Combining Rule for Molecular Interactions Derived fromMacroscopic Contact Angles and Solid-Liquid Adhesion

PatternsJunfeng Zhang and Daniel Y. Kwok*

Nanoscale Technology and Engineering Laboratory, Department of Mechanical Engineering,University of Alberta, Edmonton, Alberta T6G 2G8, Canada

Received September 4, 2002. In Final Form: March 4, 2003

We have examined a combining rule for intermolecular potentials recently proposed by Kwok et al. [J.Phys. Chem. B 2000, 104, 741] for the calculation of solid-liquid adhesion patterns using a van der Waalsmodel with a mean-field approximation. We found good agreement between the predicted and experimentaladhesion patterns. We have also employed the 9:3, Steele’s, and 12:6 combining rules for comparisonpurposes and found that they can also predict the general adhesion and contact angle patterns observedexperimentally, but with more scatter and less detail. Results suggest that macroscopic contact angle andadhesion findings can be used to infer relationships of unlike solid-fluid interactions at a molecular level.

I. IntroductionKnowledge of interfacial free energy is necessary for a

better understanding and modeling of interfacial processessuch as wetting, spreading, and flotation. However, directmeasurement of the solid-vapor (γsv) and solid-liquid(γsl) interfacial tensions is not available. Among thedifferent indirect approaches in determining solid surfacetensions, contact angle is believed to be the simplest andhence widely used approach.1,2 The possibility of estimat-ing solid surface tensions from contact angles relies on arelation known as Young’s equation3

where γlv is the liquid-vapor surface tension and θY is theYoung contact angle, that is, a contact angle that can beinserted into Young’s equation. Within the context of thiswork, we assume the experimental contact angles θ to bethe Young contact angle θY. A good indication of whetherthe contact angles truly represent θY is by contact anglehysteresis, the difference between the advancing andreceding angles. Since Young’s equation (eq 1) containsonly two measurable quantities (γlv and θ), an additionalexpression relating γsv and γsl must be sought. Such anequation can be formulated using experimental adhesionor contact angle data.

The origin of surface tensions arises from the existenceof unbalanced intermolecular forces among molecules atthe interface. Recently, Zhang and Kwok4 calculated thesolid-liquid adhesion patterns using a van der Waalsmodel with a mean-field approximation and found thatmacroscopic experimental adhesion and contact anglepatterns can, in principle, be reproduced by considerationof only intermolecular forces. The exact patterns dependon the choice of commonly used combining rules such asthe 9:3, Steele’s, and 12:6 combining rules (see later). Inaddition, starting frommacroscopicexperimentaladhesionpatterns, Kwok et al.5 modified the combining rule

originally investigated by Hudson and McCoubrey6 andproposed a new formulation that is meant to better reflectsolid-liquid interactions for intermolecular potentials.The solid-liquid work of adhesion Wsl was expressed interms of γlv and γsv as

where an empirical constant of Rk ) 1.17 m2/mJ has beendetermined. It has been illustrated that there is goodagreement between experimental adhesion data and thosepredicted from eq 2. Relating the interfacial tensions tomolecular collision diameters, Kwok5 suggested a modifiedcombining rule for solid-liquid intermolecular potentialsin the form of

where σs and σl are the solid and liquid molecular collisiondiameters, respectively; εss, εll, and εsl are respectively theintermolecular potential strengths (well depths) betweensolid-solid, liquid-liquid, and solid-liquid molecules; andK is in general a constant that depends on the Boltzmannconstant and the absolute and critical temperatures. Asthe proposed combining rule has a firm basis fromexperimental adhesion data, it would be of interest toinvestigate its usefulness and fully utilize its potentialfor molecular interaction calculations, as performed bythe procedures of Zhang and Kwok.4 Yet no further studyhas been performed to evaluate this expression (eq 3) froma molecular theory perspective. Thus, the objective of thiswork is to explore if the newly proposed expression couldindeed better describe solid-liquid interactions using thetheory of molecular interactions. We present here ageneralized van der Waals model using a mean-fieldapproximation similar to those used by van Giessen etal.7 to determine interfacial tensions. In the calculationof solid-vapor and solid-liquid interfacial tensions, a

* To whom correspondence should be addressed. E-mail:daniel.y.kwok@ualberta.ca.

(1) Kwok, D. Y.; Neumann, A. W. Adv. Colloid Interface Sci. 1999,81, 167.

(2) Sharma, P. K.; Rao, K. H. Adv. Colloid Interface Sci. 2002, 98, 341.(3) Young, T. Philos. Trans. R. Soc. London 1805, 95, 65.(4) Zhang, J.; Kwok, D. Y. J. Phys. Chem. B 2002, 106, 12594.

(5) Kwok, D. Y.; Neumann, A. W. J. Phys. Chem. B 2000, 104 (4), 741.(6) Hudson, G. H.; McCoubrey, J. C. Trans. Faraday Soc. 1960, 56,

761.

γlv cos θY ) γsv - γsl (1)

Wsl ) 2{ 4(γsv/γlv)1/3

[1 + (γsv/γlv)1/3]2}(Rkγsv)2/3

xγlvγsv (2)

εsl ) [ 4σl/σs

(1 + σl/σs)2](RkK/σs

3)2/3

xεssεll (3)

4666 Langmuir 2003, 19, 4666-4672

10.1021/la026511b CCC: $25.00 © 2003 American Chemical SocietyPublished on Web 04/22/2003

combining rule is required to relate unlike molecularinteractions (i.e. solid-fluid) to those of like pairs (i.e.liquid-liquid and solid-solid). For the sake of complete-ness, we also compare the results calculated from eq 3with those from other commonly used combining rulespreviously obtained.

II. TheoryA. Combining Rules for Solid-Liquid Intermo-

lecular Potentials. In the theory of molecular interac-tions and the theory of mixtures, combining rules are usedto evaluate the parameters of unlike-pair interactions interms of those of the like interactions.6,8-16 As with manyother combining rules, the Berthelot rule17

is a useful approximation but does not provide a securebasis for the understanding of unlike-pair interactions; εijis the potential energy parameter (well depth) of unlike-pair interactions; and εii and εjj are for like-pair interac-tions.

From the London theory of dispersion forces, theattraction potential φij between a pair of unlike moleculesi and j is given by

where I is the ionization potential, R is the polarizability,and r is the distance between the pair of unlike molecules.For like molecules eq 5 becomes

The total intermolecular potential V(ri) expressed by the(12:6) Lennard-Jones potential is in the form

where σ is the collision diameter. The attractive potentialsin eqs 6 and 7 can be equated to give

Equation 8 can be used to derive Ri and Rj; substitutingthese quantities into eq 5 yields

If we write φij in the form -4εijσij6/rij

6 such that σij ) (σi

+ σj)/2, the energy parameter for two unlike moleculescan be expressed as

This forms the basis of the so-called combining rules forintermolecular potential. The above expression for εij canbe simplified: when Ii ) Ij, the first term of eq 10 becomesunity; when σi ) σj, the second factor becomes unity. Whenboth conditions are met, we obtain the well-knownBerthelot rule, that is, eq 4.

For the interactions between two very dissimilar typesof molecules or materials where there is an apparentdifference between εii and εjj, it is clear that the Berthelotrule cannot describe the behavior adequately. It has beendemonstrated18-20 that the Berthelot geometric meancombining rule generally overestimates the strength ofthe unlike-pair interactions; that is, the geometric meanvalue is too large an estimate. In general, the differencesin the ionization potential are not large, that is, Ii ≈ Ij;thus, the most serious error comes from the difference inthe collision diameters σ for unlike molecular interactions.

For solid-liquid systems in general, the minimum ofthe solid-liquid interaction potential εsl is often expressedin the following manner8,10,12

where g(σl/σs) is a function of σl and σs; they are respectivelythe collision diameters for the liquid and solid molecules;and εss and εll are respectively the minima in the solid-solid and liquid-liquid potentials. Several other formsfor the explicit function of g(σl/σs) have been suggested.For example, by comparing εsl with the minimum in the(9:3) Lennard-Jones potential, one obtains an explicitfunction as

and the (9:3) combining rule becomes

An alternative function in the form of

has been investigated by Steele21 and others,22 suggestinga different combining rule as

For comparison purposes, we label eq 15 as the Steelecombining rule in this paper. Further, from the (12:6)Lennard-Jones potential, eq 10 implies an explicit function

(7) van Giessen, A. E.; Bukman, D. J.; Widom, B. J. Colloid InterfaceSci. 1997, 192, 257.

(8) Reed, T. M. J. Phys. Chem. 1955, 59, 425.(9) Reed, T. M. J. Phys. Chem. 1955, 59, 428.(10) Fender, B. E. F.; Halsey, G. D., Jr. J. Chem. Phys. 1962, 36, 1881.(11) Sullivan, D. E. Phys. Rev. B 1979, 20 (10), 3991.(12) Sullivan, D. E. J. Chem. Phys. 1981, 74 (4), 2604.(13) Matyushov, D. V.; Schmid, R. J. Chem. Phys. 1996, 104 (21),

8627.(14) Rowlinson, J. S.; Swinton, F. L. Liquids and Liquid Mixtures;

Butterworth Scientific: London, 1981.(15) Chao, K. C.; Robinson, J. R. L. Equation of State: Theories and

Applications; American Chemical Society: Washington, DC, 1986.(16) Steele, W. A. The Interaction of Gases with Solid Surfaces;

Pergamon Press: New York, 1974.(17) Berthelot, D. Compt. Rend. 1898, 126 (1703), 1857.

(18) Israelachvili, J. N. Proc. R. Soc. London A 1972, 331, 39.(19) Kestin, J.; Mason, E. A. AIP Conf. Proc. 1973, 11, 137.(20) Maitland, G. C.; Rigby, M.; Smith, E. B.; Wakeham, W. A.

Intermolecular Forces: Their Origin and Determination; ClarendonPress: Oxford, 1981.

(21) Steele, W. A. Surf. Sci. 1973, 36, 317.(22) Lane, J. E.; Spurling, T. H. Aust. J. Chem. 1976, 29, 8627.

εij ) xεiiεjj (4)

φij ) - 32

IiIj

Ii + Ij

RiRj

rij6

(5)

φi ) - 34

IiRi2

ri6

(6)

V(ri) ) 4εii[(σi/ri)12 - (σi/ri)

6] (7)

34

IiRi2 ) 4εiiσi

6 (8)

φij ) -2xIiIj

Ii + Ij

4σi3σj

3

rij6 xεiiεjj (9)

εij )2xIiIj

Ii + Ij[ 4σi/σj

(1 + σi/σj)2]3

xεiiεjj (10)

εsl ) g(σl/σs)xεssεll (11)

g(σl/σs) ) 18(1 +

σl

σs)3

(12)

εsl ) 18(1 +

σl

σs)3

xεssεll (13)

g(σl/σs) ) 14(1 +

σl

σs)2

(14)

εsl ) 14(1 +

σl

σs)2

xεssεll (15)

Combining Rule for Molecular Interactions Langmuir, Vol. 19, No. 11, 2003 4667

resulting in a (12:6) combining rule as

These are numerous attempts for a better representationof solid-liquid interactions from solid-solid and liquid-liquid interactions. In general, these functions are nor-malized such that g(σl/σs) ) 1 when σl ) σs; they revertto the Berthelot geometric mean combining rule (eq 4)when g(σl/σs) ) 1. Nevertheless, adequate representationof unlike solid-liquid interactions from like pairs is rare,and their validity for solid-liquid systems lacks experi-mental support.

B. Solid-Liquid Adhesion from Contact Angles.Thermodynamically, a relation of the free energy ofadhesion per unit area of a solid-liquid pair is equal tothe work Wsl required to separate a unit area of the solid-liquid interface:23

Because the free energy is directly proportional to theenergy parameter,24,20 that is, W ∝ ε, the Berthelotgeometric mean combining rule (eq 4) for the free energyof adhesion Wsl can be approximated in terms of the freeenergy of cohesion of the solid Wss and that of the liquidWll

17,24,20,25

By the definitions Wll ) 2γlv and Wss ) 2γsv, eq 19 becomes

Experimentally, one can in principle obtain the free energyof adhesion Wsl through contact angles via Young’sequation (eq 1). Combining eqs 1 and 18 eliminates γsvand γsl and yields a relation of Wsl as a function of onlyγlv and θY:

Thus, in addition to eq 18, adhesion patterns can also beobtained from experimental contact angles; that is, contactangles of different liquids on one and the same solid surfacecan be employed to study a systematic effect of changingγlv on Wsl through θY. We wish to point out that this strategyis straightforward but the underlying assumptions are,however, not trivial.1,26 For example, there exist manymetastable contact angles which are not equal to the onegiven by Young’s equation, that is, θY: the contact anglemade by an advancing liquid (θa) and that made by areceding liquid (θr) are not identical. The differencebetween θa and θr is called the contact angle hysteresis.Contact angle hysteresis can be due to roughness andheterogeneity of a solid surface. If roughness is the primarycause, then the measured contact angles are meaninglessin terms of the Young equation. Thus, experimental

determination of meaningful contact angles requirespainstaking efforts. Therefore, we have selected thecontact angle data in the literature only on very carefullyprepared solid surfaces where the hysteresis was foundto be typically less than 10°. For example, the hysteresisof water was claimed to be zero on the hexatriacontaneand cholesteryl acetate surfaces. A detailed discussion ofcontact angles is available.1

C. Calculation of Interfacial Tensions and ContactAngles. The calculations of contact angles from interfacialtensions have been described elsewhere4 and are brieflydiscussed here. We employ a mean-field approximationhere to calculate numerically the three interfacial tensionsfrom molecular interactions. In our simple van der Waalsmodel, the fluid molecules are idealized as hard spheresinteracting with each other through a potential φff(r),where r is the distance between two interacting molecules.A Carnahan-Starling model7,11,27 is adopted as the hardsphere reference system. For a planar interface formedby a liquid and its vapor, each of which occupies a semi-infinite space, z > 0 and z < 0, respectively, and the surfacetension is given by4,7,12

Here the minimum is taken over all possible densityprofiles F(z) and F is the excess free energy; φh ff representsthe interaction potential that has been integrated overthe whole x′y′ plane. For the solid-fluid (i.e., a solid-liquid or a solid-vapor) interface, the solid is modeled asa semi-infinite impenetrable wall occupying the domainof z < 0 and exerting an attraction potential V(z) to thefluid molecule at a distance z from the solid surface. Theinterfacial tension of such an interface can be obtainedfrom

where γs is the solid-vacuum surface tension, a constantthat exists in the calculations of γsv and γsl. This constant(γs) will be canceled out in the calculations of the contactangles via Young’s equation (eq 1); it has no impact on theimplication of our results, since we are interested only inthe difference between γsv and γsl. Considering a solidwith molecules interacting with fluid molecules througha potential φsf(r), we obtain easily the intermolecularpotential V(z) by integrating φsf(r) over the solid domain.We wish to point out that the above equations for γlv andγsf (eqs 22 and 23) are identical to those reported in refs7 and 12, although the forms of the equations are different.

We believe that the integral terms

in eq 9 and

(23) Dupre, A. Theorie Mecanique de la Chaleur; Gauthier-Villars:Paris, 1969.

(24) Good, R. J.; Elbing, E. Ind. Eng. Chem. 1970, 62 (3), 72.(25) Girifalco, L. A.; Good, R. J. J. Phys. Chem. 1957, 61, 904.(26) Kwok, D. Y.; Neumann, A. W. Prog. Colloid Polym. Sci. 1831,

109, 170. (27) Carnahan, N. F.; Starling, K. E. Phys. Rev. A 1970, 1 (6), 1672.

g(σl/σs) ) [ 4σl/σs

(1 + σl/σs)2]3

(16)

εsl ) [ 4σl/σs

(1 + σl/σs)2]3

xεssεll (17)

Wsl ) γlv + γsv - γsl (18)

Wsl ) xWllWss (19)

Wsl ) 2xγlvγsv (20)

Wsl ) γlv(1 + cos θY) (21)

γlv )

∫-∞

+∞dz {F[F(z)] +1

2F(z)∫-∞

+∞dz′ φhff(z′ - z)[F(z′) - F(z)]}

(22)

γsf ) γs + minF

∫0

+∞dz {F[F(z)] + F(z) V(z) +

12

F(z)∫0

+∞dz′ φhff(z′ - z)[F(z′) - F(z)] -

12

F2(z)∫-∞

0dz′ φhff(z′ - z)} (23)

- 12

F2(z)∫-∞

0dz′ φhff(z′ - z) (24)

4668 Langmuir, Vol. 19, No. 11, 2003 Zhang and Kwok

in eq 10 in ref 7 are missing. Without such integral terms,for the case of a fluid against a wall with V(z) ) 0, F(z) )Ff (the density of fluid bulk phase) would be the solutionto the Euler-Lagrange equation (eq 10 in ref 7). Thus, wewould not be finding the “drying” layer of vapor betweenthe bulk liquid (at z ) ∞) and the wall. In the expressionof the potential exerted by the solid on the fluid V(z), eq11 in ref 7, the upper integral limit should be -dsf ) -(ds

+ df)/2.28 In the expressions of all combining rules, thatis, eqs 12-14 in ref 7 and throughout that paper, all ofthe terms ds/df should be corrected as df/ds.12 We haveemployed σf/σs here, rather than df/ds.

To carry out the calculations of interfacial tensions andhence the contact angles, a given interaction potential isrequired. Here we assume a (12:6) Lennard-Jones po-tential model and consider only the attraction part. Itshould be addressed that the Lennard-Jones potentialfunction requires knowledge of two parameters: thepotential strength ε and the collision diameter σ. Thepotential strength εsf for φsf(r) is obtained from the fluidεff and solid εss potential strengths via a combining rule,such as that given by eq 3 or 15.

As mentioned above, the calculation of liquid surfacetension γlv requires two parameters εff and σf, which canbe related to the critical temperature Tc and pressure Pc

of the liquid in the following expressions for the Carna-han-Starling model7,11

where k is the Boltzmann constant and R is the van derWaals parameter, given by

The densities of the liquid Fl and vapor Fv were obtainedby requiring the liquid and vapor to be in coexistence ata given temperature T.7,11 In our calculations, we haveselected 30 liquids of different molecular structures andhave assumed T ) 21 °C, σs ) 10 Å, and Fs ) 1027 molecules/m3 for the solid surface.

In our theoretical model, a fluid (liquid or its vapor) isdetermined by the potential parameter εll and collisiondiameter σl, with the liquid density Fl and vapor densityFv found by requiring that the liquid and vapor be incoexistence at a given temperature T. A solid is modeledby its potential parameter εss, density Fs, and collisiondiameter σs. The interfacial tension was obtained byconsidering the excess free energy and interactions of everypair of molecules in both of the two bulk phases formingthe interface with a mean-field approximation. Forexample, in calculation of the liquid-vapor interfacialtension at a given temperature, only εll and σl are needed.But for the solid-vapor and solid-liquid interfacialtension (γsv and γsl), in addition to the parameters of fluidand solid, a combining rule g(σl/σs) is also involved todetermine the potential parameter εsl between the solidand fluid molecules from εll and εss by relation eq 11.Following these procedures, the three interfacial tensionsγlv, γsv, and γsl at a given temperature T can then be

expressed as

According to the model, for a given solid surface at a giventemperature with a selected combining rule and differentfluids, εss, Fs, σs, T, and the function g(σl/σs) are completelydefined and fixed, so

A first glance at the above equations might appear to bepeculiar because there seems to be no relationship withsolid properties and only the liquid properties εll and σlare involved. We shall point out that, since the solidproperties have already been used to establish thesefunctions, eqs 29-31 indeed contain the implicit (notexplicit) information on the solid properties. Thus, thereis no guarantee that the calculated curves will always besmooth; the scatter can easily arise from the differentchoices of the combining rules g(σl/σs) that might not trulyreflect the specific solid-liquid interactions, contrary tothe conclusions drawn in ref 7.

III. Results and DiscussionA. Calculated Liquid-Vapor Surface Tensions.

The calculated liquid surface tensions for the 30 liquidsselected can be found elsewhere.4 In most cases, thedifferences between the calculated and experimentalliquid-vapor surface tensions are less than 10-20%. The

-F(z)∫-∞

0dz′ φhff(z′ - z) (25)

kTc ) 0.18016R/σf3

Pc ) 0.01611R/σf6 (26)

R ) - 12∫φff(r) dr (27)

Figure 1. (a) The solid-liquid work of adhesion Wsl versus theliquid-vapor surface tension γlv and (b) cosine of the contactangle cos θ versus the liquid-vapor surface tension γlv for afluorocarbon FC722 (square), hexatriacontane (0), cholesterylacetate (]), poly(n-butyl methacrylate) ([), poly(methyl meth-acrylate/n-butyl methacrylate), (2) and poly(methyl methacry-late) (triangle pointing left) surfaces.

γlv ) γlv(εll,σl,T)

γsv ) γsv(εll,σl,εss,Fs,σs,g,T)

γsl ) γsl(εll,σl,εss,Fs,σs,g,T). (28)

γlv ) γlv(εll,σl) (29)

γsv ) γsv(εll,σl) (30)

γsl ) γsl(εll,σl). (31)

Combining Rule for Molecular Interactions Langmuir, Vol. 19, No. 11, 2003 4669

largest discrepancy comes from water with a calculatedγlv value of 93 mJ/m2 instead of an experimental value of72.8 mJ/m2. Considering the fact that we have only useda simple van der Waals model, the slightly largerdeviations for the polar liquids are indeed expected, asthe Lennard-Jones potential should not reflect the com-plicated interactions of, for example, water. It should benoted that the calculation of liquid-vapor surface tensiondoes not rely on any form of a combining rule, but the vander Waals model used here.

B. Experimental and Calculated Solid-LiquidAdhesion Patterns. Kwok et al.1,5,29 have recentlypublished experimental adhesion and contact angle pat-terns for a large number of polar and nonpolar liquids ona variety of carefully prepared low-energy solid surfaces,including fluorocarbon FC722, hexatriacontane, choles-

teryl acetate, poly(n-butyl methacrylate), PnBMA, poly-(methyl methacrylate/n-butyl methacrylate), and poly-(methyl methacrylate), PMMA. We reproduce their resultshere in Figure 1 and compare them with our patternscalculated from intermolecular potentials. Figure 1aillustrates that, for a given solid surface, say the FC722surface, the experimental solid-liquid work of adhesionWsl increases as γlv increases and up to a maximum Wsl

value identified as Wsl/ . Further increase in γlv causes Wsl

to decrease from Wsl/ . The trend described here appears

to shift systematically to the upper right for a morehydrophilic surface (such as PMMA) and to the lower leftfor a relatively more hydrophobic surface. There is alsosome indication that the location of the maximum pointWsl

/ appears to shift to the right as surface hydrophobicitydecreases. Figure 1b shows the experimental contact anglepatterns in cos θ versus γlv. We see that, for a given solidsurface, as γlv decreases, the cosine of the contact angle(cos θ) increases, intercepting at cos θ ) 1 with a “limiting”

(28) Gouin, H. J. Phys. Chem. B 1998, 102, 1212.(29) Kwok, D. Y.; Ng, H.; Neumann, A. W. J. Colloid Interface Sci.

2000, 225 (2), 323.

Figure 2. Solid-liquid work of adhesion Wsl versus the liquid-vapor surface tension γlv and cosine of the contact angle cos θ versusthe liquid-vapor surface tension γlv calculated from (a) Berthelot’s rule, (b) the (9:3) combining rule, and (c) Steele’s rule, and (d)the (12:6) combining rule. The symbols are calculated data, and the curves are the general trends of the data points.

4670 Langmuir, Vol. 19, No. 11, 2003 Zhang and Kwok

γlv value. We identify this “limiting” value as γlvc . As γlv

decreases beyond this γlvc value, contact angles become

more or less zero (cos θ ≈ 1), representing the case ofcomplete wetting. The trend described here appears tochange systematically to the right for a more hydrophilicsurface (such as PMMA) and to the left for a relativelymore hydrophobic surface (such as fluorocarbon). Chang-ing the solid surfaces in this manner changes the limitingγlv

c value, suggesting that γlvc might be of indicative value

as a solid property. In fact, Zisman labeled this γlvc value

as the critical surface tension of the solid surface γc. It isalso clear in Figure 1b that the experimental contact anglepatterns are different from what Zisman anticipated.30

The 30 polar and nonpolar liquids have been used tocalculate the solid-vapor and solid-liquid surface ten-sions for the adhesion patterns using Berthelot’s, (9:3),Steele’s, and (12:6) combining rules.4 Our calculationresults suggest that Berthelot’s rule is the worst amongall combining rules that we have considered here, and theresults are shown in Figure 2a. The calculated adhesionand contact angle patterns do not follow the generalpatterns shown in Figure 1. In fact, the discrepancies areso large that Berthelot’s rule predicts the cosine of thecontact angle to increase with larger γlv, contrary to theexperimental patterns. The fact that Berthelot’s rule couldnot give a reasonable prediction is well-known because itgenerally overestimates the strength of unlike interactionsfrom like pairs.20

The calculated adhesion and contact angle patterns forthe (9:3) combining rule are shown in Figure 2b. In thisfigure, the (9:3) combining rule reasonably predicts thegeneral trend of the adhesion patterns: Increasing γlvincreases Wsl monotonically. It is not apparent, however,that there exists a maximum Wsl

/ beyond which Wsl willdecrease when γlv increases, as in Figure 1. The generaladhesion pattern predicted from the (9:3) combining ruleis that Wsl increases with γlv. To change the hydrophilicityof the model surface and observe the change in patterns,we hypothesize that solid surface energy increases withstronger solid-solid interaction energy εss: increasing thesolid-solid interactions increases the surface free energyrequired to generate a unit interfacial surface area. Thus,we increased the solid-solid interactions systematicallytomodelhydrophilic surfacesanddecrease the interactionsfor hydrophobic ones. Increasing the interactions and,hence thesurfacehydrophilicity, shifts thecurves inFigure2b to the upper right, in good agreement with those fromFigure 1a. The lower part of Figure 2b illustrates thecalculated contact angle patterns: decreasingγlv increasescos θ for a given solid surface. Further decrease in γlv

causes cos θ to intercept at cos θ ) 1 with γlv ) γlvc ,

identifying the case of complete wetting. Increasing thehydrophilicity of the surface shifts the curves in Figure2b to the right, similar to those shown in Figure 1b. Wealso note the relatively larger scatter in Figure 2; it willbecome apparent later that the magnitude of such scatterdepends on the choice of the combining rules.

The adhesion and contact angle patterns calculated fromSteele’s combining rule are shown in Figure 2c. It appearsthat Steele’s combining rule also predicts the generaladhesion and contact angle patterns well, in good agree-ment with the experimental results shown in Figure 1.We note that this combining rule yields results whichhave significantly less scatter than those calculated fromthe (9:3) combining rule (in Figure 2b). However, the

calculated results for Wsl still increase monotonically withincreasing γlv and did not result in a maximum Wsl

/ valuethat was observed in the experimental results in Figure1.

The calculated results using the (12:6) combining ruleare shown in Figure 2d. We see that the (12:6) combiningrule appears to predict correctly the existence of amaximum Wsl

/ value as γlv increases. This maximumvalue was not observed using the (9:3) and Steele’scombining rules in parts b and c, respectively, of Figure2. The calculated contact angle patterns in the lower partof Figure 2d are also similar to those from the (9:3) andSteele’s combining rules and in good agreement with thepatterns observed experimentally. The (12:6) combiningrule, however, suffers from the same shortcoming as the(9:3) combining rule in that relatively larger scatter wasapparent.

C. Combining Rule by Kwok et al. Application ofthe combining rule proposed by Kwok et al. (eq 3) requiresknowledge of the unknown constant K, which relates solidsurface tensions to molecular collision diameters. SinceK is not readily known, we adopted here the assumptionused in ref 5 by setting K ≈ γsvσs

3 ≈ γlvσl3 and hence K/σs

3

≈ γlvσl3/σs

3, leading to a slightly different form of eq 3 as

where the empirical constant Rk is 1.17 m2/mJ.5 It shouldbe noted that K has been eliminated in favor of γlv, whichcan be obtained from the calculated γlv values. Thisrepresents a proper form of the combining rule that canbe used to evaluate the solid-fluid intermolecular po-tential strength in terms of the solid-vapor (γsv) and solid-liquid (γsl) interfacial tensions. Figure 3 shows thecalculated adhesion and contact angle results from eq 32.It is apparent that the combining rule from Kwok et al.yields much smoother curves than those from the (9:3)

(30) Zisman, W. Advances in Chemistry Series, Vol. 43; AmericanChemical Society, Washington, DC, 1964.

Figure 3. (a) Solid-liquid work of adhesion Wsl versus theliquid-vapor surface tension γlv and (b) cosine of the contactangle cos θ versus the liquid-vapor surface tension γlv calculatedfrom the combining rule by Kwok et al. (eq 32). The symbolsare calculated data, and the curves are the general trends ofthe data points.

εsl ) [ 4σl/σs

(1 + σl/σs)2](Rkγlvσl

3/σs3)2/3

xεssεll (32)

Combining Rule for Molecular Interactions Langmuir, Vol. 19, No. 11, 2003 4671

and (12:6) combining rules. The smoothness is about thesame as that predicted from Steele’s combining rule.However, this combining rule is still not capable ofpredicting the presence of a maximum Wsl

/ value as γlv

increases (Figure 3a).To make the comparison more clearly, we reproduced

in Figure 4 the calculated curves of the contact anglepattern from Figure 3 and the experimental data fromFigure 1. We note that the new combining rule in eq 32produced much steeper curves of cos θ versus γlv that aremore similar to the experimental curves in Figure 1b.However, it is not expected that the two patterns wouldmatch perfectly due to the oversimplified model used here.

IV. Conclusions

We have calculated the solid-liquid work of adhesionusing the combining rule recently derived from macro-scopic contact angle and adhesion data by Kwok et al. aswell as other commonly used combining rules. Resultssuggest that the proposed expression can predict thegeneral trends of adhesion patterns using a van der Waalsmodel with a mean-field approximation. The relationyields curves that are more similar to the experimentaladhesion and contact angle patterns for the systemsstudied. We found that the 9:3, Steele’s, and 12:6 com-bining rules also yield the general adhesion and contactangle patterns observed experimentally, but with largerscatter and less detail. The good agreement suggests thatthe newly proposed combining rule for molecular interac-tions can be used for solid-liquid intermolecular poten-tials. The proposed procedures for deriving molecularcombining rules for solid-liquid interfaces from contactangle and adhesion patterns have been shown to be useful.

Acknowledgment. This research was supported, inpart, by the Alberta Ingenuity Establishment Fund,Canada Research Chair (CRC) Program, Canada Foun-dation for Innovation (CFI), and Natural Sciences andEngineering Research Council of Canada (NSERC). J.Z.acknowledges financial support fromtheAlberta Ingenuitythrough a studentship fund. The authors also acknowledgehelpful discussion with and a program from Dr. vanGiessen.

LA026511B

Figure 4. Comparison of the calculated contact angle patterns(curves) from the combining rule by Kwok et al. (eq 32) and theexperimental data (symbols) displayed in Figure 1.

4672 Langmuir, Vol. 19, No. 11, 2003 Zhang and Kwok