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J . Phys. Chem. 1989, 93, 6441-6444
Long-Range Attraction between Hydrophobic Surfaces Phil Attardt Centre de Recherche Paul Pascal, Domaine Universitaire, Talence Cedex 33405, France (Received: October 19, 1988: In Final Form: April IO, 1989)
The long-range attractions measured between hydrophobic surfaces are explained as part of the van der Waals force. Classical continuum electrostatics is used to show that the force arising from electrostatic correlations across an electrolyte should decay exponentially with a decay length of half the Debye screening length, consistent with the experimental data. The unusually large magnitude of the measured force is attributed to an anomalous electrostatic response of the aqueous fluid which is perturbed by the adjacent hydrophobic surface, although the precise mechanism remains to be quantified. The electrostatic coupling to the other surface is mediated by the dielectric constant of the bulk water and shielded by the uniform bulk electrolyte profile that comprises almost all the interlayer separating the surfaces. The description shows how a short-range surface-induced perturbation in the fluid can give rise to measurable forces at large distances without invoking long-range structural forces.
1. Introduction Measurements of the forces between macroscopic bodies with hydrophobic (high contact angle) surfaces show remarkable long-range attractionsI4 quite unlike those seen between hydrophilic surfaces. The most recent experiments performed by Claesson, Christenson, and co-workers5v6exhibit forces measurable at 70-90 nm, which exceed the usual van der Waals force by some 2 orders of magnitude. These extraordinary results have been corroborated by Rabinovich and Derjaguin' using a different force measurement technique. Despite the considerable interest in these results, and also their profound implications, the origins of this dramatic force are not at all understood. The only attempts to formulate a theory6 has been that of Eriksson et aL9 who proposed a structural mechanism (one that propagates "layer to layer") that they described using a purely phenomenological mean-field analysis. They identified their order parameter with the local increase in hydrogen bonding of the fluid and fitted a correlation length (perpendicular to the surfaces) of 150 A! The properties of the hydrogen bond network in associated fluids were recently investigated by me in connection with the hydration force,I0 and explicit calculation for that lattice model yielded a decay length of 2-3 A. Simulations of water between inert"-I3 and mineralI4 surfaces show a very rapid decay of surface-induced effects. All of these studies agree with the notion that order in fluids persists only over molecular dimensions. In searching for an acceptable explanation of the measured data, one is forced to reconsider electrostatic forces. Analyses of the experiments rule out a double layer attraction between dissimilarly charged surfaces.6 An alternative is to consider electrostatic fluctuations between neutral bodies and in the electrolyte (van der Waals forces). These possibilities are analyzed here, and it is shown that the predicted decay length is in accord with the measured data, at least when the experiments are performed at nonvanishing electrolyte concentrations. It is also suggested that the difference between very hydrophobic surfaces and all others lies in the nature of the adjacent water and that this could give an unusually large electrostatic response. It is shown (using classical continuum electrostatics) how structural phenomena, which occur at each isolated surface, couple via the bulk water and electrolyte and give rise to the remarkable attractions measured at large separations. Note that there appear two distinct distance regimes in the experiment^^-^ and we only address the larger separation (beyond = l o nm) results here. 11. Decay Length The classical (zero-frequency) interaction free energy per unit area between electrically neutral (on average), identical, infinite 'Present address and address for correspondence: Department of Chemistry, University of British Columbia, Vancouver, British Columbia, V6T 1Y6 Canada.
0022-365418912093-6441$01SO10
planar bodies separated by a dielectric continuum is
where the temperature is T = (kBP)-',k = Ikl is the magnitude of the two-dimensional Fourier vector parallel to the planar surfaces, and h is the separation between them. This expression derives from the van Kampen modal approach,I5 which sums the free energies associated with the allowed electrostatic surface modes. The secular determinant is interpreted in terms of the reflection coefficient R ( k ) , which gives the electrostatic response of the isolated body to an externally applied p ~ t e n t i a l . ' ~ . ' ~ The electrostatic potential in the intervening uniform dielectric (no free charges or electrolyte) satisfies the source-free Laplace equation. Upon Fourier transformation, this is
and it is the exponentially decaying solution that gives rise to the factor of e-2khin the equation for the interaction free energy. For the case when an electrolyte is confined between the neutral surfaces, one has instead
(3) where K-I is the Debye screening length
(K'
= (4x/3/tl)C,q2p,,
(1) Blake, T. D.; Kitchener, J. A. J. Chem. SOC.,Faraday Trans. I 1972, 68, 1435.
(2) Israelachvili, J. N.; Pashley, R. M. J. Colloid Interface Sci. 1984, 98, 500. (3) Pashley, R. M.; McGuiggan, P. M.; Ninham, B. W.; Evans, D. F. Science 1985, 229, 1088. (4) Claesson, P. M.; Blom, C. E.; Herder, P. C.; Ninham, B. W. J . Colloid Interface Sci. 1986, 114, 234. (5) Claesson, P. M.; Christenson, H. K. J . Phys. Chem. 1988, 92, 1650. (6) Christenson, H. K.; Claesson, P. M.; Berg, J.; Herder, P. C. J . Phys. Chem. 1989, 93, 1472. (7) Rabinovich, Ya. I.; Derjaguin, B. V. Colloids Surf. 1988, 30, 243. (8) The interaction between surfaces in the presence of macroscopic vapor cavities, as distinct from the long-range attractions across water discussed here, has been addressed by: Yushenko, V. S.;Yaminsky, V. V.; Shchukin E. D. J . Colloid Interface Sci. 1983, 96, 307. (9) Eriksson, J. C.; Ljunggren, S.;Claesson, P. M. J. Chem. Soc.,Faraday Trans. I , in press. (10) Attard, P.; Batchelor, M. T. Chem. Phys. Lett. 1988, 149, 206. (1 1) Lee, C. Y.; McGammon, J. A,; Rossky, P. J. J . Chem. Phys. 1984, 80, 4448. (12) Luzar, A.; Bratko, D.; Blum, L. J. Chem. Phys. 1987, 86, 2955. (13) Valleau, J. P.; Gardner A. A. J . Chem. Phys. 1987, 86, 4162. (14) Kjellander, R.; MarEelja, S. Chem. Scr. 1985, 25, 73. (15) van Kampen, N. G.; Nijboer, B. R. A.; Schram, K. Phys. Lett. 1968, 26A, 307. (16) Pcdgornik, R.; Cevc, G.; h k S , B. J . Chem. Phys. 1987, 87, 5957. (17) Attard, P.; Mitchell, D. J. J. Chem. Phys. 1988, 88, 4391.
0 1989 American Chemical Society
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The Journal of Physical Chemistry, Vol. 93, No. 17, 1989
with ionic species a having charge qu and number density p a and with el == 80 being the dielectric constant of water). This linearized Poisson-Boltzmann equation, when used with the van Kampen modal approach, is equivelent to treating the correlations in the electrolyte at the level of Debye-Huckel t h e ~ r y . ' ~ ,The ' ~ interaction free energy (1) now becomes
+
wherep ( k 2 K ~ ) ' / ~The . reflection coefficient depends implicitly on K, being determined by whatever boundary conditions define the model one is applying in a given situation. It is independent of the separation of the bodies, since it pertains to an isolated surface. The differential equation (3) is also approximately applicable to the case of low net surface charge. In this case, the expression (4) for the interaction free energy includes only contributions arising from induced fluctuations (correlations) between the two surfaces. One expects these to be the dominant contributions to the specifically hydrophobic part of the force measurements. The asymptotic behavior of the interaction free energy is obtained by noting that for sufficiently large h the integral is dominated by the leading term of the small k Taylor expansion of the integrand. It then follows that
In fact, the preexponential coefficient should contain power laws of the separation that depend on the actual form of the reflection coefficient as a function of k . While they contribute to the leading order, the comparatively slow additional decay may not appear significant in a log-linear plot at large separations. However, the decay length K - ' / 2 is a rigorous result (for the model of planar surfaces infinite in extent) for the asymptotic behavior of the interaction free energy. Of course the screening of the electrolyte remains the Debye length. But, electrostatic fluctuations begin at one surface and induce a response at the other that returns and interacts with the original fluctuation. Hence, the distance traveled is actually twice the separation, and this is the physical reason that the van der Waals force decays twice as fast as the usual mean-field double layer repulsion. Here, only the zero-frequency van der Waals force has been treated. This is because the published experimental data show a strong salt dependence, and any higher frequency terms would be virtually independent of electrolyte (the ions are too slow moving to screen the electrodynamic modes). The higher frequency contribution, if present and measurable at low salt, would eventually dominate as the electrolyte concentration were increased. This behavior is not observed,6 and therefore only the static contribution is examined here. The expressions for the interaction free energy are general results for uncharged symmetric bodies that are equally applicable to hydrophobic and to hydrophilic surfaces. What distinguishes different situations is the reflection coefficient, and it is this that determines the magnitude of the force. For a given model, one can work out the reflection coefficient explicitly. For example, when the surfaces constitute simply a dielectric discontinuity with the aqueous phase, the calculated force turns out to be too small to account for the measured data.20,21 A generalization of the previous work for surfaces bearing adsorbed but mobile charges interacting across a dielectric continuum22again shows a force orders of magnitude too small. One can also apply the model of interacting dipolar surfacesL7to an intervening electrolyte. Unlike (18) Carnie, S. L.; Chan, D. Y. C. Mol. Phys. 1984, 5 1 , 1047. (19) Attard, P.; Mitchell, D. J.; Ninham, B. W. J . Chem. Phys. 1988,88, 4987; J . Chem. Phys. 1988, 89, 4358. (20) Gorelkin, V . N.; Smilga, V. P. Sou. Phys.-JETP (Engl. Trans/.) 1973, 36, 16 1.
(21) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic: London, 1976.
(22) Attard, P.; Kjellander, R.; Mitchell, D. J . Chem. Phys. Letr. 1988, 139, 219.
Attard TABLE I: Comparison of the Predicted Decay Length ( ~ - ' / 2 ) with That Derived from Experiment (A) concn, M K-'/2. nm A, nm ref 10-5 -4
x 10-5 10-4
1.8 x 10-4 1 x 10-3 I x 10-3 1 x 10-2
50
24 15
11 4.8 4.8 1.5
13-16 14 12-14 12 4.5
6 1.5
5 6
I 6 4 6 6
the previous two examples, there are now parameters to fit (the susceptibility or two-dimensional compressibility), but it is difficult to conceive of a physical mechanism giving the large values required if the dipoles are part of the solid or of the surfactant head-groups. In some ways, it is good that these three models cannot account for the measured data since there is nothing particularly hydrophobic about them, whereas it is clear that the measured forces are specific to hydrophobic surfaces. The experiments show forces that are almost certainly due to electrostatic fluctuations but are too large to be accounted for by any usual response of the surfaces. The possibility that remains is that the hydrophobic surface induce an anomalous electrostatic response in the adjacent aqueous fluid. One would like to suggest a specific mechanism, but the tentative nature of the speculation must be stressed. Experimentally, the fluid next to the very high contact angle hydrophobic surfaces is observed to be close to a vapor-phase transition, in the sense that spontaneous cavitation occurs on contact.23 It is conceivable that large density fluctuations are occurring in the aqueous phase along the surface, similar to the way these are known to exist at coexistence.z4-26 These density fluctuations would give an enhanced electrostatic response, since they imply large polarization or charge fluctuations. For molecules whose dipole moment has a mean orientation (such as water near a surface), part of the total polarization is directly proportional to the density, and the same relationship holds for the respective correlation^.'^ Further, since ions are asymmetrically solvated, a small effective surface charge is present in the perturbed water adjacent to the surface, and fluctuations here again directly cause charge fluctuations. Additionally, for a conventional adsorbed surface charge, the counterion profile should also fluctuate with the aqueous density. (The Born self-energy relates the ionic solubility to the aqueous dielectric constant, and Clausius-Massotti theory relates the latter to the density.) Note that these three examples all involve a mean term in directly relating density fluctuations to charge or polarization fluctuations. It should be noted that theoretically water next to inert surfaces appears to form an icelike lattice s t r ~ c t u r e .But, ~ ~one ~ ~should ~~~~ question the applicability of these analyses to the experimental situation, particularly since cavitation does not occur at small separations in these calculations. In particular, are the simple pair potentials used appropriate for water near a phase transition, and is a smooth inert surface a good model for a hydrophobic surface? Further, the simulations"J3 were implicitly carried out at elevated pressures.2x This fact (and also effects due to the finite size of the simulations) probably means that large density fluctuations or correlations parallel to the surfaces would not be seen even if the model were capable in principle of describing them. Similarly, the integral equation approach27is at the singlet level and by definition cannot include the effects of inhomogenous pair Christenson, H. K.; Claesson, P. M. Science 1988, 239, 390. Wertheim, M. S. J . Chem. Phys. 1976, 65, 2377. Kalos, M. H.; Percus, J. K.; Rao, M. J . Srat. Phys. 1977, 17, 11 1. Weeks, J. D.J . Chem. Phys. 1977, 67, 3106. Torrie, G. M.; Kusalik, P. G.; Patey, G. N. J . Chem. Phys. 1988.88, 7826. (28) The inhomogeneous system is not in equilibrium with a bulk at standard temperature and pressure, since the canonical simulations were performed with a fixed number of water molecules corresponding to bulk density over the volume. Because of the low compressibility of water, the uncertainty in the average density in the central region of the simulations precludes knowledge of the actual chemical potential or external pressure.
The Journal of Physical Chemistry, Vol. 93, No. 17, 1989 6443
Attraction between Hydrophobic Surfaces correlations that are very different from bulk. It is the aqueous correlation functions that give the electrostatic response of the fluid adjacent to a hydrophobic surface and hence determine the reflection coefficient (see the Appendix). We showed above how this then gives rise to a force between two hydrophobic surfaces, invoking only continuum classical electrostatics (Le., assuming most of the water in the interlayer to be bulk and characterized solely by its dielectric constant). The mechanisms tentatively proposed here for the unusually large electrostatic response is essentially the difference between hydrophobic surfaces and hydrophilic surfaces, namely, the nature of the fluid within several molecular layers of the interface. 111. Results The prediction of our theory for the long-range attraction between hydrophobic surfaces, that the decay length should equal K-I/2, is in good agreement with experiment, as is apparent in Table I. The data of Rabinovich and Derjaguin7 are particularly convincing. These were obtained from many different samples and with use of two different force measurement techniques. Further, those results are for purely attractive force curves, and these authors have not needed to subtract any double layer repulsion. It is impossible to microscopically quantify the preexponential coefficient, and so for the present, this entity must be regarded as phenomenological or as a measured experimental quantity. The microscopic mechanisms suggested for the magnitude of the force have not been directly observed experimentally in the present systems and are therefore somewhat speculative. This is in contrast to the rate of decay that we predict from macroscopic considerations and that is substantiated both theoretically and experimentally (Table I). The data a t low concentration of added electrolyte deserve comment. Here, the measured decay length is about 13-16 nm, considerably less than that expected from the strictly asymptotic analysis of the theory (some 50 nm, corresponding to a M impurity concentration in conductivity water). Two possibilities, that the impurity concentration is higher than this, or that the data is not yet in the asymptotic regime (the measurements were performed at about a Debye length separation), appear unlikely. Further, the force curves do not display the power law behavior expected of a van der Waals interaction in the absence of salt. However, it is possible to show how electrostatic fluctuations could give rise to exponential behavior independent of the Debye length for low enough salt concentrations. The analysis of section I1 is valid for the interaction between planar surfaces infinite in extent. Consider now the effect of the anomalous fluctuations occurring over hydrophobic patches that are of finite size =L. Then, the Fourier wave vector takes on only The integral for the discrete values k, = 2 m / L , n = 0, 1, 2, interaction free energy is replaced by a sum over the wave vectors:
....
In the limit of large separation, the sum is dominated by the largest term, which gives -keT Pt4 -lR2(l) exp[-2h(t2 + K ~ ) ~ / ~ ]h m (7) 4?r
-
where { kl = 2ir/L, the n = 0 term vanishing. These expressions represent the interaction between two domains. If one wants quantitative expressions for the total force, one should simply sum over pairs of domains. Further, it is not strictly necessary to integrate over a finite domain; one could obtain the same results for infinite areas provided the anomalous correlations were abruptly truncated at some distance L, effectively making discrete the reflection coefficient itself. There are now two lengths in the problem, and the rate of decay of the measured force is determined by the smaller of the two for a given experiment. At high salt content, the screening by the electrolyte makes the size of the hydrophobic patches irrelevant (so the analysis of section I1 is then appropriate) and the decay length is half the Debye screening length. At low salt content
-
(K 0), the limitation on the anomalous correlations is the size of the domains, and the measured decay length becomes independent of salt concentration and equal to 5'12. The experiments at vanishing salt concentrations*6are for surfactants LangmuirBlodgett-deposited on mica to have a head-group area of 60 A2 at the mica-air interface. But, since the hydrocarbon-water surface free energy is higher than that of hydrocarbon-air, when the hydrophobized mica is plunged into water, there is a surface pressure that acts to decrease the head-group area. One can expect the surfactant film to contact over more or less uniform microscopic domains, and hence these hydrophobic patches will give rise to the behavior discussed in the preceding paragraphs. If the hydrophobic surfaces are inducing an anomalous electrostatic response in the adjacent aqueous fluid (as was suggested in section II), then it is plausible that the large correlations along the surface are restricted by the boundaries of the hydrophobic patch. If this is indeed a valid model for the experiments, than the data in Table I indicate a domain size of L = lo2 nm (A i= lo5 nm2).
IV. Conclusion It is concluded that the measured decay lengths are in accord with the present prediction K-'/2, at least in nonvanishing electrolyte. Taken together with the exclusion of the normal charge and dipolar fluctuation forces and the unacceptability of very long range structural forces, it does indicate further directions for research. More extensive studies of the dependence of the decay length on the type of electrolyte and on its concentration and also on the method of surface preparation are called for. If the tentative suggestions for the mechanism of the unusual electrostatic response turn out to be correct, then one would expect the magnitude of the force to be decreased in nonaqueous fluids between surfaces with which they have a high contact angle. One would also expect that the magnitude of long-range attraction between hydrophobic surfaces would dramatically increase with increasing temperature. Further, an investigation of the forces between hydrophilic surfaces in liquids that are close to coexistence would provide much valuable information. For the present, the existing data provide some support for the putative mechanism for the measured long-range attractions: that the unusual electrostatic response of the aqueous phase adjacent to a hydrophobic surface (due perhaps to large density fluctuations that imply anomalous aqueous charge or dipole correlations) causes an enhanced attraction between the surfaces and this is mediated by the intervening bulk aqueous electrolyte. Acknowledgment. I thank John Parker for some interesting discussions and particularly for his explanations of the details of the experiments. Appendix. Polarization Fluctuations The general results obtained in section I1 are now illustrated with an explicit calculation of a simple model. Here are explored one possible microscopic mechanism that provides a link between the unusual density fluctuations that possibly occur adjacent to hydrophobic surfaces and the enhanced electrostatic response indicated by the experimental data. The main point of this Appendix is to show how the correlation functions of the fluid near an isolated surface can determine the electrostatic response of that surface. The calculations are done for nonvanishing electrolyte, and so the surfaces are taken to be infinite as is usual. Consider the interaction between two thin layers of fluid adjacent to surfaces at large separations. Rather than the modal approach of eq 4,it is more transparent to begin with the formally exact and intuitively obvious expression for the net pressure. This is -au(r; h) P = TR S d 2 r C(r; h)ah -aii(k; h) 1 = -TR S d 2 k 6 ( k ; h)(8) 472 ah Physically, this equation represents the perpendicular component of the force between two particles, weighted by the probability
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The Journal of Physical Chemistry, Vol. 93, No. 17, 1989
of them being in that position (or at that spatial frequency), integrated over all space (frequencies), and summed over all orientations, per unit area. Here, r and k are the two-dimensional radial and Fourier vectors parallel to the surfaces, G is a polarization-polarization correlation tensor between the fluids at the two different surfaces, and u is the electrostatic pair potential tensor, also between the molecules of the fluid, one in each of the two layers, in the presence of the electrolyte. The components of these Hermitian tensors represent the parallel (x and y ) and perpendicular (z) directions that characterize the symmetry of the problem (see ref 17 for more details). Here have been neglected explicit contributions due to the bulk of the water film (and also the electrolyte kinetic term), which one expects to make no contribution to the net pressure. Neither correlation terms due to the ions alone nor the effects of dielectric images have been included, although these could easily be accounted for if one required a quantitative theory. These approximations are justified since one anticipates that the dominant contribution to the hydrophobic attraction will come from the electrostatic response of the metastable aqueous surface phase, mediated by the dielectric constant of bulk water and shielded by the uniform bulk electrolyte profile. The correlation function can, according to second-order perturbation theory (valid in the limit of large separation), be approximated as
Attard to the surfaces much larger than the usual bulk correlation length. It remains to give the three archetypal components of the potential tensor in the presence of electrolyte. Making the simplest assumption that both the dielectric constant of the water and the concentration of the electrolyte are given by their bulk values right up to the surface, the Debye-Hiickel analysis (cf. eq 3 and also ref 18) gives - 8 i ~ Pkxkx €1 ( k + p)*
Gxx(k;h ) = - -e-ph
-8n P2ikx Gxz(k; h) = - -e-ph €1 (k + p)’ -8n P3 Gzz(k; h) = - -e-Ph €1 ( k + p ) 2
where each integral is over a thin layer of width A
