CAPILLARITY. Introduction. Liquid Properties

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1 CAPILLARITY 155 CAPILLARITY DOr, University of Connecticut, Storrs, CT, USA M Tuller, University of Idaho, Moscow, ID, USA ß 2005, Elsevier Ltd.All Rights Reserved. Introduction The coexistence of gaseous, liquid, and solid phases in soil pores gives rise to a variety of interfacial phenomena that, for example, lead to spreading of liquid droplets on solid surfaces, liquid rising in capillaries and soil pores, or the entrapment of liquid in crevices. These phenomena, partially attributed to capillarity, determine retention and movement of water and solutes through soils. Hence they are of great importance in a variety of environmental and agricultural problems. Liquid Properties The phenomenon of capillarity in porous media results from two opposing forces: liquid adhesion to solid surfaces, which tends to spread the liquid; and the cohesive surface tension force of liquids, which acts to reduce liquid gas interfacial area. The resulting liquid gas interface configuration under equilibrium reflects a balance between these forces. The phenomenon of capillarity is thus dependent on solid and liquid interfacial properties such as surface tension, contact angle, and solid surface roughness and geometry. Surface Tension At the interface between water and solids or other fluids (e.g., air), water molecules are exposed to different forces than are molecules within the bulk fluid. For example, water molecules in the bulk liquid are subjected to uniform cohesive forces whereby hydrogen bonds are formed with neighboring molecules on all sides. In contrast, molecules at the air water interface experience net attraction into the liquid because of lower density of water molecules on the air side of the interface, with most hydrogen bonds formed at the liquid side. The result is a membrane-like water surface that has a tendency to contract and reduce the amount of its excess surface energy. The surface tension reflects the amount of interfacial energy per unit area, or the energy required to bring molecules from the bulk liquid to increase the surface (it is also useful to express surface tension as force per unit length of interface). Different liquids vary in their surface tension (Table 1). Surface tension also depends on temperature, usually decreasing linearly as the temperature rises. Thermal expansion reduces the density of the liquid and therefore also reduces the cohesive forces at the surface as well as inside the liquid phase. Soluble substances can increase or decrease surface tension. If the affinity of the solute molecules or ions to water molecules is greater than the affinity of the water molecules to one another, then the solute tends to be drawn into the solution and to cause an increase in the surface tension. This is the effect of electrolytic solutes. For example, a 1% NaCl concentration increases the surface tension of an aqueous solution by 0.17 mn m 1 at 20 C. If, on the other hand, the cohesive attraction between water molecules is greater than their attraction to the solute molecules, then the latter tend to be relegated toward the surface, reducing its tension. That is the effect of many organic solutes, particularly detergents. Contact Angle When a liquid drop is placed on a solid surface, the angle formed between the solid liquid (SL) interface Table 1 Liquid Liquid vapor interfacial tensions for various liquids Temperature ( C) Water Methylene iodide Glycerin Ethylene glycol Dimethyl sulfoxide Propylene carbonate Methyl naphthalene Dimethyl aniline Benzene Toluene Chloroform Propionic acid Butyric acid Carbon tetrachloride Butyl acetate Diethylene glycol Nonane Methanol Ethanol Octane Heptane Ether Perfluoromethylcyclohexane Perfluoroheptane Hydrogen sulfide Perfluoropentane Surface tension (mn m 1 ) Reproduced from Adamson AW (1990) Physical Chemistry of Surfaces, 5th edn.new York: John Wiley.

2 156 CAPILLARITY and the liquid gas (LG) interface (Figure 1) is referred to as the equilibrium (or static) contact angle (). Two equivalent approaches are commonly used to describe the equilibrium contact angle on smooth and chemically homogeneous planar surfaces: (1) a force balance approach, and (2) an interfacial, free-energy minimization. The force balance formulation considers interfacial tensions ( ij ) as forces per unit length; hence the force balance at the contact line of a drop resting on a solid surface under equilibrium requires the vector sum of the forces acting to spread the drop (outward) to be equal to opposing cohesion and viscous forces. The free-energy minimization approach regards interfacial tension as energy per unit area, and calculates changes in surface free energy (F) due to infinitesimal displacement (A): F ¼ Að SL GS ÞþA cos LG ½1Š The result is identical whether considering the minimization of free energy, with F/A ¼ 0, or taking a balance of forces tangential to the solid surface; both cases yield the Young equation: LG cos þ SL GS ¼ 0 with L, G, and S indicating liquid, gas, and solid, respectively, and ij the respective interfacial surface tensions. The equilibrium contact angle is therefore: cos ¼ GS SL ½3Š LG Liquids that are attracted to solid surfaces (adhesion) more strongly than to other liquid molecules (cohesion) exhibit a small contact angle, and the solid is Figure 1 Liquid solid gas contact angles: (a) hydrophilic surface (<90 ) where liquid wets the surface; (b) hydrophobic surface (>90 ) where liquid repels the surface. ½2Š said to be wettable by the liquid (Figure 1a). Conversely, when the cohesive force of the liquid is larger than the adhesive force, the liquid repels the solid and is large (Figure 1b). Figure 2 illustrates differences in wettability of a silt soil. In Figure 2b a water droplet is resting on a soil surface that was treated to become water-repellent ( ¼ 70 ). In contrast, Figure 2a depicts a wettable soil surface. In general, the contact angle of water on clean glass, and presumably on most soil minerals, is small, and for mathematical convenience is often taken as ¼ 0. Curved Surfaces and Capillarity When the forces that spread the liquid (adhesion and spreading on solids, or gas pressure within a bubble) are in balance with surface tension that tends to minimize interfacial area, the resulting liquid gas interface is often curved. In porous media, the liquid gas interface shape reflects the need to form a particular contact angle with solid(s) on the one hand, and the tendency to minimize interfacial area within the pore. A pressure difference forms across the curved interface, where the pressure at the concave side of the interface is greater by an amount that is dependent on the radius of curvature and the surface tension of the fluid. For a hemispherical liquid gas interface having radius of curvature R, the pressure difference is given by the Young Laplace equation: P ¼ 2 ½4Š R where P ¼ P L P G when the interface curves into the gas (e.g., water droplet in air); or P ¼ P G P L when the interface curves into the liquid (e.g., air bubble in water, water in a small glass tube). In many instances a bubble may not be spherical, or an element of liquid may be confined by irregular solid surfaces, resulting in two or more different radii of curvature such as water held in pendular rings between two spherical solid particles (Figure 3). The Young Laplace equation for this case is given by: Figure 2 (a) Wettable silt surface ( 0 ); (b) treated water-repellent silt soil surface ( ¼ 70 ).(Reproduced from Bachmann J, Elliesb A, and Hartgea KH (2000) Development and application of a new sessile drop contact angle method to assess soil water repellency. Journal of Hydrology 231: )

3 CAPILLARITY 157 Figure 3 (a) Radii of curvature and shape of water held in pendular space between two spherical grains (note that for two equal spheres with radius a, the relationship between R 2 and R 1 is given as: R 2 ¼ R 2 1 [2(a R 1 )].(b) Water menisci held between three spherical glass beads at different capillary pressures. P ¼ 1 R 1 þ 1 R 2 ½5Š Note that this equation reduces to eqn [4] for spherical geometry with R 1 ¼ R 2, and the sign of R is negative for convex interfaces (R 2 < 0) and positive for concave interfaces (R 1 > 0). For an interface forming in a linear crevice or within a fracture, R 2!1, hence eqn [5] reduces to: P ¼ /R 1, where R 1 equals half the fracture aperture. The Capillary Rise Model When a cylindrical glass tube of small diameter (capillary) is dipped into free water, a meniscus forms in the tube owing to the contact angle between water and the tube walls, and minimum surface energy requirements. The smaller the tube radius, the larger the degree of curvature and the pressure difference across the air water interface (Figure 4). The pressure at the water side (P W ) is lower than atmospheric Figure 4 Capillary rise in cylindrical tubes with different radii.

4 158 CAPILLARITY pressure (P 0 ). This pressure difference causes water to rise into the capillary until the upward capillary force is balanced by the weight of the water column. In a cylindrical tube, the radius of meniscus curvature (R) is related to the tube radius r by R ¼ r/cos; consequently the equilibrium height of capillary rise in a cylindrical tube with contact angle is: h ¼ 2 cos w gr where g is the acceleration of gravity, and w is the liquid density. For water at 20 C in a glass capillary with ¼ 0, the capillary rise equation simplifies to: h(mm) ¼ 15/r(mm). Capillarity in Soils The complex geometry of soil pore space creates numerous combinations of interfaces, capillaries, and wedges in which water is retained, and results in a variety of air water and solid water configurations. Water is drawn into and/or held by these interstices in proportion to the resulting capillary forces. In Figure 5 Idealization of the soil pore space as cylindrical capillaries. ½6Š addition, water is adsorbed on to solid surfaces with considerable force at close distances. Due to practical limitations of present measurement methods, no distinction is made between the various mechanisms affecting water in porous matrices (i.e., capillarity and surface adsorption). Common conceptual models for water retention in porous media and matric potential rely on a simplified picture of soil pore space as a bundle of capillaries (See Water Retention and Characteristic Curve). The primary conceptual steps made in such models are illustrated in Figure 5. The representation of soil pores as equivalent cylindrical capillaries greatly simplifies modeling and parameterization of soil pore space and relies heavily on the capillary rise equation (eqn[6]). Capillarity in Angular Pores Cursory inspection of scanning electron micrographs of soils and other natural porous media (Figure 6) shows that pore spaces formed by aggregation of primary particles and mineral surfaces tend to be angular and slit-shaped, rarely resembling cylindrical tubes. Such observations and other shortcomings of the cylindrical capillary model have led to development of new models for capillarity in angular and slit-shaped pores. Capillarity in angular pores is quite different from the behavior in cylindrical pores with equivalent cross-sectional area. For example, when angular pores are drained, a fraction of the wetting phase (water) remains in the pore corners (Figure 7a). This aspect of dual occupancy of wetting and nonwetting phases, not possible in cylindrical tubes, more realistically represents liquid configurations and the mechanisms for maintaining hydraulic continuity in porous media. Liquid-filled corners and crevices play an important role in displacement rates of oil and in other transport processes in partially saturated porous media. For all (regular and irregular) polygons with n corners, the total water filled area (A wt )at a given matric potential is simply the sum of the Figure 6 (a) Thin section of Devonian sandstone, revealing angular pore space.(reproduced from Roberts JN and Schwartz LM (1985) Grain, consolidation and electrical conductivity in porous media. Physical Review B 31(9): ) (b) Scanning electron micrograph of calcium-saturated montmorillonite clay.

5 CAPILLARITY 159 Figure 7 (a) Dual-occupancy of wetting and nonwetting phases in triangular pores; (b) liquid vapor interfacial configuration in a triangular glass pore (2 mm). water-filled areas in each corner (Figure 7a). This sum is given by the simple equation: with FðÞ ¼ Xi n¼1 A wt ¼ rðþ 2 FðÞ ½7Š! 1 tan ð 180 iþ i where is the matric potential and F() is a shape factor dependent on pore angularity (corner angles i ) only. In contrast to a piston-like filling or emptying of circular capillaries, angular pores undergo different filling stages and spontaneous displacement in the transition from dry to wet or vice versa. Under relatively dry conditions (low chemical potentials) liquid accumulates in corners due to capillary forces. An increase in chemical potential leads to an increase in the capillary radius of interface curvature until the capillary corner menisci contact to form an inscribed circle. At this critical potential, liquid spontaneously fills up the central pore (pore snap-off). The radius of interface curvature at this critical point is equal to the radius of an inscribed circle in the pore cross-section. If an angular pore is drained, liquid is displaced from the central region first, leaving some liquid behind in corners. Subsequent decrease in chemical potential results in incrementally decreasing amounts of liquid in the corners. The critical potentials at spontaneous liquid displacement differ for imbibition and drainage. (See Water Retention and Characteristic Curve.) For completeness, one must also consider the role of liquid films due to adsorption to solid surfaces. (See Water Potential; Water Retention and Characteristic Curve.) Dynamic Aspects of Capillarity Dynamics of Capillary Rise The equilibrium height of fluid rise in a capillary (eqn [6]) does not contain any information regarding ½8Š the rate of rise and the associated time scale, which is often of significant importance in many industrial and natural processes. A simple force balance can be employed between a driving capillary force F : F ¼ 2 R cos and a retarding viscous force F (assuming Poiseuille flow): F ¼ 8x dx ½10Š dt to model the rate of capillary flow into a horizontal capillary. Inertial effects can be included, according to: m d2 x dt 2 ¼ F F ½11Š where m is the mass of the liquid in the capillary, x is distance, and t is time. Substitution of the forces (eqns [9] and [10]) into eqn [11] and integration (neglecting higher-order terms) yields the so-called Lucas Washburn Rideal (LWR) equation: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R cos x ¼ t ½12Š 2 which describes the rate of liquid penetration into a p horizontal capillary with the dependency of x on t. It is interesting to note that Washburn s neglect of inertial effects and Rideal s truncation of higher-order terms (r n, n > 2) in his series solution yield the same solution (eqn [12]). Exact solutions have been provided that fully account for inertial effects and expand the LWR expression to consider flows into horizontal grooves and other capillary shapes. Analytical solutions for dynamic capillary rise with gravity present a mathematical challenge. Several simplified analytical solutions for the rate of capillary rise in vertical capillaries have been proposed, such as the following implicit solution: gr 2 8 t ¼ zðtþ z e ln 1 zðtþ ½13Š z e ½9Š

6 160 CAPILLARITY Figure 8 (a) Comparison of measurements and theoretical models for capillary rise dynamics of silicon oil (PDMS 10) in glass capillary with r ¼ mm (calculated curve from eqn [14]; classic Washburn equation from eqn [13]); (b) inertia-induced oscillations during capillary rise of water in different glass capillary sizes (numerical simulations).note that inertial oscillations vanish for capillaries smaller than r ¼ mm according to eqn [15].(Adapted from Hamraoui A and Nylander T (2002) Analytical approach for the Lucas Washburn equation. Journal of Colloid Interface Science 250: ) The solution diverges as z(t) approaches the equilibrium capillary rise z e (eqn [6]). Another approximate solution has been proposed, based on the introduction of a retardation coefficient (): zðtþ ¼z e 1 exp cos t ½14Š z e The solution converges to the equilibrium capillary rise z e (eqn [6]) for long periods of time. Figure 8a depicts comparison of eqns [13] and [14] with capillary-rise measurements of silicon oil (PDMS 10) in a glass capillary, with r ¼ mm ( ¼ 20.1 mn m 1 ; ¼10 mpa; and ¼ kg m 3 ). The nondimensional retardation coefficient for water in glass capillaries ranges from ¼ 0.5 for large radii (r > r c ), representing friction dissipation due to contact line motion and contact angle adjustment, to ¼ 0.7 for small radii (r < r c ) representing primarily viscous dissipation. The critical radius r c is related to an interesting feature of capillary rise in the presence of gravity, namely inertia-induced oscillations in large capillaries, as depicted in Figure 8b. The inertial oscillations disappear in capillaries of radii smaller than r c : Dynamic Contact Angle r c ¼ 2 cosð Þ2 2 g 3 1=5 g ½15Š The contact angle formed between a flowing liquid front (advancing or receding) and a solid surface is not constant but reflects the interplay between capillary and viscous forces. The relative importance of these forces is often expressed by the so-called capillary number, Ca ¼ v/, with the liquid dynamic viscosity, and v the contact line velocity. The dependency of the dynamic contact angle D on the velocity of the contact line during complete wetting can be described by a nearly universal behavior according to the so-called Tanner law: 3 D ¼ ACa ½16Š where A is a constant (94 for D in radians). Eqn [16] fits the data of Hoffman for Ca < 0.1 and D < 130 (Figure 9). The complete range of Hoffman s data fitted to the empirical expression: ( " #) D ðradþ ¼cos 1 Ca 0: tanh 5:16 ½17Š 1 þ 1:31Ca 0:99 is depicted by a continuous line in Figure 9. For conditions of partial wetting ( S > 0), the relationships between contact angle and Ca are less universal. It has been postulated that at low Ca the apparent dynamic contact angle remains close to the static angle but rapidly deviates when Ca exceeds the value for S (Figure 9). This postulate is formalized by the following expression: 3 D 3 S ¼ ACa ½18Š Additional examples of advancing and receding contact angle dependency on capillary number are shown in Figure 10. Note that for receding contact angle

7 CAPILLARITY 161 there is a critical Ca above which the contact angle vanishes. The theoretical basis for the postulate in eqn [18] was first derived by Voinov, using hydrodynamic approximations near the moving contact line, resulting in: 3 D 3 S ¼ 9Ca lnðy=y mþ ½19Š where Y/Y m is a ratio of macroscopic length over which the contact angle is defined (mm) to molecular length where continuum theories fail (nm). Application of eqn [19] with Y/Y m ¼ 10 5 to the data of Hoffman is depicted in Figure 9. A key shortcoming of such hydrodynamic models for a dynamic contact angle is the lack of consideration of the effects and interactions with solid surface properties. Heterogeneous Surfaces and Microscale Hysteresis Contact Angle on Chemically Heterogeneous and Rough Surfaces Consider a chemically heterogeneous surface made up of patches of solids (or grains) with two different equilibrium contact angles a and b, and with the fraction of the area occupied by a solid given as f (Figure 11). The apparent equilibrium contact angle ( e ) for the composite surface is given by the semiempirical Cassie equation: cos e ¼ f cos a þð1 f Þcos b ½20Š An example of the Cassie law for contact angle of water on a sand surface with increasing amounts of hydrophobic grains is shown in Figure 12. The Cassie law (eqn [20]) is in remarkable agreement with experimental data for sand (Figure 12) and silt surfaces. An interesting extension of the Cassie law for porous surfaces (soil, fabric, etc.) predicts that the apparent contact angle ( e ) should be proportional to surface porosity (n): cos e ¼ð1 nþcos a n ½21Š Figure 9 Experimental results of Hoffmann fitted with eqn [17] (Hoffmann RL (1975) A study of advancing interface. Journal of Colloid Interface Science 50: ), and approximations given by eqns [16] and [19].Note that, for water flow in soils, the capillary number Ca rarely exceeds the range of values between 10 6 and 10 4 (for v ¼ 1mms 1,Ca¼ ).(Adapted from Kistler SF (1993) Hydrodynamics of wetting.in: Berg JC (ed.) Wettability, pp New York: Marcel Dekker.) The negative sign associated with porosity is due to the nonwetting properties of empty pores (i.e., air with cos air ¼ 1). These concepts of mixed wettability can be incorporated into the capillary rise model (eqn [6]) where capillary rise takes place in slits formed between two walls of different wettability. The same study applies Figure 10 Finite difference computation versus eqn [18] and parameters from Kistler (Kistler SF (1993) Hydrodynamics of wetting. In: Berg JC (ed.) Wettability, pp New York: Marcel Dekker.) for advancing (left) and receding (right) contact angle as a function of Ca.(Adapted from Hirasaki GJ and Yang SY (2002) Dynamic contact line with disjoining pressure, large capillary numbers, large angles, and prewetted, precursor, or entrained films.in: Mittal KL (ed.) Contact Angle, Wettability and Adhesion, vol.2, pp.1 30.)

8 162 CAPILLARITY the Cassie law to liquid retention in porous media and demonstrates these effects on the hydraulic properties of unsaturated porous media with varying surface wettability. In addition to surface chemical heterogeneity, the roughness of a surface is known to alter its wettability properties by increasing the wettable surface area per unit projected area, and by enabling a complex interplay between macroscopic contact angle and microscale geometry, leading to gas entrapment and a patchwork of microinterfaces underneath the wetting fluid. A spectacular demonstration of surface roughness-induced super hydrophobicity with a water drop resting on a fractal hydrophobic surface and forming a contact angle of about 170 is shown in Figure 13a. Such enhanced hydrophobicity is not only important for a variety of engineering and industrial treatments aimed at waterproofing of surfaces and fabrics, but it may also be important for explaining wettability properties of natural soil surfaces. Assuming that surface roughness only affects the solid liquid and solid vapor interfacial areas, minimization of surface free energy results in the socalled Wenzel equation: cos e ¼ rcos ½22Š where is the static contact angle for a smooth surface of similar chemical composition (see scheme in Figure 11b). The scope of surface influence is more complicated than predicted by simple expressions such as the Cassie and Wenzel equations. Other factors such as details of roughness geometry, interfacial pinning, and air trapping conspire to accentuate surface wetting properties as shown in Figure 13b. The scheme depicted in Figure 13b is based on experimental results showing the apparent contact angle on a rough surface plotted against the static contact angle on a smooth surface with similar chemical composition (to isolate the influence of surface roughness). Subsequent studies have shown a range of behaviors and asymmetry between the hydrophobic (cos<0) and hydrophilic (cos>0) sides of Figure 13b. It is interesting to note that certain roughness patterns Figure 11 Definition sketch for contact angle formation on (a) chemically heterogeneous surface and (b) rough surface with r ¼ A/ A 0, where A 0 is the projected area over a smooth surface.(reproduced from McHale G and Newton MI (2002) Frenkel s method and the dynamic wetting of heterogeneous planar surfaces. Colloids and Surfaces A 206: ) Figure 12 Application of the Cassie law to (a) experimental results of contact angle with sand surfaces containing different proportions of hydrophobic (treated) sand grains; and (b) an image of a water droplet on nonwetting sand forming a contact angle of 95.(Adapted from Bachmann J, Elliesb A, and Hartgea KH (2000) Development and application of a new sessile drop contact angle method to assess soil water repellency. Journal of Hydrology 231: )

9 CAPILLARITY 163 Figure 13 (a) Water drop (r ¼ 1 mm) resting on fractal rough surface with r ¼ 4.4 (eqn [22]); and (b) apparent contact angles as a function of surface microroughness for a range of surfaces with different wettability.(reproduced with permission from Onda T, Shibuichi S, Satoh N, and Tsujii K (1996) Super-water-repellent fractal surfaces. Langmuir 12: ) Figure 14 Two microscale mechanisms for hysteresis in capillary behavior: (a) differences between advancing and receding contact angle; and (b) the ink bottle effect depicting two different amounts of liquid retained in identical pores under the same matric potential. induce formation of air patches trapped underneath the liquid (similar to water drops resting on surfaces of some plant leaves). Hysteresis The amount of liquid retained in a porous medium is not uniquely defined by the value of the matric potential but is also dependent on the history of wetting and drying. This phenomenon, known as hysteresis, is closely related to various aspects of pore geometry, capillarity, and surface wettability. The macroscopic manifestation of hysteresis in soil water retention (or soil water characteristic) is rooted in several microscale mechanisms, including: (1) differences in liquid solid contact angles for advancing and receding water menisci (Figure 14a), which is accentuated during drainage and wetting at different rates; (2) the ink bottle effect resulting from nonuniformity in shape and sizes of interconnected pores, as illustrated in Figure 14b, whereby drainage of the irregular pores is governed by the smaller pore radius r, and wetting is dependent on the larger radius R. Additional effects stem from pore angularity; (3) differences in airentrapment mechanisms; and (4) swelling and shrinking of the soil under wetting and drying, respectively. From early observations to the present, the role of individual factors remains unclear, and hysteresis is a subject of ongoing research. See also: Water Potential; Water Retention and Characteristic Curve Further Reading Adamson AW (1990) Physical Chemistry of Surfaces, 5th edn. New York: John Wiley. Bachmann J, Elliesb A, and Hartgea KH (2000) Development and application of a new sessile drop contact angle method to assess soil water repellency. Journal of Hydrology 231: Bico J, Thiele U, and Quere D (2002) Wetting of textured surfaces. Colloids and Surfaces A 206: Blunt M and Scher H (1995) Pore-level modeling of wetting. Physical Review E 52(6): Dullien FAL, Lai FSY, and Macdonald IF (1986) Hydraulic continuity of residual wetting phase in porous media. Journal of Colloid Interface Science 109: Friedman SP (1999) Dynamic contact angle explanation of flow rate-dependent saturation-pressure relationships during transient liquid flow in unsaturated porous media. Journal of Adhesion Science Technology 13:

10 164 CARBON CYCLE IN SOILS/Dynamics and Management Haines WB (1930) Studies in the physical properties of soil. V. The hysteresis effect in capillary properties, and the modes of moisture distribution associated therewith. Journal of Agricultural Science 20: Hamraoui A and Nylander T (2002) Analytical approach for the Lucas Washburn equation. Journal of Colloid and Interface Science 250: Hirasaki GJ and Yang SY (2002) Dynamic contact line with disjoining pressure, large capillary numbers, large angles and pre-wetted, precursor, or entrained films. In: Mittal KL (ed.) Contact Angle, Wettability and Adhesion, vol. 2, pp Zeist, the Netherlands: VSP. Hoffman RL (1975) A study of advancing interface. Journal of Colloid Interface Science 50: Kistler SF (1993) Hydrodynamics of wetting. In: Berg JC (ed.) Wettability, pp New York: Marcel Dekker. Kool JB and Parker JC (1987) Development and evaluation of closed-form expressions for hysteretic soil hydraulic properties. Water Resources Research 23: Lucas R (1918) Ueber das Zeitgesetz des kapillaren Aufstiegs von Flussigkeiten. Kolloid Zeitschrift 23: Marmur A (1992) Wettability. In: Schrader ME and Loeb GI (eds) Modern Approaches to Wettability: Theory and Applications. New York: Plenum Press. McHale G and Newton MI (2002) Frenkel s method and the dynamic wetting of heterogeneous planar surfaces. Colloids and Surfaces A 206: Morrow NR and Xie X (1998) Surface energy and imbibition into triangular pores. In: van Genuchten MT, Leij FJ, and Wu L (eds) Proceedings of International Workshop on the Characterization and Measurement of the Hydraulic Properties of Unsaturated Porous Media. Riverside, CA: University of California Press. Nitao JJ and Bear J (1996) Potentials and their role in transport in porous media. Water Resources Research 32: Onda T, Shibuichi S, Satoh N, and Tsujii K (1996) Super-water-repellent fractal surfaces. Langmuir 12: Quere D, Raphael E, and Ollitrault J-Y (1999) Rebounds in a capillary tube. Langmuir 15: Rideal EK (1921) On the flow of liquids under capillary pressure. Philosophical Magazine 44: Rye RR, Mann JA Jr., and Yost FG (1996) The flow of liquids in surface grooves. Langmuir 12: Sciffer S (2000) A phenomenological model of dynamic contact angle. Chemical Engineering Science 55: Shibuichi S, Onda T, Satoh N, and Tsujii K (1996) Super water-repellent surfaces resulting from fractal structure. Journal of Physical Chemistry 100: Tuller M, Or D, and Dudley LM (1999) Adsorption and capillary condensation in porous media: liquid retention and interfacial configurations in angular pores. Water Resources Research 35(7): Ustohal P, Stauffer F, and Dracos T (1998) Measurement and modeling of hydraulic characteristics of unsaturated porous media with mixed wettability. Journal of Contaminant Hydrology 33: Voinov OV (1976) Hydrodynamics of wetting. Fluid Dynamics 11: 714. Washburn EW (1921) The dynamics of capillary flow. Physical Review 17: