Langmuir 1994,10, 1777-1779
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Thermodynamics of Microemulsions Revisited Eli Ruckenstein Department of Chemical Engineering, State University of New York at Buffalo, Buffalo,New York 14260 Received January 18,1994. In Final Form: March 24,1994" By comparing two different thermodynamic formulations, which consider a microemulsion either as a multicomponentmixture or as a dispersionof globules in a continuousmedium, one demonstrates, because they must be equivalent, that (i) the pressure in the continuous medium of a microemulsion differs from the external pressure and (ii) the Laplace equation for a microemulsion contains an additional term due to the entropy of dispersion of the globules in the continuous phase. On this basis, a choice is made between two thermodynamicapproachesto microemulsions. The derived equations explain the occurrence of a microemulsionphase in equilibrium with an excess dispersed phase, as well as the equilibriumbetween a microemulsion phase and the two excess phases, and the change in the structure of the microemulsion at the transition from two to three phases. The thermodynamic equations suggest that the middle-phase microemulsion can be described by the zero mean average curvature of the interface between the two media. The thermodynamics of microemulsions was treated in somewhat different ways by Ruckenstein' and by Overbeek et a1.2 In what follows, the approach of Ruckenstein will be denoted by I and that of Overbeek et al. by 11. The scope of this paper is to make a choice between the two. In both approaches the Helmholtz free energy F of a microemulsion is decomposed in the sum of a frozen free energy Fo, which does not include the entropy of dispersion of the globulesin the continuous medium, and a free energy AF due to the latter entropy of dispersion. The following main differences exist between the two: (i) In treatment I one demonstrates that the pressure in the continuous phase of the microemulsion differs from the external pressure, while in treatment I1 it is equated to the latter. (ii) In treatment I a modified Laplace equation, which contains, in addition to the usual terms, one of entropic origin, is derived. In treatment I1the conventional Laplace equation valid for a single droplet in a continuum is extended to microemulsions. The above differences have implications, since they affect the expressions of the chemical potentials and hence the conditions of equilibrium between a microemulsion phase and the excess dispersed phase, and between a microemulsion phase and both excess phases. In fact, only because the pressure in the continuous medium differs from the external pressure can treatment I explain the occurrence of the latter equilibrium as well as the change in the structure of the microemulsion associated with the transition from two to three phases. The choice between the two approaches will be made on the basis of formal thermodynamics, by comparing two equivalent thermodynamic equations: one in which the microemulsion is considered a single multicomponent mixture phase, and another one which models a microemulsion as a dispersion of globules of one medium in another medium. From the point of view of traditional thermodynamics, a microemulsion is a single phase which consists of the components oil, water, surfactant, cosurfactant, and an @Abstractpublished in Advance ACS Abstracts, May 15, 1994. (1) Ruckenstein, E. Ann. N. Y. Acad. Sci. 1983,404,224;Fluid Phose Equilib. 1985,20,189. For a review, see: Ruckenstein,E. In Progress in Microemdsions;Martellucci,S., Cheater,A. N., Eds.;PlenumPresa: New York, 1989, based on the proceedings of Progress in Microemulsions, Oct 26 to Nov 1, 1985, Erice, Italy; pp 3-29. (2) Overbeek,J. Th. G.;Vemoeckx, G. J.;DeBruyn,P. L.;Lekkerkerker, H. N. W. J. Colloid Interface Sci. 1987,119, 422.
electrolyte. The change a t constant temperature of the Helmholtz free energy F of such a system can consequently be written as
where pi* and Ni are the chemical potential and the number of moles of species i, respectively,p is the external pressure, Vis the volume of the microemulsion, and the summation is extended over all the components. Let us now use a more detailed representation of a microemulsion, by considering that it contains globules of water (oil) dispersed in oil (water), and that the other components are distributed among the dispersed and continuous media of the microemulsion and the interface between them. The Helmholtz free energy is the appropriate thermodynamic potential because the pressures in the globules and in the continuous medium are different. For the sake of simplicity, the globules are assumed spherical and of the same radius r. Denoting the volume fraction of the dispersed phase by 4, the interfacial area A between the two media per unit volume of microemulsion is given by
A = 34/r
(2)
The Helmholtz free energy of the microemulsion is written as the sum
F=F,,+AF
(3)
where AFis the free energy due to the entropy of dispersion of the globules in the continuous phase. For curved interfaces, characterized by the principal curvatures c1 and c2, the Gibbs thermodynamics3provides the following expression for d F 0 :
where y is a generalized interfacial tension since it includes the effect of the interactions among the globules, C1 and Cp are bending stresses associated with the (3) The Collected Works of J. Willard Gibbs; Yale University Press: New Haven, CT, 1948; Vol. I, p 225.
0143-1463/94/2410-~177$04.50/0 0 1994 American Chemical Society
1778 Langmuir, Vol. 10, No. 6, 1994
Ruckenstein
curvatures c1 and C Z , pz is the pressure in the globules, p1 is the pressure in the continuous medium, and pi are the chemical potentials of species i at the pressure p1 for the species present in the continuous medium and at the pressure pz for those present in the dispersed medium. For spherical globules c1 = c, = l / r and C, = C, = C/2
and eq 4 becomes
dF, = y d(AV) + CV d(l/r) + z
p i
Wi- pzd[V4lPI d[V(1- $13 (5)
I t is convenient to introduce the free energy density Af due to the entropy of dispersion of the globules per unit volume of microemulsion:
AF= VAf
(6)
Consequently,
dF = y d(A V) + C V d(l/r) + z
Equations 10-12 coincide with those already derived in a different way.' Because of eq 9d, the chemical potential of the continuous medium (oil or water) is equal to the chemical potential pc (i = c for the continuous medium (oil or water)), which appears in eqs 5 and 7. The latter chemical potential being expressed a t the pressure p1, one can conclude that the pressure in the continuous medium is equal to pl. Equation 12 clearly demonstrates that the pressure p1 cannot be equated to the external pressure p. The pressure p is a macroscopic quantity defined on the scale of a microemulsion (hence on a scale large compared to the size of the globules),which acta over the combination of the two media. In contrast, the pressure p1 acts on the scale of a globule, being the pressure sensed in the continuous medium by each globule. Another consequence of the above thermodynamic treatment concerns the conventional Laplace equation which is no longer valid for a microemulsion. Indeed, eliminating the pressure p between eqs 11 and 12, one obtains
p i Wi- p,
d[ V41d[V(1- 411 + d[VAfl (7)
Denoting by
Since Af depends only on r and 4l dAf =
r$)$+ (3) dr
&#I
the number of globules per unit volume of microemulsion, one can write that
Performing the differentials in eq 7, one obtains
dF= ( wr- p l ( l - $ ) - p 2 4 + A f ) d V + [ Wr + p , V - p , V +
V(z)]de+
[-y-y+
Expression 8,which is basedon a model, must be equivalent to expression 1, which is based on the traditional thermodynamics of a multicomponent mixture. Consequently, 34ylr - pl(1- 4) - pZ4+ Af = -P
(gal
3r+@1-P2)+($$ r =o
and
Equation 9c can be rewritten as
which, combined with eqs 10 and 13, yields
In addition to the first two terms, which appear in the case of a single droplet, eq 16 contains a third term which is due to the entropy of dispersion of the globules in the continuous medium. Ita presence in the modified Laplace equation is expected, because the virtual change in radius which is involved in the derivation of the Laplace equation changes the radius of the globules and hence their entropy of dispersion in the continuous phase. Finally, approach I can explain in a simple manner, in terms of pressures, the phase behavior of microemulsions. For a microemulsion to coexist with an excess dispersed phase the chemical potential pd* of the dispersed medium (water or oil) should be equal to both the chemical potential pd@2) and that of the excess dispersed phase &e@). Considering that the compositions of the dispersed medium and excessdispersed phase are the same, the above equality leads to Pz=P
and solving the system of eqs 9a and 9b, one obtains, after y is replaced with its expression given by eq 10,
(17)
The expression obtained in treatment I1for the chemical potential Pd is different from ours because of the inconsistencies which exist in ita basic thermodynamic equations. Similarly, the equilibrium between the continuous medium and the excess continuous medium leads to
and Consequently, a microemulsion will be in equilibrium
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Langmuir, Vol. 10, No. 6,1994 1779
with both excess phases when
PI = Pa = P
(19)
The condition of equilibrium in eq 19 among the three phases is responsible for the change in the structure of the microemulsion from spherical globules to a disordered one at the transition from two to three phases. This change occurs because a spherical interface cannot remain stable to thermal perturbations when the pressures inside and outside the globules are the same. The condition of thermodynamic equilibrium in eq 19 leads to the conclusion that the curvature at the transition from two to three phases is zero.' In the middle-phase microemulsion, the pressures p2 and p1 fluctuate in time and space, and intuition suggests to replace eq 19 with its average and hence to replace the condition of zero curvature with the condition of zero mean average curvature. A bicontinuous rigid sponge composed of two interpenetrating phases can have a zero average curvature?
However, the rigid structure is expected to be unstable, and a nonrigid sponge may better describe the real structure. In conclusion,the choice is in favor of approach I because (i) the pressure in the continuous medium of a microemulsion is different from the external pressure, (ii) the conventional Laplace equation is not valid for a microemulsion, and finally (iii) it can explain the occurrenceof three phases and the change in the structure of the microemulsion associated with the transition from two to three phases. Approach I1 considers the pressure in the continuousmedium equal to the external pressure, employs the conventional Laplace equation, and cannot predict the coexistence of three phases. (4) Neovius, E. R.; Minimalflachen; Frenkel, J. C.; Helsingfors, 1883. Schwartz, M. A. Gesammette Mathematkche Abhandlung; Springer: Berlin, 1890,Vol. 1. Scriven, L. E. In Micellization, Solubilization and Microemulsiom; Mithal, K. L., Ed.; Plenum: New York, 1977.
