J. Phys. Chem. 1996, 100, 15005-15010
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Computer Simulations of Water between Hydrophobic Surfaces: The Hydrophobic Force Jan Forsman,*,† Bo Jo1 nsson,† and C. E. Woodward‡ Department of Physical Chemistry 2, Chemical Centre, Lund UniVersity, POB 124 Lund S-22100, Sweden, and Department of Chemistry, UniVersity College, UniVersity of New South Wales, Canberra ACT 2600, Australia ReceiVed: February 15, 1996; In Final Form: April 23, 1996X
The force between two hydrophobic surfaces enclosing an aqueous solution has been calculated via Monte Carlo simulations using the isotension ensemble. The confined water phase is in equilibrium with a bulk at constant chemical potential, and the average density between the surfaces is gradually decreased as they are brought closer together. The density depression in the slit gives rise to a strong attractive interaction exceeding the standard van der Waals force by an order of magnitude. This force mechanism is consistent with some of the experimental results obtained for the interaction of two hydrophobized surfaces in water. The results cannot, however, be used as a comparison with the very long-range forces extending over several thousand angstro¨ms found in some experiments.
1. Introduction The strong attraction between two hydrophobic surfaces immersed in water, the so-called “hydrophobic force”, has been observed in many experiments for over a decade.1-11 It has been shown to exceed the expected van der Waals attraction at large separations by at least an order of magnitude. The range of the force has proved most difficult to explain via some microscopic mechanism, despite a number of attempts in the recent literature.12-21 These include instability of adsorbed electrolyte on the surfaces,12 anomalous polarization of water induced by the hydrophobic surfaces,13 and interactions between ordered domains of adsorbed hydrophobic molecules.14 These theories attribute the force to charge (or polarization) fluctuations on or near the surfaces, mediated by the intervening electrolyte. Orientational ordering, propagated by hydrogen bonds,15 hydrodynamic fluctuations,16 long-ranged correlations near the spinodal point,17 and microscopic cavity formation18-21 have also been suggested as possible mechanisms for the attractive force. These theories are based on some assumed microscopic behavior of the surfaces or the intervening solution, which, in just about all cases, are difficult (if not practically impossible) to ascertain experimentally. The usual theoretical tool that is often brought to bear in such situations is computer simulation. To the best of our knowledge, however, computer simulations have not as yet been used to investigate the interaction between hydrophobic surfaces that confine a nontrivial water model. Such simulations are problematic given that the fluid between the surfaces should be in chemical equilibrium with a bulk reservoir. The conventional simulation technique to use for this type of problem is the grand canonical Monte Carlo22 method. Since this technique involves particle insertion and deletions, its implementation becomes very time consuming for a dense liquid with extremely large and orientation dependent interactions, such as water. Recently, an alternative method, more suitable for dense fluids was developed by Svensson and Woodward.23 These authors used the isotension ensemble24 (IE) in which a mean lateral pressure stabilizes fluctuations in the areas of the confining surfaces. In this technique, arbitrary changes to the Hamiltonian, e.g., changes in the separation between the walls, †
Lund University. University of New South Wales. X Abstract published in AdVance ACS Abstracts, July 15, 1996. ‡
S0022-3654(96)00462-5 CCC: $12.00
are carried out at constant chemical potential by a suitable adjustment of the lateral pressure. This is accomplished by use of the so-called free energy difference method. The advantage of the IE is that continous density fluctuations are used to ensure the constancy of the chemical potential rather than discrete particle insertions and deletions as in the grand canonical or Gibbs ensemble25,26 implementations. In this article we report the results of IE simulations applied to a model of water between two hard structureless walls. With this model system we are clearly not able to test those theories purporting that structure within surfaces is responsible for the strong attraction. We are, however, able to investigate the influence of the structure of pure water on the interaction between stable hydrophobic surfaces. 2. Theory Consider a system in which a fluid is confined between two parallel walls each of area S and separated by a distance h. We define the z-axis to be perpendicular to the walls. By PL, we denote the average component of the pressure tensor acting parallel with the walls, i.e., the mean lateral pressure. It is defined by
PL ) (1/h)∫0 dz PL(z) ) -(1/h)(∂A/∂S)N,T,h h
(1)
where A and S is Helmholtz free energy and the area, respectively. N is the number of particles, and T is the temperature. In the IE, the area is allowed to fluctuate while PL is fixed. PL(z) is a local quantity, which is not uniquely defined. However, the integrated quantity, PL, is well defined and quite independent of the way by which PL(z) is chosen.30 We could obtain PL(z) from our IE simulations using the socalled Kirkwood-Buff expression had we needed it. It would then be a function dependent on both S and z, i.e., PL(z,S), which in the thermodynamic limit equals the PL(z) one would obtain in the corresponding canonical ensemble. We stress, though, that this local quantity is never used in our simulations. The partition function, Q, may be written as
Q)
Qtrans h∫dr1...drN∫dS exp(-βU(r1,...,rN) - βPLhS) N! (2)
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Forsman et al.
where U(r1,...,rN) is the total interaction energy and β-1 is Boltzmann’s constant multiplied by the temperature. Qtrans is the translational part of the partition function. The natural thermodynamic potential for this ensemble is the Gibbs free energy, G:
βG ) -ln[Q(h,N,PL,T)] ) βNµ
(3)
where µ is the chemical potential. The confined fluid, being in equilibrium with a bulk reservoir, maintains a constant chemical potential as the separation between the walls is changed. Hence, for two separations, say h1 and h2,
Q(h2)/Q(h1) ) 1
(4)
To satisfy eq 4, PL must vary with h. We may write eq 4) as
(h2/h1)N+1∫dS f(S,h1)e(S,h1,h2)exp(-RS) ) 1
(5)
where R ) β(PL(h2)h2 - PL(h1)h1). By use of the scaled coordinates
q ) (x/xS, y/xS, z/h)
(6)
explicit expressions can be obtained for the functions appearing in eq 5:
f(S,h1) ) SN∫q1...dqN exp[-βU(q1,...,qN,h1,S) - βPL(h1)h1S]
∫dq1...dqN ∫dS SN exp[-βU(q1,...,qN,h1,S) - βPL(h1)h1S] (7) e(S,h1,h2) )
∫dq1...dqN exp[-βU(q1,...,qN,h1,S)] exp[-β∆U(h1,h2)] ∫dq1...dqN exp[-βU(q1...qN,h1,S)]
〈
〉
d(PLh) 1 1 ) kT + ∫f(S)F(S)‚dS dh h〈S〉 〈S〉
(11)
where F(S) is the force acting on the wall with area S. The first term on the right-hand side of eq 11 amounts to a finite size correction that was negligible in our simulations. 3. Simulations The isotension ensemble simulations were performed with the conventional Metropolis sampling,27 and we used the simple point charge (SPC) water model.28 The walls forming the slit were hard with respect to the oxygen atoms only, and periodic boundary conditions were applied in the lateral direction. A cylindrical cutoff to the potential energy was also applied in the lateral direction with the diameter of the “cutoff cylinder” equal to the updated lateral length of the simulation box. A minimum mean lateral length of the simulation box was chosen, and the number of particles was adjusted at each separation so that this length was always exceeded. To estimate the size dependence, we compared the results using a minimum mean lateral length of 17 and 22 Å, respectively. For the larger system this requirement made the number of particles vary from 173, at 10 Å, to more than 400 at large separations. The step length by which the separation, h, was changed during the simulations was 0.1 Å. Simulations for both increasing and decreasing (negative steps) h were performed to check for systematic errors. The number of configurations per particle and separation was 1.0 × 105. During area fluctuations and virtual wall displacements only the oxygen coordinates were scaled, with the bonded hydrogens following to maintain the intramolecular geometry. We chose 10 Å separation, with a lateral pressure of 1 bar, as our thermodynamic starting point. The smaller system was simulated up to 21.3 Å, while we continued up to 23.4 Å with the larger one.
(8) ∆U(h1,h2) ) U(q1,...,qN,h2,S) - U(q1,...,qN,h1,S) where U(q1,...,qN,h2,S) is the energy evaluated after scaling the z coordinate of all particles by h2/h1. Thus, ∆U is the energy change in the system were it to undergo a uniform transverse scaling from h1to h2. The functions f(S,h1) and e(S,h1,h2) are obtained directly from simulations, and eq 5 is then solved iteratively for PL(h2). The Helmholtz free energy of the fluid between the walls is the Legendre transform of G, and thus,
A(h,N,〈S〉,T) - µN + PBh〈S〉 ) (PB - PL)h〈S〉
(9)
is the h dependent part of the free energy of the entire system, where the brackets denote an isotension ensemble average and PB is the pressure of the reservoir. The right-hand side of eq 9 becomes 2γ in the limit of large separation, where γ is the water-wall surface tension. Taking the derivative of both sides of eq 9 with respect to h at constant chemical potential, one obtains
PN )
d(PLh) - PB dh
(10)
PN is the net component of the pressure tensor normal to the walls. In practice, PB is not known and must be obtained by extrapolation or by simulating at large enough separation. The derivative on the right-hand side of eq 10 may be evaluated either by discrete differentiation of PL or by differentiating the partition function with respect to h, leading to
4. Results Figure 1 shows that PL is a monotonically increasing function of h. As the walls are pushed together, fluid molecules lose binding energy. This effect can be thought of as an increase in the relative contribution of the wall-fluid surface free energy. This effect by itself would increase the fluid chemical potential. To counteract this, PL, and thus the fluid density, is lowered, and in this way the chemical potential is kept constant. As h is decreased, one will expect that the surface free energy will eventually be so dominant as to allow for the possibility of a first-order transition to a vapor phase in the slit, sometimes denoted as cavitation. For a large enough number of sampled configurations, our simulations will in principle realize such a transition. This would be seen as a sudden jump in the area S of the simulation box. This is possible because of the finite size of our system. At coexistence, a free energy barrier exists between the gas and liquid phases, mainly owing to the surface free energy needed to create regions of vapor in the liquid phase. This barrier should scale approximately as L ) S1/2, thus becoming infinite in the thermodynamic limit. Notice that the scaling of the barrier height suggests that the transition is twodimensional in character, provided the lateral length of the simulation box is large enough. This should be most apparent near the critical temperature of the confined fluid, when the correlation length in the fluid is larger than the distance between the surfaces. At separations smaller than that at which coexistence occurs, the vapor phase becomes the thermodynamically favored state.
Water between Hydrophobic Surfaces
Figure 1. Lateral pressure, PL, as a function of slit width. Two systems have been simulated, and the solid and dashed lines represent the large and small system, respectively (see text).
This notwithstanding, free energy barriers in the microscopic nucleation process will impart an apparent metastability to the liquid phase in our simulations. The vapor is not the thermodynamically favored phase in the bulk but is stabilized in the slit purely by the lower surface-fluid free energy. Thus, one would expect that critical nucleating vapor cavities would have to span the gap between the surfaces. It is only nucleating “bubbles” of this type that would be able to grow to a vapor phase. This can be seen as follows. Suppose we treat such a nucleating bubble as a cylinder of length h and radius R. As R grows, one would obtain a negative contribution to the free energy change that would scale as R2. Opposing this is a positive liquid-vapor surface term, growing only as R. Clearly, such a cavity will reach a critical size beyond which it will grow to form an infinite vapor phase. Nucleating vapor cavities growing at a single surface will generally only increase the free energy as they grow. The free energy will not be able to decrease as long as there is a significant amount of fluid between the bubble and at least one surface. Thus, for systems that are much larger than the critical nucleating bubble, the free energy barrier to area fluctuations is given by the excess free energy of the critical bubble. This barrier height may be large but in principle always remains finite even in the thermodynamic limit, where the free energy of the vapor phase is infinitely lower than that of the liquid phase. The system will remain in the liquid phase as long as it does not experience a critical density fluctuation due to a bubble that spans the gap and is of sufficient size to irreversibly take it to the vapor phase. Depending on ambient conditions the probability of sampling such fluctuations may be quite small and the system has an inherent metastability. Hysteresis effects observed in actual surface force measurements7,20 with hydrophobic surfaces immersed in water are evidence for this metastability. In those experiments the liquid phase is observed to persist as the surfaces are brought together from large separations until the attractive force is so large that they jump into contact. When they are pulled apart, a vapor cavity forms between the surfaces. This bubble remains stable upon further separation to even beyond those where the liquid was observed on the inward path. If the experiment were to be performed using very long times at each separation, this hysteresis should essentially disappear. Similarly, a simulation with an extremely large number of sampled configurations at each h should eventually overcome the nucleation barrier. To make contact with experiments that measure on the liquid branch, it is necessary to try to maintain the liquid phase in the simulations, even at separations where the vapor phase is favored. This was
J. Phys. Chem., Vol. 100, No. 36, 1996 15007
Figure 2. Schematic drawing of the phase diagram for the confined water model at a specific separation h, where the simulated fluid is metastable (point Q). There is a free energy barrier to a more stable gas phase at point P, which will become smaller as h decreases. A small barrier may survive even when the simulated pressure becomes negative.
achieved by restricting the number of sampled conformations to 105 per particle. This number was a practical rather than a thermodynamic upper bound. Despite this, simple mechanical quantities such as the energy remained well behaved over these configurations and produced stable averages for all separations. Of course, one could argue that the liquid may not be metastable at any of these separations. However, we also observed unremarkable behavior at even smaller separations where the pressure was even (slightly) negative and where there is no doubt that the liquid is metastable. At even smaller separations the liquid eventually becomes unstable and the volume expands uncontrollably. Figure 2 is a schematic that shows how a metastable liquid phase can have a lower pressure than the coexistence value and indeed how this pressure can approach negative numbers. The liquid could in principle hop over the free energy barrier to the more stable vapor phase, denoted by P in the diagram. However, the vapor will have a lower chemical potential than that of the bulk liquid. As h approaches infinity, PL should approach the pressure of the bulk liquid. A rough extrapolation of the curves in Figure 1 gives values on the order of 1000 bar, which is much larger than in experiments. Simulations of systems approaching a bulk pressure of 1 atm would probably require millions of molecules, owing to the instability of the fluid at shorter separations. This was of course beyond our resources, and thus, all our simulations were restricted to these high bulk pressures. There is a size dependence in PL as a function of h, and the limiting pressures are clearly not equal. However, since we are really interested in the net force, the problem is not as serious as it might seem (see below). That the differences are due to size effects and not other systematic errors, such as too large step lengths, was confirmed by comparing independent test simulations of systems of equal sizes (not shown). Experimentally, the forces are often measured between crossed cylindrical surfaces with the so-called surface force apparatus.2 By use of the Derjaguin approximation, those forces are proportional to the free energy per unit area in a slit with flat surfaces, i.e., (PB - PL)h. An estimate for the free energy from our simulation results requires knowledge of the bulk pressure. We performed a least-squares fit of the lateral pressure to the approximate expression
PLh ) PBh - 2γ + A/h2
(12)
The last term emulates an effective van der Waals interaction. Fitting the simulated lateral pressure to eq 12 is one way of
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Figure 3. Net normal pressure as a function of slit width. Dashed and solid curves are d(P||h)/dh - PB for the smaller and larger system, respectively, whereas the dot-dashed one is the net pressure as obtanied from the virial eq 11 for the smaller system. All three curves are smoothed, and the values given are averages over data points within (1.5 Å. Individual numerically calculated d(P||h)/dh values (open circles) are shown for the larger system.
suppressing the statistical noise when estimating PB and γ. We have performed the fit to the two simulated systems over the entire interval (10-21.3 and 10-23.4 Å, respectively) and have also tried replacing the inverse square term by an inverse cubic one. The estimates obtained for the wall-water surface tension are similar for both densities: 75 and 83 mJ/m2, respectively, with the larger estimate for the larger system. These compare favorably with the experimentally measured surface tension at an air-water interface of 72 mJ/m2. We regard the air-water interface as the appropriate physical analogue to our model system. However, given the greater constraints on the fluid, the surface tension at an inert hard wall should be larger than at an air-water interface. The use of a cubic instead of a square term has only a marginal effect on the fitted values of the surface tension and bulk pressure. We do not attribute any physical significance to the fitted coefficient for the square term. It contains too much statistical noise and can also without loss of accuracy be exchanged for a cubic term. This is despite the fact that we expect the surface free energy per unit area to approach 2γ as 1/h2 at long range, but the separations used in the fit are too small to allow a proper estimate of the asymptotic term. The fitted bulk pressures, PB, differ between the low- and high-density systems by approximately 86 bar, with the highdensity system having the larger pressure of 1527 bar. Figure 3 gives d(PLh)/dh - PB, i.e., the net normal pressure. The curves indicate that we may have underestimated the bulk pressures and hence the attraction. This to some extent is due to the fact that we have performed the fit over the entire simulated region, and then the last (rapidly varying) term is strongly determined by the inner part of the curve. However, we believe all the data points should be used rather than some arbitarily chosen tail part to which the fit is made. It is better to view our results as a lower estimate of the attraction and bear in mind that it is likely to be stronger. The larger system with a higher density, and thus higher bulk pressure, is more attractive than the smaller one. This is partly due to the fact that the high-density system was simulated to larger separations. Fitting the high-density system data to the same region (1021.3 Å) as the one with the lower density reduces the difference to about half of that shown in Figure 3. At short separations one may anticipate that the normal pressure, PN, should have an oscillatory appearance due to packing of water molecules. There is, however, no clear evidence of such behavior in our
Forsman et al.
Figure 4. Interaction free energy, W ) (PB - PL)h - 2γ, that would be measured between two curved surfaces in a surface force experiment. Curves were estimated from the bulk pressure and surface tension obtained by fitting the data to eq 12. Solid and dashed lines denote data from the larger and smaller system, respectively. As a comparison, we have included the van der Waals attraction (thin dashed line) using a water-air Hamaker constant of 3.7 × 10-20 J given in ref 1.
Figure 5. Average density as a function of separation and the curve smoothed in the same way as the net pressure in Figure 3.
simulations. We also show in Figure 3 the result of obtaining the osmotic pressure from the virial equation. These values are subject to large statistical fluctuations. However, within these large uncertainties, they agree with those obtained by numerical differentiation. Note that this agreement is an independent check of the constancy of the chemical potential. After fitting the bulk pressures and surface tensions, we can obtain the interaction free energies as W ) (PB - PL)h - 2γ. In Figure 4 we display the interaction free energies, which all show a strong attraction, exceeding the expected van der Waals contribution by approximately an order of magnitude. We interpret this as being due to the depression of the average water density in the slit. This effect is induced by the hydrophobic surfaces and is propogated outward into the fluid. The interactions between these density tails leads to an attraction between surfaces, which seem to dominate at these separations (see Figure 5). This point of view is supported by the similarity between the density and net normal pressure curves. Note that the density depression term is neglected in the standard Hamaker approach in which the Hamaker constant is proportional to the square of the bulk density. The increased attraction at short range is consistent with some experimental measurements performed at small separations.1,3,4,9 In particular we note recent measurements by Wood et al., which showed no anomolous long-ranged attraction between stable (polymerized) hydrophobic surfaces. Those authors did observe a strong short-ranged
Water between Hydrophobic Surfaces force, which caused the surfaces to jump into contact. This occurred at distances of about 130 Å, still much too large for a direct comparison with our simulations. Those authors were able to account for their experimental results with an effective Hamaker constant of about (2-3) × 10-20 J. Assuming our results can be fitted to an effective van der Waals interaction, valid at all separations, would imply that the attraction seen in our simulations is even larger than that needed to account for these experimental results! This could be due to several factors. First, the surfaces in the experimental system may not be as hydrophobic as the perfectly hard walls used in the simulations. Second, one may not be able to trust a naive extrapolation of a simple van der Waals interaction fitted at relatively small separations. The density depression effect is expected to be largest at small separation, and the van der Waals-like contribution that survives at long range may be overestimated by the fitting. With all this taken into consideration, we suggest that our results are consistent with the observations of Wood et al.1 using polymerized hydrophobic surfaces. Any comparison with the experiments reporting a very long-ranged force is of course speculative. We merely note that in our simulations, we did not observe any indication of bubble formation nor very large surface polarization. It is plausible that simulations carried out at a lower bulk pressure would display interactions with a longer range, owing to the lower compressibility of the fluid. However, we do not expect this effect to be able to explain correlation lengths of some hundreds of angstro¨ms or more, often reported for these long-ranged forces. In Figure 6, we present some density and orientational profiles at 14 Å. These were qualitatively the same at all separations. The depletion of oxygen close to the walls is very small. A depletion is expected because of the lack of favorable waterwater interactions near the wall. Counteracting this is a hard core packing effect, which favors placing molecules close to the walls in order to maximize the available volume in the fluid. We did some small test simulations at higher densities (not shown), and at a sufficiently high density, the excluded volume effect dominates and there is an enhanced oxygen density at the walls. Note that this means that at lower pressures, the walls are effectivly more repulsive. Parts b and c of Figure 6 display the profiles of 〈cos θ〉 and 〈cos2 θ〉, respectively. Here, θ is the angle that the molecular dipole makes with the direction perpendicular to the walls, with θ ) 0 corresponding to H atoms pointing toward the walls. For water molecules adjacent to the walls, the molecular dipole is preferentially aligned parallel to the walls with a slight tendency for the hydrogen atoms to point toward the walls. This type of orientation is driven by attractive electrostatic interactions for the water molecules closest to the walls. 5. Conclusions The main qualitative result of our simulations is a strongly attractive surface free energy. The main reason for its relatively large strength is the high cohesive energy of water. In other words, the response of the force to density depression induced by the surfaces is very large. The forces observed at close separation compares favorably with our results. We do not observe any anomalous surface polarization, nor cavitation, in our simulations, and the attraction recently measured20 at 1500-2000 Å separation is probably not due to any simple solvation force. The “bubble formation” mechanism proposed might be responsible for such a force, but it is worth noting that there is no evidence that liquid water is even metastable under these conditions.29 One would expect cavitation away from contact to be a slow process, and since the
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Figure 6. Symmetrized density and orientation profiles as a function of z at h ) 14.0 Å, where the midplane of the slit is to the left and cos(u) is the projection of the water dipole vector on the z-axis, where u is defined as being zero when the vector points toward the nearest wall: (a) oxygen and hydrogen density profiles normalized to a mean value of 1; (b) cos(u); (c) cos2(u).
measurements have to be performed during a rather short period of time, the transition may never be seen. If the liquid is unstable, the process is diffusion controlled, and as the surfaces are brought closer together, liquid with a high chemical potential is pushed out into the more stable bulk. There is really no valid thermodynamic treatment of such a process, and almost any force could be obtained. To our knowledge, this article provides the first constant chemical potential simulation study of a nontrivial water model between hydrophobic surfaces. The free energy difference method, coupled with the isotension ensemble, allows us to directly determine the free energy per unit area and the net force at constant chemical potential. The computational effort grows
15010 J. Phys. Chem., Vol. 100, No. 36, 1996 very rapidly with the system size, and we were forced to simulate at rather short separations. This means that in order to stabilize the liquid state, we had to apply high pressures. Notably, at high pressures a hard wall is effectively less repulsive, since the extra volume at the walls becomes increasingly more important as the density increases. Acknowledgment. We thank Torbjo¨rn Åkesson, Sture Nordholm, and Jan-Christer Eriksson for valuable comments, suggestions, and criticisms. References and Notes (1) Wood, J.; Sharma, R. Langmuir 1995, 11, 4797. (2) Israelichvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991. (3) Israelichvili, J.; Pashley, R. M. J. Colloid Interface Sci. 1984, 98, 500. (4) Pashley, R. M.; Mcguiggan, P. M.; Ninham, B. W.; Evans, D. F. Science 1985, 299, 1088. (5) Parker, J. L.; Claesson, P. M. Langmuir 1994, 10, 635. (6) Claesson, P. M.; Christenson, H. K. J. Phys. Chem. 1988, 92, 1650. (7) Christenson, H. K.; Claesson, P. M. Science 1988, 239, 390. (8) Parker, J. L.; Claesson, P. M. Langmuir 1992, 8, 757. (9) Claesson, P. M.; Blom, C. E.; Herder, P. C.; Ninham, B. W. J. Colloid Interface Sci. 1986, 114, 234. (10) Christenson, H. K.; Claesson, P. M.; Parker, J. L. J. Phys. Chem. 1992, 96, 6725. (11) Christenson, H. K.; Fang, J.; Ninham, B. W.; Parker, J. L. J. Phys. Chem. 1990, 94, 8004.
Forsman et al. (12) Podgornik, R. J. Chem. Phys. 1989, 91, 5840. (13) Attard, P. J. Chem. Phys. 1989, 93, 6441. (14) Tsao, Y.; Evans, D. F.; Wennerstro¨m, H. Science 1993, 262, 547. (15) Eriksson, J. C.; Ljunggren, S.; Claesson, P. M. J. Chem. Soc, Faraday Trans. 2 1989, 85, 163. (16) Ruckenstein, E.; Churaev, N. J. Colloid Interface Sci. 1991, 147, 535. (17) Be’rard, D. R.; Attard, P.; Patey, G. N. J. Chem. Phys. 1993, 98, 7236. (18) Yaminsky, V. V.; Yuschenko, V. S.; Amelina, E. A.; Shchukin, D. J. Colloid Interface Sci. 1983, 96, 301. (19) Yuschenko, V. S.; Yaminsky, V. V.; Shchukin, E. D. J. Colloid Interface Sci. 1983, 96, 307. (20) Parker, J. L.; Claesson, P. M.; Attard, P. J. Phys. Chem. 1994, 98, 8468. (21) Craig, V. S.; Ninham, B. W.; Pashley, R. M. J. Phys. Chem. 1993, 97, 10192. (22) Norman, G. E.; Filinov, V. S. High Temp. Res. USSR 1969, 7, 216. (23) Svensson, B.; Woodward, C. E. J. Chem. Phys. 1994, 100, 4575. (24) Parrinello, M.; Rahman, A. Phys. ReV. Lett. 1980, 45, 1196. (25) Panagiotopoulos, A. Z. Mol. Phys. 1987, 61, 813. (26) Panagiotopoulos, A. Z. Mol. Phys. 1987, 62, 701. (27) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1953, 21, 1087. (28) Postma, J. P. M.; Berendsen, H. J. C.; Haak, J. R. Faraday Symp. Chem. Soc. 1982, 17, 55. (29) Eriksson, J. C.; Ljunggren, S. Langmuir 1995, 11, 2325. (30) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982.
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