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Langmuir 1998, 14, 2684-2692
Phase Behavior and Solubilization in Surfactant-Solute-Solvent Systems by Monte Carlo Simulations Sameer K. Talsania, L. A. Rodrı´guez-Guadarrama, Kishore K. Mohanty, and Raj Rajagopalan* Department of Chemical Engineering, University of Houston, Houston, Texas 77204-4792 Received August 4, 1997. In Final Form: March 3, 1998 We use a lattice-based Monte Carlo method presented recently for micellar solubilization (Talsania et al. J. Colloid Interface Sci. 1997, 190, 92) to determine the phase behavior in surfactant-solute-solvent systems and to examine the locus and extent of solubilization of solutes in micelles as a function of the solute hydrophobicity and chain length. A novel method, based on the distribution of solute in clusters of different sizes, is developed to study the phase behavior of the solute in the absence and presence of surfactant. The results obtained from these simulations are compared with estimates based on quasichemical theory to determine the applicability of this theory to micellar systems. Addition of surfactant to a solutesolvent system first leads to a decrease in the solubility of the solutes prior to a subsequent increase at higher surfactant concentrations. An examination of micellar shapes shows that the surfactant studied forms roughly spherical micelles. However, the presence of solute induces coexistence of spherical and oblong micelles.
1. Introduction Surfactant molecules generally have two mutually incompatible structural features: one hydrophobic and the other hydrophilic. In the presence of an appropriate solvent, they have the ability to aggregate into structures called micelles. These aggregates can be employed in improving the solubilization of hydrophobic solutes, a process known as micellar solubilization. This phenomenon is used extensively in emerging applications1,2 in several industries, particularly in the environmental,3-6 pharmaceutical,7-10 and biological11 fields. Surfactants also have an interesting effect on the phase behavior of solutions due to their tendency to aggregate into an array of microstructures. The phase behavior of surfactant-solute-solvent systems can be studied through the use of lattice theories such as the quasichemical theory12,13 and Ising models.14,15 In Ising-type models, the molecules are represented by “spins” on lattice sites * Author for correspondence. Present address: Department of Chemical Engineering, University of Florida, Gainesville, FL 326116005. (1) Elworthy, P. H.; Florence, A. T.; Macfarlane, C. B. Solubilization by Surface-Active Agents; Chapman & Hall: London, 1968. (2) Attwood, D.; Florence, A. T. Surfactant Systems: Their Chemistry, Pharmacy, and Biology; Chapman & Hall: London, 1983. (3) Dunn, R. O., Jr.; Scamehorn, J. F.; Christian, S. D. Sep. Sci. Technol. 1985, 20, 257. (4) Scamehorn, J. F.; Harwell, J. H. In Surfactants in Chemical/ Process Engineering; Wasan, D. T., Ginn, M. E., Shah, D. O., Eds.; Dekker: New York, 1988; p 77. (5) Christian, S. D.; Scamehorn, J. F. In Surfactant-Based Separation Processes; Scamehorn, J. F., Harwell, J. H., Eds.; Dekker: New York, 1989; p 3. (6) Dunn, R. O., Jr.; Scamehorn, J. F.; Christian, S. D. Colloids Surf. 1989, 35, 49. (7) Florence, A. T. In Techniques of Solubilization of Drugs; Yalkowsky, S. H., Ed.; Dekker: New York, 1981; p 15. (8) Speiser, P. In Reverse Micelles: Biological and Technological Relevance of Amphiphilic Structures in Apolar Media; Luisi, P. L., Straub, B. E., Eds.; Plenum: New York, 1984; p 339. (9) Lawrence, M. J. Chem. Soc. Rev. 1994, 23, 417. (10) Attwood, D. In Colloidal Drug Delivery Systems; Kreuter, J., Ed.; Dekker: New York: 1994; p 31. (11) Hatton, T. A. In Surfactant-Based Separation Processes; Scamehorn, J. F., Harwell, J. H., Eds.; Dekker: New York, 1989; p 57.
and/or by near-neighbor bonds. As a result, they are restricted in their ability to represent the structural details of the surfactant and, therefore, the entropic contributions to the phenomena. Phase behavior has also been studied extensively through Monte Carlo simulations based on a lattice model.16 For the most part, the works based on this model17-19 and variations of it20,21 have looked at phase boundaries at high surfactant and/or solute concentrations (i.e., over the entire range of the ternary phase diagram). However, surfactants can also have an interesting effect on phase behavior at low surfactant and solute concentrations, which is the focus of this work. Two important characteristics of micellar solubilization are the extent and the locus of solubilization. Depending on the structure and degree of hydrophobicity of the organic solutes, they can be solubilized inside the hydrocarbon core of the micelle, as in Figure 1a, or in the socalled palisade layer, as in Figure 1b.22 The solubilization inside a micelle can also affect micellar properties such as size, shape, and critical micelle concentration (cmc), which depend mostly on surfactant structure and concentration but are also affected by the structure and concentration of any solute present. For most single-tail surfactants, the shape of the micelle at the cmc is roughly (12) Fowler, R. H.; Guggenheim, E. A. Proc. R. Soc. London 1940, A174, 189. (13) Tompa, H. Polymer Solutions; Academic Press: New York, 1956. (14) Gompper, G.; Schick, M. In Phase Transitions and Critical Phenomena; Domb C., Lebowitz, J., Eds.; Academic Press: London, 1994; Vol. 16, p 1. (15) Gelbart, W. M.; Ben-Shaul, A. In Micelles, Membranes, Microemulsions, and Monolayers; Gelbart, W. M., Ben-Shaul, A., Roux, D., Eds.; Springer-Verlag: Berlin, 1995; p 1. (16) Larson, R. G.; Scriven, L. E.; Davis, H. T. J. Chem. Phys. 1985, 83, 2411. (17) Larson, R. G. J. Chem. Phys. 1988, 89, 1642. (18) Larson, R. G. J. Chem. Phys. 1989, 91, 2479. (19) Larson, R. G. J. Chem. Phys. 1992, 96, 7904. (20) Brindle D.; Care, C. M. J. Chem. Soc., Faraday Trans. 1992, 88, 2163. (21) Mackie, A. D.; Onur, K.; Panagiotopoulos, A. Z. J. Chem. Phys. 1996, 104, 3718. (22) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Dekker: New York, 1997.
S0743-7463(97)00865-2 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/17/1998
Surfactant-Solute-Solvent Systems
Figure 1. Schematic diagram of solubilization in (a) the micellar core and (b) the palisade layer.
spherical.22 However, with the presence of solutes, the aggregates can change from spherical to cylindrical, lamellar (planar), or “bicontinuous” shapes.23 Hence, in this paper, we also examine the effect of different solutes on micellization. The objective of this paper is to show how a model presented previously24 can be used to study phase behavior in a surfactant-solute-solvent system. We shall first determine which parameters of the model are essential to capture the essence of aggregation phenomena in a surfactant-solute-solvent system. Second, we shall examine the solubilization of different types of solutes in the solvent through Monte Carlo simulations using criteria we define for phase separation. These results are compared with those obtained from quasichemical theory and from a simple thermodynamic model we have proposed. Then, we examine how the introduction of surfactant molecules into a solute-solvent system affects the overall phase behavior. Finally, we examine the micellar phase of the system with respect to the types of microstructures formed by the surfactant and the locus and extent of solubilization of solutes inside these structures. The paper is organized as follows. We begin with the essential details of the simulation procedure and of the quasichemical theory (section 2). The phase behavior of a two-component solute-solvent system is discussed in section 3 using results based on the simulations. We then move on to phase behavior in the presence of surfactants and partitioning of the solutes inside of micelles. Finally, we conclude the paper with a brief summary of the results (section 4). 2. Model
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All simulations are started from a random initial configuration. Reptation is used for rearrangement of the chains on the lattice. The probability of acceptance is calculated according to the standard Metropolis algorithm.25 Characterization of the system begins once the total system energy reaches an equilibrium value, usually after 100 000 Monte Carlo cycles. Characterization includes measurement of the critical micelle concentration (cmc), the size distribution of the micelles, and the partition coefficient K, which is the ratio of the solute concentration in the micellar pseudophase to that in the solvent. Additional details of the model, simulation technique, and characterization methods can be found in ref 24. 2.2. Quasichemical Theory. In this section we present a brief outline of a lattice-based analytical theory known as the quasichemical theory (QCT). Our objective is to examine the accuracy of QCT in predicting the phase behavior of the systems of interest here. First proposed by Fowler and Guggenheim,12 QCT may be considered a first-order approximation to real fluid mixtures. The zeroth-order approximation of such a lattice approach, known as the Bragg-Williams approximation,26 corresponds to a completely random distribution of the component molecules (here, the hydrophilic and the hydrophobic beads). For example, for a collection of hydrophilic (a) and hydrophobic (b) species on a lattice, the numbers of hydrophilic-hydrophobic, hydrophilic-hydrophilic, and hydrophobic-hydrophobic pairs, that is, nab, naa, and nbb, respectively, are related to each other in the BraggWilliams approximation by
(21n )
2
ab
naanbb
)1
(1)
However, for a positive hydrophilic-hydrophobic interaction energy, distributions which have a lower lattice energy will be favored and therefore the number of hydrophilichydrophobic pairs will be less than the number for a random system. In other words, nonzero values of interaction energy introduce a certain order in the mutual distribution of the molecules and QCT is a better approximation in this case. The analogue of eq 1 in QCT is given by
(21n )
2
ab
naanbb
) exp(-2)
(2)
in which is the dimensionless energy parameter given by
2.1. Monte Carlo Simulation. The lattice used in our model is a three-dimensional simple cubic lattice with a coordination number of z ) 6; that is, only nearest neighbor interactions are taken into account. A 50 × 50 × 50 lattice has been used for all the results reported in this paper. The regular excluded volume and periodic boundary conditions are used. The lattice can be occupied by four types of beads: solvent, s; solute, c; head, h; and tail, t. Surfactant chains, hitj, and solute molecules, ck, are composed of appropriate numbers (i, j, k g 1) of h, t, and c beads. Each bead-bead pair interaction is assigned a nondimensional interaction energy, Epq, where p, q ) s, c, h, or t.
where ab, aa, and bb represent the interaction energies between the subscripted beads. Quasichemical theory leads to explicit equations for thermodynamic variables in a lattice system13 and is increasingly more accurate as decreases. Quasichemical theory is a mean field theory based on a detailed accounting of the possible configurations and the number of interactions. The basic ideas of the quasichemical approximation can be extended to multi-
(23) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1991. (24) Talsania, S. K.; Wang, Y.; Rajagopalan, R.; Mohanty, K. K. J. Colloid Interface Sci. 1997, 190, 92.
(25) Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1953, 21, 1087. (26) Hill, T. L. Statistical Mechanics: Principles and Selected Applications; McGraw-Hill: New York, 1956.
) ab -
(aa + bb) 2
(3)
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Table 1. Parameters for Eqs 4 and 5a ri qi Ai Bi
Table 2. Interaction Energies Used for the Various Systems of Simulations
i)1 (solute)
i)2 (solvent)
i)3 (surfactant)
lc [lc(z - 2) + 2]/z 0 1
1 1 1 0
l ) l h + lt [l(z - 2) + 2]/z [lh(z - 2) + 1]/[l(z - 2) + 2] [lt(z - 2) + 1]/[l(z - 2) + 2]
a l and l are the lengths of the surfactant head and tail, h t respectively, and lc is the length of the solute molecules.
component systems with molecules consisting of arbitrary numbers of hydrophobic and hydrophilic beads. We shall not go into the details of the derivation, as they are readily available in standard references,13 and merely present the pertinent equations here. We define f ) βF/L3, the dimensionless Helmholtz energy per lattice site, where β ) 1/kBT and F is the Helmholtz energy of the lattice consisting of L3 sites. The quasichemical theory leads to the following expression for f
f ) f0 +
z
[ (
2
)
1 - κB
A ln
A
(
+ B ln
)]
1 - κA B
3
qiyi ∑ i)1
(4)
with
( )]
[
ξi z yi ln Xi + qi ln 2 Xi i)1 3
f0
∑
(5)
where the number concentration of the molecules i is yi 3 (yi ) ni/L3), the volume fraction is Xi ) rini/∑i)1 rini, the 3 neighbor fraction of i is given by ξi ) qini/∑i)1 qini, the fraction of neighboring sites of a molecule i coming from 3 ξiAi, the corresponding the hydrophilic beads is A ) ∑i)1 3 ξiBi, and n1, n2, and n3 hydrophobic analogue is B ) ∑i)1 are the number of solute, solvent, and surfactant molecules, respectively. Here, ri is the length of molecule i in terms of the number of beads, Ai is the ratio of the number of neighboring sites of the hydrophilic beads of a molecule i over the total number of neighbors of the molecule, Bi is the corresponding fraction for hydrophobic beads, and qi is the ratio of the total number of neighboring sites of a molecule i to the coordination number z. Table 1 lists the expressions for these parameters. The ratio, κ, of the energy obtained from QCT to that from the BraggWilliams approximation is given by
1 - κ ) κ2AB(2 - 1)
(6)
and may therefore be regarded as a screening factor which quantifies the degree to which the solution differs from a random solution. The chemical potential of component i can be obtained by differentiating eq 4 with respect to ni, which gives
[()
(
)
ξi 1 - κB z + Ai ln + µi ) ln Xi + qi ln 2 Xi A Bi ln
(
)]
1 - κA B
(7)
The phase behavior can be determined by equating the chemical potential of each component in one phase to that of the same component in the other phase. By fixing the concentration of one of the components in one of the phases, one can obtain the three other independent concentrations by solving the three simultaneous nonlinear equations representing the equality of chemical potentials using a
system
Ehs
Ets
Ecs
Eht
Ehc
Etc
Ehh
Ett
Ecc
1 2 3 4
0 - - -
+ + + +
+ + + +
+ 0 + +
+ 0 + 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
Table 3. Variation in Micellar Properties for Different Sets of Energy Parameters for a System of h2t4 Surfactants and 250 c1 Solutes (Xc ) 0.002) with E ) 0.8 systema
Ncmc
Xcmc
Nn
ln K
1 2 3 4
70 120 180 180
0.003 36 0.005 76 0.008 64 0.008 64
40 30 25 25
3.1 3.2 3.0 3.2
a
See Table 2 for a definition of the four systems involved.
globally convergent numerical method.27 An initial guess of the three unknowns must be providedswhich must be quite close to the solution values to avoid divergencesespecially when a critical point is nearby. Once one tie line is determined, good initial guesses for the other tie lines can always be obtained simply by requiring the unknown tie line to be a sufficiently small perturbation away from the known tie line. 3. Results and Discussion 3.1. Selection of Interaction Energies. In this section we evaluate the relative importance of how various interaction energies affect micellar behavior. Many different types and magnitudes of molecular interactions have been used in the past in Monte Carlo simulations of surfactant systems. The complexity of the models varies from those which use only one parameter to designate the energy of all hydrophilic-hydrophobic interactions, such as Larson’s model,19 to models which have several energy parameters.24,28 Care29 has shown recently that the surfactant head-head interaction is not important to explain the micellization process; therefore, we have not used this interaction in what follows. We have performed simulations using four different sets of parameters (Table 2) and have measured micellar characteristicsscmc, aggregation number30 n, and partition coefficient Ksfor systems of h2t4 surfactant with 250 c1 solutes (Xc ) 0.002), as shown in Table 3. The magnitudes of all parameters shown in Table 2 were kept constant at ) 0.8. System 1 is the equivalent of Larson’s system with only repulsive hydrophilic-hydrophobic interactions between “oil-like” and “water-like” beads. System 2 is similar to that of Care and co-workers,28 who used only solvent interactions. System 3 is a combination of systems 1 and 2, and system 4 is the set of parameters used by Talsania et al.24 Figure 2 shows the number-average aggregation number Nn of the aggregates in each of the four systems as a function of the total number (or concentration) of surfactant molecules Ns (Xs). All simulations shown here are performed on a 50 × 50 × 50 lattice with h2t4 surfactant. Comparison of system 1 with system 3 shows that the presence of a head-solvent attraction deters micellization, as shown by the increase in the cmc in Table 3 and the (27) Press, W. H.; Teukolsky, S. A.; Vettering, W. T.; Flannery, B. P. Numerical Recipes, 2nd ed.; Cambridge University Press: Cambridge, 1992. (28) Desplat, J. C.; Care, C. M. Mol. Phys. 1996, 87, 441. (29) Care, C. M. J. Phys. C: Solid State Phys. 1987, 20, 689. (30) The aggregation number here refers to the number of surfactant molecules in a particular micelle.
Surfactant-Solute-Solvent Systems
Figure 2. Number average aggregation number Nn as a function of h2t4 concentration in the presence of solutes (250 c1).
Figure 3. Natural logarithm of the partition coefficient K as a function of h2t4 concentration in the presence of solutes (250 c1). The different systems correspond to different interaction energies, as in Table 2.
decrease in micelle size in Figure 2. The head-solvent attraction increases the solubility of the surfactant in the solvent and thus cmc increases. It also tends to increase the solvent contacts with the head groups. Then the surfactant molecules would prefer to have contacts with the solvent rather than with any other entity. Hence, the curvature increases, leading to smaller micelles. A comparison between systems 2 and 4 shows that a head-tail repulsion has a similar effect as the head-solvent attraction. Since there are more head-tail contacts for the surfactant in the micelle than for a single surfactant in the solvent, the inclusion of the repulsive head-tail interaction would increase the preference of the surfactant for the solvent rather than the micelle. Therefore, the head-tail repulsion leads to an increase in cmc and a decrease in micellar size. Since the solute is not involved in either of the interactions discussed above, partitioning is not affected, as can be seen in Figure 3. Finally, comparison of systems 3 and 4 in Figure 3 shows that inclusion of a head-solute repulsion does not affect the partition coefficient significantly. It indicates that this particular solute is solubilized mostly in the core of the micelle, where the number of head-solute contacts is minimal, and therefore repulsion between surfactant heads and the solute is not of major importance in the micellar solubilization process. The same conclusion was reached by Talsania et al.24 by counting the number of solvent contacts for the solute
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beads. They showed that this solubilized solute has a majority of its contacts with tail beads (as opposed to head or solvent), signifying solubilization mainly in the core of the micelle. The preceding set of data was also used to select a set of parameters to study solubilization of various types of solutes. Since all four systems presented above exhibit micellar behavior, the objective was to select the simplest system possible which would capture the essence of the micellar solubilization phenomenon. The head-tail repulsionsalong with the tail-solvent repulsionswas kept as a natural interaction stemming from the hydrophobic effect. The head-solvent attraction, for which we do not have clear criteria in order to assign a value, was not included in the subsequent simulations in this study. Hence, we are assuming that the surfactant heads have a similar behavior as the solvent molecules. Additionally, using only hydrophobic interactions allows us to compare our results with lattice theories which are available in the literature, such as the Flory-Huggins theory31 and the quasichemical theory.12,13 Therefore, we have chosen the parameters of system 1, one of the simplest systems possible, for subsequent simulations. For the simulations presented in the remainder of this paper, we have set the hydrophobicity at ) 0.7, which is above the critical value of 0.08 found experimentally for the oil-water system.32 Additionally, our value of χ ) z ) 4.2 is near that of Larson et al. (χ ) 4.0),16 whose simulations were in the micellar regime. 3.2. Solute Phase Behavior. 3.2.1. Solute Phase Behavior in the Absence of Surfactant. The solubility limit of the solutes can be determined from the simulations by examining the cluster size distributions for various concentrations (or numbers) of the solute, Xc (or Nc). Figure 4 shows distributions for c2 at two different concentrations, in the absence of surfactant. The distribution is a monotonically decreasing function of cluster size, m, for systems below the solubility limit, such as the curve for Nc ) 500 (Xc ) 0.008) in Figure 4a. The appearance of another peak at much higher cluster size indicates phase separation, as in Figure 4b (Nc ) 750, Xc ) 0.012). The aqueous solubility limit, thus derived from Monte Carlo simulations, is listed in Table 4 for different solutes and values of the interaction parameter . One can also estimate the solute solubility limit using simple thermodynamic arguments. The chemical potential of the solute in an aqueous solution, µaq c , can be approximated as
µaq c ) µ° c + kBT ln Xc
(8)
where µ°c is the chemical potential of the solute at infinite dilution, Xc is the concentration of the solute, and kBT is the Boltzmann constant multiplied by the temperature. Above the solute solubility limit X°c, a pure solute phase aq forms whose chemical potential µ∞ c, must equal µc to satisfy equilibrium conditions. Solving the above equation for X°c gives
X*c )
[
1 ∞ (µ - µ°c) kBT c
]
(9)
If the standard entropies of the solute in the two phases are assumed to be negligible, then the chemical potentials (31) Davis, H. T. Statistical Mechanics of Phases, Interfaces, and Thin Films; VCH Publishers: New York, 1996. (32) Prausnitz, J. M. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice Hall: Englewood Cliffs, NJ, 1969; Chapter 7.
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Figure 5. Solubility of c1 as calculated from quasichemical theory, from eq 10, and from the simulations.
Figure 4. Solute cluster size distributions for (a) 500 c2 solutes (Xc ) 0.008) and (b) 750 c2 solutes (Xc ) 0.012) in solvent. The ordinate is the product of Pm (the probability of finding an individual solute molecule in a cluster containing m solutes) and Nc (the total number of solutes in the system).
Figure 6. Solubility of c2 as calculated from quasichemical theory, from eq 10, and from the simulations.
Table 4. Aqueous Solubility Limit of Various Solutes (As a Function of Structure and Hydrophobicity)a 0.6 0.7 0.8
Nc Xc Nc Xc Nc Xc
c1
c2
c3
5000-6000 0.04-0.048 2500-3000 0.02-0.024
1000-1500 0.016-0.024 ∼700 0.0112 300-400 0.0048-0.0064
300-400 0.0072-0.0096 100-200 0.0034-0.0048
a The values reported are in terms of the number of soluble chains on a 50 × 50 × 50 lattice and also as volume fraction.
in the above equation can be replaced by the internal energies. Then, eq 9 can be reduced even further to
(10)
Figure 7. Solubility of c3 as calculated from quasichemical theory, from eq 10, and from the simulations.
where n∞c and n°c are the average number of solute-solvent contacts for the solutes in the pure and aqueous phases, respectively. Solutes in their own phase will have approximately zero solvent contacts while aqueous solutes have the maximum number of solvent contacts, for example, 6 for c1, 10 for c2, 14 for c3, etc. After substituting these values above, the solute solubility can be directly calculated. Figures 5-7 show the solute solubility as calculated by eq 10 for c1, c2, and c3, respectively, as compared with values from the simulation. The solubility of the solute has also been determined using the expressions for the chemical potentials deduced from QCT for a polymer solution.12,13 In Figures 5-7, we also present the results obtained from QCT, for solutes c1, c2, and c3, respectively. The solubility limits calculated from eq 10 are smaller than those from quasichemical theory, which in turn are smaller than the ones observed in the simulations. Since only internal energies were considered in calculating the values from eq 10, the
underprediction indicates that the entropic contributions are significant for the solubility of the solute. This also indicates that the quasichemical theory captures only a part of the entropic contributions. 3.2.2. Solute Phase Behavior in the Presence of Surfactant. In this section, we describe the solute solubility limit in a three-component (i.e., solute-solvent-surfactant) system. We have found through our simulation method that the cmc’s of h2t4 and h4t4 in the absence of any solutes are, respectively, 200 (Xs ) 0.0096) and 300 (Xs ) 0.0192) chains. At surfactant concentrations above the cmc, where micelles are present, it would be expected that the solubility limit of the solute would increase due to micellar solubilization. However, an interesting result found in our simulations is that the addition of a small amount of surfactant to this systemseven at concentrations slightly above the cmcsactually reduces the solute solubility. For example, the presence of h2t4 or h4t4 surfactants in the system reduces the solubility of c2
X*c ) exp[Ecs(n∞c - n°c)]
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Table 5. Solubility Limits of Various Solutes at Different Surfactant Concentrations Reported in Terms of the Number of Solutes on a 50 × 50 × 50 Lattice no surfactant 300 h2t4 500 h4t4
c1
c2
c3
5000-6000 1000-1500
700 500 300
100-200 50-100
significantly, as can be seen in Table 5. For c1, this decrease is even more dramatic; its solubility in a system of 300 h2t4 surfactants (Xs ) 0.0144) is only about one-fourth of its solubility in solvent alone. An explanation of this reduction in solubility is given later in this section. The method of determining the solute solubility limit in the presence of surfactant is slightly different from that without. Figure 8 is an example of a plot used to determine the number of phases in a system. The difference between this plot and that in Figure 4 is in the abscissa, m. Here, m is not the cluster size but only the number of solutes in a cluster, since a cluster can now contain both solute and surfactant. In this graph, a monotonically decreasing curve (with a peak only at m ) 1) still indicates, as in the case without surfactant, the presence of only an aqueous phase, representing complete solubility of the solute; however, the presence of another peak in the distribution at intermediate m (i.e., 5 < m < 50) represents solubilization inside micelles while a peak at large m (i.e., m >100) indicates formation of a separate solute (or “oil”) phase. Note that the micelles are pseudophases not actual thermodynamic phases. Figure 8 shows the solute distribution at various h4t4 concentrations at a constant solute concentration of 500 c2 chains (Xc ) 0.008). For a low number of surfactants Ns (curve b, Ns ) 50), the distribution has only the monomer peak, just as for curve a (without surfactant), indicating complete solubility. Again, for an intermediate number of surfactants, corresponding to a concentration below the cmc (curve c, Ns ) 200, Xs ) 0.0128), a distant second peak appears due to the formation of a separate solute phase. At surfactant concentration just above the cmc (curve d, Ns ) 400, Xs ) 0.0256), three peaks are present representing aqueous, micelle, and oil phases. Finally, when enough surfactant is added (curve e, Ns ) 750, Xs ) 0.048), the oil phase disappears, leaving only solute dispersed in the aqueous phase and solubilized in micelles. This figure shows that the solute solubility first decreases with addition of some surfactant. However, as the surfactant concentration is increased further, the solute once again becomes soluble. The same result is seen in Tables 6 and 7, which show the number and types of phases present as the surfactant concentration is varied for systems of 500 c2 (Xc ) 0.008) and 2000 c1 (Xc ) 0.016) solutes, respectively. Both tables show that an oil phase appears at some intermediate surfactant concentration and then disappears at some higher concentration. This result may seem unexpected but can be explained as follows. For a binary solute-solvent system, we have already shown in the previous section that both entropy and energy play a significant role in solubility. It is obvious from thermodynamic arguments that a system with solute randomly dispersed in the solvent is the most entropically favored configuration, despite its energy being at the maximum. Conversely, a system with two mutually exclusive phases of solute and solvent is the most energetically favored configuration, though it is in a state of minimum entropy. Below the solubility limit, the solutes do not form an “oil droplet” (indicating a separate phase) because the resulting decrease in their internal
energy would not be sufficient to overcome the loss of entropy of the system. However, in the presence of surfactantseven at a concentration below the cmcsthe solutes can form an “oil droplet” even below their solubility limit in the absence of surfactant. This is due to the surfactants and solutes forming an “emulsion”, where the surfactants are adsorbed on the surface of an “oil droplet”, thereby reducing the interfacial energy of the droplet. Now, the energetic driving force can overcome the entropic one, causing the phase separation of the solute at a lower solubility limit, as is the case in Figure 8c. At surfactant concentrations around the cmc (where, by definition, half of the surfactant exists in micelles), some of the surfactant will form micelles which coexist with the “emulsion” and the solutes can now reside in the emulsion or solubilize inside the micelles, as in Figure 8d. Finally, at concentrations well above the cmc, most of the surfactant goes toward the formation of micelles, causing the breakup of the “oil droplet”. The solubility limit now increases as the solutes now solubilize inside micelles (as in Figure 8e) which retain a small interfacial energy, as the solutes are still shielded from the solvent by the surfactant in the micelles. Also, the solutes now have increased entropy inside the micelles rather than when in a droplet due to the increased number of possible configurations of the solute inside the micelles. In summary, the addition of surfactant at low concentration to a binary solute-solvent system can reduce the solute solubility limit by increasing the energetic driving force of the solute to form a separate phase. However, at higher surfactant concentrations, this energetic driving force is removed by the formation of micelles and the solubility limit of the solute increases due to micellar solubilization. Figure 9 represents the phase diagram for the h4t4c2-solvent system discussed above. Figure 9a shows the entire phase diagram and Figure 9b shows an enlargement of the lower left corner of the previous diagram, (i.e., the dilute solution region). The decreasesfollowed by the increasesin solute solubility limit with increasing surfactant concentration can be seen from the curve representing the simulation data. Additionally, Figure 9b shows a comparison of the phase behavior predicted from the simulation and that predicted by quasichemical theory. Quasichemical theory cannot predict the formation of micelles because it assumes a mean-field environment for all the species; hence, there is a strong discrepancy between the simulations and QCT, especially at surfactant concentrations above the cmc. Additionally, due to the finite size of our system (50 × 50 × 50), there is a limited accuracy associated with the concentration values used in the Monte Carlo simulations. 3.3. Micellar Solubilization. The presence of solute can affect several micellar properties such as the size, the shape, and the cmc, as has been shown by Talsania et al.24 Also, the characteristics of the solute can affect the locus and extent of solubilization in micelles, as well as micellar properties. In this section, first we report an interesting observation on the change in the shape of the micelles caused by the presence of solutes. Additionally, we evaluate how two different solute characteristics, solute chain length and hydrophobicity, affect solubilization. 3.3.1. Effect of Solutes on Micellar Structure. We have investigated the shape of micelles in the absence and presence of solutes. In the absence of solutes, our simulations show that h2t4 and h4t4 surfactants form spherical micelles at concentrations near or slightly above the cmc. This conclusion was drawn by calculating the asphericity parameter As for the micelles. This parameter
2690 Langmuir, Vol. 14, No. 10, 1998
Talsania et al.
Figure 8. Solute cluster size distributions for 500 c2 solutes (Xc ) 0.008) with (a) 0, (b) 50, (c) 200, (d) 400, and (e) 750 h2t4 surfactants in solvent (ordinate as in Figure 3). Table 6. Number and Type of Phases Present for a System of 500 c2 Solutes (Xc ) 0.008) at Various Concentrations of h4t4 s
Xs
no. of phases
type
750
0.048
1 2 3 2
aq aq, oil aq, oil, mic aq, mic
Table 7. Number and Types of Phases Present for a System of 2000 c1 Solutes (Xc ) 0.016) at Various h4t4 Concentrations Ns
Xs
no. of phases
type
750
0.048
1 3 2
aq aq, oil, mic aq, mic
can be used to characterize the shape of the micelles and is defined by33 3
As )
(Ii - Ij)2 ∑ i>j)1 3
2(
Ii) ∑ i)1
(11)
2
where I1, I2, and I3 are the three principal moments of inertia of the micelle (see ref 24 for details on how to calculate these). The asphericity parameter has a value of 0 for spheres and 1 for an infinite cylinder, with values in between representing ellipsoids ranging from sphere(33) van Giessen, A. E.; Szleifer, I. J. Chem. Phys. 1995, 102, 9069.
like to elongated ones. For the surfactant-solvent systems studied, As ranges from 0.05 to 0.1, indicating roughly spherical micelles. This is also confirmed by visualization of the lattice. These results agree with those of Larson, who found that h4t4 forms spherical micelles up to a surfactant concentration of 30 vol %. However, in the presence of solutes, some of the surfactant molecules form micelles with a less spherical, more oblong structure which coexist with the spherical micelles. The micellar size distribution (Figure 10) for h2t4 at a concentration of Xs ) 0.024 with 250 c1 solutes (Xc ) 0.002) shows a second peak indicating a different type of micelle. The asphericity parameter for peak 1 is about 0.1, indicating spherical micelles; however, As ) 0.25 for peak 2, signifying a more oblong structure. Larson has also shown that cigar-shaped micelles can coexist with spherical micelles.19 Another interesting result is that the cluster size at the second peak is just about twice that of the first peak. Coupled with the fact that there is a valley between these two peaks, these data suggest that these oblong micelles are formed by the union of two spherical micelles. Micellar size distributions for other systems of surfactant and solute give similar results. 3.3.2. Effect of Solute Chain Length and Hydrophobicity on Solubilization. To study the solubilization of various types of solutes, we have employed the following twoparameter model just for this section of results: t ) Ets ) Eht and c ) Ecs ) Ehc. This allows us to study solutes with a varying degree of hydrophobicity. In the model we present, one can now look at different solutes by changing two parameters: the solute chain length k and the solute hydrophobicity c. To study the effect of solute length, we have performed three independent simulations at a constant surfactant
Surfactant-Solute-Solvent Systems
Langmuir, Vol. 14, No. 10, 1998 2691 Table 8. Variation in Solubilization Characteristics for a System of h2t4 Surfactants with Equal Volume Fraction of Three Different Lengths of Solute solute
ln K
% cov
ncs
c1 c2 c3
2.3 3.8 5.6
34 27 18
2.0 1.3 0.9
Table 9. Variation in Micellar Characteristics for a System of h2t4 Surfactants with 250 c1 Solutes (Xs ) 0.002) for Various Solute Hydrophobicities
Figure 9. (a) Phase diagram for h2t4-c2-solvent system as predicted by the quasichemical theory and by Monte Carlo simulations. (b) Enlargement of the lower left corner of the previous diagram (i.e., the dilute solution region).
c
Ncmc
Xcmc
ln K
ncs
n′cs
0.60 0.65 0.70 0.75 0.80
220 210 200 190 180
0.010 56 0.010 08 0.009 6 0.009 12 0.008 64
2.3 2.4 2.5 2.8 3.0
2.02 1.90 1.67 1.72 1.50
2.10 2.23 2.42 2.27 2.25
swell, thus increasing the radius of the micelles and allowing more surfactant chains to aggregate inside. These larger micelles then have a larger solubilization capacity. Similarly, increasing the chain length of the solute affects micellar properties, by increasing the volume of the micelle. Increasing the chain length causes the solute to go deeper into the core of the micelles, which can be seen from the data in Table 8, as a direct consequence of the strong hydrophobicity. The degree of solvent coverage of the solubilized solute (the percentage of nearest neighbors of the solubilized solute which are solvent) drops from 34% for c1 to 27% for c2 to only 18% for c3. Similarly, the number of c-s contacts per solute bead also drops. These data indicate that the solubilization occurs deeper in the core than in the palisade layer as the chain length of the solute increases. By varying the parameter c, one can examine the effect of solutes which differ in hydrophobicity. Table 9 shows how certain micellar properties are affected by c. Increasing the solute interaction has a similar effect to that of increasing the chain length of the solute in that the cmc is decreased and the partition coefficient is increased. However, the increase in partitioning is not as severe as that in the previous case with increase of solute length. In Table 9, ncs is the average number of solvent contacts per solubilized solute bead, a quantity measured directly in the simulations. The significance of entropic effects on solubilization can be examined by comparing ncs with a similar quantity n′cs (also the average number of solubilized solutesolvent contacts) estimated from the partition coefficient K by the following relation
ln K ) c[2(z - 1) + (k - 2)(z - 2) - n′cs]
Figure 10. Aggregate size distribution for a system of 500 h2t4 surfactants and 250 c1 solutes (Xs ) 0.024, Xc ) 0.002). The ordinate is the product of Pn, the probability of finding a surfactant in a micelle of aggregation number n, and Ns, the total number of surfactants in the system. Peak 1 represents spherical micelles, and peak 2 represents oblong micelles.
concentration with a constant concentration (volume fraction) Xc of either c1, c2, or c3 solute. The results are shown in Table 8. An increase in solute length leads to a large increase in the partition coefficient K, which, not surprisingly, is the same trend caused by the introduction of a solute to a surfactant-solvent system, as seen by Talsania et al.24 As is well-known and as shown by Talsania et al., solubilization of solutes causes micelles to
(12)
where k is the length of the solute chain and z is the coordination number of the lattice. The quantity within the brackets represents the difference in the numbers of solvent contacts between an unsolubilized solute chain and a solubilized one. The above equation is derived starting from the thermodynamic relation
∆µ ) µ°m - µ°aq -kBT ln K
(13)
where µ°m and µ°aq represent the standard chemical potentials of the solute in the micelle and in the aqueous phase, respectively. One obtains eq 12 when only energetic contributions are counted in eq 13 (i.e., entropic contributions are assumed to be negligible). The results show that ∆n ) n′cs - ncs is much larger than zero at larger c, thereby indicating that entropy is in fact not negligible.
2692 Langmuir, Vol. 14, No. 10, 1998
Therefore as the solubilization occurs deeper into the core of the micelles (for larger c), both energetic and entropic contributions are significant for the solute. The decrease in energy of the solute caused by the reduction in solutesolvent contacts is balanced by the increase in entropy, as there is a greater volume in the core of the micelle than in the palisade region. 4. Summary In closing, we have presented a lattice model for surfactant-solute-solvent systems which has been used to study their phase behavior. As previously known, a simple one-parameter model (i.e., only hydrophilic-hydrophobic interactions) is sufficient to exhibit micellization phenomena. In particular, a head-solvent attraction is not required to achieve aggregation of surfactant molecules, and the number of head-solute contacts is minimal regardless of interaction between the two beads. We have also established a novel method for the prediction of phase separation of the solute in a two-component solute-solvent system using a solute cluster size distribution obtained from the simulations. We have extended this method to predict phase separation of the solute in the presence of surfactant. Through this method, we have found that the solubility of the solute decreases for surfactant concentrations below the cmc and then increases at concentrations above the cmc, where the solubility limit of the solute is larger in the simulations than that predicted by quasichemical theory and by a
Talsania et al.
simple equilibrium model which we have proposed. We found that the phase behavior predicted by quasichemical theory is quite different from that predicted by our simulation method, especially at surfactant concentrations above the cmc where micelles are present. This indicates that the quasichemical theory is not appropriate for describing micellar systems, which we will show in more detail in a forthcoming publication. The addition of a small amount of surfactant to a solute-solvent (i.e., oilwater) system causes the formation of an “emulsion” as a result of the reduction in interfacial energy due to the adsorption of surfactant molecules on the oil/solvent interface, and the addition of a larger amount of surfactant increases the solubility of the solute due to micellar solubilization. From the study of solubilization inside micelles we have found that increasing the chain length of the solute increases the partitioning of the solute inside the micelle and shifts the locus of solubilization deeper into the micelle. Increasing the hydrophobicity of the solute has a similar effect on the locus and extent of solubilization, in agreement with theoretical results. The surfactants h2t4 and h4t4 were found to form coexisting cylindrical and spherical micelles, but only in the presence of solute. Acknowledgment. This work was partially funded by the Texas Higher Education Coordinating Board and the Gulf Coast Hazardous Substance Research Center. LA970865Z
