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Langmuir 2003, 19, 4666-4672
Combining Rule for Molecular Interactions Derived from Macroscopic Contact Angles and Solid-Liquid Adhesion Patterns Junfeng 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 Waals model with a mean-field approximation. We found good agreement between the predicted and experimental adhesion patterns. We have also employed the 9:3, Steele’s, and 12:6 combining rules for comparison purposes and found that they can also predict the general adhesion and contact angle patterns observed experimentally, but with more scatter and less detail. Results suggest that macroscopic contact angle and adhesion findings can be used to infer relationships of unlike solid-fluid interactions at a molecular level.
I. Introduction Knowledge of interfacial free energy is necessary for a better understanding and modeling of interfacial processes such as wetting, spreading, and flotation. However, direct measurement of the solid-vapor (γsv) and solid-liquid (γsl) interfacial tensions is not available. Among the different indirect approaches in determining solid surface tensions, contact angle is believed to be the simplest and hence widely used approach.1,2 The possibility of estimating solid surface tensions from contact angles relies on a relation known as Young’s equation3
γlv cos θY ) γsv - γsl
(1)
where γlv is the liquid-vapor surface tension and θY is the Young contact angle, that is, a contact angle that can be inserted into Young’s equation. Within the context of this work, we assume the experimental contact angles θ to be the Young contact angle θY. A good indication of whether the contact angles truly represent θY is by contact angle hysteresis, the difference between the advancing and receding angles. Since Young’s equation (eq 1) contains only two measurable quantities (γlv and θ), an additional expression relating γsv and γsl must be sought. Such an equation can be formulated using experimental adhesion or contact angle data. The origin of surface tensions arises from the existence of unbalanced intermolecular forces among molecules at the interface. Recently, Zhang and Kwok4 calculated the solid-liquid adhesion patterns using a van der Waals model with a mean-field approximation and found that macroscopic experimental adhesion and contact angle patterns can, in principle, be reproduced by consideration of only intermolecular forces. The exact patterns depend on the choice of commonly used combining rules such as the 9:3, Steele’s, and 12:6 combining rules (see later). In addition, starting from macroscopic experimental adhesion patterns, Kwok et al.5 modified the combining rule * To whom correspondence should be addressed. E-mail:
[email protected]. (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.
originally investigated by Hudson and McCoubrey6 and proposed a new formulation that is meant to better reflect solid-liquid interactions for intermolecular potentials. The solid-liquid work of adhesion Wsl was expressed in terms of γlv and γsv as
Wsl ) 2
{
4(γsv/γlv)1/3
}
(Rkγsv)2/3
[1 + (γsv/γlv)1/3]2
xγlvγsv
(2)
where an empirical constant of Rk ) 1.17 m2/mJ has been determined. It has been illustrated that there is good agreement between experimental adhesion data and those predicted from eq 2. Relating the interfacial tensions to molecular collision diameters, Kwok5 suggested a modified combining rule for solid-liquid intermolecular potentials in the form of
sl )
[
4σl/σs
]
(1 + σl/σs)2
(RkK/σs3)2/3
xssll
(3)
where σs and σl are the solid and liquid molecular collision diameters, respectively; ss, ll, and sl are respectively the intermolecular potential strengths (well depths) between solid-solid, liquid-liquid, and solid-liquid molecules; and K is in general a constant that depends on the Boltzmann constant and the absolute and critical temperatures. As the proposed combining rule has a firm basis from experimental adhesion data, it would be of interest to investigate its usefulness and fully utilize its potential for molecular interaction calculations, as performed by the procedures of Zhang and Kwok.4 Yet no further study has been performed to evaluate this expression (eq 3) from a molecular theory perspective. Thus, the objective of this work is to explore if the newly proposed expression could indeed better describe solid-liquid interactions using the theory of molecular interactions. We present here a generalized van der Waals model using a mean-field approximation similar to those used by van Giessen et al.7 to determine interfacial tensions. In the calculation of solid-vapor and solid-liquid interfacial tensions, a (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.
10.1021/la026511b CCC: $25.00 © 2003 American Chemical Society Published on Web 04/22/2003
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combining rule is required to relate unlike molecular interactions (i.e. solid-fluid) to those of like pairs (i.e. liquid-liquid and solid-solid). For the sake of completeness, we also compare the results calculated from eq 3 with those from other commonly used combining rules previously obtained.
+ σj)/2, the energy parameter for two unlike molecules can be expressed as
II. Theory A. Combining Rules for Solid-Liquid Intermolecular Potentials. In the theory of molecular interactions and the theory of mixtures, combining rules are used to evaluate the parameters of unlike-pair interactions in terms of those of the like interactions.6,8-16 As with many other combining rules, the Berthelot rule17
This forms the basis of the so-called combining rules for intermolecular potential. The above expression for ij can be simplified: when Ii ) Ij, the first term of eq 10 becomes unity; when σi ) σj, the second factor becomes unity. When both conditions are met, we obtain the well-known Berthelot rule, that is, eq 4. For the interactions between two very dissimilar types of molecules or materials where there is an apparent difference between ii and jj, it is clear that the Berthelot rule cannot describe the behavior adequately. It has been demonstrated18-20 that the Berthelot geometric mean combining rule generally overestimates the strength of the unlike-pair interactions; that is, the geometric mean value is too large an estimate. In general, the differences in the ionization potential are not large, that is, Ii ≈ Ij; thus, the most serious error comes from the difference in the collision diameters σ for unlike molecular interactions. For solid-liquid systems in general, the minimum of the solid-liquid interaction potential sl is often expressed in the following manner8,10,12
ij ) xiijj
(4)
is a useful approximation but does not provide a secure basis for the understanding of unlike-pair interactions; ij is the potential energy parameter (well depth) of unlikepair interactions; and ii and jj are for like-pair interactions. From the London theory of dispersion forces, the attraction potential φij between a pair of unlike molecules i and j is given by
φij ) -
3 IiIj RiRj 2 Ii + Ij r 6
(5)
ij
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 2
3 IiRi φi ) 4 r6
(6)
i
The total intermolecular potential V(ri) expressed by the (12:6) Lennard-Jones potential is in the form
V(ri) ) 4ii[(σi/ri)12 - (σi/ri)6]
(7)
ij )
[
]
2xIiIj 4σi/σj Ii + Ij (1 + σ /σ )2 i j
3
xiijj
sl ) g(σl/σs)xssll
(11)
where g(σl/σs) is a function of σl and σs; they are respectively the collision diameters for the liquid and solid molecules; and ss and ll are respectively the minima in the solidsolid and liquid-liquid potentials. Several other forms for the explicit function of g(σl/σs) have been suggested. For example, by comparing sl with the minimum in the (9:3) Lennard-Jones potential, one obtains an explicit function as
g(σl/σs) )
( )
σl 1 1+ 8 σs
3
where σ is the collision diameter. The attractive potentials in eqs 6 and 7 can be equated to give
and the (9:3) combining rule becomes
3 2 I R ) 4iiσi6 4 i i
sl )
(8)
Equation 8 can be used to derive Ri and Rj; substituting these quantities into eq 5 yields
2xIiIj 4σi σj Ii + I j r 6 3
φij ) -
ij
xiijj
( )x
σl 1 1+ 8 σs
(9)
If we write φij in the form -4ijσij6/rij6 such that σij ) (σi (7) van Giessen, A. E.; Bukman, D. J.; Widom, B. J. Colloid Interface Sci. 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.
(12)
3
ssll
(13)
An alternative function in the form of
g(σl/σs) )
3
(10)
( )
σl 1 1+ 4 σs
2
(14)
has been investigated by Steele21 and others,22 suggesting a different combining rule as
sl )
( )x
σl 1 1+ 4 σs
2
ssll
(15)
For comparison purposes, we label eq 15 as the Steele combining rule in this paper. Further, from the (12:6) Lennard-Jones potential, eq 10 implies an explicit function (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; Clarendon Press: Oxford, 1981. (21) Steele, W. A. Surf. Sci. 1973, 36, 317. (22) Lane, J. E.; Spurling, T. H. Aust. J. Chem. 1976, 29, 8627.
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g(σl/σs) )
[
4σl/σs
(1 + σl/σs)
Zhang and Kwok
]
3
2
(16)
resulting in a (12:6) combining rule as
sl )
[
4σl/σs
]
(1 + σl/σs)2
3
xssll
(17)
These are numerous attempts for a better representation of solid-liquid interactions from solid-solid and liquidliquid interactions. In general, these functions are normalized such that g(σl/σs) ) 1 when σl ) σs; they revert to the Berthelot geometric mean combining rule (eq 4) when g(σl/σs) ) 1. Nevertheless, adequate representation of unlike solid-liquid interactions from like pairs is rare, and their validity for solid-liquid systems lacks experimental support. B. Solid-Liquid Adhesion from Contact Angles. Thermodynamically, a relation of the free energy of adhesion per unit area of a solid-liquid pair is equal to the work Wsl required to separate a unit area of the solidliquid interface:23
Wsl ) γlv + γsv - γsl
(18)
Because the free energy is directly proportional to the energy parameter,24,20 that is, W ∝ , the Berthelot geometric mean combining rule (eq 4) for the free energy of adhesion Wsl can be approximated in terms of the free energy of cohesion of the solid Wss and that of the liquid Wll17,24,20,25
Wsl ) xWllWss
(19)
By the definitions Wll ) 2γlv and Wss ) 2γsv, eq 19 becomes
Wsl ) 2xγlvγsv
(20)
Experimentally, one can in principle obtain the free energy of adhesion Wsl through contact angles via Young’s equation (eq 1). Combining eqs 1 and 18 eliminates γsv and γsl and yields a relation of Wsl as a function of only γlv and θY:
Wsl ) γlv(1 + cos θY)
(21)
Thus, in addition to eq 18, adhesion patterns can also be obtained from experimental contact angles; that is, contact angles of different liquids on one and the same solid surface can be employed to study a systematic effect of changing γlv on Wsl through θY. We wish to point out that this strategy is straightforward but the underlying assumptions are, however, not trivial.1,26 For example, there exist many metastable contact angles which are not equal to the one given by Young’s equation, that is, θY: the contact angle made by an advancing liquid (θa) and that made by a receding liquid (θr) are not identical. The difference between θa and θr is called the contact angle hysteresis. Contact angle hysteresis can be due to roughness and heterogeneity of a solid surface. If roughness is the primary cause, then the measured contact angles are meaningless in terms of the Young equation. Thus, experimental (23) Dupre´, A. The´ orie Me´ canique 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.
determination of meaningful contact angles requires painstaking efforts. Therefore, we have selected the contact angle data in the literature only on very carefully prepared solid surfaces where the hysteresis was found to be typically less than 10°. For example, the hysteresis of water was claimed to be zero on the hexatriacontane and cholesteryl acetate surfaces. A detailed discussion of contact angles is available.1 C. Calculation of Interfacial Tensions and Contact Angles. The calculations of contact angles from interfacial tensions have been described elsewhere4 and are briefly discussed here. We employ a mean-field approximation here to calculate numerically the three interfacial tensions from molecular interactions. In our simple van der Waals model, the fluid molecules are idealized as hard spheres interacting 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 hard sphere reference system. For a planar interface formed by a liquid and its vapor, each of which occupies a semiinfinite space, z > 0 and z < 0, respectively, and the surface tension is given by4,7,12
γlv )
∫-∞+∞dz {F[F(z)] +21F(z)∫-∞+∞dz′ φh ff(z′ - z)[F(z′) - F(z)]}
(22)
Here the minimum is taken over all possible density profiles F(z) and F is the excess free energy; φ h ff represents the interaction potential that has been integrated over the whole x′y′ plane. For the solid-fluid (i.e., a solidliquid or a solid-vapor) interface, the solid is modeled as a semi-infinite impenetrable wall occupying the domain of z < 0 and exerting an attraction potential V(z) to the fluid molecule at a distance z from the solid surface. The interfacial tension of such an interface can be obtained from
∫0+∞dz {F[F(z)] + F(z) V(z) + +∞ 1 F(z)∫0 dz′ φ h ff(z′ - z)[F(z′) - F(z)] 2 0 1 2 F (z)∫-∞dz′ φ h ff(z′ - z)} 2
γsf ) γs + min F
(23)
where γs is the solid-vacuum surface tension, a constant that exists in the calculations of γsv and γsl. This constant (γs) will be canceled out in the calculations of the contact angles via Young’s equation (eq 1); it has no impact on the implication of our results, since we are interested only in the difference between γsv and γsl. Considering a solid with molecules interacting with fluid molecules through a potential φsf(r), we obtain easily the intermolecular potential V(z) by integrating φsf(r) over the solid domain. We wish to point out that the above equations for γlv and γsf (eqs 22 and 23) are identical to those reported in refs 7 and 12, although the forms of the equations are different. We believe that the integral terms
∫-∞0 dz′ φh ff(z′ - z)
1 - F2(z) 2
(24)
in eq 9 and (27) Carnahan, N. F.; Starling, K. E. Phys. Rev. A 1970, 1 (6), 1672.
Combining Rule for Molecular Interactions
∫-∞0 dz′ φh ff(z′ - z)
-F(z)
Langmuir, Vol. 19, No. 11, 2003 4669
(25)
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 solution to the Euler-Lagrange equation (eq 10 in ref 7). Thus, we would not be finding the “drying” layer of vapor between the bulk liquid (at z ) ∞) and the wall. In the expression of the potential exerted by the solid on the fluid V(z), eq 11 in ref 7, the upper integral limit should be -dsf ) -(ds + df)/2.28 In the expressions of all combining rules, that is, eqs 12-14 in ref 7 and throughout that paper, all of the terms ds/df should be corrected as df/ds.12 We have employed σf/σs here, rather than df/ds. To carry out the calculations of interfacial tensions and hence the contact angles, a given interaction potential is required. Here we assume a (12:6) Lennard-Jones potential model and consider only the attraction part. It should be addressed that the Lennard-Jones potential function requires knowledge of two parameters: the potential strength and the collision diameter σ. The potential strength sf for φsf(r) is obtained from the fluid ff and solid ss potential strengths via a combining rule, such as that given by eq 3 or 15. As mentioned above, the calculation of liquid surface tension γlv requires two parameters ff and σf, which can be related to the critical temperature Tc and pressure Pc of the liquid in the following expressions for the Carnahan-Starling model7,11
kTc ) 0.18016R/σf3
Figure 1. (a) The solid-liquid work of adhesion Wsl versus the liquid-vapor surface tension γlv and (b) cosine of the contact angle cos θ versus the liquid-vapor surface tension γlv for a fluorocarbon FC722 (square), hexatriacontane (0), cholesteryl acetate (]), poly(n-butyl methacrylate) ([), poly(methyl methacrylate/n-butyl methacrylate), (2) and poly(methyl methacrylate) (triangle pointing left) surfaces.
expressed as
γlv ) γlv(ll,σl,T) (26)
γsv ) γsv(ll,σl,ss,Fs,σs,g,T)
where k is the Boltzmann constant and R is the van der Waals parameter, given by
γsl ) γsl(ll,σl,ss,Fs,σs,g,T).
Pc ) 0.01611R/σf6
R)-
∫
1 φ (r) dr 2 ff
(27)
The densities of the liquid Fl and vapor Fv were obtained by requiring the liquid and vapor to be in coexistence at a given temperature T.7,11 In our calculations, we have selected 30 liquids of different molecular structures and have 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) is determined by the potential parameter ll and collision diameter σl, with the liquid density Fl and vapor density Fv found by requiring that the liquid and vapor be in coexistence at a given temperature T. A solid is modeled by its potential parameter ss, density Fs, and collision diameter σs. The interfacial tension was obtained by considering the excess free energy and interactions of every pair of molecules in both of the two bulk phases forming the interface with a mean-field approximation. For example, in calculation of the liquid-vapor interfacial tension at a given temperature, only ll and σl are needed. But for the solid-vapor and solid-liquid interfacial tension (γsv and γsl), in addition to the parameters of fluid and solid, a combining rule g(σl/σs) is also involved to determine the potential parameter sl between the solid and fluid molecules from ll and ss by relation eq 11. Following these procedures, the three interfacial tensions γlv, γsv, and γsl at a given temperature T can then be
(28)
According to the model, for a given solid surface at a given temperature with a selected combining rule and different fluids, ss, Fs, σs, T, and the function g(σl/σs) are completely defined and fixed, so
γlv ) γlv(ll,σl)
(29)
γsv ) γsv(ll,σl)
(30)
γsl ) γsl(ll,σl).
(31)
A first glance at the above equations might appear to be peculiar because there seems to be no relationship with solid properties and only the liquid properties ll and σl are involved. We shall point out that, since the solid properties have already been used to establish these functions, eqs 29-31 indeed contain the implicit (not explicit) information on the solid properties. Thus, there is no guarantee that the calculated curves will always be smooth; the scatter can easily arise from the different choices of the combining rules g(σl/σs) that might not truly reflect the specific solid-liquid interactions, contrary to the conclusions drawn in ref 7. III. Results and Discussion A. Calculated Liquid-Vapor Surface Tensions. The calculated liquid surface tensions for the 30 liquids selected can be found elsewhere.4 In most cases, the differences between the calculated and experimental liquid-vapor surface tensions are less than 10-20%. The
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Figure 2. Solid-liquid work of adhesion Wsl versus the liquid-vapor surface tension γlv and cosine of the contact angle cos θ versus the 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.
largest discrepancy comes from water with a calculated γlv value of 93 mJ/m2 instead of an experimental value of 72.8 mJ/m2. Considering the fact that we have only used a simple van der Waals model, the slightly larger deviations for the polar liquids are indeed expected, as the Lennard-Jones potential should not reflect the complicated interactions of, for example, water. It should be noted that the calculation of liquid-vapor surface tension does not rely on any form of a combining rule, but the van der Waals model used here. B. Experimental and Calculated Solid-Liquid Adhesion Patterns. Kwok et al.1,5,29 have recently published experimental adhesion and contact angle patterns for a large number of polar and nonpolar liquids on a variety of carefully prepared low-energy solid surfaces, including fluorocarbon FC722, hexatriacontane, choles(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.
teryl acetate, poly(n-butyl methacrylate), PnBMA, poly(methyl methacrylate/n-butyl methacrylate), and poly(methyl methacrylate), PMMA. We reproduce their results here in Figure 1 and compare them with our patterns calculated from intermolecular potentials. Figure 1a illustrates that, for a given solid surface, say the FC722 surface, the experimental solid-liquid work of adhesion Wsl increases as γlv increases and up to a maximum Wsl value identified as W/sl. Further increase in γlv causes Wsl to decrease from W/sl. The trend described here appears to shift systematically to the upper right for a more hydrophilic surface (such as PMMA) and to the lower left for a relatively more hydrophobic surface. There is also some indication that the location of the maximum point W/sl appears to shift to the right as surface hydrophobicity decreases. Figure 1b shows the experimental contact angle patterns in cos θ versus γlv. We see that, for a given solid surface, as γlv decreases, the cosine of the contact angle (cos θ) increases, intercepting at cos θ ) 1 with a “limiting”
Combining Rule for Molecular Interactions
γlv value. We identify this “limiting” value as γclv. As γlv decreases beyond this γclv value, contact angles become more or less zero (cos θ ≈ 1), representing the case of complete wetting. The trend described here appears to change systematically to the right for a more hydrophilic surface (such as PMMA) and to the left for a relatively more hydrophobic surface (such as fluorocarbon). Changing the solid surfaces in this manner changes the limiting γclv value, suggesting that γclv might be of indicative value as a solid property. In fact, Zisman labeled this γclv value as the critical surface tension of the solid surface γc. It is also clear in Figure 1b that the experimental contact angle patterns are different from what Zisman anticipated.30 The 30 polar and nonpolar liquids have been used to calculate the solid-vapor and solid-liquid surface tensions for the adhesion patterns using Berthelot’s, (9:3), Steele’s, and (12:6) combining rules.4 Our calculation results suggest that Berthelot’s rule is the worst among all combining rules that we have considered here, and the results are shown in Figure 2a. The calculated adhesion and contact angle patterns do not follow the general patterns shown in Figure 1. In fact, the discrepancies are so large that Berthelot’s rule predicts the cosine of the contact angle to increase with larger γlv, contrary to the experimental patterns. The fact that Berthelot’s rule could not give a reasonable prediction is well-known because it generally overestimates the strength of unlike interactions from like pairs.20 The calculated adhesion and contact angle patterns for the (9:3) combining rule are shown in Figure 2b. In this figure, the (9:3) combining rule reasonably predicts the general trend of the adhesion patterns: Increasing γlv increases Wsl monotonically. It is not apparent, however, that there exists a maximum W/sl beyond which Wsl will decrease when γlv increases, as in Figure 1. The general adhesion pattern predicted from the (9:3) combining rule is that Wsl increases with γlv. To change the hydrophilicity of the model surface and observe the change in patterns, we hypothesize that solid surface energy increases with stronger solid-solid interaction energy ss: increasing the solid-solid interactions increases the surface free energy required to generate a unit interfacial surface area. Thus, we increased the solid-solid interactions systematically to model hydrophilic surfaces and decrease the interactions for hydrophobic ones. Increasing the interactions and, hence the surface hydrophilicity, shifts the curves in Figure 2b to the upper right, in good agreement with those from Figure 1a. The lower part of Figure 2b illustrates the calculated contact angle patterns: decreasing γlv increases cos θ for a given solid surface. Further decrease in γlv causes cos θ to intercept at cos θ ) 1 with γlv ) γclv, identifying the case of complete wetting. Increasing the hydrophilicity of the surface shifts the curves in Figure 2b to the right, similar to those shown in Figure 1b. We also note the relatively larger scatter in Figure 2; it will become apparent later that the magnitude of such scatter depends on the choice of the combining rules. The adhesion and contact angle patterns calculated from Steele’s combining rule are shown in Figure 2c. It appears that Steele’s combining rule also predicts the general adhesion and contact angle patterns well, in good agreement with the experimental results shown in Figure 1. We note that this combining rule yields results which have significantly less scatter than those calculated from the (9:3) combining rule (in Figure 2b). However, the (30) Zisman, W. Advances in Chemistry Series, Vol. 43; American Chemical Society, Washington, DC, 1964.
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Figure 3. (a) Solid-liquid work of adhesion Wsl versus the liquid-vapor surface tension γlv and (b) cosine of the contact angle cos θ versus the liquid-vapor surface tension γlv calculated from the combining rule by Kwok et al. (eq 32). The symbols are calculated data, and the curves are the general trends of the data points.
calculated results for Wsl still increase monotonically with increasing γlv and did not result in a maximum W/sl value that was observed in the experimental results in Figure 1. The calculated results using the (12:6) combining rule are shown in Figure 2d. We see that the (12:6) combining rule appears to predict correctly the existence of a maximum W/sl value as γlv increases. This maximum value was not observed using the (9:3) and Steele’s combining rules in parts b and c, respectively, of Figure 2. The calculated contact angle patterns in the lower part of Figure 2d are also similar to those from the (9:3) and Steele’s combining rules and in good agreement with the patterns observed experimentally. The (12:6) combining rule, however, suffers from the same shortcoming as the (9:3) combining rule in that relatively larger scatter was apparent. C. Combining Rule by Kwok et al. Application of the combining rule proposed by Kwok et al. (eq 3) requires knowledge of the unknown constant K, which relates solid surface tensions to molecular collision diameters. Since K is not readily known, we adopted here the assumption used in ref 5 by setting K ≈ γsvσs3 ≈ γlvσl3 and hence K/σs3 ≈ γlvσl3/σs3, leading to a slightly different form of eq 3 as
sl )
[
4σl/σs
]
(1 + σl/σs)2
(Rkγlvσl3/σs3)2/3
xssll
(32)
where the empirical constant Rk is 1.17 m2/mJ.5 It should be noted that K has been eliminated in favor of γlv, which can be obtained from the calculated γlv values. This represents a proper form of the combining rule that can be used to evaluate the solid-fluid intermolecular potential strength in terms of the solid-vapor (γsv) and solidliquid (γsl) interfacial tensions. Figure 3 shows the calculated 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)
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IV. Conclusions
Figure 4. Comparison of the calculated contact angle patterns (curves) from the combining rule by Kwok et al. (eq 32) and the experimental data (symbols) displayed in Figure 1.
and (12:6) combining rules. The smoothness is about the same as that predicted from Steele’s combining rule. However, this combining rule is still not capable of predicting the presence of a maximum W/sl value as γlv increases (Figure 3a). To make the comparison more clearly, we reproduced in Figure 4 the calculated curves of the contact angle pattern from Figure 3 and the experimental data from Figure 1. We note that the new combining rule in eq 32 produced much steeper curves of cos θ versus γlv that are more similar to the experimental curves in Figure 1b. However, it is not expected that the two patterns would match perfectly due to the oversimplified model used here.
We have calculated the solid-liquid work of adhesion using the combining rule recently derived from macroscopic contact angle and adhesion data by Kwok et al. as well as other commonly used combining rules. Results suggest that the proposed expression can predict the general trends of adhesion patterns using a van der Waals model with a mean-field approximation. The relation yields curves that are more similar to the experimental adhesion and contact angle patterns for the systems studied. We found that the 9:3, Steele’s, and 12:6 combining rules also yield the general adhesion and contact angle patterns observed experimentally, but with larger scatter and less detail. The good agreement suggests that the newly proposed combining rule for molecular interactions can be used for solid-liquid intermolecular potentials. The proposed procedures for deriving molecular combining rules for solid-liquid interfaces from contact angle and adhesion patterns have been shown to be useful. Acknowledgment. This research was supported, in part, by the Alberta Ingenuity Establishment Fund, Canada Research Chair (CRC) Program, Canada Foundation for Innovation (CFI), and Natural Sciences and Engineering Research Council of Canada (NSERC). J.Z. acknowledges financial support from the Alberta Ingenuity through a studentship fund. The authors also acknowledge helpful discussion with and a program from Dr. van Giessen. LA026511B
