10436
J. Phys. Chem. B 2006, 110, 10436-10440
Surface Pressure Isotherm for the Fluid State of Langmuir Monolayers V. B. Fainerman† and D. Vollhardt*,‡ Medical Physicochemical Centre, Donetsk Medical UniVersity, 16 Ilych AVenue, 83003 Donetsk, Ukraine, Max Planck Institute of Colloids and Interfaces, D-14424 Potsdam/Golm, Germany ReceiVed: January 3, 2006; In Final Form: April 10, 2006
The equation of state for the monolayer comprised of the molecules of different sizes (water and biopolymers) is modified to describe the fluid (liquid-expanded and gaseous) state of the insoluble molecules monolayer. In contrast to the equation of state derived previously in ref 1, this equation does not involve either the Gibbs’ adsorption equation or the differential equation for the chemical potential of the insoluble component, but it is based only on the equations for the chemical potential of the solvent in the bulk and in the surface layer. The results calculated from the proposed equations are in perfect agreement with the experimental Π-A isotherms of the liquid-expanded state obtained for Langmuir monolayers of various types of amphiphilic compounds. The values of molecular areas of amphiphilic molecules estimated from the fitting of experimental data to the proposed equation are found to be quite similar to those measured in independent grazing incidence X-ray diffraction (GIXD) experiments.
Introduction publication,1
In our an equation of state for the Langmuir surface layer was derived by the simultaneous solution of the differential equation for the chemical potential of components within surface (Butler’s equation) and the Gibbs’ adsorption equation. In ref 1, the equation derived in this way, was then used to obtain Frumkin’s, van der Waals’, and Volmer’s equations of state. Also, in ref 1 the generalized Volmer’s equation (for the multicomponent insoluble monolayer) was derived, which was then extended to include the coexistence region between liquid-expanded and liquid-condensed phases. In subsequent publications these equations were further generalized to take into account several specific features of Langmuir monolayers such as the dissociation/association processes of amphiphilic molecules and further phase transitions between the condensed phases,2-4 and they were successfully used to describe the experimental surface pressure-area (Π-A) isotherms of monolayers for various insoluble amphiphilic compounds.6-10 However, to derive the equation of state for the insoluble monolayer, one can employ other methods, which do not involve the simultaneous solution of Butler’s and Gibbs’ equations. One such method was proposed earlier to describe the behavior of adsorbed monolayers of soluble amphiphilic molecules.11 The method involves equating (at equilibrium) the chemical potential of the solvent in the bulk phase to that in the surface layer. This is possible because in the case of an insoluble monolayer, only the solvent is the substance which is present both in the bulk phase and in the surface layer. An equation of state for the surface layer of biopolymers (proteins) which takes into account not only the difference between the molar areas of the solvent and dissolved substance (i.e., nonideality of entropy of the mixed monolayer), but also the spontaneous variation of the biopolymer molecule area with the change of adsorption * Corresponding author. Phone: -49-331-567-9258; fax: -49-331567.9202; E-mail:
[email protected]. † Medical Physicochemical Centre, Donetsk Medical University. ‡ Max Planck Institute of Colloids and Interfaces.
was derived in ref 12. So far, this equation was not yet applied to insoluble monolayers of amphiphilic molecules. However, in a recent publication, this equation (slightly corrected to account for the nonideality of enthalpy) was used to describe monolayers containing insoluble particles within a wide range of the particle diameter between 75 µm and 7.5 nm.13 The calculations do not only provide satisfactory agreement with experimental Π-A isotherms, but they also yield (i) a very realistic area per particle in a closely packed monolayer, and (ii) reasonable cross-sectional area values of the solvent (water) molecule in the range of 0.12-0.18 nm2, which is almost independent of the particle size.13 In the present study, we modify the equation of state derived in refs 12 and13 to analyze the results reported earlier for liquidexpanded monolayers of various insoluble amphiphilic molecules. It is shown below that, in many cases, this equation of state newly proposed is as capable of the description of experimental Π-A isotherms as the equation derived in ref 1. Moreover, it will be shown that the equation of state proposed in ref 1 is an approximate version that can be obtained from the new equation. Theory The derivation of the equation of state for biopolymers (proteins) was expounded in ref 12. In this publication, we present therefore, only the basic equations of the theoretical model. Chemical potentials of components within surface layer µsi depend on the composition of the layer and its surface tension γ. The dependence of µsi is given by the known Butler’s equation:12,14 S S µSi ) µ0S i + RT ln fi xi - γΩi
(1)
0S where µ0S i (T, P) ) µi are the standard chemical potentials dependent on the temperature T and pressure P. R is the gas law constant, xi ) Ni/ΣiNi are the molar surface fractions, Ni are the numbers of moles of the ith component, fSi is the activity coefficient of the ith component, and Ωi is the molar
10.1021/jp0600413 CCC: $33.50 © 2006 American Chemical Society Published on Web 05/09/2006
The Fluid State of Langmuir Monolayers
J. Phys. Chem. B, Vol. 110, No. 21, 2006 10437
ln fH0 ) aθ2, ln fHj ) a njθ02
area of the ith component. In the bulk solution the chemical potentials µRi obey the following equation: R R µRi ) µ0R i + RT ln fi xi
(2)
0R where µ0R i ) µi (T, P) are standard chemical potentials. If the molecules are insoluble, that is, a Langmuir monolayer is formed, eq 2 is suitable only for the solvent. Equations of state for the surface layer can be derived from eqs 1 and 2. The most common approach suitable for both soluble and insoluble molecules, consists of equating of the chemical potentials of the solvent (i ) 0) in the bulk solution and in the surface layer. Thus, it follows from eqs 1 and 2 that
µ00S + RT ln fS0 xS0 - γΩ0 ) µ00R + RT ln fR0 xR0
(3)
The standard state has to be formulated. For the solvent (i ) 0) usually a pure component is assumed. That means xS0 ) 1, fS0 ) 1, xR0 ) 1, fR0 ) 1, and γ ) γ0. For the standard state one obtains, from eq 3,
µ00S - γ0Ω0 ) µ00R
(4)
From eqs 3 and 4, one derives the equation of state for a surface layer with any number of components of any geometry:12
kT Π ) - (ln x0 + ln f0) ω0
(5)
where Π ) γ0-γ is the surface pressure, k is the Boltzmann constant, ω0 is the molecular area per solvent molecule. The ω0 value depends on the choice of the position of the dividing surface. Equation 5 requires positive values of ω0 and N0. In ref 12, an equation defining a dividing surface was proposed which relates the surface excesses Γi of the solvent (i ) 0) and the soluble or insoluble species (i g 1) with any molecular area ωi: n
∑ i)0
n
Γi ) 1/Nω0 + (1 - ω/ω0)
Γi ∑ i)1
(6)
where N is the Avogadro number, and ω is the average molecular area of soluble or insoluble species:
ω0 * ω )
ω1Γ1 + ω2Γ2 + ... Γ1 + Γ2 + ...
(7)
Equation 5 can be transformed into a more convenient form, if one expresses the molar fraction of the surface layer components via the surface fractions θj ) Γjωj occupied by the jth component:12
xj )
θj
∑ ig0
nj
(8) (θi/ni)
with ni ) ωi/ω0. The total value of fi can now be obtained from the additivity of the enthalpy (H) and entropy (E) contributions:12
fi ) fHi ‚fEi , or ln fi ) ln fHi + ln fEi
(9)
Approximations for the enthalpic contribution to the activity coefficients were given in ref 12 in the following way:
(10)
where a is Frumkin’s interaction parameter which accounts for nonideality. Expressions for the entropic contribution to the activity coefficients were derived from a first-order Flory-type, model for mixtures of any number of different-area molecules in ref 15:
(θi/ni) + ln[nj∑(θi/ni)] ∑ ig0 ig0
ln fEj ) 1 - nj
(11)
Introducing relations eqs 8-11 into expression eq 5, an equation of state was obtained in ref 12 which describes two-dimensional surface layers, containing the solvent and any other component (e.g., proteins, insoluble amphiphilic molecules, or solid particles), and it is valid for n ) ω/ω0 > 1:
Πω0 ) ln(1 - θ) + θ(1 - ω0/ω) + aθ2 kT
-
(12)
Here θ is the fraction of the surface covered by molecules with an average area per entity ω. The expression (12) at ω ) ω0 is transformed to Frumkin’s equation of state, and this equation was obtained in ref 1 using another method, namely the simultaneous solution of Butler’s and Gibbs’ equations. An equation similar to eq 12 for the surface tension in a mixture of molecules of different size was derived in a monograph, where the statistical mechanics formulation was applied involving the calculation of the configuration energy of a mixture, the free energy, and the chemical potentials of the components at the surface, from which the surface tension of a mixed monolayer was obtained.17 Introducing the available surface area per insoluble molecule A via the relation θ ) ω/A, and assuming that the enthalpy contribution is independent of A (i.e., taking ln fH0 to be constant, which corresponds to a liquid-expanded monolayer)1 one obtains from eq 12 an expression for the Π-A isotherm for the insoluble monolayer:
[ ( ) ( )(
)]
ω0 ω ω kT + 1Π ) - ln 1 ω0 A A ω
- Πcoh
(13)
Here Πcoh ) kT/ω0 ln fH0 is the cohesion pressure. Equation 13 yields positive values for Π, and is therefore, only applicable at surface coverages above a minimum value of (ω/A)min, determined by the condition Π ) 0. For (ω/A) < (ω/A)min, i.e., A > Amin, the contributions of ideal and nonideal entropy (the first term in the right-hand side of eq 13) are compensated by the interaction between the components (Πcoh). As one can see from eq 13, the surface pressure weakly depends on the size of the amphiphilic material but is determined mainly by the monolayer coverage ω/A. For the Π-A isotherms of insoluble (Langmuir) monolayers of amphiphilic molecules, allowing association or dissociation of these molecules in the surface layer, a generalized Volmer’s equation was derived,1-3 which has the following form:
Π)
nkT - Πcoh A-ω
(14)
In this equation, the n value accounts for the association (or dissociation) degree of amphiphilic molecules in the monolayer, or the size of these molecules. For example, for some insoluble proteins in a liquid-expanded monolayer state, a value of n )
10438 J. Phys. Chem. B, Vol. 110, No. 21, 2006
Fainerman and Vollhardt
Figure 1. Comparison between the experimental (points) and theoretical (solid lines) Π-A isotherms of THPAA monolayer for the theoretical model on the basis of eq 15 (thick solid line) and eq 13 (thick dotted line). T ) 10 °C. Values of model parameters are listed in Table 1.
TABLE 1: Parameters of Theoretical Models (13) and (15) for THPAA at 10 °C parameters
eq 13
eq 15
ω, nm2 ω0, nm2 n Πcoh, mN/m
0.240 0.160
0.247
12.60
0.714 8.060
20-100 was obtained in ref 9. Assuming an approximation in which n is equal to ω/ω0, one transforms eq 14 to the form
Π)
kT(ω/A) - Πcoh ω0(1 - ω/A)
(15)
It is seen that eq 15 which is an approximation of eq 13, assuming low monolayer coverage (i.e., - ln(1 - θ) = θ/(1 - θ)), and neglecting the nonideality of entropy, is also able to describe the behavior of insoluble monolayers comprised of molecules of any size. Similar to eq 13, this equation does not involve geometric parameters of the amphiphilic molecules, but only the coverage of the monolayer by these entities. Equation 15 provides a good description of experimental results obtained for various systems.1-10 Results and Discussion Next, consider some results reported earlier, and compare the parameters of eqs 13 and 15 which provide for the best fit with the experimental Π-A isotherms. Such a comparison would obviously be the more meaningful, the better the physical validity of the parameters of the theoretical models. Therefore, in what follows we refer mainly to publications which, in addition to the Π-A isotherms, also present experimental values of the area per one molecule ω. The molecular areas of the condensed monolayer were measured by grazing incidence X-ray diffraction (GIXD) technique. The data thus obtained, being extrapolated to the zero surface pressure, provide accurate enough approximation for the limiting area per one molecule in the liquid-expanded monolayer ω. Figure 1 illustrates the Π-A isotherm of the N-tetradecylβ-hydroxy-propionic acid amide (THPAA) at 10 °C.4 Also, in Figure 1, the results of fitting by eqs 13 and 15 in the liquidexpanded monolayer state are shown, whereas Table 1 lists the values of the model parameters. It is seen that the two models
Figure 2. Experimental (thin solid lines) and theoretical (eq 13) (thick solid line) Π-A isotherms for 14-16PE monolayers at different temperatures. Values of model parameters are listed in Table 2. The thick dotted lines in Figures 2-4 are the tangent lines to the Π-A isotherms in the liquid-condensed state.
TABLE 2: Parameters of Theoretical Models (13) and (15) for (14-16PE) at 30 °C parameters
eq 13
eq 15
ω, nm2 ω0, nm2 n Πcoh, mN/m
0.68 0.19
0.64
19.50
1.39 11.25
describe the experimental curve quite well. Table 1 shows clearly that the ω values are rather similar for both models, and agree with the GIXD data for the liquid-condensed state: 0.24 nm2 at surface pressure 6 mN/m, and ∼0.247 nm2 if extrapolated to the zero surface pressure.4 The monolayers of three chemically modified branched chain glycerophosphoethanolamines (PEs) were studied in ref 3. Figure 2 show the set of the experimental Π - A isotherms for rac-1-(2-tetradecylhexadecanoyl)-2-O-hexadecyl-glycero-phosphoethanolamine (14-16PE) at different temperatures. It has been recommended to determine the ω value from the intersection point of the tangent line to the Π-A isotherm in the liquidcondensed state (shown by the thick dotted line in Figure 2) with the abscissa axis.18 The ω value obtained in this way is 0.69 nm2 per one 14-16PE molecule. The theoretical curves for liquid-expanded monolayer region calculated from the models of eqs 13 and 15 almost coincide with each another at each temperature. The results calculated from eq 15 were presented in ref 3. Therefore, in Figure 2 we present only the theoretical curve calculated from eq 13 for 30 °C (with the most extended liquid-expanded region), whereas in Table 2 the parameter values for the two models are shown. It is seen that the model on the basis of eq 13 (and also the model on the basis of eq 15) well describe the experimental Π-A isotherm. Comparing the ω values obtained from fitting with the value ω ) 0.69 nm2 one can conclude that the calculations with eq 13 lead to a somewhat better agreement with this ω value. The same good agreement between the results calculated with eq 13 and the experimental Π-A isotherms as well as the ω values found according to the procedure proposed in ref 17 was observed also for other branched chain glycero-phosphoethanolamine (nPE) monolayers studied in ref 3. Amphiphilic melamine-type monolayers such as, 2,4-di(nalkylamino)-6-amino-1,3,5-triazine (2CnH2n+1-melamine) with the alkyl chain lengths C10H21, C11H23, and C12H25 were
The Fluid State of Langmuir Monolayers
Figure 3. Experimental (thin solid lines) and theoretical (eq 13) (thick solid line) Π-A isotherms of 2C10H21-melamine monolayers spread on water measured in the temperature range between 4.6 °C and 19.9 °C. Values of model parameters are listed in Table 3.
J. Phys. Chem. B, Vol. 110, No. 21, 2006 10439
Figure 5. Experimental (thin solid lines) and theoretical (eq 13) (thick solid line) Π-A isotherms of 2C12H25-melamine monolayers in the temperature range between 19.9 °C and 38.9 °C. Values of model parameters listed in Table 3.
TABLE 3: Parameters of Theoretical Models (13)/(15) for 2CnH2n+1-melamine
Figure 4. Experimental (thin solid lines) and theoretical using eq 13 (thick solid line) and eq 17 (thin dotted line) Π-A isotherms of 2C11H23-melamine monolayers in the temperature range between 10.2 °C and 31.9 °C. Values of model parameters are listed in Table 3.
synthesized and studied in ref 6. The Π-A isotherms of these melamine-type monolayers spread on pure water were measured in the accessible temperature range, imaging of the monolayers was performed with a Brewster angle microscope, also GIXD experiments were performed.6 For processing the experimental Π-A isotherms in ref 6, eq 15 was used for the liquid-expanded state whereas for the description of the liquid-expanded/liquidcondensed coexistence region the equation was employed which accounts for the two-dimensional condensation. A very good agreement between the theoretical and experimental isotherms was observed for all homologues. In the present publication, only the results of the calculations using eq 13 for the liquidexpanded state and the parameters of the two theoretical models are compared with the experimental data.6 The thin solid lines in Figure 3 show the experimental Π-A isotherms of 2C10H21-melamine between 4.6 and 19.9 °C, the thin solid lines of Figure 4 show the experimental Π-A isotherms of 2C11H23-melamine between 10.2 °C and 31.9 °C, and the thin solid lines of Figure 5 show the experimental Π-A isotherms of 2C12H25-melamine between 19.9 °C and 38.9 °C. The bold lines in these figures represent the theoretical curves calculated with eq 13 for the liquid-expanded state at the highest
parameters
2C10H21 at 19.9 °C
2C11H23 at 31.9 °C
2C12H25 at 38.9 °C
ω, nm2 ω0, nm2 n Πcoh, mN/m
0.400/0.360 0.16/-/1.21 26.0/17.6
0.400/0.350 0.15/-/1.39 26.5/18.4
0.385/0.360 0.15/-/1.00 26.0/14.8
temperature for each homologous 2CnH2n+1-melamine. Also at each of Figures 3-5 the dotted curve is shown which was determined according to the method proposed in ref 17 for obtaining the ω value from the intersection point between the tangent to the Π-A isotherm in the liquid-condensed state and the abscissa axis. The ω values estimated in this way (the average values within the temperature range shown in Figures 3-5) are 0.404 nm2, 0.398 nm2, and 0.395 nm2 for 2C10H21, 2C11H23, and 2C10H21-melamines, respectively. In ref 6 also the ω values obtained for 2C11H23-melamine using the GIXD experiments at 5 °C and 20 °C are presented. The extrapolation of these values to the zero surface pressure yields ω ) 0.397 nm2 and ω ) 0.414 nm2 for the respective temperature values above; it is seen that these values are almost equal to the average value ω ) 0.398 nm2 for all temperatures estimated from the intersection of the dotted line shown in Figure 4 with the abscissa axis. This correspondence can be regarded as additional experimental evidence in favor of the procedure recommended in ref 17. The comparison between the theoretical curves shown in Figures 3-5 with the experimental Π-A isotherms shows again that eq 13 provides a very good description of the observed isotherms. From Table 3, which summarizes the parameters of eqs 13 and 15 for 2C10H21, 2C11H23, and 2C12H25-melamines, one can see that the ω values obtained by fitting are quite similar for the two theoretical models. Nevertheless, the ω values which correspond to eq 13 coincide within 3% both with those obtained from GIXD experiments for 2C11H23-melamine, and those extrapolated from the intersection point of the tangent with the abscissa axis for all melamines studied. At the same time, the ω values obtained from the model on the basis of eq 15 are 10% lower than the real values of molecular area. It should be noted, however, that this difference is also quite small having in mind the approximate character of all theoretical models.
10440 J. Phys. Chem. B, Vol. 110, No. 21, 2006
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The ω0 values (molar area of the solvent) shown in Tables 1-3 were estimated from fitting of Π-A isotherms using eq 13; these values are almost independent of ω and are within the range of 0.15-0.19 nm2 for all insoluble monolayers studied. This area is rather similar to the cross-section (ca. 0.1 nm2) of water molecule. The difference could possibly be ascribed to the specific choice of the dividing surface according to eq 6. Quite similar ω0 values (0.12-0.18 nm2) were also obtained for micro- and nanoparticles within a wide range of the particle size between 75 µm and 7.5 nm.13 The ω0 values of 0.3-0.4 nm2 calculated with eq 15 are somewhat higher. However in this case, the order of magnitude is also the same. Rusanov has recently published equations of state for the fluid (G or LE) state of Langmuir monolayers.18 These equations were obtained by the simultaneous solution of the original equation proposed by him for the chemical potential of the component of surface layer accounting for the so-called excluded area (the area which is inaccessible for the motion of the centers of mass of insoluble molecules) and the Gibbs’ adsorption equation. It was mentioned in our previous publication6 that this method is similar to that proposed in ref 1. As higher approximations, Rusanov derived the following general equation of state for the fluid state of monolayer:18
Π)
{
2nk0 - 2k0 - 4 φ kT [(1 1ω (1 - k φ)n-1 (n - 1)(n - 2)k 2φ 0
0
n-1
k0φ)
}
- 1 + (n - 1)k0φ] - Πcoh (16)
kT φ(1 - 0.029φ) - Πcoh ω (1 - 1.0145φ)2
Conclusions The equation of state for monolayers comprised of molecules of different sizes (water and biopolymers) derived in ref 12 was slightly modified (assuming independence of the enthalpy contribution of the monolayer coverage) to describe the liquidexpanded state of Langmuir monolayers. In contrast to the equation of state derived previously in ref 1, this equation does not involve either the Gibbs’ adsorption equation or the differential equation for the chemical potential of the insoluble component, but it is based only on the equations for the chemical potential of the solvent in the bulk and in the surface layer. The results calculated from the proposed equations have shown perfect agreement with the experimental Π-A isotherms obtained for various types of amphiphilic compounds. The values of molecular areas of amphiphilic molecules estimated from fitting the experimental data to the proposed equation were found to be quite similar to those measured in independent GIXD experiments. An equation of state derived assuming the excluded area of the monolayer exhibits essentially worse agreement with the experimental Π-A isotherms. References and Notes
where φ ) ω/A, n g 3, and k0 is a constant. To determine k0, the data of ref 19 (Monte Carlo and molecular dynamic methods for hard disks) were used, and a value of k0 ) 1.0145 was obtained in ref 18. With this k0 value at n ) 3, eq 16 takes the following form:
Π)
better agreement between the values calculated from eq 16 and those obtained from the experiment could be achieved by additional fitting of the coefficient k0 (if considered as independent parameter).
(17)
The results calculated from eq 17 for 2C11H23-melamine are shown by the thin dotted line in Figure 4. In this case the optimum parameters involved in eq 17 are as follows: ω ) 0.367 nm2, Πcoh ) 13.5 mN/m. It is seen that the theory, which assumes the excluded area, describes the experimental Π-A isotherm rather worse than do the models leading to eqs 13 and 15. Also, the ω values determined from eq 17 are 7-10% lower than the actual molecular area value and the ω values calculated from the model described by eq 13. It can be concluded from this fact that the account for the excluded area in the equation for the chemical potential derived in ref 18 does not provide any advantages as compared with the account for the actual areas of the insoluble molecules. It is possible that a
(1) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 1999, 103, 145. (2) Vollhardt, D.; Fainerman, V. B.; Siegel, S. J. Phys. Chem. B 2000, 104, 4115. (3) Vollhardt, D.; Fainerman, V. B. J. Phys. Chem. B 2002, 106, 12000. (4) Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 2003, 107, 3098. (5) Vollhardt, D.; Fainerman, V. B. J. Phys. Chem. B 2004, 108, 297. (6) Vollhardt, D.; Fainerman, V. B.; Liu, F. J. Phys. Chem. B 2005, 109, 11706. (7) Fainerman, V. B.; Vollhardt, D.; Aksenenko, E. V.; Liu, F. J. Phys. Chem. B 2005, 109, 14137. (8) Fainerman, V. B.; Vollhardt, D. In “Organized Monolayers and Assemblies: Structure, Processes and Function”, Studies in Interface Science; Mo¨bius, D., Miller, R., Eds.; Elsevier: New York, 2002; Vol. 15, p 105-160. (9) Wu¨stneck, R.; Fainerman, V. B.; Wu¨stneck, N.; Pison, U. J. Phys. Chem. B 2004, 108, 1766. (10) Vysotsky, Yu. B.; Bryantsev, V. S.; Fainerman, V. B.; Vollhardt, D. J. Phys. Chem. B 2002, 106, 11285. (11) Lucassen-Reynders, E. H. J. Phys. Chem. 1966, 70, 1777. (12) Fainerman, V. B.; Lucassen-Reynders, E. H.; Miller, R. AdV. Colloid Interface Sci. 2003, 106, 237. (13) Fainerman, V. B.; Kovalchuk, V. I.; Lucassen-Reynders, E. H.; Grigoriev, D. O.; Ferri, J.; Leser, K. M. E.; Michel, M.; Miller, R.; Mo¨hwald, H. Langmuir 2006, 22, 1701. (14) Butler, J. A. V. Proc. R. Soc. Ser. A 1932, 138, 348. (15) Lucassen-Reynders, E. H. Colloids Surf., A 1994, 91, 79. (16) Defay, R.; Prigogine, I. Surface Tension and Adsorption; Longmans, Green and Co Ltd: London, 1966. (17) Lyklema, J. Fundamentals of Interface and Colloid Science, Volume 3: Liquid-Fluid Interfaces; Academic Press: New York, 2000. (18) Rusanov, A. I. J. Chem. Phys. 2004, 120, 10736. (19) Erpenbeck, J. J.; Luban, M. Phys. ReV. A 1985, 32, 2920.
