13190
J. Phys. Chem. 1996, 100, 13190-13199
Theory of Phase Equilibrium B. Widom Department of Chemistry, Baker Laboratory, Cornell UniVersity, Ithaca, New York 14853 ReceiVed: December 6, 1995; In Final Form: February 8, 1996X
The historical background and some of the highlights of more recent developments in the theory of phase equilibria and their critical points are described. The historical mean-field theoriesstheir strengths and certain of their deficienciessare emphasized, as well as some of their recent applications to complex systems such as polymer solutions, microemulsions, multicomponent solutions and their tricritical points, and the interfaces between phases.
Mean-Field Theories Much of the present-day physical chemistry of phase equilibria and critical points can be traced back to two phenomenal contributions by J. D. van der Waals, 20 years apart in the 19th century: his thesis of 18731 and his theory of interfaces of 1893.2 They exemplify what has since come to be called the “meanfield” approximation or theory. While designed by van der Waals as a theory of the liquid-vapor equilibrium in a onecomponent fluid, essentially equivalent approximations are basic in the theory of ferromagnetism (Curie-Weiss theory) and ferroelectricity, of order-disorder transitions in alloys (the Bragg-Williams approximation), and in many other contexts. These approximations have some notable defects, as we shall see, and much of the activity in the field in the “modern” era (the second half of this century, say) was at first devoted to understanding and correcting these defects. That is still an important activity;3 but where one seeks qualitative understanding rather than quantitative accuracy the mean-field theory remains an indispensable tool. The prototype of a mean-field theory is the van der Waals equation of state,
p)
kT a V - b V2
(1)
where p is the pressure, V the volume per molecule (V ) V/N with V the volume and N the number of molecules), T the absolute temperature, and k Boltzmann’s constant, while a and b are two constants characteristic of the fluid. (Often V is taken to be the molar volume, and then k is replaced by the gas constant R.) The first term on the right-hand side of (1) is an approximation to the pressure phs of a fluid of nonattracting hard spheres. It is exact in one dimension (hard spheres or hard rods on a line) but is a poor approximation in three dimensions; the equation of state (1) is considerably improved when 1/(V b) is replaced by a more accurate representation of phs/kT for a hard-sphere fluid in three dimensions. It is the second term in the equation of state, -a/V2, that accounts for the attractions between molecules and that is characteristic of the mean-field approximation. The fundamental physical idea is that the configurations of the molecules in the fluid are determined largely by the hard-sphere repulsions and very little by the attractions, since the attractive forces exerted by each molecule’s many neighbors largely cancel each other. At the same time the potential energies of attraction felt by each molecule are additive and are proportional to the number of X
Abstract published in AdVance ACS Abstracts, June 15, 1996.
S0022-3654(95)03646-X CCC: $12.00
Figure 1. p, V isotherms in the mean-field approximation.
molecules contributing to it and hence to the number density, N/V. Thus, the configurational Helmholtz free energy A in this approximation is
A ) -TShs - aN‚N/V
(2)
where Shs is the configurational entropy of a fluid of nonattracting hard spheres at the same density as the real fluid and -a is the constant of proportionality in the energy of attraction. The pressure is then
p)-
(∂V∂A)
N,T
) phs -
a V2
(3)
with phs the pressure of the hard-sphere fluid without attraction at the same density and temperature as the real fluid. One then obtains (1) or an improved version of (1) according to how one represents phs. The constant a, which is van der Waals’s a parameter, is given in terms of the potential energy of attraction φ between pairs of molecules, with φ a function of the locations of the two molecules of the pair relative to each other, by
a ) -1/2 ∫φ dτ
(4)
where the integral is over all relative separations of the molecules and dτ is the element of volume in the integration. The pressure-volume isotherms implied by (1) are shown schematically in Figure 1, which (except for the liberties taken in the present schematic version) is the most famous figure in the field of phase transitions. The critical isotherm T ) Tc has a horizontal tangent at the critical point (pc,Vc). Each subcritical isotherm, T < Tc, has loops, which are replaced by a horizontal © 1996 American Chemical Society
Theory of Phase Equilibrium
J. Phys. Chem., Vol. 100, No. 31, 1996 13191
Figure 3. H, M isotherms of the Ising ferromagnet in mean-field approximation.
Figure 2. µ, F isotherms in the mean-field approximation.
line segment (tie line) drawn so as to cut off equal areas above and below (the “Maxwell construction”). The locus of the end points of the tie lines, shown as the dashed inverted parabola in the figure, is the coexistence curve; inside it are the twophase, liquid-vapor equilibrium states, while in states outside it the fluid is a single homogeneous phase. There are many alternative ways of representing this phase equilibrium in the mean-field approximation; for example, as in a plot of the chemical potential-density isotherms shown schematically in Figure 2, where µ is the chemical potential and F ()1/V) is the number density. The subcritical isotherms are again reconstructed by an equal-areas rule. An alternative to the continuum picture that led to eq 1 or 3 is a lattice model that has dominated much of the theoretical work on the problem of phase transitions. In its guise as a model of ferromagnetism it is a lattice spin model known as the Ising model, while as a model of the liquid-vapor phase transition it is called the lattice gas. For the latter, one imagines the volume V to be divided into V/V0 cells each of volume V0. Two molecules are then taken to repel infinitely strongly if they are in the same cell (so a cell may be unoccupied or singly occupied only), to attract each other with an energy of attraction - (with > 0) if they are in neighboring cells, and otherwise not to interact with each other at all. This imitates the strong repulsions between real molecules at close approach, their attraction at intermediate distances, and the ultimate vanishing of their interaction at large separations. The mean-field approximation applied to this model yields, in place of (1) or (3),
p ) -(kT/V0) ln(1 - V0/V) - a/V2
(5)
independent of dimensionality. Now a ) (1/2)cV0 where c is the coordination number, i.e., the number of neighbors that each cell has. This is just what one would calculate from (4) for this model. The qualitative behavior of (5) is the same as that of (1) as shown in Figure 1, and the associated µ,F isotherms are again as in Figure 2. It was remarked above that the equivalent of this mean-field theory applies in many other contexts, and indeed, with the appropriate mapping from the variables appropriate to one kind of system to those of another, these lattice models do also. For example, in the Ising model of ferromagnetism one imagines that at each site i of a lattice there is a spin si that can take either of two values, say (1. There is a magnetic field H that interacts with each spin with energy Hsi, and spins at neighboring sites i and j interact with each other with energy -Jsisj where J is some positive energy parameter. To make the connection to the lattice gas described above, we may take the centers of
the cells in the lattice gas to be the sites of the corresponding Ising-model lattice. Then the Ising-model J, H, and magnetization M map into the lattice gas interaction energy , chemical potential µ, and density F ()1/V), via the relations
4J ) , 2H ) µ + h(T), M ) V(2F - 1/V0)
(6)
where h(T) is some function of temperature alone that depends on the precise definition of µ. The magnetic analogs of the chemical potential vs density isotherms in mean-field approximation shown in Figure 2 are the magnetic field vs magnetization isotherms in Figure 3. Critical-Point Exponents The mean-field approximation as embodied in Figures 1-3 entails universal behavior in the neighborhood of the critical point. In each figure the critical isotherm (T ) Tc) is cubic near the critical point:
p - pc ∼-(V - Vc)3, µ - µc ∼ (F - Fc)3, H ∼ M3 (T ) Tc) (7) where the subscript c indicates the critical-point value and ∼ stands for asymptotic proportionality. The coexistence curves in Figures 1 and 2 are parabolic near the critical point, so the difference F1 - Fv between the liquid and vapor densities is such that
Fl - Fv ∼ (µc - µ)1/2 ∼ (pc - p)1/2 ∼ (Tc - T)1/2
(8)
where µ, p, and T are the chemical potential, pressure, and temperature at two-phase coexistence. Analogously, the spontaneous magnetization (the value of |M| at the two ends of the tie line in a reconstructed subcritical isotherm in Figure 3) is
|M| ∼ (Tc - T)1/2
(9)
near the critical point (or Curie point, as it is called in a ferromagnet). The powers 3 in relations 7 and 1/2 in relations 8 and 9 are examples of critical-point exponents, which are quantitative characterizations of critical-point behavior and are supposedly universal, the same for all substances (with certain qualifications that are of technical interest but need not concern us here). It was already recognized in van der Waals’s time that while, indeed, the critical-point exponents are the same for all substances, the universal values of the exponents are not those of the mean-field theories.4 The critical-isotherm exponent, which in (7) has the value 3, is in reality close to 4.81. This is found both from experiment (although with less precision than this) and from exact numerical analysis of the lattice-gas or Ising model without the mean-field approximation. Likewise, the
13192 J. Phys. Chem., Vol. 100, No. 31, 1996
Widom
coexistence-curve exponent, which in (8) is 1/2, is in reality 0.32 or 0.33. In notation that is now standard, these two exponents are denoted δ and β, respectively.3 There are other critical-point exponents as well. Since (∂p/ ∂V)T ) 0 at the critical point (cf. Figure 1), the isothermal compressibility χ, defined by χ ) -V-1(∂V/∂p)T, is infinite there. On approach to the critical point, χ diverges proportionally to |T - Tc|-γ, with an exponent γ that in mean-field approximation is 1 but that is in reality approximately 1.24. In mean-field approximation the constant-volume heat capacity, CV, is finite at the critical point. Indeed, since the configurational entropy of the hard-sphere reference fluid depends only on density and not on temperature, it follows from (2) that in the one-phase region there is no configurational contribution to CV. The total CV is then entirely kinetic, and so finite, and is the same as in the corresponding ideal gas: (3/ 2)Nk for a monatomic fluid, etc. (It is a little more complicated in the two-phase region because phase transformation then makes an additional contribution to CV, but that, too, is finite at the critical point in mean-field theory.) That is in mean-field approximation; but in reality CV diverges at the critical point proportionally to |T - Tc|-R with an exponent R that is close to 0.11. With this notation, R ) 0 in mean-field theory. Later, when the theory of interfacial structure and tension is discussed, it will be seen that there is another exponent, called µ, associated with the vanishing of the surface tension σ at the critical point: σ ∼ (Tc - T)µ, with µ ) 3/2 in the mean-field theory but µ = 1.26 in reality. There is likewise an exponent ν connected with the divergence of the correlation length, ξ, which is the distance over which fluctuations of density or composition are strongly correlated and beyond which they are essentially uncorrelated: ξ ∼ |T - Tc|-ν, with ν ) 1/2 in meanfield (van der Waals plus Ornstein-Zernike) theory while more realistically ν = 0.63. It is this diverging correlation length that is responsible for critical opalescence: close to the critical point ξ becomes comparable with the wavelength of visible light, so the density fluctuations produce inhomogeneities in index of refraction on that scale, and these inhomogeneities scatter the light strongly. These are the only critical-point exponents that need concern us here (although it is recognized that there is another exponent, called η, also associated with the correlation of fluctuations3). An important part of present-day critical-point theory is a set of relations, called scaling laws,3 that show these exponents not to be all independent but to be connected by relations such as
δ ) 1 + γ/β, µ + ν ) 2 - R ) γ + 2β
(10)
dν ) 2 - R
(11)
and
where d is the dimensionality of space. It is d ) 3, and less frequently d ) 2, that is of interest, but the dimensionality plays an important role in the theory and so is left general in relations such as (11). Unlike in the mean-field theories, the exponents are in general d-dependent. The values for real fluids quoted earlier are for d ) 3. The scaling laws (10), which are independent of d, hold in every dimensionality and indeed hold also in the mean-field approximation, while those, like (11), that explicitly involve d, hold only for d up to some borderline dimensionality, d*. Beyond d* the d-dependent scaling laws such as (11) cease to hold, while the exponents then take on, and for larger d remain fixed at, their mean-field values. For the critical points of twophase equilibriaswhich are all that have been discussed here so farsd* ) 4. Thus, in four and higher dimensions the
Figure 4. Density profile of liquid-vapor interface.
exponents given by the mean-field theories are correct, but in three dimensions they are not, as we have seen. For a critical point that is the limit of three-phase coexistencesthe tricritical point discussed belowsthe borderline dimensionality d* is 3, so the exponents implied by the mean-field approximation for such a higher-order critical point are correct in d ) 3. (At the borderline dimensionality itself there are indeed non-mean-field corrections to the asymptotic critical-point behavior, but these are logarithmic and so do not alter the formal values of the exponents.) The scaling laws were originally conjectured by phenomenological arguments but were soon given a microscopic basis5 and were ultimately shown to follow from the renormalizationgroup theory.6 The latter is a powerful analytical tool for going beyond the mean-field approximation to obtain an accurate description of the neighborhood of a critical point including proper values of the exponents.3 While such corrections to the mean-field approximation constitute an important body of theory, and while they are important for making quantitative comparisons between theory and experiment, the mean-field theories, because of their simplicity and transparency, still have a prominent place in the statistical mechanics of phase equilibrium. That will be apparent in the applications that follow. Interfaces In the interface between coexisting phases one sees the transformation from one phase to the other occurring spatially, over a distance that may be identified as the interfacial “thickness”. In Figure 4 is displayed a density profile through a liquid-vapor interface. The coordinate z is in the direction perpendicular to the interfacial plane. The figure shows the mean density F(z) at the height z varying continuously from that of the bulk vapor, Fv, at z ) -∞, to that of the bulk liquid, Fl, at z ) ∞. Implicit in this picture is the presence of a gravitational field, which localizes the interface and stabilizes it against longwavelength, thermally excited “capillary waves”7swhich otherwise, for an interface of unboundedly large area, would smear out the profile F(z). It is often convenient to think of such capillary waves as undulations of an interface that, like a drumhead, has already been endowed with an intrinsic structure described by a profile such as the F(z) introduced here. That, however, is not an exact prescription for it contains an arbitrary division: assigning fluctuations that are of wavelength longer than a certain arbitrary length to “capillary waves” and taking those of shorter wavelength as contributing to the “intrinsic structure”. The simplest mean-field theory ignores these questions and imagines a well-defined profile F(z) with no explicit reference either to a stabilizing field or to fluctuations. While thereby missing some important elements of physical reality, it has nevertheless been extremely fruitful in predicting and explaining many interfacial phenomena and continues to
Theory of Phase Equilibrium
J. Phys. Chem., Vol. 100, No. 31, 1996 13193
(1/2)F(z)∫ φ(r)[F(z+ζ) - F(z)] dτ
Figure 5. Excess free energy density F(F) for Fv e F e Fl.
be widely applied. It should be remarked further that while Figure 4 shows F(z) varying monotonically between the two bulk densities, nonmonotonic profiles, even ones with many oscillations, can also occur. In the interfacial region the imagined F(z) takes on values between Fv and Fl, which are excluded for bulk phases at that temperature by the equal-areas construction in Figures 1 and 2. The fluid at such points senses the gradient F′(z) (the prime indicates differentiation), and it must be this that stabilizes the fluid at such values of the density, since there is a significant cost in free energy for maintaining a density F other than Fv or Fl. This free energy excess per unit volume of fluid of density F will be denoted F(F). At any subcritical temperature it is the integral of the difference between the analytic, unreconstructed µ,F isotherm in Figure 2scall it M(F)sand the tie line at the coexistence value of µscall it µ0sat that T:
F(F) ) ∫F
F v
or Fl
[M(F) - µ0] dF
(12)
It is a matter of indifference whether one takes the lower limit of integration to be Fv or Fl because the difference the two alternative choices would make in the value of the integral is
∫FF [M(F) - µ0] dF ) 0 l
(13)
v
which vanishes by the equal-areas rule. As a corollary, F(Fv) ) F(Fl) ) 0. The excess free energy density F(F) defined by (12) is shown in Figure 5 for just those values of F, between Fv and Fl, that occur in the interface. There is no such excess when the density is that of either of the two bulk phases, but there is a positive excess at every intermediate F. Once one has adopted an equation of state with its van der Waals loops, as in Figures 1 and 2, the associated F(F) is found by the prescription in eq 12 and is as depicted in Figure 5, but that is very much tied to the mean-field approximation that gave rise to Figures 1 and 2 in the first place. A more fundamental and rigorous definition of F(F) again presents conceptual difficulties, which continue to excite the interest of the experts but have hardly impeded the application of formulas such as those in eqs 15 and 18, below, in a great variety of physical contexts. If it were not for the gradient F′(z), the system’s free energy would be minimized by an infinitely sharp transition between Fv and Fl, i.e., by a step-function density profile in Figure 4; for then there would be no region (except one of vanishing volume) in which the excess free energy density F(F) differed from 0. But there is indeed an additional cost in free energy that comes from the inhomogeneity itself, as van der Waals recognized over 100 years agosthree years before the appearance of the first issue of The Journal of Physical Chemistry.8 Because of the inhomogeneity there is a difference F(z+ζ) F(z) between the densities at any two different levels z + ζ and z. If r is the distance between some fixed point at the level z and an arbitrary point at the level z + ζ, at which there is located an element of volume dτ, then the extra free energy density at z due to the inhomogeneity, in mean-field approximation, is
(14)
Here φ(r), as in (4), is the potential energy of attraction of a pair of molecules a distance r apart, and the integration is over all space, with r and ζ thus varying in the integration while the height z and the chosen point at that height are fixed. The total excess free energy density at height z over that in either bulk phase is the sum of F(F) in (12), evaluated at the local F ) F(z), and the nonlocal contribution (14). The total excess free energy per unit area of interface, which is the interfacial tension σ, is then the integral of this sum over all z,
σ ) ∫-∞{F[F(z)] + (1/2)F(z)∫ φ(r)[F(z+ζ) - F(z)] dτ} dz (15) ∞
This is the basic relation in the mean-field theory of interfacial structure and tension. It expresses σ as a functional of the density profile F(z); the equilibrium F(z) is that which minimizes σ [subject to the conditions F(-∞) ) Fv and F(∞) ) Fl], the minimum value being then the equilibrium interfacial tension. Near the critical point the interface is markedly diffuse: the interfacial thickness is of the order of the correlation length ξ, which, we have seen, becomes infinite at the critical point. The density gradient is then everywhere small, and the contribution (14) to the excess free energy density takes a particularly simple and revealing form. For small gradients the F(z+ζ) in the integrand of (14) may be expanded in powers of ζ,
F(z+ζ) - F(z) ) F′(z)ζ + (1/2)F′′(z)ζ2 + ‚‚‚
(16)
and the expansion truncated after the ζ2 term. The first-order term does not contribute to the integral in (14), by symmetry: the integration is over all space, and φ(r) is spherically symmetric, while ζ is odd. By the same symmetry, the secondorder term contributes exactly one-third as much as it would if ζ2 were replaced by r2. Thus, (14) becomes -(1/2)mF(z)F′′(z) with the coefficient m defined by
m ) -(1/6)∫r2φ(r) dτ
(17)
This m is seen to be proportional to the second moment of φ and is thus a measure of the range of φ. Since φ is the potential energy of attraction it is negative, so m > 0. That is an important observation, because, by integration by parts, the contribution of -(1/2)mF(z)F′′(z) to the total surface free energy σ in (15) is equivalent to that of (1/2)mF′(z)2 and is thus positive. We now have two positive contributions to the excess free energy density, F[F(z)] and (1/2)mF′(z)2. These compete to determine the minimizing profile: F[F(z)] alone, we have seen, would favor an infinitely sharp step-function profile with an infinite gradient, while (1/2)mF′(z)2 would contribute least if the interface were infinitely diffuse. The actual minimizing profile, shown schematically in Figure 4, is a compromise in which the two competing terms contribute equally to the free energy. In this small-gradient approximation to the mean-field theory the surface free energy functional σ in (15) is now
σ ) ∫-∞{F[F(z)] + (1/2)mF′(z)2} dz ∞
(18)
(In the current literature this and related expressions are often referred to as “Landau-Ginzburg” free energy functionals, but like so much else in this subject, they go back to van der Waals.) This functional is minimized by the F(z) that satisfies the associated Euler-Lagrange equation F′(F) ) mF′′(z) (where in
13194 J. Phys. Chem., Vol. 100, No. 31, 1996
Widom
Figure 6. Effective potential F′′[F(z)] in the Schro¨dinger equation.
Figure 7. Ground-state eigenfunction ψ(z) ) F′(z).
each case a prime means differentiation with respect to the indicated argument), or, from (12),
i.e., that ψ(z) ) F′(z) is an eigenfunction of the integral kernel δ2σ/δF(z)δF(ζ) belonging to eigenvalue λ ) 0.9,10 This ψ(z), obtained from F(z) in Figure 4, is shown schematically in Figure 7; it is the nodeless ground-state eigenfunction, and λ ) 0 is the “zero-point energy”. Except for this eigenvalue λ ) 0, then, all the eigenvalues of δ2σ/δF(z)δF(ζ) are indeed positive, so the equilibrium profile F(z) in Figure 4 is stable to all small perturbationssexcept ones of the form of this λ ) 0 eigenfunction, F′(z), which thus can occur with no cost in free energy. But such a perturbation corresponds not to a change in the form of the profile but rather to its translation along the z direction while its form is preserved,9 which indeed costs no free energy. That a small perturbation of F(z) proportional to F′(z) corresponds merely to uniform translation in z with no change in shape follows from F(z) + F′(z) ∼ F(z+) for small . Near the critical point the F(F) in Figure 5 becomes the quartic polynomial
M(F) - µ0 ) mF′′(z)
(19)
This has an interesting and helpful analogy in the dynamics of a particle moving on a line. If we interpret z as “time”, F as “coordinate”, m as “mass”, and -F(F) as “potential energy”, then M(F) - µ0, by (12), is the analog of “force”, (19) is “Newton’s law” equating force to mass times acceleration, the term (1/2)mF′(z)2 in the integrand in (18) is the “kinetic energy”, that integrand is the “Lagrangian”, and the principle that the F(z) that satisfies (19) minimizes the integral in (18) is “Hamilton’s principle” in mechanics. Since F[F((∞)] ) F(Fv) ) F(Fl) ) 0, the constant total “energy” -F(F) + (1/2)mF′(z)2 on the trajectory that satisfies (19) is 0, so the surface tension σ, which by (18) is the integral of the “Lagrangian”, is also the “action” in this dynamical analogy. In this mean-field theory of interfacial structure and tension, there is also an analogy to quantum mechanics.9 To be certain that the F(z) that satisfies (19) actually minimizes (18), it is necessary to verify that the second functional derivative of σ with respect to F(z), evaluated with the F(z) that satisfies (19), is always positive. Equation 19 itself expresses only the vanishing of the first functional derivative, δσ/δF(z), which is
δσ ) F′(F) - mF′′(z) δF(z)
2
(21)
where δ(z-ζ) is the Dirac delta function. The condition that this integral kernel, evaluated at the F(z) at which the first functional derivative δσ/δF(z) vanishes, be always positive is the condition that all its eigenvalues be positive. Let λ be an eigenvalue of δ2σ/δF(z)δF(ζ) and ψ(z) the associated eigenfunction. Then 2
σ ψ(ζ) dζ ) λψ(z) ∫-∞∞ δF(z)δ δF(ζ)
(22)
or, from (21),
F′′[F(z)]ψ(z) - mψ′′(z) ) λψ(z)
σ ) ∫F
Fl v
x2mF(F) dF
(26)
[x2mF(F) is the “momentum” because, as we saw, the total “energy” in the mechanical analog is 0, i.e., (1/2)mF′(z)2 - F(F) ) 0.] With (25), this yields for the surface tension near the critical point,
σ ) (1/6)x2mg(Fl - Fv)3
(27)
But as the critical point is approached, Fl - Fv vanishes proportionally to (Tc - T)1/2 in this mean-field theory according to (8), so σ vanishes as (Tc - T)3/2. This result was one of van der Waals’s great achievements.2 It implies the surface-tension exponent µ ) 3/2, as quoted earlier, which may be compared with the corrected value µ ) 1.26 obtained when one goes beyond the mean-field approximation. Tricritical Points11
(23)
with F(z) the solution of (19) that satisfies the boundary conditions F(-∞) ) Fv and F(∞) ) Fl. Equation 23 is a Schro¨dinger equation. With F(F) as in Figure 5 and F(z) as in Figure 4 it may be seen that F′′[F(z)] in (23) is as shown qualitatively in Figure 6. This is an effective potential in the Schro¨dinger equation (23). Its two asymptotes at z ) (∞ are F′′(Fv) and F′′(Fl). Between them it has a minimum that can support one or more bound states. Since the F(z) in F′′[F(z)] satisfies F′[F(z)] - mF′′(z) ) 0, it follows on differentiating this with respect to z that F(z) also satisfies
F′′[F(z)]F′(z) - mF′′′(z) ) 0
(25)
with g some constant of proportionality. Meanwhile, the minimized σ in (18), which we saw to be the “action” in the mechanical analog, is thus given also by the integral of the “momentum” over the “coordinate”,
(20)
The second functional derivative is
δσ ) F′′(F) δ(z-ζ) - mδ′′(z-ζ) δF(z) δF(ζ)
F(F) ) g(F - Fv)2(Fl - F)2
(24)
As remarked earlier, a tricritical point is a limit of threephase coexistence, just as an ordinary critical point is a limit of two-phase coexistence. One refers to a state of three-phase coexistence as a triple point. Such a state can be achieved even in a one-component system (solid-liquid-vapor, for example). When in a one-component system there is only one phase present, there are two thermodynamic degrees of freedom (pressure and temperature, say, or temperature and chemical potential, etc.). Both these must then be fixed at a triple point, which is thus a unique point in a two-dimensional thermodynamic space. With two more degrees of freedom, as in a threecomponent system, one has a whole two-dimensional surface of triple points in a four-dimensional thermodynamic space. The four variables in this space may be taken to be the temperature,
Theory of Phase Equilibrium
Figure 8. Projection onto the p,T plane of the Rβγ triple-point surface (shaded), its two terminating lines of critical end points (cep), and the tricritical point (tcp).
the pressure, and two of the chemical potentials, or the temperature and the three chemical potentials, etc., but only two (any two) of these four can vary independently on the twodimensional surface of triple points, i.e., while still having three phases in coexistence. That is the phase rule: the number of degrees of freedom f that remain when p ) 3 phases coexist in a c ) 3-component system is f ) c - p + 2 ) 2. Let the three coexisting phases be called R, β, and γ. This two-dimensional surface of triple points terminates in a linesa one-dimensional locussof critical points of the Rβ phase equilibrium. These are ordinary critical pointssthe limits of Rβ two-phase coexistencesbut are called “critical end points” because they terminate three-phase coexistence: what had been three distinct phases, R, β, and γ, are now, at any of the points of the Rβ critical end point line, only two phases, a critical Rβ phase in equilibrium with the still distinct γ phase. The two-dimensional surface of Rβγ triple points has also a second terminating edge, viz., a line of βγ critical end points, which are critical points of the βγ phase equilibrium at which the critical βγ phase still coexists with the distinct R phase. Finally, these two one-dimensional loci of critical end points, the Rβ and the βγ, may have a confluence at some point, which is then a unique point in the four-dimensional thermodynamic space at which an Rβ critical end point coincides with a βγ critical end point. This is a tricritical point. In Figure 8 are shown the two-dimensional surface of Rβγ three-phase equilibria, its two terminating lines of critical end points, and the tricritical point at which the latter two curves have a confluence (a 3/2 power tangency, according to the mean-field theory), all projected from the four-dimensional space onto a plane, the coordinates in which may be any two of the four thermodynamic variables, e.g., pressure p and temperature T, as in the figure. Note that the β phase plays a role different from that of the R and γ phases: the Rβ and βγ critical end point lines are extended, one-dimensional loci of Rβ and βγ critical points, while the only critical point of Rγ phase equilibrium in this neighborhood is the tricritical point itself; there is no line of Rγ critical end points meeting the other two at this point. Three is the minimum number of independent chemical components in a mixture in which such a tricritical point can be realized; the tricritical point is then an invariant point in a four-dimensional thermodynamic space, as we saw. Tricritical points are often observed and studied in four-component mixtures, where the three phases are all liquids. Then either the pressure is fixed (at 1 atm, say) or, if the three liquid phases of interest are simultaneously in equilibrium with vapor, the pressure is the vapor pressure. In either case the tricritical point is still an invariant point; the extra degree of freedom introduced by the fourth component is used in fixing the pressure or in specifying equilibrium with the fourth phase (the vapor).
J. Phys. Chem., Vol. 100, No. 31, 1996 13195
Figure 9. Three-phase states at a fixed temperature in a threedimensional composition space. PQRS is the locus of the vertices R, β, and γ of the three-phase triangles; PR and SQ are respectively the βγ and Rβ critical end point tie lines.
Among the earliest examples of tricritical points to be recognized in experiment are in the three-component mixture acetic acidwater-butane12 and in the four-component liquid mixture benzene-ethanol-water-ammonium sulfate.13 The recognition of tricritical points in theoryslike so much else in this subjectsgoes back to the school of van der Waals.14 Even at a fixed temperature there is a whole range of threephase states in these systems. Fixing the temperature eliminates a degree of freedom so these three-phase states may then be described in a three-dimensional thermodynamic space. If we take as the coordinates in that space the densities of the three components in the three-component mixture, or the three independent composition variables in the four-component liquid mixture, any one such three-phase equilibrium is representable in that space as a triangle, the coordinates of whose vertices give the compositions of the coexisting phases. The whole range of such three-phase states is then representable as a stack of these triangles, as in Figure 9. The figure shows two representative ones of these triangles along with a curve, PQRS, that is the locus of the triangles’ vertices R, β, and γ. The point Q, where the length of the side Rβ of the three-phase triangles vanishes, is the Rβ critical end point; S gives the composition of the distinct γ phase that is in equilibrium with the critical Rβ phase of composition Q, at this fixed temperature. Likewise, R gives the composition of the critical βγ phase at the βγ critical end point, and P gives the composition of the distinct R phase that coexists with it. The figure shows the whole three-phase region at this temperature, bounded top and bottom by the critical end point tie lines PR and SQ. As the temperature approaches that of the tricritical point, the corresponding Figure 9 shrinks, and at the tricritical temperature itself it has shrunk to a point, the coordinates of which are the tricritical composition. This is the same thermodynamic state, but seen in an isothermal composition space, as the point tcp in the different representation of Figure 8. The analog of the mean-field theory for a one-component fluid with its liquid-vapor equilibrium as described earlier, but now for a multicomponent liquid mixture with liquid-liquid equilibria, is the regular-solution theory:
µi - µj ) kT ln(yi/yj) + ∑(wih - wjh)xh
(28)
h
where µi and µj are the chemical potentials of components i and j, of mole fractions yi and yj, while the wij are interaction energy parameters (the analogs of van der Waals’s a), and the sum is over all the chemical components h. One may locate the tricritical-point temperature and composition of this mixture15 for given values of the parameters wij and then expand all the thermodynamic properties of the system in powers of
13196 J. Phys. Chem., Vol. 100, No. 31, 1996
Widom
the deviations of T and of the yi from their values at the tricritical point, thus obtaining the corresponding tricritical-point exponents.16 It is found16 that as the tricritical-point temperature T* is approached, the largest dimension of the three-dimensional object pictured in Figure 9sas measured, for example, by the length of the Rγ side of the middle triangle in the stack of triangles in the figure or by the lengths of the critical end point tie lines PR and SQsvanishes proportionally to (T* - T)1/2. This is a measure of how rapidly the difference in composition between the R and γ phases vanishes as those phases become identical at the tricritical point. The exponent 1/2 is the analog of the earlier β ) 1/2 in (8) and (9), which describe the vanishing of the difference between the liquid and vapor densities on approach to an ordinary liquid-vapor critical point or of the spontaneous magnetization on approach to the Curie point; only now, since the borderline dimensionality d* for tricritical points is 3 rather than 4, the exponent β ) 1/2 is believed to be correct in d ) 3 even beyond the mean-field approximation, as remarked earlier. (We may recall also the earlier parenthetical remark about logarithmic corrections.) The curve PQRS in Figure 9 is a segment of an analytic space curve, and it shrinks to vanishing length as T f T*. Then from elementary geometric properties of such a curve, the next-tolargest dimension of the object in Figure 9sas measured by the altitude from the β vertex to the Rγ side of the middle triangle in the stack, for examplesand the smallest of the three dimensionssas measured, say, by the thickness of the stack of trianglessvanish proportionally to the second and third powers, respectively, of the largest and hence proportionally to T* - T and to (T* - T)3/2, respectively. Since on approach to T* the distances of the β vertices of the triangles in Figure 9 from the Rγ sides of those triangles vanish more rapidly than the lengths of the Rγ sides, it means that near the tricritical point the points representing the compositions of the three coexisting phases are nearly collinear, with the β phase intermediate between the R and γ phases. Asymptotically, the mass densities of the phases are linear functions of their composition, so the β phase is also intermediate in its mass density and is therefore identifiable as the middle phase when the three phases are ordered by gravity. That the compositions of coexisting phases approach collinearity as T f T* has the important consequence that it defines a limiting direction in the composition space of Figure 9, with then a single coordinate, which we may call x, measuring distances in that direction, such that the composition of each of the coexisting phases may be specified by the value that the single variable x has in that phase: xR, xβ, or xγ. There is now an excess free energy density F(x) analogous to the F(F) in Figure 5, but for three-phase rather than two-phase equilibrium; it is as shown in Figure 10. Then from the analog of (26), which gives the surface tension of a one-component liquid against its vapor, we have now, for the tensions σRβ, σβγ, and σRγ of the three different possible interfaces in this three-phase equilibrium,
σRγ ) ∫x
xγ R
) ∫x
xβ R
x2mF(x) dx
x2mF(x) dx + ∫xβγx2mF(x) dx
) σRβ + σβγ
x
(29)
This relation among the three tensions is “Antonow’s rule”.17,18 From the analog of (19) in the present context, with the same analogy as before to the dynamics of a particle, one obtains the composition profile x(z) of any of the three interfaces as the
Figure 10. Excess free energy density F(x) as a function of a single composition variable x, for a three-phase equilibrium near the tricriticalpoint temperature.
Figure 11. Composition profiles x(z) of the interfaces in Rβγ threephase equilibrium when β wets the Rγ interface.
analog of the F(z) in Figure 4. This is sketched in Figure 11. The Rβ and βγ interfaces are of microscopic thickness, and their composition profiles, for xR < x < xβ and xβ < x < xγ, respectively, are like the density profile of the liquid-vapor interface in Figure 4. The Rγ profile, by contrast, is given by the whole of Figure 11, for xR < x < xγ, and has a section (indicated schematically by the dashed line filling in the vertical break in the solid curve at x ) xβ) where the composition is that of the bulk β phase. As x varies from xR to xγ, it inevitably passes through the composition xβ at which a macroscopic β phase is stable. The Rγ interface thus consists of a layer of the same bulk β that coexists with the phases R and γ at this triple point. If phases R and γ are put in contact, a film of β will form at the interface, and that film will constitute the equilibrium structure of that interface; the upper and lower surfaces of the film will have the composition profiles of Rβ and βγ interfaces, and the interior of the film will be like bulk β. With all three phases present in bulk, the phase β (which, near the tricritical point, we know to be the middle phase) spreads atsis said to wetsthe Rγ interface. The thermodynamic criterion for wetting is Antonow’s rule, eq 29. In a more general context, the phenomenon of wetting is one of the most active areas of current research in the statistical mechanics of phase equilibrium. It is the subject of the next section. Wetting and Prewetting19 We saw that in three-phase equilibrium in the neighborhood of the tricritical point the middle phase, β, wets the interface between the other two phases, R and γ, and that the tensions of the three possible interfaces are connected by (29). Proximity to the tricritical point is a sufficient but not a necessary condition for wetting. Indeed, proximity to a critical end point is sufficient:20 on approach to a βγ critical end point, for example, σβγ vanishes as the 1.26 power of the distance from that point (see the section Interfaces), and σRγ - σRβ vanishes more slowly than that, while the inequality σRγ > σRβ + σβγ is a thermodynamic impossibility;17 this results in one of the two nearcritical phases (the β phase in this instance) wetting the interface between the other two phases, and (29) then holds and the Rγ
Theory of Phase Equilibrium
J. Phys. Chem., Vol. 100, No. 31, 1996 13197
Figure 12. Three coexisting phases meeting at planar interfaces; in (a), β does not wet the Rγ interface, and in (b) it does.
interface remains wet by β the rest of the way to the βγ critical end point. Even this, while sufficient for wetting, is not necessary; in three-phase equilibrium one phase often wets (spreads at) the interface between the other two even when the system is far from any critical point. On the other hand, when the system is far from a critical point wetting is not guaranteed, and the three phases may meet as shown in Figure 12a. The lines in the figure are the three separate planar interfaces seen edge-on; the phases occupy the dihedral angles between planes, and the phases and the planes all meet in a line of mutual contact that is perpendicular to the plane of the figure, where it is represented by the dot. The dihedral angles are the “contact angles” between phases. If γ is a nondeformable solid, the contact angle γ is 180°. When β wets the Rγ interface, the picture is as in Figure 12b, where the contact angle β is 0. When the equilibrium configuration is that of Figure 12b, Antonow’s rule, eq 29, holds; when it is that of Figure 12a, the equality (29) is replaced by the inequality
σRγ < σRβ + σβγ
(30)
As remarked above, the reverse inequality is impossible if these are the true, equilibrium tensions. When Figure 12a and the inequality (30) apply, so that there is a line of three-phase contact, there is an excess free energy associated with the inhomogeneous region about that line. This excess, per unit length of the contact line, is a line tension. The study of this tension, the existence of which was originally recognized by Gibbs,21 is now a vigorously pursued subfield in the statistical mechanics of phase equilibrium.17,22 Sometimes Figure 12b and the corresponding equality (29) hold over one range of three-phase states while Figure 12a and the inequality (30) hold over another. The transition between these,17-20 which may occur as the temperature or other thermodynamic variable varies, is termed a wetting transition or a transition between incomplete (or imperfect) wetting and complete (or perfect) wetting. It is usually (although need not always be) a first-order transition, accompanied by an abrupt change in the structure of the Rγ interface: on one side of the transition the Rγ interface does not comprise a film of bulk β while on the other side it does. While the structure of the Rγ interface changes discontinuously at a first-order wetting transition, its tension σRγ is continuous. Suppose that at an Rβγ triple point the phase β wets the Rγ interface, so that Figure 12b describes the phase equilibrium. Then in nearby two-phase states, in which only R and γ coexist while β is of slightly too high a free energy to be present as a third bulk phase, there is nevertheless a β-like film at the Rγ interface. This film is now of only microscopic thickness and so is not truly bulk β, but it is thicker and more truly β-phaselike the closer the thermodynamic state is to the triple point where β becomes stable in bulk. It is as though the R and γ phases have a premonition of the β phase’s coming into coexistence with them and prepare for it by forming at their interface a microscopically thick layer of the anticipated phase,
Figure 13. Polymer chains on a lattice; each cell not occupied by a segment of a chain is occupied by a solvent molecule.
which then thickens into that bulk phase itself as the triple point is attained.17-20 This phenomenon is called pre- (or premonitory) wetting. When R and γ are solid and vapor while β is liquid, it is called premelting: a liquidlike layer is already present at the solid-vapor interface below the triple point, where the liquid is not stable in bulk. It is to be emphasized that prewetting or premelting occurs only when, at the triple point, the new phase wets the interface between the former two; otherwise, the new phase is not anticipated in that interface and comes more or less as a surprise. An interesting phenomenon connected with the existence of prewet (or premelted) states is a prewetting transition,17-20 analogous to the wetting transition described earlier but occurring when only two phases, R and γ, are present as bulk phases rather than three. In states far enough from a triple point at which the third phase, β, would be in coexistence with R and γ and would wet their interface, the Rγ interface will not be prewet by β; but as the triple point is neared this interface suddenly, at some moment, undergoes a first-order change to a β-like structure, this β-like layer then gradually thickening into bulk β as the triple point is even more closely approached. At the point of this transition between non-prewet and prewet states of the Rγ interface these two alternative, equal free energy structures of the interfacesone non-β-like and one β-likesmay coexist. The one-dimensional boundary between the two surface phases has a positive tension analogous to surface tension. This is phase equilibrium in a two-dimensional world. With enough thermodynamic degrees of freedom (at least two independent chemical components), there is even achievable a critical point of this phase equilibrium, where the two alternative structures of the Rγ interface become identical and the tension of the onedimensional boundary between them vanishes. Polymers and Microemulsions Much of contemporary equilibrium statistical mechanics has the ambitious object of accounting for the phase transitions that occur in complex fluids such as polymers and polymer solutions and solutions of surfactants. An early application of the meanfield approximation in the statistical mechanics of a model polymer solution is that of Flory.23 The model is a lattice model like the lattice gas described earlier; now each monomeric unit of the polymer chain occupies one lattice site (cell), the remainder being occupied by solvent molecules, as shown in Figure 13. Let φ be the fraction of the sites occupied by polymer segments (to be identified with the volume fraction of polymer in a real solution) and f the free energy per site (the free energy density multiplied by the volume of a basic cell). Flory’s meanfield approximation for f, in units of the thermal energy kT, is
13198 J. Phys. Chem., Vol. 100, No. 31, 1996
f/kT ) (1/N)φ ln φ + (1 - φ) ln(1 - φ) - χφ2
Widom
(31)
where N is the degree of polymerization (number of monomers per polymer chain) and is thus proportional to the molecular weight of the polymer, while χ, on a simple cubic lattice in three dimensions, where each site has six neighbors, is given by
χ)
6 1 w - (w + wSS) kT PS 2 PP
[
]
(32)
where wPS is the energy of interaction between a polymer segment and a neighboring solvent molecule, wPP that between two polymer segments (whether belonging to the same or different chains), and wSS that between two solvent molecules. If N were 1, the model would be that of a liquid (or solid) solution of two monomeric species and (31) with (32) would be the regular-solution theory: equivalent to (28) for a twocomponent mixture, with the chemical potential difference obtained by differentiating f with respect to the composition variable. Flory’s χ in (31) and (32) is closely related to the wij in (28), which, in turn, as noted earlier, are analogs of van der Waals’s a in (4) and (5) and so also of the one-component lattice gas and Ising model J. The behavior of the polymer solution as derived from (31) with N . 1 is profoundly different from that with N ) 1. Figure 14 shows, schematically, temperature-composition coexistence curves for N ) 1 and for some N . 1. Phase separation occurs at much higher temperatures and at much lower volume fractions of solute when N is large than when it is small. As N f ∞, the critical-solution point (consolute point), which is the critical point of the phase equilibrium, moves to φ ) 0 and to T ) θ, the latter called the theta temperature or theta point of the solution. On an absolute temperature scale θ is 4 times the critical temperature at N ) 1, in this mean-field approximation. More specifically, φc, the volume fraction of polymer at the critical point, is (1 + xN)-1 in this approximation, and so is 1/ when N ) 1 and approaches 0 proportionally to 1/xN as N 2 f ∞, while the critical value of χ in (32) (which is inversely proportional to the temperature) is χc ) (1/2)(1 + 1/xN)2 and so is 1/4 as great at N ) ∞ as at N ) 1. At any fixed N the coexistence curve in this theory is parabolic at the critical point (exponent β ) 1/2), as in all the other mean-field theories we have discussed, whereas in reality β ) 0.32 or 0.33, just as at any other critical point of real two-phase equilibrium. There is much current interest in correcting the mean-field theory by renormalization-group calculations and other modern methods.24 The nature of the interface between phases in these phaseseparated polymer solutions and, in particular, the question of how the interfacial structure and tension vary with the degree of polymerization N as well as with the temperature, are of great interest.25 One finds that the profile of the density φ(z) of polymer-chain segments is much like F(z) in Figure 4; but one may now ask, in addition, for the profiles of the densities of horizontally and vertically oriented links between adjacent monomers on a chain, and that is a question of much greater subtlety,25 for it requires understanding how the orientation of chain links couples to the composition φ(z). An orientable object may have what we may call dipolar symmetry, represented by v, which not only distinguishes vertical from horizontal but, in the vertical direction, also distinguishes up from down; or it may have what we may call nematic symmetry, represented by vV, which only distinguishes vertical from horizontal but not up from down. The orientation of objects with dipolar symmetry couples to φ′(z) because such objects can sense the sign of the gradient, whereas that of objects
Figure 14. T, φ coexistence curve for N ) 1 (regular-solution theory) and N . 1 (Flory theory). θ is the theta-point temperature.
Figure 15. Concentration profiles of orientable objects of dipolar symmetry ((a) or (b)) and of nematic symmetry ((c) or (d)), when these are coupled to a composition profile φ(z) like the F(z) in Figure 4.
with nematic symmetry couples to φ′(z)2 and to φ′′(z) because such objects cannot distinguish z from -z. The result is that, in an interface in which there is a composition profile φ(z) like the F(z) in Figure 4, objects of dipolar type are attracted to or repelled from the high-gradient region, whereas objects of nematic type are attracted to the high-φ side of the gradient and repelled from the low-φ side, or vice versa.26 Profiles of the concentration of such orientable objects coupling to a composition profile φ(z) are shown in Figure 15. In the model polymer solution the objects in question are pairs of successive links of the chain,25 some such pairs having dipolar symmetry and some nematic. The orientation profile of single links is obtained as a certain combination of those of the various pairs. The result is found to be a bias favoring horizontal orientation of links (i.e., parallel to the interface) on that side of the interface that is toward the phase with the higher concentration of polymer and favoring vertical orientation of links (i.e., perpendicular to the interface) on the side toward the more dilute phase. There is no such bias within the bulk phases themselves, beyond the purely statistical one connected with there being more ways of being oriented horizontally than vertically. Surfactant solutions are another in this class of complex fluids. Surfactants are amphiphiles, carrying both hydrophilic and hydrophobic (lipophilic) groups. They then serve to render oils and water mutually soluble through the formation of micelles, bilayers, and other such structures. The resulting solutions may be isotropicsand are then called microemulsionssor liquid crystalline, with structural periodicity in one, two, or three
Theory of Phase Equilibrium
J. Phys. Chem., Vol. 100, No. 31, 1996 13199
dimensions (lamellar, hexagonal, or cubic, respectively). The phase diagrams of oil-water-surfactant solutions are often of great complexity.27 Three-phase equilibria in which a microemulsion with high concentrations of both oil and water is in equilibrium simultaneously with a phase that is almost pure water and another that is almost pure oil are common and are, indeed, a signature of such mixtures. The coexistence of an isotropic microemulsion with a liquid-crystalline phase or of different liquid-crystalline phases with each other are also common in these solutions. All the statistical mechanical tools and principles that have been outlined above have also been used in the theoretical analysis of these surfactant-solution phase equilibria: lattice models, mean-field approximations, etc.28 An analog of van der Waals’s free energy functional (18) has proved to be a versatile tool in these studies:29
F {φ} ) ∫[c(∇2φ)2 + g(φ)|∇φ|2 + j(φ) - µφ] dτ
(33)
Here the free energy F is a functional of the composition variable φ(r), which may, for example, be the local difference, at the location r, of the concentrations of oil and water, with µ then the corresponding difference of chemical potentials. The integration is over all space with dτ the element of volume at r. The coefficient g(φ) of the square of the gradient of φ is the analog of (1/2)m in (18) while j(φ) - µφ is the analog of the earlier F(F). The square of the Laplacian of φ, with coefficient c in (33), has no analog in the earlier functional but is a characteristic element in theories of microemulsion structure and phase transformations. With this term present, with c > 0, the coefficient g(φ) of the square gradient may become negative without leading to an instability. A negative g(φ) encourages the formation of spatial structures with large gradientss characteristic of surfactant-solution phasesswhile the c(∇2φ)2 term (equivalent, after two integrations by parts, to cφ∇4φ) does not allow such gradients to become so great as to be catastrophic. Conclusion This has been a sampling of past and current work in the theory of phase equilibrium, emphasizing the classical, meanfield approximation and its applications. Many other topics could have been included and even those covered could have been elaborated in much greater detail, but neither the author’s allotment of space nor the reader’s patience would have allowed it. Probably the most striking fact about all these applications is that they would have been comprehensible and at least partly achievable in the past centurysand indeed, as we saw, many were. What has been slighted in this account are the important efforts to go beyond the mean-field approximation and to correct its defects. These developments are more adequately treated in the reviews that are among the present references, which the interested reader will find rewards in exploring. Acknowledgment. It is a pleasure to acknowledge helpful conversations with E. M. Blokhuis and D. J. Bukman. The author’s own work in the statistical mechanics of phase equilibrium has been supported by the National Science Foundation and the Cornell University Materials Science Center.
References and Notes (1) J. D. Van der Waals: On the Continuity of the Gaseous and Liquid States, Studies in Statistical Mechanics XIV; Rowlinson, J. S., Ed.; North-Holland: Amsterdam, 1988. (2) Rowlinson, J. S. “Translation of J. D. van der Waals’ ‘The Thermodynamic Theory of Capillarity Under the Hypothesis of a Continuous Variation of Density’”. J. Stat. Phys. 1979, 20, 197. (3) Sengers, J. V.; Levelt Sengers, J. M. H. In Progress in Liquid Physics; Croxton, C. A., Ed.; Wiley: New York, 1978; pp 103-174. Fisher, M. E. In Lecture Notes in Physics; Hahne, F. J. W., Ed.; Springer-Verlag: Berlin, 1983; Vol. 186, pp 1-139. (4) Levelt Sengers, J. M. H. Physica 1976, 82A, 319. (5) Kadanoff, L. P. From Order to Chaos, Series A, Vol. 1 of World Scientific Series on Nonlinear Science; World Scientific: Singapore, 1993; pp 165-174 (reprinted from Physics 1966, 2, 263). (6) Wilson, K. G. Nobel Lecture. ReV. Mod. Phys. 1983, 55, 583. Wilson, K. G.; Fisher, M. E. Phys. ReV. Lett. 1972, 28, 240. (7) Buff, F. P.; Lovett, R. A.; Stillinger, F. H., Jr. Phys. ReV. Lett. 1965, 15, 621. (8) Servos, J. W. Physical Chemistry from Ostwald to Pauling; Princeton University Press: Princeton, NJ, 1990; p 166. (9) Zittartz, J. Phys. ReV. 1967, 154, 529. (10) Wertheim, M. S. J. Chem. Phys. 1976, 65, 2377. Evans, R. AdV. Phys. 1979, 28, 143, Appendix 4. Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Oxford University Press: New York, 1982; Sections 4.5 and 4.9. Bedeaux, D.; Weeks, J. D. J. Chem. Phys. 1985, 82, 972. (11) Lawrie, I. D.; Sarbach, S. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: New York, 1984; Vol. 9, Chapter 1. Knobler, C. M.; Scott, R. L. Ibid., Chapter 2. (12) Krichevskii, I. R.; Efremova, G. D.; Pryanikova, R. O.; Serebryakova, A. V. Russ. J. Phys. Chem. 1963, 37, 1046 [Zh. Fiz. Khim. 1963, 37, 1924]. (13) Radyshevskaya, G. S.; Nikurashina, N. I.; Mertslin, R. V. J. Gen. Chem. USSR [Zh. Obshch. Khim.] 1962, 32, 673. (14) van der Waals, J. D.; Kohnstamm, Ph. Lehrbuch der Thermodynamik; Barth: Leipzig, 1912; 2nd Teil, pp 39-40. (15) Fox, J. R. J. Chem. Phys. 1978, 69, 2231. (16) Griffiths, R. B. J. Chem. Phys. 1974, 60, 195. (17) Rowlinson and Widom, loc. cit. (ref 10 above), Chapter 8. (18) Winter, A. Heterog. Chem. ReV. 1995, 2, 269. (19) Nakanishi, H.; Fisher, M. E. Phys. ReV. Lett. 1982, 49, 1565. de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 827. Dietrich, S. In Phase Transitions and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: New York, 1988; Vol. 12, pp 1-218. Schick, M. In Liquides aux interfaces/Liquids at interfaces, Les Houches, Session XLVIII, 1988; Charvolin, J., Joanny, J. F., Zinn-Justin, J., Eds.; Elsevier: Amsterdam, 1990; pp 416-497. (20) Cahn, J. W. J. Chem. Phys. 1977, 66, 3667. (21) Gibbs, J. W. In The Collected Works of J. Willard Gibbs; Longmans, Green: New York, 1928; Vol. 1, p 288, footnote. (22) Indekeu, J. O. Int. J. Mod. Phys. B 1994, 8, 309. (23) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953; Chapters XII, XIII. (24) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979; Chapters IX-XI. Freed, K. F. Renormalization Group Theory of Macromolecules; Wiley-Interscience: New York, 1987. (25) Helfand, E. J. Chem. Phys. 1975, 63, 2192. Szleifer, I.; Widom, B. J. Chem. Phys. 1989, 90, 7524. (26) Carton, J.-P.; Leibler, L. J. Phys. (Paris) 1990, 51, 1683. (27) Laughlin, R. G. The Aqueous Phase BehaVior of Surfactants; Academic Press: New York, 1994. (28) Gompper, G.; Schick, M. In Phase Transitions and Critical Phenomena, Domb, C., Lebowitz, J. L., Eds.; Academic Press: New York, 1994; Vol. 16. Laradji, M.; Guo, H.; Grant, M.; Zuckermann, M. J. AdV. Chem. Phys. 1995, 89, 159. (29) Gompper, G.; Schick, M. Phys. ReV. Lett. 1990, 65, 1116.
JP9536460
