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Articles One-Dimensional Model for Oil-Continuous Microemulsions Ricardo D. Gianotti,† Antonio E. Rodrı´guez, and Fernando Vericat*,† Instituto de Fı´sica de Lı´quidos y Sistemas Biolo´ gicos (IFLYSIB)-UNLP-CONICET, cc. 565, (1900) La Plata, Argentina Received July 17, 1995. In Final Form: October 30, 1995X A one-dimensional model for microemulsions is considered. In the model the microemulsion droplets are hard rods that interact through nearest-neighbor pair potentials. Besides the hard core repulsion the interaction presents a square well and a square barrier that have arbitrary width and strength. The thermodynamic and structure properties for this system are found via the isothermal-isobaric ensemble. The model predicts that, under certain conditions, the aggregation number increases with temperature, as is experimentally observed. When the temperature is lowered, the system structure is solid-like and it can be found in two phases with different characteristic wavenumbers.
I. Introduction When a sufficient amount of a suitable surfactant is added to a water-in-oil dispersion, the system becomes a thermodynamically stable microemulsion.1,2 Microemulsions can exist in several phases depending on the relative concentrations of their constituents. Here we are mainly interested in that phase in which water forms small droplets coated with a surfactant monolayer. The phase then comprises an oil-continuous phase and an aqueous phase dispersed into microdroplets (10-50 nm diameter), each bounded by a molecular layer of surfactant.3 For many purposes the droplets can be considered as microphases with well-defined interfaces and are reasonably well described by a hard-spheres model.4,5 There is evidence, however, of a thermodynamic equilibrium between microphases of different aggregative states for which the hard-sphere model gives just an incomplete picture.3 In particular, the model does not take into account the observed temperature dependence of the droplet’s clustering. In order to achieve a better description, improvements of the hard-sphere concept have been proposed. For example, the sticky hard sphere (SHS) model of Baxter6 allows us to introduce the temperature in the clustering of droplets.7 Nonetheless, the SHS model yields a decreasing of the aggregation with temperature whereas the experiment shows that it actually increases with T. * Corresponding author. FAX: 54-21-257317. † Also at Grupo de Aplicaciones Matema ´ ticas y Estadı´sticas de la Facultad de Ingenierı´a (GAMEFI), Departamento de Fisicomatema´tica, Facultad de Ingenierı´a, U.N. de La Plata, Argentina. X Abstract published in Advance ACS Abstracts, April 15, 1996. (1) Robb, I., Ed. Microemulsions; Plenum Press: New York, 1982. (2) Tadros, Th. F. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 3. (3) Eicke, H. F.; Kubik, R.; Hasse, R.; Zschokke, I. In Surfactants in Solution; Mittal, K. L., Lindman, B., Eds.; Plenum Press: New York, 1984; Vol. 3. (4) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: London, 1976. (5) Agterof, W. G. M.; van Zomeren, J. A. J.; Vrij, A. Chem. Phys. Lett. 1976, 43, 363. (6) Baxter, R. J. J. Chem. Phys. 1968, 49, 2770. (7) Koper, G. J. M.; Bedeaux, D. J. Chem. Phys. 1992, 96, 7193; Physica 1992, A187, 489.
S0743-7463(95)00582-8 CCC: $12.00
To obtain the correct temperature dependence, Bedeaux et al.8 suggest the use of a pair potential between droplets which, besides a hard core and a narrow square well, has a very large repulsive barrier. In this work we consider the one-dimensional version of a microemulsion model in which the droplets are hard spheres with a square well and a square barrier that have arbitrary width and strength. We get the exact thermodynamic expressions and find the correlation functions and the structure factors for this model via the isothermal-isobaric ensemble. Obviously, one-dimensional models are less realistic than their three-dimensional counterparts. However, in the present case, besides its analytical simplicity, the 1D solution shows many of the features observed in real microemulsions as well as an interesting behavior at low temperatures, where the system structure factor shows well-defined peaks like a solid with two different periods that depend on the potential parameters. II. Thermodynamic Expressions The one-dimensional model of microemulsion we consider is a system of N microdroplets modeled by hard rods of length σ which lie on a line segment [0, L]. The positions of particles are designated by xi (i ) 1, 2, ..., N; 0 e xi e L). Hereafter, we shall assume that particles interact with each other via a pairwise additive potential (see Figure 1).
{
∞ u(|xj - xi|) ) 1 2 0
if if if if
|xj - xi| e σ σ < |xj - xi| < λ λ < |xj - xi| < δ |xj - xi| > δ
(1)
where σ, λ, and δ are real positive numbers and 2 and 1 are, in principle, any real number. We also assume δ < 2σ, namely, only nearest-neighbor interactions are present. In Figure 1 the two cases ) 1/2 < 0 and > 0 are distinguished. In order to obtain the thermodynamic properties of our model, we calculated the Gibbs free energy per particle, (8) Bedeaux, D.; Koper, G. J. M.; Smeets, J. Physica 1993, A194, 105.
© 1996 American Chemical Society
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Figure 1. Pair potential for the microemulsion model: (a) ) 1/2 < 0; (b) > 0.
Figure 2. Isotherms p* vs l* at T ) 0 for (a) < 0 and (b) > 0.
g(T,p), as
g(T,p) ) -β-1[ln Ω(s) - ln Λ]
(2)
where Λ ) (h2β/2πm)1/2, β-1 is the product of temperature T and the Boltzmann constant kB, m is the mass of particles, h is the Planck constant, and
Ω(s) )
∫0 dx exp{-sx} exp{-βu(x)} ∞
with s ) βp and p the one-dimensional pressure.
(3)
In the thermodynamic limit L f ∞, N f ∞, and L/N ) l finite, the equation of state can be calculated from eq 2 as
l ) (∂g(T,p)/∂p)T
(4)
From eqs 1 and 3, we get
Ω*(s*) ) (β*p*)-1 exp{-β*(p* + )}{1 exp{-β*p*(R1 - 1)} + e-β*(1-)(exp{-β*p*(R1 - 1)} exp{-β*p*(R2 - 1)}) + exp{-β*(p*(R2 - 1) - )}} (5)
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Gianotti et al.
Figure 3. Pair distribution function for a low temperature and > 0 (a-c) and the corresponding structure factor (d-f).
where we have introduced the reduced variables p* ) pσ/2, β* ) β2, s* ) sσ ) β*p*, R1 ) λ/σ, R2 ) δ/σ, l* ) l/σ, and Ω*(s*) ) Ω(s*)/σ. The equation of state, from eqs 2 and 4, gives
l* ) 1 + (β*p*)-1 - A′1/A1
(6)
with
A′1 ) (R1 - 1)e
-β*∆1
+e
(-(R1 - 1)e
-β*∆1
-β*(1-)
(R2 - 1)e
-β*∆2
+
) - (R2 - 1)e
-β*∆3
(7)
A1 ) 1 - e-β*∆1 - e-β*(1-)(e-β*∆1 - e-β*∆2) + e-β*∆3 (8) and
∆1 ) p*(R1 - 1); ∆2 ) p*(R2 - 1); and ∆3 ) p*(R2 - 1) - (9) It is well-known9,10 that l* is an analytic function of p* for nearest-neighbor potentials, like those given by eq 1, and finite temperatures. When T f 0, the equation of state strongly depends on the sign of . In fact, the equation of state reduces to (see Figure 2a)
l* ) 1 for p* g 0
{
1 for R2 for
p*(R2 - 1) - g 0 p*(R2 - 1) - e 0
{
∞ for p* ) /(R2 - 1) for 0
for
l* e 1 1 e l* e R2
(12)
l* g R2
that shows two different phases above and below the pressure value p* ) /(R2 - 1) (Figure 2b). The nature of these phases will be more evident in the next section when we study the system structure. So far we take into account just nearest-neighbor interactions. It should be worthwhile to consider how the phase diagram changes by the addition of long range forces. For long range Kac potentials in the van der Waals limit, say a potential of infinite range and vanishingly small strength, the state equation is modified to give11
p ) p° +
RF2 plus Maxwell’s rule 2
(13)
Here R t ∫φ(r b) dr b, where φ(r b) denotes the long range perturbation and p° is the pressure in the reference system (φ(r b) ) 0).
(10) III. Structure
when < 0 and to
l* )
for > 0. Inverting this last equation, we get
(11)
(9) Van Hove, L. Physica 1950, 16, 137. (10) Lieb, E. L.; Mattis, D. C. Mathematical Physics in One-Dimension; Academic Press: New York, 1966.
It is known that the Born-Green equation gives the exact pair correlation function, F2(x1,x2), for nearestneighbor potentials with a hard core and a finite range of attraction.12 (11) Frisch, H. L., Lebowitz, J. L., Eds. The Equilibrium Theory of Classical Fluids; W. A. Benjamin Publishers: New York, 1964. (12) Sells, R.; Harris, C. W.; Guth, E. J. Chem. Phys. 1953, 21, 1422.
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Figure 4. Pair distribution function for a low temperature and < 0 (a-c) and the corresponding structure factor (d-f).
The BBGKY relation between F2(x1,x2) and the triplet correlation function F3(x1,x2,x3) is given by
lim y(r) ) 1 rf∞
∂F2(x1,x2) ∂u(x1,x2) + βF2(x1,x2) ) ∂x1 ∂x1 ∂u(x1,x3) +∞ dx3 (14) β -∞ F3(x1,x2,x3) ∂x1
∫
For the potential defined in eq 1, the Kirkwood superposition approximation may be written as (F ) 1/l)
{
F2(x) F2(r - x) FF3(x1,x2,x3) ) F2(x - r) F2(r) F2(x) F2(-r)
for for for
r>x 0 0 (a-c) and < 0 (d-f).
smaller, almost equal, and larger, respectively, than the transition value /(R2 - 1) ) 0.294. In the three cases g(x*) looks like the distribution function of a crystalline solid. At the largest pressure (Figure 3c) g(x*) presents peaks at the same position as for hard rods (without any
other interaction) of reduced length 1. For p* ) 0.2, instead, the position corresponds also to pure hard rods but now of reduced length R2 ) 1.85 (Figure 3a). For the limiting case p* ) 0.3, the curve shows peaks at positions x* ) n (n ) 1, 2 in the picture) and also at 1.85.
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It is well-known that the one-dimensional crystal cannot exist at T > 0.13 Although our system always is a fluid, we observe that when β increases, the structure factor looks more and more like a solid one, the perfect crystalline behavior being just reached in the limit T f 0. This behavior tell us about the presence of two crystalline like phases characterized each one for a different “lattice” constant, as is clearly seen in Figure 3d-f. For β* ) 100 and < 0 () -0.25), the system behavior does not depend on the pressure (Figure 4), looking always like a frozen hard-rod fluid in agreement with the phase diagram shown in Figure 1a. In Figure 5, we can see the curves g(x*) and S(k*) for β* ) 10 and ) 0.25. From these the system fluid-like behavior is apparent. By increasing the pressure, the influence of the potential details on the correlations tends to disappear, giving featureless structure factors that resemble those of simple hard-rods fluids. From the pair correlation function g(x), we define an aggregation number according to
∫σδg(x) dx
na ) 1 + 2F
(22)
In Figure 6 the curves na vs β* are drawn for the same parameters as in previous figures. We observe that, for > 0, na strongly depends on the pressure. At pressures above the transition pressure p*t ) /(R2 - 1), it monotonically increases with the inverse of temperature, going to 3 at very low temperature values. This means that on both sides of a given particle we have, on average, another two particles at distances shorter than δ. For pressures (13) Landau, L. D.; Lifshitz, E. M. Statistical Physics; Adisson-Wesley Publishing Co.: Reading, MA, 1969.
below p* t we can see that the aggregation number first increases with β* until a maximum beyond which na increases with the temperature in accord with experimental measurements on water/AOT/isooctane microemulsions.7,14,15 For < 0 the pressure dependence practically disappears, giving aggregation numbers that increase with β*, going very quickly to their saturation value 3. IV. Remarks We have considered a simple analytically tractable onedimensional model that, at not too low temperatures, shows many of the features experimentally observed in real microemulsions, particularly the increasing of the aggregation number with temperature. The model is also interesting, since, at lower temperatures, it presents a solid-like behavior with one or two different characteristic “lattice” constants depending on the system thermodynamic state. Currently we are studying this model in higher dimensions in order to obtain a more comprehensive understanding of microemulsions. Acknowledgment. Support of this work by the Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas (CONICET) and Comisio´n de Investigaciones Cientı´ficas de la Provincia de Buenos Aires (CICPBA) of Argentina is greatly appreciated. LA950582B (14) Robertus, C.; Joosten, J. G. H.; Levine, Y. K. Phys. Rev. 1990, A42, 4820. (15) van Dijk, M. A.; Joosten, J. G. H.; Levine, Y. K.; Bedeaux, D. J. Phys. Chem. 1989, 93, 2506.
