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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 Adsorbed at a Liquid-Fluid Interface by Linda Hou B.A.Sc. University of British Columbia, 1988  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES  (Department of Chemical Engineering)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA © Linda Hou, November 1993  In presenting this thesis in partial fi.ilfihlment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I fi.irther agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of Chemical Engineering University of British Columbia 2216 Main Mall Vancouver, B.C., Canada V6T 1Z4 Date: November 1993.  U  ABSTRACT  One 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 squeeze between the interfacial particles and come into contact with another droplet’s free liquid interface must be overcome and provides a barrier to the thinning of the film between droplets. In light of the work done by Denkov et al., the objective of the thesis has been to determine the effect several factors such as particle size, separation distance, wettability, fluid properties, and contact angle hysteresis have on emulsion stability, represented in terms of capillary pressures, using a two-layer particle model for coalescence. Two general models were developed, one which is based on a uniform layer of spherical particles adsorbed on a fluid-fluid interface, and the other, a two dimensional analogue, in which parallel, horizontal cylinders are situated on the interface. Numerical techniques were applied in the solution of both models since no simplification was made by the neglect of gravity as is found in similar models (8, 11, 12). As a consequence, the models can be used to describe macroscopic systems whose characteristic dimensions are well above the micron scale.  A corresponding  experimental system employing parallel cylinders was then constructed and used to generate capillary 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 the particles, the closer the packing of particles, or the rougher the solids are, the more stable the emulsions.  Furthermore, a decrease in the Bond number or a decrease in wettability of the  particles 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 angles which yield optimal stability from capillary and thermodynamic considerations lie in the range of 900 O 1800.  111  The study revealed that good agreement between the model and the experimental measurements was obtained when apparent contact angles, which take into account hysteresis, were used to generate the model profiles for each meniscus.  iv  TABLE OF CONTENTS  Abstract  .  ii  List of Figures  .ii  List of Tables  xiii  Acknowledgements  xiv  1. Introduction  1  2. Background  5  2.1 General Concepts of Solid-Liquid Interfacial Thermodynamics  5  a) Interfacial Tension and Free Energy  5  b) Contact Angle  6  c) Hysteresis  8  d) Minimization of Free Energy in Emulsions  8  2.2 Equilibrium of Particles on Horizontal Interfaces  9  2.3 Emulsions Stabilized by Finely Divided Solids  10  2.4 The Young-LaPlace Equation and the Equilibrium Shape of Menisci  12  3. Mathematical Analysis of the Axisymmetric Interface between A Sphere and its Neighboring Spheres  17  3.1 Literature Review of Related Sphere Models  18  3.2 The Sphere Model Configuration  19  3.3 Mathematical Formulation of the Differential Equations of the Meniscus Shape  23  3.3.1 Concave Menisci  24  3.3.2 Convex Menisci  26  3.4 System of Equations in Alternate Coordinates  28  3.5 Boundary Conditions  30  3.6 Effect of Contact Angle Hysteresis  31  V  3.7 Determination of Film Thickness  .34  3.8 Numerical Solution  35  4. Mathematical Analysis of the Capillary Shapes between Parallel, Horizontal Cylinders  42  4.1 Literature Review  43  4.2 System Configuration  43  4.3 Mathematical Formulation  45  4.3.1 Convex Meniscus  46  4.3.2 Concave Meniscus  50  4.4 Determination of Film Rupture Threshold  53  4.5 Numerical Computation  54  5. Theoretical Model Results  61  5.1 Relationship to Emulsion Droplet Coalescence  61  5.2 Characterization of Profile Curves  63  5.3 Cylinders Model  64  5.3.1 Effect of the Contact Angle  66  5.3.2 Effect of the Bond Number  66  5.3.3 Effect of the Separation Distance  73  5.4 Spheres Model  77  5.4.1 Effect of the Contact Angle  77  5.4.2 Effect of the Bond Number  80  5.4.3 Effect of the Separation Distance  85  5.5 Effect of Hysteresis  85  5.6 Comparison of Cylinders and Spheres  90  5.7 Relationship to Emulsion Stabilization with Finely Divided Solids  91  6. The Experimental Program  98  6.1 Meniscus Profile Image Recording Background Trials  98  6.2 Final Image Definition Technique  101  -  vi  6.3 Experimental Equipment and Setup  .101  6.3.1 The Experimental Cell  103  6.3.2 Micromanometer  104  6.3.3 Lighting  106  6.3.4 Photographic Equipment  106  6.4 Experimental Preparation of Rods  106  6.5 Experimental Procedure  107  7. Results and Discussion  112  7.1 Comparison with Theoretical Curves  112  7.2 Separation Distance Between Rods  116  7.3 Effect of Contact Angle  119  7.4 Effect of Hysteresis  121  7.5 Apparent Contact Angles  122  7.6 Hysteresis and Kinetic Forces  132  7.7 Sources of Experimental Error  137  8. Conclusions  139  Nomenclature  143  References  146  Appendix A Experimental Data  152  Appendix B Sample Calculations and Derivations  164  -  -  B. 1 Mathematical Formulation for the Negligible Gravity Case for Cylinders  165  B.2 Jacobian Matrix for the Solution of the Spheres Model  169  B.3 Capillary Pressure Measurements for Experiments and Sample Calculations  170  B.4 Digitizing Program  173  Appendix C Computer Programs -  181  CALP  182  CBO  184  vii  CD!  .187  CHYS  .189  SPALP  .192  SBO  195  SD!  198  SHYS  202  Digitizing Program  206  vu’  LIST OF FIGURES  Figure 1.1 Denkov et. al.’s one layer-particle model for coalescence  2  Figure 1.2 Two-layer particle model of present work  3  Figure 2.1 Intermolecular forces at an interface  5  Figure 2.2 Illustration of the three interfacial tensions at the s/o/w interface  7  Figure 2.3 Emulsion droplet with spherical particle on interface  10  Figure 2.4 Radii of curvature of a curved surface  12  Figure 3.1 Emulsion droplet with adsorbed layer of particles  18  Figure 3.2 Cylindrical cell model for spheres  20  Figure 3.3 Coordinate system for sphere cell model  21  Figure 3.4 Radii of curvature of surface of revolution described by the curve  21  Figure 3.5 Capillary Rise  22  Figure 3.6 Capillary Depression  22  Figure 3.7 Concave meniscus profile  25  Figure 3.8 Convex meniscus profile  26  Figure 3.9 The surface of an idealized rough solid with a sine-wave corrugation  32  Figure 3.10 Point of rupture between two emulsion droplets  34  Figure 3.11 Program flowsheet for meniscus between spheres  40  Figure 4.1 Experimental cell (simplified)  42  Figure 4.2 Coordinate system for parallel cylinders  45  Figure 4.3 Angles for convex (a) and concave (b) menisci  52  Figure 4.4 Cylinder model flowsheet of program CALP  57  Figure 4.5 Cylinder model flowsheet of program CHYS  59  Figure 5.1 Meniscus profiles between a pair of cylinders (Bo3.247, 0=90°, b/a  =  1.5)  65  Figure 5.2 Meniscus between parallel horizontal cylinders, Bo=0. 130, 0=30°  67  Figure 5.3  67  ,  Bo0.130, 0=60°  ix  Figure 5.4  “,  Bo=0.130, 0=900.67  Figure 5 5  “,  Bo=0 130, 0=120°  67  Figure 5.6  “,  Bo=0.130, 0=150°  68  “,Bo=0.130, 0=180°  68  Figure 5.7  I’  Figure 5.8 Threshold rupture pressure vs. contact angle (cylinders), b/a= 1.50  69  Figure 5.9 Meniscus profiles between parallel horizontal cylinders, Bo=1.30x10 , 0=90° 5  70  Figure 5.10  ‘I  , 0=90° 3 Bo=1.30x10  70  “,  Bo3.248, 0=90°  70  Bo=1.30x10’, 0=90°  70  Figure 5.13 Rupture capillary pressure vs. bond number (cylinders), b/a 1.5  71  Figure 5.14 Rupture capillary pressure vs. bond number (cylinders), b/a3 .0  71  H  Figure 5.11 Figure 5.12  H  u  I’ ,  Figure 5.15 Meniscus profiles between a pair of cylinders, Bo0.130, 0 =90°, b/a H  Figure 5 16  “  Figure 5 17  “  H  ‘,Bo=O 130 0=90° b/a  =  1.05  74  1 50  74  Bo0 130 0=90°, b/a = 3 00  74  Bo0.130, 0=90°, b/a  6.00  74  Figure 5.19 Rupture capillary pressure vs. separation distance (cylinders), Bo=0. 130  75  Figure 5.20 Rupture capillary pressure vs. separation distance (cylinders), Bo=3 .247  75  Figure 5.21 Meniscus profiles between spheres, Bo=0.130, 0=30°  78  I’  H  Figure 5.18  “,  H  Figure 5.22 “  Figure 5.24  “  Figure 5.25  “  Figure 5.26  “  =  Bo=0.130, 060°  78  “,Bo=O. 130, 0=90°  78  ,  Figure 5.23  =  H  Bo=0.130, 0120°  78  “,Bo0.130, 0=150°  79  H  79  ,  Bo=0.130, 0=180°  ‘I  Figure 5.27 Threshold rupture pressure vs. contact angle (spheres), b/a 1.50  81  Figure 5.28 Meniscus profiles between spheres, Bo=1.30x10 , 0=90° 5  82  Figure 5.29 Figure 5.30  H  H  I’  ,  , 0=90° 3 Bo1.30x10  82  ,  Bo=3.247, 0 =90°  82  x  Figures.31  “,Bo=l.30x10’, 0=900.82  ‘I  Figure 5.32 Rupture capillary pressure vs. bond number (spheres) , b/a=1.50  83  Figure 5.33  83  b/a3.0  ,  Figure 5.34 Meniscus profiles between spheres, b/a= 1.05, 0=90° Figure 5.35  “  Figure 5.36  “  I  H  b/a1.50, 0 =90°  86  ‘I  “,b/a=3.00, 0 =90°  86  b/a=6.00, 0 =90°  86  Figure 5.37  ‘I  Figure 5.38 Rupture capillary pressure vs. separation distance (spheres) , Bo0. 130 H  Figure 5.39  86  ‘I ,  Bo=3.247  87 87  Figure 5.40 Rupture capillary pressure vs. film thickness (cylinders) , Bo=0. 130, b/a= 1.50  88  Figure 5.41  88  “  “  “,  Bo=3.247, b/a1.50  Figure 5.42 Rupture capillary pressure vs. film thickness (spheres) , Bo=0.130, b/a=1.50  88  Figure 5.43  88  “  ‘I  ,  Bo=3.247, b/a=1.50  Figure 5.44 Particle size effect on cap (cylinders and spheres)  93  Figure 5.45 Cylinders-separation distance effects on  95  Figure 5.46 Spheres-separation distance effects on  95  Figure 5.47 Hysteresis Plots:Rupture capillary pressure vs. Film Thickness (cylinders), , b/a1.05 7 Bo1.30x10  96  , b/a=3.00 7 Figure 5.48 Hysteresis Plots (cylinders), Bo=1.30x10  96  , b/a1.05 7 Figure 5.49 Hysteresis Plots (spheres), Bo=1.30x10  96  , b/a=3.00 7 Figure 5.50 Hysteresis Plots (spheres), Bo=1.30x10  96  Figure 6.1 Original test setup  99  Figure 6.2 Negatives produced by laser light technique  99  Figure 6.3 Experimental equipment setup  102  Figure 6.4 Experimental cell  103  Figure 6.5 Plan view of the bottom cell; front view of the test cell  105  Figure 6.6 Coating of rods with polymeric resin  108  xi  Figure 6.7 Calibration sphere on microscope slide  109  Figure 6.8 Experimental program flowsheet  111  Figure 7.1 Experiment B meniscus profiles  113  Figure 7.2 Experiment C meniscus profiles  113  Figure 7.3 Experiment D meniscus profiles  114  Figure 7.4 Experiment E meniscus profiles  114  Figure 7.5 Experiment I meniscus profiles  115  Figure 7.6 Experiment J meniscus profiles  115  Figure 7.7 Model relationship between Pcap’ and q& for various b/a  117  Figure 7.8a Comparison of set 1 data Pcap’ V5. 9 c 5  118  Figure 7.8b Comparison of set 2 data  118  -  -  cap’ VS. c 3 ‘  Figure 7.9 Model relationship between capillary pressure and  cS for various 0  120  Apparent contact angles vs. a position  123  Figure 7.11 Set 2 Apparent contact angles vs. a position  123  Figure 7.10 Set 1  -  -  Figure 7.12 Experiment B comparison with model  -  Figure 7.13 Experiment C comparison with model  -  Figure 7.14 Experiment D comparison with model  -  Figure 7.15 Experiment E comparison with model Figure 7.16 Experiment I comparison with model Figure 7.17 Experiment J comparison with model  -  -  -  V5. c  124  cap’ VS. 1  124  VS. c  125  VS. øc  125  VS.  øc  126  VS.  Ø  126  Figure 7.18 Experiment B curve (B-CVXI6) fit with model curves for 0=78°  128  Figure 7.19 Experiment C curve (C-CVX16) fit with model curves for 0=88.5°  128  Figure 7.20 Experiment D curve (D-CVXI7) fit with model curves for 0=84.5°  129  Figure 7.21 Experiment E curve (CVX28) fit with model curves for 0=75.1°  129  Figure 7.22 Experiment I curve (CVX7) fit with model curves for 0=70.5°  130  Figure 7.23 Experiment J curve (CVX6) fit with model curves for 0=67.2°  130  Figure 7.24 Experiment B comparison with model a position vs. -  c  131  xii  Figure 7.25 Wilhelmy plate method  132  Figure 7.26a Force vs. depth curves for the Wilhelmy plate apparatus with hysteresis  133  Figure 7.26b Force vs. depth curves for the Wilhelmy plate apparatus without hysteresis  133  Figure 7.27 A hysteresis ioop for a system showing solid-liquid interaction  133  Figure 7.28 Hysteresis loop for experiment J  135  Figure 7.29 Typical experimental results for the dependence of the dynamic contact angle on speed of the contact line  137  xlii  LIST OF TABLES  Table 5.1 Properties of selected fluid-fluid interfaces (20°C)  63  Table 5.2 Cylinders: Capillary Pressure Dependence on Particle Size (b/a= 1.05, c= 12.99)  91  Table 5.3 Cylinders: Capillary Pressure Dependence on Separation Distance (a= 1 tm)  92  Table 5.4 Spheres: Capillary Pressure Dependence on Particle Size (b/a=1 .05, c=12.99)  92  Table 5.5 Spheres: Capillary Pressure Dependence on Separation Distance (a1 .tm)  92  Table 5.6 Comparison of Denkov et al.’s results with the present work  97  Table 7.1 Contact Angle Hysteresis  121  Table 7.2 Reproducibility of the Level Meniscus for Set 2  136  Table A. 1 Micromanometer Readings  153  Table A.2 Set 1 Compiled Experimental Data  154  Table A.3 Set 2 Compiled Experimental Data  156  Table A.4 Experiment B Comparison of -  baSed on a and c 5  158  Table A.5 Experiment C Comparison of Pcap’ based on a and  159  Table A.6 Experiment D- Comparison of  based on a and c  160  Table A. 7 Experiment E Comparison of  based on a and c  161  Table A.8 Experiment I Comparison of  based on a and cb  162  Table A.9 Experiment J Comparison of  based on a and c 5  163  -  -  -  -  xiv ACKNOWLEDGEMENTS  I wish to thank the following people who have helped in making this thesis possible, my supervisors Dr. B. Bowen and Dr. S. Levine for their expertise, direction and proofreading; the workshop crew notably John Baranowski and Chris Castles for construction of the various experimental cells, Dr. 3. Yeung for supplying the dental rubber; Horace Lam for advice on photography and darkroom film processing; various students and friends for their encouragement and 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.  1  Chapter 1  INTRODUCTION  Solids-stabilized emulsions are encountered in the mining of tar sands and the recovery of petroleum. Their presence on the most part is troublesome since the separation and removal of these emulsions is essential for the purification of the petroleum product and for the prevention of fouling of process equipment in the refining process (1, 2). The properties of these three-phase emulsions, known as Pickering emulsions, can however be desirable, as in pharmaceutical and food manufcturing for the stabilization of creams, lotions, or edibles like mayonnaise (3). It is not surprising therefore that the inordinate resilience of these emulsions formed between two immiscible 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 solidsstabilized emulsions was conducted at the turn of the century by S.U. Pickering, to whom these emulsions owe their name (4). He studied the effectiveness of a number of different metal sulfates as emulsion-stabilizing agents. Since then, investigations have covered the interfacial rheology, the interfacial structure, the kinetics of coalescence and aggregation, the formation mechanism, solidliquid interactions, the free energy and force analyses of the stabilized interface at equilibrium, and also the methods of destabilizing these emulsions.  Several comprehensive reviews have been  written on this topic (5, 6, 7). Tn summary, the literature shows that the main factors which are involved 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 both  liquids; iii) the particles must be in a state of incipient flocculation ; and iv) the increased roughness of the particles produces a higher stabilizing efficiency.  2  The physico-chemical mechanisms by which the particles stabilize emulsions were dealt with by Tadros and Vincent (10) in a simplified model, and Levine et al. (11) in a more refmed theoretical investigation. Both papers deal with the energy of adsorption of solid particles at a  liquid-liquid interface.  Levine et al. (11) hypothesized by thermodynamic arguments that the  particles partitioned on the interface of these emulsion droplets are caught in a deep free energy well (approximately 106 times the typical Brownian thermal energy) and thus they remain in that state unless sufficient energy is applied to the system for them to desorb.  Another factor for the  stabilizing effect of the adsorbed particles was pointed out by Denkov et al. (8) who reported that the capillary pressure required for the liquid between the particles to squeeze through and come into contact with another such droplet would be a barrier to the thinning of the liquid film between these droplets. For example, for a densely packed monolayer of particles of 1 tm radius adsorbed with a contact angle of 60° (measured through the film), on an interface having a tension y of 30 mN/rn, their model (see Figure 1.1) predicts that the minimum capillary pressure required for film rupture to occur would be very high, approximately  Pa.  2 p  Figure 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 of solids-stabilized emulsions in light of the paper written by Denkov et al. (8).  Their model for  3  coalescence involved the thinning of a liquid film between two similar emulsion droplets with one layer of spherical particles wetted above and below by the two droplets (Figure 1.1).  They  proceeded to prove that particle hysteresis should produce a further stabilizing effect since it would cause 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 film rupture was much lower. Furthermore, they showed that adsorption energy considerations alone predict a maximum emulsion stability at an equilibrium contact angle of 90° and a minimum stability at 0 e  =  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 0 e  90°, while 0 e  =  1800  (assuming the contact angle is measured through the dispersed phase) would yield the most stable emulsions. These two opposing effects would lead to the conclusion that the most stable films occur when the equilibrium contact angle is neither too close to 90° or 180°.  The capillary  pressure effect was found to be highest when the hysteresis is larger and when the particle size is smaller which is in agreement with experimental results. To test the idea that a principal factor for coalescence is the contact of particle-free surfaces extruded by the capillary pressure from two emulsion droplets, we investigate the necessary conditions for this to occur, in the case of a film formed by a two-monolayer particle model (Figure 1.2).  Figure 1.2 Two-layer particle model of present work  Since both colliding droplets each have an adsorbed layer of particles, the geometry depicted in Figure 1.2 is a more realistic representation than the single monolayer configuration  4  shown in Figure 1.1. The developed pressures for the two-particle layer geometry considered here are expected to be larger than those of the one-layer model of Denkov et al. The effects of varying the wettability of the solids, the interparticle distance, the particle size, and the extent of contact angle hysteresis will be investigated. Two general models are developed, one which is based on a uniform adsorbed layer of solid spherical particles at the interface of a spherical droplet dispersed in another fluid (similar to the Denkov article), and then as an extension, we look at a two-dimensional analogue, in which we consider the meniscus interactions due to our parameters on a cylindrical interface between parallel, horizontal stationary cylinders. Both models were solved by numerical techniques since simplifications generally found in other models (8, 11, 12), such as the neglect of gravity and constancy in interface curvature, were not used here. The main impetus for tackling this more complex problem is that the solutions obtained can be applied to the more macroscopic experimental systems needed to verif the results. In the accompanying experimental study, the capillary pressures were measured using a micromanometer while the equilibrium shapes of the menisci formed between two parallel cylinders were photographed and later digitized for comparison with the model-predicted profiles.  The  variables investigated in this experimental study were the separation distance between the rods, the variation in the contact angle obtained by changing the adsorbed surface film on the solids, and the effect of hysteresis.  The predictions obtained from the cylindrical interface model were then  compared with experimental results. The results of the two theoretical models are also discussed with respect to the findings by Denkov et al. (8). Then, in light of the larger picture of the stability of emulsions, our observations about capillary effects will be discussed.  5  Chapter 2  BACKGROUND THEORY  2.1 General Concepts ofSolid-Liquid Interfacial Thermodynamics  Capillary phenomena are responsible for such fundamental occurrences as tears on a wine glass, the wetting and spreading of liquids on textiles, dewdrops on grass blades, the wailcing of insects on water, and of course, the rise of liquids in capillary tubes. These capillary effects occur as a consequence of intermolecular forces of attraction which are unbalanced at the interface between two immiscible fluids (Figure 2.1).  TT  phase 2  Figure 2.1 Intermolecular forces at an interface  a) Interfacial Tension and Free Energy  A measure of this excess force for any pair of immiscible fluids is known as the interfacial tension, y or as surface tension if one of the phases is air. Surface tension is often equated with  6  the surface stress or the surface free energy which is true only in certain circumstances. It is however, more precisely defined not as a “tension”, but as the reversible work required to create a unit area of new surface. For solid-liquid systems, the surface tension is equal to the surface stress  and 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,  1 w=y+>p  w,  by the equation:  [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 liquid drops in air or drops on solids, they will take the shape which minimizes the free energy of the system.  Gibbs (1878), whose thennodynamic model of surfaces (13) is the fundamental treatise  which underpins the field of wettability, demonstrated that the minimization of free energy for a solid-fluid-fluid system (e.g. a solid/oil/water system) corresponds to the minimization of the following terms:  A+y 0 y A+ 0  where A is an area, and the subscripts  [2.2]  3 YH’S A  ow, os,  and  ws  denote the oil-water, oil-solid, and water-  solid interfaces, respectively.  b) Contact Angle  This 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 Figure 2.2):  7  row cos(6)  =  [2.3]  —  oil water  ws Figure 2.2 Illustration of the three interfacial tensions at the solidloillwater interface  This equation is applicable for surfaces which are ideally smooth, homogeneous, planar, and nondeformable and when all phases are in mutual equilibrium. Young’s equation, though generally accepted and deceivingly simple, has never been experimentally verified due to the difficulty of measuring the solid-fluid interfacial tensions (14, 15). The equilibrium contact angle that the intersection of the two fluid phases makes with the solid is a useful measure of the wettability of the solid. Gibbs’ analysis defines the existence of one stable thermodynamic contact angle. However, in practical systems, a range of different stable angles is often measured. The literature is replete with examples of uncertain, conflicting and confusing 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 under very 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 of Gibbs’ was later extended by Johnson and Dettre (19), who include hysteresis by modeling a surface with concentric sinusoidal grooves.  Bikerman (20) and Neumann and Good (21).  Other qualitative treatments have been given by  8  c) Hysteresis  There are two types of hysteresis: i) true or thermodynamic hysteresis, and ii) kinetic or time-variant hysteresis. The interpretation of thennodynamic hysteresis is generally based on the idea of metastable states caused by a number of sources such as contamination, roughness, surface heterogeneity, and “surface immobility on a macromolecular scale” (14, 22). Kinetic hysteresis is usually associated with slow equilibrium times. Furthermore, it should be noted that there is some evidence that the contact angle may also be dependent on the speed with which the contact line is advanced or receded (14, 15).  d) Minimization of Free Energy in Emulsions  Now how do these concepts relate to solids-stabilized emulsions?  In fact, all these  thermodynamic properties in some way affect how the solids remain on the interface and may explain the mechanism of stabilization. With the concept of surface free energy in mind, it is clear that in the absence of gravity, a free liquid drop will try to take on a spherical shape to minimize its free energy and thus its free surface area. Similarly, a dispersed droplet within another continuous phase will create a spherical interface, but a swarm of such droplets will be thermodynamically unstable because of its high free surface area and high free energy compared to that of coalesced droplets. The ultimate stability of such a dispersion depends on the presence of stabilizing agents such as surfactants or particles adsorbed on the interfaces. When a third phase such as a solid is situated at the fluid interface, the thennodynamic equilibrium involves many new factors.  9 2.2 Equilibrium ofParticles on Horizontal Interfaces  Under gravity, a solid particle may move through one fluid and enter the fluid/fluid interface or pass through it into the second fluid depending on the densities of the particle and the fluids, the size of the particle, the interfacial tension, and the contact angle at the three-phase boundary, 0(23, 24). If the particle density is intermediate between the two fluid densities, then the particle will always find an equilibrium position at the interface no matter what the interfacial tension, contact angle, or the size and shape of the particle (23). However, if the particle density is greater or less than both fluid densities, its final position will be determined by other factors. In the first case (i.e., particle with greater density), for a given interfacial tension, contact angle and set of densities, there exists a critical particle size above which the interface cannot support the particle and 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 an equilibrium position at the interface but, in this case, the motion of the particle is in the opposite direction. In terms of the surface energetics, those solid particles which meet the size criterion will remain at equilibrium on the planar interface only if they are partially wetted by both liquids. In other 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 considerations that condition [2.4] also applies to curved interfaces as in the case of a particle adsorbed on a droplet (subscript 2) in a continuous phase (subscript 1) (Figure 2.3).  10 1  Figure 2.3 Emulsion droplet with spherical particle on interface  On a horizontal interface, the presence of any solid particles will cause distortions to the interface which can either be convex or concave relative to fluid 1. However, in the case of spherical 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 for the droplet to exist (7).  2.3 Emulsions Stabilized by Finely Divided Solids  lii order to be effective for emulsion stabilization, the adsorbed particles must be considerably smaller than the droplets, be partially wettable by both fluids, and be in a state of incipient flocculation (6, 9).  In a gravitational field, these small adsorbed particles in close  proximity to each other will experience a mutual attraction and will tend to aggregate together on the 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 protected from coalescence with other droplets.  A complete, close-packed layer of adsorbed particles  provides steric stabilization and, if charged, also provides repulsive charge stabilization preventing coalescence of the emulsion droplets. In the absence of gravity, this capillary interaction does not exist for a perfectly planar interface (26), but becomes important again for highly curved interfaces because of the significant pressure difference which exists between the droplet and the continuous medium (11). For very fine particles, there are also other, relatively short-range forces such as van  11  der Waals attraction as well as double layer and Born repulsion, which may affect the stabilizing ability 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 contact angles 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 the thermodynamics of particles partitioned between the 01W interface of a droplet and the continuous phase. They showed that an isolated spherical particle with an appropriate contact angle is trapped in a deep energy well at the interface of the emulsion droplet. Pieranski (31) came to a similar conclusion for polystyrene beads at a water/air interface. As a result of these large adsorption energies, in order to maintain an equilibrium partition between particles on the interface and those remaining in suspension, these adsorbed particles must experience a counterbalancing repulsive force, which Levine et a!. believe to most likely be the short-range Born repulsion forces which become important at close-packing conditions (11).  The electric double layer, short-range  solvation, and solid elastic forces, as well as the van der Waals and capillary forces between the particles 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 the  particles had a contact angle close to 90°. Although this large adsorption energy is important in the stabilization of individual emulsion droplets, it does not, on its own, explain how the coalescence of such droplets can be induced 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 the particles are completely immersed in the non-wetting (disperse) phase. In this manner, liquid from the droplet is squeezed between the particles in the film thereby creating a free surface which could come into contact with a corresponding free surface on a second droplet. The strength of the  12  applied force required to achieve coalescence by this mechanism is related to the capillary pressure engendered in the droplet due to the required distortion of the free (i.e., liquid-liquid) interface. The magnitude of this pressure can be determined using the Young-LaPlace equation which relates the capillary pressure to the equilibrium shapes of the menisci formed.  2.4 The Young-LaPlace Equation and the Equilibrium Shape ofMenisci  As a consequence of surface tensions, when two phases are separated by a curved interface there is a pressure difference (zIP) across it which balances these forces. The higher pressure exists on the concave side. The relationship between the two principle radii of curvature of the surface (Rj and R ), the interfacial tension 2  (fl, and the  pressure difference can be derived from  thermodynamic concepts (14, 33). Figure 2.4 shows a segment of an arbitrarily curved surface. The surface is bound by  Figure 2.4 Radii of curvature of a curved surface  two planes which are perpendicular to each other and normal to the surface. On each plane, the surface describes an arc whose length will be denoted by x andy on the respective planes. The two radii of curvature which describe the surface are designated Rj and R 2 and lie on either plane. If  13  the surface is now displaced a small distance outward, the arclengths on each plane would be increased to x+dx and y+dy, respectively. This increase in surface area is  dA =(x +cLv)(y +)—xy  [2.5]  dA=xdy+ydx.  [2.6]  which is approximately  The free energy involved in the increase of area is  dG =ydA =y(xdy -i-ydx)  [2.7]  and the expansion of volume due to the pressure work across the interface acting on the area xy through a distance of dz, can be written  dW=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 R , we know that from similar triangles the 2 equality of the following ratios holds:  x+cfr  +dz 1 R  R  [2.10]  14  which yields  [2.11]  Furthermore,  y+dy +dzR R 2  [2.12]  which yields  [2.13]  Using these equations we can eliminate the differential elements in equation [2.9] which results in the following final form of the relationship between the pressure drop, the radii of curvature, and the 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 from sessile and pendant drops to open network soap films. The equation can be used for any curved interface in a gravitational, centrifugal, or other force field.  15  Generally, the Young-LaPlace equation yields a second-order differential equation which can only be solved analytically in very few cases. Usually, this occurs if the radii of curvature are equal, as in the case of spherical menisci, or if one radius is infinite as in the case of cylindrical interfaces. Analytical solutions are also possible for general radii of curvature if gravitational effects are either nonexistent or negligible, as in the case of very small sessile drops in a liquid of different density or menisci of microscopic dimensions (23)  .  For a planar surface, both radii of  curvature are infinite. Therefore, there is no pressure drop across the interface, i.e. zlP=O. But planar interfaces are not the only ones which have zero pressure drop. For example, curved soap  films over an open framework have the shape of a catenoid for which  1  1  [2.15]  Substituting equation [2.15] into [2.14] again produces the result that J.P=O. In this case, the opposite signs mean that the radii of curvature lie on either side of the interface thus producing a saddle-shaped interface. With emulsion interfaces adsorbed by finely divided solids, it is often reasonable to assume constant curvatures and thus constant zIP. For macroscopic systems, the radii of curvature are often not constant but vary spatially (33). For surfaces with an axis of symmetry in Cartesian coordinates, analytical geometry can be used to derive the following relationships for Rj and R 2 as functions of (xz):  z 2 d 3/2  [216 1  16  and  dz _=:j: 1 _  [217]  2 F (dz1” xI 1+1  Likewise, the pressure difference will vary along the meniscus since it is a function of capillary height, i.e.,  zSP(z)=  1  +  1  (x,z)] 2 (x,z) R 1 LR  The convention of signs adopted here for curvature will be as follows. If the  [2.18]  centre of curvature of  Rj lies in the upper phase, phase 1, then R 1 is positive. Similarly, for R R are . If both Rj and 2 2 positive, then Pj is larger than P . If the interface is saddle-shaped as it is around a sphere, then 2 1 R  2 must be opposite in signs, and the pressure difference can be either positive or negative. and R In terms of emulsion stability, we will consider two different two-dimensional models  which 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 particles dispersed in a continuous oil phase. The capillarity will be investigated around a single sphere bounded on all sides by other spheres whose saddle-shaped meniscus has circular symmetry about the vertical axis. The second model is a simplification for experimental purposes, and involves considering a cylindrical interface between two horizontal parallel cylinders of infinite length.  17  Chapter 3  MATHEMATICAL ANALYSIS OF THE AXISYMMETRIC INTERFACE BETWEEN A SPHERE AND ITS NEIGHBORING SPHERES  Except in a few cases, the meniscus shapes for which there are explicit numerical or analytical descriptions in the literature are usually two-dimensional (e.g., cylindrical interfaces) or axisymmetric (e.g., sessile drops).  Furthermore, only when gravity and other body forces are  neglected, are there convenient analytical solutions since, under these circumstances, the surfaces have 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 in the microscopic regime, it is often accepted that gravity has a negligible effect on the system and is  ignored (8, 11). In contrast, we develop two general mathematical models for macroscopic systems in which gravity plays an important role, as well as for microscopic systems where the gravitational effects are comparable or larger than the capillary effects. The reason for doing so is to extend the models into the regime where comparisons of their predictions with our macroscopic laboratory experiments become possible. The first model is an idealization of a three-dimensional case where spherical particles are adsorbed in a well-ordered array on a spherical oil-water interface.  The second model is a  simplified two-dimensional problem in which parallel cylinders of infinite length lie horizontally on a fluid interface.  The latter case was selected because it is more amenable to laboratory  investigation. In both models, we assume that the diameter of the emulsion droplet is significantly larger than the diameter of the particles sitting on its boundary such that, if we consider a small section of the 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 solids  18  (Figure 3.1).  ae>> a  Figure 3.1 Emulsion droplet with adsorbed layer of particles  The requirement that a  >>  a is one of the conditions which is necessary for solids to  stabilize emulsions as discussed earlier. The other conditions are that the solids must be partially wettable by both fluids which means the contact angle can be neither  00  or  1800  (6, 7) and that the  particles adsorbed at the interface must form a close-packed monolayer (9).  3.1 Literature Review ofRelated Sphere Models  The profile produced around an isolated sphere, a vertical rod, fluid drops or lenses, or outside a Du Nouy ring is axisymmetric and, unlike the menisci for sessile drops, does not intersect its axis of symmetry. It is considered a bounded menisci having the contact line with the solid at one end and the free level interface at the other. Several different numerical approaches for the solution of such axisymmetric menisci profiles were treated by Padday and Pitt (34), Huh and Scriven (35), and Princen et at. (36, 37). Padday and Pitt developed equations for both bounded menisci (e.g., sessile drops, vertical rod in a free surface) and unbounded menisci (e.g., liquid bridges between spheres) following the Bashforth and Adams’ approach for sessile and pendant drops except with a redefinition of the meniscus shape factor. Hartland and Hartley (38) compiled  19  numerical solutions and tabulated the results for profiles of the rod in a free surface (completely wet) case using an approach similar to that of the Bashforth and Adams’ tables. However, the case of the rod in a free surface was limited by assuming the rod to be either completely wetted or nonwetted 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 complete wetting (i.e. 0 =  00)  (39, 40). This, however, avoids the effect played by contact angle, especially  as it relates to the equilibrium position of a solid such as a sphere on the interface. Huh and Scriven formulated solutions which were general for axisymmetric interfaces of “unbounded extent” (their definition being the opposite to that of Padday and Pitt which we adopt here) which extended far from a circular contact line to a free flat interfuce. Their analysis can be applied to a floating sphere as well as spheres submerged to different extents, and includes variations in contact angle.  Princen et al. determined profiles for inuniscible liquid lenses situated at the interface  between a second liquid and air, but only for a limited range of conditions.  Unlike the above  models, we consider the presence of neighboring solid spheres such that the interface becomes flat at a finite symmetry boundary, and determine the full gamut of possible menisci as one raises or lowers the liquid volume in the space between the spheres.  3.2 The Sphere Model Configuration  The emulsions encountered in petroleum field operations are often stabilized by fine mineral and clay particles of various shapes and sizes with adsorbed asphaltenes (5). As in most instances where real systems are characterized in mathematical terms, one often finds the need to incorporate simplifications and assumptions into the model in order to make its solution manageable. Thus, for the simulation of this three-phase system, we employ idealized solids which are uniformly spherical and which have identical, homogeneous wetting characteristics. We consider a single sphere amidst a hexagonal array of identical spheres adsorbed on a  20  macroscopically planar interface between an upper fluid phase 1, and a lower fluid phase 2. One  further simplification made in the model is that there is cylindrical symmetry around the sphere such that the meniscus is axisymmetric about the sphere’s center. The meniscus profile is then essentially equivalent to that found for a sphere of radius a bounded by a coaxial circular cross0 (Figure 3.2). Such a cylindrical “cell model” has been used section ring having an inner radius a previously by Levine et al. (11) to determine the capillary interaction between neighbouring spherical particles adsorbed at an oil-water interface.  Figure 3.2 Cylindrical cell model representation of a sphere surrounded by a uniformly-spaced ring of neighbouring spheres  Each 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 exhibits a maximum or minimum. However, the dimensionless separation distance b/a or B can be no smaller 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 is horizontal and circular when viewed from above. The fluid interface has an axis of revolution around the centre of the sphere which coincides with the vertical z-axis. The interfacial profile is then a function only of the radial coordinate r measured outward from the z-axis. The horizontal  21  z=O plane of our coordinate system is to coincide with the free level interface at which the pressure  difference 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 two immiscible fluids whose densities are P1 and P2 upper and lower, respectively. z A  h 1 p  Figure 3.3 Coordinate system for sphere cell model  The pressure difference across the interface arises from the curvature of the meniscus described by the two radii of curvature Rj and R . Figure 3.4 shows a surface of revolution 2 around the z-axis described by a curve in the r-z plane. Rj is the radius of curvature of the point (rz) and R 2 is the length of the normal to the curve at (r,z) from the z-axis. The angle between 2 and the symmetry axis is qi. Rj and R R 2 are perpendicular to each other.  plane)  z  Figure 3.4 Radii of curvature of surface of revolution described by the curve  22  At the contact position (r,z) on the sphere, qS becomes q5,.  fi is the supplementary angle  of çb. The smaller of these two angles is also equal to the angle the memscus slope makes with the horizontal axis, otherwise, referred to as the “meniscus slope angle”. The contact angIe 0 is the angle between the tangent to the solid and the tangent to the liquid interface at the contact line and is measured through the lower phase. And finally, a is the angle which indicates the displacement of the sphere relative to the liquid interface. It is measured from the negative z-axis to the line of contact inscribed by the interface on the sphere. Depending upon the nature of the fluids and their contact angle with the solid, a capillary rise or a capillary depression may result between the particles. For example, if a glass capillary were 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 the characteristic convex menisci found in many thermometers and barometers (Figure 3.6).  Figure 3.5 Capillary Rise  Figure 3.6 Capillary Depression  In the former case, the water wets the glass yielding a very low contact angle while, in the latter case, mercury exhibits a very high, non-wetting contact angle with glass.  The  minimum/maximum (concave/convex) capillary height relative to the zero datum level (i.e., the z=O axis) at the centre of the tube is denoted by z , the latter being negative in the case of convex 0 menisci. Although, the pressure difference across the interface or capillary pressure varies along the meniscus radially (due to nonconstant curvature), we will concern ourselves only with the capillary pressure at z 0 as a point of comparison for different cases.  23  3.3 Mathematical Formulation ofthe Differential Equations of the Meniscus Shape  To describe the meniscus in terms of our chosen coordinates, the Young-LaPlace equation can be transformed as follows. At equilibrium, the memscus capillary pressure must equal the hydrostatic head of the meniscus (23), i.e.,  zP  )gz 1 2 —p (p =  [3.2]  where c is the capillary constant which is given by  c=  1 2 (p ) p g  r  [3.3]  and which has units of inverse length squared. Unlike the Bashforth and Adams (41) form for sessile and pendant drops, equation [3.2] is homogeneous with respect to z because it is measured from the level surface rather than a point on the meniscus. In their analysis, Bashforth and Adams chose the apex of the sessile/pendant drop as the origin because the radii of curvature are equal here. They then used the curvature value b’ to non-dimensionalize the meniscus equation and obtained the well-known shape factor:  [3.4] 7  and the following dimensionless form of the axisymmetric, Cartesian Young-LaPlace equation:  24  1  + sinçb  1b’ 1 R  x/b’  =—+2.  [3.5]  b’  Since there is no corresponding b’ for unbounded axisymmetric menisci, one often finds that the Young-LaPlace equation is non-climensionalized using the radius of a solid, or some other significant 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]  r  It represents the ratio of effects due to gravity, which tend to distort the meniscus, to the capillary forces, which act to minimize the surface area and distortions. As can be seen, this number is similar to the shape factor used by Bashforth and Adams (41) who tabulated extensive data for sessile 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 Young LaPlace 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 Menisci  For the concave meniscus cross-section (Figure 3.7), by our conventions, the principle radius of curvature Rj is positive while R 2 is negative.  25  Figure 3.7 Concave meniscus profile  The curvature K of a point on a curve is the rate of turning of the tangent line with respect to  arclength (42). Thus, the curvature K is defined mathematically as:  K=— ds  where s is the arclength of the curve and  [3.9]  w is the angle between the tangent to the curve and the  horizontal plane. In the limit of vanishingly small arclength,  a’V ds  .A’P urn ‘-°  Ss  .  =  lirn ‘-‘°  z’P  1  .  1 sin(’1’) R  =  hm ‘-‘°  (A’P) 1 R  =  —  1 R  .  [3.101  In terms of radial coordinates,  1 R  d’Vdr drds  R  dr  dP, dr  —=—————=—————cosT)  .  [3.11]  Since ‘P= r—qS,  dr  [3.12]  26  Since  d Sifl dr  is negative, then Rj is positive. Furthermore,  r sin (qS)  [3.13]  is negative. The Young-LaPlace equation therefore becomes  dsinq5sinçS dr r  [3.14]  where qS decreases from q5 at contact to 0 at r=b.  3.3.2 Convex Menisci  Figure 3.8 Convex memscus profile  Similarly, for the convex meniscus (Figure 3.8), Rj must be negative and  R  while R 2 is positive and  =  (cos’P)=  dr  dsin ‘P  dr  =  dsin dr  [3.15]  27  —  [3.16]  .  sin  Thus, the corresponding differential Young-LaPlace equation is:  clsinçS  sinç5 +—=cz  dr  which is applicable for  00  [3.17]  r  <  <  1800 as before, but in this case qS increases from q 5 to  180° at r=b. In terms of Cartesian coordinates, the Young-LaPlace equation takes the form (from equations [2.16] and [2.17]):  z 2 d  dz 2 dr + dr =cz (dzl2T 5 5 (dz21° F11+1—I I rIl+I— F [ ‘dr)] L dr  [3.18]  where 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 derivative in the denominators of both quotients.  In this case, the Young-LaPlace equation reduces to  Bessel’s equation:  ’ 2 d 2 dr  d’ rdr  [3.19]  28  which can be solved using modified Bessel functions.  3.4 System ofEquations in Alternate Coordinates  The 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 the domain of interest. For example, at çit/2, the solutions are discontinuous and undefined. An alternative approach is to make ç the independent variable which leads to the following first-order parametric equations for the convex case:  dX dqS  XcosqS  [3.20]  XZ—sin.6  and  dZ db  where X  =  and Z  =  Xsinq  [321]  XZ—sinq5  (upper-case variables will denote the dimensionless forms). However,  this approach merely shifts the discontinuity to b=0. Huh and Scriven (35), who similarly derived profiles for axisymmetric interfaces around isolated objects, used this approach but sidestepped the problem by employing Nicolson’s (44) approximation near contact angles of 90°. Yet another method which circumvents the blowup of the solution is to make the continuous, single-valued arclength function s the independent variable.  This transformation  produces the following numerically stable form of the Young-LaPlace equation with çS, z, and r as dependent variables (45):  29  dg ds  dr  sinØ  r  [3.22]  —  -=+cosq5  [3.23]  =±sinç  [3.24]  and  where the upper sign corresponds to the convex menisci, and the lower sign, to the concave. This last approach was taken by Princen and Mason (36, 37) who solved the meniscus profiles for fluid lenses 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:  dS  L=FcosqS 4  dS  X}  [3.25]  [3.26]  and =±sinçb 4  where S=s/a.  [3.27]  30  The radial location of the meniscus at the point where it contacts the sphere, X, can be related 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 outer  symmetry boundary of the cell model, the final arclength is not known a priori, but we overestimate by equating it to twice the radius of the cell. How this endpoint is dealt with will be discussed later in the numerical procedures.  3.5 Boundary Conditions  To determine the shape of the axisymmetric meniscus, the Young-LaPlace equation must be solved from the three-phase contact line on the sphere to the cell model symmetry boundary at r=b. For finite contact slopes, the boundary condition is  dZ —=±tanq5  or  dZ --=±sin(q)  [3.29]  where the upper sign is for the convex case, and the lower one, for the concave. The contact meniscus angle  can be determined from its geometric relationship with the immersion angle and  the contact angle at the three-phase contact line. For the concave meniscus, =  whereas for the convex meniscus,  r— 9— a  [3.30]  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 Z . Therefore, 0  dZ dX  dZ dS  —=0 or —=0.  [3.33]  Up to this point, we have only considered that the solids are uniform, ideal spheres whose wettability 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 the fluids may vary over several degrees. This phenomenon, known as contact angle hysteresis, must also be included in the analysis.  3.6 Effect of Contact Angle Hysteresis  The most reproducible contact angles measured for a given system are its advancing and receding angles, O and °r’ respectively (14). These angles are so named because in the sessile drop method of measuring contact angles, the advancing angle, the maximum bound for the contact angle, is measured by increasing the volume of a drop until its contact line with a planar solid just begins to move; whereas the receding angle, which is the smallest possible contact angle, is measured by causing the drop periphery to retreat over the solid. Figure 3.9 shows two liquid drops on a solid showing very different apparent contact angles, but in fact, have the same actual  32  contact angle as seen in the magnified views (Figure 3.9). The solid is an idealized surface having concentric sine-wave corrugations. This model is often used to explain the discrepancy between theory and observation as it shows hysteresis can be caused by microscopic roughness. It can also explain why the contact line does not appear to move as the drop is advanced while achieving the macroscopic advancing angle since, from a microscopic point of view, the liquid is moving down into the valleys between the ridges whilst maintaining a constant contact angle. Other sources of hysteresis are surface heterogeneity, liquid impurities, or noncleanliness of surfaces.  B  solid  liquid  Figure 3.9 The surface of the idealized rough solid has a sine-wave corrugation, as seen in the two magnified views, and has rotational symmetry about the z axis. Two static configurations are indicated by A and B with very different apparent contact angles, but in the microscopic view, the actual contact angle is the same.  The advancing and receding angles can also be determined from other contact angle measurement methods (14,19). The difference between the two angles is referred to as the contact angle hysteresis, Oj, i.e.,  6h  = 8a  [3.34]  —  Hysteresis can be quite large depending on the solid-fluid-fluid system. For example, water on mineral surfaces can have an advancing angle as much as  500  larger than the receding one (14).  33  For our experimental system, contact angle hysteresis will be measured by starting off with  a level interface between a pair of interfacial particles and then increasing the meniscus pressure until the contact line on the solid just begins to move. At the level interface obtained by withdrawing the lower fluid from the system, the contact angle is at its smallest value, 0,.. The location of the three-phase solid-fluid-fluid boundary relative to the sphere is expressed in terms of the 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 volume to occur in two stages (8). The first stage is characterized by the angle a remaining stationary while 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 will begin to move. This second stage movement of the meniscus upward along the solid is characterized by the contact angle remaining constant at its final value 61 a To include the effect of hysteresis in our previous analysis, which describes only the second stage when the contact line is moving around the sphere, we will include an initial stage in which the contact angle is increased from the receding angle to its final value while the contact line remains stationary. In other words, the contact angle hysteresis does not affect the shape of the meniscus 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 two emulsion droplets.  34  3.7 Determination of Film Thickness  The coalescence of two oil-in-water or water-in-oil droplets, stabilized by micron-sized particles, is assumed to occur when the film of continuous liquid between the two droplets thins  and the meniscus protruding between any pair of adjacent particles on one droplet touches the corresponding meniscus on the second droplet.  It is assumed for convenience that opposing  particles on the two approaching droplet surfaces are aligned. Thus, rupture occurs when the protruding menisci from the two droplets meet the line joining the points of contact between opposing particles (Figure 3.10). At any equilibrium state of the system prior to rupture, h is taken to be the distance from this line to the closest distance to each meniscus, i.e., the meniscus height at the cell model boundary, r=b. Hence, rupture takes place when h=0.  Droplet 1  h  Droplet 2  Figure 3.10 Point of rupture between two emulsion droplets  This process neglects the effects of “dimpling” (48, 49, 50) or the deformation of the approaching menisci 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 flat interface, but the line at which rupture takes place is unknown relative to the datum level, the film thickness can be determined by using the contact line as the reference height. Thus, the onset of film rupture can be followed by comparing the existing position of the meniscus relative to its  contact line to that where contact is made with the corresponding meniscus above.  The  dimensionless distance from the point of rupture H is determined by comparing the actual distance  35  of 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 the , and the contact line location Z of the actual meniscus: 0 difference between the meniscus height Z  =  0 —Z Z  [3.37]  0 and Z are determined from the simultaneous solution of the where the vertical coordinates Z equations [3.25-3.27]. The required distance for rupture, for a level meniscus having the same contact line, is determined from simple geometry:  Hr  1+cos(a)  [3.38]  The difference between Hact and Hr is to be minimized to determine the meniscus shape and capillary pressure at the threshold for film rupture.  3.8 Numerical Solution  Given the size of the solid spheres, their distance apart, the fluid densities and their interfacial 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 governing equations are the vertical coordinates of the meniscus at the contact line and at the cell model  36  boundary relative to the absolute datum line (taken to be the position of the free interface far from the influence of the particles). This is a boundary value problem which we chose to solve by the shooting method applied to the simultaneous solution of three ordinary differential equations.  We begin at the known  contact 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]. We set Z, the vertical coordinate at the contact end, initially to be zero (as for a level meniscus), then proceed to solve the equations at incremental arclength positions until the radial co-ordinate reaches the outer boundary, at a distance b/a from the centre of the sphere. In this fashion, the boundary value problem is transformed into a simpler initial value problem. If the meniscus slope happens to be zero at the outer boundary, we have achieved our desired goal. Using the initial estimate of Z, this condition will generally not be met. The determination of the true value of Z is accomplished iteratively by means of a numerical routine employing Muller’s method (DRZFUN in the UBC NLE nonlinear equation solving subroutines package (51)). Muller’s method is an iterative procedure capable of finding both complex and real roots of a single nonlinear equation efficiently and reliably. It has the advantage of not requiring a good initial estimate(s) of the root(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., the meniscus slope must be horizontal. Thus, the function to be minimized,/ is given by  f = (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 necessary before the meniscus reaches its maximum (convex), or minimum (concave) height. Likewise, if f is negative, the selected  5 chosen was too large (i.e., Z is too large) and the apex was encountered c  before the cell symmetry boundary was reached. If first one condition and then the other are obtained for two consecutive guesses of Z, the true value lies between them.  37  Initially, the ordinary differential equations (ODEs) describing the meniscus profile were solved using a Runge-Kutta or an Adams-Moulton predictor-corrector technique, but overflow conditions occurred in certain regions of the solution domain which were later attributed to the partial stiffness of the system. Stiffness behaviour apparently arises in many physical situations such as in reaction kinetics (52, 53). It occurs when the mathematical solution to the first-order system contains terms that decay at different rates and although certain terms may disappear rapidly with negligible effect on the long-term solution, they can cause numerical difficulties by requiring excessively small stepsizes in the computation of the solution (54). In order to avoid inaccuracies, we require a solution routine which has the characteristic of absolute stability in the face of relatively large stepsizes, so that the solution progresses at an acceptable rate (52). The most widely used stiff equation solvers are based on multi-step implicit backward differentiation formulas first implemented by C.W. Gear (54). The stiff ODE equation solving routine, LSODAR, used here, is also based on GEAWs method, and was obtained from NETLIB, 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] can be either stiff or nonstiff or both. The routine automatically determines whether the system is stiff or 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 numerical  means. LSODAR has the added capability of finding the roots of any given function while the solution of the n differential equations continues. This feature is especially useful for stopping the  38  solution at certain boundaries. For example, in tracking the path of a particle in an enclosed box we 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 the meniscus is unknown a priori, we tenninate the simultaneous ODE solver by continuously searching for the root of the function g, defined by  g=X—B  [3.41]  where B is the location of the outer boundary of the cell model. The stability of the resultant meniscus is not considered here, but references (55, 56) outline a methodology for testing this  aspect of the equilibrium menisci. Once the meniscus shape is known, the capillary pressure can be detennined. This pressure in terms of the capillary rise is  [3.42]  ‘capYo  or, in dimensionless form, obtained by dividing both sides of equation [3.41] through by 2y/a is  Pcap  —  2 “Z” ° 2  —  —  B’•Z0 2  [343]  The resultant capillary pressure at rupture is denoted by  Pcap —P  —  BoZ0 .  F344 L  The numerical solution of this problem is outlined in the program flowsheet shown in  Figure 3.11. The rupture capillary pressure is plotted as a function of several variables. A similar  39  analysis for the cylindrical interface was perfonned.  40  Figure 3.11 Program flowsheet for meniscus between spheres  /  [sTAj  DATA 1. Radius, distance a B 2. Physical data p,y,’O  /  /  Xc, and hysteresis Dh  nstant Xc Call HVSP (6..  Determine Zc, Zo, H, P Store H, and P in arrays Hi (i,j), P1 (i,j) Determine minimum and maximum H,  Calculate increment in +c Constant 0 and Xc —-CaIculato = Determine Zc, Zo, H, P Store in arrays H2(i,j), P2(i,j) Determine minimum and maximum H, I  Call HVSP  Xc,)  i=j±l  f  Increment initialO =0  +  dO  I  Graph all curves of film thickness vs. pressure  Call MYPLOT (Hi ,Pl ,H2,P2,M,N)  41  Figure 3.11 (continued)  Subroutine HVSP (  Xc, a, H, Pcap)  UBC DRZFUN solves for roots FCN  Determine Zo, Zc Hc=abs(Zo-Zc) Hr=(1-cosl)) H=Hr-Hc Pcap=BoZoI2  eJ Fwwt1irFCN(Z) Initialize Y(1)=øc, Y(2)Z Arclength limits T, Ta Determine o, Zo at X = B øo=Yn(1) Zo = Yn(2) 0 -0 FCN=  Call LSODAR (Y,T,To,FEX,JAC,GEX)  Return]  UBC LSODAR (Y,T,To,FEX,JAC,GEX) Partial differential equation solver’using Jacobian matrix JAC, equation derivatives in FEX, and GEX for program halt at the specified boundary.  Subroutine JAC PD(1,2)=q*cos(Y(2)) P0(2,2) =-cos(Y(2))IY(3) PD(2,1)=q*Bo P0(2,3) = sin(Y(2))/(Y(3) *( PD(3,2)-sin(Y(2))  Subroutine FEX DY(1)=q*sin(Y(2)) 9 DY(2) q*Dy(1)/\f(3)+q*y(1)*8 DY(3)=cos(Y(2))  Subroutine GEX Go(1)=Y(3)-B  [  Return1  42  Chapter 4  MATHEMATICAL ANALYSIS OF THE CAPILLARY SHAPES BETWEEN PARALLEL, HORIZONTAL CYLINDERS  The capillary interactions between two approaching emulsion droplets each with a layer of  fine particles on their respective interfaces has thus far been described by considering a small section 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 and the difficulties with visualizing the interface. Thus it was decided a simpler two-dimensional model of two macroscopic parallel cylinders lying horizontally on a liquid-liquid interface would be studied. Because of the large scale needed to allow observation of the interface, we cannot neglect the effects of gravity, and the theoretical basis takes this into consideration. Nevertheless, a system with 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/plate assemblies are sealed into a rectangular chamber such that the lower liquid enclosed in the chamber  can 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 displaced upward forming the meniscus shapes whose mathematical treatment is given in this chapter.  Fig 4.1 Experimental cell (simplified)  43  4.1 Literature Review  The meniscus profile between floating horizontal, parallel cylinders was considered by  Allain and Cloitre (57), Gifford and Scriven (26), and Chan et al. (43) who were concerned with the capillary forces of attraction that arise when gravity acts on such particles, and with the forces necessary for equilibrium and stability on an interface. However, the equilibrium shape between cylinders found by Allain and Cloitre, and Chan et al., was found only approximately by superimposing the results obtained for two individual cylinders, each on an infinitely large interface. This superposition approximation is referred to Nicolson’s method (44) and is applicable for low Bond numbers or when the cylinders are not too close together such that their profile can merely be summed. Only Gifford and Scriven supplied an exact analysis for the equilibrium shape between a pair of floating cylinders. Another fundamental analysis of this configuration is given by Princen (12) who studied the liquid bridges formed between assemblages of two or more horizontal cylinders, but in the absence of gravity which is a reasonable approximation for very thin cylinders or for two fluids of similar density.  4.2 System Configuration  As we increase the volume of liquid entering the lower chamber of the cell, the meniscus will protrude further through the slit, and its shape will change accordingly.  This capillary  pressure zIP will be proportional to the hydrostatic head measured from the line joining the centers of the two cylinders (z’) plus a contribution from the applied pressure (b’):  AP  ApgZ’,,I  r A more comprehensive derivation of this approach can be found in (58).  [4.1]  44  Alternatively, we can apply equation [3.2] to this system as well, if we change our reference line for measuring z back to the horizontal level interface whose position is determined simultaneously with the solution of the problem. This is equivalent to pushing an isolated pair of cylinders down into the interface until the capillary pressure generated by the meniscus between the cylinders 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 completely submerged below the level surface. For very narrow gaps between the cylinders, the high values of required may be physically unrealistic since the meniscus extending from the outer side of the cylinders would collapse without any vertical support. Nevertheless, the analogy still holds and  can also be likened to the cylindrical interface developed between two vertical flat plates which are inclined 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 rods can be described by one radius of curvature; since there is no axis of revolution, the other radius of curvature is infinite. The Young-LaPlace equation for cylindrical interfaces is, therefore given by:  1 rR  [4.2]  We need only look at a cross-sectional plane of the cylinders since the interface varies in only two dimensions.  We introduce Cartesian coordinates such that z=O coincides with the  horizontal plane at which the interface is not distorted by the cylinders (i.e., far from the cylinders at x=x’, where ziP=O) and the x-axis originates at the midpoint between the two cylinders (i.e., at the axis of symmetry). We define qS as the angle that the normal to the interface makes with the positive z-axis (Figure 4.2). The distance d is the separation distance between the two cylinders measured from their closest points, and d is the distance between the contact lines of the meniscus on each cylinder. The previous nomenclature for other variables holds.  45  z x  Figure 4.2 Coordinate system for parallel cylinders  Since 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  whereas for the concave meniscus, <  in the positive x direction with 0  0 decreases from 180° to  <  <900  in the same direction with 90° <ç  180°. z 0 is the height (or depression) of the meniscus at x=O. We will consider the convex  meniscus first.  4.3 Mathematical Formulation  The following derivation describing the cylindrical interface follows a similar approach taken by Princen (23) in his development of the meniscus between two vertical flat plates and by Levine et al. (58) for menisci between clamped cylinders.  46  4.3.1 Convex Meniscus  From simple geometry, the radius of curvature is  ldcos  [43]  dz  Integration of the latter equality with qS  =  0 at z  0 yields: z  z-4[ = —[cz 2 + 2(1 cos 0  [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 to  is  dqS  [4.6]  dbdz  By differentiating equation [4.4] with respect to qi one obtains  d(zJ) dq5  2 0 [cz  +2(1_cosØ)]1’2  <0  [4.7]  47  which is negative for the convex meniscus, and positive for the concave meniscus. Substituting equations [4.5] and [4.7] into [4.6] then yields  d(xJ)  dçS  cosç —  [cz +2(1—cosqS)]” 2  [ 48 I .  which supports the choice of negative sign for equation [4.4]. This also agrees with the hydrostatic model, 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 in terms of half-angles by making the following trigonometric substitution  cz +2(1—cosç5)=cz  +4sin2(-)  =cz +4[1_cos2)]  =  (cz  +4)_[4cos2()]  [4.9]  [4.10]  [4.11]  and introducing the parameter k,  k=  equation [4.8] becomes:  I  +4)  [4.12]  48  k 1_2sin2()J  =  d(x J 1 j dqS  [4.131  1/2  k2(cos2())] 2L1_  Integration of equation [4.13] and allowing  [4.14]  22  such that  [4.15]  sin2=cos2  yields  1_2sin2)) X%J  oI 2[1_k2cos2(’l 2)]  k(25jfl22_1)  /2Jr 2  d2.  2_  [i  —  2 hi k 2  [4.16]  2j1/2  The numerator can be rewritten as  2 2—1) k(2sin  =  kL_(k2 sin 2 2— i)+—-— 11  Lk2  2 k  and introducing the elliptic integrals of the first and second kind,  ]  [4.17]  49  F(k,qS)=r°  E(k,  )  =  d2 [i_i sin 2 2]  —  [4.18]  1/2  2 sin k 2 2] d2 2 ” 1  [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]  and  1/2 =  2 +_(1_cos(c))] 0 _[Z  where capitals sigrnfy dimensionless terms.  [4.22]  50  4.3.2 Concave meniscus  Now for the concave meniscus, Rj is positive such that  1 1 R  where  -  8  =  dcosqSdcos/3 dz dz  [4.23]  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  db  [4.25]  dzdb  where  [4.26]  =tan..  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 obtain  d(x./)  — —  db  —  cos  [cz +2(1+cos)] 2 ” 1  [4 27]  51  Finally, using the approach in the previous section, equation [4.27] can be integrated to yield  xiJ =  —  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 equations in terms of/) the supplementary angle to q. Thus, equation [4.24] becomes  z-4J  =  [cz +2(1_cosfi)j  1/2  [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 positional angle on the cylinder is different however for the two orientations. For the convex memscus (Figure 4.3(a)),  [4.31]  where  0  4.3(b)), itis  8  r,  0  a  r, and 0  Ø,  -  ,  whereas for the concave meniscus (Figure  52  fi = r— 0— a  [4.32]  = 9+ a  [4.331  and  where  -  qS  r and PC = r—  bC.  tangent to meniscus  tangent  tangent to meniscus tangent to cylinder  (a)  (b)  Figure 4.3 Angles for convex (a) and concave (b) menisci  To determine the correct values for the unknowns we need to satisfy the meniscus dimensions within the boundaries of our physical setup. From the physical geometry, the meniscus horizontal length must be  dr =d+2a(1—sin a)  [4.34]  and from the Young-LaPlace equation, the horizontal distance between the meniscus ends is  dC  —  =  1){F(k f) F(k, —  lr—c —  —  E(k  Jr—c  )}]• [4.35]  53  Equations [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 Threshold  In order to determine the shape of the memscus at the film rupture threshold, the vertical dimensions need to be fitted also. The dimensionless meniscus height (measured between contact  with the solid and the topmost point), H, is calculated as  [4.37]  —Z. 0 H=Z  H is then compared with the dimensionless required height Hr where  Hr=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]  54  Once the two functions, dL and dH, are minimized to within the error tolerance of 10, the rupture capillary pressure can be calculated as  cap 1  0 = P,,,, = y•cz  [4.40]  where the subscript “rup” signifies the threshold rupture point, or in dimensionless form as  Bo•Z ap  =  [4.41]  2  where the apostrophe signifies a dimensionless value.  4.5 Numerical Computation  A series of profiles were generated for the meniscus between the cylinders as its contact point was moved along their perimeter. To determine the shape of the memscus for a given set of conditions, the solution of the Young-LaPlace equation has to be forced to correspond to the actual physical dimensions of the system according to equation [4.36]. As in the previous model, we apply a root-finding method to this problem since none of the dimensions of the interface are  known at the outset. Given the Bond number, the cylinder radius, the contact angle, and the separation distance between cylinders, our unknown variables at the outset of the solution are the capillary rise z 0 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 be solved for this variable thereby allowing subsequent solutions for a,  . 0 and z  55  The concavity or convexity of a profile can be easily determined by its contact position on the cylinders and its contact angle. The turning point coincides with the location of the level meniscus which is calculated from equation [3.35]:  =  Above this level, menisci will take on a convex shape, and below this level, menisci are concave  with 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 root k of equation [4.36] by means of the numerical algorithm UBC DZERO (51) which uses a  combination of linear interpolation, rational interpolation, and bisection. between 0 and 1.  The values of k lie  The capillary rise or depression can then be calculated from the dimensionless  version of equation [4.12], i.e.,  =  [4.42]  and Z subsequently from equation [4.22]. The fiowsheet for this algorithm is shown in Figure 4.4. 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 need to be minimized, dL and dH. To reduce the number of calculations and avoid a cumbersome solution of three coupled transcendental equations, we can manipulate the equations to yield a simpler algorithm in which obtained.  is the optimizing variable, and each unknown can be successively  56  By substitution of equations [4.29], [4.37], and [4.38] into [4.39], we obtain the following 0 and relationship between Z  alone:  =  J[1:Fcos(o c)] 2  The initial step is to test qS, between  00  —  [i_coqsj]  1  Bo[lFcos(9FØj]J  [443]  and 900, and obtain Z 0 from equation [4.43]. Once Z 0 and  are known, the parameter k is determined from equation [4.12] and a is determined from [4.311 or [4.32]. Then dL [4.36] is calculated. If dL is not zero, the initial root-finding routine selects another value for  and the calculations are repeated. Z at contact then is calculated from  equation [4.29]. The programs utilizing this algorithm are used to plot rupture threshold capillary pressures as a relationship with Bond nwnber, separation distance, or contact angle (see Appendix C).  57 Figure 4.4 Cylinder model fiowsheet of program CALP generates meniscus profiles and determines rupture thresholds -  Poap Zo Ho  =  =  =  WZo/2  + 2(1-oos( O))fW) 2 -QoSqr(Zo Abs(Zo-Zo)  oos( a)) t Hr = Abs(1+Qo H = Hr Ho -  Y  58  Figure 4.4 (continued)  STOP  IR=O H=O FRI’s found  40  59  Figure 4.5 Cylinder model flowsheet of program CHYS of the capillary pressures as the menisci move further up  Part I  Part II  generates hysteresis plots (follows the progre ss the solid. -  60 Figure 4.5  (continued)  STOP  61  Chapter 5  THEORETICAL MODEL RESULTS  The meniscus shapes detennined in the model do not necessarily correspond to that formed between stationary objects floating at an interface under equilibrium conditions.  Under such  conditions, the position on the interface is governed by a balance of forces which includes the effect of 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 an applied external pressure causes the liquid to enter the capillary space between particles fixed in space. Changing the applied pressure therefore corresponds to changing the extent of immersion in the lower fluid, of which the free-floating equilibrium meniscus would be one of these profiles. No consideration is made about the critical radius of a particle above which it will not float (refer to Princen (23) for discussion of the critical radii for floating cylinders and spheres).  5.1 Relationship to Emulsion Droplet Coalescence  A spherical droplet dispersed in another immiscible fluid has a pressure of  [5.1]  ae  where ae is the radius of the emulsion droplet, and e must be larger than the pressure that exists in 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-  62  scale curvature at the interface would change and thus a new pressure drop cap would exist such that  [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 for the upper droplet than that for the lower droplet. The model solves for both the upper and lower emulsion droplets, where the upper profile is taken to be concave and the lower convex. (Also, for the 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 negligible influence, the bulk film pressure and the emulsion droplet pressure would be uniform, with no difference between the upper and the lower droplets. However, even for the macroscopic systems used in our experimental investigations, we have found that the difference in the respective capillary 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 capillary pressures calculated for corresponding curves for the upper and lower droplet were identical to three significant figures. However, a slight difference was noted when six or more significant figures were compared. Nevertheless, gravity cannot be totally neglected since it does have an effect on the curvature of the meniscus radially. The theory for the corresponding case where gravity is zero for the cylinders is developed in Appendix B. Coalescence is assumed to occur when two droplets having stabilizing particle films approach each other, and the liquid memsci protruding between the particles come into contact (we neglect 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 such droplets collide forcing the particles in contact to enter fl.irther into their respective droplet.  63  Rupture might also take place if the droplets are distorted due to shear or turbulence where the exposed menisci are momentarily in contact.  For these collision scenarios, coalescence also  requires favourable kinetics, i.e., the time for the film to rupture must be less than the time of contact between the two droplets (60).  5.2 Characterization ofProfile Curves  Four parameters need to be defined to fully describe a particular meniscus between two parallel cylinders or two adjacent spheres. These are the Bond number, the distance of separation between solids, the contact angle 0 of the solid, and the interface position a. The effect of each parameter on the shape of menisci will be investigated. We consider the simpler geometry of the parallel infinite cylinders first and then go on to discuss menisci between spheres. The example interface selected is that between carbon tetrachioride and water. Table 5.1 shows values of the physical properties, as well as the capillary constant for various fluid-fluid interfaces.  Table 5.1 Properties of Selected Fluid-Fluid Interfaces (20°C)  (dynes/cm)  Capillary Constant c ) 2 (cm  Bond Number a = 0.1 cm  Bond Number a = 0.5 cm  1.594  26.95  58.01  0.580  14.5  0.66  18.4  35.18  0.352  8.80  0/air 2 H  0.9982  72.75  13.46  0.135  3.37  IH 4 CC1 O 9  0.596  45.0  12.99  0.130  3.25  n-Hexane/H20  0.34  51.1  6.53  0.065  1.63  Fluid-fluid Interface  Density zip ) 3 (glcm  /air 4 CC1 n-Hexane/air  *  The Handbook of Chemistry and Physics (61)  64  5.3 Cylinders Model  To illustrate the effect of increasing liquid volume between a pair of parallel, horizontal cylinders on the meniscus shape, a composite plot of several menisci are shown as an example in Figure 5.1 for a system with a Bond number of 3.247, contact angle of 90°, and a dimensionless separation distance of 1.5. The profiles are shown for immersion angles starting at 10° up to in 20° increments.  1700  As with the experimental runs, which were also started with a concave  meniscus, when fluid is introduced into the cell, the meniscus moves upward becoming less and less concave until it reaches a point where it becomes a level interface and the capillary pressure is zero (represented by a dashed line in the figure). When more fluid is introduced, the meniscus continues upward but now with a convex profile.  The magnitude of the capillary pressures  generally increases as the convex profile creeps upward, but in some cases, depending on the threephase contact angle and the distance between the cylindrical rods, the pressure peaks at a maximum value and declines thereafter. The dimensionless capillary pressures are shown above the corresponding curves. The negative signs of the pressures for the convex menisci serve only to differentiate 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, capillary pressures will be referred to by their absolute values hereafter unless otherwise specified. For both the concave and convex portions of the composite plot, the rupture threshold capillary menisci were determined. These memsci are shown as dotted lines in the plots. The convex rupture meniscus corresponds to that of the lower emulsion droplet in our coalescence scenario whereas the concave rupture memscus corresponds to that of the upper emulsion droplet, but for the supplementary contact angle  u=-e  In this case, it can be seen that they are essentially mirror images with the  same capillary pressures (to three significant figures). The influence of several factors which define a particular set of profiles between a pair of cylinders will be discussed and illustrated with reference to similar plots. The factors which will be  65  x/a  Figure 5.1 Meniscus profiles between a pair of cylinders (Bo  =  3.247, 0  =  900,  b/a  =  1.5)  66  discussed are the contact angle  the Bond number Bo, and the separation distance b/a.  5.3.1 Effect of the Contact Angle Figures 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. Some of the figures represent the cylinders as noncircular profiles. This is a consequence of the size of the plot and the need to show more pronounced menisci. As the solids are made more wettable by the lower fluid, the fluid creeps up on the lower part of the cylinders requiring less pressure to do so.  However, on the upper portion of the cylinders, the increase in contact angle causes  progressively higher capillary pressures to be developed. This trend is expected since the lower concave portion is equivalent to the supplementary contact angle. For this reason we need only consider the convex case in our comparisons. The increase in the contact angle has the effect of increasing capillary pressures for curves of a specific immersion position, and of increasing the film threshold requirement as shown in Figure 5.8 for b/a  =  1.50.  5.3.2 Effect of the Bond Number The effect that the Bond number has on the meniscus profiles can be seen in Figures 5.95.12 for a separation distance b/a of 1.5 and for a contact angle of 90°. Generally, for curves having corresponding a values, the curvature of the menisci increases as the Bond number decreases. However, the dimensionless capillary pressures first increase as Bond number increases as seen in Figures 5.9-5.11 for Bond numbers 1 ,3x 1 0 to 3.25, and then decrease as seen in Figure 5.12 for the Bond number of 13.0. Comparing only the rupture threshold menisci at a fixed dimensionless separation distance b/a, Figures 5.13 and 5.14 show the relationship between  67  80  2.0  0  =  1.29910  Ba  129910  =  20  30.0  =  •  0  b/a1.50  1.5  =  —0.184  60.0  b/a=1.50 —0.165  1 0 ,.7  —0 137  0  0.15  —0.082  ..  0.3  0.0 F  —0.8  0.107  0.4  F  F 0.302  —10  —0.5  .  .  0  0.38  0  —‘.5  —.o 0.3 0.38  —2.5  —S.C  0.2  —3.0  0.297  -  —2.5  —3.5 —4.0  —3.0  0.175  4.5  I  —3.0  0.221 —3.5  I  —2.0  —1.5  0.0  1.0  2.0  I  —0.0  —1.5  1.0  0.5  x/a  re 5 2 Bo=0.130 8=30° -—Level Meniscus  Fi  0.0  0.5  1.0  1.5  0.0  x/a  ....  Figure 5.3 Bo0.130, 8=600 Numbers on profiles are dimensionlesss pressures  Film Rupture Threshold  Meniscus Profiles between Parallel Horizontal Cylinders  3.0  4.0  Ba 0 a.o  =  1.299101  30  90.0  0  =  b/a =1.50  3.0  =  =  1.29910’  120.0  —0.221  •  b/a =1.50  -  —0.205 5.5  1.5  —0.207  BE  /  1.0  .  -  a  —0.363  1.5  —0.200  0.5  L 0  .  0.0  0.000  —0.5  —0.302 0.0  0 20  —1.0  .-“  —0.107  --  0.080 —1.0  0.26  0.16 —0.0  —0.5 0.18  0.220 —2.5  I  —  —2.5  2,0  1.5  —10  I  I  0.5  0.0  I  0.5  1.0  —2.0  I  1.5  2.0  0.5  x/a  Figure 5.4 Bo0.130, 8=90° Level Meniscus  ....  Film Rupture Threshold  p  3.0  2.5  0.0  1.5  0.0 0.5 1.0 .o .o x/’a Figure 5.5 Bo=0.l30, 8=120° Numbers on profiles are dimensionlesss pressures —1.0  —0.5  Meniscus Profiles between Parallel Horizontal Cylinders  0  68  5.5  Bo  =  Bo  1.299I0  0 °  =  L29910  5.0  4.5 =  150.0  —O 122  S  •  =  180.0  b/a =1.50  b/a =1.50 4.0  3.0  _—  /  /“  —0.197  —0.268 3.0  —0.317 —0.376  10  -  —0.458  —0.444  1.0  —0.454  1.0  0.5  —0.330  00  05  -0.44  —0.156 0.0  -0.5 .,:.:0n35  —0.165  —05  —10  0.066 —5.5  —2.0  —0.044 —5.0  —L5  10  x/a  xIs  Figure 5.7 Bo=0.130, 8=180°  Figure 5.6 8o0.llO, 8=1500 —-  Level Meniscus  ....  Film Rupture Threshold  Numbers on profiles are dimensionlesss pressures  Meniscus Profiles between Parallel Horizontal Cylinders  20  30  69  0.5 -  ——Bo = 0.001 --.----Bo=0.130 103 Bo2.018 Bo = 4.035 v— Bo = 6.053  ,  —  —  —.  0.4  -  b/a  1.50  -w A  ‘1) ‘-4  0.3  /  -  / , V  0  :  0.2  0.1  0.0  -  —  -  20  40  60  80  100  Contact angle  120  140  160  0(0)  Figure 5.8. Rupture capillary pressure vs. contact angle (cylinders)  70  0.0  05.0  Do 0  =  =  129910’  Bo  90.0  0  b/a =1.50  12991O 90.0  b/a =1.50  4.0  —0.120  3.0  —0.039  7 5  =  =  2.0  5 0  —0.121  040  —  —0.11  10  2.5  . —  o  0.0  -  .  .  0.0  0. —1.0  S  0  01 -2.0  —5.0  .  .  —7.5  —3.0  10.0  —4.0  0  0  0. .036 —02.5—19.0  —7.5  —5.0  2.5  0.0  —2.5  —5.0 7.5  5.0  09.0  02.5  —4.0  —3.0  —2.0  x/a  1.0  0.0  0.0  x/a  2.0  3.0  4.0  Figure 5.9 Bo=1.3x10 , 0=900 5 Figure 5.10 Bo=l.30xl0 , 0=900 3 Level Meniscus Film Rupture Threshold Numbers on profiles are dimensionlesss pressures Meniscus Profiles between Parallel Horizontal Cylinders —  ....  2.9  -  Do  3.24810  Do  0  90.0  0  =  b/a =1.50  2.0  =  =  0 L29910 90.0  -0.179  b/a =1.50  ,  —0.225 1.0  —003  1.5  : —2 0 0.00  —1.00—0.75—050—0.25  0.25  0.50  0.75  I 00  izs  1.50  —0,057  r’r——--—,—-—-----———-1—-—-———-———r--————-’—_._. 0 l’  x/a  x/a  Figure 5.11 Bo3.247, 0=90° Level  Meniscus  ....  Film Rupture Threshold  Figure 5.12 Bo=l.30x10 , 0=90° 1 Numbers on profiles are dimensionlesss pressures  Meniscus Profiles between Parallel Horizontal Cylinders  71  a).  z  1. a)  C,  a)  C  a). C  a)  a). a)  C, C)  0 a) a)  Bond Number Bo  Figure 5.13 b/a=1.50  a). cC. a)  C.  C, a) I-.  C.  a)  a). a) C, a)  a) 0  a)  a)  Bond Number Bo  Figure 5.14 bla=3.OO  Rupture Capillaiy Pressure vs. Bond Number Cylinders  72  dimensionless rupture capillary pressures and the Bond number. For a given contact angle and separation distance, by increasing the Bond numbers, the dimensionless rupture pressures reach a  maximum and thereafter decrease in magnitude. For small particles such that the Bond numbers are less than the maximum, oil-water interfaces produce lower rupture capillary pressures than for for a similar system at its corresponding oil-air interface. Above the maximum, the opposite would be true.  The Bond number expresses the relative importance of the gravity forces and the  capillary forces operating on the three-phase system. For a single object floating at equilibrium on an infinite interface, the gravitational forces tend to distort the meniscus, while the capillary forces tend to flatten it (57). When Bond numbers are sufficiently small (Bo —* 0), the free equilibrium situation 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 the capillary forces are accentuated by the occurrence of capillary rise. It can be shown that when the Bond numbers are very small (applicable to colloidal particles) that the changes in the capillary forces are greater than the gravitational forces for the cylinders (57). Thus, the effect of gravity becomes negligible. For the series shown in Figures 5.9-5.12, the first plots of the menisci shapes resemble vertical ellipses which become more like horizontal ellipses and more oblate in the later plots as the Bond number is increased.  The transition point between the change may be the  transition between a system predominantly capillary force-controlled to gravity-controlled.  An  analysis similar to Denkov et al.’s (8) assuming gravity is absent was performed for the cylinders such that the principal radius of curvature is constant:  1 — AP —  1 R  —  —  sin + a(1 —  —  sin a)  0 2 io1 )gz (P  [5 3]  —  (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 1x10 5 for b/a=1.5, the menisci do not  73  resemble circular profiles as noted above. Thus, the assumption of constant mean curvature for memsci profiles for micron-sized particle-systems of cylinders would be in error.  5.3.3 Effect of the Separation Distance  Holding the Bond number and the contact angle constant at some reasonable value, we consider the effect of varying the distance of separation between the cylinders. Profiles are shown in Figures 5.15-5.18 for a Bond number of 0.130 and a contact angle of 900 as b/a increases from 1.05 to 6.00. It can be seen that the curvatures of all the menisci decrease as the separation  distance increases.  In terms of the rupture capillary pressure, their values decrease  correspondingly as one increases the distance between the cylinders, as shown in Figures 5.19 and 5.20 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, the overall shapes of menisci resemble vertical ellipses.  As the gap width between cylinders is  increased, the menisci become similar to horizontal ellipses. When the slit width is sufficiently large, each cylinder would deform the liquid interface  with very little interference from its neighboring cylinder. Thus, with increasing distance apart, the meniscus flattens out (see Figures 5.15-5.18), approaching the case of isolated cylinders. In terms of the governing equations (Equations[4.7]): sin(qS)  dZ  when  —-—>,  a  then  dq5  [Bo  —---*  0, and  a  .  +  2(1  —  2 cos(q5)]  [5.4]  74  3.2  Bo  1.29910 2.0  2.0  =  90.0  b/a=1.05  20  —0.425  1.5  7  —/  1.0  e  =  =  1.29I0 90.0  b/a=1.50  .  —0.265  5.5 .-o:os. .:  L.a  0.66 —0.200  0.5 0.0  -  0.0  .66  0.200  —0.5 —1.0  —L.a 0.420 —‘.5  —1.5  0.26 —2.0  0.327  —2.0 0.229  —2.5  I  —1.0  —1.5  —5.2  F  —0.5  2.5  F  2.5  0.0  1.0  L.5  2.0  r 2.0  —2.5  I  —1.5  —1.0  x/a  —0.5 0.0 x/a  0.5  1.5  Figure 5.16 bIa’I.50, 0=90° are dimensionlesss pressures Cylinders Horizontal Parallel between Meniscus Profiles  Figure 5.15 b/a1.05, 0=90°  -- Level Meniscus  ....  Film Rupture Threshold  Numbers onproffles  9.0  4.0  Bo  =  1.299101 8.0  9O.O  0  Bo  1.2991Ot  0  90.0  =  7.0  b/a =3.00  b/a =6.00  —0.103  2.0 —0.090  5.0 4.0  —0.065  ...  30  —0.030  1000102030  x/a  x/a  Figure 5.17 bla=3.00, 0=90° Level Meniscus  ....  Film Rupture Threshold  Figure 5.18 b/a6.00, 0=90° Numbers on profiles are dimensionlesss pressures  Meniscus Profiles between Parallel Horizontal Cylinders  1.5  2.0  2.5  75  a) a) 0-.  a a (-) a a) C  a)  b/a  Figure 5.19 Bo=0A30  0-.  a) a) a C. a) C-) 0,  0 a)  ‘.5  b/a  Figure 5.20 Bo=3.247  Rupture Capillary Pressure vs. Separation Distance Cylinders  76  dZ dqS  sin(b)  [5.5]  [2(1—cos(qS)] /2 1  Likewise for the horizontal coordinate, equation [4.8] becomes  dX dqS  cos(ç)  [5.6]  1/2  [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 the gap between the cylinders, the parameter k approaches its maximum value of 1 0 Z  =  (  this is true for  0). Thus, the menisci between pairs of particles far enough apart, can be estimated by  linearly 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. This superposition 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 dissimilar Bond numbers but where one of them had to be relatively small. The exact solutions for the equilibrium position of cylinders at an interface found by Gifford and Scriven (26) for this case showed 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 at all separation distances as long as the Bond number is small.  They also found that this  superposition principle of adding the fields of deformation at any point between particles is dependent on the geometry and the shapes of particles, and fails in cases where capillary rise occurs as it does between parallel plates at small distances.  77  5.4 Spheres Model  Analysis of the spherical particles case shows that such factors as the contact angle, the  Bond number, and the separation distance influence the shapes of the interfaces between the particles in a similar manner as for the cylinders.  5.4.1 Effect of the Contact Angle It has been determined experimentally that particles whose contact angle is less than  900  measured through the water phase would form oil-in-water emulsions whereas those with contact angles greater than 90° (likewise measured through the water phase) would form water-in-oil emulsions (9). However, for the spherical particles and cylinders under consideration here, the contact angle can hypothetically be above or below 90° to form both 0/W and W/0 emulsions depending on the extent of immersion of the particles and the capillary pressure between them since convex 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 contact angle is either  00  (i.e. complete wetting), or 180° (no wetting), the particles will exist entirely in  one or the other phase. When the contact angle is considerably less than 90° (measured through the lower or dispersed phase), the opportunity for the meniscus reaching the film rupture point is more likely since more than half of the spherical particle is immersed in the droplet phase and only a small capillary pressure is required to meet the criteria for rupture. Thus, a more stable position occurs 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, and  b/a = 1.50). Increasing the contact angle Oof the solids, decreases their ability to be wetted by the lower (or emulsion droplet) fluid and thus, increases the difficulty for the liquid meniscus to reach  78  r/a  Figure 5.21 Bo=0.130, 0=300 Level Meniscus —  ....  Film Rupture Threshold  r/o Figure 5.22 Bo=0.130, 6=60° Numbers on profiles are ditnensionlesss pressures  Meniscus Profiles between Spheres  i.0  • r/a Figure 5.23 Bo0.130, 8=90° Level Meniscus —-  r/a  Figure 5.24 Bo0.130, 0=120° ....  Film Rupture Threshold  Numbers on profiles are dimensionlesss pressures  Meniscus Profiles between Spheres  79  0.0  r/a  Figure 5.25 Bo0.130, 9=1500 Level Meniscus —-  r/s  Film Rupture Threshold  Figure 5.26 Bo=0.l30, 0=180° Numbers on profiles are dimensionlesss pressures  Meniscus Profiles between Spheres  80  the outer perimeter of the adsorbed particles; it requires higher capillary pressures to do so. Figure 5.27 shows the relationship between  and contact angle for a separation distance of 1.50.  5.4.2 Effect of the Bond Number  Figures 5.28-5.31 show menisci profiles become flatter (capillary pressures lower) as the Bond number increases. The relationship between the rupture capillary pressures and the Bond number of 0.13 and 3.25 at a constant separation distance of 1.50 are shown in Figure 5.32 and 5.33, 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 situation approaches that of a meniscus between two flat plates (23) of similar contact angles, i.e.,  [5.8]  —>>—  1 R  J?2  The curvature in the plane of the paper, Rj, is much smaller than the principle radius of curvature,  , perpendicular to the paper, such that equations [3.14] and [3.17] become 2 R  [5.9]  dr  which is the same equation as [4.3] and [4.23] for parallel vertical plates or parallel horizontal cylinders. Similarly, Padday and Pitt (34) found that for Bo  >  when c=10 cm 2 and the  radius of curvature of the rod is 100 cm, there is good agreement between a single rod-in-free surface profile with experimental data for the meniscus around a Willielmy flat plate.  81  0.7 ——Bo = 0.130 Bo117 —  .—  0.6  -  —-v--  Bo3.5  /  /  b/a1.5  4E..  0.4/,i  —  ...  •  .  0.3 -  ,  0.2  /1.  -  I  ::  20  40  60  80  100  I  I  120  140  160  180  200  Contact angle 0 (°) Figure 5.27. Rupture capillary pressure vs. contact angle (spheres)  82  r/a  r/a  Bo=l.30x10 9=9O , Figure 5.28 5 Level Meniscus  Figure 5.29 Bo1.30x10 , =9O° 3 Film Rupture Threshold  ....  —  Numbers on profiles are dimensionlesss pressures  Meniscus Profiles between Spheres  IC  20  r/a Figure 5.30 Bo3.247, 090° Level Meniscus —  2.0  r/a  Figure 5.31 Bo=1.30x10’, ....  Film Rupture Threshold  =9o°  Numbers on profiles are dimensionlesss pressures  Meniscus Profiles between Spheres  83  O7b/a=  1.50  O(de  o.e  Bond Number Bo Figure 5.32 b/a=1.50  0.14b/a  3.00  =  0.12 x=120  0=150 v=180  0.10 0  1:: :  r  5  .5  2.0  25  3.0  Bond Iumber Ba Figure 5.33 b!a3.OO Rupture Capillary Pressure vs. Bond Number Spheres  3.5  4.0  84  If the Bond number was minimal, i.e., Bo  —  0, two possible outcomes would occur  depending on the separation distance. If b/a was very large (approaching infinity), then equation [3.17] becomes  dsin(gS) 0 dX  [5.10]  for all radial values far from the sphere, where qS = 0, and Z  =  0.  However, if the radial  coordinate b was of the same order of magnitude as a and the dimensionless X value is finite, then equation [3.17] would reduce to  cos(c)d dX  =  sin() X  [5.11]  By separation of variables, this equation becomes  cos( S 9 )ddX sin(b) X  [5.12]  1 lnIsin(b)I =ln X+lnC  [5.13]  and integration yields  or  and similarly for the vertical coordinate Z,  X 1 C  =  sin(q5)  [5.14]  85  Z=cos(q) 1 C  [5.15]  which shows that in the limit of small separation distances, and small Bond numbers, the meniscus is described by a segment of a circle of radius 1/C . Thus, the negligible gravity assumption can 1 be used for closely-packed micron-sized spheres stabilizing emulsion droplets.  5.4.3 Effect of the Separation Distance  When the distance between the spheres increases sufficiently, the capillary pressure decreases in magnitude and the profile around each sphere becomes less curved (Figures 5.345.37).  Figures 5.38 and 5.39 summarize the convex rupture capillary pressures at various  separation distances for Bo0. 130 and Bo=3 .25. In the limiting case where the gap width between spheres becomes very small compared to a, the profiles become more circular.  This result is similar to the case for the axisymmetric  meniscus 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] would  transform into  (b—a)  =cz  [5.16]  where the top sign corresponds to the convex case, and the bottom, the concave case. This is most closely shown in Figure 5.34 for the smallest two menisci.  5.5 Effect ofHysteresis  The effect of hysteresis can be illustrated in graphical form (Figures 5.40-5.43). If we start out with the contact angle at 0 a for the level meniscus, the position of the meniscus and its  86  0o  1.29990”  9=  90.0  b/=  I OS  1.0  0.0  —1.0  -1 0  10  00  r/a  r/a  Figure 5.34 b/a°°l.05, 0=90° —  Film Rupture Threshold  Level Meniscus  Figure 5.35 b/a=l.50, 0=90° Numbers on profiles are dimensionlesss poessures  Meniscus Profiles between Spheres  r/a Figure 5.36 b/a°°3.00, 0=90° Level Meniscus  Figure 5.37 b/a6.00, 0=90° ....  Film Rupture Threshold  Numbers on profiles are dhnensionlesss pressures  Meniscus Profiles between Spheres  87  Do  0.130  =  1.4 0.  O(de  0= .3  a= 60 +=90 x=120 o=150 0=180  12 •  1.I). I  1.0  1.  a  a 0.8 LIa a 0)  00 0.6 a 6)0 B  0.4  0.2  0.0 1.0  1.5  2.0  2.1  3.0  I 3.8  4.0  4.5  8.0  5  Separation Distance b/a  Figure 5.38 Bo=’O.130  t.3 Do  1.2  3.247  =  1.1 S (dee) 0= 30  1.0  =  60  +=  90  x=120  I-.  0.9  0=150 0=180  0.8 2) I-  3..  0.7  2,  a  0.  a  0.6  LI 0I  0.5 0 0 21  0  0.4  a  B  0.3  0.2  0.1..  0.0  L  LO  Ls  .0  .5  3.0  3.5  .5  .5  Separation Distance b/a  Figure 5.39 Bo=3.247  Rupture Capillary Pressure vs. Separation Distance Spheres  88  0.  a  a. I-)  C  Film Thickness h/a  Film Thickness h/a  Figure 5.40 Bo=0.130, b/a1.50  Figure 5.41 Bo=3.247, ba=1.50 Rupture Capillary Pressure vs. Film Thicluiess Cylindess  C. 0.  a  U  C C  C  00  1.0  Film Thickness h/a  Film Thickness h/a  Figure 5.42 Bo=0.130, bIal.50  Figure 5.43 Bo=3.247, bla=l.50 Rupture Capillary Pressure vs. Film Thickness Spheres  89  corresponding pressure would be determined on the solid line for 0 a emanating from the x-axis in these 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  a equal to 1800. If the hysteresis were, for example,  with increased pressure, the meniscus  300,  to 0 which marks the boundary for  shape would move along the dotted line with the meniscus contact line remaining stationary until the hysteresis reached its limit.  The final contact angle would then be 30° greater, and the  meniscus’ shape would be shifted up to a new curve which one can see requires a higher capillary pressure at film rupture.  Notice, that the maximum possible capillary pressure does not  necessarily occur at this threshold, but can occur before, or after h = 0. If it occurs before it crosses this line, the maximum capillary pressure would still need to be overcome and would be the limiting pressure for rupture of the film. In the hysteresis plots, the dotted lines show increasing hysteresis to the extent of complete nonwetting of the solid (contact angle of 180°). One can see, that this can occur before or after  h = 0. Thus, the location of the dotted line to this point will determine the maximum possible hysteresis for a particular solid’s wettability characteristics. If the hysteresis of the solid increases, the maximum capillary pressure required for rupture 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. For both the cylinders and spheres, pushing the memscus up towards the rupture threshold usually increases the capillary pressure until a maximum pressure is attained.  From then on, further  movement of the meniscus results in lower and declining capillary pressures. The figures also show that as the Bond number increases, the maxima in the curves tend to move further towards the right of the plots (i.e. before the rupture threshold is reached). This effect is more evident in the case for the cylinders than in the case for the spheres.  90  5.6 Comparison of Cylinders and Spheres  From these graphs, it can be seen that the effect of changing the three parameters may have very different results for the case of the cylinders compared to that of the spheres, due to the form of the equations for each case. The series of meniscus profiles shown for various contact angles (Figures 5.2-5.7 and 5.21-5.26) show that the cylinders generally produce more pronounced  and concentric curves than those of the spheres whose curves at the immersion angle extremities tend to overlap adjacent menisci profiles. Increasing the contact angles has the general effect of increasing 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 and the Bond nwnber. For the cylinders case (Figure 5.13 and 5.14), the rupture capillary pressure curves show a maximum value as the Bond number is increased, whereas for the spheres (Figures 5.32 and 5.33) the capillary pressure curves show no such behaviour but decrease monotonically as the Bond number increases. cylinders produce higher  Generally, for Bond numbers above the maximum point, the values than that for the spheres. However, Bond numbers below  the maximum, result in the opposite effect. Thus, in practical emulsions, spherical particles would produce more stable systems than cylindrical particles. Increasing the distance between particles, produces the same effect for cylinders and spheres. Once again though, for Bond numbers above the maximum point, the cylinders tend to produce higher  values than the spheres (see data in Appendix A). Nevertheless, the overall  trends are similar. The decrease of hysteresis, the increase in separation distance, and the increase in Bond number, or the increase in wettability lowers the required film rupture threshold making destabilization of emulsions efficacious.  91  5.7 Relationship to Emulsion Stabilization with Finely Divided Solids  Up to this point, we have considered general trends of the effect of particular variables on the capillary pressures between particles. When we extend the model down to those sizes typically involved in emulsion stabilization, we can detennine the magnitude of the capillary pressures which must be developed to cause coalescence. Good oil and water emulsions are usually formed with particles not greater than 1 pm, and even smaller for solids of high densities (27). In contrast, for mineral 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 of varying particle sizes and separation distances in the micron range. For the cylindrical model, for a separation distance of 1.05, we vary the particle sizes from one mm to one tenth of one J.Lm for the contact angle cases of 60° and 120°. These results are given in Table 5.2. Table 5.3 shows, for the 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 and Table 5.5 show data for the spherical particle case.  The maximum pressures shown are  approximate.  Table 5.2 Cylinders: Capillary pressure dependence on particle size (b/a Particle Size (cm) 0 = 60° 0.1 0.005 0.0001 0.00001 0 = 120° 0.1 0.005 0.0001 0.00001  1.05, c  =  12.99 cm ) 2  P (Pa)  (°)  2.76E-01 8.28E-02 9.34E-03 2.30E-03  2.48E+01 1.49E+02 8.40E+02 2.07E+04  145.62 137.12 126.99 123.67  2.80E-01 8.89E-02 1.28E-02 4.06E-03  2.52E+01 1.60E+02 1.15E+03 3.65E+04  157.5 157.5  7.40E-01 2.55E-01 3.48E-02 5.15E-03  6.66E+02 4.58E+03 3.13E+04 4.63E+04  126.74 108.98 82.81 69.56  1.30E+00 3.OOE-01 4.25E-02 1.14E-02  1.17E+03 5.39E+03 3.82E+04 1.03E+05  90 90 90 90  P’  m’  max (Pa)  (°)  157.5 157.5  92  Table 5.3 Cylinders: Capillary pressure dependence on separation distance (a b/a 0  0  =  600  =  1.05 2 3 4 120° 1.05 2 3 4  Table 5.4 Particle Size (cm) 0 = 60° 0.1 0.005 0.0001 0.00001 0 = 120° 0.1 0.005 0.0001  P’  Pmp (Pa)  CL  “max’  (0)  =  1 m)  “max (Pa)  9.34E-03 3.42E-03 2.27E-03 1.76E-03  8.40E+02 3.08E+02 2.04E+02 1.58E+02  126.99 124.42 123.65 123.23  1.28E-02 8.83E-03 7.20E-03 6.24E-03  1.15E+03 7.94E+02 6.48E+02 5.61E+02  157.5 180 180 180  3.48E-02 5.34E-03 3.43E-03 2.64E-03  3.13E+04 4.81E+03 3.09E+03 2.37E+03  82.81 69.73 67.88 66.94  4.25E-02 1.17E-02 8.71E-03 7.26E-03  3.82E+04 1.06E+04 7.84E+03 6.54E+03  90 120 135 135  Spheres: Capillary dependence on particle size (b/a rup’  CL  rup (Pa)  (°)  2.87E-04 5.78E-03 2.89E-01 2.89E+00  2.58E-01 1.04E+02 2.60E+05 2.60E+07  150.84 150.81 150.81 150.81  1.O1E-03 2.04E-02 1.02E+00  9.13E-01 3.67E+02 9.17E+05  137.58 137.54 137.54  “max’  1.05, c = 12.99 cm ) 2  0  0  =  =  60° 1.05 2 3 4 120° 1.05 2 3 4  mp’  Ppjp (Pa)  CL  CL  max (Pa)  (°)  3.18E-04 6.39E-03 3.19E-01 3.19E+00  2.86E-01 1.15E+02 2.87E+05 2.87E+07  141 141 141 141  5.14E-03 1.03E-01 5.14E+00  4.63E+00 1.85E+03 4,63E+06  96 96 96  Table 5.5 Spheres: Capillary pressure dependence on separation distance (a b/a  CL (0)  “max’  (0)  1 .tm) CL  ‘max (Pa)  (°)  2.89E-01 6.48E-02 2.51E-02 1.29E-02  2.60E+05 5.83E+04 2.26E+04 1.16E+04  150.81 143.13 139.29 137.06  3.19E.-01 6.54E-02 2.86E-02 1.59E-02  2.87E+05 5.89E+04 2.57E+04 1.43E+04  141 144 150 150  1.02E+00 2.31E-01 8.87E-02 4.49E-02  9.17E+05 2.08E+05 7.99E+04 4.04E+04  137.54 119.88 110.1 104.23  5.14E+00 2.34E-01 9.IOE-02 4.92E-02  4.63E+06 2.1OE+05 8.19E+04 4.43E+04  96 114 120 120  93  Examination of the results shows that the pressures are usually much higher in the case of the spheres than the cylinders for the given separation distance. For the sizes chosen, the cylinders do not have to exceed the maximum capillary pressure before rupture takes place whereas for the spheres, the maximum occurs before rupture takes place.  Thus, the maximum point further  stabilizes against rupture if it occurs before threshold rupture. The capillary pressures increase as the particle size decreases for both cylinders and spheres (see Figure 5.44). However, for smaller particle sizes (below  1 O cm) the spheres produce higher capillary pressures than the cylinders,  and vice versa for larger particle sizes.  1010.  —°—Prup-Cy1inds(0=60°) Pmax- C’Iind (0=600) Pmp Cy1ind (0 = 120°) --o--Pnx-c’Iind(0120°) ——Pnp-ph(e’60°) Pmax- Sph (0=60°) —o—Pnip-Sph(e=120°) -—e-—Pniax-Sph(0=120°)  10’-  -  10  8 -  -  106 -  ,—.  10-  O  \\ \\  \\  \\\  .6  II ‘‘4  Particle Size (cm) Figure 5.44. Particle size effect on Pcap (Cylinders and Spheres)  94  As separation distances increase, the pressure falls (Figures 5.45 and 5.46), and the location of the maxima moves forward, occurnng much further up the particle (Table 5.3, and 5.5). 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 of close-packed, 1 pm spheres, the maximum capillary pressure occurs well before rupture is encountered, but as the separation distance increases to 3.0, the maximum capillary pressure moves 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, the stability is increased. From this perspective, the most stable position that a particle can take on an interface 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 the most thermodynamically stable position as determined by Levine et al. (11). Specifically, if the particle is immersed into the lower fluid to a depth a(1  —  cos( r— 6)), the free energy is at a  minimum, and any other immersion depth is associated with an increase in the free energy of the system. The effect of hysteresis on the particles also varies with size and distances, as the maximum extent of hysteresis varies with these changes. With increased sizes and decreased separation distances, the maximum possible hysteresis observable increases. In relation to the results of Denkov et al. (8), for the same system we obtained higher maximum capillary pressure results which is as expected due to the differences in the two models  (see Table 5.6).  95  4.Oxt  Prup (0 600) Prup(0=120°) Pmax (0 = 600) ---—-Pmax(0=120°)  3.Sxl  - -°—  -  3.Oxl  2.Sxl 4.  2.Oxl 0 0 I .SxI  4 i.0xi0  3 S.0x10  0.0 3  4  Separation Distance b/a Figure 5.45. Cylinders- Separation distance effects on Pcap  —-----Pup(e=6O) • Prup(0120°) --——Pmax(0=60°) --.—-Pmax(0= 120°)  4.OxI 0  3.0xlO-  C)  2.0xI0-  I .OxI 0’  0.0  2  3  Separation Distance b/a Figure 5.46. Spheres Separation distance effects on Pcap -  4  96  0.  C.  2.  C  2  2  0-  C  0.  C.  0  0  C  C  C  C  S  Film Thickness h/a  Figure 5.48 Bo=1.3x1(r 7 b?a=3.00 Bo=1.30x10 b/a=l.05 , Figure 5.47 7 Hysteresis Plots: Rupture Capillary Pressure vs. Film Thickness Cylinders  C. C-  0-  2  2  C-  C  C U  C. U  C C  S  C C  m  Bo=1.3x10 bia=3.00 , Figure 5.50 7 Bol.30xl0 b/a=l.05 , Figure 5.49 7 Hysteresis Plots: Rupture Capillary Pressure vs. Film Thickness Spheres  97  Table 5.6 Comparison of Denkov et al’s results with the present work b/a  Maximum capillary pressure Pmax (Pa) Denkov et al.  Present work  1.05  5 2.7x10  6 3.1x10  2.00  3 7.8x10  5 1.4x10  3.00  3 2.4x10  4 5.5x10  Thus, for a system having a  /air), 4 r= mN/rn, (assuming zlp=1.594 as for CC1 30  0= 120w, and for  a 1 pm sphere at a separation distance b/a of 1.05, results in a maximum pressure of 3. lxi 06 Pa for our model, whereas Denkov et al. obtained 2.7x10 5 Pa. Furthermore, for their one-layer model when the meniscus was advanced towards the outer perimeter of the emulsion droplet, there was no change in the direction of change in the capillary pressure, i.e., no maxima were experienced as for our model.  Their model led to the conclusion that the stablest position based on capillary  considerations was when the contact angle is 1800 which opposes the effect due to thermodynamics where the stablest position is at 90°. Similarly, we found that the most stable position occurs for a level 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 the thermodynamic and the capillary forces. The pressures developed for the spherical particles at micron-sized dimensions show extremely high pressures (up to 5 MPa for the cases chosen) which are not easily overcome, and thus show that capillary pressures constitute an important aspect of the stability of emulsion droplets covered with finely divided solids.  98  Chapter 6.  THE EXPERIMENTAL PROGRAM  The objective of the experimental program is to test some of the findings obtained from the theoretical model, namely the effect of the solid separation distance, the wettability of the solids and the injected fluid volume on the meniscus shape and the capillary pressure. The sections which follow outline the experimental background for the work, the equipment setup, the experimental procedures, and the analytical methods used for interpreting the results.  6.1 Meniscus Profile Image Recording Background Trials -  The parallel-horizontal-cylinder test cell illustrated in Figure 4.1 is a simplified version of the one actually used. Two parallel cylindrical rods connected to an enclosed glass cell with only an opening between the rods posed problems in viewing the meniscus profile, because the meniscus adjacent to the glass end wall is distorted. Depending upon the wettability of the wall material, the meniscus either crept up or receded downwards along the glass surface, thereby interfering with the end-on view of the cylinders. Ideally, the meniscus should be measured far from the effects of the container walls. To overcome this difficulty, a technique was developed in which the meniscus midway down the length of the cell was highlighted by a planar vertical beam of light and then recorded 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 a cylindrical lens and then was shone upward through the glass bottom of the cell.  This lighting configuration posed another problem because very little incident light entered directly into our photomicrographic equipment which consisted of a horizontally mounted Nikon  99  camera  —  JEEmIfLi  sheet of light  I  glass cell  çaser I  I laser housing  stereomicroscope platform jack  Figure 6.1 Original test setup  I I I I I I I I I I I I I I I I I I I I I I  liii i__IIR••*I  liii..  Ti .  iii  Figure 6.2 Negatives produced by laser light technique  i i  ii  . . . .  100  SMZ 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 the light beam or to increase the exposure density were tested. Exposure time tests showed that images appeared on the photographic film only after 2 minutes. However, the outlines were not sharp and hence were unsatisfactory. With increased exposure and/or development times, there was the hazard of fogging up the photograph.  Since the signal was quite weak, any background signals could easily mask our  results at such extended processing times (62, 63). All experiments were thus done in a darkroom to exclude any extraneous light interference. Other methods of compensation and improving the resolution 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, decreasing the distance between the sample and the camera lens, and using complementary color filters to enhance the image (64-70). A further improvement was also observed when the interface was lightly dusted with a thin layer of light reflecting particles such as pliolite or talc powder (Figure 6.2). 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 difficulties was the mode in which the image is formed.  Since the light did not directly shine into the  microscope, the image recorded was only due to light refracted and diffracted as it passed through the liquid interface.  In the formation of highly magnified images, these deletrious diffraction  effects tend to be accentuated by the use of lasers (71). Thus, the quality of the memscus images on the film, especially after enlargement, was sufficiently poor that the method had to be abandoned. An alternative approach was then devised.  101  6.2 Final Image Definition Technique  Higher quality images were obtained by using background illumination.  Under these  conditions, the meniscus shape appeared as a dark silhouette against a bright background.  To  overcome problems with end effects, the meniscus was measured away from the walls and was  visibly unobstructed because the parallel rods were slightly curved, being slightly higher in the midsection 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 created by the difference in refractive indices of the two liquids. In our experimental cell, it was necessary to photograph the cylindrical solids profile simultaneously with the liquid meniscus.  If the  meniscus were convex, the solids and meniscus profiles would have to be viewed through the top liquid whereas, if the meniscus were concave, observation would have to be through the lower liquid. 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 refractive indices (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 keep the problem as simple as possible by studying an air-water system where the image is measured through the air phase in which refraction of light is minimal.  6.3 Experimental Equipment and Setup  The overall setup is shown in Figure 6.3.  Its main components consist of the same  photomicrographic equipment described earlier, the new test cell, a light source, and a micromanometer for measurement of capillary pressures.  camera connector tube with interior lens  camera  shutter release  mercury vapor or goose-neck \ lamp  color filter / diffuser  Figure 6.3 Experimental equipment setup  platform jack and leveling platform  clamp stand  valve  capillary tube and sight glass  levelling screws and platfonu  manometer fluid  micromanometer  C  103  6.3.1 The Experimental Cell  The cell, constructed of stainless steel, is made of five separate sections which when assembled looks like Figure 6.4. The base is the main reservoir for holding the fluid, the second  layer is the rubber sheet with the two rods glued to it, the third is the upper frame which fits over top of the rubber sheet, then these pieces are clamped into place, and the caliper mechanism put in place to straddle the two rods.  distance controlling screw  clamp;  caliper compass  base cell  pipe fitting  N platform  leveling screw  Figure 6.4 Experimental cell  The rods are clamped at the ends onto a metal frame with built-in slots. The top of the liquid reservoir is covered with dental rubber stretched over the perimeter of the metal cell and clamped into place. The dental rubber sheet is slit in the middle between the two cylindrical rods  104  which are glued to it. The distance between the rods can be varied by adjusting the screw on the caliper compass mechanism which in turn pushes on either side of the rods. At each end of the metal base of the cell there is a plastic pipe thread fitted with a nut, either connected with clear Tygon tubing to measure the meniscus pressures on one side, or fitted with a septum on the other side for introducing fluid with a syringe (Figure 6.5).  6.3.2 Micromanometer  Water was used as the test liquid in order to avoid the use of solvents. To measure the small pressure differences needed to detennine the capillary pressure a micromanometer (Flow Corporation Model MM3) was used. The manometer oil employed was a water immiscible blue manometer oil having a specific gavity of 1.75 (Meriam 175 Blue Fluid). All air pockets and bubbles were meticulously removed from the tubing lines by means of suction using a syringe fitted at a Tee-valve connection (see Figure 6.3). The manometer oil proved to be an aggressive solvent such 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 to form a tight seal. Fortunately, Viton® tubing which is a thick-walled flexible tubing used for vacuum 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 either end of the tubing would cause a differential change in the height of the liquid relative to some datum line. To accentuate the meniscus between the oil and water, the glass capillary is inclined at 10° from the horizontal. Measurements are made by maintaining a point on the capillary meniscus tangent to the crosshair in the eyepiece of the micromanometer by adjusting the micrometer vertically upwards or downwards. The micrometer attachment allows meniscus height readings to an accuracy of 0.0002 inch. The large cross-section reservoir of manometer fluid cancels the effect of any volume expansion in the tubing (74).  105 tubing Teflon nut and bolt  steel rod  12.5 cm dental rubber  base frame  nut andbolt  EZ 7.5cm  upper frame  I  7.5 cm  \ 11  septum  dental rubber  hollow base cell  Figure 6.5. Plan view of the bottom cell; front view of the test cell  106  6.3.3 Lighting  To enhance the sharpness of the image, a 160 W mercury vapor lamp (short wavelengths close to ultraviolet wavelengths, 70) was used as the light source. The lamp was placed inside a box with an opening to allow the illumination of only the central area of the test cell. For the first set of experiments, the rods were made hydrophobic with a Teflon coating (Crown 6065 Permanent TFE Coating Spray) which was green in color.  Therefore, a complementary red filter was  sometimes employed to enhance the contrast between the solids and the surroundings (68, 70). The filter was placed just in front of the lamp. Adjustments were made with the height of the lamp and the size of the box opening, to control the amount of light and improve the sithouetted images of the rods and meniscus against the brighter background.  6.3.4. Photographic Equipment  The stereomicroscope allowed us to magrn1’ our subject to fill the frame of the picture. It consisted of a condenser lens (5x magnification) within the microscope tube. A manual camera equipped with a shutter release was attached to the microscope tube. The height of the microscope was adjusted to approximately the same height as the meniscus in order to photograph a full frontal backlit image. The photographic film used typically was Kodak Technical Pan for its fine grain and high contrast.  6.4 Experimental Preparation ofRods  Two sets of rods were prepared. It was necessary to render them hydrophobic to develop the required menisci visible to the camera. For the first set, the stainless steel 5 mm diameter rods  107  were 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 method outlined by Bowen (75). The polymer used in our experiments is the commercially available form known as Formvar® 15/95 E (Canadian Resins and Chemicals Limited). Two wt% of the polymer resin was prepared by dissolving the Formvar powder in reagent grade chloroform (BDH’). This concentration was found by Bowen to produce a reasonably uniform layer which had an average thickness of 200  A. The coating arrangement, shown in Figure 6.6, consisted of several large  syringes minus their plungers, a four-foot length of 1 mm i.d. glass capillary tube, several teevalves, and connecting pieces of polyvinyl tubing.  The dissolved resin was poured into the  uppermost syringe which contained the stainless steel rods to be coated. The solution then drained slowly through the long glass capillary and collected in the lower syringe flask such that the level of liquid in the upper syringe fell at a slow, steady rate.  6.5 Experimental Procedure  The test cell was filled with the test liquid, in our case, water, and all tubing was then connected and checked for leaks. With the use of the tee-valve, all noticeable air bubbles present in the lines were meticulously withdrawn using mild suction from the syringe.  The lines were  subsequently refilled with water using the same syringe. The test cell and the manometer were then levelled carefully. Once the lines were filled with water including the tubing line connected to the glass capillary of the micromanometer, the tee-valve was adjusted to connect the manometer with only the test cell. To begin the experiment, the microscope was pre-focused by focusing on a wire suspended over the centre cross-section of the cell. The water level in the cell was adjusted until the meniscus was exactly horizontal. The pressure measured at this point corresponds to a zero capillary pressure. Any pressure differences recorded as water is added or withdrawn from the chamber were then a measure of the capillary pressure at the meniscus maximum or minimum (i.e.,  108  60 ml syringe Rod  Glass capillary 1 mmi.d.  120 cm  stopcocks  Figure 6.6 Coating of Rods with Polymeric Resin  at the midplane between the two cylinders). Additional fluid was added or withdrawn using a syringe. The meniscus cross-section was then photographed using a shutter release to the camera which was supported on a platform jack.  Pressure measurements were made by controlled  adjustments to the micromanometer until the manometer oil/water meniscus in the glass capillary was observed to be tangent to the crosshair in the eyepiece.  Calibration of the objects in the  photographs was made by positioning a small 3 mm ball bearing glued to a glass microscope slide with a handle clamped on a fine-control micromanipulator supported on a stand over the crosssection of interest (Fig. 6.7).  The calibration sphere was then photographed under the same  conditions as the meniscus. Once a set is finished which includes the photographing of a level meniscus as well as concave and convex shapes, the roll of film was developed accordingly and then enlarged for further analysis.  109  3.155 mm  Figure 6.7 Calibration sphere on microscope slide  The shape of the memscus on the enlarged (13 cm x 20 cm) photographs was discretized using a digitizing tablet (Kurta Model 1000). The software program written for this application is included in Appendix C. The magnification factor for a set of photographs was detennined from the ratio of the diameter of the calibration sphere calculated by the digitizing subroutine and the actual micrometer measurement. The sphere’s diameter was determined by calculation from three points on the perimeter (Appendix B).  In the photographs, the horizontal reference level was  determined as the line that connects the centers of the two rods. This line was determined by digitizing the visible portions of each rod separately, detennining its radius, and hence, the location of its center. The rods’ center coordinates were initially estimated analytically using a method based on three points taken from the uppermost part of the profile, and then more accurately calculated using a least squares fitting of nine digitized points. The full circular rod profile could then be reconstructed from a knowledge of the center position and the radius. The horizontal line between rod centers was used to calculate the baseline orientation and thus reorient the x-y coordinates of the digitized profiles. The contact points and then the rest of the meniscus profile  110  were then digitized. Estimates were then obtained for the immersion angle and contact angle as follows. The digitized data for each meniscus was used to obtain information about its slope at the particle surface by regressing straight lines to the first and last three points at both ends of the  digitized profile. The immersion angle for both sides of each meniscus was also computed during the digitizing process. Furthermore, the immersion angle was measured by determining the center of the rods independently by finding the intersection of lines bisecting at least two chords in each circular profile of the solids and measuring with a protractor. The contact angle was detennined from the supplementary angle to x for the level meniscus. Since these measurements may be prone to error, the contact angle value was further checked by calculation using the depth of immersion of the cylinder measured from the level meniscus, t, and the cylinder radius (a) (76) from  co8)=--—1  .  [6.1]  The calculated values agreed reasonably with the measured values. The reported data are basically the measured values. The profiles were then plotted using spreadsheet and graphics software (Lotus 123 and MicroCal Origin® 2.1, respectively). Figure 6.8 shows the flowsheet describing the overall image processing from the experimental stage to the analysis of the data. The meniscus pressures cap’ were determined from fluid hydrostatic considerations based on the micromanometer readings. Detailed sample calculations are shown in Appendix B. The results of the experiments are discussed in the next chapter.  111  I—  7  Images on Negatives  Experiment  Film Processing and Image Enlargement  ‘1 Measured Capillary Pressures in Inches of Manometer Oil  [ Digitization and Computation  Calculate Pcap and compare with theoretical values  Plotting of digitized values Determination of b/a, c, 4 Incorporation of theoretical profiles  Figure 6.8 Experimental program flowsheet  112  Chapter 7  RESULTS AND DISCUSSION  The experimental data and the calculated values obtained from the experiments are included in Appendix A. Some of these numbers have been summarized and tabulated in this chapter.  Experiments were performed on two sets of differently treated stainless steel rods. One  pair 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 described previously (experiments I and J). The Teflon spray produced a visibly rough surface which was expected to result in high contact angle hysteresis.  The Formvar® coating yielded a different  contact angle and a smoother finish, and thus, should correspond to a low hysteresis system. The independent variables which were investigated experimentally are the distance of separation between the rods, the applied fluid pressure in the cell, and the surface wettability of the solids. For each run, the photographed and digitized images have been superimposed to show how the 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 shown at contact positions a from  900  and upwards in 100 increments. The theoretical menisci are  generated based on the average values for the rod radius a, the distance of separation b/a, and the contact angle measured from the location of the level meniscus.  Z 1 Comparison with Theoretical Curves  A study of the profiles shows that for the chosen contact angle there is some disparity between the experimental curves and the theoretically derived curves. This is not surprising as the  113  2  —1  -2 -3  -2  .1  0  x/a Figure 7.1. Experiment B meniscus profiles  2  I  0  —1  -2 -3  -2  -1  0  1  x/a Figure 7.2. Experiment C meniscus profiles  2  3  114  2  1  0  —1  -2 -3  -2  -1  0  1  2  3  xla Figure 7.3. Experiment D meniscus profiles  I  I  I  •  I  Calculated Cylinders Digitized Data Theoretical Profiles  2  •  1  —1  -2 -3  I  I  -2  -1  I  0  1  x/a Figure 7.4. Experiment E meniscus profiles  2  3  115  I  I  Calculated Cylinder Digitized Data Theoretical Profiles  2  1  0  —1  -2 -3  I  I  -2  -1  0  I  I  1  2  xla Figure 7.5. Experiment I meniscus profiles  I  I  I  I  I  -2  -1  0  1  2  2  1  0  —1  -2  -3  x/a  Figure 7.6. Experiment 3 meniscus profiles  3  116  model profiles here do not include any hysteresis effects. The observed menisci have more pronounced curvature than expected at a given a position, which may be evidence of some resistance of the liquid periphery to movement. The nature of this resistance will be considered later. Better agreement of the profiles, however, is obtained when higher contact angles are used for 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 (experiments B-E) shows that the dimensionless capillary pressures are of the same order of magnitude as the theoretical results for the same system. For example, for Experiment B with a separation distance of 1.5, the experimental rupture capillary pressure cap’ is -0.1674 whereas the theoretical value is -0.139 for a contact angle of 60°, and if a higher contact angle is assumed, the theoretical value would be higher. Nevertheless, when one solely considers the shape of the meniscus, regardless of its immersion angle, by comparing the contact slope angle  with that of the theoretical values there  is good agreement between their respective capillary shapes and pressures (this is similar to changing the contact angle). The contact slope angle  can then be used as a basis of comparison  as we look at the effect of changing our variables of interest.  7.2 Separation Distance Between Rods  A sample calculation for the dimensionless experimental capillary pressures used in the comparisons is shown in Appendix B. From our model results, when we compared the film rupture curves for cases where the separation distance is lengthened, with all other variables remaining constant, we see that the capillary pressure tends to decrease in magnitude (see Figures 5.19 and 5.20).  In the experimental runs, it was not possible to produce the film rupture point in all cases  because excessive bulging of the memsci at the centre of the cell sometimes led to drainage of fluid  117  from 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 as they involved visual estimation. These curves, as well as curves of similar qS,’s, are compared with each other. The expected trend for comparison of capillary pressures based on distance changes is shown in Figure 7.7 (for Bo  =  0.328, 0=  900).  as separation  The trend is similar to that  obtained when only rupture capillary pressures were compared; the absolute capillary pressures decreased as separation distance was increased.  1.0  —WaL05 Wa3.0 0.5  -  Ba = 0.328 0 = 90’  I  0.0  -  .0.5  -  I  -10 -100  —  -50  I 0  I 50  100  # (‘) Fiwe 7.7. Mc d.Iatistüp beti capillaiy pressiwe aid  fIx variws b/a  Comparison of the experimental curves having similar Ø’s show that increasing separation distance between the rods generally produces a lower capillary pressure for concave menisci but t as evident for convex menisci as seen in Figure 7.8a and 7.8b for Set 1 and 2. This result may isn be due to interference caused by increasing hysteresis for more convex menisci.  118  0.4  —  -  -— -  0.3  I  0.2  -  Experiment B (bla1 .507) Experiment C (bla=1.453) ExperimentD (bla=1.556) Experiment E (b/a1.339)  0.1  U 0.0  -  U  -0.1  -  -0.2  -0.3  -  ‘%.  •-. •,  -  -0.4 -40  —  -20  0  20  40  • (°) Figure 7.8a. Set I Effect of separation distance on Peap’ vs. -  0.5  0.4  -  -  0.3  0.2  Experiment I (b/a =1.36) Experiment 3 (b/a1 .272)  -  -  0.1  0.0  -0.1  -0.2  -  -  -0.3 -50  0  (°) Figure 7.8b. Set 2 Effect of separation distance on Pcap’ vs. -  50  119  The higher hysteresis is tantamount to increasing the apparent contact angle of each meniscus and as Figure 7.9 shows increasing contact angle would increase capillary pressures.  Z3 Effect of Contact Angle  The two sets of rods may have unique contact angles since their adsorbed surface films are not the same. The measurement of the level menisci for the first set places the contact angle of the system between 60-70°, whereas for the second set, the contact angle is smaller, at 500.  Thus, the Formvar© surface is more hydrophilic than the Teflon surface (which most likely  contains additives, binders, etc.). From the model, a comparison based on  as the contact angle is varied produces a more  complicated effect than for a comparison of the rupture curves. Figure 7.9 shows that depending on the  cS value,  for a particular contact angle its pressure may be higher or lower than another  contact angle. But if the experimental point in question is located to the right of the maximum pressure on the figure, the trend is similar to that found for the rupture curves, i.e., capillary pressures increase as the contact angle increases. Experiments E and I are essentially of the same separation distance and are compared with each other. From a plot of the theoretical Peap’ vs. cc for each run, we know that the experimental  data points occur to the right of the maximum pressure and thus, for higher contact angles, higher absolute capillary pressures are expected. If we were to superimpose the respective curves for experiment E and I from Figures 7.8a and 7.Xb together, we see that for the concave curves, the expected trend is observed. However, for the more convex curves the trend is not as apparent. For the discordant points one can see that they lie just at the bottom of the curves which show the  highest deviation from theory.  0.0  -0.1  C)  o  0.1  0.2  0.3  0.4  0.5  0.6  th  (0  0  50  Bo0.328  b/a=1.5  100  Figure 7.9. Model relationship between capillary pressure and •. for various  I  -50  I  _--..  -100  -  I  ,  -150  /  /  -  0 C  121  Z 4 Effect ofHysteresis  There are several measures of hysteresis apparent in our study, one which is the more  fmiliar, the determination of the receding and advancing values of the contact angle by slight movements of the three-phase line. This measurement was not obtainable for all runs because slight changes in position were difficult to determine objectively.  However, estimates of the  approximate measures can be obtained from the location of the curves in close proximity to the level line. These curves were advanced unidirectionally from concave to convex shapes, unless  otherwise specified. Near the level meniscus, it was apparent that the contact lines did not move very much as the meniscus was advanced such that concave and convex menisci had the same end points. From these observations we are able to measure the contact slope angles for curves having similar contact lines both above (convex) and below (concave) the level meniscus.  The differences  observed with respect to the level meniscus are a measure of the contact angle hysteresis and are summarized as follows (these are approximate):  Table 7.1 Contact Angle Hysteresis  Experimental Run  Above Level  Below Level  B  15.5°  15 5°  C  <21°  129°  D  19.2°  <26.2°  B  18.2°  13°  I  7°  15°  J  6.8°  5.8°  122  As one can see, the second set of experiments, shows low hysteresis compared to the first set. The first set showing contact angle hysteresis of 15 to 20% whereas the second set shows contact angle hysteresis between 6 to 15%.  Z 5 Apparent Contact Angles  Another manifestation of hysteresis is seen in the calculated apparent contact angles for each meniscus shown in the data in Appendix A. We expected there to be an initial change in the contact angle from the level meniscus up to the advancing angle °a and thereafter to remain constant as the depth of immersion increased.  To some degree this is seen in some of the  experiments. But the effect is masked by the presence of this insidious hysteresis.  The  experimental runs show that the effective contact angle increases initially and eventually levels off but 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 one expects it to remain constant as proposed theoretically. This change in contact angle varies from 5 to 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, the dimensionless capillary pressures for each memsci were compared with those of corresponding from the theoretical predictions taken at the corrected contact angles. Tables A.4-A.9 show that improved agreement can be obtained over comparisons based on a positions. However, the contact angle which best fits the measured data differs with each run. From Figures 7.12-7.17 which plot  Pcapvs.bc we can see that the experimental points lie within the linear portion of the curves but deviate noticeably near the higher c 3 values, being higher than expected in all runs. This serves to illustrate that hysteresis is not constant throughout each run.  And although the data do not  123  90 —  —..-.  ——  0-•  A  —°——ExptB Expt c —-°“-ExptD ---A---ExptE  60  -  -  5O  I  20 100  140  120  Figure 7.10. Set 1  -  a Apparent contact angles  //  .  o vs. a position  /  50 / /  40  :: 10122  I-•  I  124  128  126  I  I  130  132  •  a (degrees) Figure 7.11. Set 2 Apparent contact angles 0 vs. a position -  134  124  0.4  1  A  0.2  0.0  -0.2  -  -  -  -go  -60  -40  -20  0  20  40  60  ‘I’ (°) Figure 7.12. Experiment B comparison with model Pcap’ vs. -  0.6  0.4  0.2  L) 0  -0.2  -0.4 I  I  -60  -40  —  I  I  -20  0  I  20  40  (0)  , Figure 7.13. Experiment C comparison with model Peap’ vs. -  60  125  0.5  0.4  0.3  0.2  (0.l  10.0 -0.1  -0.2  A  -0.3 I  -0.4  -60  -40  .  -20  I  I  .  0  I  .  20  40  60  .c (°) Figure 7.14. Experiment D comparison with model Peap’ vs. -  0.7 0.6  8=48°  0=58° 0.5  0=68° —---—-0=78°  0.4  A  Data  0.3  0.2 o.1 0  0.0 -0.1 -0.2 -0.3 I  -60  •  I  -40  -20  0  I  I  20  40  .  (°) Figure 7.15. Experiment E comparison with model Pcap’ vs. -  I  60  126  0.7  —  0.6 0.5  II I  0.4  -  a  0.3  -  0.2  -  0.1  -  0.0  -  -0.1  -  -0.2  -  a A  -0.3  —  I  -80  .  -60  I  .  I  .  -20  -40  0  •  I  20  40  60  • (°) Figure 7.16. Experiment I comparison with model Pcap’ vs. -  0.8  —  0.6 C)  U) U)  2  0.4  L)  0.2  -  -  U) U)  0 U)  0.0  -  -0.2  I  -60  40  1___  I  -20  0 ,c  20  40  (°)  Figure 7.17. Experiment J comparison with model Pcap vs. -  60  127  follow the full theoretical curves, they do show the presence of a minimum point in the Pcap’vs.Ø plot before rising again. As in the theoretical results this shows the presence of a maximum capillary pressure point (which is not necessarily the film rupture point) above which further immersion of the rods translates into a declining capillary pressure. However, the experimental digitized data can show remarkably good agreement with model curves when the apparent contact angles are used. To illustrate the improved results one can obtain when this hysteresis is accounted for, a curve from each experiment was overlayed on capillary menisci calculated from the model for the same conditions but with its apparent contact angle as shown in Figures 7.18-7.23. For C-CVX16, shown in Figure 7.19, the experimental cap’ 1  is -0.3180 which agrees favorably with the model meniscus at the same immersion angle  having a cap of -0.2904. Comparisons based solely on a positions at the contact angle from the level meniscus have been shown to be inadequate because of the effects of hysteresis apparent in our system. This poor agreement between experiment and the model are shown in Figure 7.24 for experiment B as an example, where the immersion angle a is plotted against q. The variations encountered lead us to question what may be the cause.  128  1.5  1.0  0.5  0.0  -0.5  -1.0  -3.0  -2.0  -2.5  -1.5  -1.0  -0.5  0.0  0.5  1.0  2.0  1.5  2.5  x/a Figure 7.18. Experiment B curve (B-CVXI6) fit with model curves for o78°  2.5  2.0  1.5  1.0  0.5  0.0  -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  xfa Figure 7.19. Experiment C curve (CVXI6) fit with model curves for 0=88.5°  3.0  3.0  129  2.0  • 1.5  -  1.0  Meniscus profile (CVX17) Model profiles (10° increments)  -  0.5  -  0.0  Bo0.328 b/a1 .558 0=84.5°  -  -05  —  -1.0 -3.0  -2.5  I  I  -2.0  -1.5  I______I  I  •  -1.0  -0.5  0.0  0.5  I__•  •  1.0  I  1.5  2.0  2.5  3.0  x/a Figure 7.20. Experiment D curve (CVXI7) fit with model curves for 0=84.5°  •  1.5  Meniscus profile (CVX28) Model profiles (10° increments)  1.0  0.5  Bo0.328 b/a1 .339  0.0  0=75.10  -0.5  -  -1.0 I  -3.0  .  -2.5  I  -2.0  .  I  -1.5  .  I  -1.0  .  •  -0.5  I  0.0  .  •  0.5  I  .  1.0  I  .  1.5  x/a Figure 7.21. Experiment E curve (CVX28) fit with model curves for  0=75.10  I  2.0  .  2.5  3.0  130  1.5  1.0  0.5  0.0  -0.5  -1.0  -2.5  -2.0  -1.5  -1.0  -0.5  0.0  0.5  1.0  1.5  2.0  2.5  x/a Figure 7.22. Experiment I curve (CVX7) fit with model curves for 8=70.5°  1.5  1.0  0.5  0.0  •  Meniscus profile (CVX6) Model profiles (10° increments)  -  -  Bo0.328 bfal .279 0=67.2°  -  -0.5  -1.0 I  -2.5  -2.0  -1.5  .  I  -1.0  I  -0.5  0.0  0.5  .  I  1.0  I  .  1.5  x/a Figure 7.23. Experiment J curve (CVX6) fit with model curves for 8=67.2°  2.0  2.5  131  180  160  140 / / / / / / / / /  120  / / / / /  /  100  / / / /  /  80  / / /  60 -80  I  I  -20  0  —  -60  -40  20  40  60  (°) Figure 7.24. Experiment B comparison with model  -  a position vs.  132  Z 6 Hysteresis and Kinetic Forces  It is worthwhile to determine whether the discrepancies are caused by what we consider hysteresis or whether other factors such as surface interactions, or nonequilibrium conditions may be 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 time where a previously wetted surface shows a difference in contact angle, or if the menisci is at a metastable condition which can be overcome if the required energy level is introduced by vibrations to the system (22) we know that kinetic surface effects may be the cause. A better understanding of 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 used in this technique (Figure 7.25).  Figure 7.25 Wilhelmy plate method  A plate vertically mounted over a liquid is attached to one arm of a balance such that the force exerted on the plate can be measured while it is being immersed. When the plate is lowered so it just touches the surface, a force is exerted on the plate which is equal to the weight of the liquid in the meniscus.  As the plate is immersed further into the liquid, this force decreases due to  buoyancy. 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 to  133  z  0  2 Cl)  I  W W  I  a.  I-  FORCE  FORCE  (a)  (b)  Figure 7.26 Force vs. depth curves for the Wilhelmy plate apparatus; (a) with hysteresis; (b) without hysteresis.  z  0 Cl)  w  IL  0 I I  a.  w 0  I FORCE  Figure 7.27 A hysteresis loop for a system showing solid-liquid interaction.  134  the value of the advancing angle, whilst the measured force decreases. When 1 a has been reached it remains constant, and the force vs. immersion depth curve is a straight line with a slope due to buoyancy. When the plate is then withdrawn from the liquid, the contact angle begins to recede down to °r• Then at this value, further withdrawal results in a straight line with a slope due to buoyancy once again. This curve is the hysteresis loop and helps us to define more clearly the contact angle hysteresis using the terms outlined by Everett (76). Everett used two criteria to define the contact angle hysteresis. The first is that the hysteresis 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 to emersion at different depths. The presence of scanning curves means that all points of the loop are attainable from different directions. Variations in contact angle which do not agree with the above criteria are possible and can be caused if any of the surface energy components at the interface Ysg’ Ysl’ and hg changes during the measurement. These time-dependent effects can be isolated by testing for repeatability of the contact angle. A typical curve in which surface interactions such as desorption or adsorption occurs, causing the loop to shift to different angles is shown in Figure 7.27. These effects can be serious 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 definite variability in the results in terms of the immersion angles and to a lesser degree with the capillary pressures. Figure 7.28 shows a sample hysteresis loop which was performed for Set 2, based on experiment 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 the loop. Since experiment 3 proved to be more repeatable than experiment I, it is possible that the problem may be due to a change in surface energetics caused by wetting and may be avoided by pre-wetting the rods before taking measurements.  135  uj 8 —A—  A  Measured Data  0.2  1  0.1  2  0  9  .4  o.o  A  7  3 4 A  10 0 -  5 A  12 A  6A  JJ3  All  ‘  -20  I  I  I  -10  0  10  Figure 7.28. Hysteresis ioop for experiment 3  20  30  136  A comparison of the replicates of the level meniscus for I and J (see Table 7.2) show that there is fair agreement with the measured capillary pressure, but larger variations with respect to the 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 Meniscus  Curve  a  Type  P’  1-LEV5  126.5  adv.  0  I-LEV1O  133.0  rec.  0.009  I-LEV12  133.5  rec.  0.01404  I-LEV14  133.0  adv.  0.0227  J-LEV3  130.3  adv.  0  J-LEV7  131.6  rec.  0.01745  J-LEV9  130.5  adv.  0.00864  Average I  131.5±2.9  w/o I-LEVS  133.3±0.24  130.8±0.6 adv.= advancing rec. = receding 3  0.01144±0.008  0.008696±0.007  One other possibility for the variations in contact angle is the speed at which the three phase line is moved (15). Dynamic measurements of contact angle show that there is a relationship  with the speed of movement of the contact front. A typical experimental plot is shown below.  137  -  O (extrapolated)  r (static measurement) ‘Uc U<o  0  u>o  CONTACT LINE SPEED Figure 7.29 Typical experimental results for the dependence of the dynamic contact angle 9, on the speed of the contact line. When U>O (U<O) the contact line is advancing (receding). U denotes the slowest speed at which an experimental measurement is made (55).  It has been suggested that the advancing and receding angles should be extrapolated from dynamic contact angle measurements rather than the static contact angle generally reported which is usually measured by starting with a static meniscus and advancing it until the contact line is perceived to have moved. As one can see from the Figure 7.29, these values from the two methods tend to be different. Although there are a few cases where reproducible contact angles are the same for slow advancement and recession of the interface (77), they are not the rule. It is well known for both theoretical and experimental work that the advancement and recession (axisymmetric) of liquid on a solid with fine concentric grooves or sawtooth ridges, will produce a steady contact line motion followed by “Haines jumps” (78). This phenomenon was observed in the present experimental work.  Z 7 Sources ofExperimental Error  The use of slightly curved rods rather than horizontal rods may cause some unexpected variation in the results since the hydrostatics as well as the capillary shapes may be different along the length of the rods. This would translate into measured capillary values which might be lower  138  than the actual value for the meniscus at the center of the cell. The micromanometer readings could be erroneous if air bubbles are still present in any of the tubing between the manometer oil reservoir and the cell itself. Although no such trapped bubbles were visible in the transparent  tubing and glass fittings leading to the manometer oil tubing which was opaque.  All glass  capillaries 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 and showed no signs of adhering to the glass or resisting movement. Minimal error may be incurred when judging the line where the cross-hair coincides with the meniscus. The reproducibility error is 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. The lighting of the meniscus as well as the rods needed to be precise and was the most difficult aspect to control. However, as some of the photographs show, the lighting was at times uneven, such that the top of the rods had too much illumination making parts of the rods, and sometimes part of the menisci (near the contact line), seem to disappear into the bright background. In these cases, the missing areas were extrapolated in the digitization process. Other sources of error were the manual digitization process of images (79), and the enlargement of images by 25 times led to lines which were fuzzy and broader.  Thus, some  interpretation of the lines and extrapolation of unclear areas were made. The actual digitizing error was determined by repeatedly digitizing the calibration sphere ten times. The mean value for the diameter (calculated from the three-point method) in terms of digitized units was 3317.266 ± 8.776 or ± 2.7% standard error of the mean (SEM). The combined effect of these errors can be seen in the variability of the calculated cylinder points and the digitized points on the cylinder in the composite profiles obtained from consecutive photographs. The deviation is approximately a tenth of a millimeter. The gross error involved in the experimental reproducibility of the level meniscus as shown previously in Table 7.2 shows that the immersion angle mean for I is 131.5±2.9° (2% SEM), and for 3 is 130.8 ± 0.6° (0.5% SEM).  139  Chapter 8  CONCLUSIONS  The capillary interactions of cylindrical and spherical shapes situated on a fluid-fluid interface were considered with relation to the characterization of the capillary phenomena responsible for the stability of the disjoining film between two coalescing solids-stabilized emulsion droplets.  Both a theoretical and experimental approach were taken to determine the effect of  several factors which play a part in this process. The models represent idealized systems in which the particles are identical and homogeneous. Meniscus profiles were generated for various configurations for an ideal system without hysteresis.  Since all practical systems do contain hysteresis effects, this aspect was  incorporated into the models, and considered with respect to its effect on the coalescence between emulsion droplets. In the modelling of the coalescence process, the existence of a critical film thickness for rupture, as well as the kinetics involved, were neglected. The models represent the coalescence process by which a layer of particles on approaching emulsion droplets make contact  and the capillary menisci between particles are deformed either by pushing the particles further into their respective droplets or by squeezing liquid from within the droplets due to shear or deformation. 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 that i) the finer the particles, or ii) 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 Bond number or a decrease in the wettability to the dispersed phase (or increase in contact angle) would increase stability based on capillary pressures. These trends are similar for both the cylinders  140  model and the spheres model which nevertheless produce very different profiles and capillary pressures. The main difference between the two models is seen in the piots of the threshold rupture pressure 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 rupture pressures than the cylinders. Increasing the hysteresis of the particles in the models was also found to increase the stability of the disjoining film by increasing the required rupture capillary pressure.  Varying  hysteresis in effect changes the contact angle of the solids. The maximum observable extent of hysteresis for a given starting contact angle is determined by the position of the hysteresis curve with respect to the location of the threshold rupture line. Furthermore, we determined from the hysteresis plots how the capillary pressures can be a stabilizing mechanism to prevent the thinning of the disjoining film layer between coalescing emulsion droplets.  It was determined that the limiting pressure preventing rupture was not  necessarily 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. location of the maximum capillary pressure (as seen in the hysteresis plots  -  The  Pcap’ vs. h) can be  varied by changing the size of the solids, or the separation distance. For both the cylinders and spheres, the maximum pressure moves to the right of the film rupture line on the hysteresis plots when 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 are more effective as emulsion stabilizers than cylindrical particles. Based on capillary pressure theory, the most stable position that a particle can take on the interface is the depth to which the meniscus is level at contact (to an immersion depth of a(1—cos(r— 9))).  This position coincides with the minimum free energy of the system as  determined by Levine et al.(1 1).  Similarly to Denkov et al.’s conclusion, the stablest position  would be the point furthest from rupture which would occur if the solid is nearly completely wetted by the continuous phase (contact angle of 1800).  From a thermodynamic standpoint this position  141  occurs when the contact angle is  900.  The actual most stable position may be determined by a  combination of both thermodynamic and capillary factors such that 90° 0 . 1800. This may explain the experimental results of Schulman and Leja (27) who found that for contact angles greater than 90° (measured through the water phase), W/0 emulsions occur, and for contact angles less than 90°, 0/W emulsions are observed. Several differences were noted between Denkov et aI.’s model and our spheres model. For a similar system described in their paper, our model yielded higher maximum capillary pressure results which is as expected due to the different geometries of the two models. Furthermore, as the menisci were made to advance towards the outer perimeter of the emulsion droplet, their results produced a gradual increase in pressure whereas in our model, a maximum was often evident. One other difference between the models is in the handling of gravity. Their model neglects gravity whereas ours does not. It was determined from a theoretical basis that the constant mean curvature assumption would be satisfactory for systems of small Bond numbers and with very closely-packed particles. However, for larger systems, a general approach such as the one taken in this work would be more applicable. The experimental work revealed that comparisons based on contact angles measured from the level meniscus were in poor agreement with model-generated profiles which is not surprising as it neglects any effects of hysteresis. However, when comparisons are made based on the contact meniscus slope angle, results are better. Experiments based on this criterion showed that the expected 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 solids sprayed with Teflon than with the Formvar resin.  Hysteresis had the effect of increasing the  capillary pressures. This was most evident for the more convex curves in each run, as they showed the highest hysteresis, and produced higher capillary pressures than expected from theoretical calculations based on a contact angle measured from the level meniscus. Good agreement between  142  theory and experiment was obtained when the contribution due to hysteresis is included, by using apparent contact angles in the model. Thus, the proposed model is appropriate for describing this macroscopic system and the theory may be used to characterize effects involved in the stabilization of the disjoining film between emulsion droplets by capillary phenomena. The behaviour of the hysteresis was more complex than expected.  The hysteresis  measured at the level meniscus did not remain constant as assumed in the theoretical work, but increased with the upward movement of the interface and finally levelled off much later. The varying 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 are taken. The main sources of error occurred in the photographic recording of the images, the lack of 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 also  recommended 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.  143  NOMENCLATURE  a  radius of a solid cylinder or sphere  ae  radius of an emulsion droplet  A  area  b  half the distance between the centers of a pair of solids  b*  curvature at apex of sessile or pendant drops  B  dimensionless separation distance between adjacent cylinders/spheres (half), b/a  B’  Bashforth and Adams’ shape factor  Bo  Bond number; c a 2  )gIy 1 p 2 ( d  distance between closest points on adjacent cylinders/spheres  d  horizontal distance between contact lines on each cylinder  dr  horizontal distance determined from the physical geometry as a function of a  dH  dimensionless measure of distance from rupture; difference between actual and desired vertical coordinate of meniscus  dL  dimensionless difference between actual and desired horizontal length of meniscus  E, F  elliptic integrals of the first and second kind, respectively  Jg  function  h  distance of centre of meniscus from rupture threshold  H  h/a  Hact  dimensionless vertical distance between Z, and Z 0  Hr  required vertical distance of memscus from contact level Z to rupture line  k  modulus in the elliptic integral  K  curvature of a point on a curve  n  number of equations  O/JJ W/O  oillwater, water/oil  144  pressure  P zIP,  “2  pressure difference; pressure in the upper fluid, pressure in the lower fluid  e’ b  pressure inside emulsion droplet; pressure in the bulk phase  cap’ P,p  capillary pressure; capillary pressure at the rupture threshold  cap’ 1  dimensionless capillary pressure, P divided by 2y/a  r  radial coordinate  Rj, R 2  principle radii of curvature  s  arclength  U  speed  U  slowest measurable speed  w  specific surface free energy  W  work  X  r/a (spheres model), na (cylinders model)  Y  set of vectors  x, y, z  Cartesian coordinate axes  z  vertical coordinate measured from the free level interface  z*  “  the line joining the centres of the solids  z  level of meniscus at contact line from the free level interface  zt,  level of meniscus at its lowest or highest point from the free level interface for a capillary rise or depression, respectively.  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Thesis, University of British Columbia, 1978.  76.  Neumann, A.W. and R.J. Good, “Techniques of Measuring Contact Angles”, Surface and Colloid Science, Vol. 11, E. Matijevic, ed., Plenum Press, New York, 1979, pp. 31-91.  151 76.  Neumann, A.W. and R.J. Good, “Techniques of Measuring Contact Angles”, Surface and Colloid Science, Vol. 11, E. 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. 10771096 as cited by Johnson and Dettre.  78.  De Fazio, J.A., and D.C. Dyson, “Stability of Rectilinear Contact Lines I. Single Contact Line 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 Surface and Interfacial Tension by the Axisymmetric Drop Technique”, Colloids and Surfaces, 9, 1984, pp. 307-3 17.  -  152  APPENDIX A EXPERIMENTAL DATA -  Table A.1 Micromanometer Readings -  Table A.2 Set 1 Compiled Experimental Data -  Table A.3 Set 2 Compiled Experimental Data -  Table A.4 Experiment B: Comparison of Pcap’ based on a and -  Table A.5 Experiment C: Comparison of Pcap’ based on a and -  Table A.6 Experiment D: Comparison of Pcap’ based on a and -  Table A.7 Experiment E: Comparison of Pcap’ based on a and -  Table A.8 Experiment I: Comparison of Pcap’ based on a and -  Table A.9 Experiment J: Comparison of Pcap’ based on a and -  4  0.2903 0.5380 0.3080 0.4449 0.6060 0.5698  0.3802 0.5476 0.6557 0.7005 0.7382 0.5228 0.6960  0.4120 0.4605 0.5473 0.6720 0.7703 0.7400  0.8057 0.4921 0.6626 0.7228 0.8043  Expi B CCV13 CVX16 CCV2O LEV2 1 * CVX23 CVX24  Expt C CCV9 LEV 11 * CVX12 CVX15 CVX16 CVX19 CVX23  Expt D CCV4 CCV6 LEV7* cvxlO CVX13 CVX17  Expt E CVX23 CCV24 LEV26* CVX27 CVX28  h(in.)  1475.582 901.246 1213.505 1323.757 1473.018  754.549 843.373 1002.341 1230.720 1410.750 1355.257  696.309 1002.890 1200. 868 1282.916 1351.961 957.471 1274.675  531.664 985.309 564.080 814.803 1109.846 1043.548  SET1 P(dynlcm ) 2  Table A. 1  -262.077 312.259 0.000 -110.252 -259.513  247.792 158.968 0.000 -228.379 -408.409 -352.916  306.581 0.000 -197.977 -280.025 -349.070 45.419 -271.784  283.139 -170.506 250.723 0.000 -295.043 -228.745  -0.281 0.335 0.000 -0.118 -0.278  0.266 0.170 0.000 -0.245 -0.438 -0.378  0.329 0.000 -0.2 12 -0.300 -0.374 0.049 -0.291  0.304 -0.183 0.269 0.000 -0.3 16 -0.245  m’  0.3168 0.4142 0.4386 0.4873 0.5726 0.6579 0.7797 0.8407 0.6823 0.7091 0.7797 0.6945 0.6092 0.6823  0.5697 0.6215 0.6603 0.6733 0.7251 0.7639 0.6733 0.5697 0.6603 0.6992 0.8028 0.7639  Expt I ccvi CCV2 CCV3 CCV4 LEV5* CVX6 CVX7 CVXS LEV9 cvxi0 CVX1 1 LEV12 CCV13 LEV14  Expt J ccvi CCV2 LEV3* CVX4 cvx5 CVX6 LEV7 ccv8 LEV9 cvxi0 cvx1 1 CVX12  h(in.)  1043.365 1138.233 1209.293 1233.101 1327.969 1399.029 1233.101 1043.365 1209.293 1280.535 1470.271 1399.029  580.197 758.578 803.265 892.455 1048.676 1204.897 1427.965 1539.682 1249.584 1298.666 1427.965 1271.927 1115.706 1249.584  SET2 P(dyn/cm ) 2  Micromanometer Readings (* denotes the level meniscus standard)  ) 2 iP(dyn/cm  -  165.927 71.059 0.000 -23.809 -118.677 -189.736 -23.809 165.927 0.000 -71.243 -260.979 -189.736  468.480 290.098 245.411 156.221 0.000 -156.221 -379.289 -491.006 -200.908 -249.990 -379.289 -223.251 -67.030 -200.908  ) 2 zP(dyn/cm  0.177 0.076 0.000 -0.025 -0.126 -0.202 -0.025 0.177 0.000 -0.076 -0.278 -0.202  0.502 0.311 0.263 0.167 0.000 -0.167 -0.407 -0.526 -0.215 -0.268 -0.407 -0.239 -0.072 -0.215  m’  153  -29.8 22.5 -28 0 35.5 32.5  B=1.460 -25 0 23.7 32.7 38.3 3 40  B1.558 -31.7 -14.5 0 20 35 39  112.5 124.3 123.8 123.8 133.5 137.8  a0.156 118.3 119.7 120 125 129.8 130.7 134  a0.156 111.5 108.5 111.7 114.7 132.3 134.5  CCV13 CVX16 CCV2O LEV21 CVX23 CVX24  ExptC  CCV9 LEV11 CVX12 CVX15 CVXI6 CVX19 CVX23  ExptD  CCV4 CCV6 LEV7 CVX1O CVX13 CVX17  -  B1.503  a=’0.156  -  ExptB  Table A.2  36.8 57 68.3 85.3 82.7 84.5  36.7 60.3 83.7 87.7 88.5 52.3 86  37.7 78.2 28.2 56.2 82 74.7  Apparent 0  0.266 0.1704 0 -0.245 -0.438 -0.3784  0.3287 0 -0.2123 -0.3002 -0.3743 0.0487 -0.2914  0.301 -0.181 0.266 0 -0.313 -0.243  Set 1 Compiled Experimental Data  0.2262 0.1197 0.0000 -0.1863 -0.6121 -0.7981  0.1675 0.0000 -0.1580 -0.3112 -0.3351 -0.1437 -0.5745  0.3617 -0.1752 0.1745 0.0000 -0.4249 -0.4498  Au  0.0380 0.0201 0.0000 -0.0313 -0.1028 -0.1341  0.0281 0.0000 -0.0265 -0.0523 -0.0563 0.0000 -0.0965  0.0608 -0,0294 0.0293 0.0000 -0.0714 -0.0756  0.2280 0.1503 0.0000 -0.2137 -0.3352 -0.2443  0.3006 0.0000 -0.1858 -0.2479 -0.3180 0.0487 -0.1949  0.2402 -0.1516 0.2367 0.0000 -0.2416 -0.1674  Porn’  154  -  83.8 73.8 58.7 76.7 85.5  -0.2792 0.3326 0 -0.1175 -0.2765  Set 1 Compiled Experimental Data (continued)  46.3 13.3 0 18.2 35.3  142.5 119.5 121.3 121.5 129.8  CVX23 CCV24 LEV26 CVX27 CVX28  Table A.2  B1.339  a=O.156  ExptE -0.6562 0.1072 0.0000 -0.1205 -0.3214  -0.1103 0.0180 0.0000 -0.0202 -0.0540  -0.1689 0.3146 0.0000 -0.0973 -0.2225  155  123±1.3 124.5±1.9 126.9±4.6 129.6±3.3 128.4±1 124.6±3 133.5±1 132.3±1.2 129.4±.55 131.9±4.2 133.52±2.5 134.01±4.3 134.6±3.7 132.8±8  1.345 1.337 1.340 1.301 1.364 1.404 1.328 1.409 1.403 1.377 1.342 1.343 1.356 1.346 1.357 0.027  0.16 0.159 0.159 0.164 0.157 0.152 0.159 0.15 0.152 0.153 0.157 0.154 0.154 0.156 0.156 0.003  CCV#1 CCV#2 CCV#3 CCV#4 LEV#5 CVX#6 CVX#7 CVX#8 LEV#9 CVX#10 CVX#11 LEV#12 CCV#13 LEV#14  Average Std. Dev.  m’  0.4693 0.3803 0.2370 0.1340 0.0000 -0.1427 -0.2636 -0.2960 0.0053 -0.0134 -0.1682 -0.0059 0.1189 0.0047 18.2 22 33.2 40.7 53.5 59 70.5 70.5 49.5 52 64.8 46.5 57.8 47  -38.5 -34.5 -22.5 -15 0 6.5 21 24 0 5 18.3 0 11 0  123.3 123.5 124.3 124.3 126.5 127.5 130.5 133.5 130.5 133 133.5 133.5 133.2 133  (measured)  pp (°) i0  (°)  a (°) (manual)  Table A.3 Set 2 Compiled Experimental Data -  a (°)  B  a (cm)  Expi I  0.2558 0.1584 0.1340 0.0853 0.0000 -0.0853 -0.2071 -0.2681 -0.1097 -0.1365 -0.2071 -0.1219 -0.0366 -0.1097  0.0419 0.0259 0.0219 0.0140 0.0000 -0.0140 -0.0339 -0.0439 -0.0180 -0.0224 -0.0339 -0.0200 -0.0060 -0.0180  0.4274 0.3544 0.2151 0.1201 0.0000 -0.1287 -0.2296 -0.2521 0.0233 0.0090 -0.1343 0.0141 0.1249 0.0227  cap’  156  1.304 1.276 1.27 1.254 1.216 1.278 1.307 1.298 1.268 1.262 1.273 0.019  0.153 0.154 0.156 0.156 0.159 0.155 0.153  0.155 0.156  0.157  0.155 0.001  CCV#1 CCV#2 LEV#3 CVX#4 CVX#5 CVX#6 LEV#7 CCV#8 LEV#9 CVX#10 CVX#11 CVX#12  Average Std. Dev.  Table A.3  B  a (cm)  Expt J  -  (°)  -9.5 -5.8 0 6.8 15 18 4.8 14.8 0 6.5 22.5 18  a (°) (manual)  128.8 129.5 130.3 131 130.8 130.8 131.6 130.3 130.5 130.8 131.3 131.5  m’ (mSUd)  0.1697 0.0970 0.0000 -0.0399 -0.1500 -0.2592 0.0153 0.2470 0.0086 -0.0503 -0.3297 -0.1942  app (°) 0  41.7 44.7 49.7 55.8 64.2 67.2 53.2 64.5 49.5 55.7 71.2 66.5  Set 2 Compiled Experimental Data (continued)  125.9±.7 130±1.3 128.9±2.6 130.9±2.4 133.±1 131±1.5 131.6±.1 130.3±.7 130.5±1 129.±4.6 131.3±.3 128.8±.2  a (°)  0.0906 0.0388 0.0000 -0.0130 -0.0648 -0.1036 -0.0130 0.0906 0.0000 -0.0389 -0.1425 -0.1036  0.0148 0.0064 0.0000 -0.0021 -0.0106 -0.0170 -0.0021 0.0148 0.0000 -0.0064 -0.0233 -0.0170  0.1548 0.0906 0.0000 -0.0377 -0.1394 -0.2422 0.0174 0.2322 0.0086 -0.0439 -0.3063 -0.1772  cap’  157  158  Table A.4 Experiment B Comparison of cap’ based on a and cb -  a=0.156  B  =  1.503 Meniscus Curve Number  Parameter  Experimental a (°) • (°) “cap’ Model 0 = 56° abased %Diff based %il3jff  0  =  46° abased %]3jff  based  %Djff 0  =  66° abased %Diff Øbased O/Jjff  CCV: concave CVX: convex  CCV13  CVX16  CCV2O  LEV21  CVX23  CVX24  112.5 -26 0.2402  124.3 22.5 -0.1516  123.8 -22 0.2367  123.8 0 0.0000  133.5 35.5 -0.2416  137.8 32.5 -0.1674  cap 0.1222 -49.1 0.2708 12.7  -0.0087 -94.3 -0.1341 -11.5  -0.0045 -101.9 0.2381 0.6  -0.00452 0.0 0.0000 0.0  -0.0696 -71.2 -0.1640 -32.1  -0.0960 -42.7 -0.1567 -6.4  0.2111 -12.1 0.2662 10.8  0.0894 -159.0 -0.1134 -25.2  0.1026 -56.7 0.2267 -4.2  0.1026 0.0 0.0000 0.0  0.0109 -104.5 -0.1388 -42.6  -0.0240 -85.7 -0.1356 -19.0  0.0119 -95.1 0.2844 18.4  -0.0775 -48.9 -0.1550 2.3  -0.0655 -127.7 0.2385 0.8  -0.0655 0.0 0.0000 0.0  -0.1388 -42.6 -0.1900 -21.3  -0.1590 -5.1 -0.1837 9.7  159  Table A.5 Experiment C Comparison of Pcap’ based on a and -  a=0.156  •(°) “cap’  =  1.460  Meniscus Curve Number  Parameter  Experimental a(°)  B  CCV9  LEV11  CVX12  CVX15  CVX16  CVX19  CVX23  118.3 25 0.3006  119.7 0 0.0000  120 23.7 -0.1858  125 32.7 -0.2479  129.8 38.3 -0.3180  130.7 3 0.0487  134 40 -0.1949  Model 0 = 60° abased % Diff •based % Diff  0.0189 -93.7 0.2974 -1.0  0.0005 0.0 -0.0028 0.0  0.0024 -101.3 -0.1530 -17.6  -0.0440 -82.2 -0. 1732 -30.1  -0.0831 -73.9 -0.1833 -42.4  -0.0904 -285.7 -0.0250 -151.4  -0.1070 -45.1 -0.1854 -84.5  50° abased % Diff • based % Diff  0.1259 -58.1 0.2944 -2.1  0.1057 0.0 -0.0010 0.0  0.1016 -154.7 -0.1302 -29.9  0.1050 -142.3 -0.1470 -40.7  0.0020 -100.6 -0.1556 -51.1  -0.0042 -108.6 -0.0178 -136.6  -0.0302 -84.5 -0. 1583 -18.7  70° abased % Diff based % Diff  -0.0858 -128.6 0.3016 0.3  -0.0964 0.0 -0.0007 0.0  -0. 1031 -44.5 -0.1816 -2.3  -0. 1394 -43.8 -0.2077 -16.2  -0.1672 -47.4 -0.2139 -32.7  -0.1694 -447.9 -0.0271 -155.7  -0.1828 -6.2 -0.2160 10.8  0  0  =  “cap’  160  Table A.6 Experiment D Comparison of Pcap’ ba.sed on a and -  a=0J56  B= 1.558 Meniscus Curve Number  Parameter  Experimental a (°) (°) “cap’  CCV4  CCV6  LEV7  CVX1O  CVX13  CVX17  111.5 -31.7 0.2280  108.5 -14.5 0. 1503  111.7 0 0.0000  114.7 20 -0.2137  132.3 35 -0.3352  134.5 39 -0.2443  Model o = 48° abased % Diff 4based % Diff  0.1875 -17.8 0.2981 30.7  0.2197 46.2 0.1290 -14.1  0.1858 0.0 0.0000 0.0  0.1557 -172.8 -0.1014 -52.5  -0.0022 -99.4 -0.1335 -60.2  -0.0121 -95.1 -0.1359 -44.4  8=58° a based % Diff based % Diff  0.0978 -57.1 0.3028 32.8  0. 1276 -15.1 0. 1451 -3.5  0.0978 0.0 0.0000 0.0  0.0635 -129.7 -0.1175 -45.0  -0.0720 -78.5 -0.1543 -54.0  -0.0860 -64.8 -0.1562 -36.1  68° a based % Diff based % Diff  0.0078 -96.6 0.2998 31.5  0.0377 -74.9 0. 1514 0.7  0.0024 0.0 0.0000 0.0  -0.0256 -88.0 -0. 1409 -34.1  -0.1422 -57.6 -0.1795 -46.4  -0.1499 -38.7 -0.1833 -25.0  78° abased % Diff based % Diff  -0.0843 -137.0 0.2750 -100.5  -0.0583 -138.8 0.1514 -101.0  -0.0902 0.0 0.0000 0.0  -0.1141 -46.6 -0.1610 -99.9  -0.2001 -40.3 -0.2102 -99.7  -0.2062 -15.6 -0.2122 -99.7  o  8  =  =  ’cap 1  161  Table A.7 Experiment E Comparison of Pcap’ based on a and -  a0.156  B=1.339  Meniscus Curve Number  Parameter  CVX23  CCV24  LEV26  CVX27  CVX28  142.5 46.3 0.1689  119.5 13.3 0.3146  121.3 0 0.0000  121.5 18.2 -0.0973  129.8 35.3 4).2225  Experimental  a(°) “cap’ Model o = 48° abased % Diff Øbased % Diff  o  o  O  =  =  =  “cap’ -0.0799 -52.7 0.1713 -201.4  0.1479 -53.0 -0.1687 -153.6  0.1337 0.0 0.0013 0.0  0.1196 -222.9 -0.1230 26.5  0.0227 -110.2 -0.1613 -27.5  58° abased % Diff based % Diff  -0.1572 -6.9 0.1965 -216.3  0.0311 -90.1 -0.1943 -161.8  0.0110 0.0 0.0035 0.0  0.0110 -111.3 -0.1487 52.9  -0.0829 -62.7 -0.1891 -15.0  68° abased % Diff Øbased % Diff  -0.2203 30.4 0.2087 -223.5  -0.0866 -127.5 -0.2323 -173.8  -0.1072 0.0 0.0557 0.0  -0.1125 15.7 -0.1752 80.1  -0.1689 -24.1 -0.2253 1.3  78° a based % Diff based % Diff  -0.2794 65.4 0.2206 -230.6  -0. 1998 -163.5 -0.2799 -189.0  -0.2200 0.0 0.0582 0.0  -0.2200 126.2 -0.2067 112.5  -0.2587 16.3 -0.2752 23.7  162  Table A.8 Experiment I Comparison of Pcap’ ba.Sed on a and -  a=0.157  B=1.36  Meniscus Curve Number  Parameter Ccvi  CCV2___CCV3  123.3 -38.5 0.4297  123.5 -34.5 0.3562  CCV4___LEV5  CVX6___CVX7___CVX8  124.3 -15 0.1207  126.5 0 0.0000  127.5 6.5 -0.1294  130.5 21 -0.2309  133.5 24 -0.2535  Experimental  cx(°) (°) “cap’ Model 0=43° a %jjff  0 %Diff 0=53° a O/,4jff  %Diff 0=63° a %jjff  0 %Diff  124.3 -22.5 0.2162  “cap’ 0.1540 64.15 0.4817 -12.11  0.1483 58.36 0.4338 -21.79  0.1394 35.51 0.2722 -25.90  0.1394 -15.52 0.1709 -41.58  0.1141 0.00 0.0000 0.00  0.0995 176.89 -0.0441 65.93  0.0627 127.16 -0.1164 49.57  0.0399 115.74 -0.1259 50.33  0.0424 90.13 0.4917 -14.42  0.0424 88.09 0.4553 -27.83  0.0310 85.66 0.3036 -40.42  0.0310 74.31 0.1973 -63.45  0.0114 0.00 0.0000 0.00  -0.0001 99.95 -0.0561 56.66  -0.0343 85.14 -0.1429 38.12  -0.0546 78.46 -0.1524 39.88  -0.074 134.23 0.3201 -48.05  -0.074 161.30 0.2213 -83.33  -0.06601 -0.06601 118.53 115.36 0.4458 0.4722 -25.16 -9.90  -0.09455 -0.10197 -0.12765 -0.14591 0.00 44.72 42.44 21.20 0 -0.06807 -0.16935 -0.1838 0.00 47.40 26.66 27.49  163  Table A.9 Experiment 3- Comparison of Pcap’ baSed on a and  a=0.155  Photo  Parameter  Experimental a (°) (0)  cap’ Model 0=43° a %Diff %Diff 0=53° a %Diff %Diff 0=63° a 4jff 1 O/  0 %Diff  B=1.273 Number  Ccvi  CCV2  LEV3  CVX4  CVX5  CVX6  128.8 -9.5 0.1548  129.5 -5.8 0.0906  130.3 0 0.0000  131 6.8 -0.0377  130.8 15 -0.1394  130.8 18 -0.2422  “cap 0.1081 30.21 0.1197 22.71  0.1018 -12.28 0.0711 21.53  0.0892 0.00 0.0070 0.00  0.0829 319.81 -0.0510 -35.06  0.0892 164.00 -0.1057 24.15  0.0892 136.83 -0.1146 52.68  -0.0050 103.20 0.1473 4.85  -0.0140 115.46 0.0894 1.39  -0.2029 0.00 0.0000 0.00  -0.0301 20.33 -0.0661 -75.16  -0.0329 76.44 -0.1303 6.56  -0.0329 86.44 -0.1485 38.69  -0.12765 182.45 0.17783 -14.87  -0.13278 246.48 0.10913 -20.39  -0.14306 0.00 0 0.00  -0.14819 -292.75 -0.0813 -115.47  -0.14591 4.67 -0.15739 -12.90  -0.14591 39.76 -0.17522 27.66  164  APPENDIX B SAMPLE CALCULATIONS AND DERIVATIONS -  B. 1 Mathematical Formulation for the Negligible Gravity Case for Cylinders B.2 Jacobian Matrices for the Spheres Model Solution B.3 Capillary Pressure Measurement and Sample Calculation B.4 Digitizing Program i) Determination of the Diameter of a Circle from Three Points ii) Least Squares Minimization of 9 Points on a Circle  165  APPENDIX B.1. Mathematical Formulation For the Negligible Gravity Case For Cylinders z  For parallel, horizontal cylinders, the meniscus between them is described by only one radius of curvature since the other is infinite. Assuming negliglible gravity, there is constant mean curvature:  =  1 R  dx  [B 111  y  where 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]  yr  or  Fsin(4)=  P•a cap 2y  (2 I \a)  (b—r)I  —  [B.1.3]  Upon introduction of the dimensionless variable p  Pa cap  equation [B. 1.3] then becomes  [B.1.4]  166  +sin(çb)=—(b—r). a  —.  [13.1.5]  After rearrangement, the equation for pressure as a function of 4) and horizontal separation distance is:  —  asin()  [13.1.6]  2(b—r) 13  To relate the pressure to the vertical film thickness between two emulsion droplets, we use:  =Ftan().  [13.1.7]  dr  Integration of this equation yields:  b Zb_Zr=IXZ=f  :F(br)dr 1/2  2  [13.1.8]  2 —(b—r)  The last equation expresses the difference between the level at the meniscus centre and the level at the contact 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  ) 0 z  [B. 1.9]  b) Convex  h = a+acos(x)—(z 0 —zr)  [B.1.1OJ  —  —  —  or in dimensionless form:  H=!-i=FFcos(a)±(ZoZc)  [13.1.11]  167  As described in Chapter 3, taking into account hysteresis, the motion of the contact line takes place in two stages. Starting with a level meniscus, the capillary fluid volume is increased. The initial contact angle 3 e is measured at the level meniscus. The immersion angle at the start is just the supplementary angle to the contact angle. This stage is characterised by a fixed contact line on the solid, and the contact angle will increase until the advancing angle 3 a is reached. From this point forward, the contact angle will remain at a and increasing pressure will cause the meniscus to move up the cylinders. This is the second stage. 0 Partl  -constantrcandcz -  contact angle (9 changes from 0 e to  -  contact angle hysteresis is the difference between 9 a e and 0  .Gc P*  P=o  =0 (X  lr—Oe  Capillary pressure increases from 0 to P’  i,  =+a(1—sin(cx))  asin()  2(b—r)  [B.1.12]  [B.1.131  168  a sin() -  [B. 1.14]  2(--a(1+sin(8e)))  In dimensionless form, this final pressure at 3 a is  sin(4)  rd 21 __+(1+SIfl(Oe))] L 2a  [B.1.15]  Part II contact angle constant at 3 a -  -  movement of meniscus, a changes  a sin(4) 2(b—rj  [B.1.16]  d r =—+a(l+sin(Oa—)) c2  sin() +(1+sin(O 1 —2  [2a  [B.1.17]  [B.1.181  _.))]  169  APPENDIX B.2 Jacobian Matrix for the Solution of the Spheres Model  The three variables z, x and  are represented as: 1 y  =  z  [B.2.1] [B .2.21  Y2  3 y  =  x.  [B.2.3J  The derivatives with respect to the arclength s are:  .1;  -=Qsin(4) 4 =  [B.2.4]  f ==Q[Bo.z_”] 2  3 f where  Q is -1  thc =  -  =  [B.2.5]  cos()  [B.2.6J  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 on the derivatives of these functions with respect to the previous three variables:  L=O  -=Q.Bo  i-o 1 dy  L=Qcos(çj)  —  Q•cos() x  —sin()  2  sin()  ‘9:3  2 x  3 dy  170 APPENDIX B.3 Capillary Pressure Measurements from Experiments and Sample Calculations  Manometer Model  Pu C  e  dOevel)  b*  Nomenclature: Pr = Pressure above reservoir (atmospheric) Pu  =  Pressure above test cell meniscus (atmospheric)  Pb  =  Pressure below meniscus interface  = density of manometer oil 0 p Pw density of water c  =  level of oil in reservoir (unknown)  d = position of interface at the level meniscus a, b e  =  =  measured levels of the oillwater interface (b corresponds to the level case)  measured height of the concave or convex meniscus  Pcap  =  capillary pressure = Pu Pb -  -1-  171 Level Case:  g(c-b)-pg(d-b)=O 0 Pu-Pr=p  [B.3.1j  (Pu Pa)+(Pa Pr)=O  [B.3.2]  g(c-a)O 0 -iPcap-pg(e-a)+p  [B.3.3]  Convex Case:  -  -  (-a + b)J 0 p  [B.3.4]  0 -p)g(b-a)-pg(e-d) iPcap(p  [B.3.5]  Pcap  =  g[ Pw(a b -  +  d e) -  +  Concave meniscus:  (Pu Pa’)  +  -  (Pa’ Pc) -  =  0  [B.3.6]  tPcap pg (e’ a’)+ p g (c a’) 0 -  -  -  =  0  [B.3.7]  L\Pcap =g[p(e’ a’ d+ b)+ p 0 (a’ b) -  APcap  =  Pw g (d e’) -  -  -  -  + (0  -  =  p)g (a’ b) -  0  [B.3.8J 0  [B.3.9]  In terms of our measured data read from the micromanometer (whose scale is in the opposite direction), we replace (a  -  b) with (hlev  -  hcp,,) which is negative for convex menisci and positive for  concave menisci. The capillary pressure is comprised of two terms. The first term is the larger one, and the second term due to the hydrostatic height above the level meniscus is smaller.  172  Sample Calculations:  First Term, the measured pressure Pm, Pm=(0.735)*981*(2.54)*(h hiev) -  [dynes/cm2]  [B 3.10]  dimensionless  [B.3. 11]  where h is in inches  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:  [B.3.13]  APcap’=z\Pm’ + APh’  The calculated apparent contact angles for each meniscus were determined using the following equations  and the measured values for  Convex  6  Concave  6= it-a  =  it  -  a+  3 f  [B.3.14j  [B.3.15j  173  APPENDIX B.4 Digitizing Program I) Determination of the Diameter of a Circle from Three Points  1 ,y (x ) 1 (X2  ‘“2  (X3  ,)  The equation for a circle whose center is at (xe, Yc) and has a radius of rc:  2 +(y—yj (x—xj 2  =,2  [B.4. 1]  expanding into the form  2 +y -(x )=ax+by+c 2  [B.4.2J  the equation of the circle becomes:  —(x 2 +y 2 )=(—2.x)x+(—2.yjy+(x  2  2 +y  2  —re)  [B .4.3]  Knowing three points on the circle:  )=R 2 +y 21 1 ax +by 1 +c=—(x  [13.4.4]  2 +c 2 +by ax  [B.4.51  =  2 +y —(x ) 2  =  )=R 2 +y by 3 + ax + 3 2 c=—(x  [B.4.61  174 Using Cramer’s rule for solving three simultaneous equations using the determinant method: the detenninant of the coefficient matrix:  1 x 2 D= x 3 x  Da  where a =  y .1 1 1 1 y 3 y  [B.4.7]  1  1 R  1 y  1  1?2  2 Y  1  J?3  3 y  1  [B.4.81  D D  Similarly for b and C: 1  2 x  R 1 2 R  3 x  J?3  1  1 x  1  D  D 1 x  1 y  1 R  2 X  2 Y  2 R  3 x  3 y  D  [B.4.9]  [13.4.10]  D  solving these determinants:  Da  =RIV2  _RzV’I  +R3[1’i  ) Da=Ri(Yi_Y ) ( 3 ) 2 YiY +R R  B.4.11]  [B.4.12]  175  2 x Db=—Rl  3 x  1 1  x‘  1  3 x  1  x  3 —R  1 ‘  2 x  [B.4.13]  1  — Db=—R(x ) ) ( 3 ) 2 xI—x xl—x —R +R x  ) 1 y 2 =R ) — ) ( 3 2 y —R +R x x —x D1  3 y 1 2 —y y 1 2 —(x x 1 D=x  3 y 2 ) + (x 1 y 3 x  —  —  ) 2 y 3 x  [B.4.14]  [B.4.15]  [B.4.16J  The radius of the circle is r, and the center coordinates are (xe, Yc)  0 a=—2•x  [B.4.17]  b=—2•y  [B.4.18]  ) 2 c=(x+y—r  [B.4.19]  (a2  2 (b  —r2  [B.4.20]  2 +b 4c=a 2 —(2rj 2  [B.4.21]  c=—)  i—)  and the diameter of the circle is twice the radius:  0 2r  = g2  2 —4c +b  [B.4.22J  176  ii) Least-Squares Minimization of Nine Points on a Circle  2 2 +(y—b) (x—a) 2 =r  Ypred  [B.4.23]  =b±gr2 —(x—a) 2  [13.4.24]  where the upper sign is for the upper hemisphere of circle  OBJECTiVE:  Find parameters for the least-squares fit of digitized points to the circle equation. The terms a, b, r are estimated using the three-point method outlined earlier, and used as initial guesses The minimizing function is:  2= s = (yred _j  —  2 —y) a)  [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:  177  s 2 a  s 2 1a  aiab  s 2 a  s 2 a  Idbda  I  t9bdr s 2 a 2 dr  idbi  Ij I asi  s 2 a  db 2 s 2 a  Ia2s  d S 2 1 dadr  drab  [B.4.27]  Ias  1dm  The derivatives of equation [B.4.25] are:  _a)2) _1121 j{2(x ” _ye][.(r2 —(x 2 —(x _a)  =  2[b +(r2  =  2Eb+((r  ’ 1 _(x_a)2)  2  =  2{b +(r2  0  [B.4.28]  Y)j = 0  L  db  —a)]  [B.4.29]  2 _ye][T((r2 _(x_a)2) _(x_a)2)  —1/2’]  )j = 0  [B.4.301  -  F  (x—a)(b + (r2  2 2(x—a)  = [(r2 _(x_a)2)  —  2 (x— a)2)”  Ye)  312 (r2 _(x_a)2)  —  2[b + (r2 —  _(x_a)2)  1-1  1/2  Yej  (r2 _(x_a)2) 2  [B.4.3 1]  d S 2 daab  =  2[(r2  —  (x a) 2 —  F[(2 _(x_a)2)1(x_a)j+2[b+(r t9adr  L I.  2  )1/2  (x a)]  [B.4.32]  —  2\h!’2  -(x-a))  j  _ye][_r(r2  _(x_a)2)  3/2  (x_a)jj  [B.4.33]  178  S 2 9 t  S 2 d  [B.4.34]  =2  2(x—a)  —  [B.4.35]  22 dbda(r2_(x_a)j  S 2 d  2r  [B 4.36]  2\hl’2 db(r2_(x_a))  S 2 d drda  2r[b +(r2  2(x—a)r = (r2 _(x_a)2)  2r2[b+(r2  2 2r = (r2 _(x_a)2)  2 (x _a)2)”  (r2 _(x_a)2) 2 ” 3  2r (r2_(x_a)  2 (xa)2)U  [B.4.38]  2’)hl’2  Yej + 2[b+(r2  2 _(x_a)2Y’  (r2  2 (xa)2)u’  (r2 _(x_a)2) 312  —  _ye](a)  [B.4.37] —  d S 2 drab  S 2 d  —  Ye]  [B.4.39]  Variables used in the program:  D1 = x  —  a  D2=(r2_(x_a)2)=r2D12  D3 =  [B.4.40]  [B.4.41]  [B.4.42]  D4  =  179  2 ’ 3 D2  [13.4.43]  D5=—y+b+D3  [13.4.44]  Augmented matrix (AM) coefficients:  N  r2.D12  AM(11)=[  D2  —  AM(1,2)  N  AIv[(1,3)  =[  =  AM(2,2)  AM(2,3)  =  [13.4.46]  D3  D2  =  [B.4.45]  2•D1  2r•D1  AM(2,1)  1 2D5  D1 • 2 D5 D4  2r•D5•D1 1 D4  j  —  2•D1  [13.4.47]  [B.4.48]  D3  N  [B.4.49] i= I  =  r2r.D1 AM(31)=[ D2  r  N  —  [B.4.50]  2r•D5•D1l D4  j  [B.4.51]  180  N  AM(3,2)  2r  [B.4.521  =  Alvf(3 3) =  12 L  2r D 2 5 2D5 1 +— I D4 D3]  [B.4.53]  181  APPENDIX C COMPUTER PROGRAMS -  CYLINDER MODEL CALP Meniscus Profiles -  CBO Rupture Pressure vs. Bond numbers -  CDI Rupture Pressure vs. Separation Distance -  CHYS Hysteresis Plots -  SPHERES MODEL SPALP Meniscus Profiles -  SBO- Rupture Pressure vs. Bond Number SDI  -  Rupture Pressure vs. Separation Distance  SHYS Hysteresis Plots -  *  ***fl**  C  C  **fl*  IMPLICIT REAL8(A-H,O-Z) PARAMETER (tw’9,N”20,MPM+2) DIMENSION X(M,N),Y(M,N),XC(N),YC(N),Q(N),V(N) DIMENSION PCAP(MP),HTci),FX(2,N),FY(2,N) EXTERNAL FCN LOGICAL 12 CHARACTER4O arlTLl,crrrrL2,XnTL.YrITL COMMON/PARJC,Dl,Pl,PCR,W COMMON/PAR2/PHI,PR,R COMMON/CATIPC,Pl,DP COMMONICORDIXMIN,XMA)cYMIN,YMAX COMMONJPLaT/GTrrLl,GrrrLzxlTrL,YTITL COMMONIINPISPT,SD COMMON/CIRCIXC,YC,FX,FY COMMONIZERO/AC)BCY COMMON/PRESS/PCAP,MX  The shape of the meniscus changes as one causes the the interface to move up along the cylinder. As P increases the capillaxy pressure also changes. The critical point ofrupture is also determined for both convesc and concave mertisci.  Meniscus ProEles between Two Parallel Horizontal Cylinders as a Function of Position CALP’-dimensionless  DP-5.dO P0—I lO.D0 PC-66.DO PI=DACOS(-LD0) PCR,.PC*PI/l80.D0 R.AD=P11180.D0 EPS—1.D-6 IR=l NF=0  G—981.D0 C—(.59600)GISFr 0-13.4600 W’=CRR  SFr—72.75D0  C C Graph headings act C G1’I’I’L1—Meniscsas ProfileS cDTrL2=’For Parallel CylindeisS’ XlIThcJa YTfl’L’zIa’ C C initialize constants and properties C R-0.156D0 C DI.lll6D0 C BD1/(2.D0R) + I.D0 B=l.50300 DI”(B-1.D0)2.D0*R C SFr=45.D0  C C C C C C C C C C C C C  S  *5*  T=tJl C C Determine the dimensionless capillaty pressure for the meniscus C IF (PR .NE. ACR) THEN Z0-DSQRT(4.D0/W(l.DOfr-l.00))  DO30I’I.M lM=I-l 11=1 35 P”PO+DPDBLEQh RAD 5 PR’P C C If position is above the critical value, the meniscus c is convex, ifit is below, it is concave. C IF (PR .GT. ACR) THEN PiII=PR+PCR-PI IDX-l QC-IDO ELSEIF (PR .LT. ACR) THEN PIII’=PI-(PR+PCR) IDX=0 Q0--l.D0 ELSE ZO-0.DO PHI-0.D0 PCAP(IQ—0.D0 MX=M-l IMl C GOTO 40 ENDIF C C SolveferFCNtodetennineTforthespecjfied C contactphiTisgreaterthanobutlessthanl. C Ul-0.0000000lDO 50 U2-0.99999999D0 CALL ZEROI(tJl,U2,FCN,EPS,LZ)  MXM C C Print results in table C WPJTE (6,5) 5 FORMAT (s)c’Caplllasy Pressure Effects as Interface’, + IX,Position Changes) wRITE (6,7) PC,W,B 7 F0RMAT (/s)c’C.A.=’,Fs.l,’ Bo’*’,F5.3,’b/r’,F5.3) WRITE (6,15) 15 FORMAT (J46XPCAF,7c2a,73c’PHp,73cpf,) C C The cntical point at which the level nieniscus occurs C isatACR. C ACRPI-PCR C C Increment PR to follow progress of meniscus around C cylinder C  40  GO’IO 55 ELSE NF-NF+I IR-0 H-O.D0 lff(l)-H ENDIF ENDIF ENDIF PHPIlT/RAD  C C Change in signs, signals Slits nipture points C IF (HO*H UT. 0.00) THEN IR-2 GOTO 40 HNEW-HT(l) 25 HOLD—HT(I.1) P AD D Pl-’PHI.QC R 5 P2—Pm PHI-HNEW(P2.Ply(HOLD-HNEW)+P2 55 PR-Pl-PCR+QC*PHl G0’IO 50 IF (DABS(H) .GT. EPS) THEN 60 IF (IS HNEW .GI’. aDO) THEN 5 P2-PHI HNEW-H ELSE HOLD-H P1—Pill ENDIP  HRDABS(l.D0+QC*DCOS(PR)) H=HR-HC HT(1)=H IF (IR .EQ. 2) GOTO 60 IF (IR EQ. 1 AND. NT .LT. 2) THEN HO-Fff(I-l) IF (I EQ. 1) HO=HT(l)  C C Calculate fIlm thickness C C HI))) SQRT(ZOZGI-2.D0/W 1.D0.COS(P ZC—Q D ( 5 HC=DABS(ZO-ZC)  ENDIF ENDIF  ELSE 11=1 PCAP(ll)—WZO/2.D0  TX”Dif(ZOR) Z001R 5 TY=C IF (IDX EQ. 1) ZO=-ZO IF (lR EQ. 2) THEN llM+NF+l ZOf2.D0 5 PCAP(Il>W  182  ENDTF CONTINUE IF OR ,EQ. 0) ThEN OELH=FYtNF,N)-BCY GOl’O 33 ENDIF DELH=ECY-Y(I,N) PHS=WDELHI2.D0 PTOT=PCAP(I)+PHS WRITE (6,97) DELH, PHS YTOT FORMAT(lX DELH— ‘,E15,3,’ PHS=’,E15.3,’ FTOT=cFlO.5) IF OR £Q. 2) GOTO 25 IF (NF .LE. 2) lIt—I IF (Y(I$) CI’. YMAX) YMAX—Y(I,N) IF (Y(I,N) .LT. YMIN) YMIN-Y(I,N) CONTINUE  30 C C Deterntine the limits ofgeaphing taking into C account that the whole profile and both cylinders C wlllbedrawn. C  26  C 97  33  20  C C Graph the nreniscus profile between the two C cylinders in dimensionless coordinates C CALL MENISC(PHI,ZO,T,W,Q,V,N) IF (I EQ. 1) THEN C1=Q(1)+DSJN(PR) ACX=Ct-DSIN(ACR) ECY=-DCOS(ACR) DA=Pl/DFLOAT(N’l) DOIOK-I,N DFLOAT(K-l) 5 A=DA XCQQ=Cl.DSIN(A) YCQQ=-DCOS(A) CONTINUE 10 YMIN-YC(l) YMAX-’YC(N) ENOIF C C Allothernseniscihavelobesbiftedtobeshown C ontheaaanecylindem C ZC=-DCOS(PR) YD=V(1)-ZC C C Store meaaiaci profile in matrices C DO 20 J—l,N IF (Ill. .EQ. 0) THEN Fx(NF,frQQ) FY(NF,frV(J)-YD ELSE X(IM=QQ) Y(I,J)=V(fl-YD  99  WRITE (6,99) PCAPtH),Z0,PH,PR/RAD FORMAT (lx, 2F10.5,2F10.3) WRiTE (6,) FILM H’, H  IMPLICIT REALS(A-H,O-Z) COMMONIPAPJC,DI,PI,PCR,W COMMONIPA1L2DçA,R COMMONIOUI’IZO FX-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))) +  This function describes the horizontal width of the meniscos and is used toth the specifications of the separation dimension Dl and the contact geometry PHI.  C C Determine meniscras profile by solving equations for C anglesftomPRtoo. C BT-P/DFLOAT(N) PH=P DO 10 I—l,N ANG=<PI-PH)f2.D0 (T ,IND) F—DELIPK T 5 E=DELIPE(PT,IND) F1=DELLIF(l’,ANG,IND) El=DELLIE(T,ANG,IND) f(T (r (2.DO )-l.D0)(F’FI> XD(I)( t 1 5 (2.D0ffl)(E’El)))/DSQRT(B) + IF .Lt 0.D0) THEN  —  •  DISSPLA Graphing Routine  aa**aa*aaaa.aa.aaaa*aaaaaaaa.aa...aa*sa  SUBROUI’INE MYPLOT(XI,X2,M,N) C C This subroutine plots the aneniscas profile between C two parallel horizontal cylinders (dimensionless) C IMPLICIT REAIfl(O-Z) PARAMETER (lM’9,lN—20,MP—at÷2) DIMENSION Xl(MX),21,N),X3(IN),X4(IN),SX(2,IN),SY(2,IN) DIMENSION PCAP(MP) REAL4U(IN),V(IN),YON),Z(IN),U1(IN),Y1(IN),APOM) *4 CASW)flON)flON)YXWN) CHARACI’ER40 GflTLl,GTITL2,2GTfl,,YTITL COMMON/PAR,sqSD,Pl,PCR,W COMMONIPAR2/PIII,PR,R COMMoN/C0RD,xMncxrciAicYMlN,YMAx COMMON/PLOTIGflfll,G1Tlt2,XTfl1,fllTh COMMON/CAT/Pcl,TP COMMON/ZEROI3OcTY COMMONICIRC/X3,X4,SX,SY COMMON/PRESSIPCAP,MX C C Defines the borders, titles, heading, etc. C CALL DSPDEVQPLOT) CALL UNITSQCEN’l CALL NOBRDR CALL PAOE(21.,26.) CALLAREA2D(14.0,17.) CALL HEIGHT(0.30) CALL GRACE(.0) CALL SCMPLX CALL MXIALF(STANDARD’;&) CALL MX2ALF(LICSTIY,’j CALL MIGALF(GREEIC’,’l) CALL MX4ALF(INSTRUCTION’,’@) CALL MXSALFçLfCGR’#’) CALL PMYSOR(3,l.) CALL YAXANG(0) CALL FRAME C C Convect data to single precision values C AX-SNGL(XMIN) RX=SNGL(XMA2Q AY-’SNGL(YMIN) BT-SNGL(YMAX9,IDO)  C C  C  C  ‘  FUNCTION FCN(I)  END C C Meniscas Profile between Two Parallel C Plates or Horizontal Cylinders C C—---————— C C Determines the shape of the meniscus between the C Iwo cylinders C StJBROIJDNE MENISCtfl,,TflD,1D,N) IMPLICIT REALS(A-H,ODIMENSION XD(20),ZD(20) COMMON/PAItJC,D1,PI,PCR,W C  C C C C C C C  C C  ZD(fl—(DSQRTtZZ+2.D0fE°O.D0DCOStPN)D) ELSE ZD(fl=IDSQRT(ZZ+2.DOIB(l.D0’DCOS(PHX)) ENDIF PH=PH-BT 10 CONTINUE 20 RETURN END C  XMAX-Cl IF (YMAX OF. 3O.IAX) XMAXYMAX9S.DWl7.D0 IF (XMAX .01’. YMAX) YMAX-XMAXI7.D0/15.D0 XMIN=’-XMAX CALL MYPLOT(Xj,M,N) STOP END  183  C C Meniscusprofilebyreflection C CALL CURVE(tJI,V,N,0) CALL CURVE(BV,N,0)  91 99 10  FY(J)’-SNGL(SY(lJ)) FXIØ=SNGL(.SXtl,I)) ENDIF IF (I .EQ. I) THEN WRITE (6,98) V(J),Urn,Ut(B),Z(s),Yrn).Yt(B) ELSE WRITE (6,’) WRITE (6,99) V(ThIJ(I),UlQ) ENDIF FORMAT (1X,6F10.3) FORMAT (1X,IFlO3) CONTINUE CALL MARXER(I)  FXØ=SNGL(SX(I,J))  C C Reflect semi-profile around Y-axis to obtain complete C profile C CX=SNGL(XMIN-XMAX) OX=(BX.A5C)’5.D0 DY-(BY-AYflDO C CALL A5cSPLT(XBXl4.0,ORlGSTEF,A) C CALL AXSPLT(AY,BY,t7.,YOR,YST,B) CALL XNAME(XTITL,14) CALL ENAME(fl1TL,17) CALL GRAF(AX,’SCALE,BX,AY,’SCALE’,BY) CALL HEADIN(OTITLI,16,3,2) CALL HEADIN(GTITL2,22,3,2) CALL HEIGHT(OM CALL SCLPIC(O.7) C C CALL XPCKS(2) CALL YTICKS(2) C CALL XGRAXS(olUG,SThP.B)ct4.,XTITL,4,0.,0.) C CALL YGRAXStyOR,YST,BY,t7.,YITIL,l,0O,O.0) CALL HEIGHT(O25) C C Plot interpolated line C DO 20 I—l,MX API3)SNGL(PI) CAP(B-SNGLtPCAP(B) DO1OJ=t,N U(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) THEN CAP(M+I)-SNGLlPCAP(M+Il)  DP=SNGL(TP) Dl=SNGL(SDIR) BO=SNGL(W)  END  C C Film rupture menisci are dotted lines C IF (I LE. 2) THEN CALL DOT CALL CURVE(FXFY,N,0) CALL CURVE(FXI,FY,N,0) CALL RESETCDOT) CALL RLREAL(CAP(IW-l),3,-Q.2,FY(N)+0.03) ENDIF CALL RASFLN(2) C C Draw both cylinders by reflection of curves C and print caplllaay pressure for each curve C CALL CURVEIYtAN,0) CALL CURVE(Y,Z.N,0) IF (CAD)) EQ. 0) GOlD 20 CALL RLREAL(CAP(B,3,-.2,V04)+êO)) Ft=Pl+DP 20 CONTINUE C C Draw straight line apecifjing location of Pcar0 C CALL DASH ACX=SNGL(XX) BCYSNGL(YY) CALL RLVEC(-ACX,BCY,ACX,BCY,0100) CALL HEIGHT(03) C C Point contact angle distance from center of C cylinder to renter of asrnisrus (B. dimensionless), and Bond number C CALL HEIGHT(030) B=t.+Dl/2. CALL RLMESSQBo —‘,4,-B’0.95,0.3) CALL RLREAL(BO,3,-B’0.00,0.3) CALL RLMESS#q .F,4,B*.95,0.) CALL RLREAL(FC,1,-B’0.I5,0.) C CALL MESSAGD/R -‘,5,t.0,7.) C CALL REALNO(B,2,247) CALL RLMESSQbIa ‘,6,-B.95,-3) CALL RLREAL(B,2,.B’.75,-.3) CALL HEIGHT(0.t5) CALL RLMESSQo’,I,-B’.62,0.09) CALL ENDPL(0) CALL DONEFL C C C C C C C  C  C C C C C C C C C C C C C C CBO  Interface  Bond vs Pcap (dimensionless) Two Horizontal Cylinders on an *  ELSEIF (IbID .EQ. 1) THEN CHTFLI-’PL4NSCENDENTAL F1JNUDON OF PHI’ IF (ITXI’ EQ. 0)TIIEN GTITL2-’AT VARIOUS D/R’ ELSE Gl’flt2tAT VARIOUS CONTACT ANGLES’  nm. n  IF (ND .EQ. 0) THEN GTIThI-’ MENISCUS BETWEEN TWO PARALLEL CYLINDERS ON AN INTERFACE t G1TIL2XflTL X/Po  nr-o  ND-I IDX-0  Set titles and labels for graph. Two options: One graphs aneniscus profile (0) whereas the other graphs a flmctionva. one variable (1,2). If ITXI’ is 1 legend is respect to contact angIe otherwise his D/R If lOX -0, then convex menincus desired.  COMMON/PARIC,DI,PI,PCR,W COMMON/CORD/XMIN,XMAX,YMIN,YMAX,SI COMMONIPLOT/GTITLI,GTITL2,X’UETJTITL COMMON/CIRC/XC,YC,PC,IC COMMON/OUr/ZZ,T,PA COMMON)INP/SFI’,SD,R,TIXI’ COMMON/SHAPE/lOX COMMONtLEGIPCL,GP,SP COMMON/PRESS/PCAP COMMON/BDIM/B  184  IMPLICIT AL’fl(A-M,O-Z) PARAMETER (M-5,MD-3%N-20) CIIARACI’ER’40 GTIThI,GDTL2,XT1TL,Tl]TL CHARACTER’15 GP CHARACTER’4 PCL(M) CHARACTER’3 DUM CHARACTER’t Dl CHARACrER’2 D2 CHARACI’ER’4 SPQ,t) DIMENSION X(M,N),YQ,tN),Q(N),V,)OC(2) DIMENSION XC(M,N),YC(MJ,PCAPQ4MD),D(M)CQ4TBO(MD) DATA ltIlOl,RHO2,RNO3/22D0,l.594D0,0.99g2D0/ EXTERNAL FCN LOGICAL 12  To detennine the required capillary pressure to cause the menincus level to reach the height of thr cylinder as a flnactioa’r of Bond mother. Both convex and concave fonts may be obtained. It plots Bond vs. Peap era fimetion vs. any variable.  ‘  C C Store text for plot legends, choice of varying C contact angle or D/R C IF (OTXT EQ. 1) THEN GP—’Contact AnglerS’ ELSE GP Distance B/KS’ ENDIF C C Initialize cylinder radius, separation distance, C contact angle, and physical properties C R0.lD0 SFr—45.lD0 PCI-30.00O1D0 G98l.D0 PI=DACOS(-l.DO) RADPI/l80.D0 SD-R1102-RHO3 C=0.1DO CI=C EPS=l.D-5 YMAX-O.D0 YMIN-0.D0 DI-0.4D0 B’.-DIf(2.D0R) +1.1)0 C C Print headings and other info for printout C WRITE (6,5) 5 FORMAT (1 5)ç’RUPl,JRE PRESSURE VS. BOND NUMBER) WRflE (6,12) 12 FORMAT (30) ‘CYLINDER’,i) WRITE (6,6) B 6 FORMAT (6X,’Separation Distance b/a -‘F5. I,)) WRITE (6,7) 7 FORMAT (6X, ‘Bo’,8)çPHI’,8)ç’A’,12)çPCAr’,8)ç + ‘PCAP[PaJ) 98 FORMAT (3X,’C.A. = ‘,FI 0.3) C C Increment contact angle and apply analysis to obtain C results for each case C rr=l Vs=0.DO  C  -  ENDIF YflTL=”Meniscus Fitting Function’ Xrm,’’Phi Contact meniscus angle’ ELSEIF (IND .EQ. 2) THEN GTITLI=’RUPFURE CAPILLARY PRESSURE VS.’ GTITL2=’BOND NUMBER AND CONTACT ANGLE’ YflTL=’Dimersienless Pressure Pnip’ YmV’PREssURE PCAP (Par IF (tXT .EQ. I) THEN XTITL-’Bond Number Bo’ ELSE XTITLContact Angle (degy ENDIF ENDIF  8  XI’-O.DO  C C Convert contact angles into character shings for C use as labels in legend C NPT’NINT(PC(I)) CALL B’m(Nvr,DUM,3,ND,”) PCL(l>DUMu/’S’ C C Increment Bond numbers W at a constant B C DO 30 J=l,MD C’Cl+DB’DBLE(J-I) WC’K’R THO(JP-W C C C Determine contact PHI byrootfinding. There are C two roots of which one is correct; error criterion C and sign of ZZ should be checked C Ul=0.000 U2PI/2.D0 CALL ZEROI(U1,U2,FCN,EPS,LZ) PHI=Ul IF (IDX .EQ. 0) THEN IF (ZZ .LE. 0.D0) GOTO 8 ELSE U2’-(PI-PCR)0.999D0 IF (ZZ GE. Otto) 00108 ENDIF Ul-I,0003D0’UI Ii U2-P112.DO CALL ZEROI(Ul,U2,FCN,EPS,LZ) IF (NOT. 12) ThEN WRITE (6,’) ‘FALSE’ GOTO 11 ENDIF IF (IDX .EQ. O.D0) THEN U2’(PI.PCR)’O.999D0 WRITE (6,’) ‘#66’ IF (ZZ ar. O.DO) aro 11 ELSE IF (ZZ .LT. Otto) GOTO 11 ENDIF PHI’tJl IF (IDX .EQ. O.DO) THEN IF (ZZ ar. O.D0) GOTO 11 ELSE IF (ZZ .LT. 0.1)0)001011 ENDIF  VO0.DO DP30.D0 DB65O.DO/DBLE(MD-l) DO 40 I-1,M PC(I)=PCI + DPaDFLOAT(Il) PCRPC(I)*P,AD WRITE (6,’)’ WRITE (6,98) PCQ) CALL MISC(PHL,ZOj,W,Q,V,N) C1(l>fDSIN(PR) Sl-V(I)+DCOS(PR) )CD-(CI.Q(1))/20.D0 DA-P1/(DFLOAT(I’t-l)) DO 10 fl—IN A’DA’DFLOAT(ll-I) XQ.lI)-Q(II) Y(3,fl>V(Il) XC(3,1l)-Cl-DSIN(A) YC(3.ll)—Sl-DCOS(A) COWONTJE  C C C  20  IF (I EQ. I AND.! EQ. 1) ThEN YMIN-PCAP(l,I) a7diAX.’PCAP(l,l) ELSE  Determine max, and rain. X and Y for this plot  ENDIF CONTINUE ENDIF IF (IND .EQ. 2) THEN  YMAX—Y(3K) VS-YMAX  10 C C Detcmiine the maximum X and Y values for C adjusting the limits of the plot C IF (1 EQ. 1) THEN XMlN-X(1N) XMAX’.Cl YMIN-YC(J,I) YMAX-YQN) ENDIF IF (Cl ar. XMAX) XMAX—Cl IF (YC(3,N) ar. YMAX) YMAX-YC(3,N) IF (X(J,N) .LT. XMIN)XMIN=XQN) IF (YCQ,l) .LT. YMIN) YMIN=YC(3.l) ENDIF IF (IND .EQ. 1) THEN DO20K-lN DQC)-XI/RAD Y(I,IQ-FCN(XI) )U-X1+0.ISDO IF (YQ,K) .LT. VO) THEN YMIN—Y(3,K) VO-YMIN ENDW IF (Y(I,K) ar. VS) THEN  25  PPACSFr’ZO’RJIO.DO PCAP(I,J)-DABS(W’ZO/(2.D0)) WRITE (6,25) W,PHI/RAD,P,PCAP(IJ),PPA FORMAT (IclElO.3,2FI03,2FI5i) IF (IND EQ. 0) THEN  ZO-ZZ PR’PA P=PR/RAI)  185  ENDIF ENDIF 30 CONTINUE 40 CONTINUE IF (IND EQ. 2) THEN IF (ITXT EQ. 1) THEN XMIN=TBO(I) mom) CALL MYPLOT(rBO,PCAP,M,MD) ELSE nxri XMJNPC(l) XMAX”PC(M) CALL MYPLOT(PC,PCAP,M,M) ENDIF ENDIF STOP END C *.*.*fl. ** C C FUNCTION FCN C C C C The transcendental equation in phi whose root must be C determined C FUNCTION FCN(X) 8(A.H,O-Z) 5 IMPLICIT REAL’ COMMON/PARIC,DI,PI,PCR,W C0MMON/oUrlZoj,A C0MMONIINPISFr.SD,R,rr)cr COMMON/SHAPE/IDX COMMON/BDIM/B ************ C C C If IDX is 0 then the capillasy is convex, otherwise C itis concave C IF (IDX .EQ. 0) THEN A=PI-(PCR-X) ZO-(1.D0.DCOSQQy(W°(l.DG+DCOS(A)))+ + (I.D0+DCOS(A)) 5 0.5D0 ELSE A”PI-(PCR+)Q ZO=(l.D0DCOSQç)ftW*(l.D0DCOS(A))). 05D0(l.D0-DCOS(A)) + ENDIF T”4.D0/(WZOZOl-4D0) FXDELIPK(TT,JND) T,IND) 5 EDELIPE(I (FX5 FCN2.D0T/DSQRTtW)((2Do/(I1>l.D0) + DELLIF(T,(I’I-X)/2.D0,IND))+ (2.D0/)(E.DELLIE(r,(PI.)QfLD0,IND))) + -2.D0(B-DSIN(A)) 10 END  YMINMINlPCAPtl,J),YMtN) TIvtAXMAX4PCAPtU),YMAX)  •  0*5***  DISSPLA Graphing Routine  SUBROOJIINE MYPLLYT(XI,X2,M,N) C C This eabroutine graphs the caplllasypressure C vs.theBondnunrbet C C IMPLICIT REAL8(O-Z) PARAMETER (lM-5,IN-30) DIMENSION Xl(N).X2(l1,N),PC(IM) ppJ,4 U(IN),V(IN),Y(IN),Z(IN),A(IhQ,IPACIC(2000) CHARACIER40 G1TILI,GTITh2,XrITL,Y’TITL CHARACrEPl5 TLEG CJ{ARACTER*4 PCL(I0, CHARACTER4 SP(IM) COMMON/CORD/XMIN,)OciAXYMIN,YMAX,ZO  C C  C  C  C C Detetmine sneniscus profile by solving equations for C angIesontPRto0. C BTP/DFLOAT(N) DO1OI=l,N ANG—(PI.P)/2.D0 FDELIPKCrT,IND) EDELIPE(T*T,IND) Fl=OELLIF(r,ANG,IND) EIOELLIE(r,ANG,tND) ((2.D0f(rT)-1.Do)(F-Fr)5 XD(rtçr + iE-E(I))/DSQRTtB) 5 2.Dof(r1)) IF (lOX EQ. 0) THEN (l.D0-DCOSQ’)))) 5 ZD((I=-(DSQRT(Z*Z+2.DOIB ELSE RT(Z ZD((l=(DSQ Z 5 +2D0fB(1.D0-DCOS(I’)))) ENIJIF P-’P-BT 10 CON1INUE 20 REI1JRN END C  C C C Meniscus Profile between Two Parallel Plates or Horizontal Cylinders C C C C This program determines the graphical profile C pointsofthemeniscusatagivencontactP C andmeniscusdepthL C SC(P,Z,T,B,XD,ZD,N) SUBROUTINE IMPLICIT p,*j*8(A4{,Q) DIMENSION XD(50),ZD(50) COMMON/PARJC,DI,PI,PCR,W COMMON/SHAPEIIDX C COMMONfSHAPE/rnX C Defines the borders, titles, heading, etc. C CALL DSPDEVçPLOT’) CALL UNDSCCENT) CALL NOBRDP. CALL PAGE(21.,26.) CALL ARE.A20(lS.0,l7.) CALL HEIGIflXO.25) CALL GRACE(0.0) CALL COMPLX CALL MXIALFCSTA}IDARJY) CALL hALFcIJCsTD:, CALL MX3ALFGREEK’,’t CALL MX4ALFCSPECIAL’,’@ CALL MX5ALFCLICGR’,’#) CALL PHYSOR(3.,l.) CALL YAXA14G(0) CALL THIFRM(0.025) CALL THKFRM(0.05) CALL FRAME C C Convert data to single precision values C AX”SNGL(XMIN) BX-SNGL(XMAX) AY-SNGL(YMIN) BYSl’0GL(YMA)Q DX-(BX-AX)/3.D0 DY=(BY-AV5.D0 CW-SNGL(W) CB-SNGL(IB) CALL AXSPL1AXBXI50,ORlG,sTEP,A) CALL AXSPLTtAY,BY,l7.,YOR,YST,B) CALL GF(AcDX,BX,AY.DY,By) CALL HEADIN(GflTLI,29,2,2) CALL HEADIN(GTITE2,29,2,2) CALL HEIGHT(0.3) CALL SCLPIC(0.1) C C CALL XIICKSCZ) CALL XGRAxS(ORIGSTEP,B3cI5,xITrL,3o,o.,o.) CALL YGRAXS(YOR,YST,BY,l7.,YflTL,30,0.0,0.0) C C Plot interpolated line C D020 l—l,M DO lOi’l,N IF (I EQ. I)THEN U(J)SNGL(XlQ)) ENDIF V(J)”SNGL(X2(I3))  C  COMMON/PLITr/GTITL1,GITrL2,XTITL,’rrrrL COMMON/LEG/PCL,TLEGSP COMMON/INP/SFr,SD,R,mcr COMMON/PARJrC,TDI,PI,PCR,W COMMON/BDIM/IB COMMON/SIOAPE/IDX  186  CONTINUE CALL MAR.XER(1) CALL RASPLN(3) CALL CURVE(U,V,N,1) 20 CONTINUE CALL HEIGHT(0.25) C C Write properties on graph C CALL MESSAGCb/a -‘,5,I0.l,16.2) CALL REALNO(CB,2,l 1.2,16.2) IF (IDX EQ. 0) THEN CALL MESSAGCCONVEX’,6,6,17.5) ELSE CALL MESSAGCCONCAVE,7,6,17.5) ENDIF IF (lOX EQ. 0) THEN CALL MESSAGCCONVEX’,6,6,17.5) ELSE CALL MESSAGCONCAVE,7,6,l7.5) ENDIF C C Preparelegendblock C MAXLIN-LINEST(lPACK,2000,26) 0030 l-l,M IF (IXT .EQ. 1) THEN CALL LTNES(PCL(I),IPACIçt) ELSE CALL UNES(SP(l),IPACIcI) ENDIF 30 CONTINUE XL=XLEGND(IPACIçM) YL-YLEGND(IPACK,M) IF (flXF .EQ. 1) ThEN CALL MYLEGNCrLEC,l4) ELSE CALL MYLEGN(TLEG,3) ENDIF CALL LEOEND(IPACK,M,lO.5,l3.) CALL BLREC(l0.0,l2.5.XL÷2.5,YL+1.O,0.Ol) CALL ENDPL(0) CALL DONEPL RETURN END C  10  IMPLICIT REAL8(A-H,O-Z) PARAMETER (M-5,N-lO,MP=M+2) DIMENSION Xt),Y(M.lCC,YC(N),Q(ll),V(N) DIMENSION PCAP(M,N),}IT(M),FX(2,N),FY(2,N) DIMENSION PC(lvt),D(l) EXTERNAL FCN LOGICAL Li Cl{ARACTER1$GP CRARACTER4 PCW4) CHARACTER3 DUM CHARACI’ER40 GUrLl,GTITL2,XflTL,YnTh COMMON/PARJC,DIJ’l,PCR,W COMMON/PAR2/PHI,PR,R COMMON/CAT/PC,Pi,DP COMMON/CORD/XMIN,XMAX,YMJN,YMAX CoMMONIPLoTfGTITLl,GTITL2,xrm,TflTh COMMON/lNp/sFr,SD COMMON/LEG/PCL,OP COMMON/CIRC/)CC,YC,FX,FY COMMON/ZERO/ACX,BCY COMMON/PRESSiCAP,MX  The supture capillary pressures are calculated for the parafleihorizontal cylinders case as the separation distance is varied. These variables are then graphe& These curves are determined for a constant Bond number and contact angle. Convex and concave results may be obtained.  Threshold Rupture Capillary Pressures as a Function of Separation Distance (Cylinders) CDF• dimensionless variables  C C Graph headings set C GflTLIRupture Capillary Pressure’ GTrrL2—’vs. Separation Distance’ b/a’ X1TFL’ YflTL-’Dirrrcnsionless Pressure Pnp@,’ GP=’Contact Angles (degy C C Initialize constants and properties C R’.5D0 SFT-72.75D0 c SFT-45.D0 YMIN-0.D0 YMAX-0.D0 0-981.00 C—(l.594D0-0.9982D0)G/SFT C C%99S2D0/SFr WCRR C P0—10.00 PCI-30.D0 DC-30.D0  C C C C C C C C C C C C C C  .  0030 I-1,M IM—l-l It—I P-ACRJRAD+0P0BLE(IM) 35 PR-PRAD C C Ifpositionrs above the critical value, the meniscus C isconvex,ifitisbeIow,jtisconcav C IF (PR LIT. ACR) THEN PHIPR+PCR-PI IDX-1 QC—L00 ELSEIF (PR .LT. ACR THEN PHI-PI-(PR+PCR)  C C The crirical point at which the level mesiscus occurs C is atACR, C ACR—PI-PCR C C Increment PRto follow progress ofmenisctrs around C cylinder C  WRiTE (6,7) PCQQ NPT-NINT(PC(K)) CALL BT04PT,DUM,3,N0,’) PCL(K)-DUM//’S’ 0090 KK-i,N IR=1 NF-0 B=BI+0.200DFLOAT(iCX.1) DI-(B-l.D0)(2.00R) D(KK)B  C C Printresults intable C WRITE (6,5) 5 FORMAT (5X,Capillary Pressure Effects as Distance’, + 1c’Changes [Cylindetj) WRITE (6,8) W 7 FORMAT (/4XC.A. - ‘,F5.l,/) 8 FORMAT(ll,23c’BO - ‘,El0.3) WRITE (6,15) ’A’,1lX,PCAP ,9c ‘,7c’PHr,73ç ’ 15 FORMAT (16X,’b/a 5 + PCAP[PaJ) DO 100 K-l,M PC(K)-PCI+DCDFLOAT(K-I) DP=PC(KYDFLOAT(M) PCR-PC(K)RAD  R.AD”PI/tSO.DO BI=l.0000100 EPS-1.D-S rn-i NF-0 MX=M  PI=DACOS(-1.D0)  187  C C C Change in signs, signals film rupture points C H .LT. 0.1)0) THEN 5 IF (HO IR-2 cEYIO 40 HNEWHT(l) 25 HOLD=HT(J-l) RAD 5 Pl=PHl-QCDP P2—PHI PH-HNEW(P2-Pl)fcHOLD-HNE÷P2 55 PHI 5 PR—PI-PCR+QC  HR-DABS(l.DG4QCDCOS(J’R)) H=HR-l-IC HT(J1=H IF (JR EQ. 2) GOTO 60 IF (JR .EQ. t) THEN HO=lff(J-l) IF (I EQ. 1) HO-HT(J) WRE (6,99) ZO,PHJ/RAD,PRfRAD  C C Calculate filon thickness C ZC—QCDSQRT(ZOZO+2.D0/W(l .D0-COS(J’HQ)) HC=DABStZO-ZC)  ENDIF  ELSE 11=1  IF (DX .EQ. 1) ZO=-ZO IF (1k .EQ. 2) ThEN ll—M+NF+l  T-Ul C the dimensionless capillary pressure for the meroiscus Detemoine C C ZO=DSQRT(4.D0/W(1.D01F-l.D0))  C C SolveforFCNtodetenuineTfocthespecified C con tphiTisgreatcrthanobutlessthanl. C U1=0.0000000IDO 50 U2=0.99999999D0 CALL ZERO1lUlU2,FCN,EPS,LZ)  C  IDX=O QC=-1.D0 ELSE PHI=O.D0 PCAP(fl)—0.D0 MX=M-1 TM-I GOT0 35 ENDIF GOTO 55 ELSE NF—NF+l IR-0 H.D0 HT(J1=H pCAP(IcKK)-DAllS(wzo/2.Do) ppA—CSFrzOR/lo.Do WRiTE (6,99)3, PHIIRAD,PRJRAD,PCAP(K,KK),PPA IF (PCAP(JEK) Or. YMAX) YMAX—PCAP(K,KIC) IF (PCAPQKK) IT. YMIN) YMIN=PCAP(K,KK) GOlD 90 ENDIF ENDIF ENDIF PH-PHI/RAD  GOTO 50 IF (DABSQS) .GT. EPS) THEN IF (HHNEW .GT. 0.1)0) THEN P2=PHI HNEW-H ELSE HOLD-H Pl=PHI ENDIF  99  FORMAT (lx, 3F10.3, 2Fl5. IF(IR EQ. 2) GOTO 25 26 IF (NF I.E. 2) IR—I IF (Y(I,N) or. YMAX) YMAX-Y(I,N) IF (Y(I,N) .LT. YMIN) YMIN-Y(l,N) 30 CONTINUE 90 CONTINUE 106 CONTINUE C C Determine the Emits of graphing taking into C account that the whole profile and both cylinders C willbe drawn. C XMAX—B XMIN-BI CALL MYPLOT(D,PCAP,M,N) STOP END C C aflflfl*****flfl*flflfl**fl***flflflfl.s*flflfl.*flfl FUNCTION FCN(T) can... a. C C This function describes the horizontal width of C C the meniscus and is used to fit the specifications C ofthe separation dimension DI and the contact C geometyPHL C IMPLICIT REALS(A-H,O-Z COMMON/PARJC,DI,I°I,PCR,W COMMON/PAR2/X,A,R C COMMON/OUT/ZO FX-DELIPK(rT,IND)  40  60  nfl*nnflsnSn*nflnflnnnnnnnn  DISSPLA Graphing Routine C • a Ca •.*****.ae....annn,.fl*....*.....*. C C SUBROUTINE MYPLOT(x1,X2,M,N) C C This subroutine graphs either the capillary pressure C orthe function FUN vs. an independent variable C i.e.D/R) C IMPLICIT REAL8(O-Z) PARAMETER (IM-5,114=lO)  C  C C Determine meniocus profle by solving equations for C anglesfromPRtoo. C BT-P/DFLOAT(J1) PH-P TOO l0I=l,N .ANG—(PI-PH)o2.D0 F-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) THEN ZD—lDSQRT(ZZ+2.D0/B(l.D0-DCOS(PH)))) ELSE ZD(JO-(DSQRT(ZZ+2.D0/B(I.D0-DCOS(PIQ))) ENDIF PH=PH-BT 10 CONTINUE 20 RETURN END C  Meniscus Profile between Two Parallel C Plates or Horizontal Cylinders C C C C Determines the shape of the meniaeus between the C two cylinders C SUBROIJI1NE MENISC(P,4T,B,XO,ZD,N) IMPLICIT REAL8(A-H,O-Z) DIMENSION XD(20),ZD(20) COMMON7PARIC,DI,PI,PCR,W C  c  C  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)))  en.  188  C C C  C  DO 20 I’1,M DO l0J=l,N IF (I EQ. 1) ThEN  Plot interpolated line  CALL XTICKS(2) CALL XGRAXS(ORIG,STEP,BX,15.,XTITL,30,0.,0.) CALL YGRAXS(YOR,YST,DY.l7..Yrn’L,30.o.o.o.o)  C C Defines the borders, titles, heading, etc. C CALL DSPDEVCPLOI’) CALL UNrrSCCENT) CALL NOBRDR CALL PAGE(21.,26.) CALL AREA2D(15.0,17.) CALL HEIGHT(0.30) CALL GRACE(0.0) CALL COMPLX CALL MXIALFQSTANDARD,’& CALL M)C2ALFCL/CSTD,e) CALL M)I3ALFCGREEIC,!) CALL MX4ALFCSPEC,’) CALL MXSALFCL/CGR’,’W) CALL PHYSOR(3.,l.) CALL YAXANG(0) CALL THKFRM(0.025) CALL FRAME C C Convert data to single precision values C AX-SNGL(XMIN) BXSNGWcMA)Q AY’SNGL(YMlN) EY-SNGL(YMAX) DX-(BX-AX)/5.DO DY-(BY-AY)15.Dfl CWSNGL(Y CALL AXSPLT(A)cB)cl5,0,ORIG,STEP,A) CALL AXSPLT(AYBY,i7.,YOR,YST,B) CALL opF(A,cD)cBX,AY,DY,B) CALL HEADTN(GTrFLI,35,2,2) CALL HEADIN(GTITL2,29,2,2) CALL HEIGHT(O.3) CALL SCLPlC(0.  DIMENSION Xl(N),X2Qci,N),PC(UQ REAL4 U(IN),VIIN),Y(IN),Z(IN),AII?vO,IPACK(2000) CHARACTER4O GTITLI,GTIU2,3lTfll,YDTL CHARACTERI5 TLEG Cl{ARACTER4 PCL(IM) 4 SP(IN) 5 Cl{ARACTER COMMON/CORD/XMIN,XMAX,YMIN,YMAX COMMON/PLOTIGTrrLI,Gm12,Xrnt,YrIm COMMON/LEGIPCL,ThEG COMMONIINP/SFr,SD COMMON/PAWrC,IDI,PI,PCR,W COMMON/PAR2IPHI,PR,R  END  UQ)SNGLtXl(3t) ENDIF V(J>=SNGLX2(l,i)) 10 CONTINUE CALL MARKER(l) CALL R.ASPLN(3) CALL CURVE(UV,N,l) 20 CONTINUE CALL 1{EIGHT(O.2 C C Write properties on graph C CALL MESSAGCB0nd Number —,l3,l0.,i5.2) CALL REALNO(CW,3,l3.3,l5.2) C IF (IDX EQ. 0) THEN CALL MESSAGCCONVE)C,6,7,172) C C ELSE CALL MESSAGCONCAVE’,7,7,l7.2) C C ENDIF C C Prepare legend block C MAXLIN=LINEST(lPAClç2000,2 DO 30 l—l,M CALL LINES(PCL(l),IPACK,l) 30 CONTINUE XL=XLEGND(IPACK,M) YL-YLEGND(IPACIçM) CALL MYLEGN(TLEO,l4) CALL LEGEND(IPACK,M,l0.S,ll.) CALL BLREC(l0.0,t0.5,XL+2.5,YL+1.0,0.Ol) CALL ENDPL(0) CALL DONEPL ****  R’5.D-l DI”0.6D0 DB=Dl/(iD0R)÷lD0 DB’l.50d0 DI’(DB-l.D0)1D0 R 5 SPT—t5.dO G98l.D0 PI-’DACOS(-l.DO) RAD’PI/l80.D0  IMPLICffREALS(A-H,O-Z) PAAMEtR(M”6N=l0,NPN+l) CHAEACTER4 PCL(M) CHARACTER*3 DUM cHARAcrER1 DD DIMENSION Pl(M,N),P2(M,NP),lll(M,N),H2(M,NP) EXrERNAL FCN LOGICAL Li COMMONILIMIT/XMIN,XMAX,YMIN,YMAX COMMON/PAR/C,PI,PCR,W COMMON/PAR2/PHR,ACR,R,DlDB COMMONIINFO/DP,SFr,P COMMON/LEG/PCL  •S*e**fl*fl.**************Ss****S*.+.,*fl*  Objective: To plot the effect ofhysteresis on the mardmum pressine when one increaser pressurefronrotothatatwbichthefllm thickness reaches zero (convex shapes)  CONTACTANGLE HYSTERESIS  *  189  C C Density of flinds, interthcial tension, profile C lintlts and tolerance limit set C C—(l.594D0-0.9982D0)’GISFr C C-0.596d0GfsFr W-CRR C C Iitltialize contact angle (relative tolrppcr fluid) and C the contact angle hysteresis C PCl0D0RAD PHI—I0.DORAD DP-30.D0 PO-0.D0 C A-0.DO EPS=l.D-5 XMIN-0.D0 XMAX-0.D0 YMIN’O.DO YMAX=0.D0 C C Print headings C wRrrs ( 22 FORMAT (l5)çCAPILLAR.Y PRESSURE VS. FILM ThtCKNESS  C C  C  C  C • C C • C • c  C  WRITE (6,23) FORMAT (3o5c’CYLINDERSyI) WRITE (6,2’O W,DB 24 FORMAT (231, ‘Do “,E9.3,’ b/a ,F10.3 ,i) WRITE (6,26) aJ fIr,63cH’ycpcAp ,lo3c’PcAp[p (o3c’A’,o2cp ’ 26 FORMAT 5 C  C C C  C C C  C C C  C C C C C C  UI-0.000000000lOO U2-o.999999999900 CALL ZEROl(Ul,U2,FCN,EPS,L i-Ut  Solve foc T for the specified case  PHR—AlI  Meniscus slope angle is the same as the CA. hysteresis  IF (3 .EQ. I) THEN Pl(I,3)—PO P-Pl(L3) Dz=0.D0 ELSE MI=DHOFLOAT(3)  -  Starting position Level meniscus  OII=PCRJDFLOATQ4) IF (PCR EQ. 0.00) OOTO 15 0010 3-l,N ACR=PCR  PARTI For different extents of hysteresis, detennine the pressure as a thnction of the film thickness as contact angle changes with ACR position constant  B-0U2.D0+R0.D0-DSINIPCR))  00301=l,M pCR=PCl+DP*RADaDFLOAT(I1) C C Change contact angles (ant lower angles) to labels for use C in legend C IF (PCR .NE. 0.00) THEN CC NFr-NINT((PI-PCR)/RAD) CALL Bm(HPT,ouM,3No;) PCL=DUM/P$’ ELSE CC NPT-NINT(PI-PCR) CC CALL BTD(NPT,00 ,l,ND;0 CC PCLQDD/flHf0’//’S’ CC ENDIF CC C C Meniscus contact point on horizontal scale determined C  C  C Increment starting contact angles  23  Hl(I,3)-l.O0+OCOS(ACR)-OZ H-HItJ,3) IF (H CT. XEtAX) XMAX-H IF (H Li XMIN) XMIN-H CONTINUE  Determine the film thickness  zo=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-ZN ENDIF IF (P .af. YMAX) YM.AX=P IF (P Li YMIN) YMIN-P  Oeteeenine the depth and dimensionless pressure of the meniscus  10 C PAItT2 C C With the contact angle constant at TR, the C matiscus begins to move along the cylinder C HS=0.D0 15 IR—l TR-=PCR NM=N-l DH=(PI-PCR)/OBLEQtM-I) 0020 3-l,NM 33=1 3M-I-I IF (IR .EQ. 0) 33—3+1 PHR-DHDFLOAT(3M) ACR-TR+PHR 25 RC_OI/(2.oo)+Ra(t.DoOSIN(ACR)) IF (3 EQ. 1) THEN P2(I,i)—PO P-P2Q,3) DZ=0.D0 ELSE C C Solve for the dimensionless pressure P2 C Ul-.00000000100 50 U2-0.9999999900 CALL ZEROI(Ul,U2,FCN,EPS,LZJ T’Ul ZO—DSQRT(4.Do/W(t.oorr-l.o0)) P2tJ,33>DABS(Z0aW/(2.D0)) P—p2(I,J3) FC_DABS(zoaCaSFrRylo.oo IF (POT. YMAX) YMAX—P IF (P Li YMIN) YMIN—P ZH—DSQRflOZO+2.D0/W(l.D0-DCOS(PHR))) DZ-ZO-ZN ENDIF H2(I,3fl-I.D0+DCOS(ACR)-OZ H=H2(I,Jl) C  C C C  C C C  HS-H CONTINUE CONTINUE CALL MYPLOT(Hl,PI,H2,P2,M,N) STOP END  H2Ø,33+l)=H2t3,33) P2Q,31+l)=P2(I,33) HNEW-H HOLD-ItS PA-PHR-OH FB=PHR pHR=HNEWa(PBpA)/(HoLoHNEwPB ACR=PCR+PHR WRiTE (6,) ‘flflIIRJRAD,ACRJR.AD,H,ZO OOTO 50 IF(OABS(H) CT. EPS)THEN IF (It HNEW Or. 0.00) THEN 5 PB-PHR HNEW=H ELSE HOLD-H PA=PHR ENDIF OOTO 55 ELSE IR—0 WRITE (6,9 ‘SUCCESS’ ENDIF ENDIF ENDIF  33—N  Store previous meniseiin slot after FRP  IF (IR .EQ. 1) THEN IF (HS H .LT. 0.00) THEN 5 lR-2  Findfllmnaptueepoint  IF (H Of. XMAX) XMAX=H IF (H .Lt XMIN) XMIN=H IF (P .EQ. 0.0) THEN ZO=0 ppA_C*z0*RaSF-rflo.D0 WRITE (6,109) ACRJRAD,PHRJRAD,H,P,PPA FORMAT (lx, 2F10.2,3F15.t IF (IR .EQ. 2)00TH 60  UselinearintcspolationtoobtainvalueatH-0  C C aaaaa***ae*aaae.saa FCN a*eneaaa.eaaes.. aaaa.e C DOUBLE PRECISION FUNCFION FCN(T) C C •aaeeaaa*aaaaamaeeee.fleeesa..eeaee,aa..,,ea.a,a, Inputs: Pbr, Arm radians C C  20 30  60  C  55  C C C C  C C  109  C C  190  C C C C  C C C  C C C  C  C C  •  ***************fl*******  DY=(BY.AY)15D0  DX=f,BX-AX)/5.D0  AX”SNGL(XMIN) BX=SNGL(XMAX) AY=SNGL(YMTN) BY’SNGL(YMA)Q  Convert to single precision values  20  CALL HEIGHT(0.25) CALL ORACE(0.0) CALL COMPLX CALL MX1ALPCSTANDARtY,’&) CALL MX4ALFCL/CGRf,’#) CALL MX5ALFCSPECIAL’,’@) CALL PHYSOR(3.,l.) CALL YAXANG(0) CALL THKFRM(0.02) CALL FRAME  30  C C C  C C C  C  lit  C C C  CALL DSPDEVCPLOT) CALL UNrrSçCEN’r) CALL NOBRDR CALL PAGE(21.,26.) CALL AREA2O(15.0,17.)  Set up dimensions, title, headings, etc.  IMPLICrr REAL*8(OZ) PARAMETER (IM%INl0) DIMENSION XIN),X2(MN),Y1tM,N),Y2(M,N) R5AL4 Ut)N),VQN),Y(IN),ZIIN),IPACK(2000) CHARACTER*40 G1TrL,’T1TL,XT1TL,GTTL2 CHARACTER4 PCL(lM) COMMONJLIMrrJXMIN,XMA)cyMlN.YMAX COMMON/PAR2IPHI,XR,R,PDI,PDB COMM0N/INF0IPDP,SFr,P COMMONIPARJWC,Pl,PC,W COMMONILEGJPCL  SUBROUTINE MYPL0rQcI,Yl,x2,Y2,M,N)  *fl**fl*C******  lMPLlCrr REAL8(A.H,O-Z) PARAMETER (M=6) COMMON/PAPJC,PI,PCR,W COMMON/PAR2/X,A,R,D1,DB FX=DELIPK(r*T,IND) E=DELIPE(PT,IND) FCN2.DODSQRT(C)((2D0/(rI)-l.D0)(FX + -DELUF(T,(Pl-X)12.D0,INO)) -{2.Dw(ro)’(E-DELLlE(r.tPI-XtI2.DO,tND))) + + .(DI+2.DO*Re(l.D0DStN(A))) RETURN END  DO 30 i”IM CALL LINES(PCL(l),IPACJçI) WRiTE (6,) PCLQ) CONTINUE 3-XLEGNP(IPACIçM) yL=YLEGNO(lPAC1c1  Legend for identiEcation of starting contact angle  CALL MESSAGCOond Number ‘.‘,13,l,l6.) CALL MESSAG(Ws “,5,l.,15.2) CALL REALNO(DW,-3,4.6,16.) CALL REALNOtOB,2,2.4,1i2)  Write properties on graph  0020 l-l,M 0010 J”l,N UQ)=SNGLtX1IIM) V(31=SNGL(X2II,J)) Y(1t=SNGL(Yl(I,3)) Z(J)SNGL(Y2(I,J)) CONTINUE CALL MARKER(I-l) CALL SPLINE CALL RASPLN(2) CALL DOT CALL CURVE(U,Y,N,0) CALL RESET COOP) CALL CURVEtV,Z,N,l) CONTINUE CALL DASH CALL RLVEC(0.0,AY,0.O,BY,0100) CALL RLVEC(AX,0.0,BX,0.0,0l00) CALL HEIGHT(03)  Plot the interpolated line  CALL XGRAXS(ORIG,SrEP,B)c15.,Xnm,lg,o.,o.) CALL YGRAXS(yOR,YST,BY,17.,YrITL,39,0.0,0.0) CALL MESSAGC#_ &Constant #ql5,l.,14.3) CALL MESSAGt’#... &Constant #5,17,1,13.5)  36,3,2) 5 CALL HEADIN(G1T11 JflL2-Cylindrrs CALL HEADIN(GTIL2,15,2,2) CALL HEIGHT(0.3) CALL SCLPIC(0.9) CALL XTICKS(2) CALL YTICKS(2) )CflTL’Fihn Thickness Wa’ YTITL*’Dirnensjonless CapillasyPressureprup,  DBSNGL(PDB) DFT=SNGL(SFI’) DW=SNGL DR”SNGL(R) CALL AxSPLT(AXB)cl5..ORIG,STEp,A) CALL AXSPLT(AY,BY,17,YOR,YST,B) CALL GRAr(A,cDcBXAYDY,BI) GrIv’Meniscus Pressure vs. Flint Thickness’ C CALL ENDPL(0) CALL DONEPL RETURN END  CALL MYLEGNC#q &(deg)$’,9) CALL LEGEND()PACIçM,Lo,9.5) CALL BLREC(0.5,9XL+l.2,YL+1.0,0.01)  191  IMPLICIT P.LdS(AH,OZ)  The shape of the meniscus changes as one causes the theinterfacetomoveupalongthesphere.A5P increases the capillary pressure also changes. This program can be used for either convex (IDX=l) or concave (IOX)) menisci. The film rupture points are determined.  -  Meniscus Profiles for a Sphere Surrounded by other spheres Meniscus Profiles SPALP  PARAMETER (MM=6, MY=10, NN=3, MPMY+2, ND’20) DIMENSION Y(HN),YC(HYNiNN),CX(NO),CY(HD),FX(2.ND),FY(2,ND) DIMENSION PllMP),P2(MM,MP),Ftl(HP),H2(MM,MP) DIMENSION HX(lOO),HY(l00),PCAP(MP),QM(2,20),QX(MY,20),QY(MY,20) EXTERNAL F.PCN,FEX,GEX CHARACTER9 DD CHARACFER3 DUM CHARACI’ER4 PCL(MM) INTEGER FLAG LOGICAL LZ COMMON/LI?cr/XMJN,XMA)cYMlN,YMAX COMMON/YLIM/YL COMMON/PROP/C,SFr,IcIDX COMMON/GEO/PHIC,PC,A.XC,B COMMON/CONST/EPS,PI,N,M COMMON/PARA/PCI,W,Q COMMON/OtThXO,ZO COMMON/LEG/PCL COMMON/ODE/H COMMON/CAT/AO,DA COMMON/BAT/IFLAG COMMON/ARRJQM COMMON/PRESS/PCAP,MX COMMON/STOP/DEG,IERR COMMON/CIRC/CX,CY COMMON/ZER.O/AC)cBCY,F)cFY C C C PART 1- Initialize constants and properties C M’MM N’NN PI=DACOS(-LDO) RAD-PI/180.DO SFT=72.7500 SFr45.D0 G98l.OD0 G/SFr 5 C.(l.594D0-0.9982Oo) C.-O.99s2DoG/SFr C R’0.lO0 B1.05Od0 W=CCRCR  C C C C C C. C C C C C C C C C  WRFE (6,11) FORMAT (5)c’PHl’,5X,3cC,7)çzC’,8,c’ZO,7XH, 7X,PCAP,9XA/) +  GOTO 8 ELSE HOLD=H PH1=PHIC  XCOSlN(A) IF (DABS(A-ACR) .LT. EPS) THEN HOLD=H PHI=PHIC 0+Q Cos(A)) HOABs(1.D D 5 HlQ)’H PHJC0.D0 IF (NF .EQ. 1) ThEN  C C Calculate location of level meniscus C ACR=PI-PCI ACXOS1N(ACR) BCY”-DCOS(ACR) C C Loop for varying imenersronal angle A C MXMY 0020 Jl,MY JM=J-l jPJ+1 A’AO+DADBLE(3M) 7  11  C Set initial contact angle PCI and increment DC C PCI=90.O0RAD DA20.DORAO AO=lO.DOCRAD C AAO*ltAD XMD4’O.DO XMAX=0.O0 YMIN=O.O0 YMAX0.OO YL-0.O0 MX”O IDX’0 NF=1 IRcYrI C C Titles for printout C WRiTE (6,3) 3 FORMAT (20X, riENISCUS PROFILES BETWEEN SPHERES) WRITE (6,9)W,PCI/RAD,B 9 FORMAT (5X’Bl2.7’CA7.t,2X,’bfa’,F7.2//)  C  EPS’l.D-5 IERR=0  CON11NUE IF (A OT. ACR) IROT-l IF (HF .EQ. 3) (101020 IF (I .EQ. 1) (101020 C C Find film iupttcu point where H-0 C IF (IROT .EQ. 1) THEN IF (H*HOLD .LT. 0.00) THEN IINEW-Hl(J) 8 PH2—PHIC 10 PHIC-I8NEW(PH2-PHl(HOLO-HNEW)+PH2 AI.PCI+Q*PHIC XC-DSJN(A) CALL HVSP(PHIC,XC.A,ZC,P) HR-DABS(l.D0+QDCOS(A)) HC-DABS(ZD-ZC) H=HR-HC IF (DABS(H) Or. EPS) THEN IF (HI4NEW .GT. 0.00) THEN PH2—PHIC HNEW-H  5  C C Determine imlesowna ZC and ZO front solution of a system C of ODEs C CALL HVSP(PHIC,XC,A,ZC,P) HR_OABS(I.00+Q*DCOS(A)) HC=DABS(ZO-ZC) H’HR-HC WRiTE (6,12) PIIICJRAD,XC,ZC,Z0,H,P,A/RAI) Hl(l)H P1(J)—P PCAPP IF (P .GT. YMAX) YMAX-P IF (P .LT. YMIN) THIN-P IF (H Or. XMAX) XMAX-H IF (H .LT. XMIN) XMIN-H 12 FORMAT (l.3F63,lX,F3.3,2X,F8.3,2X,F7.3, + 2X,F7.3,4X,F7.3) ZT—DCOS(A) YO-QM(z1)-zr D051C-l,NO QX(J,K)-QM(l,K) QY(3,K)-QM(2,K)-YD IF (QY(1,K) .LT. YL) YL-QY(J,K)  GOTO2O ENDIF ELSEIF (A .GT. ACR) THEN PHIC=-(PI-PCI-A) IDX’l Q-1i)0 ELSEIF (A IT. ACR) THEN PHIC=PI-PCI-A IDX”O Q—I.D0 ENOIF IF (A GE. P1) GOTO 30 HOLDH  192  0035 K=1,ND FX(NF,K)=QM(l,K) Y(I4F,K)=QM(2,K)-YD IF (FY(NF,K) .LT. YL) YL=FYtNF,K)  Film rupture nienisci (upper and lower) stored in FX,FY (scaled).  This subroutine solves for the unknowns 20 & ZC for the tneniscua and then calculates the capillary pressure and vertical distance of the rneniscus from the top  ofthe cylinders.  C C  *.*.*,,**,,**.*,,,,,fl.*,****,**fl*******n***,*.*,**  C C C  C  SUBROUTINE HVSP(PHI,XC,A,ZC,PCAP)  *C********a**qsss*****SI***5*****  CALL MNPLOT(Q)çQY,MY,ND)  ISO STOP END C  35  CONTINUE IROT=0 NF—NF+l ENDIF ELSEIF (WHOLD or. 0 DO) THEN PH1-PHIC ENDIF ENDIF 20 CONTINUE C C Profile for spheres in CX,CY C 30 DRPI/DBLE(ND-t) 0015 K=l,ND DP-DRDBLE(K-l) CXQC)=DSIN(DP) CYQQ—DCOS(DP) IS CONTINIJE C C Plot the meniscus profiles as liquid moves up C  C C C C  ELSE HOLD—H PH1—PHIC ENDIF GOTO 10 ELSE H1(MY+NF)=H P1IMY+NF)=P PCAPiMY+NF)-P WRITE (6,12) PHIC/RAO,XC,ZC,ZO,H,P,AJRAD H=0.D0 zr=-DCOS(A) YD=QM(2,1)-ZT 35-1+1  C C C C C C C C  C C  too  70  C C  CS 5 C  #****sfl**s**S***  This fimction describes the determination of the the contact position on the spheres by fitting the differential equations until conditions st both ends where X=Xc and X-B are satisfied. 20 and ZC are unknown and X is the independent vasiabIe  FUNCTION FCNIZ)  RETIJEN END  *****S*****e  IMPLICIT REAL8(A-H,O-Z) INTEGER FLAG PARAMETER (NN-3,LRW-80,L1W-25,NG-2) DIMENSION Y(NN),S(NN),YN(5O),Y1(NN),QM(2,20) DIMENSION ATOL(NN),RWORK(LRW),IWORX(LIW),JROOT(NG) COMMON/GEO/PHIC,PC,A,XC,B COMMON/CONST/EPS,Pl,N,M COMMON/Our/XO,Zo COMMON/LENOIWXOLD,YOLD.S1,X COMMON/ARRIQM  +  XT(l)—0.D0 CALL DRZFUN(FCN,l,100,XT,IND,5.E-7,EPS,I.E40, 1.E-3) IF (IND .EQ. 0) GOl’O 5 WRiTE (6,*) ‘DRZFUN FAILS’ IERk1 0010 100 WRITE (6,) ROOT IS,XT(1) ZC—Xr(1) IERR-l Go’lO 70 ENDIF ZC-U1 ELSE PCAP-0.D0 ZO=0.D0 ZC-0.D0 ENDIF PCAP—DABS(P.1ZO)l(2.D0)  COMMON/STOP/DEG,IERR C C Calculate ZC and ZO using the rootsolverUllC ZEROI C IF (PHI NE. 0.00) ThEN  C  IMPLICIT REAL8(A-H,O. DIMENSION XT(I) LOGICAL Li EXTERNAl. F,FCN,FEX,GEX COMMON/PROP/C,SFT,R,IDX COMMON/CONST/EPS,PI,N,M COMMONIOUr/X0,ZO COMMON/ODE/HOUr COMMON/BAT/1FLAG COMMON/PARA/PCI,W,Q  C C Setloopforllstepsancldetetmineequationroots C DO 40 IOUT-i,i CALL 10 LSODAR(FE3cN,YTour,noL,RToL4ToLjrAStc1STATE, + IOPT,RWORcLRW,WORXV,JAC,JT.GE,cNG,JROol) C C Save the nteniscus shape in an array C C WRITE (6,) Y(2YRAD,Y(3),Y(t) FORMAT (IX,S— ,EI2.4PH- ‘,E12.4,’X- ,Et2.4,Z- ‘,E12.4) 20 IF STpit .EQ. -1) THEN ISTATE-3 ZWORK(6)l200 0010 10 ENDIF IF (ISTATE .LT. 0)001080 IF (ISTATE EQ. 2) GOTO 40 C WRII’E(6,30) 3ROOT(1) ,JROOT(2) dO FORMAT (5X ThE ABOVE LINE ISA ROOT, JROOTh’,lIS) C C If the dent equations root is found (X-B) then stop integration C IF (JROOT(l) .EQ. I) GOlD 50  EXrERNAL FEX,GEX,JAC C C Setargumentstosolvefor300Esandlequation C RAD=PI/180.00 Y(2)=PHIC Y(I)=Z Y(3)=XC C C Save initial values for plotting C YI(l)Z YI(2)PH1C Yl(3)=XC T’O.DO TOUT2.D0B TO1oUr C C Set tolerances, and parameters for LSODAR C IFOL=2 RTOL-i.D4 A’10L(l)-l.D-6 ATOL(2)-lD-6 ATOL(3)=1.D-6 ITASK-1 ISTATE=l lOFT-I DOS 1—5,10 RWORK(l)=O IWOR.K(l)”O 5 CONTINUE 1WORK(6)-i000 IT-i  193  C C C  80 90  100  C  END  wRrrE (6,90) ISTATE FORMAT(fu/5XERROR HALT... ISTATE  WRITE (6,20) T, Yl(2)/RAD,YIt3),YIO) QM(l,IOLTI)-Yl(3) QM(2,IOITI)-YIQ) IF (ISTATE .EQ. -1) THEN !STATE=3 IWORK(6t-’1200 GOTO 75 ENDIF IF (ISTATE .LT. 0) GOTO SO IF(ISTATE.EQ.2)GOTOIOO lSTAlt2 GOTO 75 TOTJP’IODBLE(IOIJJ) ENDIF  ‘,13)  00100 IOUT=2,20 75 CALL LSODAEcN,yI,r,vUr,rrOL.R1oL.ATOL,rrASlctSTAm, IOPT,RWORIcLRW,IWORIcLIW,JAC,JT,GEX,NG,JROOI) + C Savethemeniscusshapeinanarray C C  C  IF (DABS(FCN) .LE. EPS) THEN TOTIl9.D0 TOUT=1O T=0.D0 ISTATE”l QM(l,l)=’Ylc3) QM(2,l)-’YI(l)  C 70 C C Determine meniscus shape for plotting, reset values C  FCN=PH-0,D0 WRITE (6,70) ZC,ZO,FCN FORMAT (2X,FCN’,3E12.4)  zC=z  ISTATE=2 GOTO 10 TOlJP’.TODBLE(IOUl) 40 WRITE (6,60) IWORK(ll),rWORK(l2),r.vORK(l3)JwORK(lo). C50 IWORK(19),RWORK(l5) C + FORMAT(/3X.’NO. SThPS’,l4,5)ç’NO. F-S=çI4,5)çNO, J-S”.’, C60 14,5X,’NO. GS.d.l4J,3,c’MEwOD LAST USED-çl2,5ç C + ‘LAST SWITCH WAS ATT,El2.4) C + C C 50 X0’B PH=Y(2) ZO=Y(l)  IMPLICITREALS(A-HODIMENSION Y(I).GOUr(I4G) COMMON/GEO/PHIC,PC,XC,A,B COMMON/ODE/H  SUBROUTINE GEX(N,T,Y,NGGOUl)  END  S(A-H,O-Z) 5 IMPLICrr REAL DIMENSION Y(I4),DYtN) COMMON/PROP/C,SFT,R,IDX COMMON/PARA/PCI,W,Q COMMONILENGTH/XOLD,YOLD,SO,X DStNtY(I)) 5 DY(I)’Q DY(2)’-QDY(l)Y(3)4-QY(1)W DY(3)DCOS(Y(2))  Function supplied for use with LSODAP.  SUBROIJrINE FEX(N,T,Y,DY)  PD(1,2)’QDCOS(Y(2)) PD(2,2)=-DCOS(Y(2))/Y(3) PD(2,l)’Q W 5 pD(2,3)DSrN(y(2)y(Y(3)y(3)) PD(3,2)=-DSINçf(2)) RWEN END  **  ..*•.fl  DISSPLA Graphing Routine  *****fl.flflflfl*,fls*fl!...  SUBROtJIINE MNPLOT(Xl,X2,M,N)  •  •  .flefleeflfl*fl  C C This subroutine plots the meniscus profiles between C two spheres at different imniersional angles,IM is C M+2forthenapturemenisci. C IMPLICIT REAL8(0-Z) PARAMETER (IM—l2JN’ZO)  C C  C  C  REWEN END  C C C Evaluate the 2 equations where roots .re wanted C GOt.Tr(l)=Y(I)-B  C C  C C  C C C  C C  C  SUBROIJflNE JAC(N,T,Y,ML,MU,PDNRPD) AL (A-H,O-z 5 IMPLICrrRE S DIMENSION YtN),PD(NRPDN) COMMON/PROP/C,SFT,R,IDX COMMON/PARA/PCI,W,Q  C  DY=(BY-AY)f5.D0 DX=DY DD<BY-A1’)14Jl7. BX-I.05+DD/2. AX-I.05-DD/2.  DX=(BX-AXy6DQ  CX”SNGL(X?.flN.XMAX)  C C Reflect semi-profile around Z-ads to obtain complete C profile C  AX—0.O DB—SNGL(rB) BX-2.OSNGL(TB) AY-SNGL(YMIN) BY2.O DPSNGL(TDP) DW’.SNGL(W) DPC-SNGL(PCI)  C C Defines the borders, titles, heading, etc. C CALL DSPDEV(PLO’l CALL UNITSCCENT) CALL NOBRDR CALL PAGE(21.,26.) CALL AREA2D(l4,0,17.) CALL HEIGHT(0.25) CALL GRACE(S) CALL Contplx CALL MXlALF(STA}DARD) CALL MX2ALF(VCSTD’7) CALL MX3ALF(GREEK’,’I) CALL MX4ALFCSPECIAL’,’@) CALL MX5ALF(LICG1t CALL PHYSOR(3.,l.) CALL YAXANG(0) CALL THKFIUd(O.02) CALL FRAME C C Convert data to single precision values C  DIMENSION X1(M,N),X2(M,N),X3(tN),X4(IN),SX(2,IN),SY(2,IN) DIMENSION PCAJ’(Ih REL4U(1N),V(IN),Y(IN),Z(,UI(IN),Yl(IN),AP(IM) REAL*4 CAP(IM),FXØN),FY(IN),FXI(IN),FYI(IN) CHARACTER4O Grfltl,GITILIXflTI YflTL 5 COMMON/YLIMIYMIN YITI1, 5 COMMON/PLOTIGTITL1,GTITLZXflTI COMMON/ZER000çTY,sIçSY COMMON/CIRC/X3,X4 COMMON/PRESS/PCAP,MX COMMONIGEO/PHIC,PC,TA,XC,TB COMMONICONS’DTPS,Pl,NO,MO COMMON/PARA/PCIW,Q COMMON/CAT/P1,TDP  194  xTrrL=”r/a’  Reflected menisci  DO 50 It,MX CAPtQ=SNGLtPCAP(Q) DO 40 J=1,N U(31=SNGLIXI(lM) V(Ji=SNGL(X2(l,i)) Y(J)SNGLtX3(J)) ZQ)’SNGLQC4(Jl)  Plot interpolated line  yrtTh=’zla’ CALL XGRAXS(ORIG,STEP,BX14.,XTlTL,3,0.,0.0) CALL YGRAXS(YOR,YST,BY,17.,YflTL,3,0.0,0.0) CALL HElGHT(0.2  CALL XTICKS(2) CALL YflCKS(2)  DX=(BX-AX)/5.D0 DDBY-A14J17. BX=’DD DD=(BXal7Jl4.) BY=DD/2.)al.2 AY”(-BY)f1.3 CALL AXSPLT(A5çBX,14.0,ORIG,STEP,A) CALL AXSPLT(AY,BY,17.,YOIt,YST,B) CALL GRAF(ADAY,DY,R) GT11t1’Fig. : Meniscus Between Spheres GTITL2’at Various Positions’ CALL HEADIN(GITrLI,32,3,2) CALL HEADIN(GflTL2,25,3,2) CALL HEIGHT(0.3) CALL SCLPIC(0.7)  FORMAT (l)c4Flo.3) CONTINUE CALL MARXER(I)  CALL CURVEtUl,VN,0)  C C Meniscus proifie by reflection C  99 40  Ul(J)SNGL(2.D0TB-XlQM) YlQ)SNGL(2.D0TB-X3(S)) C C Film rupture C IF (I .LE. 2) ThEN FX(S)’SNGL(SX(lM) FY(J)=SNGL(SY(T,J)) FXl(J)=SNGL(2.D0*TBSX(I,i)) ENDIF WRiTE (6,99)U(J),VQ),Y(3),Z(J) C  C C C  C C C  C  C C  CC CC CC CC CC  C  END  50 CONTINUE C C Draw straight line specifying location of Pcap=Q C CALL DASH ACX’SNGLtXX) BCY=SNGL(YY) CALL RLVEC(ACX,BCY,21JB-ACX,BCY,0100) CALL RESETCDASH) C C Drawboth spheres by reflection of curves C CALL CURVtuyI,Z,N,0) CALL CURVE(Y,4N,0) CALL HElGwr(o.25) C C Print contact angle C CALL MESSAOBo ‘,4,t0.0,16.5) CALL REALNO(DW,-3,l l.00,t6. CALL MESSAGC#q=’,3,l0.0,15.8) CAD=SNGL(PCI*lS0.D0/Pl) CALL REALNO(CAD,l,l 1.0,15.8) CALL MESSAG(b/a’’,4,l0.0,l5.0) CALL REALNO(DB,2,tl.0,15.0) CALL HEIOHT(0.l5) CALL MESSAGo’,l,l2.3,l60) CALL ENDPL(0) CALL DONEPL  C C Drew film rupture nieniscus as dotted line C 30 IF (I .LE. 2) THEN CAP(IM-2+l)SNGL(PCAP(IM-2÷I)) CALL DOT CALL CURVE(FXFY,N.0) CALL CURVE(FXI,FY,N,0) CALL RESETCDOT) CALL RLREAL(CAP(IM-2+I),3,DB-0.I,FY(N)4-0.02) ENDIF P1Pl+DP  CALL CURVE(U,V,N,0) CALL POLY5 IF (CAP(I) EQ. 0.0)001030 CALL RLREAL(CAPtl),3,DB-0.1,V+0.02)  C C Conyex (IDX=1) and concave (IDX.’O) C lOX-I IF DX EQ. 1) ThEN Q=l.D0 ELSE Q-’l.DO ENDIF  PI=DACOS(.l.D0) RAD=PI/180D0 EPS’l.D-S  C C • PART 1- Initialize constants and properties C M-MM N.’NN SFr45.D0 G.98l.D0 C_(l.594D00.9982D0)*G/SFr C C.lDO CS-’DSQRT(C) R-0.1DO B=’O.lSDO/R. WI-CRR  PARAMETER (MM”6, MY”lO, NN-3, MP=MY+I) DIMENSION Y4N),YC’JN),YN(NN) DIMENSION PI(MMMP),P2(MM,MP),Hl(MM,MP),H2(MM,MP) DIMENSION HX(l00),HY(l00),PCA,PBQsIM,MY),BO(MY) EXTERNAL F,FCN,F2,F3,PDF,FZERO,FEXGEX CHAiACltRl DD CHARACFER3 DUM CHARACrERa4 PCL(MM) INTEGER FLAG LOGICAL 12 COMMON/LIMT/XMIN,XMAX,YMIN,YMAX COMMON/PROP/C,SFT,R,IDX COMMONIGEOIPHIC,PC,A,XC,B COMMON/CONST/EPS,Pl,N,M COMMON/PARA/PCI,W,Q COMMON/OUT/XO,ZO COMMON/LEG/PCI, COMMON/ODE/H COMMON/CAT/PP COMMON/BAT/IFLAG  C C Adzymntetric Saddle-shaped Meniscus of a Sphere C in a Field of Spheres (Gravitational) Rupture Pressure vs. Bond Number C ‘BO’-DTMENSIONLESS C C C C Plots the relationship between capillasypressure with C Bond rsimber as the fluid type changes. C C Th4PLTCfl’ REAL8(A-H,O-Z)  195  ntheingsforthetableofressilts  PP=PC  -  WRITE (6,34) PC/RAD F0RMAT(/,1X,’CA ‘,FlO.3) IF (A .EQ. P1) ThEN WRITE (6,*) ‘Level meniscus occurs at 180’ GOTO 100 ENDIF XC=DSIN(A)  C C The meniscus moves along the cylinder at a constant  34  C Calculate location oflevel meniscus C A’-PI-PC  C C Prepare labels for legend in graph C tF (PC NE. 01)0) THEN NFf’N1NT(PC/RAD) CALL BTD(HPT,DUM,3,ND, ) PCL(H=DUM/PT ELSE NPTNTNT(PC) CALL BTD(NPT,DD,1,ND,’O) PCL(H..DD/P.’/PO/PS ENDIF C C • PART 3Hysteresis Profile C i) Determine cueves for different extents of hysteresis at a constant contact line, C A-const and XCconst C C  22 23  WRITE (6,22) WRiTE (6.23) FORMAT (I5)çRUpIIJRE PRESSURE VS. BOND NUMBER) FORMAT (30X, SPHERES WRITE (6,53) B 33 FORMAT(f463(F,8XPHr,8)ç’AçI0X,PCAP.9)ç’PCAP[Pay/) WRITE (6.33) 55 FORMAT (1Xb/a “,F7.3) C C Set initial contact angle PCI and increment DC C PCF-30.D0 DC”.(180.D0-PCI)IDBLE(M-I) PO”.O.DO XMIN”O.DO XMAX=0.D0 YMtN=0.D0 YMAX=0.D0 1)090 I=1,M C C PART 2- Determine profiles for several contact angles C PC.PCI+DCeDBLE(I1))*RAD  C C C  C  17  C C  C  18  12  16  C C C  GOlD 17 ELSE H2(I,SP)=H P2(lJlP PCAP(I)-P PB(l,K)=P  XCDSIN(A) CALL HVSP(PHtC,XC,A.ZC,H,P) IF (DABS(H) (Fl’. EPS) THEN IF (H HNEW (Fr. ODO) THEN 5 PH2—PHIC NNEW=H ELSE 000LD”H PH1’PHIC ENDIF  PH1’PHIC-DP P}I2PHIC PHIC..HNEW(I’112-PHI)/(HOLD-HNEW)+PH2 API-PC+QPH1C  HNEW=H2(t,3) HOLD’H2(I,JM)  IF (tROT EQ 1) THEN IF (H .LT. 01)0) ThEN  Find root of H  1)020 J’-I,MY JM=J.1 IP’,J+I PHIC=DPDBLE(3M) A”PI-PC+PHIC IF (IDX .EQ. 0) A=PI-PC-PHIC XC=DSIN(A) CALL HVSP(PHIC,XC,A,ZC,H,P) FORMAT (1X 3F10.3,2F14.7) H2ØJ)”H P2(I,3)=P  DP=CM/D8LE(M) DB3.5D0fDBLE(MY-1) DO 50 K”l.MY IROT1 W=WI+DBDBLE(IC-1)  CMPC  contactangle(PC),PfllandAchange.  *  IMPLICIT REAL8(A-H,O-2) DIMENSION 3CF(1) LOGICALLZ EXTERNAL F,FCN,PEX,GEX COMMONIPROPIC,SFF,P.,IDX COMMON/GEO/PHIC,PC,AT,XC,B COMMON/CONST/EPS,Pl,N,M COMMON/OUT/X0,Z0 COMMON/ODE/Hour COMMON/BAT/IFLAG COMMON/PARAIPCI.W,Q COMMON/CAT/PP  of the cylinders.  This subroutine solves for the unlasowns ZO & ZC for the meniscus and then calculates the capitlasy pressure and vertical distance of the rneniscus from the top  *5*  SUBROUTINE HVSP(I’Hl,XC,A,ZC,H,PCAP)  a*****fl*a*e.a**********a.*******a***s.ea*********...  C C Calculate ZC and ZO using the rootsolveruflC ZEROI C IF (PHI NE. 0.1)0) ThEN XT(l)..0.D0 CALL DRZCN,I,100r,IND,5.E-7EPS,I.E-I0, l.E-3) + IF(IND .EQ. 0) GOTO 5 WRiTE (6.*) VRZFUN FAILS’ 0010100 ZC-’Xr(l) 5 GOTO 70  C  C C  C C C C  C  50 CONTINUE 90 CONTINUE )4IN=WI XMAX=W IF (IPLOT .EQ. 1) THEN CALL BPLOT(BO,PB,M,MY) ENDIF 31 WRITE (6,) 1)4 100 STOP END C  C PPAsthecaplllayprcssureinPascals C PPA-CSFTZOPJ10D0 IF (P or. YMAX) YMAX-P IF (P .LT. YMIN) YMIN=P BOQQ’W WRiTE (6,12) BO(lC),PHICIRAD,A,P,PPA IROT-0 001018 C IF (IPLOT EQ. 1)001050 ENDIF ENDIF ENDIF CONTINUE 20  196  ******  *  *0  IMPLICIT REALO(A-H,O.Z) INTEGER FLAG PARAMEIER (NN’-3,LRW=S0,LIW=25,NG=2) DIMENSION YtNN),S(NN),YN(50),YItNN),Q(NN,20) DIMENSION ATOL(NN),RWORKLRV/),IWORK(LIW),JROOT(NG) COMMON/GEOIPHIC,PC,A,XC,B COMMON/CONSTIEPS,PI,N,M COMMON/ODE/H COMMON/OTJT/XO,ZO COMMON/LENGTHIXOLD,YOLD,SI,X EXrERNAL FEXGE/cJAC  This flmction describes the determination of the the contact position on the spheres by fitting the differential equations until conditions at both ends whereX=XcandX=BaresutisfiedZOandZCara unknown and T the arclength is the independent variable.  •  FUNCTION FCN(Z)  C C 5.1 arguments to solve for 3 ODEs and 1 equation C RAD=PI/180.DO Y(2)=PHIC Y(1)=Z Y(3)=XC T=0.D0 TIYfl=200B 10=TOUT C C Set tolerances, and parameters for LSODAR C rroL=2 RTOLID-4 ATOL(I)-1D-6 ATOL(2)=1.D-6 A’ItiL(3)’=1.D-6 ITASK=i ISTATE-! ioir=i  C C C C C C C  C  C  -  ELSE PCAP=ODO Z0=0.D0 ZC=0.D0 ENDIF HC=DABS(ZO-ZC) 70 IF (IDX EQ. I) THEN HR=DABS(l.D0 + DCOS(A)) ELSE HR=DABS(1.D0 DCOS(A)) ENDIF i-IT=HR-HC H=HT HOIJr=H PCAP=DABS(WZO)f(2.D0) 100 RETURN END C  C  C C C SUBROTJIINE 3ACQ’1,T,Y,ML,MU,PDNRPD) IMPLICiT REAL8(A-H,O-Z) DIMENSION Y(34),PD(34RPD,N) COMMONIPROP/C,SFI’,R,IDX COMMONIPARA/PCI,W,Q  END  C C Set loop foci! steps and determine equation roots C DO 40 lOUT”!,! 10 CALL LSODAR(FE/cN,Yj,TOURDL,ATOLJTASK,1STATE. 1Opr,RWoRcLRW,WoR3cLIwJAC,jT,GE3çNG,JRoO1) + WRiTE (6,20)T, Y(2)IRPD,Y(3),Y(1) C FORMAT (!X,’S— ‘,E12.4,’PH= ‘,E!2,4,’X— ‘,E12.4,’Z— ‘,E12.’O C20 IF (ISTATE EQ. -1) THEN ISTATE=3 IWORE(6t=1200 GOlD 10 ENDIF IF (ISTATE .LT. 0) GOTO 80 IF (ISTATE EQ. 2)001040 WRITE (6,30) JROOT(l), JROOT(2) C FORMAT (D ‘THE ABOVE LINE ISA ROOT, 3ROOT=’,2I5) C30 C C If the first equation’s root ii found (31—B) then stop integration C IF (JROOT(1) EQ. 1) GOTO 50 ISTATE=2 0010 10 DBLE(3Otn) 0 Totrr=T0 40 WRITE (6,60) 1WORX(I 1),IWORK(i2),IWORX(13),IWORK(iO), C50 IWORK(19),RWORK(l5) C + FORMAT(/3X,’NO. STEPS-I4,5X,’NO. F-S-J4,5)çNO. 3-5=’, Coo I4,5XN0. G-5=ç14/,3X,?,4ETHOD LAST USED=’,Izs3c C + ‘LASTSWrrCHWASATT=E12.4) C + C C X0=B 50 PH=Y(2) ZO=Y(I) zC—z FCNPH-O.D0 WRiTE (6,70) ZC,ZO,FCN C 70 FORMAT (2X,’FCN’,3E12.4) RETURN WRITE (6,90) 1STATE 80 FORMAT(//I5çERROR HALT... ISTATE - ‘,13) 90  DO 31=5,10 RWORK=0 IWORK(1)=0 5 CONTINUE IWORK(6t=l000 11=1 END  IMPLICIT REAL8(A-H,ODIMENSION Y(N).GOUrING) COMMONIGEO/PHIC,PC,XC,A,B COMMON/ODE/it  SUBROUTINE GEX(I4j,Y4G,00Ul)  END  C  C C C C C C C  C  fPLICrrREALs(o-z PARAMETER (IM-6,IN—10) DIMENSION X(N),Y(M,N) REAL’4 U(IN),V(IN),Z(IN),IPACIC(2000) CHAP.ACTER4O GTTrL,YTITL,xmL,GTIL2 CHARACTER4 PCL(IM) COMMON/LIMT/XMIN,3IMAX,YMIN,YMAX C0MMON/PROP/QC,SFT,R,IDX COMMON/PARA/PC,W,QS  This subroutine plots meniscus presswe vs. film thickness for several contact angles, and also includes dashed lines representing hysteresis  SUBROUTINE BPLOT(x,Y,M,N)  END  C C Evaluate the 2 equations where roots are wanted C GOUT(1)=Y(3)-B  C C  C C  SUBROUTINE EX(Nj,Y,D’t) C C Function supplied foe us. with LSODAR C IMPLICrrREALS(A-H,O.Z) DIMENSION Y(N),DY(I4) COMMON/PROP/C,SFr,R,IDX CDMMON/PARA/PCI,W,Q COMMON/LENGTH/XOLD,YOLD,SO,X DY(1)=QDSIN(Y(2)) DY(2)’—QDY(1)/Y(3)+QY(l)W DY(3)=DCOS(Y(2))  C C  PD(1,2)=QDC0S(Y(2)) PD(2,2)=-DCOS(Y(2))/Y(3) PD(2,i)=QW Y(3)) 5 PD(2,3DSIN(Yl2))ItY(3) PD(3,2)=-DSINIY(2))  197  C C C  C C C C  Do 103=1,N  DO2OI—l,M  Plot the inteapolated line  CALL HEADIN(GflTL,40,20,2) GTIL2=’Between Spheres CALL HEADIN(GTIL2,37,2.0,2) CALL HEIGHT(0. CALL SCLPIC(0.9) CALL )CDCKS(2) CALL YTICKS(2) X1Tlt’Bond Number Bo yrrrL=’Dimensioniess Capillaty Pressure Pcap@, CALL XGRAXS(ORiG,SCALE’,BX14.,)CIITL,l9,0.,0.) CALL YGRAXS(YO?,’SCALE’,BY.17.,rrrn,39,o.o,0.o)  CALL AXSPLT(A3cB)c14.,ORIG,STEP,A) CALL AXSPLT(AY,BY,17.,YOR,YST,B) CALL GRAF(A)cD)cBx,AY,DY,Bt) GTrrL-CAPILLARY PRESSURE VS BOND NUMBER’  AX”SNGL(XMIN) BX=SNGL(XMAX) AY=SNGLIYMJN) BY=SNGL(YMA)Q DX”tBX-AX)/5.D0 DYBY-A’O/5.D0 Dl=SNGL(TB) DFP-SNGL(SFI) DW’.SNGL(V) DR’-SNGLiR)  Convert to single precision values  CALL UNfl’SCCENT) CALL NOBRDR CALL PAGE(21.,26.) CALL AREA2D(l 40.17.) CALL HEIGHT(0.2) CALL GRACE(00) CALL COMPLX CALL MXIALFCSTANDARD,’&) CALL MX2ALFCL/CGR’,’#) CALL MX4ALFCSPECIAL’,’@) CALL PHYSOR(3.,l.) CALL YAXANG(0) CALL THKFRM(0025) CALL FRAME  CALL DSPDEVCPLOl’)  C C Setup dimensions, title, headings, etc. C  COMMONJGEO)PHIC,PCD,QA,XC,TB COMMON/LEG/PCL  C  C  30  C C C  C C C  20  10  END  DO 30 I=l,M CALL LINEStPCL(1),IPACK,l) WRITE (6,e) PCL(D CONTINUE XL=XLEGND(IPACIçM) YL’YLEGND(IPACK,M) CALL MYLEGN(#q &(degl,9) CALL LEGENPACK,M,1l.5,0. CALL LEGEND(IPACE,M,9i0,12.0) CALL BLREC(0.5,05,XL+l.2,YL+t,O,O.0l) CALL BLREC(9.,l l.5,XL+l.2,YL+l.0,0.0l) CALL ENDPL(0) CALL DONEPL  Legend for identitlcation of starting contact angle  CALL MESSAGCb/a ‘‘,5.9.0,l6.0) IF (IDX EQ. 1) THEN CALL MESSAGCCONVE)C,6,6,175) ELSE CALL MESSAGCC0NCAVE’,76,l7.5) ENDIF CALL REALNO(D1,2,ll.,16.)  Wiite properties on graph  CALL HEIGHT(0.3)  CALL CURVE(U,Z,N.l) CONTINUE  UQ)SNGL(X(l)) Z(1)=SNGLIYM) CONTINUE CALL MARXER CALL RASPLN(2)  34  33  23 24  C WRiTE (6,23) WRITE (6,2 FORMAT(1SX,RUPflJRE PRESSURE VS. DISTANCE7 FORMAT (30X,’SPHERES’,//) WRiTE (6,33)W Bo-’,El0.3,,) FORMAT WRflt ( FORMAT(a’PHr,8’,tEtcPCAcPCAi’[Pa]’) PI’.VACOS(.l.DO) RAD’-PI/lSO.DO  CSDSQRT(f R”.O.IDO BI0.l05D0I0.lD0 W-CR’R  C c aPART 1-Initialize constants and properties C M=MM N’NN SFF—45.Do Gl.DO 998200) ISFT O.(1594D0-0 G 5  PARAMEFER (MM=6, MY’l0, NN’3, MP=MY+l) DIMENSION Y4N),YC(NN),YNalN) DIMENSION PlMM,MP),P2MP),HiMM,MP)H2(MMMP) DIMENSION HX(lOO),HY(100),PCAP(MM),PB(MM,MY),BD(MY) E)CTER)AL FFCN,F2,F3,PDF,FZERO,FE/çGEX Cl{ARACTER1 DO Ct{ARACrER3 DUM CRARACTER4 PCL(M!e INTEGER FLAG LOGICAL 12 COMMON/LlliffI7aelIN,XMAXYMD4,yMAx COMMON/PROP/C,SFr,R,IDX COMMON/GEO/PHIC,PC,A,XC,B COMMON/CONST/EPS,P1,N,M COMMON/PARA/PCI,W,Q COMMONIOUr/xo,zo COMMONILEG/PCL COMMON/ODE/H COMMON/CAT/PP COMMON/BAT/IFLAG  C A.sisymntetric Saddle-shaped Mcniscus of a Sphere C C in a Field of Spheres (Gravitational) C C ‘SDI’-DIMENSIONLESS C C C C C Plots the relationship between capillasy pressure with C separation distance between spheres at a constant Bond C number and contact angle. Both convex anc concave cases C can be obtained. C IMPLICrr REL8(A-H,o-Z)  198  22  WRITE (6,22) PC/RAn FORMAT QC.A. “,IFlO.3) IF (A .EQ. P1) THEN WRITE (6,) ‘Level eneniscus occurs at 180’ 0010100 ENDIF XC=DS1N(A) IF (IPLOT EQ. 1)0010 15  C Calculate location oflevel meniscus C A=Pl-PC  C C  ENDIF  C C Prepare labels for legend in graph C IF (PC NE. 0.00) THEN NPT=NlNT(PC/pjj) CALL B1])(NPT,DU,ND”) PCLDj’ ELSE NFr=NINr(pq CALL BTD(PlJ’F,DDI,NDO)  D090I1,M C C CPALT 2- Detennjze profiles for several contact angles C  EPS1.D-5 C C Cottve (lDXl) and concave (IDX’O) C DX1 IF (IDX EQ. 1) ThEN Q1.D0 ELSE Q’-1.D0 ENDIF C C Set type ofplot squiresL IfIPLOTisO then hysteresis C analysis to be done, if IPL0T is 1 then enessiscus profiles C to be plotted C IPLOT’=j C C Set initial contact angle PCI and increment DC C PCI3o.D0 DC’=(180 DO-PC1)IDDLE(M1) P.D0 XMIN”O DO XMAX=oJ)o 4INflo  IF (PC CIT. P1/21)0) THEN CMPC ELSE CM=PC ENDIF DPCMftJBLE(p, DO .000 LE(My-1) 0050 K=1,My = IR0T BRI+D*DBLEl)  17  -  GOTO 17 ELSE B2(I,JP>=H P2(IJp)P PCAP(1)p PB(I,K>”p  XCDSlN(A) CALL HVSP(PHIC,XC,aç,p) IF (DABS(lt) Or. EPS) THEN IF(HHNEW.Or 0.00) THEN PH2”PJUC HNEW-H ELSE H0LH PHI=PH[C ENDIF  PH1PJdIC.flp Pfl2Pfl1C yGIoLpH PHtCHNEw*(pH2pHl 2 A_PlPC+Q*pHlC  RNEWH2(J,3) H0LH23  0020 J’=l,My JM3-l JP-’J+I PHlC.DpaDoLE(Jp 16 A”PI-PC+pHfC IF (IDX EQ. 0) A=PI-PC-PHIC XCDSIN(A) CALL HVSP(PIncxc,,çpp) C WRITE (6,12) 12 F0RMAT(j)FlO3,2FI 4 H2(IJt’=H P2(I,1)=p 18 Cl8 lF(P Or.YMAJQojjp IF (P .LT YMll YMflip C C IF GILT, C )PT=H C C FmdrootofH IF (IROT EQ. 1) THEN IF (II .LT. 0.00) THEN  13  C0MM0N/CoNs’i-/EpS,p C0MMoN/Jo,ZO C0MMoNfoDo COMONATfp C0MMoN/PAJsvpCJW,Q COMMON/CAT/pp  ,M  PLICITRL*A DIMENSION XT(t) LOGICAL 12 EXTERNAL F,FCN,FEX,GEX C00NRop/C,5 x 0  of the epdem  and vertical ditsoce of the xneniscus flora thelop  This subroutjne solves for the unIusosa ZO &ZC for the  C C Calculate ZC and 21) using the rooloolyor hOC ZEROI C IF (PRI .m. 0.00) ThEN Xr(l)o.D0 CALL ORZFuN(PCN,llooxr,1N0,7EpslEIo L.E-3) + IF (IND .EQ. 0)00105 WRiTE (6,*) ‘DRZFUN FAILS’ GOb 100 5 ZC-XT(i) GOH) 70 ELSE PCAP—Q.00  C  C C  C C C C  ED(IC) PPA*Z0*RaS/ 1000 IF (P Or. YMAX) YMAX=p IF (P .LT. Thll WR (6,12) B,PNIppA 1Ror0 C GOTO 18 IF (IPLOT EQ. 1)001050 ENDIF ENTJIF ENDIF 20 CON’DNUE 25 WRITE (6,*) ‘pp’p 50 C0NTflJtj 90 CONUNIJE XMINEI XMAXB IF (IPLOT EQ. 1) THEN CALL ENDIF 31 WP.rfE (6) ‘FIN’ 100 op END C C R0t HVSP(PHI,2CC,p)  199  *  IMPLICIT REAL8(A-H,0-Z) INTEGER FLAG PARAMETER NN-3,LRW=80,LIW=25,NG-2) DIMENSION YQN),S,YN(5r’IlNN),Q(NN,20) DIMENSION A1DL(NN),RW0RX(LRW),1WORKtLPV),JR00TtNG) COMMONIGEO/PHIC,PC,A,XC,B COMMON/CONSTIEPS,PI,N,M COMMON/ODE/H COMMON/OUT/Xo,Z0 COMMON/LENGTH/XOLD,YOLD,Sl,X EXTERNAL FE)GEX,3AC  This function describes the determinatIon of the the contact position on the spheres by fitting the differential equations until conditions at both ends where X=Xc and X=B are satisfied. 10 and 72 are unlcnown and Xis the independent variable.  en..,  *  C C Setargurnentstosolvefor3ODEsandlequation C RAD=PI/l80.D0 Y(2)=PNIC Y(1)-Z Y(3)=XC T’-O.DO TOIJ1=2.DOB To-Tour C C Set tolerances, and parameters for LSODAR C ITOL—2 RTOL-l.D-4 ATOL(1)’l.D-ti ATOI.(2)1 .0-6 ATOL(3)1.D-6 ITASK=I TSTATE=1 loFT-I 0051—5,10 RWORK(1)=0  C C C C C C C  C  -  10=0.00 ZC=0.D0 ENDIF 70 HC’DABS(Z0-ZC) IF (IDX EQ. I) THEN HR-DABS(1 .00 + DCOS(A)) ELSE HR-DABS(I .00 DCOS(A)) ENDIF lrr=HR-HC I4,’HT HOUr=H PCAP=DABStWZ0)f(2.D0) 100 RETURN END C C FUNCTION FCN(Z)  IWORKtI)=0 CONTINUE IWORK(6)=l000 rrt  C C Setloopforllstrpsanddetennineequationroots C 0040 IOtJr=i,i 10 CALL LSODARFE)cN,Yj,eur,rroL,RToL,A1O1,rrASK,TSTAm, IOFRW0RcLRW,IWORIcLIW,JAC,rr,GE3cNG,JRool) + C WRITE (6,20) T, Y(2)/RAD,Y(3),Y(l) C20 FORMAT (lx,’S= ‘,E12.4,’PH= ‘,E12.4,’X— ,E12.4,’Z— ‘,E12.4) IF (ISTATE EQ. -1) THEN ISTATE=3 IWORK(6)=1200 GOb 10 ENDIF IF (ISTATE iT. 0) GOTO SO IF (ISTATE EQ. 2) GOb 40 WRITE (6,30) .TROOT(l) , JROOT(2) C FORMAT (5X ‘rIlE ABOVE LINE ISA ROOT, 3R0OT-’,2I C30 C C If the first equations root is found (X=B) then stop integration C IF (JROOT(1) EQ. 1) GOTO 50 ISTATE=’Z GOTO 10 1ODBLE(IOU1) 40 WRITE (6,60) IWORK(l 1),IWORX(12),IWOR}C(13),JWORX(10), C50 IWORK(19),RWORX(15) C + FORMAT(0X’NO. ErEPS.-cI4,5x’NO. F-S-I4,5X,34O. 3-S-’, C60 HOD LASTUSED-’,lb,SX, 14,SX,b40. G-S=’,W,3 C + ‘LASTSWITOHWASATT-’,E12.4) C + C C 50 X0=B PH=Y(2) l0—Y(1) 72=1 FCN=PH-0.D0 WRITE (6,70) ZC,ZO,FCN C 70 FORMAT (2X,’FCN’,3E12.4) RETURN 80 WRITE (6,90) ISTATE FORMAT(1115X,’ERROR HALT... ISTATE - ‘.13) 90 RETURN END C C C SUBROUtINE JAC(Nj,Y,ML,MU,PD,NRPD) IMPLICIT REAL’8(A-H,O-l) DIMENSION Y4),PD(NRPD,N) COMMON/PROP/C,SFr,R,IDX COMMON/PARAJPCI,W,Q C PD(1,2)=QDCOS(Y(2)) PD(2,2)=-DCOS4YI2)YY(3)  5  END  IMPLICIT REALS(A-Ii,O-Z) DIMENSION Y(N),GOur(IG) COMMON/GEOIPHIC,PC,XC,A.,B COMMON/ODE/H  SUBROUrINE GEX(N,T,Y,NGGOUI)  END  END  *  *0  DISSPLA Graphing Routase C • C. C C SUBROUDNE MNPLor(XI,,c2,MN) C C This subroutine plots the meniscus profile between C two parallel horizontal cylinders C IMPLICIT REAL’8(O-z PARAMETER (IM-9,IN-20) DIMENSION Xl(M,N),X2(M,N),3C3(IN),X4(IN) DIMENSION PCAP(IM) 4 UV(IN),Y,U1(IN),YI(IN),AP(IM) 0 REAL REAI.4 CAP(JM) 4O GITILl,GTITL2XITI’L,YflTh 0 CRARAcrER COMMON/LIMrOD.flN,XMAX,YMII4,YMAX COMMON/PLOT/GITIL1,GTFIL2,XTFTL,YTITL C GOMMONICAT/PC,Pl,TP COMMON/ZEROIXX,YY  C  C  C C Evaluate the 2 equations where roots are wanted C GOUr(1)-Y(3)-B  C C  C C  SUBROIJDNE FEX(N,T,Y,D) C C Fwrction supplied foe use with LSODAR C IMPLICIT REAL8(A-H0-Z) DIMENSION Y(N),DY(N) COMMON/PROP/C,SFr,R,IDX COMMON/PARA/PCI,W,Q COMMON/LENGflI/XOLD,YOLD,S0,X DY(1)-QDSIN(Y(2)) DY(2)-.QDY(IyY(3)+QY(1)W DY(3)=DCOS(Y(2))  C C  PD(2,1>’*W PD(2,3)=DSINtY(2))/lY(3)*Y(I)) PD(I,2)=-DSINIY(2))  200  C C Defines the borders, titles, heading, etc. C CALL DSPDEVCPLO’P) CALL UN1TSCENT) CALL NOBRDR CALL PAGE(2l.,26.) CALL AREA2D(14.0,17.) CALL HEIGHT(0.23) CALL GRACE(.0) CALL COMPLX CALL MXlALFSTANDARD,&) CALL MX2ALFCLICSflY,’) CALL MX3ALFCGREEK,’r) CALL MX4ALFCINSTRUCTION’,@) CALL MX5ALFCUCGR,#’) CALL PHYSOR(3.,I.) CALL YAXANG(0) CALL T1tKFRM(0.02 CALL FRAME C C Convert data to single precision values C AX=’SNGL(XMIN) BX=’SNOL(XMAX) AY=’SNGL(YMIN) BY=SNGL(YMAX*l.D0) DPSNGL(TP) C C Reflect semi-profile around Y-axis to obtain complete C profile C CXSNGL(XMtN-XMAx) DX’(BX-AX)/5.D0 DY=’(BY.A’i)/5.DO CALL AXSPLT(Ax,BX,14.0,ORIG,STEP,A) CALL AXSPLT(AY,BY,17.,YOR,YsT,B) CALL GRAF(Ax,D)çBXAY,DYBY) CALL HEADIN(GTITL1,16,3,2) CALLHEADIN(GTrrL2,29,3,2) CALL HEIGHT(0.5) CALL SCLPIC(0.7) C C CALL XflCICS(2) CALL YIICKS(2) CALL XGRAXS(ORIG,STEP,Bc14.,xTTrL,4,o,o.) CALL YGRAXS(YOR,YST,BY,l7.,YrITL,l,o.o,oo) CALL HEIGHT(0.3) C C Plot interpolated line C DO 20 l’-l,M AP(l)=SNGL(Pl) CAP(l)=SNGL(PCAP(l)) DO 10 3’l,N U(J)=SNGL(XlO,J)) V(J)’SNGL(X2(l,J))  COMMON/CIRC/X3,X4 COMMONIPRESSJPCAP  C C C C C C C  C *S*S*fl**fl**flØ*flqflfl  lMPLlCrrREALs(o-z PARAMETER (1M%IN’iO) DIMENSION X(N)Y(M,N) U(IN),V(IN),Z(IN),IPACIC(2000) R m11,YITII)clTrL,GTlLz O CHARACTE 4 5 CHARACFER*4 PCL(IM) MA5cyMIN,YM&Jc ftff/XMlN,X COMMON/L COMMON/PROP/QC,SFT,R.,IDX COMMONIPAP,AJPC,W,QS COMMONIGEO/PHIC,PCD,QA,XC,TB  This subroutine plots rneniscus pressure vs fibs thiclatess for several contact angles, and also includes dashed lines representing hysteresis  ‘  SUBROIJflNE BPL0T2cYM,N)  Y(J)=SNGLtX3(Jt) Z<11=SNGLQC4(J)) U1(J)=SNGL(-Xla,J)) YIQ)=SNGL(-X3(fl) C WRITE (6,99) U(J,V(1),YQ),Z(J) FORMAT (lX,4F10.3) 99 10 CONTINUE CALL MARKER(l) C ectin C Mersiscus profile C CALL CIJRVE(tJ1,V,N,0) CALL CURVE(tJ,V,N,0) CALL SPLINE C C Draw both spheres by reflection of curves C CALL CURVE(Yl,Z,N,0) CALL CURVE(Y,Z,N,0) CALL RLREAL(CAP(l),l,..2,V) Pl=Pl+DP 20 CONTINUE C C Draw straight line specifying location of Pcap’0 C CALL DASH ACX=SNGL(XX) BCY=SNGL(YY) CALL RLVEC(-ACXBCY,ACXBCY,0l00) CALL HEIOHT(0.4) C C Print contact angle C CALL MESSAG#q”,3,2.,7.) CALL REALNO(PC,l,2.7,7.) CALL ENDPL(0) CALL DONEPL RETURN END C  C C C  C C C C  C C C  DO 10 i-l,N UQ)-SNGL(X1))) -SNGL(Y(l,J))  DO 20 I-l,M  Plot the interpolated fine  CALL HEADIN(GTITL,40,2.0,2) GTIL2=’SetweenSpheres’ CALL HEADIN(GflL2,37,i0,2) CALL HEIGHT(D.3) CALL SCLPIC(O.9) CALL XTICKS(2) CALL YIICKS(2) XITIL.”Separation Distance b/a’ YTflDbsensioniess Capillary Pressure Pcap@,’ CALL XGRAXS(ORIG,’SCALE,EXl4.,XflTL,25,O,o.) CALL YGRAXS(YOR,’SCALE’,BYj7.,yTrrL,39,o.o,o.o)  CALL AXSPLT(A5cBX,l4.,ORIG,sTEp,A) CALL AXSPLT(AY,BY,17.,YOR,YST,B) CALL GRAF(AxD)cBX,Ay,DY,B (rlTll—’CAPILLARYPRESSUREVSDJSTMCE  AX’SNGL(xM1N) BX=SNOL(XMAX) AY-SNGL(YMIN) BY=SNGL(YMAX) DX-(BX-AX)/5.Do DY-(BY-A’t5DO DI=SNGL(TB) DFr.SNGL(SFl) DW’SNGL(W) DR-SNGLQt)  Convett to single precision values  CALL DSPDEVCPLOT) CALL UNSCCENT) CALL NOBRDR CALL PAGE(21,26.) CALL AREA2D(14.0,17.) CALL HEIGHT(02) CALL GRACE(0.0) CALL COMPLX CALL MXIALPCSTAi’IDAIW,’&) CALL MX2ALFçUCGR#) CALL MX4ALFCSPECIAL,’@) CALL PHYSOR(3.,l.) CALL YAXANG(0) CALL THKFRM(Q.025) CALL FRAME  Set up dmensions, title, headings, etc.  COMMONILEGIPCL  201  C  C  30  C C C  C C C  20  10  DO 30 l-1,M CALL LINES(PCL(I),IPACIcI) WitrrE (6,) PCL(I) CONTINUE XL-XLEGND(IPACK,M) YL-YLEGND(IPACK,M) CALL MYLEGN#q &(desJ’) CALL LEGEND(IPACK,1i.5t. CALL LEGEND(IPACK,M,9.50,l 10) CALL BLREC(0.5,0.5,XL+l.ZYL+l.00.0l) CALL BLREC(9.,l0.5,XL+l.2,YL+l.0,0.0I) CALL ENDPL(0) CALL DONEPL RETURN END  Legend for identification of starting contact angle  CALL MESSAGBo ,5,9.0,l6.0) IF (IPX .EQ. 1) THEN CALL MESSAGCCONVEX6,6,l7.5) ELSE CALL MESSAGCCONCAVE’,7,6j7.5) ENDIF CALL REALNO(DW,3,ll.,l6.)  Write properties on graph  CALL HEIGHT(0.3)  CALL CURVE(U,Z,N,t) CONTINUE  CONTINUE CALL MARICER(I) CALL RASPLN(2)  C  C C C  C C C C C C C C C C  C  C C C C C C  • -  PI=DACOS(-i.D0) RAD..Pt/180.DO EPS=i.D-5  M—MM N=NN SFr—45.D0 0-901.D0 C—(l.594D0-0.9982D0)G/SFr CS=DSQRT(C) P-l.D-4 B=3.OODO W—CR’R WRTE(6,’)’BO—’W IERR=0  PART I Initialize constants and properties  PARAMETER (IdM=6, MY-20, NN=3, MP=MY+l) DIMENSION YcNN),YC(I1N),ThNN) DIMENSION Pl(MM,MP)P2(MM,MP),Hl(MMMP),H2Q.tM,MP) DIMENSION HX(l00),HY(ltO),PCAP(MM) EXTERNAL F,FCN,F2,F3,PDF,FZERO,FEX,GEX CHARACIER1 DD CHARACTER3 DUM CHARACTER4 PCL(MM) INTEGER FLAG LOGICAL 12 COMMON/LIMT/XMIN,XMA)cYMIN,YMAX COMMON/PROP/CSFTR,!DX COMMON/GEO/PHIC,PCA,XC,B COMMON/CONSTIEPSPI,N,M COMMON/PARA/PCI,W,Q COMMON/OIJr/XOZO COMMON/LEG/PCL COMMON/ODE/H COMMON/CAT/PP COMMON/BAT/IFLAG COMMON/STOP/DEG,IERP.  IMPLICrr REALB(A-H,O-Z)  The effect ofhysteresis is determined as position of the interface moves up along the sphere. As position increases the capillary pressure also changes. This program can be used for either convert (IDX=l) or concave (IDX—0) menisci. This shows the hysteresis plot for spheres (dimensionless). For profiles use ‘SPALP.  ‘SHYS’-Hysteresis (dimensionless)  Hysteresis Plots for the Spheres Model (Gravitational effects considered)  WRtIE (6,) PCPORAD IF (A EQ. P1) ThEN WRflE(6,*)LcveImeniscus occem at ISO’ 0010) 100 ELSEIF (A .EQ. 0.00) THEN  C Calculate locationoflevelmeniscus C A-PI-PC  -  C Convex (IDX—1) and concave (IDX—0) C lOX-i IF flX EQ. 1) THEN Q=1.D0 ELSE Q—.l.D0 ENDIF C C Set type ofplot requireri If IPLOT isO then hysteresis C analysis to be done, if IPLOT is 1 then merriscus profiles C to be plotted C IPLOT-1 C C Set initial contact angle PCI and increment DC C PCI—30.l)Q DC-(l80.D0.PC1)IDBLECM.1) DEG-2.DORAD P0-0.00 XMIN=0.D0 XMAX’-O.DO YMIN=0.D0 YMAX=0.DO DO 30 I-l,M C C PART 2- Determine profiles for several contact angles C PC-(PCI+DCDBLEØ.l))RAD PP-PC C C Prepare labels for legend in graph C IF (PC NE. 0.1)0) THEN NFr—NINT(I’C/RAD) CALL BTh(IWr,DUM,3,ND.”) PCL41)=DUM/f$’ ELSE NPT-NINT(PC) CALL B1D(NPT,DD1,ND,0 PCL(1)-DDM.Yi0Y/’ ENDIF C C • PART 3-Hysteresis Profile C i) Determine curves for different extents of hysteresis at a constant contact line, C C A-vonst and XC-const C  202  PHIC=AIt  i)Increment hysteresis forgiven XC and A The meniscus slope angle is eqaivalenl to the hysteresis  12  16  IS  CALL HVSP(PHIC,XC,A,ZC,H,P) IF (IERR. EQ. 1)001025 WRiTE (6,12) PHICIRAD,XC,ZC,Z0,H,P,A/RAD FORMAT (t) F7.2,F7.2,2E10.2,lt= ,F7.2,  A=P1-PC+PHIC IF (lOX EQ. 0) A=PI-PC-PHIC XC=DSIN(A)  DP=PC/DBLE(MX) IROT=1 0020 3=l,MY JM=3-l JP=3+l IF (IROT EQ. 0) THEN 33=3+1 ELSE 33=3 ENDJF PHIC=DP’DBLEQM)  DEG—2.DO9LAD C C C Determine unlinowus ZC and 10 from solution of a system C of ODEs C CALL HVSP(PHIC,XC,A,ZC,H,P) WRITE (6,12) PHICJRAD,XC,ZC,ZO,H,P,A/RAD Hl(1,3)=H Pl(1,3)=P IF (P ar. YMAX) YMAX=P IF (P .LT. YMJN) YMIN=P IF (H Of. YMAX) XMAX=t-t IF (H UT. XMIN) XMIN=H CONTINUE 10 IERR’=O C C ii) The meniscus moves along the cylinder at a constant contact angle (PC), PHI and A change. C c C Ifcontactsngleisabove9o,thenlimitis 180 C  C C C C  GOTO15 ENDIF XC=DSIN(A) CMIN=A H=0.D0 DH=’CMIN/DBLE(MY) 0010 3=I,MP 3M=J-l AJt=OFI1JBLE(3M)  P=’,E12.5,’A=’,F7.2) H2(1,J3)=H P2(1,33)-P IF (P .GT. YMAX) YMAX=P IF (P UT. YMIN) YMIN=P IF (H .GT. XMAIQ XMAX=’H IF (II LT. XMIN) XMIN=H  PHI=PHTC-DP PH2=PHIC PHIC=HNEW(PH2-PHIy(1IOLD-HNEVQ)+PH2 A=PI-PC+QPHIC  C C C  This subroutine solves for the unlcnowns 10k ZC for the nreniscus and then calculates the capillatypressure  XC=DSIN(A) CALL HVSP(PHIC,XC,A,ZC,H,P) IF (DABS(1l) .GT. EP THEN IF(15HNEW.GT. ODO)THEN PH2”PNIC HNEW=H ELSE HOLDH PH1=PHIC ENDIF 001017 ELSE H2(I,J)-H P2(1,3)—P PCAP(I)”P WRITE (6,12) PIIIC/RAD,XC,ZC,ZO,H,P,A/RAD IROT=0 GOTO 18 ENI)IF ENDIF ENDIF 20 CONTINUE WRrrE(6,) 25 30 CONTINUE 31 CALL MYPLOT(Hl,PI,1t2,P2,M,MP) 100STOP END C ** C SUBROiJDNE HVSP(PHI,XC,A,ZC,H,PCAP)  17  C C FindrootofH C IF (IROT .EQ. I) THEN IF (H UT. 0.00) THEN C C Store preslous meniseus in next address C H2(1,JP)=H2(1,3) P2(1,JP)=P2(1,J) HNEW=H2(I,J) HOLDH2(I,3M)  18  +  of the cylinders.  C C  C  C C C C C C C C  c  IMPLICIT REAL*8(A-H,O-Z INTEGER FLAG  This thnction describes the deteimination of the the contact position on the spheres by fitting the differentialequationsuntllcondidons atbothends where X=Xc and X=B are satisfied. 10 and ZC are unknown and Xis the independent variable.  s*******fl**Se*******n***a..*...*......*.........*s.  FUNCTION FCN(  -  C C Calculate ZC and lOusing the rootsolver UBC ZERO1 C IF (PHI .NE. 0.1)0) THEN Xr(l)=o.Do CALL DRZFUN(PCN.1.loo.Xr.TNO,5.E-7,EPS,l.E-lO, I.E-3) + IF (IND EQ. 0) GOTO 5 WRITE (6,) ‘L)R2UN FAILS’ ZC-Ul C GOb 100 ZC-XT(1) 5 001070 ELSE PCAP=0.D0 Z0=0.D0 ZC=0.D0 ENDIF 70 HC=DABS(ZO-ZC) IF(IDX.EQ. 1)THEN HR-DABS(l.D0+DCOS(A)) ELSE HR-DAS(I.D0 DCOS(A)) ENDIF HT-HR-HC H’HT HOUPH PCAP-DABS(WZOY(2.D0) 100 RETURN END  IMPLICIT REAL8(A-H,O.Z) DIMENSION XT(l) LOGICAL 12 EXTERNAL F,FCN,FE)çGEX COMMON/PROPIC,SFr,R,IDX COMMON/CONST/EPSI,N,M COMMONIOIJr/X0,ZO COMMON/ODE/Hour COMMON/BAT/IFLAG COMMON/ST0PIDEG,IERR. COMMON/PARA/PCI,W,Q COMMON/CAT/PP  ndverticsldistanceofthemeniscusftomthetop  C  203  C C Set loop for 11 steps and determine equation roots C DO 40 IOUT’l,l CALL 10 LSODARIFEX,N,Y,T,Tour,ITOL,RTOL,ATOL,rrASK,ISTATE, + lopT,RWORIcLRW,IWORJçLIW.JAC,JT,GE3çNG.SROO’l) WRiTE (6,20)T, Y(2)/R.AD.Y(3),Y(1) C FORMAT (1X,’S” ‘,E12.4,’PH= ‘,E12.4,’X= ‘,El2.4,Z” ‘,E12.4) C20 IF (ISTATE EQ. -1) ThEN ISTATh3 lWORK(wi200 GOTO 10 ENDIF IF (JSTATE .LT. 0) GOTO 80 IF (ISTATE EQ. 2) GOD) 40 WRITE (6,30) .IROO’IXI) , JROOr12) C FORMAT (5) ThE ABOVE LINE IS A ROOT, JROOT’.ç215) C30 C C If the first equation’s root is found (XB) then stop integration C IF (JROOT(l) EQ. 1) GOlD 50 ISTATE”2  C C Setazgtmsentstosolvefor3ODEsandleqoafiosi C RAD=P11180.D0 Y(2)=PHIC Y(l)-Z Y(3)=XC T’=O.DO TOtJr_2.D0*B To=TOTJr C C Set tolerances, and parameters for LSODAR C rrOL=2 RTOL-l.D-4 ATOL(l)”1.D-6 ATOL(2)=l.D-6 ATOL(3)’1D-6 fl’ASK=t ISTATF”l IOPTl DO5I=5,l0 RWORK(l)=0 lWORK(l)0 5 CONTINUE IWORK(l000 rrl  PARAMETER (NN-3,LRW=S0,LIW=25,NG=2) DIMENSION Y(NN),SiNN),YN(50),Y1(NN),Q(NN,20) DIMENSION ATOLN),RWORK(LRW),IWORK(LIW),JROOT(NG) COMMON/GEO/PHIC,PC,A,XC,B COMMON/CONSTIEPS,PI,N,M COMMON/ODE/H COMMON/OUT/XO,ZO COMMON/LENGIWXOLD,YOLD,SI,X EXrERNAL FEX,GEX,3AC  C C  C C  C C C  C  C C C  80 90  IMPLICiT REAL8(A-H.C)-Z) DIMENSION Y(I4),GOUTING)  SUBROtJITNE GEX(N.r,Y,No,00ur)  END  IMPLICIT pu.sT,’LC8(A4{O DIMENSION Y(N),DY(N) COMMON/PROP/C,SFT,R,IDX COMMON/PARA/PCI,W.Q COMMON/LENGTH/XOLD,YOLD,SO,X DY(1)=QDSIH(Y(2)) DY(2).Q*DY(l)/Y(3)÷QsY(l)aW DY(3)”DCOSIY(2))  Function supplied for use with LSODAIt.  SUBROIJflNE FEXCN,T,Y.DY)  END  PD(l,2>=QDCOS(Y(2)) PD(2,2)=’-DCOS(Y(2))/Y(3) PD(2,l)—QW PD(2,3)—DSIN(Y(2))/(Y(3)Y(3)) PD(3,2.DS1N(Y(2))  SUBRO{JflNE JACO4.T,Y,ML,MU.PD,NRPD) IMPLICIT P8(A.Ho) DIMENSION Y(I4),PD(NRPD,N) COMMON/PROP/C,SFT,R,IDX CQMMON/PARA/PCI,W,Q  END  WRITE (6,90) ISTATE FORMAT(//15X,’ERROR HALT... ISTATE - ‘.13)  GOTO 10 TOUTTO*DBLE(IOIJI) 40 WRITE (6,60) JWORK(ll),IWORK(12),IW0RK(13),IWORK(lO), C50 IWORK(19),RWORK(15) C + FORMAT(/3X,340. EPS=’,I4,5XO. F-S-,I4,5X’NO. 1-S-p, C60 14,5X,’NO. G-S’,14/,3X,’MErHOD LAST USED.d,l2,5X C + ‘LAST SWITCH WAS ATT.d,El24) C + C C 50 X0=B PH=Y(2) ZOY(l) ZC=Z FCN”PH-O.DO WRiTE (6,70) ZC,ZO,FCN C 70 FORMAT (2X,PCN’,3E12.4)  C C C C  C C C  C C C C C C C  C C  AX=SNGL(XMIN) BX-SNGL(XMAX) AY-SNGL(YMIN) BY-SNGL(YMA)Q DX-(BX-AXy5.D0 DY-(BY-AYYS.D0  Convect to single precision values  CALL DSPDEV(’PLOT) CALL UNTSCCENT) CALL NOBRDR CALL PAGE(2l.,26.) CALL AREA2D(l5.0,19.) CALL HEIGHT(0.2) CALL GRACE(0.0) CALL COMPLX CALL MXIALFCSTANDARD’,’&) CALL MX4ALF(L!CGH,’#) CALL MX5ALFSPECIAL’,’@ CALL PHYSOR(3.,l.) CALL YAXANG(0) CALL Thl(FRM(0.02) CALL FRAME  Set up dimensions, title, headings, etc.  IMPLICIT REAL8(O-Z) PARAMETER (IM=6JN’-21) DIMENSION XI(M,bX2(N),Yl(M,N),Y2(M,N) PL4U(lN),V(IN),Y(IN),Z(IN),IPAClC(2000) CHAR.ACTER4O G’ITfl,,YITrL,XDTL,GTIL2 CHARACTER4 PCL(]1 COMMON/LIMT/XMIN,XMAJçYMIN,YMAX COMMON/PROP/QC,SFT,R,IDX COMMON/PARAIPC,W,QS COMMON/GEO/PHIC,PCD,QA,XC,PDI COMMON/LEG/PCL  This subroutine plots meniscus pressure vs. film thickness for several contact angles, and also inchides dashed lines representing hysteresis  SUBROUTINE MYPLOT(Xl,Yl,XZY2,M,N)  END  (2 C Evaluate the 2 equations where roots are wanted C GOur(l)-Y(3)-B  COMMON/GEO/PHIC,PC,XC,A,B COMMON/ODE/H  204  C C C  C C C  20  10  C C C  Legend for identification of starting contact angle  CALL MESSAG(Do ‘‘,4,I0.7,18.) CALL MESSAOçb/a ‘,5,I0.7,17.2) IF (X EQ. 1) THEN CALL MESSAGCCONVEX’,6,619.3) ELSE CALL MESSAGCCONCAVE,7,6,19.5) ENDIF CALL REALNO(DW,-3,12.0,t8.) CALL REALNO(DI,2,12.0,17.2)  Write properties on graph  CONflNUE CALL MARKER(I) CALL RASPLN(2) CALL DOT CALL CURVE(U,YN,0) CALL RESET (DOT) CALL CURVE(V,4N,l) CONTINUE CALL DASH CALL RLVEC(0.0,AY,0.0,BY,0 100) CALL RLVEC(A)cO.o,BX,o.o,0100) CALL HEIGHT(0.3)  DO 20 T=I,M DO1OJ-l,N UQ)SNGLQCl(l,J)) V(J>SNGL(X2tI,3t) Y(J>=SNGLtYIII,i)) ZQ’SNGL(Y2(I,Jt)  Plot the interpolated line  CALL HEAIJIN(GTITL,40,2.0,2) G11L2=’Capillasy Pressure vs. Film Thickness’ CALL HEADIN(GTIL2,37,2.0,2) CALL HEIOHT(0.3) CALL SCLPIC(0.9) CALL )CflCKS(2) CALL YflCKS(2) XTIU=’Film Thickness h/a’ YflTL’DimensionIess Capillasy Pressure Pcap@,’ CALL XORAXS(ORIG,STEP.BX,15.,)CrITL,Is,o.,0.) CALL YGRAXS4YOR,YST,BY,19.,YTITL,39, CALL HETGHT(0.3) ,l6.) 7 CALL MESSAG#_ &Constant lq’,l5,t0. CALL MESSAO#... &Constant #a’,17,l0.7,15.2)  DI=SNGLtPDI) DFTSNGL(SFI) DW=SNGL(W) DR=SNGL(R) CALL AXSPLTçk)cB)c15.,ORIG.STEP,A) CALL AXSPLT(AY,BY,19.,YOR,YST,B) CALL GRAF(AX,D)cBX,AY,DY,EY) G11TL=’MENISCUS BE1WEEN SPHERES ON AN INTERFACE C  30  CALL ENDPL(0) CALL DONEPL REI1JRN END  DO 30 I=l,M CALL LINES(PCLCI),IPACJcI) WRITE (6,a) PCLQ) CONTINUE XLXLEGNDOPACK,M) YL=YLEGND()PACIçM) CALL MYLEGN#q &(deg)$’,9) CALL LEGEND(IPACK,M,1.,15.0) CALL LEGENDQPACK,M,L0,1.0) CALL BLREC(.50,14.5,XL+1.2,YL+t.0,0.01)  205  CLS ° *CDIGITIzING FNDORAN PRINT “ FOR NENISCI**************” PRINT * This BASIC program was written for the objective of” “ PRINT digitizing photographic images of nenisci for” PRINT * comparison with theoretical curves. The eenisci points” PRINT “ are in dimensionless form by division with the radius” REM of the cylindrical rods. 80 PRINT C 5*0*6* eoaaa*osssooa*o CCC *0*0*55*0*5*55*560* ** *5*60*505*” * 90 PRIN Corr. JUN. 4/93 too PRINT 110 PRINT “ 120 DIM X(200),Y(200),XM(200),YN(200),XN(200),YN(200) 130 DIM XC(3),Yc(3),CD(4),ANG)2),CA(2),XPC(2),XPM(2),XPN)2) 135 DIM YPC(2),YPN(2),YPN(2),CX)20),CY(20),CAVG(2),COTD)2) 137 DIM B(3,4),o(3),00(3),D(2),A24(3,4) ,NI(3),XT(4,3),YT(4,3) 140 NPICSO 150 MAXIT300 160 ALPN1.7 170 EPSo. 0001 175 P1—3.14159271 180 RD=O 190 OPEN “B:NEN.DIG” FOR OUTPUT AS 11 200 000UB 2000 210 PRINT “NON MANY NENISCUS PROFILES”; 220 INPUT NPICS$ 230 PRINT #1, NPIDS$ 240 PRINT 245 PRINT #1, “ 250 PRINT “SIX—LETTER NAME OF SAMPLE 260 INPUT NAS 265 PRINT #1, NM 270 NPTSO 280 N=0 290 NPICSNFICS+1 300 CEO 310 PRINT “*s****aesssases**es**6655a500**s*000soe****u” 320 IF NPIS > 1 GOTO 330 ELSE 370 735 PRINT “DO YOU NION TO USE TNE PREVIOUS ICALINc FACTOR (V/N)”; 340 INPUT Cs 350 IF CS—V’ OR C$”y” GOTO 350 360 IF C$=”N” OR C$”n° GOTO 370 370 005UB 4000 380 REM *6*6 DETERMINE RADIUS OF CYLINDERS *66*66 395 CLI *66* DETERMINE TNE DIAMETER OF TNE 400 PRINT CYLINDERS *6*0” 410 PRINT ***5 AND TNE LOCATION OF TNE CONTACT POINT *0*0” 420 ND1 425 N3 427 NPN+1 440 ST—2 450 FOR M1 TO ST 485 ODOUR 5000 450 001UB 8000 540 XPC(MfrS)1) 550 YFC(M)=I(2) 560 D(M)=S(3)5SF52 580 PRINT “CYLINDER DIA.=”,D(N) PRINT “DO YOU MISN TO REDO TNE CYLINDER DIAMETER?”; 890 600 INPUT AS 610 IF AS=”Y” DR AS—”y” GOTO 480  00 20 30 40 00 60 70  5  *555*50*0*0*  BASELINE RE—ORIENTATION  XFN(I)(3CpC(I)—XO)eRF/R YFN(I)=(YFC(I) —YO) *IFfR NEXT I FOR I—i TO 2 XFN(I)=XPN(I) *CO$(BXTA) —YFN(I) *IIN(BETA) YFN(I)XFN(I)*EIN(BETA)+YFN(I)*COS(BETA) NEXT I DX2*FI/20 FOR K1 TO 2 FOR J1 TO 20 OX(J)—XPN(K)+IIN(DX6J) CV (J) =YFN (K) +001 (DX*J) PRINT #1, CX(J),cY(J) NEXT S NEXT K CLI REM *50* DIDITIIE CONTACT FDINTS AND INCLUDE IN PROFILE PRINT “DIGITIIE CONTACT POINTS ALTERNATELY 3 TINES * N—i FOR N1 TO 6 001UB 3000 X(N)=X:Y)N)—Y X(N)=X(N)*COS(SETA)_Y)N)SSIN)BETA) 955 957 Y(N) “X)N) 5SIN(BETA) +Y(N) COS (BETA) 960 IF X)N)X)N—1) AND Y)N)—Y)N-1) TNEN NN—1 970 IF (X=11 AND Y=0) TNEN N=N-3 980 NEXT N 990 REEF 1000 X1AVO=(X(t)+X(3)+X(5flf3 1010 X1SD—()X)1)—X1AVO)”2+)X(3)—X1AVG)”2+)X)5)—X1AVG)”2)”.5 1020 X2AVG—(X(2)+X(4)+X(6))/3 1030 X2SD=UX(2)—X2AVD)”2+(X(4)—X2AVGY3+(X(6)—X2AVGy2)”.5 1040 Y1AVO(Y(1)+Y(3)+Y(5))/3 1050 Y100=UY(i)—Y1AVG)”2+(Y(3)—Y1AVGY2+(Y(5)—nAVGy2)”.s 1060 Y2AVG=(Y(2)+Y(4)+Y(Gfl/3 1075 Y210—i.)Y(2)—Y2AVO)”2+(Y(4)—Y2AVG)”2+(Y)6)—Y2AVG)”2)”.S 1072 X)1)X1AVO  877 878 890 900 910 920 930 940 950  876  778 780 790 800 810 871 872 873 874 875  777  776  774 YO—(YFC(i)+YPC(2))/2 778 FOR 1=1 TO 2  770 BETA——ATN((YFC(2)—YFC(i))/(XFC(2)—XPC)1))) 772 XO—(XPC(i)+XPC(2))/2  760 PRINT “CENTRES OF CYLINDERS ARE TNE BASELINE”  PRINT PRINT  CLI  PRINT 11, “CYLINDER#” ,M, “DIANETER=”, D)M) NEXT N PRINT #1,” REM 05*6*05*5* RE—ORIENT BASELINE ****s*00000**s*  715 FOR Nt TO ST  716 717 718 720 730 740 750  *6*6  206  *55*5*6*05*6*  R=(D)1)+D)2))/4 RSTD—( (D(1) /2—R) “2t(D(2) /2—R) “2) “.0 PRINT “AVG. CYLINDER RADIUI—”,R,”ITD DEV—”,RSTD PRINT “DO YOU MOON TO REDO TNE CYLINDER DIAMETERS?”; INPUT A$  700 IF A$—”Y” ON A$”’”y” GOTO 390 710 IF A$—”N” OR A$—”n” GOTO 715  640 645 650 680 690  620 IF A$=”N” DR A$=”n” GDTD 625 625 IF N—i TNEN PRINT “DIGITIIE NEXT CYLINDER” 630. NEXT N  1074 1076 1078 1080 1090 1100 1110 1120 1130 1140 1180 1159 1160 1165 1170 1172 1174 1176 1177 1178 3179 1180 1181 1182 1184 1186 1188 1190 1200 1210 1220 1220 1227 1230 1240 1270 1200 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 1460 1470 1480 1485 14E7 1490 -  X(2)X2AVG Y(i)=Y1AVG Y(2)Y2AVG PRINT “X1AVG”,XiAVO, “ST.DEV.”,XISO PRINT “Y1AVG=”,Y1AVO,”ST.DEV.”,YiSD PRINT “X2AVG” ,X2AVG, “ST.OEV.=”,X250 PRINT “Y2AVO” ,Y2AVG, “ST.DEV.”,YlSD N0 POR 1=1 TO 2 T_(Y(I)YPC)Ifl*2*SP/(O(I)) ALP——ATN(T/SQR)—TT+lfl+1.5707633# ANO(I)ALP*180/PI PRINT “ALPNA”, I, ANG)I) CA(I)=180—ANG)I) NEXT I AG—(ANO)1)tANG)2))/2:CAO—(CA)i)+CA(2))/2 ASTD—UANO)1)—AG)2+(ANG(2)—AO)”2)”.5 CASTDUCA)i)—CAG)2+)CA(2)—CAG)”2).5 PRINT “ALPNA=”,AG, “lTD. OEV.=”,ASTD PRINT #1,” PRINT #1, AG,CAO,CASTO PRINT #1,” PRINT “CONTACT ANOLE (LEVEL NEN.)—”,CAG,”STD.DEV.—”,CAlTO PRINT “DO YOU WISN TO REDO TNE CONTACT POINTS?”; INPUT 0$ IP O$”Y” OR O$—”y” THEN GOTO 900 IP O$”N” OR O$”n” THEN GOTO 1190 REM **** DETERMINE WNETNER CONVEX OR CONCAVE ***** PRINT *** DIGITIIE APEX POIN GOSUE 3000 X(3)—X:Y(3)=Y XM)3)=)X)3)—XO)*SP/R YM(3))Y)3)—YO)*SP/R XN(3)XM)3)*COS)BETA)_YN)3)*SIN)EETA) YN)3)=XN)3)*SIN)BETA)+YN)3)*COS(EETA) PRINT TAE)10) “X(A)=”;XN(3);”Y)A)—”;YN)3) YRAXA8E(YN)3)) PRINT “YNAX—”,YMAX PRINT XA=X(3):YA=Y(3) CLI IF YA > Y)i) THEN IDX=0:RER *** CONVEX NENIECUS *** IP YA < Y)1) TNEN IDX—1:REM *** CONCAVE NENIECUE *** PRINT “IDX”,IOX ****** RER *“““ DIGITIZE DROPLET PROFILE PRINT “***** DIGITIZE DROPLET PROFILE (BETWEEN 30—70 POINTS) SO IT IS” PRINT * ) THE CONTACT POINTS ARE INCLUDED IN TNE PILE NOT NECESSARY TO RE—DIGITIZE THEN)” PRINT “ PRINT * TO END OIOITZZINO PROFILE CHOOSE ANY OF TNE RENU KEYS” PRINT “ PRINT “ NN+1 IF R<2 OOTO 1465 IF R>2 GOTO 1460 GOSUB 3000 X)M)=X:Y)R)—Y IF )X=11 AND Y=0) THEN N—N—1:GOTO 1460 XN)N) — )X (N) —XO) “5F/R YN)N)=)Y(N)—YO)*SF/R XN)N)_XN(N)*Coo)BETA)—YN)N)*SZN)BETA) *  PRINT #1,  IF )X—12 AND Y=0) THEN N—N—i GOSUB 5000 DS(N)D PRINT “OZA—” , D NEXT N DAVG)DS)i)+DS)2)+DS)3))/31 DSTUDS)i)—DAVGV2+)DS(2)—DAVOY2+(DS)3)—DAVG)2).5 PRINT “ DAVG—” ,DAVG, “ST. DEV” , DET PRINT DI. 3155 PRINT BEEP PRINT K44*44*4*******444*****K 4100 4110 4120 4130 4i40 4150 4160 4170 4180 4190 4200 4210 4220  207  (REPEAT 3X TO OBTAIN AVERAGE)”  PRINT Mi ST3 ND—i FOR N—i TO ST 4050 4060 4070 4080 4090  —  END RESt *8*4 ENABLE DIGITIZER “““*4 OPEN “CON2:9600,O,7,i” AS #2 CLS RETURN REM ““ SUBROUTINE GET X AND Y *4*4 A$INPUT$(i2,#2) B$NID$(A$,2,S) C$=MID$(A$,?,S) X—VAL(B$) :Y—VAL(C$) IF (Y’”G) THEN GOTO 3060 PRINT N, “X= “;X, “Y= “;Y PRINT RETURN REM **** SUBROUTINE GET SCALING FACTOR *4*4 CLS ** PRINT “ * PRINT “ CALCULATION OF SCALING FACTOR BY DIGITIIING 3 POINTS” 4040 PRINT “OF THE CALIBRATION SPHERE  1720 2000 2010 2020 2030 3000 3010 3020 3030 3040 3050 3060 3070 3080 4000 4010 4020 4030  1700 CLOSE 2 1710 END  1680 PLAY “CEDC” 1690 CLOSE 1  i670 IF DR$’””N” OR DN$”n” TNEN 1680  1640 PRINT “. DO YOU WISH TO CONTINUE (Y/N)”; 1650 INPUT DN$ 1660 IF DN$’””Y” OR DN$—”y” THEN 250  XN)J),YN)J)  YN)M)XN(N) *SIN)BETA)+Y74(R) *COS)BETA) IF (N < >2) TNEN GOTO 1930 REM IF N—i THEN GOTO 1570 IF X(N)X(N—1) AND Y(R)’Y)R-l) THEN N—N-i:GOTO 1430 IF Y0 THEN GOTO 1570 IF Y > 0 OR Y < 0 GOTO 1430 BEEP NUTS—N—i REM **** WRITE DATA TO FILE **** PRINT #1, SF, IDE, HPTS PRINT #1,” FOR 1=1 TO RUTS  1630 NEXT J 1635 PRINT #1,”  1620  1500 1510 1530 1540 1550 1660 1570 1980 1590 1600 1605 1610  “; SF *******  *5<5*  GET 3 POINT  OR  (1=12 ANO 5=0) TNEN OOTO 5050  ODOUR 3000 6020 6030 IF 1=10 AND 1=0 TNEN CLOSE#1:005UR 7000:END 6040 IF 1=13 AND 1=0 TNEN PRINT “REDO TNIS FOINT”:JJ—1:OOTO 6020 IF 1=12 AND 1=0 TNEN PRINT “REDO PREVIOUS CIRCLE”: OOTO 6010 6050 6055 IF 1=13 AND 1=0 TNEN PRINT “REDO INS CYLINDSR”:OOTO 450 6060 XC(J)=X:YC(J(1 6070 J0+1 6080 IF J<4 TNSN OOTO 6020 6090 IF X=11 AND 1=0 TNSN FRINT “RSDO”:J=,7-1:OOTO 6020 6100 RSSP:RSTURN 7000 RE CALCULATE DIAMETS 7010 FOR 1=1 TO 3 7020 R(I)=—)XC)I)2+YC(I)3) 7030 NEXT I 7040 DS=XC(1)*(YC(2)_YC(3)(=YC(1)*(XC(2)_XC(3fl+XC(2)*YC(3)_YC(2)SXC(2) 7050 IF 05=0 TNSN RSSP,BEEP:PRINT”RSDO TNIS CIRCLS”:RSTURN 7060 DAR(1)”(YC(2)—YC(3fl—0C(1)”(R(2)—R(3))+R(2)*YC(3)=YC)2)*R(3) 7070 DRXC(1)*(R(2)—R(3)(=R(1)*(XC(2)—XC(3))+XC(2)*R(3)_XC(3)*R(2) 7080 DC=XC(1)*(YC(2)*R(3)YC(3)*R(2))YC(1(*(XC(2)*R(3)R(2)*XC(3fl÷R(1(*(XC(2 YC)3)—YC(2) *XC(3( 7090 ACDA/DS 7100 BC=DB/0S 7110 CCDCfDE 7120 DIA=AC”2+RC24CC:IP DIA <=0 TNSN RSSP;BSSP:PRINT”RSDO TNIS CIRCLS”:RSTU 7130 D=SQR(DIA) 7140 PRINT “CYLINDSR#:”;M “DIAMETER :“;D 7150 PRINT 7160 XP=—AC/2:YP=—RC/2 7170 RA=D/2 7190 RETURN 8000 REM *°* LEAST SQUARES FInING OF CYLINDSR PROFILE 8010 REM *** CALCULATION OF AUOMSNTSD MATRIX ““5 8011 FOR J=1 TO 3 8012 XT(4,J)=XC(J( 8013 YT(4,J(=YC(J) 8014 D5(J(=0! 8015 NEXT J 8017 S(1)=XP 8020 S(2)=YP  6010 J1  6000 REM  5080 RETURN  5010 5020 5030 5040 RCLE” 5050 005UR 6000 5060 IF (X11 AND 10) 5070 005UR 7000  5000  PRINT “DO YOU NISN TO REDO SCALING FACTOR POINTS (YIN)”; INPUT 0$ ND1 IF O$=”Y” OR G$=”y” TNEN OOTO 4000 IF O$=”N” OR O$=”n” TNEN OOTO 4310 RETURN REM **** CALCULATE DIAMETER OF CIRCLES **** PRINT PRINT PRINT “OBTAIN DIAMETER BY TOUCNING 3 POINTS ON CIRCLE” PRINT “FO,TO QUIT”:PRINT “F2:TO REDO TNE CIRCLE”:PRINT “F2:REOO PREVIOUS  *  4260 4270 4280 4291 4310 4310  **  PRINT  4250  *  4230 SFDI/DAVG 4240 PRINT “SCALING FACTOR IS:  N is the no.  DIFMAXO! RAS(3) FOR 11 TO 4  AM(2,4)=AM(2,4)=2*D5 AM)3,4)=AM(3,4)=2*D5”RA/D3  (DY > DIFMAX)  TNEM DIFMAX=DY  8435 8437 PRINT “iter=”,KK,S(l),S)2),S(3) 8440 REM 5* Verisnoe of fit *5 8480 VAR=0  IF )DIFMAX < SF5) 0010 8435 NEXT KR PRINT “MARNING—CDNVSRGSNCS FAILURE” PRINT flu, “ITSR=”,KK,S(l),S)2),S(3)  IF 8420 8430 8432  NEXT I  8400  DY=ABS)ALPN”DS(I)/S(I))  NEXT 2 REXT I CDSUR 9000 IF (IERROR = 1) 0010 8370 PRINT “ISRROR” , IERJ8OR FOR 1=1 TO N S)I)S)I) + ALPN*DS)I)  8410  8380 8290  8340 8350 8360 8265 8367 8370  8320 8330  8270 REM AM(3 , l)=AM)3, 1) +2*RAeD1fD2=2*RA*D5*Dl/D4 8275 AM(3,l)=AM)l,3) 8280 AM)3,2)—AM)3,2)+2*RA/D3 REM 8285 AM)3,2)=AM)2, 3) 8290 AM(3 , 3)=AM)3 ,3) +2*05/D2=2*RA”RA*D5/D4+2*RA*RA/D2 8300 REM *55* Coeffioients of RNS of equations *e* 8310 AM(l,4)=AM)l,4)=2*DS5D1/n2  AM(l,3)=AI4)l,2)+)2*RA*D1/D2)=2*RA*D15D5/D4 AM(2,1)=AM)2,1)+2*Dl/D3 AM)2,l)=AM(l,2) AM(2,2)=AM(2,2)+2 AM(2,3)=AM(2,3)+2*RA/D3 REM  AM)l,l)A11(l,1)+(2*D1501/D2)=2e)D15D1*D5/04)—2*D5/D2 AM)l,2)=AM(l,2)+2*D1/D3  8230 8240 8245 8250 8260  208  of ego’s.  8210 8220  8150 FOR Jl TO 3 8160 Dl—XT(I,J)=S(l) 8170 D2=(RA5RA)-)015D1) 8174 PRINT “02”,55,02 8176 IF (02 < 0) TNEN GDTO 8029 8180 n3=SQR)D2) 8190 04)D2))l.5) 8200 n5—=YT(I,J)+s(2)+D3  8130 8125 8140  8030 FOR 1=1 TO 3 8040 GOSUB 6000 8050 FOR J=1 TO 3 8060 XT)I,J)=XC(J) 8070 YT)I,J)=YC(J) 8080 AM(I,J)=0 8100 NEXT 2 8105 AM)I,NP)0 8110 NEXT I 8115 REM ** Applies only if top pert of oylinder digitized 8120 FOR 55=1 TO MAXIT  MN12 PRINT “DIGITIZE CYLINDER PROFILE NITN 9 PDINTS”  8028 8029  8025 S(3)—R.A 8026 NRITE #1,”ROOTS”,S(1),S(2),S(2) 8027 REM * MN is the no. of digitized points,  0)IFIVOT,J)=8)X,J)  in the diagonal  in which case a return  YP=S)2) EFS)1) WRITE ‘N,N”,NN,N FOR 1=1 TO 4 FOR J1 TO 3 VAR=VAR+)1T)I,J)—IP—SQR)RARA—)XT)I,J)—XP)’2)’2) NEXT J NEXT I VAR=VAN/ (MN-N) FRINT “TNE VARIANCE OF TRE LON FIT IS “VAR RETURN REM *°ss GAUSS-JORDAN ELIMINATION METNOO*ee* REM as Prepare working matrix 9 ** N3 NF4 FOR 1=1 TO N FON Jl TO NF 8)I,J)=AN)I,J) NEXT J NEXT I REM ae Ssarch for largest coefficient in column K, for rows K REM ** through N. IPIVOT is the row index of the largest REM *5 coefficient. FOR K1 TO N XF=K+1 IPIVOTK FOR I=K TO N NIT’A80)0)I,I)) FOR J=KP TO N 99A8S)B)I,J)) IF (00 > NIT) THEN NIT=BN NEXT 0 NI)I)NIT NEXT I REM e* Search for largest scaling factor SI in rows K to N 010=ANS)N)K,R)/NI)K)) FOR I=KP TO N SI=ASS)8)I,K)/NI)I)) SF (SI <= BIG) THEN GOTO 9230 010=81 IFIVOTI NEXT I REM aa Interchange rows K and IFIVOT if IPIVOT is not K IF )IFIVOT = K) TNEN GOTO 9310 FOR J=X TO NP TEMP=B)IPIVOT,J)  9290 N)K,J)=TEMF NEXT 0 9300 9310 REM ° Check diagonal for a zero entry, 9320 REM *5 is performed with IENROR=2 IF )8)K,K) 5> 0) THEN OOTO 9360 9330 IEKRON=2 9340 9350 RETURN 9360 REN 5* Eliminate all terms except terms FOR 2=1 TO N 9370 9380 IF (I = K) GOTO 9440 9390 QUOT8)I,K)/0)K,X) 9400 0)I,K)=0 9410 FOR JXF TO NP  9280  9007 9008 9010 9020 9030 9040 9050 9005 9056 9057 9060 9070 9080 9090 9100 9110 9120 9130 9140 9100 9160 9165 9170 9180 9190 9200 9210 9220 9230 9240 9250 9260 9270  9005  5455 0457 8458 8460 8470 9480 8490 8500 8005 8510 8560 9000  RETURN END  9660  normal return with  Check last diagonal element for a zero entry  IF )R)N,N) <> 0) TNEN GOTO 9030 IRRROR2 RETURN REM * Calculate norm of residual vector, REM ** IERROR—l RSQO FOR 1=1 To N SUMO FOR J=l TO N SUM=SUM+A24(I,J)eDS(J) NEXT J RSQ=RSQ+(ZiM(I,NF) —SUM) 2 NEXT I RNORHSQR)RSQ) IENROR1  9650  9540 9550 9560 9570 9580 9590 9600 9610 9620 9630 9640  9530  9000 9510 9520  **  B)I,J)”B(I,J)-R)K,J)*QUOT NEXT J NEXT NEXT K FOR K1 TO N DS(KfrB(K,NP)/B(K,K) NEXT K  9490 REM  9420 9430 9440 9400 9460 9470 9480  209  


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