Capillarity Revised01

8/11/2019 Capillarity Revised01

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CAPILLARITY 1

CAPILLARITY(Revised 09/13/2003)

Dani Or , Department of Civil and Environmental EngineeringUniversity of Connecticut, Storrs, Connecticut, USA

Markus Tuller , Department of Plant, Soil & Entomological SciencesUniversity of Idaho, Moscow, Idaho, USA

Introduction

The coexistence of gaseous, liquid, and solid phases in soil pores gives rise to a varietyof interfacial phenomena that, for example, lead to spreading of liquid droplets on solidsurfaces, liquid rising in capillaries and soil pores, or the entrapment of liquid increvices. These phenomena, partially attributed to capillarity, determine retention andmovement of water and solutes through soils. Hence they are of great importance in avariety of environmental and agricultural problems.

Liquid Properties

The phenomenon of capillarity in porous media results from two opposing forces, liquidadhesion to solid surfaces that tends to spread the liquid, and the cohesive surfacetension force of liquids that acts to reduce liquid-gas interfacial area. The resultingliquid-gas interface configuration under equilibrium reflects a balance between theseforces. The phenomenon of capillarity is thus dependent on solid and liquid interfacial

properties such as surface tension, contact angle, and solid surface roughness andgeometry.

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 cohesiveforces whereby hydrogen bonds are formed with neighboring molecules on all sides. Incontrast, molecules at the air-water interface experience net attraction into the liquidbecause of lower density of water molecules on the air side of the interface, with mosthydrogen bonds formed at the liquid side. The result is a membrane-like water surfacehaving a tendency to contract and reduce the amount of its excess surface energy(Hillel, 1998). 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 (itis also useful to express surface tension as force per unit length of interface). Differentliquids vary in their surface tension (Table 1).


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Table1: Liquid-vapor interfacial tensions for various liquids (Adamson, 1990).

LiquidTemp.

[oC]

SurfaceTension[mN/m]

LiquidTemp.

[o C]

SurfaceTension[mN/m]

Water 20 72.94 Butyric acid 20 26.51

25 72.13 Carbon tetrachloride 25 26.43

Organic Compounds Butyl acetate 20 25.09

Methylene iodide 20 67.00 Diethylene glycol 20 30.9

Glycerin 24 62.6 Nonane 20 22.85

Ethylene glycol 25 47.3 Methanol 20 22.50

Dimethyl sulfoxide 20 43.54 Ethanol 20 22.39

Propylene carbonate 20 41.1 Octane 20 21.62

1-Methyl naphthalene 20 38.7 Heptane 20 20.14

Dimethyl aniline 20 36.56 Ether 25 20.14

Benzene 20 28.88 Perfluoromethylcyclohexane 20 15.70

Toluene 20 28.52 Perfluoroheptane 20 13.19

Chloroform 25 26.67 Hydrogen sulfide 20 12.3

Propionic acid 20 26.69 Perfluoropentane 20 9.89

Surface tension also depends on temperature, usually decreasing linearly as thetemperature rises. Thermal expansion reduces the density of the liquid, and thereforealso reduces the cohesive forces at the surface of (as well as inside) the liquid phase.

Soluble substances can increase or decrease surface tension. If the affinity of the solutemolecules or ions to water molecules is greater than the affinity of the water moleculesto one another, then the solute tends to be drawn into the solution and to cause anincrease in the surface tension. This is the effect of electrolytic solutes. For example, a1% NaCI concentration increases the surface tension of an aqueous solution by 0.17mN/m at 20C. If, on the other hand, the cohesive attraction between water molecules isgreater than their attraction to the solute molecules, then the latter tend to be relegatedtoward 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 formedbetween the solid-liquid (S-L) interface and the liquid-gas (L-G) interface (see Fig. 1) is

referred to as the equilibrium (or static) contact angle ( ). Two equivalent approachesare commonly used to describe the equilibrium contact angle on smooth and chemicallyhomogeneous planar surfaces: (1) a force balance approach, and (2) an interfacial freeenergy minimization (McHale and Newton, 2002). The force balance formulationconsiders interfacial tensions ( ij) as forces per unit length; hence the force balance atthe contact line of a drop resting on a solid surface under equilibrium requires the vectorsum of the forces acting to spread the drop (outward) to be equal to opposing cohesionand viscous forces. Free energy minimization regards interfacial tension as energy per


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unit area, and calculates changes in surface free energy ( F) due to infinitesimaldisplacement ( A):

LGGSSL cos A)( AF += (1)

The result is identical whether considering the minimization of free energy withF/ A=0, or taking a balance of forces tangential to the solid surface; both cases yieldthe Young equation (Adamson, 1990):

0cos GSSLLG =+ (2)

with L,G, and S indicating liquid, gas, and solid, respectively, and ij the respectiveinterfacial surface tensions. The equilibrium contact angle is therefore:

LG

SLGScos

= (3)

Liquids that are attracted to solid surfaces (adhesion) more strongly than to other liquidmolecules (cohesion) exhibit a small contact angle, and the solid is said to be "wettable"by the liquid (Fig.1a). Conversely, when the cohesive force of the liquid is larger thanthe adhesive force, the liquid "repels" the solid and is large (Fig.1b).

Figure 1: Liquid-solid-gas contact angles, (a) hydrophilic surface ( 90 o) where liquid repels the surface.

Figure 2 illustrates differences in wettability of a silt soil (Bachmann et al., 2000). InFig.2b a water droplet is resting on a soil surface that was treated to become waterrepellent ( =70). n contrast, Fig.2a depicts a wettable soil surface. In general, thecontact angle of water on clean glass, and presumably on most soil minerals, is small,and for mathematical convenience is often taken as =0.


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Figure 2: (a) Wettable silt soil surface ( ~ 0 o). (b) Treated water-repellant silt soil surface ( = 70 o)(Bachmann et al., 2000).

Curved Surfaces and Capillarity: When the forces that spread the liquid (adhesion andspreading on solids, or gas pressure within a bubble) are in balance with surfacetension that tends to minimize interfacial area, the resulting liquid-gas interface is oftencurved. In porous media, the liquid-gas interface shape reflects the need to form aparticular contact angle with solid(s) on the one hand, and the tendency to minimizeinterfacial 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 isdependent on the radius of curvature and the surface tension of the fluid. For ahemispherical liquid-gas interface having radius of curvature R, the pressure differenceis given by the Young-Laplace equation:

R2

P

= (4)

where P=P liq-P gas when the interface curves into the gas (e.g., water droplet in air); orP=P gas -P liq when the interface curves into the liquid (e.g., air bubble in water, water ina small glass tube). In many instances a bubble may not be spherical, or an element ofliquid may be confined by irregular solid surfaces resulting in two or more different radiiof curvature such as water held in pendular rings between two spherical solid particles(Fig.3). The Young-Laplace equation for this case is given by:

+=

21 R1

R1

P (5)

Note that this equation reduces to Eq. 4 for spherical geometry with R 1=R 2, and the signof R is negative for convex interfaces (R 20).For an interface forming in a linear crevice or within a fracture, we can consider R 2 ,hence Eq. 5 reduces to: 1R/P = where R 1 = half the fracture aperture.


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Fig. 3: (a) Definition sketch of radii of curvature and shape of water held in pendular space between twospherical grains [note that for two equal spheres with radius a, the relationships between R 2 andR 1 is given as: R 2=R 1

2/2(a-R 1)]; (b) photographs of water menisci held between three sphericalglass beads at different capillary pressures.

The Capillary Rise Model

When a cylindrical glass tube of small diameter (a capillary) is dipped into free water, ameniscus forms in the tube due to the contact angle between water and the tube walls,and minimum surface energy requirements. The smaller the tube radius, the larger thedegree of curvature and the pressure difference across the air-water interface (Fig.4).The pressure at the water side (P W) is lower than atmospheric pressure (P 0). Thispressure difference causes water to rise into the capillary until the upward capillaryforce is balanced by the weight of the water column. In a cylindrical tube, the radius ofmeniscus curvature (R) is related to the tube radius r by r=R/cos , consequently theequilibrium height of capillary rise in a cylindrical tube with contact angle is:


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r gcos2

hw

= (6)

where g is the acceleration of gravity, and w is the liquid density. For water at 20 0C ina glass capillary with =00 the capillary rise equation simplifies to: h(mm)=15/r(mm).

Figure 4: Capillary rise in cylindrical tubes with different radii.

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-waterand solid-water configurations. Water is drawn into and/or held by these interstices inproportion to the resulting capillary forces. In addition, water is adsorbed onto solidsurfaces with considerable force at close distances. Due to practical limitations ofpresent measurement methods, no distinction is made between the variousmechanisms affecting water in porous matrices (i.e., capillarity and surface adsorption).Common conceptual models for water retention in porous media and matric potentialrely on a simplified picture of soil pore space as a bundle-of-capillaries (see article onRetention of Water in Soil and the Soil Water Characteristic Curve in thisEncyclopedia). The primary conceptual steps made in such models are illustrated inFig.5. The representation of soil pores as equivalent cylindrical capillaries greatlysimplifies modeling and parameterization of soil pore space and relies heavily (capillaryrise).


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Figure 5: Idealization of the soil pore space as cylindrical capillaries.

Capillarity in Angular Pores:

Cursory inspection of scanning electron micrographs of soils and other natural porousmedia (Fig.6) shows that pore spaces formed by aggregation of primary particles andmineral surfaces tend to be angular and slit-shaped, rarely resembling cylindrical tubes.Such observations and other shortcomings of the cylindrical capillary model have ledto development of new models for capillarity in angular and slit-shaped pores (Tuller etal., 1999).

Figure 6: (a) Thin section of Devonian Sandstone [Roberts and Schwartz, 1985] revealing angular

pore space. (b) Scanning electron micrograph (SEM) of calcium-saturated montmorilloniteclay.

Capillarity in angular pores is quite different from the behavior in cylindrical pores withequivalent cross-sectional area. For example, when angular pores are drained, afraction of the wetting phase (water) remains in the pore corners (Fig.7a). This aspect ofdual occupancy of wetting and non-wetting phases (Morrow and Xie, 1998), notpossible in cylindrical tubes, more realistically represents liquid configurations and themechanisms for maintaining hydraulic continuity in porous media (Dullien et al., 1986).


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Liquid-filled corners and crevices play an important role in displacement rates of oil(Blunt et al., 1995) and in other transport processes in partially saturated porous media.Tuller et al. (1999) have shown that for all (regular and irregular) polygons with ncorners, the total water filled area (A wt) at a given matric potential is simply the sum ofthe water-filled areas in each corner (Fig.7a). This sum is given by the simple equation:

( ) F)(r Aw2

t = (7)with

( ) ( )=

=i

1n

i

i 360180

2tan

1F

(8)

where is the matric potential and F( ) is a shape factor dependent on pore angularity(coner angles i) only (Tuller et al., 1999).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 dryto wet or vice versa. Under relatively dry conditions (low chemical potentials) liquidaccumulates in corners due to capillary forces. An increase in chemical potential leadsto an increase of the capillary radius of interface curvature until the capillary cornermenisci contact to form an inscribed circle. At this critical potential, liquid spontaneouslyfills up the central pore (pore snap-off). The radius of interface curvature at this criticalpoint is equal to the radius of an inscribed circle in the pore cross section (Tuller et al.,1999). If an angular pore is drained, liquid is displaced from the central region first,leaving some liquid behind in corners. Subsequent decrease in chemical potentialresults in incrementally decreasing amounts of liquid in the corners. The criticalpotentials at spontaneous liquid displacement differ for imbibition and drainage (seearticle on Retention of Water in Soil and the Soil Water Characteristic Curve in this

Encyclopedia).

Figure 7: (a) Conceptual sketch of dual occupancy of wetting and non-wetting phases in triangularpores. (b) An image of liquid-vapor interfacial configuration in a triangular glass pore (~2 mmsize).

For completeness, one must also consider the role of liquid films due to adsorption tosolid surfaces as described in article on Retention of Water in Soil and the Soil WaterCharacteristic Curve in this Encyclopedia, and reviews by Tuller et al. (1999) and Nitaoand Bear (1996).


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Dynamic Aspects of Capillarity

Dynamics of Capillary Rise:

The equilibrium height of fluid rise in a capillary (Eq.6) does not contain any informationregarding the rate of rise and the associated time scale, which is often of significantimportance in many industrial and natural processes. Lucas (1918) and Washburn(1921) employed a simple force balance between a driving capillary force F

)cos(R2F = (9)

and a retarding viscous force F (assuming Poiseuille flow):

dtdx

x8F = (10)

to model the rate of capillary flow into a horizontal capillary. Rideal (1921) includedinertial effects according to:

FFdt

xdm 2

2= (11)

where m is the mass of the liquid in the capillary, x is distance, and t is time.Substitution of the forces (Eqs. 9 and 10) into Eq.11, and integration (neglecting higherorder terms) yields the so-called Lucas-Washburn-Rideal (LWR) equation:

t2

)(CosRx

= (12)

that describes the rate of liquid penetration into a horizontal capillary with the

dependency of x on t . It is interesting to note that Washburns (1921) neglect ofinertial effects and Rideals (1921) truncation of higher order terms (r -n, n>2) in hisseries solution yield the same solution (Eq.12). Rye et al. (1996) provide exact solutionsthat fully account for inertial effects, and expand LWRs expression to consider flowsinto horizontal grooves and other capillary shapes.

Analytical solutions for dynamic capillary rise with gravity present a mathematicalchallenge. Several simplified analytical solutions for the rate of capillary rise in verticalcapillaries have been proposed such as the following implicit solution (Marmur, 1992):

=e

e

2

z

)t(z1lnz)t(zt

8

gR

(13)

The solution diverges as z(t) approaches the equilibrium capillary rise z e (Eq.6). Anotherapproximate solution was proposed by Hamraouni and Nylander (2002) based on theintroduction of a retardation coefficient ( ):


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=

tz

)cos(

eeexp1z)t(z

(14)

The solution converges to the equilibrium capillary rise z e (Eq.6) for long times. Figure.

8a depicts comparison of Eqs. (13) and (14) with capillary rise measurements of siliconoil (PDMS 10) in a glass capillary with r=0.315 mm ( =20.1 mN/m; =10 mPas, and =0.935 kg/m 3).Hamraouni and Nylander (2002) have shown that the nondimensional retardationcoefficient for water in glass capillaries ranges from =0.5 in large radii (r>r c)representing friction dissipation due to contact line motion and contact angleadjustment; to =0.7 for small radii (r


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Dynamic Contact Angle

The contact angle formed between a flowing liquid front (advancing or receding) and asolid surface is not constant but reflects the interplay between capillary and viscousforces. The relative importance of these forces is often expressed by the so-calledcapillary number vCa = , with the liquid dynamic viscosity, and v the contact linevelocity. The dependency of the dynamic contact angle D on the velocity of the contactline during complete wetting can be described by a nearly universal behavior accordingto the so-called Tanners law:

Ca A3D = (16)

where A is a constant (~94 for D in radians). Kistler (1993) has shown that Eq. (16) fitsthe data of Hoffman (1975) for Ca


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capillary number are less universal. Hoffman (1975) postulated that at low Ca theapparent dynamic contact angle remains close to the static angle but rapidly deviateswhen Ca exceeds the value for S (see Fig.9 above). This postulate is formalized by thefollowing expression:

Ca A3S3

D = (18)

Additional examples of advancing and receding contact angle dependency on capillarynumber are shown in Fig.10 (Hirasaki and Yang, 2002). Note that for receding contactangle, there is a critical Ca above which the contact angle vanishes.

Figure 10: Finite difference computation vs. Eq. (18) and parameters from Kislter (1996) for advancing(left) and receding (right) contact angle as a function of Ca (Modified from Hirasaki andYang, 2002}

The theoretical basis for the postulate in Eq. (18) was first derived by Voinov (1976)using hydrodynamic approximations near the moving contact line resulting in:

)Ym/Yln(Ca93S3

D = (19)

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 Eq.19with Y/Y m=10 5 to the data of Hoffman (1975) is depicted in Fig.9. A key shortcoming ofsuch hydrodynamic models for a dynamic contact angle is the lack of consideration ofthe effects and interactions with solid surface properties (Sciffer, 2000).

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 areaoccupied by a solid given as f. The apparent equilibrium contact angle ( e) for thecomposite surface is given by the semi-empirical Cassies equation (McHale and


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Newton, 2002):

bae cos)f 1(cosf cos += (20)

Figure 11: Definition sketch for contact angle formation on (a) chemically heterogeneous surface and(b) rough surface with r= A/ A0 where A 0 is the projected area over a smooth surface[McHale and Newton, 2002].

An example of Cassies law for contact angle of water on a sand surface with increasingamounts of hydrophobic grains is shown in Fig.12 (Bachmann et al., 2000). Cassieslaw (Eq. 20) was in remarkable agreement with the experimental data of Bachmann etal. (2000) for sand (Fig.12) and silt surfaces.

Figure 12: Application of Cassies law to (a) experimental results of contact angle with sand surfacescontaining different proportions of hydrophobic (treated) sand grains; and (b) an image of awater droplet on nonwetting sand forming a contact angle of 95 0 [Modified from Bachmann etal., 2000].

An interesting extension of Cassies law for porous surfaces (soil, fabric, etc) predictsthat the apparent contact angle ( e) should be proportional to surface porosity (n)according to (Adamson, 1990):


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ncos)n1(cos ae = (21)

The negative sign associated with porosity is due to the non-wetting properties of emptypores (i.e., air with cos air =-1).These concepts of mixed wettability can be incorporated into the capillary rise model

(Eq. 6 above) such as described by Ustohal et al. (1998) who derived expressions forcapillary rise in slits formed between two walls of different wettability. The same studyapplies Cassies law to liquid retention in porous media and demonstrated these effectson the hydraulic properties of unsaturated porous media with varying surface wettability.

In addition to surface chemical heterogeneity, the roughness of a surface is known toalter its wettability properties by increasing the wettable surface area per unit projectedarea, and by enabling a complex interplay between macroscopic contact angle andmicroscale geometry leading to gas entrapment and a patchwork of micro-interfacesunderneath the wetting fluid (Bico et al., 2002). Onda et al. (1996) and Shibuichi et al.(1996) provided a spectacular demonstration of surface roughness-induced superhydrophobicity with a water drop resting on a fractal hydrophobic surface and forming acontact angle of about 170 0 (Fig.13a). Such enhanced hydrophobicity is not onlyimportant for a variety of engineering and industrial treatments aimed at water-proofingof surfaces and fabrics, but it may also be important for explaining wettability propertiesof natural soil surfaces.

Assuming that surface roughness only affects the solid-liquid and solid-vapor interfacialareas, minimization of surface free energy results in the so-called Wenzels equation:

cosr cos e = (22)

where is the static contact angle for a smooth surface of similar chemical composition(see scheme in Fig. 11b).

The scope of surface influence is more complicated than predicted by simpleexpressions such as Cassies and Wenzels equations. Other factors such as details ofroughness geometry, interfacial pinning and air trapping conspire to accentuate surfacewetting properties as schematically shown in Figure 13b (Onda et al. 1996; and Bico etal. 2002). The scheme depicted in Fig.13b is based on experimental results showing theapparent contact angle on a rough surface plotted against the static contact angle on asmooth surface with similar chemical composition (to isolate the influence of surfaceroughness). Subsequent studies by Bico et al. (2002) have shown a range of behaviorsand asymmetry between the hydrophobic (cos 0) sides of

Fig. 13b. It is interesting to note that certain roughness patterns induce formation of airpatches trapped underneath the liquid (similar to water drops resting on surfaces ofsome plant leaves).


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Figure 13: (a) water drop (r=1 mm) resting on fractal rough surface with r=4.4 (Eq. 22); and (b) apparentcontact angles as a function of surface micro-roughness for a range of surfaces with different wettability[Onda et al, 1996]

Hysteresis

The amount of liquid retained in a porous medium is not uniquely defined by the valueof matric potential but is also dependent on the history of wetting and drying. Thisphenomenon, known as hysteresis, is closely related to various aspects of poregeometry, capillarity, and surface wettability. The macroscopic manifestation ofhysteresis in soil water retention (or soil water characteristic) as discussed in the article

on Retention of Water in Soil and the Soil Water Characteristic Curve in thisEncyclopedia is rooted in several microscale mechanisms including: (i) differences inliquid-solid contact angles for advancing and receding water menisci (Fig.14a, Hillel,1998), that is accentuated during drainage and wetting at different rates (Friedman,1999); (ii) the "ink bottle" effect resulting from nonuniformity in shape and sizes ofinterconnected pores as illustrated in Fig.14b, whereby drainage of the irregular pores isgoverned by the smaller pore radius r, and wetting is dependent on the larger radius R.

Additional effects stem from pore angularity discussed in the article on Retention ofWater in Soil and the Soil Water Characteristic Curve in this Encyclopedia; (iii)differences in air entrapment mechanisms; and (iv) swelling and shrinking of the soilunder wetting and drying, respectively. From the early observations of Haines (1930) to

the present (e.g., Kool and Parker, 1987), the role of individual factors remain unclear,and hysteresis is a subject of ongoing research.


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Figure 14: Two microscale mechanisms for hysteresis in capillary behavior: (a) differences betweenadvancing and receding contact angle; and (b) the ink bottle effect depicting two differentamounts of liquid retained in identical pores under the same matric potential (see text).

Further Reading

Adamson, A.W., 1990. Physical chemistry of surfaces , 5 th edition, John Wiley and Sons, NewYork.

Bico, J., U. Thiele and D. Quere, 2002. Wetting of textured surfaces. Colloids and Surfaces A206, 41-46.

Blunt, M., and H. Scher, 1995. Pore-level modeling of wetting, Physical Review E , 52(6), 6387-6403.

Bachmann, J., A. Elliesb, and K.H. Hartgea, 2000. Development and application of a newsessile drop contact angle method to assess soil water repellency. J. Hydrology 231 6675.

Dullien, F.A.L., F.S.Y. Lai, and I.F. Macdonald, 1986. Hydraulic continuity of residual wettingphase in porous media, J. Colloid Interface Sci., 109:201-218.

Friedman, S.P., 1999. Dynamic contact angle explanation of flow rate-dependent saturation-pressure relationships during transient liquid flow in unsaturated porous media, J.

Adhesion Sci. Technol. 13:1495-1518

Haines, W. B. 1930. Studies in the physical properties of soil. V. The hysteresis effect incapillary properties, and the modes of moisture distribution associated therewith, J.

Agric. Sci., 20:97-116.

Hamraoui, A., and T. Nylander, 2002, Analytical Approach for the LucasWashburn Equation. J.Colloid and Interface Sci ., 250, 415421.

Hillel, D., 1998. Environmental Soil Physics, Academic Press, San Diego.

Hirasaki, G.J., and and S.Y. Yang, 2002. Dynamic contact line with disjoining pressure, largecapillary numbers, large angles and pre-wetted, precursor, or entrained films. Contact

Angle, Wettability and Adhesion , Mittal, K.L. (ed.). Vol. 2, pp. 130

Hoffman, R. L., 1975, A study of advancing interface: J. Colloid Interface Sci ., 50:228-241.

Kistler, S. F., 1993. Hydrodynamics of wetting, in Wettability , J. C. Berg (Ed.), pp. 311-429,


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CAPILLARITY 17

Marcel Dekker, New York.

Kool, J. B., and J. C. Parker. 1987. Development and evaluation of closed-form expressions forhysteretic soil hydraulic properties, Water Resour. Res ., 23:105-114.

Lucas R., 1918. Ueber das Zeitgesetz des kapillaren Aufstiegs von Flussigkeiten. Kolloid Z.,23:15-22.

Marmur, A. 1992. Wettability, In: Modern Approaches to Wettability: Theory and Applications,Schrader M. E. and G. I. Loeb ( eds.), Plenum Press, New York.

McHale G., and M.I. Newton, 2002, Frenkels method and the dynamic wetting ofheterogeneous planar surfaces. Colloids and Surfaces A 206:193201

Morrow, N.R., and X. Xie, Surface energy and imbibition into triangular pores, 1998. In: M.Th.van Genuchten, F.J. Leij, and L. Wu (eds.), Proc. Int. Workshop on the C haracterizationand Measurement of the Hydraulic Properties of Unsaturated Porous Media , Universityof California, Riverside, CA.

Nitao, J.J., and J. Bear, 1996. Potentials and their role in transport in porous media, WaterResour. Res. , 32:225-250.

Onda T., Shibuichi S., Satoh N. and Tsujii K., 1996. Super-water-repellent fractal surfaces.Langmuir , 12:2125-2127.

Quere, D., E. Raphael, and J-Y. Ollitrault, 1999. Rebounds in a capillary tube. Langmuir ,15:3679-3682 .

Rideal, E. K. Phil. Mag . 1921, 44:1152.

Rye, R. R., J. A. Mann Jr., and F. G. Yost, 1996, The flow of liquids in surface grooves.Langmuir , 12:555-565.

Sciffer, S., 2000. A phenomenological model of dynamic contact angle. Chem. Eng. Sci.55:5933-5936.

Shibuichi,S., T. Onda, N. Satoh, and K. Tsujii, 1996. Super water-repellent surfaces resultingfrom fractal structure J. Phys. Chem . 100:1951219517.

Tuller, M., D. Or, and L.M. Dudley, 1999. Adsorption and capillary condensation in porousmedia: Liquid retention and interfacial configurations in angular pores, Water Resour.Res ., 35(7), 1949-1964.

Ustohal, P., F. Stauffer, and Th. Dracos, 1998. Measurement and modeling of hydrauliccharacteristics of unsaturated porous media with mixed wettability, J. Cont. Hydrology,33, 5-37.

Voinov, O. V., 1976. Hydrodynamics of wetting. Fluid Dyn . 11:714.

Washburn, E.W., 1921. The dynamics of capillary flow. Phys. Rev., 17, 273-283.

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Capillarity Revised01