• (°) -26 22.5 -22 0 35.5 32.5“cap’ 0.2402 -0.1516 0.2367 0.0000 -0.2416 -0.1674Model cap0 = 56°abased 0.1222 -0.0087 -0.0045 -0.00452 -0.0696 -0.0960%Diff -49.1 -94.3 -101.9 0.0 -71.2 -42.7based 0.2708 -0.1341 0.2381 0.0000 -0.1640 -0.1567%il3jff 12.7 -11.5 0.6 0.0 -32.1 -6.40 = 46°abased 0.2111 0.0894 0.1026 0.1026 0.0109 -0.0240%]3jff-12.1 -159.0 -56.7 0.0 -104.5 -85.7based 0.2662 -0.1134 0.2267 0.0000 -0.1388 -0.1356%Djff 10.8 -25.2 -4.2 0.0 -42.6 -19.00 = 66°abased 0.0119 -0.0775 -0.0655 -0.0655 -0.1388 -0.1590%Diff -95.1 -48.9 -127.7 0.0 -42.6 -5.1Øbased 0.2844 -0.1550 0.2385 0.0000 -0.1900 -0.1837O/Jjff 18.4 2.3 0.8 0.0 -21.3 9.7CCV: concaveCVX: convex159Table A.5 Experiment C - Comparison of Pcap’ based on a anda=0.156 B = 1.460Parameter Meniscus Curve NumberCCV9 LEV11 CVX12 CVX15 CVX16 CVX19 CVX23Experimentala(°) 118.3 119.7 120 125 129.8 130.7 134•(°) 25 0 23.7 32.7 38.3 3 40“cap’ 0.3006 0.0000 -0.1858 -0.2479 -0.3180 0.0487 -0.1949Model “cap’0 = 60°abased 0.0189 0.0005 0.0024 -0.0440 -0.0831 -0.0904 -0.1070% Diff -93.7 0.0 -101.3 -82.2 -73.9 -285.7 -45.1•based 0.2974 -0.0028 -0.1530 -0. 1732 -0.1833 -0.0250 -0.1854% Diff -1.0 0.0 -17.6 -30.1 -42.4 -151.4 -84.50 = 50°abased 0.1259 0.1057 0.1016 0.1050 0.0020 -0.0042 -0.0302% Diff -58.1 0.0 -154.7 -142.3 -100.6 -108.6 -84.5• based 0.2944 -0.0010 -0.1302 -0.1470 -0.1556 -0.0178 -0. 1583% Diff -2.1 0.0 -29.9 -40.7 -51.1 -136.6 -18.70 70°abased -0.0858 -0.0964 -0. 1031 -0. 1394 -0.1672 -0.1694 -0.1828% Diff -128.6 0.0 -44.5 -43.8 -47.4 -447.9 -6.2based 0.3016 -0.0007 -0.1816 -0.2077 -0.2139 -0.0271 -0.2160% Diff 0.3 0.0 -2.3 -16.2 -32.7 -155.7 10.8160Table A.6 Experiment D - Comparison of Pcap’ ba.sed on a anda=0J56 B= 1.558Parameter Meniscus Curve NumberCCV4 CCV6 LEV7 CVX1O CVX13 CVX17Experimentala (°) 111.5 108.5 111.7 114.7 132.3 134.5(°) -31.7 -14.5 0 20 35 39“cap’ 0.2280 0. 1503 0.0000 -0.2137 -0.3352 -0.2443Model 1’capo = 48°abased 0.1875 0.2197 0.1858 0.1557 -0.0022 -0.0121% Diff -17.8 46.2 0.0 -172.8 -99.4 -95.14based 0.2981 0.1290 0.0000 -0.1014 -0.1335 -0.1359% Diff 30.7 -14.1 0.0 -52.5 -60.2 -44.48=58°a based 0.0978 0. 1276 0.0978 0.0635 -0.0720 -0.0860% Diff -57.1 -15.1 0.0 -129.7 -78.5 -64.8based 0.3028 0. 1451 0.0000 -0.1175 -0.1543 -0.1562% Diff 32.8 -3.5 0.0 -45.0 -54.0 -36.1o = 68°a based 0.0078 0.0377 0.0024 -0.0256 -0.1422 -0.1499% Diff -96.6 -74.9 0.0 -88.0 -57.6 -38.7based 0.2998 0. 1514 0.0000 -0. 1409 -0.1795 -0.1833% Diff 31.5 0.7 0.0 -34.1 -46.4 -25.08 = 78°abased -0.0843 -0.0583 -0.0902 -0.1141 -0.2001 -0.2062% Diff -137.0 -138.8 0.0 -46.6 -40.3 -15.6based 0.2750 0.1514 0.0000 -0.1610 -0.2102 -0.2122% Diff -100.5 -101.0 0.0 -99.9 -99.7 -99.7161Parameter Meniscus Curve NumberCVX23 CCV24 LEV26 CVX27 CVX28Experimentala(°) 142.5 119.5 121.3 121.5 129.846.3 13.3 0 18.2 35.3“cap’ 0.1689 0.3146 0.0000 -0.0973 4).2225Model“cap’o = 48°abased -0.0799 0.1479 0.1337 0.1196 0.0227% Diff -52.7 -53.0 0.0 -222.9 -110.2Øbased 0.1713 -0.1687 0.0013 -0.1230 -0.1613% Diff -201.4 -153.6 0.0 26.5 -27.5o = 58°abased -0.1572 0.0311 0.0110 0.0110 -0.0829% Diff -6.9 -90.1 0.0 -111.3 -62.7based 0.1965 -0.1943 0.0035 -0.1487 -0.1891% Diff -216.3 -161.8 0.0 52.9 -15.0o = 68°abased -0.2203 -0.0866 -0.1072 -0.1125 -0.1689% Diff 30.4 -127.5 0.0 15.7 -24.1Øbased 0.2087 -0.2323 0.0557 -0.1752 -0.2253% Diff -223.5 -173.8 0.0 80.1 1.3O = 78°a based -0.2794 -0. 1998 -0.2200 -0.2200 -0.2587% Diff 65.4 -163.5 0.0 126.2 16.3based 0.2206 -0.2799 0.0582 -0.2067 -0.2752% Diff -230.6 -189.0 0.0 112.5 23.7Table A.7 Experiment E - Comparison of Pcap’ based on a anda0.156 B=1.339162Table A.8 Experiment I - Comparison of Pcap’ ba.Sed on a anda=0.157 B=1.36Parameter Meniscus Curve NumberCcvi CCV2___CCV3 CCV4___LEV5 CVX6___CVX7___CVX8Experimentalcx(°) 123.3 123.5 124.3 124.3 126.5 127.5 130.5 133.5(°) -38.5 -34.5 -22.5 -15 0 6.5 21 24“cap’ 0.4297 0.3562 0.2162 0.1207 0.0000 -0.1294 -0.2309 -0.2535Model“cap’0=43°a 0.1540 0.1483 0.1394 0.1394 0.1141 0.0995 0.0627 0.0399%jjff 64.15 58.36 35.51 -15.52 0.00 176.89 127.16 115.740 0.4817 0.4338 0.2722 0.1709 0.0000 -0.0441 -0.1164 -0.1259%Diff -12.11 -21.79 -25.90 -41.58 0.00 65.93 49.57 50.330=53°a 0.0424 0.0424 0.0310 0.0310 0.0114 -0.0001 -0.0343 -0.0546O/,4jff 90.13 88.09 85.66 74.31 0.00 99.95 85.14 78.460.4917 0.4553 0.3036 0.1973 0.0000 -0.0561 -0.1429 -0.1524%Diff -14.42 -27.83 -40.42 -63.45 0.00 56.66 38.12 39.880=63°a -0.06601 -0.06601 -0.074 -0.074 -0.09455 -0.10197 -0.12765 -0.14591%jjff 115.36 118.53 134.23 161.30 0.00 21.20 44.72 42.440 0.4722 0.4458 0.3201 0.2213 0 -0.06807 -0.16935 -0.1838%Diff -9.90 -25.16 -48.05 -83.33 0.00 47.40 26.66 27.49163Table A.9 Experiment 3- Comparison of Pcap’ baSed on a anda=0.155 B=1.273Parameter Photo NumberCcvi CCV2 LEV3 CVX4 CVX5 CVX6Experimentala (°)(0)cap’Model0=43°a%Diff%Diff0=53°a%Diff%Diff0.108130.210.119722.71-0.0050103.200.14734.85-0.12765182.450.17783-14.87128.8-9.50.1548129.5 130.3 131 130.8 130.8-5.8 0 6.8 15 180.0906 0.0000 -0.0377 -0.1394 -0.2422“cap0.1018 0.0892 0.0829 0.0892 0.0892-12.28 0.00 319.81 164.00 136.830.0711 0.0070 -0.0510 -0.1057 -0.114621.53 0.00 -35.06 24.15 52.68-0.0140 -0.2029 -0.0301 -0.0329 -0.0329115.46 0.00 20.33 76.44 86.440.0894 0.0000 -0.0661 -0.1303 -0.14851.39 0.00 -75.16 6.56 38.69-0.13278 -0.14306 -0.14819 -0.14591 -0.14591246.48 0.00 -292.75 4.67 39.760.10913 0 -0.0813 -0.15739 -0.17522-20.39 0.00 -115.47 -12.90 27.660=63°aO/14jff0%Diff164APPENDIX B - SAMPLE CALCULATIONS AND DERIVATIONSB. 1 Mathematical Formulation for the Negligible Gravity Case for CylindersB.2 Jacobian Matrices for the Spheres Model SolutionB.3 Capillary Pressure Measurement and Sample CalculationB.4 Digitizing Programi) Determination of the Diameter of a Circle from Three Pointsii) Least Squares Minimization of 9 Points on a Circle165APPENDIX B.1. Mathematical Formulation For the Negligible Gravity Case For CylinderszFor parallel, horizontal cylinders, the meniscus between them is described by only one radius of curvaturesince the other is infinite. Assuming negliglible gravity, there is constant mean curvature:=[B 111R1 dx ywhere the upper sign is for concave; the lower, for convex if 4>O.Integration of [B. 1.11 by separation of variables yield±$°dsin() = fL$’dx [B.1.2]• yrorP•a (2Fsin(4)= cap (b—r)I— I [B.1.3]2y \a)Upon introduction of the dimensionless variable pPacap [B.1.4]equation [B. 1.3] then becomes166—.+sin(çb)=—(b—r). [13.1.5]aAfter rearrangement, the equation for pressure as a function of 4) and horizontal separation distance is:— asin()132(b—r) [13.1.6]To relate the pressure to the vertical film thickness between two emulsion droplets, we use:=Ftan(). [13.1.7]drIntegration of this equation yields:b :F(br)drZb_Zr=IXZ=f2 1/2[13.1.8]—(b—r)2The last equation expresses the difference between the level at the meniscus centre and the level at thecontact line. The distance from rupture (or film thickness h) for the upper emulsion droplet (concave case)and the lower emulsion droplet (convex case) are as follows:a) Concave h = a — a cos(a)— (z — z0) [B. 1.9]b) Convex h = a+acos(x)—(z0—zr) [B.1.1OJor in dimensionless form:H=!-i=FFcos(a)±(ZoZc) [13.1.11]167As described in Chapter 3, taking into account hysteresis, the motion of the contact line takes place in twostages. Starting with a level meniscus, the capillary fluid volume is increased. The initial contact angle 3eis measured at the level meniscus. The immersion angle at the start is just the supplementary angle to thecontact angle. This stage is characterised by a fixed contact line on the solid, and the contact angle willincrease until the advancing angle 3a is reached. From this point forward, the contact angle will remain at0a and increasing pressure will cause the meniscus to move up the cylinders. This is the second stage.Partl-constantrcandcz- contact angle (9 changes from 0e to- contact angle hysteresis is the difference between 9e and 0a.GcP=o P*=0(X lr—OeCapillary pressure increases from 0 to P’i, =+a(1—sin(cx)) [B.1.12]asin()2(b—r) [B.1.131168a sin() [B. 1.14]- 2(--a(1+sin(8e)))In dimensionless form, this final pressure at 3a issin(4) [B.1.15]rd21__+(1+SIfl(Oe))]L 2aPart II - contact angle constant at 3a- movement ofmeniscus, a changesa sin(4)2(b—rj [B.1.16]dr =—+a(l+sin(Oa—)) [B.1.17]c2sin()—21+(1+sin(O_.))][B.1.181[2aThe three variables z, x and are represented as:y1 = zY2y3 = x.The derivatives with respect to the arclength s are:.1;=4-=Qsin(4)-=Q.Boi-ody1— Q•cos()x2 sin()‘9:3 x2dy3169APPENDIX B.2 Jacobian Matrix for the Solution of the Spheres Model[B.2.1][B.2.21[B.2.3J[B.2.4]f2 ==Q[Bo.z_”] [B.2.5]thcf3 = - = cos() [B.2.6Jwhere Q is -1 or 1 if the meniscus is concave or convex, respectively.These three equations are the set of functions to be solved. The Jacobian matrix for this system is based onthe derivatives of these functions with respect to the previous three variables:L=O L=Qcos(çj)—sin()170APPENDIX B.3 Capillary Pressure Measurements from Experiments and Sample CalculationsManometer ModeldOevel)eNomenclature:Pr = Pressure above reservoir (atmospheric)Pu = Pressure above test cell meniscus (atmospheric)Pb = Pressure below meniscus interfacep0= density of manometer oilPw density of waterc = level of oil in reservoir (unknown)d = position of interface at the level meniscusa, b = measured levels of the oillwater interface (b corresponds to the level case)e = measured height of the concave or convex meniscusPcap = capillary pressure = Pu - PbCPub*-1-171Level Case:Pu-Pr=p0g(c-b)-pg(d-b)=O [B.3.1jConvex Case:(Pu - Pa)+(Pa - Pr)=O [B.3.2]-iPcap-pg(e-a)+p0g(c-a)O [B.3.3]Pcap = g[ Pw(a - b + d - e) + p0(-a + b)J [B.3.4]iPcap(p0 -p)g(b-a)-pg(e-d) [B.3.5]Concave meniscus:(Pu - Pa’) + (Pa’ - Pc) = 0 [B.3.6]tPcap-pg (e’ - a’)+ p0g (c - a’) = 0 [B.3.7]L\Pcap =g[p(e’ - a’ - d+ b)+ p0 (a’ - b) = 0 [B.3.8JAPcap =- Pw g (d - e’) + (0 - p)g (a’ - b) 0 [B.3.9]In terms of our measured data read from the micromanometer (whose scale is in the oppositedirection), we replace (a - b) with (hlev - hcp,,) which is negative for convex menisci and positive forconcave menisci.The capillary pressure is comprised of two terms. The first term is the larger one, and the secondterm due to the hydrostatic height above the level meniscus is smaller.172Sample Calculations:First Term, the measured pressure Pm,Pm=(0.735)*981*(2.54)*(h- hiev) [dynes/cm2] [B 3.10]where h is in inchesdimensionless [B.3. 11]Second Term, the hydrostatic contribution,APh’ = (0.9982)*981*(ed*)*(a)/(2*72.75) dimensionless [B.3.12]The dimensionless capillary pressure is then the sum of the two terms:APcap’=z\Pm’ + APh’ [B.3.13]The calculated apparent contact angles for each meniscus were determined using the following equationsand the measured values forConvex 6 = it - a + [B.3.14jConcave 6= it-a f3 [B.3.15j173APPENDIX B.4 Digitizing ProgramI) Determination of the Diameter of a Circle from Three Points(X2 ‘“2(x1 ,y1)(X3 ,)The equation for a circle whose center is at (xe, Yc) and has a radius of rc:expanding into the form(x—xj2+(y—yj2=,2 [B.4. 1]-(x2+y)=ax+by+cthe equation of the circle becomes:2 2 2 2 2—(x +y )=(—2.x)x+(—2.yjy+(x +y —re)Knowing three points on the circle:ax +by1 +c=—(x12+y12)=Rax2 +by2 +c = —(x2+y2)=[B.4.2J[B .4.3][13.4.4][B.4.51ax3+byc=—(x2+y32)=R [B.4.61174Using Cramer’s rule for solving three simultaneous equations using the determinant method:the detenninant of the coefficient matrix:x1 y1 .1D= x2 y1 1 [B.4.7]x3 y3 1R1 y1 1Da 1?2 Y2 1 [B.4.81J?3 y3 1Dwhere a =DSimilarly for b and C:x1 R1 1x2 R2 1x3 J?3 1 [B.4.9]D Dx1 y1 R1X2 Y2 R2x3 y3 [13.4.10]D Dsolving these determinants:Da =RIV2_RzV’I+R3[1’i B.4.11]Da=Ri(Yi_Y3)(Yi+R [B.4.12]175x2 1 x 1 x 1Db=—Rl‘—R3 ‘ [B.4.13]x3 1 x3 1 x2 1Db=—R(x—x)+(xI xl [B.4.14]D=R1(x2y3x)—R+ —x2y1) [B.4.15]D = x1y2 —y1x2—(x1y3—x3y1)+ (x2y3 —x3y2) [B.4.16JThe radius of the circle is r, and the center coordinates are (xe, Yc)a=—2•x0 [B.4.17]b=—2•y [B.4.18]c=(x+y—r2) [B.4.19](a2 (b2 2c=—) i—) —r [B.4.20]4c=a2+b2 —(2rj2 [B.4.21]and the diameter of the circle is twice the radius:2r0 = g2 +b2 —4c [B.4.22J176ii) Least-Squares Minimization of Nine Points on a Circle(x—a)2+(y—b)2=r2 [B.4.23]Ypred =b±gr2 —(x—a) [13.4.24]where the upper sign is for the upper hemisphere of circleOBJECTiVE:Find parameters for the least-squares fit of digitized points to the circle equation. The termsa, b, r are estimated using the three-point method outlined earlier, and used as initial guessesThe minimizing function is:s = (yred_j2 = — a)2 —y) [B.4.25]of which[B.4.26]By Newton’s method of least-squares minimization, we can obtain more accurate values for a, b, r.We solve for three simultaneous nonlinear equations in matrix form:1771a2s a2s d2S1 Ia2s a2s a2s I asi [B.4.27]Idbda db2 t9bdr idbiaiab dadr IjIa2s a2s a2s Iasdrab dr2 1dmThe derivatives of equation [B.4.25] are:_1121= 2[b +(r2 —(x _a)2”_ye][.(r2 —(x _a)2) j{2(x —a)] 0 [B.4.28]2= 2Eb+((r _(x_a)2)1’ Y)j = 0 [B.4.29]db L—1/2’]-= 2{b +(r2 _(x_a)2) _ye][T((r2 _(x_a)2) )j = 0 [B.4.3011/2 1-1F 2(x—a) (x—a)(b + (r2 — (x— a)2)”2 Ye) 2[b + (r2 _(x_a)2) Yej= [(r2 _(x_a)2)— (r2 _(x_a)2)31 — (r2 _(x_a)2) j[B.4.3 1]d2S= 2[(r2— (x — a)2 )1/2 (x — a)] [B.4.32]daab3/22 2\h!’2F[(2_(x_a)2)1(x_a)j+2[b+(r-(x-a)) _ye][_r(r2 _(x_a)2) (x_a)jjt9adr L I.[B.4.33]178t92S=2 [B.4.34]d2S — 2(x—a)22[B.4.35]dbda(r2_(x_a)jd2S 2r2\hl’2[B 4.36]db(r2_(x_a))d2S 2(x—a)r 2r[b +(r2 — (x_a)2)”2_ye](a)drda = (r2 _(x_a)2) — (r2 _(x_a)2)3”[B.4.37]d2S 2r2’)hl’2 [B.4.38]drab (r2_(x_a)d2S 2r2r2[b+(r2 (xa)2)U2 Yej + 2[b+(r2 _(x_a)2Y’ Ye]= (r2 _(x_a)2)—(r2 _(x_a)2)31 (r2 (xa)2)u’2[B.4.39]Variables used in the program:D1 = x — a [B.4.40]D2=(r2_(x_a)2)=r2D12 [B.4.41]D3 = [B.4.42]179D4 = D23’2 [13.4.43]D5=—y+b+D3 [13.4.44]Augmented matrix (AM) coefficients:N r2.D12 D12•D5 2D51AM(11)=[D2 — D4[B.4.45]AM(1,2) 2•D1= [13.4.46]D3AIv[(1,3)N 2r•D1 2r•D5•D11=[ D2 — D4 j [13.4.47]AM(2,1) 2•D1= [B.4.48]D3NAM(2,2) = [B.4.49]i= IrAM(2,3) = [B.4.50]N r2r.D1 2r•D5•D1lAM(31)=[D2 — D4 j [B.4.51]N 2rAM(3,2)=[B.4.521=2rD5 2D51Alvf(3 3) +— I [B.4.53]L12 D4 D3]180181APPENDIX C - COMPUTER PROGRAMSCYLINDER MODELCALP - Meniscus ProfilesCBO - Rupture Pressure vs. Bond numbersCDI - Rupture Pressure vs. Separation DistanceCHYS - Hysteresis PlotsSPHERES MODELSPALP - Meniscus ProfilesSBO- Rupture Pressure vs. Bond NumberSDI - Rupture Pressure vs. Separation DistanceSHYS - Hysteresis PlotsC****fl****fl*S*5*CMeniscusProElesbetweenTwoParallelHorizontalCCylindersasaFunctionofPositionCCALP’-dimensionlessC C C CTheshapeofthemeniscuschangesasonecausestheCtheinterfacetomoveupalongthecylinder.AsPCincreasesthecapillaxypressurealsochanges.CThecriticalpointofruptureisalsodeterminedforCbothconvescandconcavemertisci.CIMPLICITREAL8(A-H,O-Z)PARAMETER(tw’9,N”20,MPM+2)DIMENSIONX(M,N),Y(M,N),XC(N),YC(N),Q(N),V(N)DIMENSIONPCAP(MP),HTci),FX(2,N),FY(2,N)EXTERNALFCNLOGICAL12CHARACTER4OarlTLl,crrrrL2,XnTL.YrITLCOMMON/PARJC,Dl,Pl,PCR,WCOMMON/PAR2/PHI,PR,RCOMMON/CATIPC,Pl,DPCOMMONICORDIXMIN,XMA)cYMIN,YMAXCOMMONJPLaT/GTrrLl,GrrrLzxlTrL,YTITLCOMMONIINPISPT,SDCOMMON/CIRCIXC,YC,FX,FYCOMMONIZERO/AC)BCYCOMMON/PRESS/PCAP,MXC CGraphheadingsactCG1’I’I’L1—MeniscsasProfileScDTrL2=’ForParallelCylindeisS’XlIThcJaYTfl’L’zIa’C CinitializeconstantsandpropertiesCR-0.156D0CDI.lll6D0CBD1/(2.D0R)+I.D0B=l.50300DI”(B-1.D0)2.D0*RCSFr=45.D0SFr—72.75D0G—981.D0CC—(.59600)GISFr0-13.4600W’=CRRCDP-5.dOP0—IlO.D0PC-66.DOPI=DACOS(-LD0)PCR,.PC*PI/l80.D0R.AD=P11180.D0EPS—1.D-6IR=lNF=0DO30I’I.MlM=I-l11=135P”PO+DPDBLEQhPR’P5RADC CIfpositionisabovethecritical value,themeniscuscisconvex,ifitisbelow,itisconcave.CIF(PR.GT.ACR)THENPiII=PR+PCR-PIIDX-lQC-IDOELSEIF(PR.LT.ACR) THENPIII’=PI-(PR+PCR)IDX=0Q0--l.D0ELSE ZO-0.DOPHI-0.D0PCAP(IQ—0.D0MX=M-lCIMlGOTO40ENDIFC CSolveferFCNtodetennineTforthespecjfiedCcontactphiTisgreaterthanobutlessthanl.C 50Ul-0.0000000lDOU2-0.99999999D0CALLZEROI(tJl,U2,FCN,EPS,LZ)CT=tJlCDeterminethedimensionless capillatypressurefor themeniscusCIF(PR.NE.ACR)THENZ0-DSQRT(4.D0/W(l.DOfr-l.00))TX”Dif(ZOR)TY=C5Z001RIF(IDXEQ.1)ZO=-ZOIF(lREQ.2) THENllM+NF+lPCAP(Il>W5ZOf2.D0ELSE 11=1PCAP(ll)—WZO/2.D0ENDIFENDIFMXMC CPrintresultsintableCWPJTE(6,5)5FORMAT(s)c’CaplllasyPressureEffectsasInterface’,+IX,PositionChanges)wRITE(6,7)PC,W,B7F0RMAT(/s)c’C.A.=’,Fs.l,’Bo’*’,F5.3,’b/r’,F5.3)WRITE(6,15)15FORMAT(J46XPCAF,7c2a,73c’PHp,73cpf,)C CThecnticalpointatwhichthelevelnieniscusoccursCisatACR.CACRPI-PCRC CIncrement PRtofollowprogressofmeniscusaroundCcylinderCC CCalculatefIlmthicknessCZC—QC5DSQRT(ZOZGI-2.D0/W(1.D0.COS(PHI)))HC=DABS(ZO-ZC)HRDABS(l.D0+QC*DCOS(PR))H=HR-HCHT(1)=HIF(IR.EQ.2)GOTO60IF(IREQ.1AND.NT.LT.2)THENHO-Fff(I-l)IF(IEQ.1)HO=HT(l)C CChangeinsigns, signalsSlitsnipturepointsCIF(HO*HUT.0.00)THENIR-2GOTO4025HNEW-HT(l)HOLD—HT(I.1)Pl-’PHI.QC5DPRADP2—Pm55PHI-HNEW(P2.Ply(HOLD-HNEW)+P2PR-Pl-PCR+QC*PHlG0’IO5060IF(DABS(H) .GT.EPS)THENIF(IS5HNEW.GI’.aDO)THENP2-PHIHNEW-HELSE HOLD-HP1—PillENDIPGO’IO55ELSE NF-NF+IIR-0H-O.D0lff(l)-HENDIFENDIFENDIF40PHPIlT/RAD182WRITE(6,99)PCAPtH),Z0,PH,PR/RAD99FORMAT(lx,2F10.5,2F10.3)WRiTE(6,)FILMH’,HC CGraphthenreniscusprofilebetweenthetwoCcylindersindimensionlesscoordinatesCCALLMENISC(PHI,ZO,T,W,Q,V,N)IF(IEQ.1)THENC1=Q(1)+DSJN(PR)ACX=Ct-DSIN(ACR)ECY=-DCOS(ACR)DA=Pl/DFLOAT(N’l)DOIOK-I,NA=DA5DFLOAT(K-l)XCQQ=Cl.DSIN(A)YCQQ=-DCOS(A)10CONTINUEYMIN-YC(l)YMAX-’YC(N)ENOIFC CAllothernseniscihavelobesbiftedtobeshownContheaaanecylindemCZC=-DCOS(PR)YD=V(1)-ZCC CStoremeaaiaciprofileinmatricesCDO20J—l,NIF(Ill..EQ.0)THENFx(NF,frQQ)FY(NF,frV(J)-YDELSE X(IM=QQ)Y(I,J)=V(fl-YDENDTF20CONTINUEIFOR,EQ.0)ThENOELH=FYtNF,N)-BCYGOl’O33ENDIFDELH=ECY-Y(I,N)33PHS=WDELHI2.D0PTOT=PCAP(I)+PHSCWRITE(6,97)DELH,PHSYTOT97FORMAT(lXDELH—‘,E15,3,’PHS=’,E15.3,’FTOT=cFlO.5)IFOR£Q.2)GOTO2526IF(NF.LE.2)lIt—IIF(Y(I$)CI’.YMAX)YMAX—Y(I,N)IF(Y(I,N).LT. YMIN)YMIN-Y(I,N)30CONTINUEC CDeterntinethelimitsofgeaphingtakingintoCaccountthatthewholeprofileandbothcylindersCwlllbedrawn.CXMAX-ClIF(YMAXOF.3O.IAX)XMAXYMAX9S.DWl7.D0IF(XMAX.01’.YMAX)YMAX-XMAXI7.D0/15.D0XMIN=’-XMAXCALLMYPLOT(Xj,M,N)STOPENDC CFUNCTIONFCN(I)C‘C CThisfunctiondescribesthehorizontalwidthofCthemeniscosandisusedtoththespecificationsCoftheseparationdimensionDlandthecontactCgeometryPHI.CIMPLICITREALS(A-H,O-Z)COMMONIPAPJC,DI,PI,PCR,WCOMMONIPA1L2DçA,RCOMMONIOUI’IZOFX-DELIPIC(I’T,IND)E—DEUPE(TT,IND)FCN=2.O0’T/DSQRT(W)2.D0/(I’fl)-l.D0)tF2C-+DELLlF(Y,tPl3Qf2.D0,IND))-÷(2.D0f(Tfl))(E’DELLIE(r,(PI’2QIZ.D0,IND)))+-(DIJR÷2.DO(I.D0-DSIN(A)))ENDC C CMeniscasProfilebetweenTwoParallelCPlates orHorizontalCylindersC—---——————C CDeterminestheshapeofthemeniscusbetweentheCIwocylindersCStJBROIJDNEMENISCtfl,,TflD,1D,N)IMPLICITREALS(A-H,O-DIMENSIONXD(20),ZD(20)COMMON/PAItJC,D1,PI,PCR,WC C CDeterminemeniscras profilebysolvingequationsforCanglesftomPRtoo.CBT-P/DFLOAT(N)PH=PDO10I—l,NANG=<PI-PH)f2.D0F—DELIPK(T5T,IND)E=DELIPE(PT,IND)F1=DELLIF(l’,ANG,IND)El=DELLIE(T,ANG,IND)XD(I)-(rt((2.DOf(T51)-l.D0)(F’FI>+(2.D0ffl)(E’El)))/DSQRT(B)IF.Lt0.D0)THENZD(fl—(DSQRTtZZ+2.D0fE°O.D0DCOStPN)D)ELSE ZD(fl=IDSQRT(ZZ+2.DOIB(l.D0’DCOS(PHX))ENDIFPH=PH-BT10CONTINUE20RETURNENDC Caa**aa*aaaa.aa.aaaa*aaaaaaaa.aa...aa*saC•DISSPLAGraphingRoutineC CSUBROUI’INEMYPLOT(XI,X2,M,N)C CThissubroutine plotstheaneniscasprofilebetweenCtwoparallelhorizontalcylinders(dimensionless)CIMPLICITREAIfl(O-Z)PARAMETER(lM’9,lN—20,MP—at÷2)DIMENSIONXl(MX),21,N),X3(IN),X4(IN),SX(2,IN),SY(2,IN)DIMENSIONPCAP(MP)REAL4U(IN),V(IN),YON),Z(IN),U1(IN),Y1(IN),APOM)*4CASW)flON)flON)YXWN)CHARACI’ER40GflTLl,GTITL2,2GTfl,,YTITLCOMMON/PAR,sqSD,Pl,PCR,WCOMMONIPAR2/PIII,PR,RCOMMoN/C0RD,xMncxrciAicYMlN,YMAxCOMMON/PLOTIGflfll,G1Tlt2,XTfl1,fllThCOMMON/CAT/Pcl,TPCOMMON/ZEROI3OcTYCOMMONICIRC/X3,X4,SX,SYCOMMON/PRESSIPCAP,MXC CDefinestheborders,titles,heading,etc.CCALLDSPDEVQPLOT)CALLUNITSQCEN’lCALLNOBRDRCALLPAOE(21.,26.)CALLAREA2D(14.0,17.)CALLHEIGHT(0.30)CALLGRACE(.0)CALLSCMPLXCALLMXIALF(STANDARD’;&)CALLMX2ALF(LICSTIY,’jCALLMIGALF(GREEIC’,’l)CALLMX4ALF(INSTRUCTION’,’@)CALLMXSALFçLfCGR’#’)CALLPMYSOR(3,l.)CALLYAXANG(0)CALLFRAMEC CConvectdatatosingleprecisionvaluesCAX-SNGL(XMIN)RX=SNGL(XMA2QAY-’SNGL(YMIN)BT-SNGL(YMAX9,IDO)183C C C CDP=SNGL(TP)Dl=SNGL(SDIR)BO=SNGL(W)Reflectsemi-profilearoundY-axistoobtaincompleteprofileCX=SNGL(XMIN-XMAX)OX=(BX.A5C)’5.D0DY-(BY-AYflDOCCALLA5cSPLT(XBXl4.0,ORlGSTEF,A)CCALLAXSPLT(AY,BY,t7.,YOR,YST,B)CALLXNAME(XTITL,14)CALLENAME(fl1TL,17)CALLGRAF(AX,’SCALE,BX,AY,’SCALE’,BY)CALLHEADIN(OTITLI,16,3,2)CALLHEADIN(GTITL2,22,3,2)CALLHEIGHT(OMCALLSCLPIC(O.7)C CCALLXPCKS(2)CALLYTICKS(2)CCALLXGRAXS(olUG,SThP.B)ct4.,XTITL,4,0.,0.)CCALLYGRAXStyOR,YST,BY,t7.,YITIL,l,0O,O.0)CALLHEIGHT(O25)C CPlotinterpolatedlineCDO20I—l,MXAPI3)SNGL(PI)CAP(B-SNGLtPCAP(B)DO1OJ=t,NU(fl-SNGL(XttI,t)V(frSNGLIX2(lJt)Y(J)=SNGL(X3Q))ZØ=SNGLIX4(J))ul(3)=SNGL(-Xl(I,I))YIQ)-SNGL(-X3(J))IF(I.LE.2)THENCAP(M+I)-SNGLlPCAP(M+Il)FXØ=SNGL(SX(I,J))FY(J)’-SNGL(SY(lJ))FXIØ=SNGL(.SXtl,I))ENDIFIF(I.EQ.I)THENWRITE(6,98)V(J),Urn,Ut(B),Z(s),Yrn).Yt(B)ELSEWRITE(6,’)WRITE(6,99) V(ThIJ(I),UlQ)ENDIFFORMAT(1X,6F10.3)FORMAT(1X,IFlO3)CONTINUECALLMARXER(I)CMeniscusprofilebyreflectionCCALLCURVE(tJI,V,N,0)CALLCURVE(BV,N,0)C CFilmrupturemenisciaredottedlinesCIF(ILE.2)THENCALLDOTCALLCURVE(FXFY,N,0)CALLCURVE(FXI,FY,N,0)CALLRESETCDOT)CALLRLREAL(CAP(IW-l),3,-Q.2,FY(N)+0.03)ENDIFCALLRASFLN(2)C CDrawbothcylindersbyreflectionof curvesCandprintcaplllaaypressureforeach curveCCALLCURVEIYtAN,0)CALLCURVE(Y,Z.N,0)IF(CAD))EQ.0) GOlD20CALLRLREAL(CAP(B,3,-.2,V04)+êO))Ft=Pl+DP20CONTINUEC CDrawstraightlineapecifjinglocationof Pcar0C CCALLDASHACX=SNGL(XX)BCYSNGL(YY)CALLRLVEC(-ACX,BCY,ACX,BCY,0100)CALLHEIGHT(03)CPointcontactangledistancefromcenter ofCcylinder torenterofasrnisrus(B.dimensionless),andBondnumberCCALLHEIGHT(030)B=t.+Dl/2.CALLRLMESSQBo—‘,4,-B’0.95,0.3)CALLRLREAL(BO,3,-B’0.00,0.3)CALLRLMESS#q.F,4,B*.95,0.)CALLRLREAL(FC,1,-B’0.I5,0.)CCALLMESSAGD/R-‘,5,t.0,7.)CCALLREALNO(B,2,247)CALLRLMESSQbIa‘,6,-B.95,-3)CALLRLREAL(B,2,.B’.75,-.3)CALLHEIGHT(0.t5)CALLRLMESSQo’,I,-B’.62,0.09)CALLENDPL(0)CALLDONEFLENDC‘BondvsPcap(dimensionless)TwoHorizontalCylindersonanInterface*CBOTodetenninetherequiredcapillarypressuretocausethemenincusleveltoreachtheheightofthrcylinderasaflnactioa’rofBondmother.Bothconvexandconcavefonts maybeobtained. ItplotsBondvs.Peaperafimetionvs.anyvariable.IMPLICITAL’fl(A-M,O-Z)PARAMETER(M-5,MD-3%N-20)CIIARACI’ER’40GTIThI,GDTL2,XT1TL,Tl]TLCHARACTER’15GPCHARACTER’4PCL(M)CHARACTER’3DUMCHARACTER’tDlCHARACrER’2D2CHARACI’ER’4SPQ,t)DIMENSIONX(M,N),YQ,tN),Q(N),V,)OC(2)DIMENSIONXC(M,N),YC(MJ,PCAPQ4MD),D(M)CQ4TBO(MD)DATAltIlOl,RHO2,RNO3/22D0,l.594D0,0.99g2D0/EXTERNALFCNLOGICAL12COMMON/PARIC,DI,PI,PCR,WCOMMON/CORD/XMIN,XMAX,YMIN,YMAX,SICOMMONIPLOT/GTITLI,GTITL2,X’UETJTITLCOMMON/CIRC/XC,YC,PC,ICCOMMON/OUr/ZZ,T,PACOMMON)INP/SFI’,SD,R,TIXI’COMMON/SHAPE/lOXCOMMONtLEGIPCL,GP,SPCOMMON/PRESS/PCAPCOMMON/BDIM/BSettitlesandlabelsforgraph.Twooptions:Onegraphsaneniscusprofile(0)whereastheothergraphsaflmctionva.onevariable(1,2).IfITXI’is1legendisrespect tocontactangIeotherwisehisD/RIflOX-0,thenconvexmenincusdesired.ND-IIDX-0nr-oIF(ND.EQ. 0)THENGTIThI-’MENISCUSBETWEENTWOPARALLELG1TIL2-tCYLINDERSONANINTERFACEXflTLX/Ponm.nELSEIF(IbID.EQ.1)THENCHTFLI-’PL4NSCENDENTALF1JNUDONOFPHI’IF(ITXI’EQ.0)TIIENGTITL2-’ATVARIOUSD/R’ELSE Gl’flt2tATVARIOUSCONTACTANGLES’C C C C C C C C C C C C C C91 99 10 CC C C C C C C184ENDIFVO0.DOZO-ZZYflTL=”MeniscusFittingFunction’DP30.D0PR’PAXrm,’’Phi-Contactmeniscusangle’DB65O.DO/DBLE(MD-l)P=PR/RAI)ELSEIF(IND.EQ.2)THENDO40I-1,MGTITLI=’RUPFURECAPILLARYPRESSUREVS.’PC(I)=PCI+DPaDFLOAT(Il)PPACSFr’ZO’RJIO.DOGTITL2=’BONDNUMBERANDCONTACTANGLE’PCRPC(I)*P,ADPCAP(I,J)-DABS(W’ZO/(2.D0))YflTL=’DimersienlessPressurePnip’WRITE(6,’)’WRITE(6,25) W,PHI/RAD,P,PCAP(IJ),PPACYmV’PREssUREPCAP(ParWRITE(6,98)PCQ)25FORMAT(IclElO.3,2FI03,2FI5i)IF(tXT.EQ.I)THENCIF(INDEQ.0)THENXTITL-’BondNumberBo’CConvertcontactanglesintocharactershingsforELSECuseaslabelsinlegendCALLMISC(PHL,ZOj,W,Q,V,N)XTITLContactAngle(degyCC1(l>fDSIN(PR)ENDIFNPT’NINT(PC(I))Sl-V(I)+DCOS(PR)ENDIFCALLB’m(Nvr,DUM,3,ND,”))CD-(CI.Q(1))/20.D0CPCL(l>DUMu/’S’DA-P1/(DFLOAT(I’t-l))CStoretextforplotlegends,choiceofvaryingCDO10fl—INCcontactangleorD/RCIncrementBondnumbersWataconstantBA’DA’DFLOAT(ll-I)CCXQ.lI)-Q(II)IF(OTXTEQ.1)THENDO30J=l,MDY(3,fl>V(Il)GP—’ContactAnglerS’C’Cl+DB’DBLE(J-I)XC(3,1l)-Cl-DSIN(A)ELSEWC’K’RYC(3.ll)—Sl-DCOS(A)GPDistanceB/KS’THO(JP-W10COWONTJEENDIFCCCCCDetcmiinethemaximumXandYvaluesforCInitializecylinderradius,separationdistance,CDeterminecontactPHIbyrootfinding.ThereareCadjustingthelimitsoftheplotCcontactangle,andphysicalpropertiesCtworootsofwhichoneiscorrect;errorcriterionCCCandsignofZZshouldbecheckedIF(1EQ.1)THENR0.lD0CXMlN-X(1N)SFr—45.lD0Ul=0.000XMAX’.ClPCI-30.00O1D0U2PI/2.D0YMIN-YC(J,I)G98l.D0CALLZEROI(U1,U2,FCN,EPS,LZ)YMAX-YQN)PI=DACOS(-l.DO)PHI=UlENDIFRADPI/l80.D0IF(IDX.EQ.0) THENIF(Clar.XMAX)XMAX—ClSD-R1102-RHO3IF(ZZ.LE.0.D0)GOTO8IF(YC(3,N)ar.YMAX)YMAX-YC(3,N)C=0.1DOELSEIF(X(J,N).LT.XMIN)XMIN=XQN)CI=CU2’-(PI-PCR)0.999D0IF(YCQ,l).LT.YMIN)YMIN=YC(3.l)EPS=l.D-5IF(ZZGE.Otto)00108ENDIFYMAX-O.D0ENDIFIF(IND.EQ.1)THENYMIN-0.D0IiUl-I,0003D0’UIDO20K-lNDI-0.4D0U2-P112.DODQC)-XI/RADB’.-DIf(2.D0R)+1.1)0CALLZEROI(Ul,U2,FCN,EPS,LZ)Y(I,IQ-FCN(XI)CIF(NOT.12)ThEN)U-X1+0.ISDOCPrintheadingsandotherinfoforprintoutWRITE(6,’)‘FALSE’IF(YQ,K).LT.VO)THENCGOTO11YMIN—Y(3,K)WRITE(6,5)ENDIFVO-YMIN5FORMAT(15)ç’RUPl,JREPRESSUREVS. BONDNUMBER)IF(IDX.EQ.O.D0)THENENDWWRflE(6,12)U2’(PI.PCR)’O.999D0IF(Y(I,K)ar.VS)THEN12FORMAT(30)‘CYLINDER’,i)WRITE(6,’)‘#66’YMAX—Y(3K)WRITE(6,6)BIF(ZZar.O.DO) aro11VS-YMAX6FORMAT(6X,’SeparationDistanceb/a-‘F5.I,))ELSEENDIFWRITE(6,7)IF(ZZ.LT.Otto) GOTO1120CONTINUE7FORMAT(6X,‘Bo’,8)çPHI’,8)ç’A’,12)çPCAr’,8)çENDIFENDIF+‘PCAP[PaJ)PHI’tJlIF(IND.EQ. 2)THEN98FORMAT(3X,’C.A.=‘,FI0.3)IF(IDX.EQ.O.DO)THENCCIF(ZZar.O.D0)GOTO11CDeterminemax,andrain.XandYforthis plotCIncrementcontactangleandapplyanalysistoobtainELSECCresultsforeachcaseIF(ZZ.LT.0.1)0)001011IF(IEQ. IAND.!EQ. 1)ThENCENDIFYMIN-PCAP(l,I)rr=la7diAX.’PCAP(l,l)Vs=0.DO8XI’-O.DOELSE185YMINMINlPCAPtl,J),YMtN)TIvtAXMAX4PCAPtU),YMAX)ENDIFENDIF30CONTINUE40CONTINUEIF(INDEQ.2)THENIF(ITXTEQ. 1)THENXMIN=TBO(I)mom)CALLMYPLOT(rBO,PCAP,M,MD)ELSE nxriXMJNPC(l)XMAX”PC(M)CALLMYPLOT(PC,PCAP,M,M)ENDIFENDIFSTOPEND***.*.*fl.FUNCTIONFCNThetranscendentalequationinphiwhoserootmustbedeterminedFUNCTIONFCN(X)IMPLICITREAL’58(A.H,O-Z)COMMON/PARIC,DI,PI,PCR,WC0MMON/oUrlZoj,AC0MMONIINPISFr.SD,R,rr)crCOMMON/SHAPE/IDXCOMMON/BDIM/BC************C CIfIDXis0thenthecapillasyisconvex,otherwiseCitisconcaveCIF(IDX.EQ.0) THENA=PI-(PCR-X)ZO-(1.D0.DCOSQQy(W°(l.DG+DCOS(A)))++0.5D05(I.D0+DCOS(A))ELSE A”PI-(PCR+)QZO=(l.D0DCOSQç)ftW*(l.D0DCOS(A))).+05D0(l.D0-DCOS(A))ENDIFT”4.D0/(WZOZOl-4D0)FXDELIPK(TT,JND)EDELIPE(I5T,IND)FCN2.D0T/DSQRTtW)((2Do/(I1>l.D0)5(FX-+DELLIF(T,(I’I-X)/2.D0,IND))-+(2.D0/)(E.DELLIE(r,(PI.)QfLD0,IND)))+-2.D0(B-DSIN(A))10ENDC C CMeniscusProfilebetweenTwoParallelCPlatesorHorizontalCylindersC C CThisprogramdeterminesthegraphicalprofileCpointsofthemeniscusatagivencontactPCandmeniscusdepthLCSUBROUTINESC(P,Z,T,B,XD,ZD,N)IMPLICITp,*j*8(A4{,Q)DIMENSIONXD(50),ZD(50)COMMON/PARJC,DI,PI,PCR,WCOMMON/SHAPEIIDXC C CDetetmine sneniscusprofilebysolvingequationsforCangIesontPRto0.CBTP/DFLOAT(N)DO1OI=l,NANG—(PI.P)/2.D0FDELIPKCrT,IND)EDELIPE(T*T,IND)Fl=OELLIF(r,ANG,IND)EIOELLIE(r,ANG,tND)XD(rtçr5((2.D0f(rT)-1.Do)(F-Fr)-+2.Dof(r1))5iE-E(I))/DSQRTtB)IF(lOXEQ.0)THENZD((I=-(DSQRT(Z*Z+2.DOIB5(l.D0-DCOSQ’))))ELSE ZD((l=(DSQRT(Z5Z+2D0fB(1.D0-DCOS(I’))))ENIJIFP-’P-BT10CON1INUE20REI1JRNENDC C0*5***C•DISSPLAGraphingRoutineC CSUBROOJIINEMYPLLYT(XI,X2,M,N)C CThiseabroutinegraphsthecaplllasypressureCvs.theBondnunrbetC CIMPLICITREAL8(O-Z)PARAMETER(lM-5,IN-30)DIMENSIONXl(N).X2(l1,N),PC(IM)ppJ,4U(IN),V(IN),Y(IN),Z(IN),A(IhQ,IPACIC(2000)CHARACIER40G1TILI,GTITh2,XrITL,Y’TITLCHARACrEPl5TLEGCJ{ARACTER*4PCL(I0,CHARACTER4SP(IM)COMMON/CORD/XMIN,)OciAXYMIN,YMAX,ZOCOMMON/PLITr/GTITL1,GITrL2,XTITL,’rrrrLCOMMON/LEG/PCL,TLEGSPCOMMON/INP/SFr,SD,R,mcrCOMMON/PARJrC,TDI,PI,PCR,WCOMMON/BDIM/IBCOMMON/SIOAPE/IDXCOMMONfSHAPE/rnXDefinestheborders,titles,heading, etc.CALLDSPDEVçPLOT’)CALLUNDSCCENT)CALLNOBRDP.CALLPAGE(21.,26.)CALLARE.A20(lS.0,l7.)CALLHEIGIflXO.25)CALLGRACE(0.0)CALLCOMPLXCALLMXIALFCSTA}IDARJY)CALLhALFcIJCsTD:,CALLMX3ALFGREEK’,’tCALLMX4ALFCSPECIAL’,’@CALLMX5ALFCLICGR’,’#)CALLPHYSOR(3.,l.)CALLYAXA14G(0)CALLTHIFRM(0.025)CALLTHKFRM(0.05)CALLFRAMEC CConvertdatatosingleprecisionvaluesC C CAX”SNGL(XMIN)BX-SNGL(XMAX)AY-SNGL(YMIN)BYSl’0GL(YMA)QDX-(BX-AX)/3.D0DY=(BY-AV5.D0CW-SNGL(W)CB-SNGL(IB)CALLAXSPL1AXBXI50,ORlG,sTEP,A)CALLAXSPLTtAY,BY,l7.,YOR,YST,B)CALLGF(AcDX,BX,AY.DY,By)CALLHEADIN(GflTLI,29,2,2)CALLHEADIN(GTITE2,29,2,2)CALLHEIGHT(0.3)CALLSCLPIC(0.1)CALLXIICKSCZ)CALLXGRAxS(ORIGSTEP,B3cI5,xITrL,3o,o.,o.)CALLYGRAXS(YOR,YST,BY,l7.,YflTL,30,0.0,0.0)C CPlotinterpolatedlineCD020l—l,MDOlOi’l,NIF(IEQ. I)THENU(J)SNGL(XlQ))ENDIFV(J)”SNGL(X2(I3))C C CC C C C C C C C C18610CONTINUEPI=DACOS(-1.D0)CALLMAR.XER(1)CCALLRASPLN(3)CThresholdRuptureCapillaryPressuresasaR.AD”PI/tSO.DOCALLCURVE(U,V,N,1)CFunctionofSeparationDistance(Cylinders)BI=l.000010020CONTINUECCDF•dimensionlessvariablesEPS-1.D-SCALLHEIGHT(0.25)Crn-iCCNF-0CWritepropertiesongraphCMX=MCCThesupturecapillarypressuresarecalculatedfortheCCALLMESSAGCb/a-‘,5,I0.l,16.2)CparafleihorizontalcylinderscaseastheseparationCPrintresultsintableCALLREALNO(CB,2,l1.2,16.2)Cdistance isvaried.Thesevariablesarethengraphe&CIF(IDXEQ.0)THENCThesecurvesaredeterminedforaconstant BondnumberWRITE(6,5)CALLMESSAGCCONVEX’,6,6,17.5)Candcontactangle.Convexandconcaveresultsmaybe5FORMAT(5X,CapillaryPressureEffectsasDistance’,ELSECobtained.+1c’Changes[Cylindetj)CALLMESSAGCCONCAVE,7,6,17.5)CWRITE(6,8)WENDIFIMPLICITREAL8(A-H,O-Z)7FORMAT(/4XC.A.-‘,F5.l,/)IF(lOXEQ.0)THENPARAMETER(M-5,N-lO,MP=M+2)8FORMAT(ll,23c’BO-‘,El0.3)CALLMESSAGCCONVEX’,6,6,17.5)DIMENSIONXt),Y(M.lCC,YC(N),Q(ll),V(N)WRITE(6,15)ELSEDIMENSIONPCAP(M,N),}IT(M),FX(2,N),FY(2,N)15FORMAT(16X,’b/a‘,7c’PHr,73ç’A’,1lX,PCAP5’,9cCALLMESSAGCONCAVE,7,6,l7.5)DIMENSIONPC(lvt),D(l)+PCAP[PaJ)ENDIFEXTERNALFCNDO100K-l,MCLOGICALLiPC(K)-PCI+DCDFLOAT(K-I)CPreparelegendblockCl{ARACTER1$GPDP=PC(KYDFLOAT(M)CCRARACTER4PCW4)PCR-PC(K)RADMAXLIN-LINEST(lPACK,2000,26)CHARACTER3DUM0030l-l,MCHARACI’ER40GUrLl,GTITL2,XflTL,YnThWRiTE(6,7)PCQQIF(IXT.EQ.1)THENCOMMON/PARJC,DIJ’l,PCR,WNPT-NINT(PC(K))CALLLTNES(PCL(I),IPACIçt)COMMON/PAR2/PHI,PR,RCALLBT04PT,DUM,3,N0,’)ELSECOMMON/CAT/PC,Pi,DPPCL(K)-DUM//’S’CALLUNES(SP(l),IPACIcI)COMMON/CORD/XMIN,XMAX,YMJN,YMAX0090KK-i,NENDIFCoMMONIPLoTfGTITLl,GTITL2,xrm,TflThIR=130CONTINUECOMMON/lNp/sFr,SDNF-0XL=XLEGND(IPACIçM)COMMON/LEG/PCL,OPB=BI+0.200DFLOAT(iCX.1)YL-YLEGND(IPACK,M)COMMON/CIRC/)CC,YC,FX,FYDI-(B-l.D0)(2.00R)IF(flXF.EQ.1)ThENCOMMON/ZERO/ACX,BCYD(KK)BCALLMYLEGNCrLEC,l4)COMMON/PRESSiCAP,MXCELSECCThecrirical pointatwhichthelevelmesiscusoccursCALLMYLEGN(TLEG,3)CGraphheadingsset.CisatACR,ENDIFCCCALLLEOEND(IPACK,M,lO.5,l3.)GflTLIRuptureCapillaryPressure’ACR—PI-PCRCALLBLREC(l0.0,l2.5.XL÷2.5,YL+1.O,0.Ol)GTrrL2—’vs.SeparationDistance’CCALLENDPL(0)X1TFL’b/a’CIncrementPRtofollowprogressofmenisctrsaroundCALLDONEPLYflTL-’DirrrcnsionlessPressurePnp@,’CcylinderRETURNGP=’ContactAngles(degyCENDCCCInitializeconstantsandproperties0030I-1,MCIM—l-lR’.5D0It—IcSFT-72.75D035P-ACRJRAD+0P0BLE(IM)SFT-45.D0PR-PRADYMIN-0.D0CYMAX-0.D0CIfpositionrsabovethecriticalvalue,themeniscus0-981.00Cisconvex,ifitisbeIow,jtisconcavC—(l.594D0-0.9982D0)G/SFTCCC%99S2D0/SFrIF(PRLIT.ACR) THENWCRRPHIPR+PCR-PICIDX-1P0—10.00QC—L00PCI-30.D0ELSEIF(PR.LT.ACRTHENDC-30.D0PHI-PI-(PR+PCR)187IDX=OQC=-1.D0ELSE PHI=O.D0CPCAP(fl)—0.D0MX=M-1TM-IGOT035ENDIFC CSolveforFCNtodetenuineTfocthespecifiedCcontphiTisgreatcrthanobutlessthanl.C 50U1=0.0000000IDOU2=0.99999999D0CALLZERO1lUlU2,FCN,EPS,LZ)ELSE 11=1ENDIFC CCalculatefilonthicknessCZC—QCDSQRT(ZOZO+2.D0/W(l.D0-COS(J’HQ))HC=DABStZO-ZC)HR-DABS(l.DG4QCDCOS(J’R))H=HR-l-ICHT(J1=HIF(JREQ.2)GOTO60IF(JR.EQ.t)THENHO=lff(J-l)IF(IEQ.1)HO-HT(J)CWRE(6,99)ZO,PHJ/RAD,PRfRADC CChangeinsigns,signalsfilmrupturepointsCIF(HO5H.LT.0.1)0)THENIR-2cEYIO4025HNEWHT(l)HOLD=HT(J-l)Pl=PHl-QCDP5RADP2—PHI55PH-HNEW(P2-Pl)fcHOLD-HNE÷P2PR—PI-PCR+QC5HIGOTO5060IF(DABSQS).GT. EPS)THENIF(HHNEW.GT. 0.1)0)THENP2=PHIHNEW-HELSE HOLD-HPl=PHIENDIFGOTO55ELSE NF—NF+lIR-0H.D0HT(J1=HpCAP(IcKK)-DAllS(wzo/2.Do)ppA—CSFrzOR/lo.DoWRiTE(6,99)3,PHIIRAD,PRJRAD,PCAP(K,KK),PPAIF(PCAP(JEK)Or.YMAX)YMAX—PCAP(K,KIC)IF(PCAPQKK)IT.YMIN)YMIN=PCAP(K,KK)GOlD90ENDIFENDIFENDIF40PH-PHI/RAD99FORMAT(lx,3F10.3, 2Fl5.IF(IREQ.2)GOTO2526IF(NFI.E.2)IR—IIF(Y(I,N)or.YMAX) YMAX-Y(I,N)IF(Y(I,N).LT.YMIN)YMIN-Y(l,N)30CONTINUE90CONTINUE106CONTINUEC CDeterminetheEmitsofgraphingtakingintoCaccountthatthewholeprofileandbothcylindersCwillbedrawn.CXMAX—BXMIN-BICALLMYPLOT(D,PCAP,M,N)STOPENDaflflfl*****flfl*flflfl**fl***flflflfl.s*flflfl.*flflFUNCTIONFCN(T)can...a.CThisfunctiondescribesthehorizontalwidthofCthemeniscusandisusedtofitthespecificationsCoftheseparationdimensionDIandthecontactCgeometyPHLCIMPLICITREALS(A-H,O-ZCOMMON/PARJC,DI,I°I,PCR,WCOMMON/PAR2/X,A,RCCOMMON/OUT/ZOFX-DELIPK(rT,IND)E-DELIPE(TT,IND)FCN=2.D0T/DSQRTtW)(t2.D0fCIl)-1.D0)(J°X-+DELLIF(r,(PI-5Q12.D0,IND))-+(200fcrl))(E-DELLIE(r,1PI.,cy2.Do,IND)))+-(DI/R+2.D0(l.D0-DSIN(A)))C c CMeniscusProfilebetweenTwoParallelCPlatesorHorizontalCylindersC C CDeterminestheshapeofthemeniaeusbetweentheCtwocylindersCSUBROIJI1NEMENISC(P,4T,B,XO,ZD,N)IMPLICITREAL8(A-H,O-Z)DIMENSIONXD(20),ZD(20)COMMON7PARIC,DI,PI,PCR,WC C CDeterminemeniocusproflebysolvingequationsforCanglesfromPRtoo.CBT-P/DFLOAT(J1)PH-PTOOl0I=l,N.ANG—(PI-PH)o2.D0F-DELIPK(rT,IND)E—DELIPEçFT,IND)FI—DELLIF(r,ANcUND)EI-DELLIE(T,ANG,IND)XD(l)=(r*((2.D0/(rwl).LDO).(F.FI).+(2.D0f(F’I))E-EQ))/DSQRT(B)IF(1.LT.O.D0)THENZD—lDSQRT(ZZ+2.D0/B(l.D0-DCOS(PH))))ELSE ZD(JO-(DSQRT(ZZ+2.D0/B(I.D0-DCOS(PIQ)))ENDIFPH=PH-BT10CONTINUE20RETURNENDC Cnfl*nnflsnSn*nflnflnnnnnnnnen.C•DISSPLAGraphingRoutineCaaC•.*****.ae....annn,.fl*....*.....*.CSUBROUTINEMYPLOT(x1,X2,M,N)C CThissubroutinegraphseitherthecapillary pressureCorthefunctionFUNvs.anindependent variableCi.e.D/R)CIMPLICITREAL8(O-Z)PARAMETER(IM-5,114=lO)C C CT-UlDetemoinethedimensionlesscapillarypressureforthemeroiscusZO=DSQRT(4.D0/W(1.D01F-l.D0))IF(DX.EQ.1) ZO=-ZOIF(1k.EQ.2) ThENll—M+NF+lC C C C188DIMENSIONXl(N),X2Qci,N),PC(UQREAL4U(IN),VIIN),Y(IN),Z(IN),AII?vO,IPACK(2000)CHARACTER4OGTITLI,GTIU2,3lTfll,YDTLCHARACTERI5TLEGCl{ARACTER4PCL(IM)Cl{ARACTER54SP(IN)COMMON/CORD/XMIN,XMAX,YMIN,YMAXCOMMON/PLOTIGTrrLI,Gm12,Xrnt,YrImCOMMON/LEGIPCL,ThEGCOMMONIINP/SFr,SDCOMMON/PAWrC,IDI,PI,PCR,WCOMMON/PAR2IPHI,PR,RC CDefinestheborders,titles,heading,etc.CCALLDSPDEVCPLOI’)CALLUNrrSCCENT)CALLNOBRDRCALLPAGE(21.,26.)CALLAREA2D(15.0,17.)CALLHEIGHT(0.30)CALLGRACE(0.0)CALLCOMPLXCALLMXIALFQSTANDARD,’&CALLM)C2ALFCL/CSTD,e)CALLM)I3ALFCGREEIC,!)CALLMX4ALFCSPEC,’)CALLMXSALFCL/CGR’,’W)CALLPHYSOR(3.,l.)CALLYAXANG(0)CALLTHKFRM(0.025)CALLFRAMEC C C CConvertdatatosingleprecisionvaluesAX-SNGL(XMIN)BXSNGWcMA)QAY’SNGL(YMlN)EY-SNGL(YMAX)DX-(BX-AX)/5.DODY-(BY-AY)15.DflCWSNGL(YCALLAXSPLT(A)cB)cl5,0,ORIG,STEP,A)CALLAXSPLT(AYBY,i7.,YOR,YST,B)CALLopF(A,cD)cBX,AY,DY,B)CALLHEADTN(GTrFLI,35,2,2)CALLHEADIN(GTITL2,29,2,2)CALLHEIGHT(O.3)CALLSCLPlC(0.CALLXTICKS(2)CALLXGRAXS(ORIG,STEP,BX,15.,XTITL,30,0.,0.)CALLYGRAXS(YOR,YST,DY.l7..Yrn’L,30.o.o.o.o)C CPlotinterpolatedlineCDO20I’1,MDOl0J=l,NIF(IEQ.1)ThENUQ)SNGLtXl(3t)ENDIFV(J>=SNGLX2(l,i))10CONTINUECALLMARKER(l)CALLR.ASPLN(3)CALLCURVE(UV,N,l)20CONTINUECALL1{EIGHT(O.2C CWritepropertiesongraphCCALLMESSAGCB0ndNumber—,l3,l0.,i5.2)CALLREALNO(CW,3,l3.3,l5.2)CIF(IDXEQ.0)THENCCALLMESSAGCCONVE)C,6,7,172)CELSECCALLMESSAGCONCAVE’,7,7,l7.2)CENDIFC CPreparelegendblockCMAXLIN=LINEST(lPAClç2000,2DO30l—l,MCALLLINES(PCL(l),IPACK,l)30CONTINUEXL=XLEGND(IPACK,M)YL-YLEGND(IPACIçM)CALLMYLEGN(TLEO,l4)CALLLEGEND(IPACK,M,l0.S,ll.)CALLBLREC(l0.0,t0.5,XL+2.5,YL+1.0,0.Ol)CALLENDPL(0)CALLDONEPLENDCCONTACTANGLEHYSTERESIS****C•Objective:ToplottheeffectofhysteresisConthemardmumpressinewhenoneincreaserC•pressurefronrotothatatwbichthefllmC•thicknessreacheszero(convexshapes)c C•S*e**fl*fl.**************Ss****S*.+.,*fl**CIMPLICffREALS(A-H,O-Z)PAAMEtR(M”6N=l0,NPN+l)CHAEACTER4PCL(M)CHARACTER*3DUMcHARAcrER1DDDIMENSIONPl(M,N),P2(M,NP),lll(M,N),H2(M,NP)EXrERNALFCNLOGICALLiCOMMONILIMIT/XMIN,XMAX,YMIN,YMAXCOMMON/PAR/C,PI,PCR,WCOMMON/PAR2/PHR,ACR,R,DlDBCOMMONIINFO/DP,SFr,PCOMMON/LEG/PCLR’5.D-lCDI”0.6D0CDB=Dl/(iD0R)÷lD0DB’l.50d0DI’(DB-l.D0)1D05R SPT—t5.dOG98l.D0PI-’DACOS(-l.DO)RAD’PI/l80.D0C CDensityofflinds,interthcialtension,profileClintltsandtolerancelimitsetCC—(l.594D0-0.9982D0)’GISFrCC-0.596d0GfsFrW-CRRC CIitltializecontactangle(relativetolrppcrfluid)andCthecontactangle hysteresisCPCl0D0RADPHI—I0.DORADDP-30.D0PO-0.D0CA-0.DOEPS=l.D-5XMIN-0.D0XMAX-0.D0YMIN’O.DOYMAX=0.D0C CPrintheadingsCwRrrs(22FORMAT(l5)çCAPILLAR.YPRESSUREVS.FILMThtCKNESS 189WRITE(6,23)23FORMAT(3o5c’CYLINDERSyI)WRITE(6,2’OW,DB24FORMAT(231,‘Do“,E9.3,’b/a,F10.3,i)WRITE(6,26)26FORMAT(o3c’A’,o2cpfIr,63cH’ycpcAp5’,lo3c’PcAp[paJC CIncrementstartingcontactanglesC00301=l,MpCR=PCl+DP*RADaDFLOAT(I1)C CChangecontactangles(antlowerangles) tolabelsforuseCinlegendC CCIF(PCR.NE.0.00)THENNFr-NINT((PI-PCR)/RAD)CALLBm(HPT,ouM,3No;)PCL=DUM/P$’ELSE NPT-NINT(PI-PCR)CALLBTD(NPT,00,l,ND;0PCLQDD/flHf0’//’S’ENDIFCMeniscuscontactpointonhorizontalscaledeterminedCB-0U2.D0+R0.D0-DSINIPCR))C CPARTICFordifferentextentsofhysteresis,detennineCthepressureasathnctionofthefilmthicknessCascontactanglechangeswithACRpositionconstantCOII=PCRJDFLOATQ4)IF(PCREQ.0.00)OOTO1500103-l,NACR=PCRC CStartingposition-Level meniscusCIF(3.EQ.I)THENPl(I,3)—POP-Pl(L3)Dz=0.D0ELSE MI=DHOFLOAT(3)C CMeniscusslopeangleis thesameastheCA.hysteresisCPHR—AlIC CSolvefoc TforthespecifiedcaseCUI-0.000000000lOOU2-o.999999999900CALLZEROl(Ul,U2,FCN,EPS,Li-UtCUselinearintcspolationtoobtainvalueatH-0CIF(HOf.XMAX)XMAX=HIF(H.LtXMIN) XMIN=HIF(P.EQ.0.0)THENZO=0ppA_C*z0*RaSF-rflo.D0WRITE(6,109)ACRJRAD,PHRJRAD,H,P,PPA109FORMAT(lx, 2F10.2,3F15.tIF(IR.EQ.2)00TH60CFindfllmnaptueepointCIF(IR.EQ.1)THENIF(HS5H.LT.0.00)THENlR-2C CStorepreviousmeniseiinslotafter FRPC C33—NH2Ø,33+l)=H2t3,33)P2Q,31+l)=P2(I,33)HNEW-HHOLD-ItSPA-PHR-OHFB=PHR55pHR=HNEWa(PBpA)/(HoLoHNEwPBACR=PCR+PHRCWRiTE(6,)‘flflIIRJRAD,ACRJR.AD,H,ZOOOTO5060IF(OABS(H)CT.EPS)THENIF(It5HNEWOr.0.00)THENPB-PHRHNEW=HELSE HOLD-HPA=PHRENDIFOOTO55ELSE IR—0WRITE(6,9‘SUCCESS’ENDIFENDIFENDIFHS-H20CONTINUE30CONTINUECALLMYPLOT(Hl,PI,H2,P2,M,N)STOPENDC Caaaaa***ae*aaae.saaFCNa*eneaaa.eaaes..aaaa.eCDOUBLEPRECISIONFUNCFIONFCN(T)C C•aaeeaaa*aaaaamaeeee.fleeesa..eeaee,aa..,,ea.a,a,CInputs:Pbr,ArmradiansCCC CC CC CC CC CC COeteeenine thedepthanddimensionlesspressureofthemeniscusCzo=DSQRT(4.oo/Wa(Loofrl.Do))Pl(I,frOABS(ZOW/(2.D0))P=Pl(I,J)zN=.oSQRT(zO*zo+2.Do/wa(LDoDCoS(PHR)))DZ=ZO-ZNENDIFIF(P.af.YMAX)YM.AX=PIF(PLiYMIN)YMIN-PC CDeterminethefilmthicknessCHl(I,3)-l.O0+OCOS(ACR)-OZH-HItJ,3)IF(HCT. XEtAX)XMAX-HIF(HLiXMIN)XMIN-H10CONTINUEC CPAItT2CWiththecontactangleconstantat TR,theCmatiscusbegins tomovealongthecylinderC 15HS=0.D0IR—lTR-=PCRNM=N-lDH=(PI-PCR)/OBLEQtM-I)00203-l,NM33=13M-I-IIF(IR.EQ. 0)33—3+1PHR-DHDFLOAT(3M)25ACR-TR+PHRRC_OI/(2.oo)+Ra(t.DoOSIN(ACR))IF(3EQ. 1)THENP2(I,i)—POP-P2Q,3)DZ=0.D0ELSEC CSolveforthedimensionlesspressureP2C 50Ul-.00000000100U2-0.9999999900CALLZEROI(Ul,U2,FCN,EPS,LZJT’UlZO—DSQRT(4.Do/W(t.oorr-l.o0))P2tJ,33>DABS(Z0aW/(2.D0))P—p2(I,J3)FC_DABS(zoaCaSFrRylo.ooIF(POT. YMAX)YMAX—PIF(PLiYMIN)YMIN—PZH—DSQRflOZO+2.D0/W(l.D0-DCOS(PHR)))DZ-ZO-ZNENDIFH2(I,3fl-I.D0+DCOS(ACR)-OZH=H2(I,Jl)C190lMPLlCrrREAL8(A.H,O-Z)DBSNGL(PDB)CALLMYLEGNC#q&(deg)$’,9)PARAMETER(M=6)DFT=SNGL(SFI’)CALLLEGEND()PACIçM,Lo,9.5)COMMON/PAPJC,PI,PCR,WDW=SNGLCALLBLREC(0.5,9XL+l.2,YL+1.0,0.01)COMMON/PAR2/X,A,R,D1,DBDR”SNGL(R)CFX=DELIPK(r*T,IND)CALLAxSPLT(AXB)cl5..ORIG,STEp,A)CALLENDPL(0)E=DELIPE(PT,IND)CALLAXSPLT(AY,BY,17,YOR,YST,B)CALLDONEPLFCN2.DODSQRT(C)((2D0/(rI)-l.D0)(FXCALLGRAr(A,cDcBXAYDY,BI)RETURN+-DELUF(T,(Pl-X)12.D0,INO))GrIv’MeniscusPressurevs.Flint Thickness’END+-{2.Dw(ro)’(E-DELLlE(r.tPI-XtI2.DO,tND)))+.(DI+2.DO*Re(l.D0DStN(A)))CALLHEADIN(G1T11536,3,2)RETURNJflL2-CylindrrsENDCALLHEADIN(GTIL2,15,2,2)CCALLHEIGHT(0.3)CCALLSCLPIC(0.9)C*fl**fl*C*********************fl*******CALLXTICKS(2)SUBROUTINEMYPL0rQcI,Yl,x2,Y2,M,N)CALLYTICKS(2)C)CflTL’FihnThicknessWa’CYTITL*’DirnensjonlessCapillasyPressureprup,CCALLXGRAXS(ORIG,SrEP,B)c15.,Xnm,lg,o.,o.)•IMPLICrrREAL*8(OZ)CALLYGRAXS(yOR,YST,BY,17.,YrITL,39,0.0,0.0)PARAMETER(IM%INl0)CALLMESSAGC#_&Constant #ql5,l.,14.3)DIMENSIONXIN),X2(MN),Y1tM,N),Y2(M,N)CALLMESSAGt’#...&Constant#5,17,1,13.5)R5AL4Ut)N),VQN),Y(IN),ZIIN),IPACK(2000)CCHARACTER*40G1TrL,’T1TL,XT1TL,GTTL2CPlottheinterpolatedlineCHARACTER4PCL(lM)CCOMMONJLIMrrJXMIN,XMA)cyMlN.YMAX0020l-l,MCOMMON/PAR2IPHI,XR,R,PDI,PDB0010J”l,NCOMM0N/INF0IPDP,SFr,PUQ)=SNGLtX1IIM)COMMONIPARJWC,Pl,PC,WV(31=SNGL(X2II,J))COMMONILEGJPCLY(1t=SNGL(Yl(I,3))CZ(J)SNGL(Y2(I,J))CSet updimensions,title,headings, etc.litCONTINUECCALLMARKER(I-l)CALLDSPDEVCPLOT)CCALLSPLINECALLUNrrSçCEN’r)CALLRASPLN(2)CALLNOBRDRCALLDOTCALLPAGE(21.,26.)CALLCURVE(U,Y,N,0)CALLAREA2O(15.0,17.)CALLRESETCOOP)CALLCURVEtV,Z,N,l)CALLHEIGHT(0.25)20CONTINUECALLORACE(0.0)CALLDASHCALLCOMPLXCALLRLVEC(0.0,AY,0.O,BY,0100)CALLMX1ALPCSTANDARtY,’&)CALLRLVEC(AX,0.0,BX,0.0,0l00)CALLMX4ALFCL/CGRf,’#)CALLHEIGHT(03)CALLMX5ALFCSPECIAL’,’@)CCALLPHYSOR(3.,l.)CWritepropertiesongraphCALLYAXANG(0)CCALLTHKFRM(0.02)CALLMESSAGCOondNumber ‘.‘,13,l,l6.)CALLFRAMECALLMESSAG(Ws“,5,l.,15.2)CALLREALNO(DW,-3,4.6,16.)CCALLREALNOtOB,2,2.4,1i2)CCCConvert tosingleprecisionvaluesCLegendforidentiEcationofstartingcontactangleCCAX”SNGL(XMIN)DO30i”IMBX=SNGL(XMAX)CALLLINES(PCL(l),IPACJçI)AY=SNGL(YMTN)WRiTE(6,)PCLQ)BY’SNGL(YMA)Q30CONTINUEDX=f,BX-AX)/5.D03-XLEGNP(IPACIçM)DY=(BY.AY)15D0yL=YLEGNO(lPAC1c1191C C CMeniscusProfilesforaSphereSurroundedbyCotherspheresCMeniscusProfiles-SPALPC.C C CTheshapeofthemeniscuschanges asonecausestheCtheinterfacetomoveupalongthesphere.A5PCincreasesthecapillarypressurealsochanges.ThisCprogramcanbeusedforeitherconvex(IDX=l)orCconcave(IOX))menisci.ThefilmrupturepointsareCdetermined.CIMPLICITP.LdS(AH,OZ)PARAMETER(MM=6,MY=10,NN=3,MPMY+2,ND’20)DIMENSIONY(HN),YC(HYNiNN),CX(NO),CY(HD),FX(2.ND),FY(2,ND)DIMENSIONPllMP),P2(MM,MP),Ftl(HP),H2(MM,MP)DIMENSIONHX(lOO),HY(l00),PCAP(MP),QM(2,20),QX(MY,20),QY(MY,20)EXTERNALF.PCN,FEX,GEXCHARACTER9DDCHARACFER3DUMCHARACI’ER4PCL(MM)INTEGERFLAGLOGICALLZCOMMON/LI?cr/XMJN,XMA)cYMlN,YMAXCOMMON/YLIM/YLCOMMON/PROP/C,SFr,IcIDXCOMMON/GEO/PHIC,PC,A.XC,BCOMMON/CONST/EPS,PI,N,MCOMMON/PARA/PCI,W,QCOMMON/OtThXO,ZOCOMMON/LEG/PCLCOMMON/ODE/HCOMMON/CAT/AO,DACOMMON/BAT/IFLAGCOMMON/ARRJQMCOMMON/PRESS/PCAP,MXCOMMON/STOP/DEG,IERRCOMMON/CIRC/CX,CYCOMMON/ZER.O/AC)cBCY,F)cFYC CCPART1-InitializeconstantsandpropertiesCM’MMN’NNPI=DACOS(-LDO)RAD-PI/180.DOSFT=72.7500SFr45.D0G98l.OD0C.(l.594D0-0.9982Oo)5G/SFrCC.-O.99s2DoG/SFrR’0.lO0B1.05Od0W=CCRCRCEPS’l.D-5IERR=0CSetinitialcontactanglePCIandincrementDCCPCI=90.O0RADDA20.DORAOAO=lO.DOCRADCAAO*ltADXMD4’O.DOXMAX=0.O0YMIN=O.O0YMAX0.OOYL-0.O0MX”OIDX’0NF=1IRcYrIC CTitlesforprintoutCWRiTE(6,3)3FORMAT(20X,riENISCUSPROFILESBETWEENSPHERES)WRITE(6,9)W,PCI/RAD,B9FORMAT(5X’Bl2.7’CA7.t,2X,’bfa’,F7.2//)WRFE(6,11)11FORMAT(5)c’PHl’,5X,3cC,7)çzC’,8,c’ZO,7XH,+7X,PCAP,9XA/)C CCalculate locationoflevelmeniscusC CACR=PI-PCIACXOS1N(ACR)BCY”-DCOS(ACR)CLoopforvaryingimenersronal angleACMXMY0020Jl,MYJM=J-ljPJ+17A’AO+DADBLE(3M)XCOSlN(A)IF(DABS(A-ACR).LT. EPS)THENHOLD=HPHI=PHICHOABs(1.D0+Q5DCos(A))HlQ)’HPHJC0.D0IF(NF.EQ.1)ThENGOTO8ELSE HOLD=HPH1=PHICGOTO2OENDIFELSEIF(A.GT.ACR)THENPHIC=-(PI-PCI-A)IDX’lQ-1i)0ELSEIF(AIT.ACR)THENPHIC=PI-PCI-AIDX”OQ—I.D0ENOIFIF(AGE.P1)GOTO30HOLDHC CDetermineimlesownaZCandZOfrontsolutionofasystemCofODEsCCALLHVSP(PHIC,XC,A,ZC,P)HR_OABS(I.00+Q*DCOS(A))HC=DABS(ZO-ZC)H’HR-HCWRiTE(6,12) PIIICJRAD,XC,ZC,Z0,H,P,A/RAI)Hl(l)HP1(J)—PPCAPPIF(P.GT.YMAX) YMAX-PIF(P.LT.YMIN) THIN-PIF(HOr.XMAX)XMAX-HIF(H.LT.XMIN) XMIN-H12FORMAT(l.3F63,lX,F3.3,2X,F8.3,2X,F7.3,+2X,F7.3,4X,F7.3)ZT—DCOS(A)YO-QM(z1)-zrD051C-l,NOQX(J,K)-QM(l,K)QY(3,K)-QM(2,K)-YDIF(QY(1,K) .LT.YL)YL-QY(J,K)5CON11NUEIF(AOT.ACR) IROT-lIF(HF.EQ.3)(101020IF(I.EQ.1)(101020C CFindfilmiupttcupointwhereH-0CIF(IROT.EQ. 1)THENIF(H*HOLD.LT.0.00)THEN8IINEW-Hl(J)PH2—PHIC10PHIC-I8NEW(PH2-PHl(HOLO-HNEW)+PH2AI.PCI+Q*PHICXC-DSJN(A)CALLHVSP(PHIC,XC.A,ZC,P)HR-DABS(l.D0+QDCOS(A))HC-DABS(ZD-ZC)H=HR-HCIF(DABS(H) Or.EPS)THENIF(HI4NEW.GT.0.00)THENPH2—PHICHNEW-H192EXrERNALFEX,GEX,JACC CSetargumentstosolvefor300EsandlequationCRAD=PI/180.00Y(2)=PHICY(I)=ZY(3)=XCC CSaveinitialvaluesforplottingCYI(l)ZYI(2)PH1CYl(3)=XCT’O.DOTOUT2.D0BTO1oUrC C C C CSetloopforllstepsancldetetmineequationrootsCDO40IOUT-i,i10CALLLSODAR(FE3cN,YTour,noL,RToL4ToLjrAStc1STATE,+IOPT,RWORcLRW,WORXV,JAC,JT.GE,cNG,JROol)C CSavethenteniscusshapeinanarrayC CWRITE(6,)Y(2YRAD,Y(3),Y(t)20FORMAT(IX,S—,EI2.4PH-‘,E12.4,’X-,Et2.4,Z-‘,E12.4)IFSTpit.EQ.-1) THENISTATE-3ZWORK(6)l200001010ENDIFIF(ISTATE.LT.0)001080IF(ISTATEEQ.2)GOTO40CWRII’E(6,30) 3ROOT(1),JROOT(2)dOFORMAT(5XThEABOVELINEISAROOT, JROOTh’,lIS)C CIfthedentequationsrootisfound(X-B)thenstopintegrationCIF(JROOT(l).EQ.I) GOlD50ELSEIMPLICITREAL8(A-H,O.HOLD—HDIMENSIONXT(I)PH1—PHICLOGICALLiENDIFEXTERNAl.F,FCN,FEX,GEXGOTO10COMMON/PROP/C,SFT,R,IDXELSECOMMON/CONST/EPS,PI,N,MH1(MY+NF)=HCOMMONIOUr/X0,ZOP1IMY+NF)=PCCOMMON/ODE/HOUrPCAPiMY+NF)-PCOMMON/BAT/1FLAGWRITE(6,12)PHIC/RAO,XC,ZC,ZO,H,P,AJRADCOMMON/PARA/PCI,W,QH=0.D0zr=-DCOS(A)COMMON/STOP/DEG,IERRYD=QM(2,1)-ZTC35-1+1CCalculateZCandZOusingtherootsolverUllCZEROICCCFilmrupturenienisci(upperandlower)storedinFX,FYIF(PHINE.0.00)ThENC(scaled).CXT(l)—0.D00035K=1,NDCALLDRZFUN(FCN,l,100,XT,IND,5.E-7,EPS,I.E40,FX(NF,K)=QM(l,K)+1.E-3)Y(I4F,K)=QM(2,K)-YDIF(IND.EQ.0)GOl’O5IF(FY(NF,K).LT.YL)YL=FYtNF,K)WRiTE(6,*)‘DRZFUNFAILS’IERk135CONTINUE0010100IROT=0CSWRITE(6,)ROOTIS,XT(1)NF—NF+l5ZC—Xr(1)ENDIFCIERR-lELSEIF(WHOLDor.0 DO)THENGo’lO70PH1-PHICCENDIFENDIFCZC-U1ENDIFELSE20CONTINUEPCAP-0.D0CZO=0.D0CProfileforspheresinCX,CYZC-0.D0CENDIF30DRPI/DBLE(ND-t)70PCAP—DABS(P.1ZO)l(2.D0)0015K=l,NDDP-DRDBLE(K-l)tooRETIJENCXQC)=DSIN(DP)ENDCYQQ—DCOS(DP)CISCONTINIJECCFUNCTIONFCNIZ)CPlotthemeniscusprofilesasliquidmovesupC#****sfl**s**S********S*****eCC CThisfimctiondescribesthedetermination oftheCALLMNPLOT(Q)çQY,MY,ND)CthecontactpositiononthespheresbyfittingtheCdifferentialequationsuntilconditionsstbothendsISOSTOPCwhereX=XcandX-Baresatisfied.20andZCareENDCunknownandXistheindependentvasiabIeCC*C********a**qsss*****SI***5*****IMPLICITREAL8(A-H,O-Z)SUBROUTINEHVSP(PHI,XC,A,ZC,PCAP)INTEGERFLAGPARAMETER(NN-3,LRW-80,L1W-25,NG-2)C*.*.*,,**,,**.*,,,,,fl.*,****,**fl*******n***,*.*,**DIMENSIONY(NN),S(NN),YN(5O),Y1(NN),QM(2,20)CThissubroutinesolvesfortheunknowns20&ZCfortheDIMENSIONATOL(NN),RWORK(LRW),IWORX(LIW),JROOT(NG)CtneniscuaandthencalculatesthecapillarypressureCOMMON/GEO/PHIC,PC,A,XC,BCandverticaldistanceoftherneniscusfromthetopCOMMON/CONST/EPS,Pl,N,MCOMMON/Our/XO,ZoCofthecylinders.COMMON/LENOIWXOLD,YOLD.S1,XCCOMMON/ARRIQMSettolerances,andparameters forLSODARIFOL=2RTOL-i.D4A’10L(l)-l.D-6ATOL(2)-lD-6ATOL(3)=1.D-6ITASK-1ISTATE=llOFT-IDOS1—5,10RWORK(l)=OIWOR.K(l)”O5CONTINUE1WORK(6)-i000IT-i193C 50X0’BPH=Y(2)ZO=Y(l)zC=zFCN=PH-0,D0CWRITE(6,70)ZC,ZO,FCN70FORMAT(2X,FCN’,3E12.4)C CDeterminemeniscusshapeforplotting,resetvaluesC CIF(DABS(FCN) .LE.EPS)THENTOTIl9.D0TOUT=1OT=0.D0ISTATE”lQM(l,l)=’Ylc3)QM(2,l)-’YI(l)00100IOUT=2,2075CALLLSODAEcN,yI,r,vUr,rrOL.R1oL.ATOL,rrASlctSTAm,+IOPT,RWORIcLRW,IWORIcLIW,JAC,JT,GEX,NG,JROOI)C CSavethemeniscusshapeinanarrayC CWRITE(6,20)T, Yl(2)/RAD,YIt3),YIO)QM(l,IOLTI)-Yl(3)QM(2,IOITI)-YIQ)IF(ISTATE.EQ.-1)THEN!STATE=3IWORK(6t-’1200GOTO75ENDIFIF(ISTATE.LT.0)GOTOSOIF(ISTATE.EQ.2)GOTOIOOlSTAlt2GOTO75100TOTJP’IODBLE(IOIJJ)ENDIF80wRrrE(6,90)ISTATE90FORMAT(fu/5XERRORHALT...ISTATE‘,13)ENDAX—0.ODB—SNGL(rB)BX-2.OSNGL(TB)AY-SNGL(YMIN)BY2.ODPSNGL(TDP)DW’.SNGL(W)DPC-SNGL(PCI)C CReflectsemi-profile aroundZ-adstoobtaincompleteCprofileCCX”SNGL(X?.flN.XMAX)CDX=(BX-AXy6DQDY=(BY-AY)f5.D0DX=DYDD<BY-A1’)14Jl7.BX-I.05+DD/2.AX-I.05-DD/2.ISTATE=2GOTO1040TOlJP’.TODBLE(IOUl)C50WRITE(6,60)IWORK(ll),rWORK(l2),r.vORK(l3)JwORK(lo).C+IWORK(19),RWORK(l5)C60FORMAT(/3X.’NO.SThPS’,l4,5)ç’NO.F-S=çI4,5)çNO,J-S”.’,C+14,5X,’NO.GS.d.l4J,3,c’MEwODLASTUSED-çl2,5çC+‘LASTSWITCHWASATT,El2.4)CDIMENSIONX1(M,N),X2(M,N),X3(tN),X4(IN),SX(2,IN),SY(2,IN)DIMENSIONPCAJ’(IhREL4U(1N),V(IN),Y(IN),Z(,UI(IN),Yl(IN),AP(IM)REAL*4CAP(IM),FXØN),FY(IN),FXI(IN),FYI(IN)CHARACTER4OGrfltl,GITILIXflTI5YflTLCOMMON/YLIMIYMINCOMMON/PLOTIGTITL1,GTITLZXflTI5YITI1,COMMON/ZER000çTY,sIçSYCOMMON/CIRC/X3,X4COMMON/PRESS/PCAP,MXCOMMONIGEO/PHIC,PC,TA,XC,TBCOMMONICONS’DTPS,Pl,NO,MOCOMMON/PARA/PCIW,QCOMMON/CAT/P1,TDPC CDefinestheborders,titles,heading, etc.CCALLDSPDEV(PLO’lCALLUNITSCCENT)CALLNOBRDRCALLPAGE(21.,26.)CALLAREA2D(l4,0,17.)CALLHEIGHT(0.25)CALLGRACE(S)CALLContplxCALLMXlALF(STA}DARD)CALLMX2ALF(VCSTD’7)CALLMX3ALF(GREEK’,’I)CALLMX4ALFCSPECIAL’,’@)CALLMX5ALF(LICG1tCALLPHYSOR(3.,l.)CALLYAXANG(0)CALLTHKFIUd(O.02)CALLFRAMEC CConvertdatatosingleprecisionvaluesCSUBROIJflNEJAC(N,T,Y,ML,MU,PDNRPD)IMPLICrrREAL5S(A-H,O-zDIMENSIONYtN),PD(NRPDN)COMMON/PROP/C,SFT,R,IDXCOMMON/PARA/PCI,W,QCPD(1,2)’QDCOS(Y(2))PD(2,2)=-DCOS(Y(2))/Y(3)PD(2,l)’Q5W pD(2,3)DSrN(y(2)y(Y(3)y(3))PD(3,2)=-DSINçf(2))RWENENDC CSUBROIJrINEFEX(N,T,Y,DY)C CFunctionsuppliedforusewithLSODAP.CIMPLICrrREAL5S(A-H,O-Z)DIMENSIONY(I4),DYtN)COMMON/PROP/C,SFT,R,IDXCOMMON/PARA/PCI,W,QCOMMONILENGTH/XOLD,YOLD,SO,XDY(I)’Q5StNtY(I))DY(2)’-QDY(l)Y(3)4-QY(1)WDY(3)DCOS(Y(2))ENDC CSUBROUTINEGEX(N,T,Y,NGGOUl)C CIMPLICITREALS(A-HO-DIMENSIONY(I).GOUr(I4G)COMMON/GEO/PHIC,PC,XC,A,BCCOMMON/ODE/HC CEvaluatethe2equationswhereroots.rewantedCGOt.Tr(l)=Y(I)-BREWENENDC.flefleeflfl*fl*****fl.flflflfl*,fls*fl!...C•DISSPLAGraphingRoutineC•**..*•.flCSUBROtJIINEMNPLOT(Xl,X2,M,N)C CThissubroutineplotsthemeniscusprofilesbetweenCtwospheresatdifferentimniersionalangles,IMisCM+2forthenapturemenisci.CIMPLICITREAL8(0-Z)PARAMETER(IM—l2JN’ZO)C C C194DX=(BX-AX)/5.D0CCDDBY-A14J17.CCBX=’DDCCDD=(BXal7Jl4.)CCBY=DD/2.)al.2CCAY”(-BY)f 1.3CALLAXSPLT(A5çBX,14.0,ORIG,STEP,A)CALLAXSPLT(AY,BY,17.,YOIt,YST,B)CALLGRAF(ADAY,DY,R)GT11t1’Fig.:MeniscusBetweenSpheresGTITL2’atVariousPositions’CALLHEADIN(GITrLI,32,3,2)CALLHEADIN(GflTL2,25,3,2)CALLHEIGHT(0.3)CALLSCLPIC(0.7)C CCALLXTICKS(2)CALLYflCKS(2)xTrrL=”r/a’yrtTh=’zla’CALLXGRAXS(ORIG,STEP,BX14.,XTlTL,3,0.,0.0)CALLYGRAXS(YOR,YST,BY,17.,YflTL,3,0.0,0.0)CALLHElGHT(0.2C C CPlotinterpolatedlineCDO50It,MXCAPtQ=SNGLtPCAP(Q)DO40J=1,NU(31=SNGLIXI(lM)V(Ji=SNGL(X2(l,i))Y(J)SNGLtX3(J))ZQ)’SNGLQC4(Jl)C CReflectedmenisciCUl(J)SNGL(2.D0TB-XlQM)YlQ)SNGL(2.D0TB-X3(S))C CFilmruptureCIF(I.LE.2)ThENFX(S)’SNGL(SX(lM)FY(J)=SNGL(SY(T,J))FXl(J)=SNGL(2.D0*TBSX(I,i))ENDIFCWRiTE(6,99)U(J),VQ),Y(3),Z(J)99FORMAT(l)c4Flo.3)40CONTINUECALLMARXER(I)C CMeniscusproifiebyreflectionCCALLCURVEtUl,VN,0)CALLCURVE(U,V,N,0)CALLPOLY5IF(CAP(I)EQ.0.0)001030CALLRLREAL(CAPtl),3,DB-0.1,V+0.02)C CDrewfilmrupturenieniscusasdottedlineC 30IF(I.LE. 2)THENCAP(IM-2+l)SNGL(PCAP(IM-2÷I))CALLDOTCALLCURVE(FXFY,N.0)CALLCURVE(FXI,FY,N,0)CALLRESETCDOT)CALLRLREAL(CAP(IM-2+I),3,DB-0.I,FY(N)4-0.02)ENDIFP1Pl+DP50CONTINUEC CDrawstraightlinespecifyinglocationofPcap=QCCALLDASHACX’SNGLtXX)BCY=SNGL(YY)CALLRLVEC(ACX,BCY,21JB-ACX,BCY,0100)CALLRESETCDASH)C CDrawbothspheres byreflectionofcurvesCCALLCURVtuyI,Z,N,0)CALLCURVE(Y,4N,0)CALLHElGwr(o.25)C CPrintcontactangleCCALLMESSAOBo‘,4,t0.0,16.5)CALLREALNO(DW,-3,ll.00,t6.CALLMESSAGC#q=’,3,l0.0,15.8)CAD=SNGL(PCI*lS0.D0/Pl)CALLREALNO(CAD,l,l1.0,15.8)CALLMESSAG(b/a’’,4,l0.0,l5.0)CALLREALNO(DB,2,tl.0,15.0)CALLHEIOHT(0.l5)CALLMESSAGo’,l,l2.3,l60)CALLENDPL(0)CALLDONEPLENDCAdzymntetricSaddle-shapedMeniscusofaSphereinaFieldofSpheres(Gravitational)RupturePressurevs.BondNumber‘BO’-DTMENSIONLESSPlotstherelationshipbetweencapillasypressurewithBondrsimberasthefluidtypechanges.Th4PLTCfl’REAL8(A-H,O-Z)PARAMETER(MM”6,MY”lO,NN-3,MP=MY+I)DIMENSIONY4N),YC’JN),YN(NN)DIMENSIONPI(MMMP),P2(MM,MP),Hl(MM,MP),H2(MM,MP)DIMENSIONHX(l00),HY(l00),PCA,PBQsIM,MY),BO(MY)EXTERNALF,FCN,F2,F3,PDF,FZERO,FEXGEXCHAiACltRlDDCHARACFER3DUMCHARACrERa4PCL(MM)INTEGERFLAGLOGICAL12COMMON/LIMT/XMIN,XMAX,YMIN,YMAXCOMMON/PROP/C,SFT,R,IDXCOMMONIGEOIPHIC,PC,A,XC,BCOMMON/CONST/EPS,Pl,N,MCOMMON/PARA/PCI,W,QCOMMON/OUT/XO,ZOCOMMON/LEG/PCI,COMMON/ODE/HCOMMON/CAT/PPCOMMON/BAT/IFLAGC C•PART1-Initializeconstantsand propertiesCM-MMN.’NNSFr45.D0G.98l.D0C_(l.594D00.9982D0)*G/SFrCC.lDOCS-’DSQRT(C)R-0.1DOB=’O.lSDO/R.WI-CRRPI=DACOS(.l.D0)RAD=PI/180D0EPS’l.D-SC CConyex(IDX=1)andconcave(IDX.’O)ClOX-IIFDXEQ.1)ThENQ=l.D0ELSE Q-’l.DOENDIFC C C C C C C C C C C195CCcontactangle(PC),PfllandAchange.CPPAsthecaplllayprcssureinPascalsCntheingsforthetableofressiltsCCCCPPA-CSFTZOPJ10D0WRITE(6,22)IF(Por.YMAX)YMAX-PWRiTE(6.23)CMPCIF(P.LT. YMIN)YMIN=P22FORMAT(I5)çRUpIIJREPRESSUREVS.BONDNUMBER)BOQQ’W23FORMAT(30X,SPHERESWRiTE(6,12)BO(lC),PHICIRAD,A,P,PPAWRITE(6,53)BIROT-033FORMAT(f463(F,8XPHr,8)ç’AçI0X,PCAP.9)ç’PCAP[Pay/)DP=CM/D8LE(M)C001018WRITE(6.33)DB3.5D0fDBLE(MY-1)IF(IPLOTEQ.1)00105055FORMAT(1Xb/a“,F7.3)DO50K”l.MYENDIFCIROT1ENDIFCSetinitialcontactanglePCIandincrement DCW=WI+DBDBLE(IC-1)ENDIFC20CONTINUEPCF-30.D0DC”.(180.D0-PCI)IDBLE(M-I)1)020J’-I,MY50CONTINUEPO”.O.DOJM=J.190CONTINUEXMIN”O.DOIP’,J+I)4IN=WIXMAX=0.D016PHIC=DPDBLE(3M)XMAX=WYMtN=0.D0A”PI-PC+PHICIF(IPLOT.EQ.1)THENYMAX=0.D0IF(IDX.EQ.0)A=PI-PC-PHICCALLBPLOT(BO,PB,M,MY)1)090 I=1,MXC=DSIN(A)ENDIFCCALLHVSP(PHIC,XC,A,ZC,H,P)31WRITE(6,)1)4CPART2-Determine profilesforseveralcontactangles12FORMAT(1X3F10.3,2F14.7)100STOPCH2ØJ)”HENDPC.PCI+DCeDBLE(I1))*RAD18P2(I,3)=PCPP=PCCCa*****fl*a*e.a**********a.*******a***s.ea*********...CCFindroot ofHSUBROUTINEHVSP(I’Hl,XC,A,ZC,H,PCAP)CPreparelabelsforlegendingraphCCIF(tROTEQ1)THENC*5**tF(PCNE.01)0) THENIF(H.LT.01)0)ThENCThissubroutinesolvesfortheunlasownsZO&ZCfortheNFf’N1NT(PC/RAD)CmeniscusandthencalculatesthecapitlasypressureCALLBTD(HPT,DUM,3,ND, )HNEW=H2(t,3)CandverticaldistanceoftherneniscusfromthetopPCL(H=DUM/PTHOLD’H2(I,JM)ELSECofthecylinders.NPTNTNT(PC)CCALLBTD(NPT,DD,1,ND,’O)PH1’PHIC-DPIMPLICITREAL8(A-H,O-2)PCL(H..DD/P.’/PO/PSP}I2PHICDIMENSION3CF(1)ENDIF17PHIC..HNEW(I’112-PHI)/(HOLD-HNEW)+PH2LOGICALLZCAPI-PC+QPH1CEXTERNALF,FCN,PEX,GEXC•PART3HysteresisProfile-COMMONIPROPIC,SFF,P.,IDXCi)DeterminecuevesfordifferentextentsofXCDSIN(A)CCOMMON/GEO/PHIC,PC,AT,XC,BChysteresisataconstantcontactline,CALLHVSP(PHtC,XC,A.ZC,H,P)COMMON/CONST/EPS,Pl,N,MCA-constandXCconstIF(DABS(H)(Fl’.EPS)THENCOMMON/OUT/X0,Z0CIF(H5HNEW(Fr. ODO)THENCOMMON/ODE/HourPH2—PHICCOMMON/BAT/IFLAGNNEW=HCOMMON/PARAIPCI.W,QCCalculatelocationoflevelmeniscusELSECOMMON/CAT/PPC000LD”HCA’-PI-PCPH1’PHICCCalculateZCandZOusingtherootsolveruflCZEROIENDIFCWRITE(6,34)PC/RADIF(PHINE.0.1)0)ThEN34F0RMAT(/,1X,’CA‘,FlO.3)XT(l)..0.D0IF(A.EQ.P1)ThENGOlD17CALLDRZCN,I,100r,IND,5.E-7EPS,I.E-I0,WRITE(6,*)‘Levelmeniscusoccursat180’ELSE+l.E-3)GOTO100H2(I,SP)=HIF(IND.EQ.0)GOTO5ENDIFP2(lJlPWRiTE(6.*)VRZFUNFAILS’XC=DSIN(A)PCAP(I)-P0010100CPB(l,K)=P5ZC-’Xr(l)CThemeniscusmovesalongthecylinder ataconstantCGOTO70196ELSE PCAP=ODOZ0=0.D0ZC=0.D0ENDIF70HC=DABS(ZO-ZC)IF(IDXEQ.I)THENHR=DABS(l.D0+DCOS(A))ELSE HR=DABS(1.D0-DCOS(A))ENDIFi-IT=HR-HCH=HTHOIJr=HPCAP=DABS(WZO)f(2.D0)100RETURNENDC C******FUNCTIONFCN(Z)C•**0C CThisflmctiondescribesthedeterminationoftheCthecontactpositiononthespheres byfittingtheCdifferentialequationsuntilconditionsatbothendsCwhereX=XcandX=BaresutisfiedZOandZCaraCunknownandTthearclengthistheindependentvariable.CIMPLICITREALO(A-H,O.Z)INTEGERFLAGPARAMEIER(NN’-3,LRW=S0,LIW=25,NG=2)DIMENSIONYtNN),S(NN),YN(50),YItNN),Q(NN,20)DIMENSIONATOL(NN),RWORKLRV/),IWORK(LIW),JROOT(NG)COMMON/GEOIPHIC,PC,A,XC,BCOMMON/CONSTIEPS,PI,N,MCOMMON/ODE/HCOMMON/OTJT/XO,ZOCOMMON/LENGTHIXOLD,YOLD,SI,XEXrERNALFEXGE/cJACC C5.1argumentstosolvefor3ODEsand1equationCRAD=PI/180.DOY(2)=PHICY(1)=ZY(3)=XCT=0.D0TIYfl=200B10=TOUTC CSettolerances,andparametersforLSODARCrroL=2RTOLID-4ATOL(I)-1D-6ATOL(2)=1.D-6A’ItiL(3)’=1.D-6ITASK=iISTATE-!ioir=iDO31=5,10RWORK=0IWORK(1)=05CONTINUEIWORK(6t=l00011=1C CSetloopfoci!stepsanddetermineequationrootsCDO40lOUT”!,!10CALLLSODAR(FE/cN,Yj,TOURDL,ATOLJTASK,1STATE.+1Opr,RWoRcLRW,WoR3cLIwJAC,jT,GE3çNG,JRoO1)CWRiTE(6,20)T,Y(2)IRPD,Y(3),Y(1)C20FORMAT(!X,’S—‘,E12.4,’PH=‘,E!2,4,’X—‘,E12.4,’Z—‘,E12.’OIF(ISTATEEQ.-1)THENISTATE=3IWORE(6t=1200GOlD10ENDIFIF(ISTATE.LT.0)GOTO80IF(ISTATEEQ.2)001040CWRITE(6,30)JROOT(l),JROOT(2)C30FORMAT(D‘THEABOVELINEISAROOT,3ROOT=’,2I5)C CIfthefirstequation’srootii found(31—B)thenstopintegrationCIF(JROOT(1)EQ.1)GOTO50ISTATE=200101040Totrr=T00DBLE(3Otn)C50WRITE(6,60)1WORX(I1),IWORK(i2),IWORX(13),IWORK(iO),C+IWORK(19),RWORK(l5)CooFORMAT(/3X,’NO.STEPS-I4,5X,’NO.F-S-J4,5)çNO.3-5=’,C+I4,5XN0.G-5=ç14/,3X,?,4ETHODLASTUSED=’,Izs3cC+‘LASTSWrrCHWASATT=E12.4)C C 50X0=BPH=Y(2)ZO=Y(I)zC—zFCNPH-O.D0CWRiTE(6,70)ZC,ZO,FCN70FORMAT(2X,’FCN’,3E12.4)RETURN80WRITE(6,90)1STATE90FORMAT(//I5çERRORHALT...ISTATE-‘,13)ENDC C CSUBROTJIINE3ACQ’1,T,Y,ML,MU,PDNRPD)IMPLICiTREAL8(A-H,O-Z)DIMENSIONY(34),PD(34RPD,N)COMMONIPROP/C,SFI’,R,IDXCOMMONIPARA/PCI,W,QPD(1,2)=QDC0S(Y(2))PD(2,2)=-DCOS(Y(2))/Y(3)PD(2,i)=QWPD(2,3DSIN(Yl2))ItY(3)5Y(3))PD(3,2)=-DSINIY(2))ENDSUBROUTINEEX(Nj,Y,D’t)Functionsuppliedfoeus.withLSODARIMPLICrrREALS(A-H,O.Z)DIMENSIONY(N),DY(I4)COMMON/PROP/C,SFr,R,IDXCDMMON/PARA/PCI,W,QCOMMON/LENGTH/XOLD,YOLD,SO,XDY(1)=QDSIN(Y(2))DY(2)’—QDY(1)/Y(3)+QY(l)WDY(3)=DCOS(Y(2))ENDSUBROUTINEGEX(I4j,Y4G,00Ul)IMPLICITREAL8(A-H,O-DIMENSIONY(N).GOUrING)COMMONIGEO/PHIC,PC,XC,A,BCOMMON/ODE/itC CEvaluatethe2equationswhererootsarewantedCGOUT(1)=Y(3)-BEND SUBROUTINEBPLOT(x,Y,M,N)Thissubroutine plotsmeniscuspresswevs.filmthickness forseveralcontactangles, andalsoincludesdashedlinesrepresentinghysteresisfPLICrrREALs(o-zPARAMETER(IM-6,IN—10)DIMENSIONX(N),Y(M,N)REAL’4U(IN),V(IN),Z(IN),IPACIC(2000)CHAP.ACTER4OGTTrL,YTITL,xmL,GTIL2CHARACTER4PCL(IM)COMMON/LIMT/XMIN,3IMAX,YMIN,YMAXC0MMON/PROP/QC,SFT,R,IDXCOMMON/PARA/PC,W,QSC C C C C C C C C C C C C C C C C CC197C CPlottheinteapolatedlineCDO2OI—l,MDo103=1,NCALLMESSAGCb/a‘‘,5.9.0,l6.0)IF(IDXEQ.1)THENCALLMESSAGCCONVE)C,6,6,175)ELSE CALLMESSAGCC0NCAVE’,76,l7.5)ENDIFCALLREALNO(D1,2,ll.,16.)C CLegendforidentitlcationofstartingcontactangleCDO30I=l,MCALLLINEStPCL(1),IPACK,l)WRITE(6,e)PCL(D30CONTINUEXL=XLEGND(IPACIçM)YL’YLEGND(IPACK,M)CALLMYLEGN(#q&(degl,9)CCALLLEGENPACK,M,1l.5,0.CALLLEGEND(IPACE,M,9i0,12.0)CCALLBLREC(0.5,05,XL+l.2,YL+t,O,O.0l)CALLBLREC(9.,ll.5,XL+l.2,YL+l.0,0.0l)CALLENDPL(0)CALLDONEPLENDPlotstherelationshipbetween capillasypressure withseparationdistancebetweenspheresataconstantBondnumberandcontactangle.Bothconvexancconcavecasescanbeobtained.IMPLICrrREL8(A-H,o-Z)PARAMEFER(MM=6,MY’l0,NN’3, MP=MY+l)DIMENSIONY4N),YC(NN),YNalN)DIMENSIONPlMM,MP),P2MP),HiMM,MP)H2(MMMP)DIMENSIONHX(lOO),HY(100),PCAP(MM),PB(MM,MY),BD(MY)E)CTER)ALFFCN,F2,F3,PDF,FZERO,FE/çGEXCl{ARACTER1DOCt{ARACrER3DUMCRARACTER4PCL(M!eINTEGERFLAGLOGICAL12COMMON/LlliffI7aelIN,XMAXYMD4,yMAxCOMMON/PROP/C,SFr,R,IDXCOMMON/GEO/PHIC,PC,A,XC,BCOMMON/CONST/EPS,P1,N,MCOMMON/PARA/PCI,W,QCOMMONIOUr/xo,zoCOMMONILEG/PCLCOMMON/ODE/HCOMMON/CAT/PPCOMMON/BAT/IFLAGC caPART1-InitializeconstantsandpropertiesCM=MMN’NNSFF—45.DoGl.DOO.(1594D0-0998200)5GISFTCSDSQRT(fR”.O.IDOBI0.l05D0I0.lD0W-CR’RCWRiTE(6,23)WRITE(6,223FORMAT(1SX,RUPflJREPRESSUREVS. DISTANCE724FORMAT(30X,’SPHERES’,//)WRiTE(6,33)W33FORMATBo-’,El0.3,,)WRflt (34FORMAT(a’PHr,8’,tEtcPCAcPCAi’[Pa]’)PI’.VACOS(.l.DO)RAD’-PI/lSO.DOC C C C C C CUQ)SNGL(X(l))Z(1)=SNGLIYM)10CONTINUECALLMARXERCALLRASPLN(2)CALLCURVE(U,Z,N.l)20CONTINUECALLHEIGHT(0.3)C CWiitepropertiesongraphCA.sisymntetricSaddle-shapedMcniscusofaSphereinaFieldofSpheres(Gravitational)‘SDI’-DIMENSIONLESSC C C C C C C C C C C C C CCOMMONJGEO)PHIC,PCD,QA,XC,TBCOMMON/LEG/PCLSetupdimensions,title,headings,etc.CALLDSPDEVCPLOl’)CALLUNfl’SCCENT)CALLNOBRDRCALLPAGE(21.,26.)CALLAREA2D(l40.17.)CALLHEIGHT(0.2)CALLGRACE(00)CALLCOMPLXCALLMXIALFCSTANDARD,’&)CALLMX2ALFCL/CGR’,’#)CALLMX4ALFCSPECIAL’,’@)CALLPHYSOR(3.,l.)CALLYAXANG(0)CALLTHKFRM(0025)CALLFRAMEConverttosingleprecisionvaluesAX”SNGL(XMIN)BX=SNGL(XMAX)AY=SNGLIYMJN)BY=SNGL(YMA)QDX”tBX-AX)/5.D0DYBY-A’O/5.D0Dl=SNGL(TB)DFP-SNGL(SFI)DW’.SNGL(V)DR’-SNGLiR)CALLAXSPLT(A3cB)c14.,ORIG,STEP,A)CALLAXSPLT(AY,BY,17.,YOR,YST,B)CALLGRAF(A)cD)cBx,AY,DY,Bt)GTrrL-CAPILLARYPRESSUREVSBONDNUMBER’CALLHEADIN(GflTL,40,20,2)GTIL2=’BetweenSpheresCALLHEADIN(GTIL2,37,2.0,2)CALLHEIGHT(0.CALLSCLPIC(0.9)CALL)CDCKS(2)CALLYTICKS(2)X1Tlt’BondNumberBoyrrrL=’DimensioniessCapillatyPressurePcap@,CALLXGRAXS(ORiG,SCALE’,BX14.,)CIITL,l9,0.,0.)CALLYGRAXS(YO?,’SCALE’,BY.17.,rrrn,39,o.o,0.o)198EPS1.D-513IF(PCCIT.P1/21)0) THENED(IC)CCMPCPPA*Z0*RaS/ 1000CCottve(lDXl)andconcave(IDX’O)ELSEIF(POr.YMAX) YMAX=pCCM=PCIF(P.LT. ThllDX1ENDIFWR(6,12) B,PNIppAIF(IDXEQ.1) ThENDPCMftJBLE(p,1Ror0Q1.D0DO.000LE(My-1)CGOTO18ELSE0050K=1,MyIF(IPLOTEQ. 1)001050Q’-1.D0IR0T=ENDIFENDIFBRI+D*DBLEl)ENTJIFCENDIFCSet typeofplotsquiresLIfIPLOTisOthenhysteresis20CON’DNUECanalysistobedone, ifIPL0Tis1 then enessiscusprofiles0020J’=l,My25WRITE(6,*)‘pp’pCtobeplottedJM3-l50C0NTflJtjCJP-’J+I90CONUNIJEIPLOT’=j16PHlC.DpaDoLE(JpXMINEICA”PI-PC+pHf CXMAXBCSet initial contactangle PCIandincrement DCIF(IDXEQ.0) A=PI-PC-PHICIF(IPLOTEQ. 1) THENCXCDSIN(A)CALLPCI3o.D0CALLHVSP(PIncxc,,çpp)ENDIFDC’=(180 DO-PC1)IDDLE(M1)CWRITE(6,12)31WP.rfE(6) ‘FIN’P.D012F0RMAT(j)FlO3,2FI4100opXMIN”ODOH2(IJt’=HENDXMAX=oJ)o18P2(I,1)=pC4INfloCl8lF(POr.YMAJQojjpCCIF(P.LTYMllYMflipR0tHVSP(PHI,2CC,p)D090I1,MCCCIFGILT,)PT=HCCCPALT2-Detennjze profilesfor severalcontactanglesC-CThissubroutjne solves fortheunIusosaZO&ZCfortheCCFmdrootofHCIF(IROTEQ. 1) THENCandvertical ditsoceof thexneniscusflorathelopIF(II.LT. 0.00)THENCCof theepdemCPreparelabelsfor legendingraphRNEWH2(J,3)CCH0LH23PLICITRL*AIF(PCNE. 0.00)THENDIMENSIONXT(t)NPT=NlNT(PC/pjj)LOGICAL12CALLB1])(NPT,DU,ND”)PH1PJdIC.flpEXTERNALF,FCN,FEX,GEXPCLDj’Pfl2Pfl1CC00NRop/C,50xELSE17PHtCHNEw*(pH2pHlyGIoLpH2CNFr=NINr(pqA_PlPC+Q*pHlCC0MM0N/CoNs’i-/EpS,p,MCALLBTD(PlJ’F,DDI,NDO)C0MMoN/Jo,ZOXCDSlN(A)C0MMoNfoDoENDIFCALLHVSP(PHIC,XC,aç,p)COMONATfpCIF(DABS(lt)Or.EPS)THENC0MMoN/PAJsvpCJW,QCIF(HHNEW.Or 0.00)THENCOMMON/CAT/ppPH2”PJUCCHNEW-HCCalculate ZCand21) usingthe rooloolyor hOCZEROICCalculate locationof levelmeniscusELSECCH0LHIF(PRI.m.0.00) ThENA=Pl-PCPHI=PH[CXr(l)o.D0ENDIFCALLORZFuN(PCN,llooxr,1N0,7EpslEIoWRITE(6,22) PC/RAn+L.E-3)22FORMATQC.A.“,IFlO.3)IF(IND.EQ. 0)00105IF(A.EQ. P1)THENGOTO17WRiTE(6,*)‘DRZFUNFAILS’WRITE(6,)‘Level eneniscusoccursat180’ELSEGOb1000010100B2(I,JP>=H5ZC-XT(i)ENDIFP2(IJp)PGOH) 70XC=DS1N(A)PCAP(1)pELSEIF(IPLOTEQ. 1)001015PB(I,K>”pPCAP—Q.0019910=0.00ZC=0.D0ENDIF70HC’DABS(Z0-ZC)IF(IDXEQ.I)THENHR-DABS(1.00+DCOS(A))ELSE HR-DABS(I.00-DCOS(A))ENDIFlrr=HR-HCI4,’HTHOUr=HPCAP=DABStWZ0)f(2.D0)100RETURNENDC C*FUNCTIONFCN(Z)Cen..,*C CThisfunctiondescribesthedeterminatIonoftheCthecontactpositiononthespheresbyfittingtheCdifferentialequationsuntilconditionsatbothendsCwhereX=XcandX=Baresatisfied.10and72areCunlcnownandXistheindependentvariable.CIMPLICITREAL8(A-H,0-Z)INTEGERFLAGPARAMETERNN-3,LRW=80,LIW=25,NG-2)DIMENSIONYQN),S,YN(5r’IlNN),Q(NN,20)DIMENSIONA1DL(NN),RW0RX(LRW),1WORKtLPV),JR00TtNG)COMMONIGEO/PHIC,PC,A,XC,BCOMMON/CONSTIEPS,PI,N,MCOMMON/ODE/HCOMMON/OUT/Xo,Z0COMMON/LENGTH/XOLD,YOLD,Sl,XEXTERNALFE)GEX,3ACC CSetargurnentstosolvefor3ODEsandlequationCRAD=PI/l80.D0Y(2)=PNICY(1)-ZY(3)=XCT’-O.DOTOIJ1=2.DOBTo-TourC CSettolerances,andparametersforLSODARCITOL—2RTOL-l.D-4ATOL(1)’l.D-tiATOI.(2)1.0-6ATOL(3)1.D-6ITASK=ITSTATE=1loFT-I0051—5,10RWORK(1)=0IWORKtI)=05CONTINUEIWORK(6)=l000rrtC CSetloopforllstrpsanddetennineequationrootsC0040IOtJr=i,i10CALLLSODARFE)cN,Yj,eur,rroL,RToL,A1O1,rrASK,TSTAm,+IOFRW0RcLRW,IWORIcLIW,JAC,rr,GE3cNG,JRool)CWRITE(6,20)T,Y(2)/RAD,Y(3),Y(l)C20FORMAT(lx,’S=‘,E12.4,’PH=‘,E12.4,’X—,E12.4,’Z—‘,E12.4)IF(ISTATEEQ.-1)THENISTATE=3IWORK(6)=1200GOb10ENDIFIF(ISTATEiT.0)GOTOSOIF(ISTATEEQ.2)GOb40CWRITE(6,30).TROOT(l),JROOT(2)C30FORMAT(5X‘rIlEABOVELINEISAROOT,3R0OT-’,2IC CIfthefirstequationsrootis found(X=B)thenstopintegrationCIF(JROOT(1)EQ.1)GOTO50ISTATE=’ZGOTO10401ODBLE(IOU1)C50WRITE(6,60)IWORK(l 1),IWORX(12),IWOR}C(13),JWORX(10),C+IWORK(19),RWORX(15)C60FORMAT(0X’NO.ErEPS.-cI4,5x’NO.F-S-I4,5X,34O. 3-S-’,C+14,SX,b40.G-S=’,W,3HODLASTUSED-’,lb,SX,C+‘LASTSWITOHWASATT-’,E12.4)C C 50X0=BPH=Y(2)l0—Y(1)72=1FCN=PH-0.D0CWRITE(6,70)ZC,ZO,FCN70FORMAT(2X,’FCN’,3E12.4)RETURN80WRITE(6,90)ISTATE90FORMAT(1115X,’ERRORHALT...ISTATE-‘.13)RETURNENDC C CSUBROUtINEJAC(Nj,Y,ML,MU,PD,NRPD)IMPLICITREAL’8(A-H,O-l)DIMENSIONY4),PD(NRPD,N)COMMON/PROP/C,SFr,R,IDXCOMMON/PARAJPCI,W,QCPD(1,2)=QDCOS(Y(2))PD(2,2)=-DCOS4YI2)YY(3)PD(2,1>’*WPD(2,3)=DSINtY(2))/lY(3)*Y(I))PD(I,2)=-DSINIY(2))ENDSUBROIJDNEFEX(N,T,Y,D)FwrctionsuppliedfoeusewithLSODARIMPLICITREAL8(A-H0-Z)DIMENSIONY(N),DY(N)COMMON/PROP/C,SFr,R,IDXCOMMON/PARA/PCI,W,QCOMMON/LENGflI/XOLD,YOLD,S0,XDY(1)-QDSIN(Y(2))DY(2)-.QDY(IyY(3)+QY(1)WDY(3)=DCOS(Y(2))ENDSUBROUrINEGEX(N,T,Y,NGGOUI)IMPLICITREALS(A-Ii,O-Z)DIMENSIONY(N),GOur(IG)COMMON/GEOIPHIC,PC,XC,A.,BCOMMON/ODE/HC CEvaluatethe2equationswhererootsarewantedCGOUr(1)-Y(3)-BENDC C**0C•DISSPLAGraphingRoutaseC.C CSUBROUDNEMNPLor(XI,,c2,MN)C CThissubroutineplotsthemeniscusprofilebetweenCtwoparallelhorizontalcylindersCIMPLICITREAL’8(O-zPARAMETER(IM-9,IN-20)DIMENSIONXl(M,N),X2(M,N),3C3(IN),X4(IN)DIMENSIONPCAP(IM)REAL04UV(IN),Y,U1(IN),YI(IN),AP(IM)REAI.4CAP(JM)CRARAcrER04OGITILl,GTITL2XITI’L,YflThCOMMON/LIMrOD.flN,XMAX,YMII4,YMAXCOMMON/PLOT/GITIL1,GTFIL2,XTFTL,YTITLCGOMMONICAT/PC,Pl,TPCOMMON/ZEROIXX,YYC C C C C C C C C200COMMON/CIRC/X3,X4Y(J)=SNGLtX3(Jt)COMMONILEGIPCLCOMMONIPRESSJPCAPZ<11=SNGLQC4(J))CCU1(J)=SNGL(-Xla,J))CSetupdmensions, title,headings,etc.CDefinestheborders,titles, heading, etc.YIQ)=SNGL(-X3(fl)CCCWRITE(6,99)U(J,V(1),YQ),Z(J)CALLDSPDEVCPLOT)CALLDSPDEVCPLO’P)99FORMAT(lX,4F10.3)CALLUNSCCENT)CALLUN1TSCENT)10CONTINUECALLNOBRDRCALLNOBRDRCALLMARKER(l)CALLPAGE(21,26.)CALLPAGE(2l.,26.)CCALLAREA2D(14.0,17.)CALLAREA2D(14.0,17.)CMersiscus profileectinCALLHEIGHT(02)CALLHEIGHT(0.23)CCALLGRACE(0.0)CALLGRACE(.0)CALLCIJRVE(tJ1,V,N,0)CALLCOMPLXCALLCOMPLXCALLCURVE(tJ,V,N,0)CALLMXIALPCSTAi’IDAIW,’&)CALLMXlALFSTANDARD,&)CALLSPLINECALLMX2ALFçUCGR#)CALLMX2ALFCLICSflY,’)CCALLMX4ALFCSPECIAL,’@)CALLMX3ALFCGREEK,’r)CDrawbothspheres byreflectionof curvesCALLPHYSOR(3.,l.)CALLMX4ALFCINSTRUCTION’,@)CCALLYAXANG(0)CALLMX5ALFCUCGR,#’)CALLCURVE(Yl,Z,N,0)CALLTHKFRM(Q.025)CALLPHYSOR(3.,I.)CALLCURVE(Y,Z,N,0)CALLFRAMECALLYAXANG(0)CALLRLREAL(CAP(l),l,..2,V)CALLT1tKFRM(0.02Pl=Pl+DPCCALLFRAME20CONTINUECCCCConvett tosingleprecisionvaluesCConvert datatosingleprecisionvaluesCDrawstraightlinespecifyinglocationofPcap’0CCCAX’SNGL(xM1N)AX=’SNGL(XMIN)CALLDASHBX=SNOL(XMAX)BX=’SNOL(XMAX)ACX=SNGL(XX)AY-SNGL(YMIN)AY=’SNGL(YMIN)BCY=SNGL(YY)BY=SNGL(YMAX)BY=SNGL(YMAX*l.D0)CALLRLVEC(-ACXBCY,ACXBCY,0l00)DX-(BX-AX)/5.DoDPSNGL(TP)CALLHEIOHT(0.4)DY-(BY-A’t5DOCCDI=SNGL(TB)CReflectsemi-profilearoundY-axistoobtaincompleteCPrint contactangleDFr.SNGL(SFl)CprofileCDW’SNGL(W)CCALLMESSAG#q”,3,2.,7.)DR-SNGLQt)CXSNGL(XMtN-XMAx)CALLREALNO(PC,l,2.7,7.)DX’(BX-AX)/5.D0CALLENDPL(0)CALLAXSPLT(A5cBX,l4.,ORIG,sTEp,A)DY=’(BY.A’i)/5.DOCALLDONEPLCALLAXSPLT(AY,BY,17.,YOR,YST,B)CALLAXSPLT(Ax,BX,14.0,ORIG,STEP,A)RETURNCALLGRAF(AxD)cBX,Ay,DY,BCALLAXSPLT(AY,BY,17.,YOR,YsT,B)END(rlTll—’CAPILLARYPRESSUREVSDJSTMCECALLGRAF(Ax,D)çBXAY,DYBY)CCALLHEADIN(GTITL1,16,3,2)CALLHEADIN(GTITL,40,2.0,2)CALLHEADIN(GTrrL2,29,3,2)CGTIL2=’SetweenSpheres’CALLHEIGHT(0.5)SUBROIJflNEBPL0T2cYM,N)CALLHEADIN(GflL2,37,i0,2)CALLSCLPIC(0.7)CCALLHEIGHT(D.3)CC‘*S*S*fl**fl**flØ*flqflflCALLSCLPIC(O.9)CCCALLXTICKS(2)CALLXflCICS(2)CThissubroutine plotsrneniscuspressurevsCALLYIICKS(2)CALLYIICKS(2)Cfibsthiclatessforseveralcontactangles, andXITIL.”SeparationDistanceb/a’CALLXGRAXS(ORIG,STEP,Bc14.,xTTrL,4,o,o.)CalsoincludesdashedlinesrepresentinghysteresisYTflDbsensioniessCapillaryPressurePcap@,’CALLYGRAXS(YOR,YST,BY,l7.,YrITL,l,o.o,oo)CCALLXGRAXS(ORIG,’SCALE,EXl4.,XflTL,25,O,o.)CALLHEIGHT(0.3)lMPLlCrrREALs(o-zCALLYGRAXS(YOR,’SCALE’,BYj7.,yTrrL,39,o.o,o.o)CPARAMETER(1M%IN’iO)CPlotinterpolatedlineDIMENSIONX(N)Y(M,N)CCU(IN),V(IN),Z(IN),IPACIC(2000)CPlotthe interpolatedfineDO20l’-l,MCHARACTER54Om11,YITII)clTrL,GTlLzCAP(l)=SNGL(Pl)CHARACFER*4PCL(IM)DO20I-l,MCAP(l)=SNGL(PCAP(l))COMMON/Lftff/XMlN,XMA5cyMIN,YM&JcDO103’l,NCOMMON/PROP/QC,SFT,R.,IDXDO10i-l,NU(J)=SNGL(XlO,J))COMMONIPAP,AJPC,W,QSUQ)-SNGL(X1)))V(J)’SNGL(X2(l,J))COMMONIGEO/PHIC,PCD,QA,XC,TB-SNGL(Y(l,J))20110CONTINUECCConvex(IDX—1)andconcave(IDX—0)CALLMARICER(I)CHysteresisPlotsfortheSpheresModelCCALLRASPLN(2)C(Gravitationaleffectsconsidered)lOX-iCIFflXEQ.1) THENCALLCURVE(U,Z,N,t)CQ=1.D020CONTINUEC‘SHYS’-Hysteresis(dimensionless)ELSEC______________________________________________Q—.l.D0CENDIFCCCALLHEIGHT(0.3)CTheeffectofhysteresisisdeterminedaspositionCSet typeofplotrequireriIfIPLOTisOthenhysteresisCCoftheinterfacemovesupalongthesphere.AspositionCanalysistobedone, ifIPLOTis1thenmerriscusprofilesCWritepropertiesongraphCincreasesthecapillarypressurealsochanges.ThisCtobeplottedCCprogramcanbeusedforeitherconvert(IDX=l)orCCconcave(IDX—0)menisci.ThisshowsthehysteresisIPLOT-1CALLMESSAGBo,5,9.0,l6.0)Cplotforspheres(dimensionless).ForprofilesuseCIF(IPX.EQ.1)THENC‘SPALP.CSet initialcontactanglePCIandincrementDCCALLMESSAGCCONVEX6,6,l7.5)CCELSEIMPLICrrREALB(A-H,O-Z)PCI—30.l)QCALLMESSAGCCONCAVE’,7,6j7.5)DC-(l80.D0.PC1)IDBLECM.1)ENDIFPARAMETER(IdM=6,MY-20,NN=3,MP=MY+l)DEG-2.DORADCALLREALNO(DW,3,ll.,l6.)DIMENSIONYcNN),YC(I1N),ThNN)P0-0.00CDIMENSIONPl(MM,MP)P2(MM,MP),Hl(MMMP),H2Q.tM,MP)XMIN=0.D0CLegendforidentificationofstartingcontactangleDIMENSIONHX(l00),HY(ltO),PCAP(MM)XMAX’-O.DOCEXTERNALF,FCN,F2,F3,PDF,FZERO,FEX,GEXYMIN=0.D0DO30l-1,MCHARACIER1DDYMAX=0.DOCALLLINES(PCL(I),IPACIcI)CHARACTER3DUMDO30I-l,MWitrrE(6,)PCL(I)CHARACTER4PCL(MM)C30CONTINUEINTEGERFLAGCPART2-Determine profilesforseveralcontactanglesXL-XLEGND(IPACK,M)LOGICAL12CYL-YLEGND(IPACK,M)COMMON/LIMT/XMIN,XMA)cYMIN,YMAXPC-(PCI+DCDBLEØ.l))RADCALLMYLEGN#q&(desJ’)COMMON/PROP/CSFTR,!DXPP-PCCCALLLEGEND(IPACK,1i.5t.COMMON/GEO/PHIC,PCA,XC,BCCALLLEGEND(IPACK,M,9.50,l10)COMMON/CONSTIEPSPI,N,MCPreparelabelsforlegendingraphCCALLBLREC(0.5,0.5,XL+l.ZYL+l.00.0l)COMMON/PARA/PCI,W,QCCALLBLREC(9.,l0.5,XL+l.2,YL+l.0,0.0I)COMMON/OIJr/XOZOIF(PCNE.0.1)0)THENCALLENDPL(0)COMMON/LEG/PCLNFr—NINT(I’C/RAD)CALLDONEPLCOMMON/ODE/HCALLBTh(IWr,DUM,3,ND.”)RETURNCOMMON/CAT/PPPCL41)=DUM/f$’ENDCOMMON/BAT/IFLAGELSECOMMON/STOP/DEG,IERP.NPT-NINT(PC)CCALLB1D(NPT,DD1,ND,0C•PARTI-InitializeconstantsandpropertiesPCL(1)-DDM.Yi0Y/’CENDIFM—MMCN=NNC•PART3-HysteresisProfile-SFr—45.D0Ci)Determinecurvesfordifferentextentsof0-901.D0Chysteresisataconstant contactline,C—(l.594D0-0.9982D0)G/SFrCA-vonstandXC-constCS=DSQRT(C)CP-l.D-4B=3.OODOW—CR’RCCalculate locationoflevelmeniscusWRTE(6,’)’BO—’WCIERR=0A-PI-PCWRtIE(6,)PCPORADPI=DACOS(-i.D0)IF(AEQ.P1)ThENRAD..Pt/180.DOWRflE(6,*)LcveImeniscusoccematISO’EPS=i.D-50010) 100CELSEIF(A.EQ.0.00)THEN202GOTO15+P=’,E12.5,’A=’,F7.2)CndverticsldistanceofthemeniscusftomthetopENDIFH2(1,J3)=HXC=DSIN(A)P2(1,33)-PCofthecylinders.CMIN=A18IF(P.GT.YMAX)YMAX=PCH=0.D0IF(PUT.YMIN)YMIN=PIMPLICITREAL8(A-H,O.Z)DH=’CMIN/DBLE(MY)IF(H.GT.XMAIQXMAX=’HDIMENSIONXT(l)00103=I,MPIF(IILT.XMIN)XMIN=HLOGICAL123M=J-lCEXTERNALF,FCN,FE)çGEXAJt=OFI1JBLE(3M)CFindrootofHCOMMON/PROPIC,SFr,R,IDXCCCOMMON/CONST/EPSI,N,MCi)IncrementhysteresisforgivenXCandAIF(IROT.EQ.I)THENCOMMONIOIJr/X0,ZOCThemeniscusslopeangleiseqaivalenltothehysteresisIF(HUT.0.00)THENCOMMON/ODE/HourCCCOMMON/BAT/IFLAGPHIC=AItCStore preslousmeniseusinnextaddressCOMMON/ST0PIDEG,IERR.CCOMMON/PARA/PCI,W,QDEG—2.DO9LADH2(1,JP)=H2(1,3)COMMON/CAT/PPCP2(1,JP)=P2(1,J)CCHNEW=H2(I,J)CCalculateZCandlOusingtherootsolverUBCZERO1CDetermineunlinowusZCand10fromsolutionofasystemHOLDH2(I,3M)CCofODEsIF(PHI.NE.0.1)0)THENCXr(l)=o.DoCALLHVSP(PHIC,XC,A,ZC,H,P)PHI=PHTC-DPCALLDRZFUN(PCN.1.loo.Xr.TNO,5.E-7,EPS,l.E-lO,WRITE(6,12)PHICJRAD,XC,ZC,ZO,H,P,A/RADPH2=PHIC+I.E-3)Hl(1,3)=H17PHIC=HNEW(PH2-PHIy(1IOLD-HNEVQ)+PH2IF(INDEQ.0)GOTO5Pl(1,3)=PA=PI-PC+QPHICWRITE(6,)‘L)R2UNFAILS’IF(Par.YMAX)YMAX=PCZC-UlIF(P.LT.YMJN)YMIN=PXC=DSIN(A)GOb100IF(HOf.YMAX)XMAX=t-tCALLHVSP(PHIC,XC,A,ZC,H,P)5ZC-XT(1)IF(HUT.XMIN)XMIN=HIF(DABS(1l).GT.EPTHEN00107010CONTINUEIF(15HNEW.GT.ODO)THENELSEIERR’=OPH2”PNICPCAP=0.D0CHNEW=HZ0=0.D0Cii)ThemeniscusmovesalongthecylinderataconstantELSEZC=0.D0Ccontactangle(PC),PHIandAchange.HOLDHENDIFcPH1=PHIC70HC=DABS(ZO-ZC)CIfcontactsngleisabove9o,thenlimitis180ENDIFIF(IDX.EQ.1)THENC001017HR-DABS(l.D0+DCOS(A))ELSEELSEH2(I,J)-HHR-DAS(I.D0-DCOS(A))P2(1,3)—PENDIFISDP=PC/DBLE(MX)PCAP(I)”PHT-HR-HCIROT=1WRITE(6,12)PIIIC/RAD,XC,ZC,ZO,H,P,A/RADH’HT00203=l,MYIROT=0HOUPHJM=3-lGOTO18PCAP-DABS(WZOY(2.D0)JP=3+lENI)IF100RETURNIF(IROTEQ.0)THENENDIFEND33=3+1ENDIFELSE20CONTINUEC33=325WRrrE(6,)cENDJF30CONTINUEFUNCTIONFCN(16PHIC=DP’DBLEQM)31CALLMYPLOT(Hl,PI,1t2,P2,M,MP)Cs*******fl**Se*******n***a..*...*......*.........*s.100STOPCA=P1-PC+PHICENDCThisthnctiondescribesthedeteiminationoftheIF(lOXEQ. 0)A=PI-PC-PHICCCthecontactpositiononthespheresbyfittingtheXC=DSIN(A)C**CdifferentialequationsuntllcondidonsatbothendsSUBROiJDNEHVSP(PHI,XC,A,ZC,H,PCAP)CwhereX=XcandX=Baresatisfied.10andZCareCALLHVSP(PHIC,XC,A,ZC,H,P)CunknownandXistheindependentvariable.IF(IERR.EQ.1)001025CCWRiTE(6,12)PHICIRAD,XC,ZC,Z0,H,P,A/RADCThissubroutinesolvesfor theunlcnowns10kZCfortheIMPLICITREAL*8(A-H,O-Z12FORMAT(t)F7.2,F7.2,2E10.2,lt=,F7.2,CnreniscusandthencalculatesthecapillatypressureINTEGERFLAG203PARAMETER(NN-3,LRW=S0,LIW=25,NG=2)DIMENSIONY(NN),SiNN),YN(50),Y1(NN),Q(NN,20)DIMENSIONATOLN),RWORK(LRW),IWORK(LIW),JROOT(NG)COMMON/GEO/PHIC,PC,A,XC,BCOMMON/CONSTIEPS,PI,N,MCOMMON/ODE/HCOMMON/OUT/XO,ZOCOMMON/LENGIWXOLD,YOLD,SI,XEXrERNALFEX,GEX,3ACC CSetazgtmsentstosolvefor3ODEsandleqoafiosiCRAD=P11180.D0Y(2)=PHICY(l)-ZY(3)=XCT’=O.DOTOtJr_2.D0*BTo=TOTJrC CSettolerances,andparametersforLSODARCrrOL=2RTOL-l.D-4ATOL(l)”1.D-6ATOL(2)=l.D-6ATOL(3)’1D-6fl’ASK=tISTATF”lIOPTlDO5I=5,l0RWORK(l)=0lWORK(l)05CONTINUEIWORK(l000rrlC CSetloopfor11stepsanddetermineequationrootsCDO40IOUT’l,l10CALLLSODARIFEX,N,Y,T,Tour,ITOL,RTOL,ATOL,rrASK,ISTATE,+lopT,RWORIcLRW,IWORJçLIW.JAC,JT,GE3çNG.SROO’l)CWRiTE(6,20)T,Y(2)/R.AD.Y(3),Y(1)C20FORMAT(1X,’S”‘,E12.4,’PH=‘,E12.4,’X=‘,El2.4,Z”‘,E12.4)IF(ISTATEEQ.-1)ThENISTATh3lWORK(wi200GOTO10ENDIFIF(JSTATE.LT.0)GOTO80IF(ISTATEEQ.2)GOD)40CWRITE(6,30) .IROO’IXI),JROOr12)C30FORMAT(5)ThEABOVELINEISAROOT,JROOT’.ç215)C CIfthefirstequation’srootisfound(XB)thenstopintegrationCIF(JROOT(l)EQ.1)GOlD50ISTATE”2GOTO1040TOUTTO*DBLE(IOIJI)C50WRITE(6,60) JWORK(ll),IWORK(12),IW0RK(13),IWORK(lO),C+IWORK(19),RWORK(15)C60FORMAT(/3X,340.EPS=’,I4,5XO.F-S-,I4,5X’NO.1-S-p,C+14,5X,’NO.G-S’,14/,3X,’MErHODLASTUSED.d,l2,5XC+‘LASTSWITCHWASATT.d,El24)C C 50X0=BPH=Y(2)ZOY(l)ZC=ZFCN”PH-O.DOCWRiTE(6,70)ZC,ZO,FCN70FORMAT(2X,PCN’,3E12.4)80WRITE(6,90)ISTATE90FORMAT(//15X,’ERRORHALT...ISTATE-‘.13)ENDC C C C C C CSUBRO{JflNEJACO4.T,Y,ML,MU.PD,NRPD)IMPLICITP8(A.Ho)DIMENSIONY(I4),PD(NRPD,N)COMMON/PROP/C,SFT,R,IDXCQMMON/PARA/PCI,W,QPD(l,2>=QDCOS(Y(2))PD(2,2)=’-DCOS(Y(2))/Y(3)PD(2,l)—QWPD(2,3)—DSIN(Y(2))/(Y(3)Y(3))PD(3,2.DS1N(Y(2))ENDSUBROIJflNEFEXCN,T,Y.DY)FunctionsuppliedforusewithLSODAIt.IMPLICITpu.sT,’LC8(A4{ODIMENSIONY(N),DY(N)COMMON/PROP/C,SFT,R,IDXCOMMON/PARA/PCI,W.QCOMMON/LENGTH/XOLD,YOLD,SO,XDY(1)=QDSIH(Y(2))DY(2).Q*DY(l)/Y(3)÷QsY(l)aWDY(3)”DCOSIY(2))ENDSUBROtJITNEGEX(N.r,Y,No,00ur)IMPLICiTREAL8(A-H.C)-Z)DIMENSIONY(I4),GOUTING)COMMON/GEO/PHIC,PC,XC,A,BCOMMON/ODE/H(2 CEvaluatethe2equationswhererootsarewantedCGOur(l)-Y(3)-BEND SUBROUTINEMYPLOT(Xl,Yl,XZY2,M,N)Thissubroutine plotsmeniscuspressurevs.filmthickness forseveralcontactangles, andalsoinchidesdashedlinesrepresentinghysteresisIMPLICITREAL8(O-Z)PARAMETER(IM=6JN’-21)DIMENSIONXI(M,bX2(N),Yl(M,N),Y2(M,N)PL4U(lN),V(IN),Y(IN),Z(IN),IPAClC(2000)CHAR.ACTER4OG’ITfl,,YITrL,XDTL,GTIL2CHARACTER4PCL(]1COMMON/LIMT/XMIN,XMAJçYMIN,YMAXCOMMON/PROP/QC,SFT,R,IDXCOMMON/PARAIPC,W,QSCOMMON/GEO/PHIC,PCD,QA,XC,PDICOMMON/LEG/PCLC CSetupdimensions, title,headings, etc.CCALLDSPDEV(’PLOT)CALLUNTSCCENT)CALLNOBRDRCALLPAGE(2l.,26.)CALLAREA2D(l5.0,19.)CALLHEIGHT(0.2)CALLGRACE(0.0)CALLCOMPLXCALLMXIALFCSTANDARD’,’&)CALLMX4ALF(L!CGH,’#)CALLMX5ALFSPECIAL’,’@CALLPHYSOR(3.,l.)CALLYAXANG(0)CALLThl(FRM(0.02)CALLFRAMEConvecttosingle precisionvaluesAX=SNGL(XMIN)BX-SNGL(XMAX)AY-SNGL(YMIN)BY-SNGL(YMA)QDX-(BX-AXy5.D0DY-(BY-AYYS.D0C C C C C C C C CC C C CC C C C204DI=SNGLtPDI)DO30I=l,MDFTSNGL(SFI)CALLLINES(PCLCI),IPACJcI)DW=SNGL(W)WRITE(6,a)PCLQ)DR=SNGL(R)30CONTINUECALLAXSPLTçk)cB)c15.,ORIG.STEP,A)XLXLEGNDOPACK,M)CALLAXSPLT(AY,BY,19.,YOR,YST,B)YL=YLEGND()PACIçM)CALLGRAF(AX,D)cBX,AY,DY,EY)CALLMYLEGN#q&(deg)$’,9)G11TL=’MENISCUSBE1WEENSPHERESONANINTERFACECALLLEGEND(IPACK,M,1.,15.0)CCALLLEGENDQPACK,M,L0,1.0)CALLHEAIJIN(GTITL,40,2.0,2)CALLBLREC(.50,14.5,XL+1.2,YL+t.0,0.01)G11L2=’CapillasyPressurevs.FilmThickness’CALLHEADIN(GTIL2,37,2.0,2)CALLENDPL(0)CALLHEIOHT(0.3)CALLDONEPLCALLSCLPIC(0.9)REI1JRNCALL)CflCKS(2)ENDCALLYflCKS(2)XTIU=’FilmThicknessh/a’YflTL’DimensionIessCapillasyPressurePcap@,’CALLXORAXS(ORIG,STEP.BX,15.,)CrITL,Is,o.,0.)CALLYGRAXS4YOR,YST,BY,19.,YTITL,39,0.0.0.0)CALLHETGHT(0.3)CALLMESSAG#_&Constantlq’,l5,t0.7,l6.)CALLMESSAO#...&Constant#a’,17,l0.7,15.2)C CPlottheinterpolatedlineCDO20T=I,MDO1OJ-l,NUQ)SNGLQCl(l,J))V(J>SNGL(X2tI,3t)Y(J>=SNGLtYIII,i))ZQ’SNGL(Y2(I,Jt)10CONflNUECALLMARKER(I)CALLRASPLN(2)CALLDOTCALLCURVE(U,YN,0)CALLRESET(DOT)CALLCURVE(V,4N,l)20CONTINUECALLDASHCALLRLVEC(0.0,AY,0.0,BY,0100)CALLRLVEC(A)cO.o,BX,o.o,0100)CALLHEIGHT(0.3)C CWritepropertiesongraphCCALLMESSAG(Do‘‘,4,I0.7,18.)CALLMESSAOçb/a‘,5,I0.7,17.2)IF(XEQ.1)THENCALLMESSAGCCONVEX’,6,619.3)ELSE CALLMESSAGCCONCAVE,7,6,19.5)ENDIFCALLREALNO(DW,-3,12.0,t8.)CALLREALNO(DI,2,12.0,17.2)C CLegendforidentificationofstartingcontactangleC20500CLS20PRINT“°*CDIGITIzINGFNDORANFORNENISCI**************”30PRINT*ThisBASICprogramwaswrittenfortheobjectiveof”40PRINT“digitizingphotographicimagesofneniscifor”00PRINT*comparisonwiththeoreticalcurves.Theeeniscipoints”60PRINT“areindimensionlessformbydivisionwiththeradius”70REMofthecylindricalrods.80PRINTC5*0*6*eoaaa*osssooa*oCCC*0*0*55*0*5*55*560****5*60*505*”90PRINCorr.JUN.4/93*tooPRINT110PRINT“120DIMX(200),Y(200),XM(200),YN(200),XN(200),YN(200)130DIMXC(3),Yc(3),CD(4),ANG)2),CA(2),XPC(2),XPM(2),XPN)2)135DIMYPC(2),YPN(2),YPN(2),CX)20),CY(20),CAVG(2),COTD)2)137DIMB(3,4),o(3),00(3),D(2),A24(3,4),NI(3),XT(4,3),YT(4,3)140NPICSO150MAXIT300160ALPN1.7170EPSo.0001175P1—3.14159271180RD=O190OPEN“B:NEN.DIG”FOROUTPUTAS11200000UB2000210PRINT“NONMANYNENISCUSPROFILES”;220INPUTNPICS$230PRINT#1,NPIDS$240PRINT245PRINT#1,“250PRINT“SIX—LETTERNAMEOFSAMPLE260INPUTNAS265PRINT#1,NM270NPTSO280N=0290NPICSNFICS+1300CEO310PRINT“*s****aesssases**es**6655a500**s*000soe****u”320IFNPIS>1GOTO330ELSE370735PRINT“DOYOUNIONTOUSETNEPREVIOUSICALINcFACTOR(V/N)”;340INPUTCs350IFCS—V’ORC$”y”GOTO350360IFC$=”N”ORC$”n°GOTO370370005UB4000380REM*6*6DETERMINERADIUSOFCYLINDERS*66*66395CLI400PRINT*66*DETERMINETNEDIAMETEROFTNECYLINDERS*6*0”410PRINT***5ANDTNELOCATIONOFTNECONTACTPOINT*0*0”420ND1425N3427NPN+1440ST—2450FORM1TOST485ODOUR5000450001UB8000540XPC(MfrS)1)550YFC(M)=I(2)560D(M)=S(3)5SF52580PRINT“CYLINDERDIA.=”,D(N)890PRINT“DOYOUMISNTOREDOTNECYLINDERDIAMETER?”;600INPUTAS610IFAS=”Y”DRAS—”y”GOTO480620IFA$=”N”DRA$=”n”GDTD625625IFN—iTNENPRINT“DIGITIIENEXTCYLINDER”630.NEXTN640R=(D)1)+D)2))/4645RSTD—( (D(1)/2—R)“2t(D(2)/2—R) “2)“.0650PRINT“AVG.CYLINDERRADIUI—”,R,”ITDDEV—”,RSTD680PRINT“DOYOUMOONTOREDOTNECYLINDERDIAMETERS?”;690INPUTA$700IFA$—”Y”ONA$”’”y”GOTO390710IFA$—”N”ORA$—”n”GOTO715715FORNtTOST716PRINT11,“CYLINDER#”,M,“DIANETER=”,D)M)717NEXTN718PRINT#1,”720REM05*6*05*5*RE—ORIENTBASELINE****s*00000**s*730CLI740PRINT5*555*50*0*0*BASELINERE—ORIENTATION*55*5*6*05*6*750PRINT760PRINT“CENTRESOFCYLINDERSARETNEBASELINE”770BETA——ATN((YFC(2)—YFC(i))/(XFC(2)—XPC)1)))772XO—(XPC(i)+XPC(2))/2774YO—(YFC(i)+YPC(2))/2778FOR1=1TO2776XFN(I)(3CpC(I)—XO)eRF/R777YFN(I)=(YFC(I)—YO) *IFfR778NEXTI780FORI—iTO2790XFN(I)=XPN(I)*CO$(BXTA) —YFN(I)*IIN(BETA)800YFN(I)XFN(I)*EIN(BETA)+YFN(I)*COS(BETA)810NEXTI871DX2*FI/20872FORK1TO2873FORJ1TO20874OX(J)—XPN(K)+IIN(DX6J)875CV(J) =YFN(K)+001(DX*J)876PRINT#1,CX(J),cY(J)877NEXTS878NEXTK890CLI900REM*50*DIDITIIECONTACTFDINTSANDINCLUDE910PRINT“DIGITIIECONTACTPOINTSALTERNATELY3920N—i930FORN1TO6940001UB3000950X(N)=X:Y)N)—Y955X(N)=X(N)*COS(SETA)_Y)N)SSIN)BETA)957Y(N) “X)N)5SIN(BETA) +Y(N)COS(BETA)960IFX)N)X)N—1)ANDY)N)—Y)N-1)TNENNN—1970IF(X=11ANDY=0)TNENN=N-3980NEXTN990REEF1000X1AVO=(X(t)+X(3)+X(5flf31010X1SD—()X)1)—X1AVO)”2+)X(3)—X1AVG)”2+)X)5)—X1AVG)”2)”.51020X2AVG—(X(2)+X(4)+X(6))/31030X2SD=UX(2)—X2AVD)”2+(X(4)—X2AVGY3+(X(6)—X2AVGy2)”.51040Y1AVO(Y(1)+Y(3)+Y(5))/31050Y100=UY(i)—Y1AVG)”2+(Y(3)—Y1AVGY2+(Y(5)—nAVGy2)”.s1060Y2AVG=(Y(2)+Y(4)+Y(Gfl/31075Y210—i.)Y(2)—Y2AVO)”2+(Y(4)—Y2AVG)”2+(Y)6)—Y2AVG)”2)”.S1072X)1)X1AVOINPROFILE*6*6TINES*2061074X(2)X2AVG1076Y(i)=Y1AVG1078Y(2)Y2AVG1080PRINT“X1AVG”,XiAVO,“ST.DEV.”,XISO1090PRINT“Y1AVG=”,Y1AVO,”ST.DEV.”,YiSD1100PRINT“X2AVG”,X2AVG,“ST.OEV.=”,X2501110PRINT“Y2AVO”,Y2AVG,“ST.DEV.”,YlSD1120N01130POR1=1TO21140T_(Y(I)YPC)Ifl*2*SP/(O(I))1180ALP——ATN(T/SQR)—TT+lfl+1.5707633#1159ANO(I)ALP*180/PI1160PRINT“ALPNA”,I,ANG)I)1165CA(I)=180—ANG)I)1170NEXTI1172AG—(ANO)1)tANG)2))/2:CAO—(CA)i)+CA(2))/21174ASTD—UANO)1)—AG)2+(ANG(2)—AO)”2)”.51176CASTDUCA)i)—CAG)2+)CA(2)—CAG)”2).51177PRINT“ALPNA=”,AG,“lTD.OEV.=”,ASTD1178PRINT#1,”3179PRINT#1,AG,CAO,CASTO1180PRINT#1,”1181PRINT“CONTACTANOLE(LEVELNEN.)—”,CAG,”STD.DEV.—”,CAlTO1182PRINT“DOYOUWISNTOREDOTNECONTACTPOINTS?”;1184INPUT0$1186IPO$”Y”ORO$—”y”THENGOTO9001188IPO$”N”ORO$”n”THENGOTO11901190REM****DETERMINEWNETNERCONVEXORCONCAVE*****1200PRINT***DIGITIIEAPEXPOIN1210GOSUE30001220X(3)—X:Y(3)=Y1220XM)3)=)X)3)—XO)*SP/R1227YM(3))Y)3)—YO)*SP/R1230XN(3)XM)3)*COS)BETA)_YN)3)*SIN)EETA)1240YN)3)=XN)3)*SIN)BETA)+YN)3)*COS(EETA)1270PRINTTAE)10)“X(A)=”;XN(3);”Y)A)—”;YN)3)1200YRAXA8E(YN)3))1290PRINT“YNAX—”,YMAX1300PRINT1310XA=X(3):YA=Y(3)1320CLI1330IFYA>Y)i)THENIDX=0:RER***CONVEXNENIECUS***1340IPYA<Y)1)TNENIDX—1:REM***CONCAVENENIECUE***1350PRINT“IDX”,IOX1360RER*“““DIGITIZEDROPLETPROFILE******1370PRINT“*****DIGITIZEDROPLETPROFILE(BETWEEN30—70POINTS)1380PRINT*)THECONTACTPOINTSAREINCLUDEDINTNEPILE-SOITIS”1390PRINT“NOTNECESSARYTORE—DIGITIZETHEN)”1400PRINT1410PRINT“*TOENDOIOITZZINOPROFILECHOOSEANYOFTNERENUKEYS”1420PRINT“*1430NN+11440IFR<2OOTO14651450IFR>2GOTO14601460GOSUB30001470X)M)=X:Y)R)—Y1480IF)X=11ANDY=0)THENN—N—1:GOTO14601485XN)N)—)X(N)—XO)“5F/R14E7YN)N)=)Y(N)—YO)*SF/R1490XN)N)_XN(N)*Coo)BETA)—YN)N)*SZN)BETA)1500YN)M)XN(N)*SIN)BETA)+Y74(R)*COS)BETA)1510IF(N<>2)TNENGOTO19301530REMIFN—iTHENGOTO15701540IFX(N)X(N—1)ANDY(R)’Y)R-l)THENN—N-i:GOTO14301550IFY0THENGOTO15701660IFY>0ORY<0GOTO14301570BEEP1980NUTS—N—i1590REM****WRITEDATATOFILE****1600PRINT#1,SF,IDE,HPTS1605PRINT#1,”1610FOR1=1TORUTS1620PRINT#1,XN)J),YN)J)1630NEXTJ1635PRINT#1,”1640PRINT“.DOYOUWISHTOCONTINUE(Y/N)”;1650INPUTDN$1660IFDN$’””Y”ORDN$—”y”THEN250i670IFDR$’””N”ORDN$”n”TNEN16801680PLAY“CEDC”1690CLOSE11700CLOSE21710END1720END2000RESt*8*4ENABLEDIGITIZER“““*42010OPEN“CON2:9600,O,7,i”AS#22020CLS2030RETURN3000REM““SUBROUTINEGETXANDY*4*43010A$INPUT$(i2,#2)3020B$NID$(A$,2,S)3030C$=MID$(A$,?,S)3040X—VAL(B$):Y—VAL(C$)3050IF(Y’”G)THENGOTO30603060PRINTN,“X=“;X,“Y=“;Y3070PRINT3080RETURN4000REM****SUBROUTINEGETSCALINGFACTOR*4*44010CLS4020PRINT“***4030PRINT“CALCULATIONOFSCALINGFACTORBYDIGITIIING3POINTS”4040PRINT“OFTHECALIBRATIONSPHERE—(REPEAT3XTOOBTAINAVERAGE)”4050PRINT4060Mi4070ST34080ND—i4090FORN—iTOST4100IF)X—12ANDY=0)THENN—N—i4110GOSUB50004120DS(N)D4130PRINT“OZA—”,D4i40NEXTN4150DAVG)DS)i)+DS)2)+DS)3))/314160DSTUDS)i)—DAVGV2+)DS(2)—DAVOY2+(DS)3)—DAVG)2).54170PRINT“DAVG—”,DAVG,“ST.DEV”,DET4180PRINT4190DI.31554200PRINT4210BEEP4220PRINTK44*44*4*******444*****K2074230SFDI/DAVG4240PRINT“SCALINGFACTORIS:“;SF4250PRINT***********4260PRINT“DOYOUNISNTOREDOSCALINGFACTORPOINTS(YIN)”;4270INPUT0$4280ND14291IFO$=”Y”ORG$=”y”TNENOOTO40004310IFO$=”N”ORO$=”n”TNENOOTO43104310RETURN5000REM****CALCULATEDIAMETEROFCIRCLES****5010PRINT5020PRINT5030PRINT“OBTAINDIAMETERBYTOUCNING3POINTSONCIRCLE”5040PRINT“FO,TOQUIT”:PRINT“F2:TOREDOTNECIRCLE”:PRINT“F2:REOOPREVIOUSRCLE”5050005UR60005060IF(X11AND10)OR(1=12ANO5=0)TNENOOTO50505070005UR70005080RETURN6000REM*5<5*GET3POINT6010J16020ODOUR30006030IF1=10AND1=0TNENCLOSE#1:005UR7000:END6040IF1=13AND1=0TNENPRINT“REDOTNISFOINT”:JJ—1:OOTO60206050IF1=12AND1=0TNENPRINT“REDOPREVIOUSCIRCLE”:OOTO60106055IF1=13AND1=0TNENPRINT“REDOINSCYLINDSR”:OOTO4506060XC(J)=X:YC(J(16070J0+16080IFJ<4TNSNOOTO60206090IFX=11AND1=0TNSNFRINT“RSDO”:J=,7-1:OOTO60206100RSSP:RSTURN7000RECALCULATEDIAMETS7010FOR1=1TO37020R(I)=—)XC)I)2+YC(I)3)7030NEXTI7040DS=XC(1)*(YC(2)_YC(3)(=YC(1)*(XC(2)_XC(3fl+XC(2)*YC(3)_YC(2)SXC(2)7050IF05=0TNSNRSSP,BEEP:PRINT”RSDOTNISCIRCLS”:RSTURN7060DAR(1)”(YC(2)—YC(3fl—0C(1)”(R(2)—R(3))+R(2)*YC(3)=YC)2)*R(3)7070DRXC(1)*(R(2)—R(3)(=R(1)*(XC(2)—XC(3))+XC(2)*R(3)_XC(3)*R(2)7080DC=XC(1)*(YC(2)*R(3)YC(3)*R(2))YC(1(*(XC(2)*R(3)R(2)*XC(3fl÷R(1(*(XC(2YC)3)—YC(2)*XC(3(7090ACDA/DS7100BC=DB/0S7110CCDCfDE7120DIA=AC”2+RC24CC:IPDIA<=0TNSNRSSP;BSSP:PRINT”RSDOTNISCIRCLS”:RSTU7130D=SQR(DIA)7140PRINT“CYLINDSR#:”;M“DIAMETER:“;D7150PRINT7160XP=—AC/2:YP=—RC/27170RA=D/27190RETURN8000REM*°*LEASTSQUARESFInINGOFCYLINDSRPROFILE8010REM***CALCULATIONOFAUOMSNTSDMATRIX““58011FORJ=1TO38012XT(4,J)=XC(J(8013YT(4,J(=YC(J)8014D5(J(=0!8015NEXTJ8017S(1)=XP8020S(2)=YP8025S(3)—R.A8026NRITE#1,”ROOTS”,S(1),S(2),S(2)8027REM*MNistheno.ofdigitizedpoints,Nistheno.ofego’s.8028MN128029PRINT“DIGITIZECYLINDERPROFILENITN9PDINTS”8030FOR1=1TO38040GOSUB60008050FORJ=1TO38060XT)I,J)=XC(J)8070YT)I,J)=YC(J)8080AM(I,J)=08100NEXT28105AM)I,NP)08110NEXTI8115REM**Appliesonlyiftoppertofoylinderdigitized8120FOR55=1TOMAXIT8130DIFMAXO!8125RAS(3)8140FOR11TO48150FORJlTO38160Dl—XT(I,J)=S(l)8170D2=(RA5RA)-)015D1)8174PRINT“02”,55,028176IF(02<0)TNENGDTO80298180n3=SQR)D2)819004)D2))l.5)8200n5—=YT(I,J)+s(2)+D38210AM)l,l)A11(l,1)+(2*D1501/D2)=2e)D15D1*D5/04)—2*D5/D28220AM)l,2)=AM(l,2)+2*D1/D38230AM(l,3)=AI4)l,2)+)2*RA*D1/D2)=2*RA*D15D5/D48240REMAM(2,1)=AM)2,1)+2*Dl/D38245AM)2,l)=AM(l,2)8250AM(2,2)=AM(2,2)+28260AM(2,3)=AM(2,3)+2*RA/D38270REMAM(3,l)=AM)3, 1)+2*RAeD1fD2=2*RA*D5*Dl/D48275AM(3,l)=AM)l,3)8280REMAM)3,2)—AM)3,2)+2*RA/D38285AM)3,2)=AM)2, 3)8290AM(3,3)=AM)3,3)+2*05/D2=2*RA”RA*D5/D4+2*RA*RA/D28300REM*55*CoeffioientsofRNSofequations*e*8310AM(l,4)=AM)l,4)=2*DS5D1/n28320AM(2,4)=AM(2,4)=2*D58330AM)3,4)=AM(3,4)=2*D5”RA/D38340NEXT28350REXTI8360CDSUR90008265IF(IERROR=1)001083708367PRINT“ISRROR”,IERJ8OR8370FOR1=1TON8380S)I)S)I)+ALPN*DS)I)8290DY=ABS)ALPN”DS(I)/S(I))8400IF(DY>DIFMAX)TNEMDIFMAX=DY8410NEXTI8420IF)DIFMAX<SF5)001084358430NEXTKR8432PRINT“MARNING—CDNVSRGSNCSFAILURE”8435PRINTflu,“ITSR=”,KK,S(l),S)2),S(3)8437PRINT“iter=”,KK,S(l),S)2),S(3)8440REM5*Verisnoeoffit*58480VAR=0208SIinrowsKtoNIPIVOTisnotKB)I,J)”B(I,J)-R)K,J)*QUOTNEXTJ942094309440NEXT9400NEXTK9460FORK1TON9470DS(KfrB(K,NP)/B(K,K)9480NEXTK9490REM**Checklastdiagonalelementforazeroentry9000IF)R)N,N)<>0)TNENGOTO90309510IRRROR29520RETURN9530REM*Calculatenormofresidualvector,normalreturnwith9540REM**IERROR—l9550RSQO9560FOR1=1ToN9570SUMO9580FORJ=lTON9590SUM=SUM+A24(I,J)eDS(J)9600NEXTJ9610RSQ=RSQ+(ZiM(I,NF)—SUM) 29620NEXTI9630RNORHSQR)RSQ)9640IENROR19650RETURN9660END5455YP=S)2)0457EFS)1)8458WRITE‘N,N”,NN,N8460FOR1=1TO48470FORJ1TO39480VAR=VAR+)1T)I,J)—IP—SQR)RARA—)XT)I,J)—XP)’2)’2)8490NEXTJ8500NEXTI8005VAR=VAN/(MN-N)8510FRINT“TNEVARIANCEOFTRELONFITIS“VAR8560RETURN9000REM*°ssGAUSS-JORDANELIMINATIONMETNOO*ee*9005REMasPrepareworkingmatrix9**9007N39008NF49010FOR1=1TON9020FONJlTONF90308)I,J)=AN)I,J)9040NEXTJ9050NEXTI9005REMaeSsarchforlargestcoefficientincolumnK,forrowsK9056REM**throughN.IPIVOTistherowindexofthelargest9057REM*5coefficient.9060FORK1TON9070XF=K+19080IPIVOTK9090FORI=KTON9100NIT’A80)0)I,I))9110FORJ=KPTON912099A8S)B)I,J))9130IF(00>NIT)THENNIT=BN9140NEXT09100NI)I)NIT9160NEXTI9165REMe*Searchforlargestscalingfactor9170010=ANS)N)K,R)/NI)K))9180FORI=KPTON9190SI=ASS)8)I,K)/NI)I))9200SF(SI<=BIG)THENGOTO92309210010=819220IFIVOTI9230NEXTI9240REMaaInterchangerowsKandIFIVOTif9250IF)IFIVOT=K)TNENGOTO93109260FORJ=XTONP9270TEMP=B)IPIVOT,J)92800)IFIVOT,J)=8)X,J)9290N)K,J)=TEMF9300NEXT09310REM°Checkdiagonalforazeroentry,inwhichcaseareturn9320REM*5isperformedwithIENROR=29330IF)8)K,K)5>0)THENOOTO93609340IEKRON=29350RETURN9360REN5*Eliminatealltermsexcepttermsinthediagonal9370FOR2=1TON9380IF(I=K)GOTO94409390QUOT8)I,K)/0)K,X)94000)I,K)=09410FORJXFTONP209

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