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J. Phys. Chem. B 2006, 110, 13062-13067
Selective Coalescence of Bubbles in Simple Electrolytes Stjepan Marcˇ elja* Department of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National UniVersity, Canberra 0200, Australia ReceiVed: February 17, 2006; In Final Form: May 16, 2006
Simple ions in electrolytes exhibit different degrees of affinity for the approach to the free surface of water. This results in strong ion-specific effects that are particularly dramatic in the selective inhibition of bubble coalescence. I present here the calculation of electrostatic interaction between free surfaces of electrolytes caused by the ion accumulation or depletion near a surface. When both anion and cation are attracted to the surface (like H+ and Cl- in HCl solutions), van der Waals attraction facilitates approach of the surfaces and the coalescence of air bubbles. When only an anion or cation is attracted to the surface (like Cl- in NaCl solutions), an electric double layer forms, resulting in repulsive interaction between free surfaces. I applied the method of effective potentials (evaluated from published ion density profiles obtained in simulations) to calculate the ionic contribution to the surface-surface interaction in NaCl and HCl solutions. In NaCl, but not in HCl, the double-layer interaction creates a repulsive barrier to the approach of bubbles, in agreement with the experiments. Moreover, the concentration where ionic repulsion in NaCl becomes comparable in magnitude to the short-range hydrophobic attraction corresponds to the experimentally found transition region toward the inhibition of coalescence.
Introduction One of the most striking manifestations of ion-specific effects in electrolytes is the selective inhibition of bubble coalescence. Sea waves create much more foam than their freshwater counterparts, and the difference is easily demonstrated on the laboratory scale by using pure water and NaCl solutions. Such differences in bubbling characteristics of simple liquids have long attracted attention. For example, in 1911, Pollock1 extensively photographed bubble formation and noticed the difference in behavior between dilute solutions of acetic acid and sulfuric acid. In 1923, Cohen2 reported that electrochemically formed bubbles are small in KOH and large in H2SO4. Comparing freshand saltwater, Monahan and Zietlow3 measured bubble spectra, while Scott4 presented pictures showing dramatic differences between bubble columns. Hofmaier, Yaminsky, and Christenson5 performed a detailed study of bubble interaction at the capillary orifice. They reported videotapes showing bubbles in NaCl solutions bouncing off of each other rather then coalescing as they do in pure water. This latter article also contains a more extensive historical survey. A systematic investigation by Craig, Pashley, and Ninham,6,7 who compared the inhibition of coalescence for a number of electrolytes, brought significant insight into the mechanisms controlling the bubble coalescence. The apparatus consisted of a vertical glass column with a high density of rising nitrogen bubbles produced at a sinter at the base of the column. The coalescence was monitored by a collimated light source passing through the column and recorded by a photodiode. The authors discovered the combination rules that assigned to each cation or anion a tag R or β. Electrolytes comprising an R-R or β-β combination exhibit inhibition of coalescence, while the combinations R-β or β-R have no effect on the coalescence. No exceptions to the rules were found. With increasing electrolyte * Corresponding author. E-mail:
[email protected].
concentration, inhibition of coalescence develops over a fairly narrow range, typically between 50 and 150 mM. The nature of the inhibition and the logic behind the combination rules remained unexplained during the ensuing decade. But with new advances in the knowledge of the propensity of different simple ions for the aqueous surface,8 it has now become possible to explore ion-specific effects in the electrostatic part of the bubble-bubble interaction in simple electrolytes. In a recent conference report,9 I argued that the combination rules arise from the affinity of simple monovalent ions for the free surface of water: R anions (Cl-, Br-, OH-, NO3-, ...) and β cations (H+, (CH3)4N+, (CH3)3NH+, (CH3)2NH2+, CH3NH3+, ...) concentrate at the surface, while β anions (ClO3-, ClO4-, CH3COO-, ...) and R cations (Na+, K+, Li+, Cs+, NH4+, ...) avoid the surface. R-R or a β-β pairs thus create charge separation at the free surface, and the resulting electrostatic double-layer interaction adds a repulsive contribution to the force between the bubbles. No simple divalent ion is likely to be attracted to the surface, and the charge separation in these less-studied ions should include the classical image charge repulsion of divalent ions from the surface. While the picture of R and β ions with different propensity for the free surface explains the behavior of solutions with simpler ions, more complex ions need to be explored individually. An example here is acetate, which is classified as β and should accordingly avoid the surface, in disagreement with simulations and experiments. Although acetate is attracted to the surface, it is a larger, anisotropic ion that is on average strongly oriented with the methyl end right on the surface and the carboxyl end about 2 Å further inside.10 In experiments, magnesium acetate shows no coalescence inhibition. Most likely, the reason is the absence of significant charge separation between magnesium and acetate ions at the surface.10 In this work, we consider simple monovalent electrolytes and examine the contribution of solute ions to the interaction
10.1021/jp0610158 CCC: $33.50 © 2006 American Chemical Society Published on Web 06/15/2006
Coalescence of Bubbles in Simple Electrolytes
J. Phys. Chem. B, Vol. 110, No. 26, 2006 13063
between free surfaces. One test case for each type of behavior was selected for detailed study: HCl (no effect on bubble coalescence) and NaCl (inhibits coalescence). This is the simplest pair of electrolytes with opposing behavior, and relatively good simulation data about constituent ions are available both in bulk water and at the free surface. In the following section, we look at a simple qualitative behavior of the interaction by using the primitive model electrolytes. Subsequently, we attempt a state-of-the-art calculation of the electrostatic interaction in aqueous solvent by using the method of effective potentials between ions in solution and ions and the surface. We end the article with a discussion of the mechanism controlling the coalescence. A Simple Model While the coalescence rules depend on the specific nature of the aqueous solvent, it is nevertheless instructive to review in a much simpler model different types of electrostatic interaction of ionic layers at the surface of two bubbles. Ion-specific surface layers cannot form within the primitive model of electrolytes where hard-core ions move in the dielectric continuum, but we could evaluate the interaction created by such layers by setting them up artificially. The layers were chosen to mimic the surface-specific effects of aqueous solvent. In the simulations of HCl solutions, both hydronium and chloride ions showed preference for the free surface. In contrast, in NaCl solution, chloride ions were attracted to the surface, while sodium ions were excluded from the immediate surface zone. The resulting ion density profiles were recently approximately fitted11 within the primitive model by assuming simple square-well potentials at the surface for chloride and hydronium ions and an exclusion zone for the sodium ion. In a qualitative illustration of the primitive model results, we slightly simplified the potential fit of ref 11 and used it to calculate the interaction of free surfaces in NaCl and HCl solutions. We assumed planar surfaces and 2.5 Å wide squarewell potentials at each surface. For the hydronium and chloride ions, the depth of the well was 2.5 kBT, while the sodium ion was excluded from the 2.5 Å wide surface zone. The system was assumed to be in equilibrium with a bulk reservoir at 0.1 M, which is the concentration where NaCl solutions already exhibit strong inhibition of coalescence. Except for the ions moving freely in and out of the potential wells, there was no other surface charge in the picture. The method of calculation was an anisotropic hypernetted chain scheme described in our earlier work.12 The hard-core contact separations in Å units were set as: Na+-Cl- 2.6 Å, H+-Cl- 2.9 Å, Na+-Na+ 3.0 Å, Cl-Cl- 4.4 Å, H+-H+ 2.8 Å, and the temperature was 25 °C. The results for the electrostatic pressure between such free surfaces, shown in Figure 1, bring no surprises. In hydrochloric acid solutions, both H+ and Cl- enter the potential well at the surface. The interaction between such charge-neutral layers is dominated by the attractive contribution arising from correlations in positions of nearby ions.13 The decay of the interaction is fast (∼2 λD), and the result conforms to the known asymptotic form for the interaction free energy. In NaCl solutions, chloride ions form a charged layer at the surface, while sodium ions are excluded. The electric double layer forms, and the interaction between surfaces is repulsive. The influence of ion correlations is small, and the pressure shown in Figure 1 is very close to the Poisson-Boltzmann result for the identical model. As the overall surface charge density due to specific ion preferences for the surface or otherwise is fairly low (from ion
Figure 1. Pressure between free surfaces with adsorption as calculated in the primitive model electrolyte. When both anion and cation adsorb,the interaction is attractive (e.g., HCl), and when a single species adsorbs, an electric double layer forms and the interaction is repulsive (e.g., NaCl).
density profiles in ref 8, we calculate the order of magnitude as one elementary charge per 1000 Å2), repulsive interaction within the primitive model can be evaluated using the PoissonBoltzmann theory and attractive interaction can be evaluated using analytical approximations. The pressure example was shown here as a reminder that, already within the very simple primitive model, one finds attractive or repulsive forces depending on the preferential positioning of ions near the surface. Method of Calculation Including the Structure of Aqueous Solvent Theoretical Background. As typical electrolyte concentrations showing the inhibition of coalescence are around 0.1 M or higher, the screening is strong and electrostatic interaction between ions in surface layers occurs at surface separations of less than about 30 Å. At such small separations, solvent structure is important and the primitive model cannot be expected to provide accurate results. The theoretical basis for the present calculation including aqueous solvent structure is described in my recent work14 and will only be briefly recounted here. Ionic solution between the two surfaces is treated with the McMillan-Mayer15 theory adapted to describe the inhomogeneous ion distribution modified by the presence of the surfaces. Within the classical statistical mechanics, the theory is an exact expansion. The grand partition function of the inhomogeneous system containing water and different ion species between the surfaces is expressed as a product of the partition function for the reference system (water between surfaces) and the partition function for the dissolved ions interacting with the potentials of mean force evaluated in the reference system. The formal expression for the inhomogeneous system has the standard McMillan-Mayer form
Ξ(z) Ξ(z*)
)
[ ( )] 1
zs - z/s
s
γ/s
∑ ∏m!
mg0
ms
∫×
/ ({m})] d{m} (1) exp[-βw{m}
where quantities with the asterisk correspond to the reference state. For the set of (m1, m2, ...) ) {m} molecules, z denotes the activities, zs and γs the activity and the activity coefficient of ionic species s, respectively, and w(m)({m}) the corresponding potential of mean force defined as
13064 J. Phys. Chem. B, Vol. 110, No. 26, 2006
w{/ m}
[ ]
({ R} ) ) -kBT ln
Marcˇelja
F{/ m}({ R} )
∏
(2)
(F/s )ms
where {R} specifies the positions for the set {m}. In the inhomogeneous system, the many-particle density / F{m} in the reference state includes contribution from the interaction of each ion with the surfaces. From the form of the partition function, the pressure between the surfaces is immediately seen to consist of two contributions
P ) -kBT
∂ log Ξ(z) ) Pref + Pions ∂x
(3)
where x is the separation of the surfaces. In the present case, the reference state is pure water between the surfaces, and the full system is electrolyte solution between the surfaces. From each pressure term, one must subtract the pressure in the bulk reservoir. To evaluate the sums involving many-particle potentials of mean force, the usual method is the Kirkwood superposition approximation that reduces the problem to pairwise potentials. We argued that, because of the truncation of higher-order terms in the interactions, it is better to replace the potentials of mean force appearing in eq 2 with the effectiVe potentials evaluated in a reverse Monte Carlo procedure. While potentials of mean force approximately evaluated from simulations correspond to dense systems that include screening, effective potentials relate to two-particle correlation functions (or the density profiles for single particle functions) at vanishing concentration and thus reproduce correct asymptotic behavior. In the present work, we do not attempt to evaluate the reference pressure between two nearby free surfaces in pure water (hydrophobic interaction between free surfaces) and concentrate instead on the ionic contribution to the pressure. In the Discussion Section, the reference system term will be estimated from the experimental information on the interaction of hydrophobic surfaces in water. In the calculation, we use the single and pairwise terms of the expansion and evaluate ion-ion effective potentials from the available bulk solution simulations, disregarding the expected weak dependence of the potentials on system geometry and the distance from the surface. Ion-surface effective potentials are obtained from published simulations of polarizable aqueous systems with free surfaces as described below. The present calculation of the interaction of ionic surface layers is particularly interesting, as we believe that this is the first application of the effective potential method in the presence of specific ion-surface interactions. In earlier work, the method was shown to correctly reproduce bulk activity and osmotic coefficients16 and provide new insights into the double-layer interaction at short surface separations.17-20 Effective Ion-Surface Potentials. As described above, the calculation of surface interaction with the effective potential method requires two sets of potentials: (i) ion-ion effective potentials for all ion pairs in the problem; (ii) ion-surface effective potentials for each kind of ion. The ion pair potentials can be approximately evaluated from the simulations of bulk ionic aqueous solutions via either reverse Monte Carlo (MC) or Ornstein-Zernike equations with a suitable closure.21 They are required in the subsequent evaluation of ion-surface potentials from the density profile data calculated at finite electrolyte concentrations.
Figure 2. Short-range part of the ion-ion effective potential in aqueous solutions. Red: H+-Cl-, violet: Na+-Cl-, green: Na+-Na+, yellow: Cl--Cl-, blue: H+-H+. At separations closer than those shown for each curve, we assumed hard-core repulsion.
Ion-ion effective potentials used were the sum of the shortrange components shown in Figure 2 and the Coulomb force (not shown). Na+-Cl-, Na+-Na+, and Cl--Cl- potentials were evaluated in our earlier work.20 The H+-H+ potential between two hydronium ions consists of the Coulomb repulsion and the short-range part, which is similar to the potential between two water molecules. It was approximated by the potential of mean force for SPC water kindly supplied by A. Lyubartsev. The short-range part of the H+-Cl- (more accurately H3O+-Cl-) potential is expected to be similar to the short-range part of the H2O-Cl- potential and was approximated as the short-range potential of mean force between chloride and water oxygen from a recent work.22 For the case under study where the surface charge is relatively small, there is no crowding of ions in the Stern layer and all the listed short-range corrections to the Coulomb potential are relatively unimportant. They are only significant for the opposite-charge pairs Na+-Cl- and H+Cl- because other pairs seldom approach sufficiently closely. Having selected ion-ion effective potentials, we evaluated ion-surface effective potentials in a reverse MC procedure. The simulation system was set up with the same geometry and electrolyte concentration (1.2 M) as the original polarizable simulation of the full aqueous solution.8 In the reverse MC procedure, effective ion-surface potentials were adjusted until the results accurately reproduced the ion density profiles found in the original work with aqueous solvent. Ion-surface potentials obtained through this procedure are shown in Figure 3 for NaCl solution and in Figure 4 for HCl solution. We also show the corresponding potentials of mean force calculated as Vpmf(x) ) -kBT log[F(x)], where F(x) is the relative density of a given species. Ion-surface potentials evaluated with the reverse MC procedure are similar to the potentials of mean force, but with the potential wells near the surface enhanced for Cl- in NaCl and for H+ in HCl. Without attractive electrostatic correlation with other ions, a deeper well is required to attract ions to the surface, where the dielectric constant is low on the vapor side. While reverse MC results for effective ion-surface potentials reproduce the original aqueous simulations, it should be noted that the simulated systems are not large enough for accurate evaluation of the potentials. As shown in the next section, most important events occur at surface separations around 25 Å. Available results from aqueous system simulations in narrow
Coalescence of Bubbles in Simple Electrolytes
Figure 3. Effective ion-surface potentials for NaCl calculated in the reverse Monte Carlo procedure (symbols) and as the potential of mean force (continuous lines); yellow: Cl-, green: Na+. Water density is shown in blue.
Figure 4. Effective ion-surface potentials for HCl calculated in the reverse Monte Carlo procedure (symbols) and as the potential of mean force (continuous lines); yellow: Cl-, red: H+. Water density is shown in blue.
water slabs become less reliable away from the surface, and the resulting error is the largest source of uncertainty in the present work. Electrostatic Interaction between the Surfaces We will now attempt to calculate as accurately as possible the electrostatic part of the interaction between free surfaces in aqueous electrolytes. The method adopted in the previous section reduces the problem of evaluating interaction between bubble surfaces in the full aqueous solution to two simpler tasks: evaluating the pressure of pure water reference system and calculating the pressure of ions using the effective potentials. Calculation of the effective ion-surface potentials completed the preparation for the task of calculating electrostatic contribution of ions to the pressure. The calculated potentials were included into the anisotropic HNC integral equation scheme that was previously shown23 to produce very accurate results for electrostatic interactions. Before the calculation of the interaction pressure, we tested the accuracy of the effective potential method in the anisotropic HNC calculation by setting the system width to
J. Phys. Chem. B, Vol. 110, No. 26, 2006 13065
Figure 5. Ionic contribution to the pressure between free surfaces in NaCl solutions at the concentrations of 0.1 M (green symbols) and 0.05 M (red symbols). Calculation for separation less than about 15 Å (dashed or dotted lines) is unrealistic because the solution film between free surfaces will collapse.
Figure 6. Ionic contribution to the pressure between free surfaces in 0.1 M HCl solutions at the concentrations of 0.1 M (green symbols) and 0.2 M (red symbols).
30 Å and evaluating ionic density profiles at 1.2 M concentration. The results (not shown) reproduced accurately the input profiles taken from the full aqueous system calculation.8 Next, we adjusted the input activity coefficients to the values corresponding to the desired bulk electrolyte concentrations. Temperature was kept at 25 °C. Electrostatic images were not included because the effect is accounted for in the effective ionsurface potentials. The results for the ionic contribution to the pressure between free surfaces in NaCl and HCl solutions are shown in Figures 5 and 6. Unlike the results in the primitive model, solvent structure included in the effective potentials introduces oscillatory behavior of the interaction pressure. The major difference between the two studied systems is the repulsive barrier in NaCl solution for surface separations between about 20 and 25 Å. In NaCl solutions, the experimentally determined concentration at the midpoint of the transition toward the inhibition regime varies slightly depending on the experimental setup but always remains in the vicinity of 0.08-0.09 M. The calculations show weak repulsive barrier at 0.05 M, increasing to the repulsive pressure of almost 0.1 MPa at 0.1 M and rapidly increasing further at concentrations beyond 0.1 M (not shown).
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Figure 7. Ion density profiles between free surfaces in 0.1 M NaCl solutions at the surface separation of 33 Å; yellow: Cl-, green: Na+.
Figure 8. Ion density profiles between free surfaces in 0.1 M NaCl solutions at the surface separation of 23 Å; yellow: Cl-, green: Na+.
In HCl at 0.1 M, calculation shows no repulsive barrier at surface separations larger than 15 Å. This is maintained at higher concentrations, and at 0.2 M, the oscillations are magnified but there is still no barrier. The results are not sensitive to the details of the ion-ion effective interaction potentials in the bulk but are sensitive to details of the ion-surface effective potentials. Aqueous system simulations of ion density distributions near a surface in larger systems will make the surface-surface calculations more reliable. Figure 7 shows ionic density profiles in 0.1 M NaCl solution at a separation where there is no significant interaction between the surfaces, and Figure 8 shows the profiles at maximum repulsion. In the latter case, it is interesting that both Na+ and Cl- concentrations in the middle of the water layer are above the bulk value of 0.1 M. Discussion The mechanism of ion-specific inhibition of bubble coalescence was the subject of a number of earlier studies, resulting in very different proposals. The change of surface tension with concentration correlates weakly with the inhibition of coalescence. This led Christenson and Yaminsky to suggest that the inhibition is controlled by interface elasticity.24 But already in 1913, it was noted by Pollock (for acetic acid, ref 1 p 217) that it is hard to imagine that a relatively very small change in surface tension (1.5%) can lead to such a dramatic effect. Nevertheless,
Marcˇelja the dynamics of the coalescence effect, not considered in the present work, must also be important and have an effect on the outcomes. While the average surface separations where coalescence is normally observed are of the order of tens of nanometers, details of the coalescence event including the effect of capillary waves are not known. For electrolyte solutions at the transition concentration where coalescence inhibition first appears, the controlling effect of surface-adsorbed ions cannot have a long range. Typical ranges of several Debye lengths (electrostatic interaction) or several molecular diameters (structural effects) do not extend beyond 2-3 nm. In 1995, Weissenborn and Pugh25 examined a number of mechanisms and noted that the combining rules may be explained by the positive or negative adsorption of a given ion at the surface. At that time, the distinct propensity of polarizable anions for approach to the surface was not known, so anions were classified incorrectly and the authors did not propose a specific mechanism. Two features of the inhibition of coalescence point to the controlling role of the ionic contribution to the interaction between free surfaces. The effect is strongly ion-specific, and the transition concentration is well defined. As explained in the section on effective potentials, the interaction between surfaces can be separated into two additive contributions: (i) interaction between the surfaces in pure solvent and (ii) ionic contribution evaluated with the effective potentials between ions in the solvent. When free surfaces in pure water approach, very high surface free energy leads to the attractive interaction. The result should be similar to the shortrange interaction between hydrophobic surfaces in water (the “true” hydrophobic interaction). When ionic contribution is repulsive, or shows a repulsive barrier, the inhibition will first appear at the electrolyte concentration when the two opposing contributions become comparable in magnitude. If the concentration is further increased, interaction in the solutions where ionic contribution is attractive remains attractive. When ionic contribution is repulsive, the repulsion increases with increasing electrolyte concentration. The end result at high concentrations is an all or nothing effect in coalescence inhibition, as found in the experiments.7 Short-range hydrophobic force between free surfaces is not well-known, and it depends sensitively on the concentration of dissolved air.26 Nevertheless, we can estimate the order of magnitude from a number of available results on mica surfaces coated with hydrophobic layers. The pressure between the surfaces in the surface force apparatus can be calculated in the Derjaguin approximation as P ) -(dF/dx)/(2πR), where F is the measured force, R the radius of curvature of the surfaces, and x the separation of the surfaces. Let us consider NaCl solution at surface separation of 23 Å, where we find a maximum in the ionic repulsion of about 0.9 MPa. We use the accurate recent study by Lin et al.27 on mica sheets coated with double-chain surfactants to estimate the hydrophobic attraction. From their Figure 4 at 23 Å, we find the total attraction of approximately 2.0 MPa. Part of this can be ascribed to the long-range electrostatic interaction between the coating layers, and subtracting this part using the authors’ fit formula leads to the true short-range hydrophobic interaction that is only half as large, about the same as the calculated ionic repulsion. Given the overall level of accuracy, the very good agreement is rather fortuitous. Nevertheless, it suggests that the two opposing contributions are of a comparable magnitude. The sequence of events leading to the collision and eventual coalescence of two bubbles can now be envisaged with the help
Coalescence of Bubbles in Simple Electrolytes of a recently described and filmed model colloidal system near a critical point28 where the dynamics is much slower. The sequence begins with the hydrodynamic drainage of the fluid between the surfaces, which becomes slower in the later stages. After the approach of surfaces roughened by capillary waves, the final coalescence event is controlled by the short-range interaction that is dependent on the surface affinity of each ionic species present in the solution. Acknowledgment. I am greatly indebted to Barry Ninham who first suggested this problem to me and, together with Werner Kunz, co-organized the important Regensburg workshop on ion-specific interactions, and to Vince Craig and Casuarina Dalton for their enthusiasm, help, and valuable experimental information. References and Notes (1) Pollock, J. A. R. Soc. N.S.W. J. Proc. 1911, 45, 204, abbreviated in Philos. Mag. 1912, 24 (S.6), 189. (2) Cohen, A. Z. Electrochem. 1923, 29, 1. (3) Monahan, E. C.; Zietlow, C. R. J. Geophys. Res. 1969, 74, 6961. (4) Scott, J. C. Deep Sea Res. 1975, 22, 653. (5) Hofmaier, U.; Yaminsky, V. V.; Christenson, H. K. J. Colloid Interface Sci. 1995, 174, 199. (6) Craig, V. S. J.; Ninham, B. W.; Pashley, R. M. Nature 1993, 364, 317. (7) Craig, V. S. J.; Ninham, B. W.; Pashley, R. M. J. Phys. Chem. 1993, 97, 10192.
J. Phys. Chem. B, Vol. 110, No. 26, 2006 13067 (8) Mucha, M.; Frigato, T.; Levering, L. M.; Allen, H. C.; Tobias, D. J.; Dang, L. X.; Jungwirth, P. J. Phys. Chem. 2005, 109, 7617. (9) Marcˇelja, S. Curr. Opin. Colloid Interface Sci. 2004, 9, 65. (10) Minofar, B.; Wahab, A.; Mahiuddin, S.; Kunz, W.; Jungwirth, P. J. Phys. Chem. B, submitted for publication. (11) Manciu, M.; Ruckenstein, E. Langmuir 2005, 21, 11312. (12) Kjellander, R.; Marcˇelja, S. J. Chem. Phys. 1985, 82, 2122. (13) Ke´kicheff, P.; Spalla, O. Phys. ReV. Lett. 1995, 75, 1851. (14) Marcˇelja, S. Langmuir 2000, 16, 6081. (15) Mayer, J. E. Equilibrium Statistical Mechanics; Pergamon: Oxford, 1968; Hill, T. L. Statistical Mechanics; McGraw-Hill: New York, 1956. (16) Lyubartsev, A. P.; Laaksonen, A. Phys. ReV. E 1997, 55, 5689. (17) Marcˇelja, S. Nature 1997, 385, 689. (18) Otto, F.; Pattey, G. N. J. Chem. Phys. 2000, 112, 8939; 2001, 113, 2851. (19) Burak, Y.; Andelman, D. J. Chem. Phys. 2001, 114, 3271. (20) Kjellander, R.; Lyubartsev, A. P.; Marcˇelja, S. J. Chem. Phys. 2001, 114, 9565. (21) Lyubartsev, A. P.; Marcˇelja, S. Phys. ReV. E 2002, 65, 041202-1. (22) Khalack, J. M.; Lyubartsev, A. P. Condens. Matter Phys. 2004, 7, 683. (23) Kjellander, R.; Åkesson, T.; Jo¨nsson, B.; Marcˇelja, S. J. Chem. Phys. 1992, 97, 1424. (24) Christenson, H. K.; Yaminsky, V. V. J. Phys. Chem. 1995, 99, 10420. (25) Weissenborn, P. K.; Pugh, R. J. Langmuir 1995, 11, 1422. (26) Craig, V. S. J.; Ninham, B. W.; Pashley, R. M. Langmuir 1999, 15, 1562. (27) Lin, Q.; Meyer, E. E.; Tadmor, M.; Israelachvili, J. N.; Kuhl, T. L. Langmuir 2005, 21, 251. (28) Aarts, D. G. A. L.; Schmidt, M.; Lekkerkerker, H. N. W. Science 2004, 304, 847.
