Contact Angle Measurements on Particulate Systems

Transcription

1 Contact Angle Measurements on Particulate Systems By Nathan ael I. Stevens Bachelor of Applied Science (Chemistry and Chemical Process Technology) Bachelor of Applied Science, Honours (Chemical Technology) A thesis submitted for the degree of Doctor of Philosophy of Applied Science (Minerals and Materials) Ian Wark Research Institute University of South Australia August 2005

2 Declaration I declare that this thesis does not incorporate without acknowledgment any material previously submitted for a degree or diploma in any university; and that to the best of my knowledge it does not contain any materials previously published or written by another person except where due reference is made in the text. Nathanael I. Stevens August

3 Abstract Conventional techniques for contact angle measurement do not perform well for small particles. The equilibrium capillary pressure technique (ECP) consists in measuring the pressure required to prevent liquid penetration into a packed bed of particles and calculating the contact angle from a simple model, namely the equivalent capillary model. The ECP is well suited for the measurement of advancing contact angles. In its most convenient version the capillary pressure is measured for two different liquids (one of which is fully wetting and thus allows the calculation of the effective capillary radius). The use of ECP to obtain the receding contact angle on powders has been developed. The major difference between a liquid penetrating a porous bed and retreating from it is that liquid pockets are left behind in the receding case. Effectively, this reduces the porosity of the packed bed. The volume fraction of the retained liquid apparently depends on the surface tension of the liquid but is only marginally affected by the wettability and size fraction of the particles. Therefore a simple procedure for the determination of the receding contact angle, based on the use of a calibrating liquid, is outlined and verified. The approach gives realistic values for the receding contact angle.

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5 Table of Contents DECLARATION ABSTRACT HI ACKNOWLEDGMENTS TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES IV V VIII XIV GLOSSARY INTRODUCTION XVI XIX CHAPTER 1: LITERATURE REVIEW 1 CAPILLARITY 1 THE CONTACT ANGLE 2 The Young Equation 2 Wenzel Equation 3 Cassie Equation 4 WORK OF ADHESION AND COHESION, SPREADING COEFFICIENTS AND EQUILIBRIUM FILM PRESSURE 6 CONTACT ANGLE HYSTERESIS 8 SOLID SURFACE ENERGETICS 12 Empirical Approach 12 Theoretical Approach 13 WETTING PHENOMENA IN POROUS MEDIA 15 Capillary Rise 15 Capillary Hysteresis 16 EFFECTIVE CAPILLARY RADIUS 21 CONTACT ANGLE MEASUREMENT 24 Contact Angle Measurements on Plates 24 Contact Angles in Uniform Capillaries 26 Contact Angle Measurements on Multiple Particles 27 Washburn Method 29 Equilibrium Capillary Pressure (ECP) Technique 32 SINGLE PARTICLE CONTACT ANGLES 43 Optical Techniques 43 Force Techniques 45 V

6 CHAPTER 2: MATERIALS AND METHODS 56 MATERIALS 56 CLEANING GLASSWARE AND METHYLATION PROCEDURE 56 Glassware and Particles 56 Methylation 57 SURFACE TENSION MEASUREMENTS 58 Wilhelmy Balance Method 58 Drop Weight Method 59 PARTICLE BED CHARACTERISATION 60 Solid Packing Fractions and Liquid Retention 60 Measurement of the Density of Particles 61 Effective Capillary Radius from Flow Measurements 62 CONTACT ANGLE MEASUREMENTS 63 Sessile Drop Method 63 Equilibrium Capillary Pressure Apparatus 63 Sphere Tensiometty 67 SURFACE CHARACTERISATION 68 Particle Sizing 68 Scanning Electron Microscopy 69 Tomography 69 CHAPTER 3: GEOMETRY AND PHYSICS OF PARTICLE BEDS 73 PARTICLE CHARACTERISATION-SIZING AND SURFACE FEATURES 73 PACKING OF DRY PARTICLE BEDS 78 RETAINED LIQUID IN PARTICLE BEDS 82 EFFECTIVE RADIUS OF PACKED PARTICLE BEDS 91 CHAPTER 4: A THEORY FOR THE EFFECTIVE RADIUS IN POROUS MEDIA WITH RETAINED LIQUID 97 THERMODYNAMIC MODEL 101 DISCUSS ION 105 CONCLUSION 109 CHAPTERS: EQUILIBRIUM CAPILLARY PRESSURE MEASUREMENTS 112 INTRODUCTION 112 ADVANCING AND RECEDING CONTACT ANGLES IN MODEL CAPILLARIES 112 EQUILIBRIUM CAPILLARY PRESSURE (ECP) MEASUREMENTS 118 The Effect of Particle Size and Shape 120 Contact Angles for Mixtures of Model Particles 121 Contact Angles on Mineral Particles 125 SUMMARY OF EQUILIBRIUM CAPILLARY PRESSURE TECHNIQUE MEASUREMENTS 127 Vi

7 CHAPTER 6: CONTACT ANGLES ON SINGLE PARTICLES 130 CONTACT ANGLE MEASUREMENTS ON SINGLE PARTICLES 130 Summary of Single Particle Contact Angle Measurements 139 CHAPTER 7: CONCLUSIONS AND FURTHER WORK 141 APPENDIX A EQUILIBRIUM CAPILLARY PRESSURE APPARATUS OPERATING PROTOCOL 149 vii

8 List of Figures Figure 1. A liquid drop on an ideal solid surface at equilibrium 2 Figure 2. A liquid drop at equilibrium on a rough surface. The real surface area includes the area provided by the physical defects. 4 Figure 3. Liquid drop at equilibrium on a chemically heterogeneous surface. 5 Figure 4. The Lotus leaf is a super hydrophobic surface that exhibits self-cleaning properties. Figure 5. Conceptual representation of (A) the work of adhesion, (B) the work of cohesion and (C) the spontaneous spreading coefficient. (Adamson 1982). 6 Figure 6. Advancing (A) and receding (B) liquid interfaces on a model sinusoidal rough surface. Oa and 0, are the advancing and receding contact angles respectively and O is the contact angle with respect to the local surface (Dettre and Johnson 1965). 9 Figure 7. Advancing (A) and receding (B) liquid interfaces on a model heterogeneous surface. Oa and Or are the advancing and receding contact angles respectively and 0' and 0" are the contact angles of the surface where 0' > 0" (Dettre and Johnson 1965).9 Figure 8. The surface free energy of a three phase system vs. the experimental contact angle; the shape of the local minima is important in the effect on the apparent mobility of the meniscus. Image reproduced from Della Volpe et. a/.(2001). 10 Figure 9. The Zisman plot of contact angles of homologous series of organic liquids on Teflon: 0, RX; 0, alkylbenzenes; cro, n-alkanes;, dialkyl ethers; El, siloxanes; A, miscellaneous polar liquids. Image reproduced from Zisman (1964). 12 Figure 10. Classical case of capillary rise for water (A) and capillary depression for mercury (B) contacting a glass capillary. The capillaries are magnified in relation to the reservoir. 15 Figure 11. Scenario of capillary rise in two different capillaries of radius r and r'. Capillaries are enlarged for illustration purpose. 16 Figure 12. Equilibrium during penetration a well-defined capillary and an irregular capillary. Figure 13. Equilibrium during drainage in well-defined and irregular model capillaries

9 Figure 14. A hysteresis loop of saturation in a model particle bed. Inside the main loop is a scanning curve which shows the path followed when the process is reversed before reaching the edge of the loop. Image reproduced from Marmur (1990). 19 Figure 15. Metastable states predicted by the first-order model. The dashed line is for an ideal single capillary and the solid line is for the modelled porous particle bed. Image reproduced from Marmur (1970). 20 Figure 16. A modern sessile drop apparatus. With (a) CCD camera, (b) a remote controlledmicrometer syringe and (c) a diffuse light source. 24 Figure 17. A typical wilhelmy balance apparatus consisting of (A) an electronic balance, (B) motorised stage and (C) a computer. 25 Figure 18. A solution of polyethylene glycol placed on a particle cake of aluminium glass ballotini. (A) t = 0 s, (B) t = 2 s and (C) t = 5 s. Image reproduced from Hapgood (2002). 28 Figure 19. The displacement cell used for determining advancing capillary pressures by Bartell and co-workers. The apparatus consisted of (A) a measurement cell, (B), perforated plungers for containing the particle bed, (C) mercury manometer and (D) indicator tube. Image reproduced from Diggins (1990). 32 Figure 20. The experimental apparatus as used by Dunstan and White (1982) to measure advancing and receding capillary pressure in packed particle beds. The apparatus consists of (A) powder cell, (B) pressure transducer, (C) bulk liquid reservoir, (D) manometer, (E and f) regulating valves and (G, H, I and J) shut off valves. Image reproduced from Diggins (1990). 34 Figure 21. Capillary pressure, APE, as a function of 1/rep- as calculated from the specific surface area using equation (25). Image reproduced from Dunstan and White (1985). 35 Figure 22. The Equilibrium Capillary Pressure Apparatus. The apparatus consists of (A) liquid reservoir U-tube, (B) sample holder, (C) reducing union, (D) pressure transducer and (V1 and V2) isolation valves. 36 Figure 23. Capillary Pressure as a function of particle size. Powder: hydrophilic quartz. Liquid: water. The capillary pressure as a function of the equivalent spherical particle diameter for quartz. Image reproduced from Diggins (1990). 37 Figure 24. Capillary pressure for mixtures of hydrophobic and hydrophilic particles. Measured pressure is shown as a function of surface area fraction of the ix

10 hydrophilic particles ( um). Image reproduced from Diggins et al. (1990). 38 Figure 25. Equilibrium capillary pressures for water-wetted galena particles plotted versus the specific surface area, A, of the particles. Figure taked from Prestidge and Ralston (1995). 40 Figure 26. Equilibrium capillary pressure for (A) cyclohexane wetted and (B) water wetted, ethyl xanthate coated galena particles, plotted against the reciprocal of the average particle diameter. Figure reproduced from Prestidge and Ralston (1996). 41 Figure 27. Flotation recovery against the powder contact angle of ethyl xanthate treated galena particles. Figure reproduced from Prestidge and Ralston (1996). 42 Figure 28. Aveyard's solution for the contact angle of a particle in a pendant drop. Image reproduced from Aveyard (1996). 44 Figure 29. Schematic of an AFM colloid particle at a liquid vapour interface 45 Figure 30. Sequence of immersion (A-D) and final emersion (E) of a sphere. 46 Figure 31. The ECP apparatus configured to measure the capillary pressure of an advancing liquid front in a particle bed. The apparatus consists of a glass reservoir U-tube (A) which has a valve to control the liquid flow (V2), glass sample holder (B), reducing union (C) and a pressure transducer (D) which is connected to the transducer isolating valve (V2). 64 Figure 32. The ECP apparatus configured for receding pressure measurements. The sample holder (B), reducing union (C) and pressure transducer (D) are carried over from the advancing configuration and a micrometer syringe (E) and a beaker containing the analysing liquid (F) were added. 66 Figure 33. Illustration of the projection scheme at the APS 2-BM beamline. Image taken from Wang et. al. (2001). 69 Figure 34. Custom made PMMA sample holder for tomography experiments. 70 Figure 35.The particle size distribution of glass spheres as measured by the Malvern Mastersizer. 73 Figure 36. Size distribution of model crushed quartz particles, as measured by the Malvern Masters izer. 74 Figure 37. Size distribution of um chalcopyrite particles use for contact angle measurements. 76 Figure um glass spheres at 150x magnification. Note the irregular particles at the centre of the image. 77

11 Figure pm crushed quartz particles at 150x magnification 77 Figure pm chalcopyrite particles used for contact angle measurements. 78 Figure 41. Comparison of compaction of pm glass spheres using mechanical and manual agitation. manual tapping (number of taps) and mechanical agitation (s). 79 Figure 42. X-Ray tomography cross section of (A and B) gm methylated glass spheres and (C and D) gm methylated quartz particles. 81 Figure 43. Retention as a function of surface tension for pm glass spheres. Key: hydrophilic and 0 methylated spheres. 83 Figure 44. Three liquid bonds between two touching spheres for and vet= 50, 60 and 80. Figure reproduced from Kramer Figure 45. Contours of static liquid hold-up per particle and the contact angle, 0, as a function of wetting angle, g, and the capillary pressure, P. The arrow indicated isochoric retreat of a pendular bond after becoming isolated from other liquid bonds for systems with non-zero contact angles. Figure reproduced from Kramer (1998).87 Figure 46. X-Ray tomography cross section of a 2.5 M CsC1 solution drained from pm methylated glass spheres (A and B) and gm methylated quartz particles (C and D). 88 Figure 47. X-Ray tomography cross sections of a 2.5 M CsC1 solution drained from a pm packed bed of glass spheres. (A) and (B) show drainage in methylated particle beds, (C) and (D) show drainage from hydrophilic particle beds. 89 Figure 48. The effective capillary shape in a packed particle bed of pm glass spheres. The small White spheres in the image are the air bubbles in the glass spheres. 90 Figure 49. Permeability of packed particle beds of hydrophilic glass spheres. Key: porous plug, V pm, p.m and pm. 92 Figure 50. Unit cell for packing of spherical particles in a particle bed. 93 Figure 51. Schematic of capillary rise in a porous plug used for White's determination of the effective radius. 97 Figure 52. Retained liquid in a packed bed of pm glass spheres. The water has been dyed with 1 M potassium permanganate. 100 Figure 53. Liquid penetration into a porous medium (or packed bed of particles): (A) the porous plug is dry; (B) the porous medium is fully imbibed with liquid; (C) the xi

12 liquid has receded and pockets of liquid are left behind the receding front (located at height h). 101 Figure 54. Initial and final state of wetting of the average unit cell of a porous medium by a liquid: (A) imbibition in a dry bed; (B) imbibition in a pre-wet bed (V - total volume, Vs - volume of the solid, As - total area of the solid, Asi: - area of the solid/liquid interface in the pre-wet case, AIN* - area of the liquid/vapour interface in the pre-wet case). 102 Figure 55. Model of the porous medium: cubic solid particles (size a) arranged in a cubic lattice (spacing a + 2reif ): (A) the porous plug is dry; (B) the porous medium is fully imbibed with liquid; (C) the liquid has receded and the vertical slots are empty while the horizontal ones are filled with liquid (the liquid front is located at height h). 106 Figure 56. Advancing and receding pressure measurements of ethylene glycol in 550 pm diameter methylated glass capillary. Ethylene glycol was used because of its higher viscosity, thus small changes in the capillary height could be easily observed. 113 Figure 57. 'Haines jump' type spike in pressure. The liquid meniscus moves from one stable configuration to the next as a result of applying pressure above the meniscus to obtain a receding capillary pressure measurement. 114 Figure 58. Displacement of liquid in an 8 mm capillary. The capillary walls exhibit the natural hydrophobicity of glass. The water has been coloured with KMn04. (Note the marks on the glass for reference) 115 Figure 59. Gas pressure registered by the pressure sensor during liquid penetration (advancing liquid front) and during drainage (receding liquid front). 118 Figure 60. Contact angle measurements on mixtures of hydrophilic and hydrophobic quartz particles plotted against the area fraction of hydrophobic material in the particle bed for: (A) spheres and (B) quartz mixtures. advancing contact angles and 0 receding contact angles. 123 Figure 61. Advancing () and receding (0) contact angles measured on mixtures of chalcopyrite and quartz particles. 126 Figure 62. A 40 pm particle mounted to a 60 lam glass fibre for contact angle measurement. Figure 63. Phases of immersion of a 5 mm diameter Teflon AF1600 coated sphere xii

13 Figure 64. The measured capillary force acting on the 5 mm diameter Teflon AF1600 coated ball bearing vs. the immersion position in water. 132 Figure 65. Capillary force vs. immersion position for a Teflon AF1600 coated 180!Am diameter glass sphere immersed in water. 133 Figure 66. Zisman plot for advancing contact angles measured on Teflon AF1600 coated quartz rod and glass spheres. Key: 2 mm rod, pm glass sphere, 100 p.m glass sphere and V 50 pm glass sphere. 135 Figure 67. Capillary force vs. immersion position for a 180 pm glass sphere treated with 10-1M TMCS. The measurement liquid is water. The data shows the same points of interest (A-I) as was observed on the 5 mm Teflon AF1600 coated sphere. 136 Figure p.m un-oxidised chalcopyrite particle 138 Figure 69. Force vs. position plot of chalcopyrite particle being wet with water. The dashed line shows the height of the particle. 139

14 List of Tables Table 1. Surface tension parameters (in mj/m2) of some liquids. Table reproduced from Good (1992) 14 Table 2. Advancing and receding pressure measurements for angular particles <75 [tm. 38 Table 3. Advancing and receding contact angles on hydrophobic glass plates. 39 Table 4. Mean particle diameter and specific surface area of the particles as measured by the Malvern Mastersizer. 75 Table 5. Volume fraction of solid material, Os, in packed particle beds. 80 Table 6. Retention as a function of particle bed length for hydrophobic pm glass spheres and the associated time taken to drain the particle bed. 82 Table 7. Retained liquid in particle hydrophilic and methylated packed beds of glass spheres 84 Table 8. Comparison of liquid retained, 0;, in spherical glass and crushed quartz particle beds. 85 Table 9. Solid packing fraction, Os, and the retained liquid measured for hydrophilic and hydrophobic chalcopyrite ( Jim). 85 Table 10. Volume fraction of packed particles for gm particles, Ø, and retained liquid, 0;, calculated from three dimensional reconstructed X-ray tomography images. 91 Table 11. Calculated number of effective capillaries, n. 93 Table 12. Comparison of determined effective radii using the Kozeny-Carman, White and Hagen-Poiseuille models for capillary radius. 94 Table 13. Comparison of effective radii calculated for hydrophilic and methylated spheres and quartz particles with the same size fraction. 95 Table 14. Parameters predicted by the model shown in Figure 55 and equations (81) and experimental values from Morrow (1970). 107 Table 15. Capillary pressure and the associated advancing and receding contact angles for water in well-defined glass capillaries and on flat glass plates. 116 Table 16. Capillary pressure and the associated advancing and receding contact angles for ethylene glycol in well-defined glass capillaries and on flat glass plates. 116 xiv

15 Table 17. Advancing and receding pressure measurements and contact angles calculated with equation. The surface tension of water and cyclohexane are 72.8 mn/m and 25.5 mn/m, respectively. 119 Table 18. Advancing and receding water contact angles determined by the ECP technique on methylated glass spheres. Contact angles were measured on three different size fractions 120 Table 19. Advancing and receding contact angles determined by ECP on methylated crushed quartz particles. Contact angles were measured on three different size fractions 121 Table 20. Advancing and receding contact angles, measured for 50% by weight mixtures of glass spheres and quartz particles. 122 Table 21. Calculated contact angles using the Cassie equation. The first contact angle is the calculated contact angle of the hydrophobic material in the particle mixture; the contact angle in brackets is the actual contact angle measured on a solely hydrophobic particle bed. 123 Table 22. Advancing and receding capillary pressure for mixtures of chalcopyrite and quartz particles of [tm distribution. 125 Table 23. ECP advancing and receding contact angles for chalcopyrite-quartz mixtures of size fraction gm with respect to the fraction of hydrophobic material,fchp. 126 Table 24. Predicted chalcopyrite contact angle from the contact angle of mixtures of chalcopyrite and quartz. 127 Table 25. Advancing contact angles measured on Teflon AF1600 coated rods and particles. 134 Table 26. Receding contact angles for Teflon AF1600 coated rods and particles. 134 Table 27. Advancing and receding contact angles on methylated glass spheres with heterogeneous surface coverage of trimethylchlorosilane. 137 XV

16 Glossary Greek Symbols a - Ratio of the solid-liquid interfacial area to the total solid surface area - Ratio of the liquid-vapour interfacial area to the total solid surface area 74, Surface tension of liquid-vapour interface, mn/m yci - Interfacial tension of solid-liquid interface, mn/m, Surface tension of solid-vapour interface, mn/m - Acid-base component of surface energy, mn/m y'w -Liftshitz-van der Waals component of surface energy, mn/m - Zisman constant ii - Liquid viscosity, Pa.s 0 - Equilibrium contact angle, deg Oa- Advancing contact angle, deg O- Cassie contact angle, deg Or- Receding contact angle, deg Ow- Wenzel contact angle, deg ire- Equilibrium film pressure, mn/m - Angle determining the position of the wetting line with respect to the centre of a particle, deg 1- sly Adhesion tension, mn/m r - Tortuosity factor for effective capillaries - Porosity (volume fraction of void spaces) Ø5-Volume fraction of solid Volume fraction of retained liquid xvi

17 Alphabetic Symbols A - Surface area, m2 A - Specific surface area, m2/kg - Residual area of the solid-liquid interface, m2 - Residual area of the liquid-vapour interface, m2 a - capillary constant c - Geometric correction factor for capillary shape D - Diameter of particles, m FB- Buoyancy force, N Fc - Capillary force, N Fg- Weight, N f 1- Surface area fraction of species g - Acceleration of gravity, m/s2 n - Number of capillaries P - Perimeter, m AP,- Capillary pressure, Pa APh- Hydrostatic pressure, Pa Pt01- Total pressure, Pa Volumetric flow rate, m3/s Particle radius, m r - Capillary radius, m reff - Effective radius of capillary in porous media, m rh- Hydraulic radius, m r,- Radius of the meniscus wetting a particle, m rw- Wenzel roughness factor xvii

18 S- Saturation g Spreading coefficient Wa Work of adhesion, mn/m W.- Work of cohesion, mn/m xviii

19 Introduction The understanding and measurement of the contact angle of liquids on small particles leads to improvements in flotation recovery of valuable minerals, better water quality, higher extraction of mineral oil from rock beds and more effective pharmaceuticals. Different techniques have been used to measure the contact angle on particles e.g. sphere tensiometry and colloid probe microscopy (for single particles); skin flotation, the Washburn and capillary pressure methods (for multiple particles and packed beds). Single particle contact angle measurements give specific wettability information, while a measurement on many particles yields an average value of the contact angle. Contact angle measurements on particle beds is often favoured for complex mineral systems. All many-particle techniques provide a meaningful measure of the advancing contact angle only. However, receding contact angles are particularly important in the recovery of valuable minerals by flotation. When a bubble and a particle come into contact the thin film between them drains, ruptures and the liquid recedes along the particle surface. Thus a value of the receding contact angle is very useful, especially when predicting flotation recovery (Pyke et al. 2001), in understanding pigment flushing or in improving oil extractability. The challenge, therefore, was to develop a technique for obtaining receding contact angles in particle beds. The technique chosen was the equilibrium capillary pressure technique (ECP) pioneered by Bartell. The ECP was chosen because it has been developed into a robust technique by Diggins (Diggins 1990). The technique has been validated by contact angle measurements on model quartz particles (Diggins et al. 1990) and successfully used for advancing contact angles on metal sulphide particles from flotation processes by Prestidge and Ralston (1995; 1996; 1996). The advancing contact angle is calculated using the Laplace equation; 2 APEy_ lv cos 0 reff where is the surface tension of the liquid, APc is capillary pressure and 0 is the contact angle on the particles. Because of the complex geometry of the capillaries in a particle bed the radius is an effective one. White (1982) derived an expression for the effective radius in terms of particle density, specific surface area and volume fraction of the solid phase. xix

20 2(1-05) r, = P Advancing contact angles are determined by measuring the pressure required to prevent capillary rise in the packed bed of particles. For two liquids, the contact angle is expressed as cos, = --LAP AP2 Receding contact angles were determined by measuring the capillary pressure after the liquid had receded. The volume fraction of retained liquid, o*l, must be accounted for. For two liquids, the expression becomes.a.p 01*,) cosi9 = 2 AP2y, (1-0,-0*,) The full expression for the effective radius includes the retained liquid, Ø, and the changes in solid-liquid, A:, and liquid-vapour, Al*,, interfacial areas. Analysis shows that the most significant correction to consider is with respect to the retained liquid fraction as shown in the above equation. The ECP was validated by measuring advancing and receding contact angles on hydrophobic and hydrophilic particle mixtures of model spheres and irregular particles. Contact angles measured on particle beds showed a close correlation with contact angles measured on identically treated flat plates. Advancing and receding contact angles were also convincingly measured on metal sulphide particles mixed with crushed quartz. Milestones in this thesis include quantitative measurements of static liquid hold-up in packed particle beds and an investigation into the capillary structure and drainage behaviour using high resolution X-ray tomography. The most significant achievement in this thesis is the development of the first tool that can be used "on site" for both advancing and receding contact angle measurement in particle beds. XX

21 Chapter 1: Literature Review The way a liquid behaves when in contact with a solid surface has been of interest for many years. Aristotle and Archimedes made observations of a gold leaf floating on water. Most of the understanding of wetting has occurred over the last 200 years since Young, in 1805, first postulated the relationship between interfacial tensions and the angle of the liquid resting on a solid surface. Since then, the importance of wettability has been identified in many different systems including textiles and dyeing, lubrication, insecticides, pharmaceuticals and mineral flotation. Capillarity The pressure difference across a curved fluid interface is called the capillary pressure and is described by the Laplace (Laplace 1806) equation: AP = 2,1,,( j ri r2 (1) where yh, is the surface tension of the liquid and r I and r2 are the principal radii of curvature of the surface, perpendicular to each other. If the radii are equal then the Laplace equation reduces to: 27, r (2) For planar surfaces the radii of curvature are infinite and therefore there is no pressure difference. 1

22 The Contact Angle The contact angle is the angle that a liquid surface makes with the solid surface. The degree of wettability can be characterised by the contact angle. Surfaces exhibiting contact angles of less than 90 are classified as hydrophilic and are said to be wettable (Blake 1984). Surfaces with contact angles of are typically termed hydrophobic (Blake 1984) and with contact angles greater than 140, super-hydrophobic (Blake 1984). The Young Equation When a liquid drop is placed on a solid surface the liquid will spread across the solid surface. It can form a wetting film or remain as a drop having a finite contact angle. 7Iv 7SN, Figure 1. A liquid drop on an ideal solid surface at equilibrium The equilibrium contact angle was first described as the balance of interfacial tensions at the contact line (Young 1805). Dupre (1869) formulated the relationship between the interfacial surface tensions formally as: COS e=y,-7,/ (3) where ysi, YIN, and ys, are the interfacial tensions between the solid-liquid, liquid-vapour and solid-vapour interfaces and 0 is the equilibrium contact angle measured through the liquid phase. Equation (3) is known as the Young equation. 2

23 The Young equation is based on the assumption that the droplet is resting on a chemically and physically homogeneous surface and the whole system is at equilibrium. In reality, most surfaces are less than ideal and exhibit hysteresis, i.e. a difference between the advancing and receding contact angles. The adhesion tension describes the net force exerted by the solid on the liquid at the three phase contact line (Adamson 1982), given by: rsiv = rs, = 71, cos 8 (4) The solid-vapour and solid-liquid interfacial tensions (unlike ylv) are not easily measured experimentally (Adamson 1982). Wenzel Equation Surfaces that have an inherent roughness can be described by a roughness ratio which is the ratio of the actual surface area to the apparent surface area (Wenzel 1936), given as rw =- Aactual I Aapparent The equilibrium contact angle is then given by: cos Ow = rw cos (5) where Ow is the contact angle on the rough surface and 0 is the contact angle on a chemically identical but smooth surface. This is shown in Figure 2. 3

24 sv Figure 2. A liquid drop at equilibrium on a rough surface. The real surface area includes the area provided by the physical defects. Because rw > 1, for contact angles greater than 900, roughness will increase the measured contact angle and for contact angles less than 900 roughness will decrease the measured contact angle. Cassie Equation The equilibrium contact angle of a liquid drop on a physically smooth but chemically heterogeneous surface can be described as: coo, -E icosoi and Efi =1 (6) where 0, is the measured contact angle, f, is the surface area fraction the chemical domains on the surface having contact angle 0, (See Figure 3). 4

25 Figure 3. Liquid drop at equilibrium on a chemically heterogeneous surface. The Cassie equation is based on the assumption that the surface area fractions are large compared with the molecular scale and small compared with the size of the droplet (Cassie and Baxter 1944). Super-hydrophobic surfaces, such as lotus leaves, are rough and hydrophobic so that the liquid droplet rests on caps of the surface, resulting in small pockets of air being trapped under the droplet as shown in Figure 4 (Blake 1984). Figure 4. The Lotus leaf is a super hydrophobic surface that exhibits self-cleaning properties. In this case, the Cassie equation becomes: cos 0, = fi (cos A + 1) (7) where fi is the area fraction of the droplet in contact with solid surface. The interest in super hydrophobic surfaces largely due to the low adhesion and self cleaning properties exhibited. 5

26 Work of Adhesion and Cohesion, Spreading Coefficients and Equilibrium Film Pressure The work of adhesion, Wa, was defined by Dupre (1869) as the reversible work necessary to separate the solid and liquid phases into vapour in the absence of an adsorbed vapour on the solid surface and is given by: W.= 7h, (8) The work of adhesion is illustrated in Figure 5A. Using Dupre's equation for the work of adhesion, we can determine the work required to separate two identical phases, the work of cohesion (see Figure 5B). The work of cohesion, Wc, is given by: W, = 27h, (9) (A) V ;<\.\\"\ N^^\':;>",.\\`:, \\\\`:,.\\ Figure 5. Conceptual representation of (A) the work of adhesion, (B) the work of cohesion and (C) the spontaneous spreading coefficient. (Adamson 1982). 6

27 The initial spreading coefficient, S, for a liquid on a solid defines whether the liquid will spontaneously spread. Cooper and Nuttall (1915) showed this to be: (10) The spreading coefficient can be used to predict if a liquid will spontaneously wet a surface or form a finite contact angle on the surface. The spreading coefficient is defined as the work required to spontaneously remove the solid-liquid and liquid-vapour interfaces leaving the solid surface as shown in Figure 5C. If the spreading coefficient is negative the liquid will form a droplet with a finite contact angle, if the coefficient is positive the liquid will spontaneously spread on the surface. Harkins and Feldman (1922) stated that if the work of adhesion between the solid and the liquid was greater than the work of cohesion for the liquid, then the liquid would spread. Indeed, S =Wa (11) Hence if S is positive, the liquid will spread on the surface. For equilibrium conditions, it is possible for equations (8) and (10) to be combined with the Young equation giving respectively: and, W, = + cose) (12) = (cos 6' 1) (13) The equilibrium film pressure describes the pressure at the interface of two more components. At equilibrium, the film pressure can be described as r esv sv (14) where res, is the equilibrium film pressure of the adsorbate and ys is the specific interfacial energy of the pure solid in the absence of any adsorbed vapour. The surface coverage is generally a monolayer or less coverage for a solid having contact angle greater than zero 7

28 (Good 1992). It is also worth noting that the equilibrium film pressure can only be grater than or equal to zero. The equilibrium film pressure term, a-es), has been considered in great detail and the term is usually ignored as adsorption of liquid vapour on a low energy solid surface is assumed to be negligible. Contact Angle Hysteresis The Young equation predicts the contact angle of a droplet at equilibrium. Experimentally we find that the measurement of a single equilibrium contact angle is impossible and it is more common to measure a range of static apparent contact angles. The maximum static contact angle is known as the advancing contact angle, Oa, and the minimum static contact angle is known as the receding contact angle, Or. The difference between the advancing and receding contact angles is called hysteresis. The contributions of hysteresis are accepted as (1) surface heterogeneity, (2) surface roughness and (3) surface mobility (Johnson and Dettre 1964; Adamson 1982; Blake 1984). The influence of surface roughness and surface heterogeneity on hysteresis was first discussed by Derjaguin (1946) and also Shuttleworth and Bailey (1948). They introduced the concept that the contact angle of a liquid could have many values where the contact line was in a metastable state. Metastable states are separated by energy barriers and, when a force is applied to the wetting line, the liquid would move across the surface, jumping between metastable states. When the force was removed the wetting line would come to rest in a new metastable state, having a suitable low energy somewhere near the equilibrium contact angle. Therefore the interface advances or recedes in a series of thermodynamically irreversible steps. Johnson and Dettre (1964) further analysed the influence of roughness on heterogeneity. They modelled the behaviour of a liquid drop placed on a rough surface consisting concentric sinusoidal corrugations (See Figure 6). 8

29 (A) (B) Figure 6. Advancing (A) and receding (B) liquid interfaces on a model sinusoidal rough surface. and Of are the advancing and receding contact angles respectively and a is the contact angle with respect to the local surface (Dettre and Johnson 1965). The model showed that the local contact angle, 0, could be much higher or lower than the contact angle at the wetting line because of the influence of the slope of the solid surface. It is therefore apparent that the value of the local contact angle is dependent on the position of the wetting line on the surface and therefore the local contact angle is affected. In a similar fashion Johnson and Dettre considered a liquid drop placed on a heterogeneous surface having concentric hydrophilic and hydrophobic rings (see Figure 7). (A) (B) 0' " Figure 7. Advancing (A) and receding (B) liquid interfaces on a model heterogeneous surface. a, and 0, are the advancing and receding contact angles respectively and 0' and 0" are the contact angles of the surface where 0' > 0" (Dettre and Johnson 1965). 9

30 Hysteresis on a chemically heterogeneous surface can be explained by the interactions around the three phase line, as shown in Figure 7. Figure 7A shows a droplet on a heterogeneous surface pinned at a hydrophilic-hydrophobic interface. If a force is applied to the droplet, the contact angle will increase to keep thermodynamic equilibrium. In the receding case, the contact angle will decrease and remained pinned due to the attraction to the hydrophilic surface. Surface roughness and heterogeneity models have been developed and lead to a series of metastable states that have different contact angles (Johnson and Dettre 1964; Della Volpe et al. 2001). Each metastable state is separated by an energy barrier that hinders movement of the wetting line across the surface (see Figure 8). - Low High, Contact Angle Figure 8. The surface free energy of a three phase system vs. the experimental contact angle; the shape of the local minima is important in the effect on the apparent mobility of the meniscus. Image reproduced from Della Volpe et. a/.(2001). When that energy barrier is overcome the liquid drop will move to the next metastable state. Thus, along the wetting line of a drop there would be many different contributions to the macroscopic contact angle. The maximum and minimum contact angles can therefore be described as the equilibrium contact angle plus or minus the maximum slope of the roughness. For a chemically heterogeneous surface, the maximum and minimum contact 10

31 angles are governed by the hydrophobicity of the individual components present on that surface. The third cause of hysteresis can be attributed to surface mobility. The adsorption at the three phase line of vapour species, or surfactants to the surface of the solid, can modify the wettability of a solid surface. At equilibrium, these species will be constantly adsorbing and desorbing at the contact line. The rate of these processes is governed by the affinity of the adsorbed species for the solid surface. If the contact line is then moved a change in solidliquid and liquid-vapour interfacial tensions can occur, due to the slow rate of desorption of the species. Therefore the contact angle will change to maintain the balance of interfacial tensions. 11

32 Solid Surface Energetics Empirical Approach For many applied purposes it is useful to have an understanding of the wettability of a particular solid surface, therefore it would be practical to have a measure of the free energy of its surface. An empirical approach is to measure the contact angle of a surface utilising an array of liquids with different surface tensions. Zisman (1964) found that a homologous series of organic liquids displayed essentially a linear response when plotted against cos (Figure 9). cos 8 Figure 9. The Zisman plot of contact angles of homologous series of organic liquids on Teflon: o, RX; 0, alkylbenzenes; cto, n-alkanes;, dialkyl ethers; El, siloxanes; A, miscellaneous polar liquids. Image reproduced from Zisman (1964). Each series of results can be extrapolated to 0 = 00. The intercept with the x-axis where the contact angle is zero was termed by Zisman to be the critical surface tension of wetting, y, and considered to be a measure of the surface free energy of the solid. Therefore if yl < y, the liquid will spontaneously spread, completely wetting the solid in question. The Zisman method also allows prediction of contact angles of liquids on a solid surface. The relationship between the predicted contact angle and the critical surface tension of wetting is given by: cos0 =I (50'1, re) (15) 12

33 where 6 is a constant and typically has a value of It should be stated that the Zisman method is not recommended for use with binary solutions, e.g. water-ethanol mixtures, as these are not pure liquids. However binary solutions are sometimes employed, for some pure organic liquids can have adverse effects on the surface being investigated. Theoretical Approach Young's equation is phenomenological and is therefore limited in its interpretation of contact angle data, as it does not consider explicitly intermolecular forces between the solid, liquid and vapour phases. Early approaches to consider contributions of the intermolecular forces attempted to describe the surface energy of a phase as being made up from dispersion and polar components. Dispersion interactions are generally well-understood and arise from dipole-dipole, dipole-induced dipole and induced dipole-induced dipole interactions. Polar contributions include contributions due to orientation, induction and hydrogen bonding. Fowkes (1964) described the solid-liquid interfacial tension for non-polar phases as the geometric mean of the solid and liquid components interfacial energies and the contribution of the dispersion interactions at the interface. =r, + (16) The most popular theory was derived by van Oss, Good and Chaudhury (1986). They considered two contributions to the interfacial energy: the Lifshitz-van der Waals (LW) component, ytiv, and an acid-base (AB) component, 7,JAB. Thus, the interfacial energy is defined as: =W yijab (17) The LW component encompasses orientation forces between permanent dipoles, induction forces between permanent and induced dipoles and dispersion forces between fluctuating dipoles and is calculated according to Fowkes (1964). The AB component includes interactions between hydrogen ion donors and acceptors, electron pair acceptors and donors and hydrogen bonding. The AB component is calculated as: 13

34 All r ± 2(\1777.; (18) where the acid-base component is broken down into acidic (y+) and basic (yr) parameters. The acid component were treated as Lewis acids (electron acceptor) and the base components as a Lewis base (electron donor). The LW and AB contribution to )'s, and ysi can then be substituted into the Young equation (3) which gives rh, (1+ cos 0) = 2 (, y slw ILW + Y 77 r; (19) Using equation (19) Good reported experimental and theoretical values of surface tension for a series of liquids, shown in Table 1. Table 1. Surface tension parameters (in mj/m2) of some liquids. Table reproduced from Good (1992) Liquid 7h, 7uf )/ Water Glycerol Formamide Ethylene Glycol Chloroform The calculated values of interfacial surface energy showed good agreement with the experimental surface energies; hence these results support the Fowkes treatment of interfacial surface energetics. 14

35 Wetting Phenomena in Porous Media Capillary Rise Capillary rise is due to capillary pressure. The shape of the meniscus will determine whether the liquid will rise or depress depending on the liquid contact angle in the capillary. If the contact angle is less than 900, liquid will enter the capillary and rise, and if it is greater than 900, capillary depression will result. Equilibrium is achieved in the capillary when the liquid in the capillary rises to a level where the capillary pressure is equal to the hydrostatic pressure. If this condition is met (Figure 10A) the liquid will penetrate into the capillary such that (Marmur 1990): APc = APh 2y, cos 0 = Apgh (20) where APc is the difference in pressure at the meniscus, Ap is the difference between the liquid and vapour density, g is the acceleration of gravity, h is the height of the liquid column, r is the radius of the capillary and 0 is the local contact angle of the meniscus in the capillary. Figure 10. Classical case of capillary rise for water (A) and capillary depression for mercury (B) contacting a glass capillary. The capillaries are magnified in relation to the reservoir. 15

36 The capillary radius has to be sufficiently small such that the capillary pressure is relatively important. This is determined by the capillary constant, a. Capillary pressure is important when r <a where a is a _ \12y,,,/ /.6.pg (21) The height of the liquid penetration, h, in two capillaries having the same contact angle, 0, radii r and r', where r < r', placed in a reservoir of liquid can be determined according to equation (20) as shown in Figure 11 and is a very good way of determining surface tension (Adamson 1982). Figure 11. Scenario of capillary rise in two different capillaries of radius r and r'. Capillaries are enlarged for illustration purpose. Capillary Hysteresis The movement of a liquid in porous media is, of course more complex than penetration and displacement in a uniform capillary with a well-defined radius. The capillary structure of a porous bed is complex, as capillaries are not uniform and run the length of the bed as well as transversely. Capillary rise in porous media may also be arrested due to shape factors in the particle bed that flatten the meniscus. This flattening of the meniscus can occur at several heights in a porous bed above and below the stable equilibrium height, which indicates the presence of metastable states and therefore hysteresis (Marmur 1990). Therefore there are many more possible equilibrium positions, as well as capillary hysteresis. The complex 16

37 behaviour of liquid in irregular capillaries will be modelled by examination of a model irregular capillary. As a simplified model, consider a capillary with a single bulge (Marmur 1990). When liquid penetrates into the capillary from the reservoir beneath it, the meniscus equilibrates in the lower part of the expansion of the irregular capillary (See Figure 12). This occurs because the meniscus shape changes and therefore the capillary pressure changes due to the change in curvature of the meniscus. The radius of curvature changes until the interfacial pressure of the meniscus (or capillary pressure) is in equilibrium with the hydrostatic pressure of the liquid in the capillary. The liquid in the cylindrical capillary equilibrates at a height higher than in the irregular capillary, because there is no geometric influence in the cylindrical capillary to change the meniscus shape, hence the rise is defined by equation (20). Figure 12. Equilibrium during penetration a well-defined capillary and an irregular capillary. Liquid drainage in the capillaries is shown in Figure 13. The liquid will drain reversibly until the meniscus comes to the same level as in the uniform capillary. If the liquid is caused to flow through the bulged capillary, the liquid will descend rapidly until the meniscus reaches C (Morrow 1970). During this rapid drainage there is a fluctuation in capillary pressure due to the rapid change of meniscus radii. The resulting fluctuations of capillary pressure are called Haines jumps and were first observed in 1925 (Haines 1925). 17

38 Figure 13. Equilibrium during drainage in well-defined and irregular model capillaries. These fluctuations of pressure in the particle bed are the result of the liquid interface jumping from one stable configuration to the next. Observations of liquid penetrating into soil samples concluded that the pressure jumps during capillary rise are caused by sudden changes of the menisci radii in the particle bed, as the menisci follows the geometry of the effective capillaries (Haines 1925). It was also concluded that there were a range of stable heights for capillary rise. Experimental observations of wetting fronts in porous media have shown that the wetting front in a porous bed fluctuates in local capillaries, depending on the local effective radius (Hackett and Strettan; 1925; Haines 1925; 1927)). Repetitive imbibition and drainage in particle beds result in hysteresis loops, as shown in Figure

39 100 to SO PRESSURE OFFICIDICY. arbitrary units Figure 14. A hysteresis loop of saturation in a model particle bed. Inside the main loop is a scanning curve which shows the path followed when the process is reversed before reaching the edge of the loop. Image reproduced from Marmur (1990). The capillary hysteresis loop can be explained as follows: once a particle bed has been imbibed, the liquid can be removed by changing the external hydrostatic pressure such that liquid drains out of the particle bed. Since the largest capillaries have the lowest capillary pressure, they drain first, followed by smaller capillaries (See Figure 11). As the liquid leaves the particle bed small pendular rings of liquid are retained between the particles. Further increases in pressure cannot remove the liquid, as it is no longer part of the receding wetting front. Thus the saturation of the bed stays constant. If the pressure is then changed such that the liquid re-enters the porous particle bed, the smallest capillaries are filled first since they exhibit the highest capillary pressure followed by the larger capillaries (Figure 11). The saturation does not reach 100% after complete imbibition because of the entrapment of gas bubbles. Hence it can be concluded that the repetitive imbibition and penetration of a real porous medium does not occur reversibly (Marmur 1990). Contact angle hysteresis also present when measuring contact angles in porous media. The contributors to contact angle hysteresis described above include physical and chemical heterogeneities. The individual contributions of the heterogeneities to the hysteresis is difficult to distinguish from the capillary hysteresis, as they are present with the capillary hysteresis. Theoretical models for equilibrium in porous media must discuss the criteria for penetration and predict the dependence of saturation on pressure. To date, only partial fulfilment of these 19

40 goals has been achieved. Many models predict only a single equilibrium height largely because they do not account for local features in the porous bed hence they do not predict capillary hysteresis. Smith, Foote and Busang (1931) proposed the first order model to predict the capillary rise in a porous bed. The model considers the local values of porosity, specific surface area and the apparent contact angle in the porous medium. These are treated as averages over the cross section of the porous medium, perpendicular to the direction of penetration. Their model also acknowledges that characterising the local details of structure is limited. The graphical solution to the first order model for a spherical particle bed (solid line) is shown in Figure 15, as well as the case for a microscopically uniform capillary (dashed line). The y-axis represents the predicted equilibrium positions and the x-axis shows the arbitrary height of capillary rise. Figure 15. Metastable states predicted by the first-order model. The dashed line is for an ideal single capillary and the solid line is for the modelled porous particle bed. Image reproduced from Marmur (1970). The case for the uniform capillary predicts only one equilibrium position where w = 0 and x = I. In contrast, there are many predicted equilibrium points that result from the local variations in the porous medium. The model also gives a minimum height of capillary penetration (x = 0.6) and a maximum height for capillary penetration (x = 1.2) The difference in height is the capillary hysteresis range. The hysteresis range is dependent upon the specific surface area of the particles and the intrinsic contact angle. 20