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Notes Predicting Wetting Behavior from Initial Spreading Coefficients Harvey Dobbs†,‡ and Daniel Bonn*,† Laboratoire de Physique Statistique, Ecole Normale Supe´ rieure, 24, rue Lhomond, 75231 Paris Cedex 05, France, and INFM-Dipartimento di Fisica “G. Galilei”, Via Marzolo 8, I-35131 Padova, Italy Received November 29, 2000. In Final Form: April 5, 2001
Introduction Wetting behavior is illustrated by the fate of a liquid drop in equilibrium with a surrounding vapor, when placed onto a substrate. Either the drop forms a cap with nonvanishing contact angle θ and does not spread over the substrate, or the contact angle vanishes and the drop spreads to form a uniform film. In the former case (partial wetting), Young’s equation gives the contact angle1,2
γsv ) γsl + γlv cos θ
(1)
Here, γ’s indicate various interface tensions, and subscripts s, l, and v refer to substrate, liquid, and vapor. In the latter case (complete wetting),2
γsv ) γsl + γlv
(2)
because for a wet surface the solid-vapor interface is a combination of solid-liquid and liquid-vapor interfaces. In neither case does γsv exceed the sum γsl + γlv. The difference between these quantities defines the equilibrium spreading coefficient
Seq ) γsv - γsl - γlv
(3)
(4)
Because Si > Seq, Si can be positive or negative, but whenever Si < 0, Seq is also negative and the liquid does not spread. On the other hand, if Si > 0, there is no direct argument whether Seq is less than or is equal to zero, and a theory is needed to calculate the difference between γsv and γs0. † ‡
Cahn Theory In Cahn theory, the surface free energy of an impenetrable substrate exposed to a vapor of density cv is expressed as a functional of the density profile c(z) of the adsorbate,3
γsv[c(z)] ) γs0 +
∫0∞ {∆f(c) + (m/2)(dc/dz)2} dz + φ(cs) (5)
which is never greater than zero.2 For partial wetting, Seq < 0, and for complete wetting Seq ) 0. However, the spreading coefficient may be positive initially.1 The difference between initial and equilibrium spreading coefficients is due to molecules from the vapor that adsorb to the substrate-vapor interface and lower its free energy, so γsv is less than the surface free energy in the absence of the vapor, γs0. The initial spreading coefficient follows by replacing γsv by γs0 in eq 3:
Si ) γs0 - γsl - γlv
Ragil et al. recently applied the Cahn theory of wetting3 to the adsorption of n-alkanes on water and were able to predict spreading coefficients and first-order wetting transition temperatures.4 In this implementation of the theory, surface pressure isotherms were used to determine the various parameters required. Unfortunately, such isotherms are not available in the literature for many systems, and moreover, Pethica has suggested that these data are not always reliable.5 On the other hand, values of Si are more easily obtained, as they demand only knowledge of the surface tensions of the components. In this note, we show how Cahn theory can be used to estimate Seq from Si and determine whether a liquid is wetting or not. An estimate can also be made of the wetting temperature. Our method applies equally whether the substrate is a liquid, immiscible with the adsorbing fluid, or a solid, and we apply our method to the wetting of about a dozen hydrocarbons on water. For aromatic as well as aliphatic hydrocarbons, we find values of the equilibrium spreading coefficient that agree with experiment to within 1-3 mJ m-2. For the temperature of the thin-thick transition of n-alkanes on water,6-8 our method gives results comparable to (or better than) those from the original approach using the full surface pressure isotherms.4
Laboratoire de Physique Statistique, Ecole Normale Supe´rieure. INFM-Dipartimento di Fisica “G. Galilei”.
(1) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (2) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982.
where z is the distance from the substrate. The functional is minimized with respect to profiles which tend to cv for large z.1 The substrate/liquid interface tension γsl is found from the same functional by minimizing with respect to profiles that tend to the liquid density cl. We use the notation of ref 4, where each term in (5) is explained. In brief, ∆f is a free energy density with two minima corresponding to the density of the liquid and gas phases, calculated from the equation of state of the adsorbate. For the hydrocarbon compounds discussed below, we use the Peng-Robinson equation9 (3) Cahn, J. W. J. Chem. Phys. 1977, 66, 3667. (4) Ragil, K.; Bonn, D.; Broseta, D.; Meunier, J. J. Chem. Phys. 1996, 105, 5160. (5) Pethica, B. A. Langmuir 1996, 12, 5851. (6) Ragil, K.; Meunier, J.; Broseta, D.; Indekeu, J. O.; Bonn, D. Phys. Rev. Lett. 1996, 77, 1532. (7) Shahidzadeh, N.; Bonn, D.; Ragil, K.; Broseta, D.; Meunier, J. Phys. Rev. Lett. 1998, 80, 3992. (8) Bertrand, E.; Dobbs, H.; Broseta, D.; Indekeu, J.; Bonn, D.; Meunier, J. Phys. Rev. Lett. 2000, 85, 1282. (9) Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59.
10.1021/la001668u CCC: $20.00 © 2001 American Chemical Society Published on Web 06/27/2001
Notes
Langmuir, Vol. 17, No. 15, 2001 4675
p)
ckBT ac2 1 - cb 1 + cb(2 - cb)
(6)
where p is the pressure, kB is Boltzmann’s constant, and T is the temperature. The values of a and b are determined from the critical temperature and pressure.9 The “influence parameter” m is deduced from the surface tension:10
γlv ) (2m)1/2
∫c
cl v
(∆f)1/2 dc
(7)
The parameters a, b, and m describe the adsorbate. In Cahn theory, the interaction with the substrate is modeled as a “contact” energy φ, which depends on the density of the adsorbate at the surface, cs ) c(z ) 0). An expansion
φ(cs) ) -h1cs - 1/2gcs2
(8)
is commonly assumed,1 which was demonstrated recently to be sufficient over the range of density relevant to determine the wetting transition.11 The term h1 is sometimes called the “surface field”, although strictly that name is reserved for the leading term in an expansion of φ about the density at the bulk critical point in the style of Landau, and g is called the “surface coupling enhancement”. Given the bulk properties and surface tension of the adsorbate, h1 and g are the only parameters to be determined. To proceed on the basis of a single input related to the substrate, Si, we must estimate g. This represents the effect that a fluid molecule close to the substrate has fewer neighboring fluid molecules than a molecule in the bulk, and it is expected to be approximately independent of the substrate.12 If we assume the dominant interaction is between nearest neighbors (approximately 12 in the bulk and 9 at the surface, assuming closepacking),
g ≈ (-1/2)aσ ) -0.39ab1/3
(9)
where σ ≈ (3b/2π)1/3 is the hard-sphere diameter.13 This estimate compares well with the value in ref 11 for n-pentane, g/ab1/3 ) -0.45,14 and similar values for other n-alkanes.15 Generally, the numerical factor in (9) depends on the type of fluid-fluid interaction, so we expect that the dimensionless ratio g/ab1/3 is similar for many hydrocarbons and use g ) -0.4ab1/3 as a working approximation. The final parameter h1 can now be determined, thus: After minimization of the surface free energy with respect to the density profile c(z), both γsv and γsl can be written as functions of the surface density cs in each case,
γsv,l(cs) ) γs0 + |
∫c
cs ∞
(2m∆f)1/2 dc| + φ(cs)
(10)
where c∞ is the density of the vapor or liquid, depending on the required interface tension. Minimizing with respect to cs gives the boundary condition1
-φ′(cs) ) {2m∆f(cs)}1/2 sign(cs - c∞)
(11)
where the prime (′) indicates a derivative. By solving (11) for cs and substituting the value to (10), the free energy γsl can be calculated numerically as a function of the (10) Carey, B. S.; Scriven, L. E.; Davis, H. T. AIChE J. 1978, 24, 1076. (11) Indekeu, J. O.; Ragil, K.; Bonn, D.; Broseta, D.; Meunier, J. J. Stat. Phys. 1999, 95, 1009. (12) Maritan, A.; Langie, G.; Indekeu, J. O. Physica A 1991, 170, 326.
Table 1. Initial Si and Equilibrium Seq Spreading Coefficients at 20 °C (mJ m-2) and Temperatures Tw1 (°C) of the First-Order Transitiona Seq Si n-pentane n-hexane n-heptane n-octane n-decane n-tetradecane cyclohexane benzene toluene o-xylene ethyl-benzene chloroform carbon tetrachloride
7.9 3.3 2.2 0.2 -2.3 -5.1 -2.9
EXP
PR
Alkanes 0 0 -0.5 -1.8 -0.2 -2.5 -1.2 -3.8 -5.6 -7.5 -4 -7.2
Tw1 QA
EXP
PR
0 -1.6 -2.8 -4.5 -6.5 -8.7 -7.8
25 73 138
8 -13 48 42 59 62 80 92 109 140 153 210 108 105
Aromatics 8.9 -0.4 -0.2 0.0 8.2 -0.0 -0.2 0.0 6.6 -1.0 -2.1 5.2 -1.7 -2.1 Halogenates 14.0 0.0 0.0 0.0 0.8 -2 -4.9 -5.3
23 23 35 43
QA
16 20 40 51
-8 -29 80 79
a Values of S are deduced from data in ref 16. Experimental i values (EXP) for Seq for heptane, octane, and benzene are from ref 20, and the rest are from Figure 20 of ref 21. Experimental estimates of Twl are from ref 8. Theoretical values are calculated using ∆f derived from the Peng-Robinson equation (PR) or the quartic approximation (QA) (eq 12).
surface field h1, which is then determined by matching γs0 - γsl - γlv with the experimental initial spreading coefficient, Si. The minimized substrate-vapor interface free energy and equilibrium spreading coefficient are calculated, using this value for h1. To calculate the wetting transition temperature, determined by Seq ) 0, we assume h1 is not a function of temperature, as was found for n-alkanes on water.4 Finally, a semianalytic approximation can be made by assuming a simple quartic form for the free energy function, as an alternative to using the Peng-Robinson equation (eq 6), ∆f(c) ∝ (c - cl)2(c - cv)2. The constant of proportionality is determined from the surface tension γlv, so that
(2m∆f)1/2 ) (6γlv/∆c3)|(c - cl)(c - cv)|
(12)
where ∆c ) cl - cv. With this quartic approximation (QA), the boundary condition (11) is a quadratic equation for cs and the surface free energy (10) is a cubic polynomial, so the determination of h1 and Seq is a simple numerical procedure. Results We applied the method above to a selection of hydrocarbons for which values of the surface and interface tensions at 20 °C are given in ref 16. Our choice is restricted to fluids that are not (or hardly) miscible in water, so that the water can be viewed as an impenetrable substrate. Values for critical temperatures and pressures were taken from standard sources.17 Our results for Seq and Tw1 using the free energy density ∆f calculated from the PengRobinson equation (PR) and the approximation (QA), eq 12, are shown in Table 1, together with experimental values in the literature. (13) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: London, 1992. (14) In ref 11, a unit λPc ≈ 0.083ab-5/3 is used for surface tensions. (15) Dobbs, H. Langmuir 1999, 15, 2586. (16) Girifalco, L. A.; Good, R. J. J. Chem. Phys. 1957, 61, 904. (17) Handbook of Chemistry and Physics; Weast, R. C., Ed.; CRC Press: Boca Raton, FL, 1985.
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Although the theory systematically underestimates Seq, the error is typically between 1 and 3 mJ m-2. Values obtained with the approximation (QA) are generally slightly less than those found using the Peng-Robinson equation. The predicted differences between initial and equilibrium spreading coefficients are more accurate in cases where Si is positive. For negative initial spreading coefficients, the error is more significant. To some extent, this can be attributed to the two-term approximation for the surface field (8), which is valid for large values of the surface density cs4,11 and not so reliable for cases of low surface pressure. However, because the wetting behavior is not in question for cases with Si < 0, this failure is unlikely to be crucial. Turning to the wetting temperatures, for each system here, the Cahn theory predicts a first-order wetting transition1 at a temperature Tw1. This might not be the whole story, as recent experiments6-8 show that two different things may happen at Tw1. If the Hamaker constant, which gives the net effect of the van der Waals interactions in the system, is negative at Tw1 (favoring wetting), the first-order wetting transition is allowed to occur. On the other hand, if the Hamaker constant at the predicted temperature is positive, the wetting transition is prevented, and instead there is a first-order thin-thick transition to a state of “frustrated-complete” wetting. However, the effect on spreading coefficients and contact angles is very small,8 and as reliable methods to evaluate the Hamaker constant are well-known,13 we do not address this issue here. For the n-alkanes, we find values of Tw1 with our method in modest agreement with experimental estimates for the
Notes
thin-thick transition,7,8 although systematically lower. However, our simple treatment, using either form for ∆f, gives better agreement than the original implementation of the Cahn theory using surface pressure isotherms.4 Discussion In this note, we have shown how to estimate equilibrium spreading coefficients from initial spreading coefficients, as an improved guide to whether a liquid wets a substrate. An extension of this work would be to determine the initial spreading coefficient solely from the substrate and liquidvapor surface tensions, γs0 and γlv, by using combining laws to calculate γsl.16,18,19 If these methods can yield values of Si accurate to within a few mJ m-2, then it would be possible to estimate equilibrium spreading behavior for immiscible fluids using only data for the pure liquids in isolation. Acknowledgment. We are grateful to V. Bergeron, E. Bertrand, D. Broseta, J.O. Indekeu, and J. Meunier for useful discussions and Y. H. Mori for spreading coefficient data. H.D. thanks the EC for funding an enjoyable visit to the ENS and the ENS for hospitality. This work was supported by EC TMR Network No. FMRX-CT98-0171 and MURST (COFIN 99). LA001668U (18) Fowkes, F. M. J. Phys. Chem. 1963, 67, 2538. (19) van Oss, C. J.; Good, R. J.; Chaudhury, M. K. Langmuir 1988, 4, 884. (20) Akatsuka, S.-Y.; Yoshigiwa, H.; Mori, Y. H. J. Colloid Interface Sci. 1995, 172, 335. (21) Hirasaki, G. J. J. Adhes. Sci. Technol. 1993, 7, 285.
