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Understanding and explaining surface tension and capillarity:

an introduction to fundamental physics for water professionals

Marios Sophocleous

Abstract Understanding and explaining surface tensionand capillarity are not easy tasks. In this manuscript, anattempt is made to explain such phenomena for thegenerally educated professional. Following a conceptualexplanation of the molecular basis of surface tension, it isshown that the true force exerted by the solid walls of acapillary tube on the liquid causing capillary rise is thedifference in the surface tensions of the solid-liquid and

solid

vapor interfaces leading to free surface energyreduction on wetting the solid walls of the capillary bore,and not the pressure drop at the curved interface. It is alsoshown that the water in the capillary tube is suspendedfrom the meniscus and is under tension, rather than being

pushed up the capillary from below. The energy necessaryfor capillary rise, being derived from the diminution ofsurface energy that accompanies the movement of liquidin the capillary, emphasizes the important role of wettingin characterizing surface phenomena. A recommendedexplanation of capillary rise incorporates the reduction infree surface energy on wetting the inside walls of thecapillary bore, where the opposition offered to spreading

by the weight of the raised liquid is smaller (because ofthe narrowness of the bore) than that of the outside walls.

Keywords Capillary rise . Unsaturated zone . Wettingphenomena . Young's equation . Interfacial energy

Introduction

The subject of surface tension and capillarity is complexwith sophisticated mathematics, and is thus difficult for

both students and water professionals to understand. As a

result, most university physics/soil physics textbooks(such as Tipler 1991; Giancoli 1988; Miller and Schroeer1987; Snow and Shull 1986; Kane and Sterheim 1984;Sears et al. 1982; Jury and Horton 2004; Hillel 1998;Marshall and Holmes 1988; Bear1979; Kirkham 2005, toname a few among others) present elementary andsimplified explanations of such phenomena, which oftenlead to confusion and frustration, as also experienced by

this author. The conceptual diffi

culty with understandingand explaining the capillary-rise equation as presented inthese textbooks is the following: on the one hand, surfacetension, g, is portrayed as a net inward force at the air-water interface, and on the other hand, the surface tensionforce is portrayed as an upward force that counter-

balances the downward gravity force in deriving thecapillary-rise equation. Further confusion is encounteredwhen considering whether the risen water column is

pushed up or pulled up the capillary and what actually isthe driving force for capillary rise. As other generallyeducated professionals, and especially water-related pro-fessionals, may have encountered similar problems under-standing such issues, the author has attempted in this

paper to present a critical review and synthesis of selectedaspects of the multidisciplinary subject matter of surfacetension and capillarity in a comprehensive, logical, andmore easily understandable manner. The explanatoryapproach is purposely kept largely qualitative andmechanical, avoiding the formalism of thermodynamicsso as to reach a larger water-professional audience.

Why are surface phenomena, which may sound esoteric,of interest to water professionals? Surface phenomenacomprise an area that is generally neglected by water

professionals, as a quick look at the standard hydrogeologytextbooks (Freeze and Cherry 1979; Domenico andSchwartz 1998; Fetter 2001) can attest. However, in thisage of broadened horizons, nanotechnology, and interdisci-

plinary science, surface-science topics related to interfacialand capillary phenomena constitute an indispensable basic

background for understanding vadose hydrology, ecohy-drology, geomicrobiology, petroleum hydrogeology, multi-

phase transport through porous media, and other disciplinarythemes. Common conceptual models for water retention in

porous media and matrix potential rely on a simplifiedrepresentation of pore space as a bundle of capillaries resulting in the representation of soil and rock pores asequivalent cylindrical capillarieswhich greatly simplifies

Received: 19 May 2009 /Accepted: 30 November 2009Published online: 15 January 2010

Springer-Verlag 2009

M. Sophocleous ())Kansas Geological Survey,University of Kansas,1930 Constant Ave., Lawrence, KS 66047, USAe-mail: [email protected].: +1-785-8642113Fax: +1-785-8645317

Hydrogeology Journal (2010) 18: 811821 DOI 10.1007/s10040-009-0565-5


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modeling and parameterization of porous media pore space(Or and Tuller2005). It is hoped that this overview articlewill both educate and stimulate the interest of water

professionals, creating a better understanding and appreci-ation of the complexities of surface-science phenomena asrelated to their particular disciplines.

The molecular basis of

surface tension

The molecules in a fluid experience attractive andrepulsive forces in all directions due to the surroundingmolecules. Because the repulsive forces have a muchshorter range than the attractive forces, and are importantonly at extremely high external pressures (Hunter 2001),attractive forces generally predominate. The resultantforce on such molecules in the bulk of the fluid, averagedover a macroscopic time (a time which is much longerthan that spent in collisions between molecules), will bezero (Isenberg 1992). The interaction of a given moleculewith its (nearest) neighbors leads to a reduction of its

potential energy, i.e., intermolecular forces act to stabilizethe system (Pellicer et al. 1995). Such forces give rise tocohesion among the molecules of a liquid and toadsorption or adhesion between molecules of that liquidand any bounding solid surface. The molecules at thesurface region of the fluid have a smaller number ofnearest neighbors (as the density of the gaseous regionabove the surface is considerably smaller than that of the

bulk fluid), and therefore their potential energy is notdecreased as much as in the interior of the fluid.Molecules near the surface of the fluid experience aweaker force from the gaseous region above the surfacethan they would if the gaseous region was replaced byfl

uid. Thus, such molecules will experience, on average, aforce, normal to the surface, pulling them back into thebulk of the fluid. It is therefore necessary to do workagainst this force to take a molecule from the interior tothe surface, and consequently the surface molecules

possess greater energy than those inside the liquid (Brown1947). In other words, considering the potential energy ofwater molecules inside bulk water as zero, the moleculesat the interface have a positive potential energy. This ischaracterized by the surface tension, which is the interface

potential energy divided by the interface area (J/m2),which manifests itself as a force per unit length (N/m), thetwo units being the same (in SI units they correspond to

Nm/m2 and N/m, respectively). According to Hunter(2001), only a few attempts have been made to explainhow such an apparently unbalanced intermolecular forcenormal to the surface can be responsible for a stress

parallel to that surface, giving rise to surface tension.In accordance with the principle that every system

moves towards a state of minimum potential energy, iffree to do so, the surface of a liquid shows a tendency tocontract (Brown 1947). The smallest surface area for agiven mass of water is that of a sphere. This explains theformation of spherical droplets of rain, sprays, etc. Anyother shape would represent a larger surface area and thus

a higher energy content. For the same reason, themeniscus of water in a capillary generally assumes thespherical shape.

Hunter (2001) presents a qualitative outline of Orowans(1970) more mathematical argument for the existence of asurface tension as a consequence of the attractive (andrepulsive) forces between molecules, based on the balanceof forces on infinitesimal cubic elements offluid near the

surface. The forces arise from the pressure within the fluid,so an understanding of the molecular basis of pressure isneeded (Hunter 2001). It will be shown here how this

pressure is anisotropic near the surface, which leads to atension in the surface. In equilibrium, the pressure in a fluidis the time-averaged normal force per unit area exerted byall the molecules on one side of an imaginary test surfaceon all the molecules on the other side of the test surface.This pressure can be separated into two parts (Berry 1971):

1. A kinetic contribution, pk, due to the transport ofmomentum by molecules moving across the testsurface. This is the familiar pressure term in the kinetic

theory of a perfect gas. It is the same for a liquid and isalways positive (Hunter 2001).2. A cohesive contribution to the pressure due to the time

average of the net attractive and repulsive forcesbetween molecules on opposite sides of the imaginarytest surface in the body of the fluid. This secondcontribution is called the static pressure, ps, isnormally negative (i.e., the attractive force, conven-tionally given a negative sign, dominates) and is

particularly important for dense gases or liquids. Thetotal pressure, P pk ps

, is the sum of these two

contributions and is thus less than the kinetic pressure,and must be equal to the applied pressure, that is the

vapor pressure for a one-component system (Hunter2001).

Neglecting the repulsive forces when considering thestatic pressure (because the repulsive forces, as mentionedearlier, have a much shorter range than the attractiveforces and are important only at extremely high external

pressure), the intermolecular forces can be considered tohave a sphere of influence beyond which they arenegligiblean idea introduced by Laplace in 1806(Hunter 2001). The intermolecular forces extend onlyover a short range of the order of 10 (Angstrom) forsimple molecules (Jaycock and Parfitt 1981). For mole-cules further from the surface than the diameter of thissphere, the pressure must be isotropic because of thesymmetry of their surroundings.

Near the surface between the two bulk phases, thetangential and normal contributions to the static pressureare not the same because of the asymmetric distribution ofmolecules within the sphere of influence. When the sphereof influence cuts the surface (Fig. 1a), fewer and fewer

pairs of molecules attract across an imaginary test planenear a liquid surface as the center of the sphere approachesthe surface; hence, the magnitude of the static-pressurecontribution decreases for both the pressure across a plane

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parallel to the surface and for the pressure across a planenormal to the surface (Hunter 2001). It is apparent thatwhen the test plane is in the surface (Fig. 1b), the decreasein magnitude of the normal static pressure is greater thanthe decrease in magnitude of the tangential static pressure

because of the smaller number of interactions remaining inthe former case.

Now consider the consequences of the fact that theforces on opposite sides of an infinitesimally small cube offluid must be equal and opposite for mechanical equili-

brium to exist. Referring to Fig. 2, which is taken fromBerrys (1971) paper and also summarized in Temperleyand Travena (1978), in the bulk vapor (A) or bulk liquid(C), there is directional symmetry in the distribution of themolecules and so the pressure has the same value p0irrespective of the orientation of the test surface used todefine it. However, in the surface region (B), the tangential

pressure pt and the normal pressure pn will not be the

same because there is no longer symmetry of direction,there being depletion in the vertical z direction but not inthe horizontal x direction. However, because the fluid is inequilibrium, the forces on the opposite faces of the threesmall cubes shown must be equal and opposite. Thus,neglecting gravity changes between A and C, the normal

pressure pn has the same constant value p0 between A andC, right through the surface layer. On the other hand, thetangential pressure pt is equal to p0 in the bulk vapor at Aand in the bulk liquid at C, but is not equal to p0 in thesurface layer. The resultant net stress is the surfacetension. Thus energy is stored in the form of surfacetension as in a stretched spring. Berry (1971) concludes

that

the surface layer of liquid, in contrast with the bulk,must possess rigidity in order to resist the shear stress thatresults from pt differing from pn; this is the basis for thestatement appearing in older textbooks that liquids behaveas if their surfaces are covered by an elastic skin.

Objects do not remain in positions of high potentialenergy unless the force by virtue of which they possess thatenergy is balanced by some other equal and opposite force.Following Browns (1947) argument, suppose that thesurface has reached its minimum area and that theconcentration of molecules near the surface is the same asit is in the interior. This condition cannot persist becausethere is no force to balance the inward attraction experi-

enced by the surface molecules. Therefore, the surfacemolecules will continue to pass into the interior until thereduced concentration in the surface sets up a pressure

Fig. 1 a Schematic of the sphere of influence of radius of amolecule B below a plane and curved surface (XX and SSrespectively) with imaginary test planes parallel and normal to thesurface (shown as short bold lines at the center of the sphere ofinfluence). CD is the diameter of the sphere of influence of moleculeB and is parallel to the plane surface XX. CD is the region ofintersection of the sphere of influence of molecule B with the planesurface XX. A represents a molecule in the bulk of the liquid. bSphere of influence around imaginary planes in the liquid surfacewith test planes parallel and normal to the surface. Note theattraction between molecules in quadrants (I + II) and (III + IV) inthe diagram for normal forces (i) will be very small because there

are very few molecules in quadrants III and IV. Attraction betweenmolecules in quadrants (I + IV) and (II + III) in the diagram fortangential forces (ii) will be much greater as quadrants I and II aredensely populated. (a Adapted from Temperley and Travena 1978;and b from Hunter 2001)

Fig. 2 Schematic of equilibrium of elementary fluid cubes in liquid, vapor, and surface (adapted from Berry 1971)

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gradient opposing the inward attraction (Brown 1947).This explains why the number of molecules in the surface,and hence the surface area itself, tends to decrease. Ofcourse, this process of contraction is eventually opposed bythe repulsive forces between the molecules, which preventa complete inward collapse of the liquid (Temperley andTravena 1978). An example of this is the increasedhydrostatic pressure within an isolated liquid droplet that

opposes the contractile tendency of the surface (surfacetension) and thus provides the necessary restoring force. Asa result, when equilibrium is reached, there are fewermolecules per unit volume in the surface than in theinterior, resulting in a smaller liquid density, which means asmaller pressure. Thus, the surface layer is in a state oftension resulting from the decrease in pressure in thesurface layer compared with the condition of the interior(Brown 1947).

As Brown (1947) stresses, the fluid character of thesubstance is essential to this argument. No shearing forcesexist within the liquid itself to oppose the inward flow ofmolecules until the lack of molecules near the surface

establishes an opposing force. The lower molecularconcentration in the surface is an example of the generalprinciple that molecules distribute themselves with smallerconcentrations at places of higher potential energy (Brown1947). A good deal of evidence shows that, as might beexpected, the surface layer of a liquid is depleted. That is,the number density of the molecules is less than that in the

bulk liquid. Experiments using polarized light suggest thatthis decreased-number density is confined to a layerwhose thickness is just a few molecular diameters(Temperley and Travena 1978).

Pressure difference across an interfaceand contact angle

Following Temperley and Travena (1978), consider amolecule B, part of whose sphere of influence of radius alies above the surface, as shown in Fig. 1a under a plane(XX) and under a curved (SS) surface. The resultantforce on B due to its neighboring molecules will then bedirected inward () along the normal to the surface, asthere are a greater number of molecules in the hemisphere

below CD than in the volume CDDCC above CD; thenearer B is to the surface XX, the greater the inwardresultant force on B. As the distance y of a molecule

below XX increases from zero to a, this inward force onB changes from its maximum value to its minimum valueof zero and remains zero for all y>a, as in molecules likeA in Fig. 1a.

Now compare the downward force on molecule Bunder a curved liquid surface, SSconcave towards the

bulk of the liquid (Fig. 1a)with what it would be if theliquid surface were the plane surface XX (Temperley andTravena 1978). In the case of the curved surface, there is agreater volume of the sphere of influence above the liquidsurface (shown as a hatched area in the figure) and so,from the reasoning above, there is a greater downward

force on B in this case. Hence, it follows that the intrinsicpressure (due to the cohesive forces) at a distance y belowthe upper point of the curved surface is greater than that atthe same depth below a plane surface. Further, the

pressure at this point below a curved surface increases asy increases to y=a, and is constant for all y>a. Similarlywhen the liquid surface is convex towards the liquid, the

pressure below the surface is less than that below a plane

surface.Thus, when the liquid (L)vapor (V) interface is curvedrather than planar (flat), the resultant surface tension forcenormal to the liquidvapor interface creates a pressuredifference across the interface. The pressure is greater atthe concave side of the interface by an amount that isdependent on the radius of curvature and the surfacetension of the fluid. The pressure difference across theinterface is given by the Young-Laplace equation (Or andWraith 2002; Or and Tuller 2005)

$P gLV1

R1

1

R2

; 1

where gLV is the interfacial surface tension and the interfacecurvature is specified by the radii of curvature along twoorthogonal directions, say R1 and R2, and DP PL PVwhen the interface curves into the gas phase (water dropletin air) orDP PV PL when the interface curves into theliquid (air bubble in water, water in a small glass tube); PLand PV are the liquid and vapor pressures, respectively. Thesign of Ri is negative for convex interfaces (R2 < 0) and

positive for concave interfaces (R1 > 0). For a hemi-spherical liquidvapor interface having radius of curvature

R, R1 = R2 = R, and Eq. (1) reduces to

DP 2gLV

R: 2

A short proof of Eq. (2) may be adduced here(Bikerman 1970). Imagine a spherical bubble of radius Rimmersed in a liquid but still connected to a source of air.If more air is reversibly pumped into the bubble toincrease the radius to R + dR, the work done against the

pressure P inside the bubble is P4R2dR. At the sametime, the area of the bubble increases by 8RdR and thework performed against the surface forces is g8RdR. As

both expressions physically represent an identical amountof energy, they must be equal, i.e., P4R2dR = g8RdR.Hence, Eq. (2) is the result.

The existence of surface tension is a direct con-sequence of the facts that (1) intermolecular forces havea short range of action and (2) the concentration ofmatter in the gas phase, as a rule, is very much smallerthan in a liquid (Bikerman 1970). The asymmetric fieldof force across the vaporliquid interface also causes achange in the internal pressure across the interface, if itis curved.

When a liquid is brought into contact with a solidsurface, adhesion of the liquid with the solid and with the

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ambient air and cohesion of the liquid become interactingforces. The contact angle, (i.e. the angle between the

liquid surface and the solid, measured in the liquid, at thecontact linethe location where the liquid surface meetsthe solid surface) is often seen as a result of the balance ofthese forces. This is shown in Fig. 3.

Following the arguments given by Thomas Youngabout 200 years ago, the contact angle is the result of amechanical equilibrium of the forces acting on a unitlength of the contact line (Roura 2007). One obvious forceis the liquid surface tension gLV (Fig. 3). Its componentnormal to the surface, gLV sin , would be equilibrated bythe adhesion of the liquid molecules to the solid surface.On the other hand, to equilibrate the tangential compo-nent, gLV cos , a surface tension for the solidliquid and

solidvapor interfaces (gSL and gSV, respectively) must beintroduced. The equilibrium of these forces parallel to thesolid surface leads to the famous Youngs equation orYoung-Dupr equation (Fig. 3):

gLV cos gSV gSL; 3

which states that the contact angle depends exclusively onthe surface tensions and that it is independent of the

particular solid-liquid system (such as drops or capillaries)and gravity. Typical values of the interfacial tensions fromBailey and Kay (1967) as presented in Jaycock and Parfitt(1981) are reproduced in Table 1 to give an idea of their

numerical orders of magnitude.

Equation (3) assumes that (1) the solid phase is an idealone, hence it should be smooth, flat, rigid, chemicallyhomogeneous, insoluble, and non-reactive; (2) the liquid

phase is pure, hence no mixture of liquids can be used; and(3) vapor adsorption to the solid phase is negligible (Gilboaet al. 2006; Kwok and Neumann 1999). Thus, Eq. (3) hasonly limited practical significance in soil-water studies

because soils are rough, heterogeneous, anisotropic, elastic,

deformable, and may react with a wettingfluid (Bachmann etal. 2003); however, Youngs Eq. (3) is the starting point forthe investigation of soil surface properties.

The surface tension of a solid surface can besignificantly reduced as a result of the adsorption of thevapors of the wetting fluid onto the solid surface, andwhen the solid surface is in equilibrium with the liquidvapor, such reduction is termed the equilibrium spreading

pressure (orfilm pressure), e. Thus, Youngs equationwas modified as:

gLVcos gS gSL pe

where pe gS gSV , and gS is the surface tension of thesolid in vacuum (Gilboa et al. 2006). However, if thecontact angle is greater than zero, the spreading pressuremay be regarded as negligible for low-energy solidssolids that resist wetting (Good 1993); that is, vaporadsorption is considered to be negligible, and hence +SV =+S. In many studies, e is ignored or kept at low andconstant levels by conducting measurements of contactangle on air-dry soils (Gilboa et al. 2006).

Generally, it is extremely difficult to verify Youngsequation because the values of gSV and gSL are verydifficult to measure, especially for irregularly shaped solid

particles such as soil particles. A conceptual difficulty

with the mechanical justifi

cation of Young

s equation ishow a surface tension such as +SV can develop on a flatsolid surface that is undeformable, and how it can act on acontact line that is allowed to move freely above it (Roura2007). Orowan (1970) pointed out that the difference ofsurface tensions on the right-hand side of Eq. (3) should,strictly speaking, be interpreted as a difference in surfaceenergies (more properly free surface energies in order todenote the mechanical energies stored per unit area ofsurface and available to perform mechanical work), whichcorrespond to the tangential stresses exerted by the solidon the wedge of liquid adjacent to it.

Capillary rise

Adhesion and cohesion forces cause water to rise in glasstubes and in soil pores. Adhesion, attaching watermolecules to the solid surface, and cohesion, which joinsthe water molecules together, combine to allow the waterto rise. The adhesion causes the rise in a capillary, and thecohesion makes all the water molecules follow the upward

pull (Ehlers and Goss 2003). The concave curvatureindicates the presence of a pressure below the watersurface being smaller than the surrounding normal,

Fig. 3 A liquid drop schematic in a a wetting situation withcontact angle < 90, and b a non-wetting situation with > 90 ona planar solid surface. g is the interfacial tension between the phasesindicated by the subscripts. Thus, subscripts SV, SL, and LV standfor solidvapor, solidliquid, and liquidvapor, respectively

Table 1 Parameters in the Young equation in milliNewtons/m atroom temperature. The solid (S) employed is mica, and the subscriptV represents the vapor of the specified liquid (L) (adapted fromJaycock and Parfitt 1981)

Liquid gSV gSL gLV

Water 182.8 107.3 72.8Hexane 271.0 255.0 18.4

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atmospheric pressure. Therefore, the concave shapesignifies a sub-atmospheric or negative pressure in thewater in the capillary tube. Is the water column in thecapillary tube the result of this sub-atmospheric pressureor is the very capillary water column the cause of this sub-atmospheric pressure? In other words, what is the trueforce causing the liquid column to rise in the capillary?This question is investigated in the following.

According to McCaughan (1987), Tabor (1991) andothers, the proper way to treat the rise of water in thecapillary tube is in terms of the pressure difference acrossthe meniscus. Following McCaughan (1987) and ignoringair and vapor pressure, PA = PC = 0, and by Pascals

principle, PC = PD = 0 (Fig. 4a), where PA is the pressureat point A, just below the meniscus in the capillary, and

PC and PD are the pressures at the liquid reservoir leveloutside and inside the capillary, respectively. Now PB PB and a pressure gradient exists producing a forceacting vertically upwards over the entire bulk of the liquid

column in opposition to the weight of the straight-wallcolumn. In quantitative terms, by Newtons second law:

PD PB gh pr2 0;

where is the liquid density, g is the acceleration ofgravity, r is the capillary radius, and h is the height of theliquid drawn up the capillary tube.

From Eq. (2) and the above reasoning, PA PB 2gLV

R PD PB. On substitution

2gLV

R gh

pr2 0 is

obtained. From Fig. 4 it can be seen that R cos = r, orR = r/cos . Then, substituting that result

h 2g

LVcos

gr : 4

This treatment has ignored the liquid that stands higherthan point B in the curved portion of the meniscus

(Fig. 4a). Bikerman (1970) provides a more precisetreatment that incorporates the curved portion of themeniscus that stands higher than point B.

The same result can be obtained for a conical capillary(Fig. 4b) in agreement with experimental observation (Tabor1991). It should be clear, however, that in the case of aconical or other shaped capillary, the liquid will not rise on itsown accord to heighth because it will not be able to traverse

the wider parts of the tube. The situation in the conicalcapillary mentioned above is stable if the liquid is sucked upto height h and then the suction is removed. Thus, it is notnecessary for the capillary to have a small radius throughoutits length. Only the capillary radius at the meniscus isrelevant (Fig. 4) in determining the height of the column thatcan be supported in a capillary tube (Hunter2001).

In a further comment on capillary rise, McCaughan(1992) provided the following explanation of that phe-nomenon. At first, with the curved surface at the bottomof the capillary tube (the lower end of which is placed

just below the free surface of a large reservoir of liquid),the gradient extends outside the bottom of the tube into

the liquid reservoir because the lower pressure beneath thecurved surface is at the same level as the flat surface of theliquid in the reservoir, which is at the same pressure asthat above the curved surface. This external gradientdrives liquid into the tube and pushes the columnupward. Even if one were to suppose that the walls were

pulling up the liquid column through a perimeter forcethat results from the upward pull by the glass around theupper edge of the column of liquid (i.e., the verticalcomponent of the adhesive force exerted by the glass), onethen would have to explain how the liquid column issliding over the same walls where their effect is to opposethe motion. According to McCaughan (1992), sliding

upwards must mean a push upward. In support of thisidea, Peiris and Tannakone (1980) show that capillary risemoves according to the laws of viscous flow, andPoiseuilles equation also assumes a pressure gradient asthe driving force.

Fig. 4 Schematic of the rise of liquid a in a straight and b in a conical capillary tube of radius r. A, B are points just above and just belowthe central surface of the meniscus respectively. C, D are points at the liquid reservoir level outside and inside the capillary, respectively. his the height of rise of liquid in the capillary above the reservoir level, is the contact angle, and R is the radius of curvature of the meniscusin the capillary

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Thus, in accordance with the arguments presented in thepreceding, the rise of a liquid in a capillary occurs becausethe pressure under the convex side of the meniscus is lessthan that along the opposite side of the interface. Bikerman(1970) poses the question, Why does the higher pressureabove the liquid in the capillary tube not push the liquiddown? This certainly would have happened if the liquidvapor interface were analogous to a piston.

According to Bikerman (1970) two satisfactory answersare available. One of them is based on the analogy betweenmolecular and gravitational forces. Because of the gravita-tional forces, the deeper layers of our planet Earth aresubjected to a high pressure, but this pressure has notendency to push the Earthatmosphere interface towardthe atmosphere. The other uses the analogy between surfacetension and the tension of an inflated rubber balloon. Theexcess pressure inside the balloon is not felt outside it anddoes not push the walls out because it is exactlycompensated by the stresses in the rubber envelope. Thusa drop, of radius R, has no tendency to swell although the

pressure in it is greater by 2g/R than the outside pressure; an

increase in volume would entail an increase in the area,which is energetically unfavorable (Bikerman 1970).Capillary rise is sometimes explained in terms of the

surface forces around the periphery of the meniscus(Tabor1991; Fig. 5c). The meniscus has peripheral length2r, and for contact angle , the upward force is2rgLV cos . This is balanced for straight-wall capillaries

by the downward weight of the liquid column, mg =Vg= hr2g, where m is the mass of the fluid column, Vis its volume, and all other symbols are defined previouslyand/or in the last section (see Appendix). Thus, h g 2gLV cos

r, a result already arrived at in Eq. (4). The

(mechanical) interpretation given previously, and also in

many textbooks that consider the topic of capillary rise(Tipler 1991; Giancoli 1988; Miller and Schroeer 1987;Snow and Shull 1986; Kane and Sterheim 1984; Sears etal. 1982; Jury and Horton 2004; Hillel 1998; Marshall andHolmes 1988; Bear 1979; Kirkham 2005; and others), isthat only gLV is invoked and the equilibrium is formulatedas a balance between the weight of the liquid column anda force related to gLV (Pellicer et al. 1995). Most authorsthen introduce the force exerted by the solid walls on theliquid column as the reaction of the force exerted by

the liquidvapor interface on the line of contact (Fig. 5c).The vertical component of this reaction force is then saidto cancel the weight of the liquid column (Pellicer et al.1995), and the equation of capillary rise is written as

2prgLV cos g pr2h;

from which Eq. (4) results. Although Eq. (4) is correct,

Tabor (1991) points out that this is largely fortuitous. Forexample, if the capillary was conical in shape (Fig. 4b),the capillary rise would still be given by Eq. (4), where ris the capillary radius at the point where the meniscusformed. Clearly, 2rgLV cos could be considerably lessthan the weight of the conical column of liquid. Addi-tional weak points are present in the derivation of Eq. (4),which will be outlined following the presentation of whatPellicer et al. (1995) consider as the proper mechanicalinterpretation of capillary rise.

According to Pellicer et al. (1995), the mechanicalinterpretation of the capillary-rise equations can be done

properly by considering that the forces involved are the

surface tensions of the different interfaces, which act on theline of contact, and the weight of the liquid column(Figs. 5a, b). Two conditions of mechanical equilibriumare needed: that of the liquid column and that of the line oftriple contact. For example, the force exerted by the liquidvapor interface on the line of contact is 2rgLV and actsalong this interface, i.e. forming an angle with the solidwalls. The equation for capillary rise is then obtained intwo steps (Pellicer et al. 1995). First, consider the liquidcolumn (Fig. 5b) and note that the force exerted by the solidtube on the liquid column, (gSVgSL) 2r, must compen-sate for the weight of the liquid column mg, giving Eq. (5):

gSV gSL 2pr mg: 5(When gSL < gSV there is capillary rise; when gSL > gSVthere is capillary depression). Expressing the fluid mass mof density and heighth in the capillary tube of radius rasm = r2h, Eq. (5) can be expressed as

gSV gSL 2pr pr2hg; thus

2 gSV gSL

gr h:

6

It is thus evident that the surface free energy differenceDgs gSV gSL is the driving force causing the liquidcolumn to rise ifDs

g

> 0, that is, if it is favorable for theliquid to wet the solid wall (Henriksson and Eriksson2004). At equilibrium, this surface free energy differenceequals a mechanical tension, the adhesion tension, actingon the three-phase contact line.

It is noteworthy that the resulting equilibrium condition(Eq. 6) does not involve the curvature of the meniscus orthe surface tension of the liquid, gLV. In other words,following this route of capillary-rise interpretation, thequestion as to why and how the meniscus is curved hasnot been addressed (Henriksson and Eriksson 2004).Thus, treating the subject matter along this route shows

Fig. 5 Schematic of the balance of forces in capillary rise: aequilibrium of the line of triple contact (Eq. 3); b equilibrium of theliquid column (Eq. 5); and c the situation usually depicted inphysics textbooks (Eq. 4) (Adapted from Pellicer et al. 1995)

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that the capillary rise is not caused by the pressure drop atthe curved interface, as indicated previously.

To introduce the contact angle and the surface tensionof the liquid, the second step in Pellicers et al. (1995)analysis is followed (the first step, as stated previously,

being the equilibrium of the liquid column), whichinvolves considering the equilibrium of the line of triplecontact (which, in turn, involves changes in the contact

angle but not in the mass/volume of the liquid in thecolumn), and Youngs equation (Eq. 3) is obtained.Introducing Eq. (3) into Eq. (6) gives

2prgLV cos gpr2h 7

or

2gLV cos

gr

2 gSV gSL

gr h; 8

assuming that the volume of the spherical cap at the top

of the liquid column is negligible. Thus, the left-handside of Eq. (8) indicates that the effects of the surfaceenergies of the SV and SL interfaces are hidden in thecontact angle .

According to Pellicer et al. (1995), there are severalweak points in the derivation of Eqs. (4) or (7). First,nothing is said about the physical magnitudes determin-ing the value of the contact angle. Second, the studentis left with the idea that capillary rise is directly relatedto gLV, while Fig. 5b and Eqs. (5) and (6) state clearlythat the force exerted by the solid walls on the liquidcolumn is related to the difference in the surfacetensions of the liquidsolid and solidvapor interfaces.The direction of the force in Fig. 5c is also misleading

because the solid walls can only exert forces in thedirection of the solid-liquid interface and cannot forman angle with it (Pellicer et al. 1995). The surfacetension, gLV, considered as a force pulling the liquidcolumn up along a capillary wall is a misleadingconcept. Wetting does not appear in the derivation ofEq. (4), even though the wetting of the capillary wall

plays an important role in capillary rise. This isevidenced by the fact that when the wall is wetted with aliquid, the liquid rises up in the capillary tube, and when thewall is not wetted by the liquid, the liquid is depressed in thecapillary tube, and also by the common observation thatthe capillary rise is complete and reproducible only if theliquid first wets the walls above its surface (McCaughan1992). Finally, the mechanical interpretation does notexplain where the energy necessary for capillary rise comesfrom, which is further expanded on in the next section.

Is the liquid then pushed up in a capillary tube from thebottom, as McCaughan (1987, 1992) and others contend,or is it drawn up at the top by a climbing meniscus? It isdefinitely known that the liquid pulls on the tube andtherefore the tube must pull on the liquid. Chapin (1959)has shown that the capillary rise h (Eqs. 4 or7) is obtainedeven when the hydrostatic pressure is far too low (i.e.,

much less than gh) to account for it, and he advocatesabandoning the notion that the capillary rise can beexplained by hydrostatic pressure (i.e., by equating thehydrostatic pressure of a column of liquid of depth h tothe reduction in pressure at the top of the column due tothe curvature of the meniscus). The analyses shown above

based on Pellicer et al. (1995), Henriksson and Eriksson(2004), and others, show that the wall of the capillary tube

does exert an upward force on the liquid through thesurface free energy difference (gSVgSL). According toDingman (1984), the water in the capillary tube issuspended from the meniscus, which in turn is attachedto the walls by hydrogen bonds. Thus, the water is undertension, which is defined as negative pressure. Denny(1993) points out that because the water in the tube is, inessence, hanging from the airwater interface, it is at alower pressure than that of the surrounding air. As a result,if one were to poke a hole in the side of the tube, airwould be drawn in rather than water being forced out.This is in contrast to a column of water that has beenforced up from below, as argued by McCaughan (1987,

1992).

Source of the energy for capillary rise

Consider a tube of radius r. The liquid rises to a height hgiven by Eq. (4). The energy required to establishcapillary rise is derived from the diminution of surfaceenergy which accompanies the movement of the liquid inthe capillary (Brown 1941). It follows therefore that theequation relating capillary rise with the surface tensionand the radius of the tube (Eq. 4) may be established byequating the loss of free surface energy to the work doneduring the motion of the liquid. The potential energy ofthe raised column, Eg, gained by the liquid is equivalentto the raising of a mass of liquid, m = r2h through aheight, hcm 1=2h, where hcm represents the center ofmass of the liquid column with respect to the free surfacein the container. Hence,

DEg mghcm 1=2pr2h2g: 9

This energy comes from the wetting of the walls of thetube by the liquid (Tabor 1991).

Consider a liquid with contact angle . The equilibriumcondition from Eq. (3) is

gSV gLV cos gSL: 10

Following Tabor (1991), if the liquid advances alongthe tube so as to cover 1 m2 of the surface, 1 m2 ofgSV isdestroyed and 1 m2 ofgSL gained. The area of the liquidvapor meniscus has no change for a constant capillarycross section. The energy given up by the system or theloss of surface energy that occurs when an area 2rh ofthe wall of the tube is wetted is 2rh (gSVgSL) or, byYoungs equation, 2rhgLVcos . Thus, each m

2 of

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capillary surface wetted releases energy gLVcos. Theenergy, E, released for a rise of h is then

DEg gLV cos 2prh: 11

On the outside surface of the capillary, there is noinherent physical possibility to reduce the airwaterinterfacial area because further spreading up the outside

of the tube is prevented by the weight of the raised liquid.However, inside the capillary, the interfacial area would bereduced if the capillary would fill with water (Koorevaaret al. 1983) because the opposition offered to spreading bythe weight of the raised liquid is smaller as a result of thenarrowness of the capillary bore. The reduction in air(vapor)water interfacial energy can be used to drawwater up into the capillary against gravity. Above and

below the water level in the capillary, the walls of thecapillary are covered with a water film (Koorevaar et al.1983). Thus, after the initial wetting of the capillary walls,the energy of the solidwater interfaces remains the sameindependent of the height of liquid rise inside the

capillary. Thus, water will continue to enter the capillaryuntil the reduction in the airwater interfacial energy

brought about by an additional incremental volume ofwater has become equal to the increase in gravitationalenergy of that incremental volume in the capillary(Koorevaar et al. 1983).

Substituting from Eq. (4) above forgLV cos , for theloss of surface energy, Eg:

DEg pr2h2rg: 12

This is exactly twice the gain in potential energy givenpreviously by Eq. (9). This implies that if the liquid werenon-viscous, it would rise to a height 2h and oscillate

between 0 and 2h with its mean position at h (Tabor1991). In practice viscosity consumes the excess energyvery quickly. As Tabor (1991) explains, the behavior islike that of a vertical spiral spring onto the end of which aweight is suddenly attached. The weight falls instanta-neously to double its equilibrium and oscillates betweenthis and zero until the kinetic energy is consumed byfriction and static equilibrium is achieved.

Alternative derivation of the source of energyfor capillary rise

Instead of considering that the height h of the liquidcolumn inside a thin capillary is the result of a mechanicalequilibrium of the net upward force of the surface tensionof the free liquid surfacethat equals the weight of theliquid column, (Eq. 7)its value can be determined byimposing a minimum total energy of the system (Mark-worth 1971). The energy consists of the total surfaceenergies of the interfaces (LV, SL, and SV) and the

potential energy of the liquid inside the tube.

The starting point is the relation

DE DEg DEg; 13

where E is the change in total energy of the raised liquidcolumn, Eg is the work done against gravity to constructthe liquid column, and Eg is the corresponding changein interfacial energy which arises from the fact that when

the liquid column is formed within the tube of radius r, aninterface characterized by gSV is changed to gSL.

DEg

ZV

rgzdV

Zh

0

rgzpr2dz rgpr2h2

2

1

2prgr2h2; 14

where V = volume offluid in the capillary;

DEg gSL gSV 2prh: 15

This last value can be easily understood if it is realizedthat the wetted surface area has increased by 2rh at theexpense of an equal amount of dry surface area.

Using the familiar Young equation (Eq. 3): gSV gSL gLV cos; thus gSL gSV gLVcos, and thus,DEg gLV cos2prh. Therefore,

DE DEg DEg 1

2pgr2h2 2prhgLV cos : 16

The equilibrium height of the liquid column correspondsto that value of h for which E has its minimum

value, i.e.@DE

@h 0; thus@

@h1

2 pgr2

h2

2prhgLVcos

0; thus, pgr2h 2prgLVcos 0; which results inEq. (4).

From Eqs. (16) and (4) it can be shown that theminimum value of E can be expressed as DE min 1

2pgr2 2

gLVcosgr

2

2pr 2gLVcosgr

gLVcos

1

2pgr2 2gLVcos

gr

2

pgr2 2gLVcosgr

2

;which results in

DE min 2pg2LVcos

2

g; 17

with Eg = 2Eg at this particular value of E. Asexpected, (E)min is negative, regardless of whether the

liquid column is elevated or depressed and, in addition, isindependent ofr. As Markworth (1971) also points out, thisapproach lends considerable insight into the details of the

physical changes that occur as the liquid column is formed.

Concluding comments

It should have become evident by now that equating theforce of surface tension gLV around the circumference ofthe meniscus in a capillary tube to the weight of the

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column of liquid is misleading and unnecessary asphysical principles are obscured. Based on the expositionof surface phenomena outlined in this paper, the followingqualitative explanation of capillary rise is recommended.When a small diameter, capillary tube is first immersed ina beaker with a liquid that has a tendency to adhere to thewalls of the capillary, molecules of vapor, escaping fromthe liquid, adsorb to the solid surface of the capillary tube,

forming an extremely thin capillary layer just above theliquidvapor interface. The mass of the liquid thenattempts to cohere to this layer and a raised meniscusforms, both in the inside and outside walls of the capillarytube, creating a new larger interfacial area. This largerliquidvapor interfacial area of the adsorbed layer filmrepresents relatively high energy that the system wouldattempt to minimize by moving spontaneously to a state oflower potential energy if the liquidvapor interface areacould be reduced. On the outside surface of the capillary,there is no inherent physical possibility to reduce theliquidvapor interfacial area as further spreading up theoutside of the tube is prevented by the weight of the raised

liquid. However, inside the tube, the opposition offered tospreading by the weight of the raised liquid is smaller onaccount of the narrowness of the bore. Consequently, for asmall-enough bore diameter of the capillary tube, aconsiderable elevation occurs, and the rise ceases whenthe weight of the raised column is equal to the forcecausing spreading. (The volume of liquid lifted up abovethe liquid reservoir level outside the capillary tube is thesame as that inside the capillary tube but in the confined

bore space of the tube that liquid volume is lifted higher.)Furthermore, inside the capillary, the interfacial areawould be reduced if the capillary would fill with theliquid as gSL < gSV, as the liquid tends to wet the capillary

tube, and the energy of the system is decreased byincreasing the solidliquid interface, i.e., by producing acapillary rise effect. In short, although the capillary riseequation (Eq. 4) was derived directly from the mechanicalequilibrium condition that must exist across any curvedinterface, the reason why the liquid rises in the narrowcapillary tube against gravity is not because of thatequilibrium condition but instead because the interfacialenergy of the solidliquid interface (gSL) is appreciably lowerthan that of the solidvapor interface (gSV). This leads to afree surface energy reduction on wetting the solid walls of thecapillary tube that is balanced by the gain in gravitational

potential energy represented by the capillary rise.Despite the fact that conceptual difficulties still remain

such as questions related to solidvapor and solidliquidinterfaces of undeformable solid surfaces in Youngsequation, briefly mentioned earlier in this paper (see textfollowing Table 1)this exposition is a current attempt tostress the importance of wetting in capillary rise and toclarify some of the points which arise from the failureto understand and explain the fundamentals.

Acknowledgements Detailed review comments by one eponymousreviewer (Arye Gilboa) and three anonymous reviewers were helpful inrevising the manuscript. Previous discussions on aspects of surface

tension and capillarity with Lajpat Ahuja (USDA-ARS AgriculturalSystems Research Unit Leader, Fort Collins, Colorado), T.N. Nar-asimhan (Department of Materials Science and Engineering, Univer-sity of California-Berkeley, Paul Willhite (Chemical and PetroleumEngineering, University of Kansas), and Scott Tyler (Department ofGeological Sciences and Engineering, University of Nevada-Reno)were helpful in developing this manuscript.

Appendix: notation

Symbol description Radius of sphere of influence of

intermolecular forces [L]g Interfacial surface tension [M T2]gLV, gSL, gSV Interfacial surface tension

between different phases: liquid(L) and gas/air/vapor (V) subscriptsSL and SV stand for solidliquid andsolidvapor interfaces, respectively[M T2]

gS Surface tension of a solid in vacuum [M T

2

]gs Surface free energy difference (= gSV gSL)[M T2]

E Change in total energy of a raisedliquid column [M L2 T2]

(E)min Minimum value ofE [M L2 T2]

Eg Change in interfacial energygained by a raised liquidcolumn [M L2 T2]

Eg Change in potential energygained by a raised liquidcolumn [M L2 T2]

g Acceleration of gravity [L T2]h Height of liquid in a capillary tube [L]hcm Height of center of mass

of a liquid column [L] Contact angle []m Mass of a fluid column of density

and height h in a capillary tubeof radius r (= r2h) [M]

pk Kinetic contribution to fluidpressure (always positive) [ML1 T2]

ps Static or intrinsic pressure (cohesive contributionto fluid pressure normally negative) [ML1 T2]

po Pressure in the bulk of the fluid [ML1 T2]

pn Normal pressure [ML1 T2]pt Tangential pressure [ML1 T2]P Total pressure (=pk+ps ) [ML1 T2]PL Liquid pressure [ML

1 T2]PV Vapor/air/gas pressure [ML

1 T2]P Pressure difference across a liquidvapor inter-

face (= PV PL orPL PV) [ML1 T2]

e Equilibrium spreading pressure orfilm pressure(= gS gSV) [M T

2]R Radius of curvature or radius of a drop [L1 or L]R1, R2 Radii of curvature along two orthogonal directions

[L1]r Radius of a capillary column [L]

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Fluid density [M L3]V Volume offluid in a capillary column [L3]y Distance of a molecule from the interface within the

radius of influence [L]

References

Bachmann J, Woche SK, Goebel M-O, Kirkham MB, Horton R(2003) Extended methodology for determining wetting proper-ties of porous media. Water Resour Res 39(12):1353.doi:10.1029/2003WR002143

Bailey AI, Kay SM (1967) A direct measurement of the influence ofvapour, of liquid, and of oriented monolayers in the interfacialenergy of mica. Proc R Soc Lond A301:4756

Bear J (1979) Hydraulics of groundwater. McGraw-Hill, New YorkBerry MV (1971) The molecular mechanism of surface tension.

Phys Educ 6(2):7984Bikerman JJ (1970) Physical surfaces. Academic, New YorkBrown RC (1941) Note on the energy associated with capillary rise.

Proc Phys Soc 53:233234Brown RC (1947) The fundamental concepts concerning surface

tension and capillarity. Proc Phys Soc 59:429446Chapin EK (1959) Two contrasting theories of capillary action. Am

J Phys 27(9):617

619Denny MW (1993) Air and water. The biology and physics of lifesmedia. Princeton Univ Press, Princeton, NJ

Dingman LS (1984) Fluvial hydrology. Freeman, New YorkDomenico PA, Schwartz FW (1998) Physical and chemical hydro-

geology, 2nd edn. Wiley, New YorkEhlers W, Goss M (2003) Water dynamics in plant production.

CABI, Cambridge, UKFetter CW (2001) Applied hydrogeology, 4th edn. Prentice Hall,

Englewood Cliffs, NJFreeze RA, Cherry JA (1979) Groundwater. Prentice Hall, Engle-

wood Cliffs, NJGiancoli DC (1988) Physics for scientists and engineers, 2nd edn.

Prentice Hall, Englewood Cliffs, NJGilboa A, Bachmann J, Woche SK, Chen Y (2006) Applicability of

interfacial theories of surface tension to water-repellent soils.Soil Sci Soc Am 70:14171429

Good RJ (1993) Contact angle, wetting, and adhesion: a criticalreview. In: Mittal KL (ed) Contact angle, wettability andadhesion. VSP, Utrecht, The Netherlands, pp 336

Henriksson U, Eriksson JC (2004) Thermodynamics of capillaryrise: Why is the meniscus curved? J Chem Educ 81(1):150154

Hillel D (1998) Environmental soil physics. Academic, San Diego, CA

Hunter RJ (2001) Foundations of colloid science, 2nd edn. OxfordUniv Press, Oxford

Isenberg C (1992) The science of soap films and soap bubbles.Dover, New York

Jaycock MJ, Parfitt GD (1981) Chemistry of interfaces, 1st edn.Ellis Horwood series, Wiley, Chichester, UK

Jury WA, Horton R (2004) Soil physics, 6th edn. Wiley, Hoboken,NJ

Kane JW, Sterheim MM (1984) Physics, 2nd edn. Wiley, New YorkKirkham MB (2005) Principles of soil and plant water relations.

Elsevier, AmsterdamKoorevaar P, Menelik G, Dirksen C (1983) Elements of soilphysics. Developments in Soil Science 13. Elsevier, Amsterdam

Kwok DY, Neumann AW (1999) Contact angle measurement andcontact angle interpretation. Adv Colloid Interface Sci 18:167249

Markworth AJ (1971) Liquid rise in a capillary tube: a treatment onthe basis of energy. J Chem Educ 49(8):528

Marshall TJ, Holmes JW (1988) Soil physics, 2nd edn. CambridgeUniv. Press, New York

McCaughan JBT (1987) Capillarity: a lesson in the epistemology ofphysics. Phys Educ 22:100106

McCaughan JBT (1992) Comment on The upward force on liquid in acapillary tube, by Scheie, P.O. Am J Phys 60(1):87

Miller F Jr, Schroeer D (1987) College physics, 6th edn. Harcourt,New York

Orowan E (1970) Surface energy and surface tension in solids and

liquids. Proc R Soc Lond A316(1527):473

491Or D, Tuller M (2005) Capillarity. In: Hillel D (ed) Encyclopediaof soils in the environment. Elsevier, Amsterdam, pp 155164

Or D, Wraith JM (2002) Soil water content and water potentialrelationships. In: Warrick AW (ed) Soil physics companion.CRC, Boca Raton, FL

Peiris MGC, Tannakone K (1980) Rate of rise of a liquid in acapillary tube. Am J Phys 48

Pellicer J, Manzanares JA, Mafe S (1995) The physical descriptionof elementary surface phenomena: thermodynamics versusmechanics. Am J Phys 63(6):542547

Roura P (2007) Contact angle in thick capillaries: a derivation basedon energy balance. Eur J Phys 28:L27L32

Sears FW, Zemanski MW, Young HD (1982) University physics,6th edn. Addison-Wesley, Reading, UK

Snow TP, Shull JM (1986) Physics. West, St Paul, MNTabor D (1991) Gases, liquids and solids and other states of matter,

3rd edn. Cambridge Univ Press, Cambridge, UKTemperley HNV, Travena DH (1978) Liquids and their properties.

Ellis Horwood series, Wiley, Chichester, UKTipler PA (1991) Physics for scientists and engineers, 3rd edn.

Worth, New York

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