4672
Langmuir 1996, 12, 4672-4680
On the Distribution of Surfactants among Mixed Micelles Alexander V. Barzykin† and Mats Almgren* Department of Physical Chemistry, University of Uppsala, Box 532, S-751 21 Uppsala, Sweden Received February 5, 1996. In Final Form: June 19, 1996X Composition distributions of mixed micelles are analyzed using basic principles of classical statistical mechanics with special emphasis on potential application to fluorescence quenching experiments in such systems. The problem is formulated as a lattice model with nearest-neighbor interactions between surfactant hydrophilic moieties. Firstly, the mean-field approximation is discussed in some detail with focus on its general character. Deeper analysis, involving direct evaluation of the lattice partition function, shows how substantial the deviations from the mean-field predictions can be. The situation is dramatically improved when the truncated cumulant expansion is introduced. Two special cases are distinguished, where the so-called interchange energy takes positive or negative values. In the former case, a binary mixture can separate into two phases of different composition, while in the latter micelles can undergo structural ordering. Finally, mixtures with excess of one component are considered and a useful approximation for the grand partition function is derived.
I. Introduction Mixed micelles of two surfactants, distinguished as type 1 and type 2 hereafter, are usually treated within a pseudophase-separation model using a regular solution approach.1-5 The result of such a treatment is a relation between concentrations c1 and c2 of the surfactants in the aqueous pseudophase at the cmc of the mixed micellar solution and the composition, xj1 and xj2 ) 1 - xj1, of the micelles
ci ) c0i fixji
(i ) 1, 2)
(1)
where c0i is the cmc of the ith surfactant and the activity factor fi is given by
ln fi ) R(1 - xji)2
(2)
The interaction parameter R is zero in the ideal case, giving activity factors equal to unity. Equations 1 and 2 are used in practice to interpret the cmc measurements. Actually, there appears to be some confusion in terminology. Strictly speaking, eq 2 is not a regular solution result. As defined by Guggenheim,6 the regular solution model is based on the following assumptions:7 (1) The two kinds of molecules have comparable volumes and similar manners of packing so that a rigid quasilattice is assumed with each molecule occupying a single lattice site. All molecules are interchangeable. As a result, the volume is proportional to the total number N of molecules and the excess volume is zero. (2) The free energy of mixing is assumed to depend only on the configurational partition function which, in turn, † On leave from the Institute of Chemical Physics, Russian Academy of Sciences. X Abstract published in Advance ACS Abstracts, September 1, 1996.
(1) Clint, J. H. J. Chem. Soc., Faraday Trans. 1 1975, 71, 1327. (2) Rubingh, D. N. In Solution Chemistry of Surfactants; Mittal, K. L., Ed.; Plenum Press: New York, 1979; Vol. 1, p 337. (3) Holland, P. M.; Rubingh, D. N. J. Phys. Chem. 1983, 87, 1984. (4) Nguyen, C. M.; Rathman, J. F.; Scamehorn, J. F. J. Colloid Interface Sci. 1986, 112, 438. (5) Hoffmann, H.; Po¨ssnecker, G. Langmuir 1994, 10, 381. (6) Guggenheim, E. A. Mixtures; Clarendon Press: Oxford, 1952. (7) The term regular solution was actually introduced by Hildebrand in 1929 to distinguish mixtures with experimentally determined zero excess entropy. Guggenheim defined a model for the so-called strictly regular solution, which could then be analyzed by the methods of statistical thermodynamics.
S0743-7463(96)00107-2 CCC: $12.00
is evaluated under the assumption that all molecules are at rest at their lattice sites. (3) Traditionally, another simplification is introduced by the assumption that only those molecules interact which are nearest neighbors. The model is thus restricted to short-range interactions. In order to evaluate the lattice partition function, further simplifications have to be made. The most familiar approach, due to Bragg and Williams, assumes a random arrangement of the molecules on a lattice. It is this approximation that leads to eq 2. The weakness of such a treatment is its macroscopic character. The energy of a molecule is actually assumed to depend on the mean field created by the whole lattice and not on the neighboring molecules. However, the weakness of this theory turns out also to be its power. The mean-field approximation is not restricted to the nearest-neighbor interactions nor to the regular solution model itself, and eq 2 is one of its general consequences.8 We shall return to this point later in more detail. It is understandable that the pseudo-phase-separation model based on formal classical thermodynamic relations cannot have a predictive power for micellar systems of practical interest nor can it provide any information about the aggregation number of mixed micelles as a function of the composition. At best, it can describe available experimental results by fitting the measured cmc data. Other theoretical efforts have led to the development of a molecular theory of mixed surfactant aggregates9-12 that, based on fundamental statistical mechanical principles, is capable of predicting the size and composition distributions. This theory originates from the earlier treatment of aggregation in single-component micelles.13-15 (8) The fact that eq 2 is not restricted to the regular solution theory has been emphasized already in the literature.5 The authors have traced it back to a more general, albeit empirical, approach due to Porter who assumed, as early as in 1920, a special form for the molar excess free enthalpy, symmetric for both components. No assumptions were made about the excess entropy. It has been suggested in ref 5 to use the expression “symmetrical mixture” instead of “regular solution”. As regards the terminology, we feel that the mean-field approximation is still more appropriate, as it appears all throughout the statistical thermodynamics. (9) Stecker, M. M.; Benedek, G. M. J. Phys. Chem. 1984, 88, 6519. (10) Nagarajan, R. Langmuir 1985, 1, 331. (11) Ben-Shaul, A.; Rorman, D. H.; Hartland, G. V.; Gelbart, W. M. J. Phys. Chem. 1986, 90, 5277. (12) Puvvada, S.; Blankschtein, D. J. Phys. Chem. 1992, 96, 5567. (13) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525.
© 1996 American Chemical Society
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The molecular model accounts for hydrophobic interactions between surfactant hydrocarbon chains and water, curvature-dependent interfacial effects due to creation of the micellar core-water interface, conformal effects arising from hydrocarbon chain packing in the micellar core, entropic mixing effects as well as repulsive steric and electrostatic interactions between surfactant hydrophilic moieties. No doubt, the appearance of the molecular theory was a breakthrough in the field. However, a salient deficiency of the treatments currently available in the literature is that they still use the mean-field approximation when it comes to calculating the free energy change due to interaction between surfactants. At the same time, while being primarily focused on the size distribution and its variation with the overall composition of the solution, the authors do not specially concern themselves with the composition distribution of individual micelles. In most cases, for their needs it is enough to simply regard all micelles as having the most probable but constant composition, which suffices as long as the aim is to interpret the cmc or light scattering measurements. Another way to study mixed micelles is based on the fluorescence quenching method.16-19 Briefly, a very low concentration of a hydrophobic fluorescent probe is added to a system where one of the surfactant components can also act as a quencher of the probe fluorescence. After exciting the probe by a short laser pulse, one monitors its deactivation kinetics in a single photon counting experiment. It is not difficult to choose a system where quenching occurs independently in different micelles. Then, if we take a monodisperse ensemble of micelles for simplicity, we can write the observed signal as N
Pn(n j ,N) Qn(t) e-t/τ ∑ n)0
I(t) ) I(0)
(3)
where I(t) and I(0) is the intensity of fluorescence (the number of counts) at time t and at time zero (at the maximum), respectively, τ is the self-decay lifetime, Qn(t) is the probability to find the probe excited at time t in a j ,N) is the micelle with n cohabitant quenchers, and Pn(n distribution of quenchers among micelles with the mean n j . Qn(t) is usually assumed known and approximated by the pseudo-first-order decay law.16 At very long times only those excitations survive which were initially created in micelles without quenchers. Therefore, we have for the amplitude I∞ of the final stage of the decay
I∞/I(0) ) P0(n j ,N)
(4)
Whether fitting the whole decay curve or determining only the amplitude of the long-time stage, we must know j ,N) before we can draw any conclusion. It is our main Pn(n goal here to determine the distribution of surfactants among mixed micelles and to explore how it is influenced by molecular interactions. Related problems addressed in the literature are the occupancy statistics of interacting guest molecules in zeolite cavities20,21 and the distribution of solubilizates among normal micelles.22,23
If we assume that all micelles have the same aggregation number N, then in the ideal case the surfactants should be distributed according to the binomial law, which for large N and xji , 1 is well approximated by the Poisson law for the ith species. There is of course no reason for micelles to have the same size. In an ideally mixed system, in which the pure components form micelles of similar size, the size distribution should be similar to that in the pure systems. In a nonideal mixture, both the size of the micelles and the size distribution should be expected to vary with the mean composition. Here we shall assume, for the sake of simplicity, that the micelles at a given mean composition all have the same total aggregation number (but not necessarily the same at all compositions). We seek the composition distribution of the micelles for fixed xji and N. We formulate our problem within the lattice (regular solution) model using the grand canonical approach. Although the lattice model is certainly a simplification, it captures the basic features of the collective interactions between the surfactants in a micelle. We further restrict ourselves to nearest-neighbor interactions, with an idea of going beyond the mean-field approximation at least in this simple case. The results may find their application for mixtures of nonionic surfactants. Firstly, the meanfield approximation is discussed in some detail with an emphasis on its general character. We thus interpret the interaction parameter in eq 2. Comparison of the meanfield predictions with some recent experimental data on fluorescence quenching in several mixed micellar systems is presented elsewhere.19 In order to understand the value of the results obtained, we reduce our problem to the Ising model of a ferromagnet in a field. Analytical solution of the latter is not available. However, comparison with numerical results shows serious deficiencies of the mean-field approximation. The situation is dramatically improved when we introduce the truncated cumulant expansion. This method also allows us to correct the expression for the activity factor. We distinguish between positive and negative values of R. If R > 0, increasing R eventually causes the binary mixture to separate into two phases of different composition. This is not a macroscopic separation into two micellar solutions but almost pure micelles of each component surfactant coexisting, as in mixtures of hydrocarbon and fluorocarbon surfactants.24 On the other hand, if R < 0, increasing |R| leads to structural ordering of micelles in the mixture. Finally, we focus on mixtures with excess of one component and derive useful approximations for the activity factor as well as for the grand partition function in this case. The methods used in this paper and even some of the results derived are not new in statistical physics, though may not be well-known in the physical chemistry of micelles. An important distinction is that here we have an infinite ensemble of essentially finite systems (N is order of 100), while classical thermodynamics normally deals with one system in the limit of N f ∞. II. General Formulation
(14) Nagarajan, R.; Ruckenstein, E. J. Colloid Interface Sci. 1979, 71, 580. (15) Puvvada, S.; Blankschtein, D. J. Chem. Phys. 1990, 92, 3710 and references therein. (16) Tachiya, M. In Kinetics of Nonhomegeneous Processes; Freeman, G. R., Ed.; Wiley: New York, 1987; p 575. (17) Almgren, M. In Kinetics and Catalysis in Microheterogeneous Systems; Gra¨tzel, M., Kalyanasundaram, K., Eds.; Marcel Dekker: New York, 1991; p 63. (18) Gehlen, M. H.; De Schryver, F. C. Chem. Rev. 1993, 93, 199 and references therein. (19) Almgren, M.; Hansson, P.; Wang, K. Langmuir 1996, 12, 3855. (20) Gu¨e´mez, J.; Velasco, S. Physica A 1988, 152, 226.
We aim at finding the distribution of surfactant molecules among binary mixed micelles assuming that the aggregation number, N, is independent of the composition. A satisfactory approach (albeit simplified) lies (21) Chmelka, B. F.; Raftery, D.; McCormick, A. V.; de Menorval, L. C.; Levine, R. D.; Pines, A. Phys. Rev. Lett. 1991, 66, 580. (22) Barzykin, A. V. Chem. Phys. 1992, 161, 63. (23) Bales, B. L.; Stenland, C. J. Phys. Chem. 1993, 97, 3418. (24) Asakawa, T.; Hisamatsu, H.; Miyagashi, S. Langmuir 1995, 11, 478 and references therein.
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within the framework of the lattice model. We focus on micellized surfactant molecules only. Thus, consider a binary system of N particles of which on the average n j1 are of type 1 and n j 2 are of type 2. These particles are distributed on lattice sites and interact in a pairwise fashion, provided they are nearest neighbors. The grand partition function of our system is25
Ξ)
∑ ∑ exp[β(µ1n1 + µ2n2 - Um)]
(5)
n1+n2)N m
where β ) (kBT)-1, µi denotes the chemical potential and ni the number of ith species; m numbers the states of the system for fixed ni, and Um is the total energy of the system in the state m, given by
Um ) n11u11 + n12u12 + n22u22
(6)
Here nij is the number of ij pairs, where one member of the pairs is of the ith species and the other is of the jth species, and uij is the corresponding interaction energy. The chemical potentials must be chosen in such a way that the system gives the preassigned composition
∑ ∑ni exp[β(µ1n1 + µ2n2 - Um)]
n j i ) Ξ-1
(7)
It is straightforward at this stage to generalize this formulation, if necessary, to include interactions between pairs of molecules which are not nearest neighbors. We assume periodic boundary conditions on our lattice which seems intuitively reasonable if surfactant head groups form a spherical interface. Let the coordination number of the lattice be z. Using the relation
zni ) 2nii + n12
(8)
and introducing the definitions
(9)
1 λi ) exp β µi - zuii , 2
(10)
)
[(
)]
λn1 λn2 (n )∑g(n12,n1,N) exp(-n12w) ∑ 1 n n +n )N 1
1
2
2
N
III. Mean-Field Approximation The simplest way to analyze thermodynamic behavior of a nonideal mixture is by taking advantage of the BraggWilliams (mean-field) approximation. Thus, we assume a random arrangement of the particles and obtain
g(n12,n,N) exp(-n12w) ) exp(-〈n12〉w) ∑ n
(14)
〈n12〉 ) zn(N - n)/(N - 1)
(15)
12
where
()
Pn(n j ,N) ) Ξ-1
N n N-n λ λ × n 1 2 exp[-zwn(N - n)/(N - 1)] (16)
It can be readily seen that upon increasing the interaction energy (decreasing temperature) our binary mixture separates into two phases of different composition. This is because for w > 0, 2u12 > u11 + u22 and twin pairs 11 and 22 are energetically favored over mixed 12 pairs. The coexistence curve is determined from the minima of the free energy of mixing (see Figure 1)
()
N + zwn(N - n)/(N - 1) n
(17)
βΦ/N ) x ln x + (1 - x) ln(1 - x) + zwx(1 - x) (18)
(11)
where g(n12,n1,N) is the statistical weight of configurations of n1 particles of the first species and n2 ) N - n1 particles of the second species with n12 mixed pairs. The binomial N! N in eq 11 accounts for the total coefficient n ) n 1 1!n2! number of configurations. Finally, we can write down the probability distribution for a micelle to have certain composition n1 ) n, n2 ) N - n as follows
( )
()
(13)
where x ) n/N. The coexistence curve is thus described by equation
12
j ,N) ) Ξ-1 Pn(n
N n xj (1 - xj)N-n n
Assuming that n and N are large enough, we can use Stirling’s approximation and rewrite eq 17 as
we rearrange the partition function in eq 5 as
Ξ)
()
with xj ) n j /N. In general, however, we have to calculate g(n12,n,N) somehow and then evaluate λi using eq 7. While the second part of this task is relatively easy to accomplish, the first problem is very complicated.
βΦ ) -ln
1 1 w ) β u12 - u11 - u22 2 2
(
Pn(n j ,N) )
j ,N) is, therefore, completely deterThe distribution Pn(n mined
n1+n2)N m
) kBT ∂ ln Ξ/∂µi
In the ideal case where w ) 0, eq 12 reduces to the binomial distribution
N n N-n λ λ × n 1 2 g(n12,n,N) exp(-n12w) (12)
∑ n 12
(25) See, e.g.: Hill, T. L. Introduction to Statistical Thermodynamics; Addison Wesley: Reading, MA, 1960.
zw(1 - 2x) ) ln
1-x x
(19)
corresponding to the critical point where the two phases become identical at
2 1 wc ) , xc ) z 2
(20)
In Figure 2 we compare the coexistence curve calculated from eq 19 to that obtained for finite values of N. Below the critical temperature, the average composition of the two coexisting phases can be found in a standard fashion using the lever rule, as illustrated in Figure 1. The chemical potential, µi, may be obtained by differentiating Φ with respect to ni. As a result, we find for the activity factor
( )
ln fi ) ln
λi
λ0i
xji
) zw(1 - xji)2
(21)
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Langmuir, Vol. 12, No. 20, 1996 4675
Pn(n j ,N) ) Ξ ˜ -1
()
N n xj (1 - xj)N-n exp[RNx(x - 2xj)] (22) n
where we have combined all inessential constants into the normalization factor Ξ ˜ . Equation 22 proves to be a reasonable approximation far from the critical region, as compared to the mean-field solution for finite but not very small N. It is worthwhile to emphasize that the mean-field theory is actually more general and not restricted to the regular solution model. Consider the grand partition function in its simplified form neglecting orientations and structural details of individual micelles
Ξ)
∑ n +n )N 1
eβ(µ1n1+µ2n2) n1!n2!
2
∫Ve-βU(R ) dRN N
(23)
where U(RN) is the total potential energy depending on the spatial configuration RN of the N molecules; the integration is performed over the micelle volume V. The mean-field approximation is equivalent to setting 〈exp(-βU)〉 ≈ exp(-β〈U〉). Assuming pairwise interactions we have
〈U〉 )
n2(n2 - 1) n1(n1 - 1) 〈u11〉 + n1n2〈u12〉 + 〈u22〉 2 2 (24)
where
〈uij〉 ) Figure 1. Free energy of mixing, φ(x) ) βΦ/N, coexistence curve, R(x) ) zw(x), and the composition distribution, P(x) ) j ,N), for N ) 100 and xj ) 0.3 within the mean-field Pn(n approximation. The relationship between the two coexisting phases is determined using the lever rule, i.e., the ratio of the fraction X1 of micelles with the average composition of x0 to the fraction X2 of micelles with the average composition of 1 - x0 is given by X1/X2 ) (1 - x - x0)/(x - x0).
∫
1 uij(|ri - rj|) d3ri d3rj V2
(25)
If we further define the average interchange energy as
1 1 R ) (N - 1) 〈u12〉 - 〈u11〉 - 〈u22〉 2 2
(
)
(26)
and combine individual molecular properties into
[(
ξi ) V exp β µi -
N-1 〈uii〉 2
)]
(27)
we finally arrive at
Ξ)
∑ n +n )N 1
Figure 2. Coexistence curve, w(x), for the Ising model on a square lattice with periodic boundary conditions (IM) as compared against the mean-field approximation (MF) for N ) 64 (solid curves) and infinite (dotted) number of lattice sites.
We can now interpret the interaction parameter R introduced in eq 2. It is given by R ) zw and has a physical meaning of an interchange energy (in units of kBT) such that if we start with a pure micelle of type 1 and interchange one of its molecules with a molecule from a pure micelle of type 2, the total increase of energy is 2R.6 The expression obtained for the activity coefficient can be used in eq 16 to approximate the probability distribution as19
2
(ξ1)n1 (ξ2)n2 n1!
n2!
(
exp -R
n1n2
)
N-1
(28)
Equation 28 predicts essentially the same form of the distribution as that obtained earlier in this section for the lattice with nearest-neighbor interactions. At the same time, we are led back to eq 2 for the activity factor but with a generalized interpretation of the interchange energy parameter. It should be noted that the approximation considered actually assumes the interactions to be weak. The situation can be improved if one uses the virial expansion to derive the partition function. The result can be reduced again to eq 28 in the second order but with a new definition for R in terms of a straightforward combination of second virial coefficients. IV. Ising Model It is important to realize what deviations from the results above are to be expected if we take a model somewhat more realistic than a simple mean-field approximation. It is well-known that the lattice model of a binary mixture is mathematically equivalent to the Ising model of a ferromagnet in a field.26 We just represent the
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Barzykin and Almgren
state of the jth lattice point by the spin variable σj ) (1, the plus sign taken if the site is occupied by a particle of type 1 and minus if it is occupied by a particle of type 2. Starting with eq 5, after straightforward transformations we arrive at
Ξ ) exp(Nγ)Ξ0
(29)
where γ ) 1/2 ln(λ1λ2) - 1/4zw and Ξ0 is the standard partition function of the Ising problem with the interaction parameter K ) 1/2w and the magnetic field parameter H ) 1/2 ln(λ1/λ2). The constant γ has no effect upon any average. Therefore, all important thermodynamic functions of the Ising problem and those of the binary mixture are essentially the same. Exact analytical expressions for Ξ0 are available only for two-dimensional lattices in the absence of external magnetic field. Approximating a micelle by a closed twodimensional lattice is certainly an idealization, but it retains basic physics of the many-body interaction between well-packed surfactant head groups forming a micellar interface. Zero magnetic field corresponds to a mixture with xj ) 1/2. In general, the composition is related to the unit average magnetization 〈σ〉, as follows
xj ) 1/2(1 - 〈σ〉)
(30)
It is known that the Ising ferromagnet in two (and higher) dimensions has a spontaneous magnetization. Having the value of 〈σ〉 for H ) 027
〈σ〉 )
[
]
cosh2 w (sinh2 w - 1) sinh4 w
lattice type
z
Rc
honeycomb quadratic triangular
3 4 6
3 asinh x3 ≈ 3.9509 4 asinh 1 ≈ 3.5255 3 ln 3 ≈ 3.2958
mean-field
∞
2
at the same distance from the corresponding critical points, the difference is less conspicuous, but the mean-field distribution is still narrower reflecting the shape of the coexistence curve. We have also performed calculations for a triangular lattice (z ) 6). Principal conclusions are all the same here: the mean-field theory is not satisfactory when it comes to describing the phase separation. It predicts a universal value for the interchange energy parameter at the critical point, Rc ) 2, independent of the lattice structure and dimensionality. In a certain sense, the mean-field approach applies to a hypothetical lattice with an infinite coordination number. It can be seen from Table 1 that the exact value of Rc increases considerably as z is decreased. Ultimately, it becomes infinite for z ) 2, that corresponds to a one-dimensional chain which exhibits no phase transition. V. Cumulant Expansion In a further attempt to approximate the observed distribution, we turn to Kirkwood’s cumulant expansion method. We write
[∑
1/8
(31)
one deduces the critical composition (i.e., the coexistence curve) from eq 30. Equation 31 was derived for a simple square lattice in the limit of N f ∞. The critical point now appears at
wc ) asinh 1 ≈ 0.8814,
Table 1. Interchange Energy Parameter at the Critical Point, rc, for Several Infinite Two-Dimentional Lattices As Compared against the Mean-Field Approximation
xc )
1 2
∑ n 12
(26) Newell, G. F.; Montroll, E. W. Rev. Mod. Phys. 1953, 25, 353. (27) Yang, C. N. Phys. Rev. 1952, 85, 808.
(-w)j
j)1
j!
]
κj(n,N)
(33)
where
κ1 ) 〈n12〉 κ2 ) 〈(n12)2〉 - 〈n12〉2
(32)
which corresponds to appreciably lower Tc than that predicted by eq 20 for z ) 4. The coexistence curve becomes very flat in the vicinity of this point, as shown in Figure 2. Unfortunately, this is about all the advantage that we can get by reducing our problem to the Ising model. Exact analytical solution for the composition distribution is not available. However, we are quite fortunate that our practical problem is concerned with low values of N and, therefore, may be handled by simple numerical analysis involving direct evaluation of g(n12,n,N). Comparison against numerical results is not in favor of the mean-field approximation, as shown in Figure 3, although it correctly pictures the general tendency of the distribution to become bimodal at high values of w. The mean-field theory also accounts for the decrease of the critical value of w with decreasing N. This effect is actually much more pronounced when evaluated exactly (see Figure 2). Apparently, the principal fault of the mean-field approximation is that it fails to determine the coexistence curve correctly. This leads to huge discrepancies in Figure 3, since the form of the distribution undergoes most drastic changes right upon approaching and crossing the phase separation boundary. If one compares the distributions
∞
g(n12,n,N) exp(-n12w) ) exp
κ3 ) 〈(n12)3〉 - 3〈(n12)2〉〈n12〉 + 2〈n12〉3
(34)
etc., are the cumulants of g(n12,n,N). Although in principle κj can be calculated up to any order and for any lattice, the process is laborious. Expressions for the first few cumulants are available in the literature.6 For example, the second (variance) and the third cumulants are given by
κ2 ) 2zNx2(1 - x)2
(35)
κ3 ) -4zNx2(1 - x)2(1 - 2x)2
(36)
in the limit of very large N. Actually, for N as large as 100 (a typical value of the micelle aggregation number) there is already almost no appreciable difference between eqs 35-36 and numerically evaluated cumulants. Calculations for finite values of N are exceedingly tedious. Here we present only an approximate expression for the variance
κ2 ) 2zN
n(n - 1)(N - n)(N - n - 1)(N - z - 1) N(N - 1)2(N - 2)(N - 3) (37)
Note how x2 splits into n(n - 1)/N(N - 1) ensuring that
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Figure 3. Probability distribution P(x) ) Pn(n j ,N) of the micelle composition x ) n/N as a function of the interaction energy parameter (the values of w are attached to the plots) for xj ) 0.3 (left column) and xj ) 0.5 (right column); 2D periodic square lattice with N ) 64 sites. Mean-field approximation (dotted curves), second-order (dashed), and third-order (solid) truncated cumulant expansion, compared with numerical results (open circles). Numerical evaluation of g(n12,n,N) was performed by averaging over up to 108 random configurations for each value of n e N/2 (Monte Carlo method).
κ2 is exactly zero for n ) 1. Higher cumulants will also contain this factor. In Figure 3 we compare the cumulant expansion with numerically calculated distributions. Inclusion of only the second term in eq 33 leads to a dramatic improvement of the mean-field approximation. In fact, the second-order truncated cumulant expansion allows us to describe the composition distribution with sufficient accuracy in the one-phase region far from the phase separation boundary (i.e., small w). For higher values of w, especially in the vicinity of the critical point, more terms are required. However, as far as the distribution is concerned, just an extra third-order term suffices for all our needs. There still remain certain deviations from the exact distribution but we must ask ourselves what does “exact” actually mean here. Having started with a spherical shape of a micelle, we have assumed that all sites of our lattice are equivalent and had thus to impose periodic boundary conditions. However, the only rigorous topological closure of a finite two-dimensional lattice with its sites all equivalent is a torus, and not a sphere. A lattice formed on a spherical surface can never be perfect and that, no doubt, will have its effect on the critical behavior. We can safely neglect such effects if we restrict ourselves to the accuracy of the third-order truncated cumulant expansion. The first three cumulants are universal in the sense that they only depend on the lattice coordination number (average if not exact). Lattice-specific structural peculiarities manifest themselves in higher cumulants. There is no need to go beyond the third order, and there is even a strong argument against it, for the lattice model itself is already a rather crude approximation of a micelle.
The cumulant expansion allows us to write the free energy of mixing in the form
()
βΦ ) -ln
N n
∞
∑ j)1
(-w)j j!
κj(n,N)
(38)
Using Stirling’s formula and eq 35 for the variance, we obtain the second-order approximation
βΦ/N ) x ln x + (1 - x) ln(1 - x) + zwx(1 - x) - zw2x2(1 - x)2 (39) Now we take a step from eq 39 to derive an expression for the activity factor
3 ln fi ) zw(1 - xji)2 - 2zw2xji(1 - xji)2 1 - xji 2
(
)
(40)
The last term in the right-hand side of eqs 39 and 40 is due to the second cumulant. It corrects the mean-field results of eqs 17 and 21. As we have seen by comparing the distributions, this correction is significant. One can include another term, if necessary, as well as one can approximately calculate the distribution in a way similar to how eq 22 has been obtained. We will not pursue this task here. An alternative way to account for nonrandom arrangements within the lattice model is based on Guggenheim’s quasichemical approximation.6 This method was applied to mixed nonionic-anionic micelles,28 where conformational entropy changes in the polyethylene oxide head
4678 Langmuir, Vol. 12, No. 20, 1996
Figure 4. Probability distribution P(x) ) Pn(n j ,N) of the micelle composition x ) n/N as a function of the interaction energy parameter w ) -1, -0.6, 0, 0.6, 1 (top to bottom, according to the arrow) for xj ) 0.05 on a periodic 2D square lattice with N ) 64 sites.
group chains were also considered and a good agreement with experimental data was obtained while the regular solution approximation was found to be inadequate. VI. Structural Ordering So far we have tacitly assumed that w is positive. This is not necessarily the case. If the components of the mixture are oppositely charged, w may take only negative values. The system then exhibits quite different behavior from that considered above. Instead of phase separation, a kind of structural ordering may take place with the two types of particles alternating on the lattice. For instance, when a long-chain alcohol (nonionic cosurfactant) is added to a ionic micellar solution, it acts as a spacer that dilutes the charge density at the micelle-water interface, thereby reducing the electrostatic repulsion between the ionic head groups. For a mixture of equal composition (xj ) 1/2), ordering at high values of |w| may bring the system to a state where all micelles are equally composed. It should be fairly obvious that in our case we cannot speak of the formation of a completely ordered state with a regular alteration of lattice sites, which is possible only in a true crystal. But this is a direct consequence of the model that we have chosen. In most experimental situations, however, one component of the mixture is normally in excess, and the system may thus exhibit only partial ordering. The lattice model should be expected to give reasonable results then. Theoretically, order-disorder transition in binary alloys is analyzed on the basis of the dual lattice formalism. We shall not go into details here as our interest is concentrated on the composition distribution only. Thus we can simply employ the results of the previous sections, the only difference being that w may now take also negative values. Decreasing w leads to narrowing of the distribution, as shown in Figure 4, but the changes for negative values of w are much less appreciable than those for positive values. In the limit of very large |w| (w < 0), an ultimate equienergetic distribution of the micelle composition is achieved which is determined purely by combinatorics. We must admit that nearest-neighbor model is not adequate for mixed micelles with at least one ionic component due to the long-range nature of electrostatic interactions. The generalized mean-field or quasichemical approximation may work better in this case. (28) Osborne-Lee, I. W.; Schechter, R. S.; Wade, W. H.; Barakat, Y. J. Colloid Interface Sci. 1985, 108, 60.
Barzykin and Almgren
Figure 5. Plot of ln Ξ/n j versus xj for a periodic 2D square lattice with N ) 64 sites as a function of the interaction energy parameter w ) -1, -0.3, 0, 0.2, 0.4, 0.6, 0.8, 1 (top to bottom): numerical results (open circles) and third-order cumulant expansion (solid curves), as compared against dilute solution approximation (dotted curves).
VII. Mixtures with Excess of One Component Our original interest in the composition distribution of mixed micelles was stimulated by recent studies on fluorescence quenching in such systems, where one surfactant component acts as a quencher.19 As mentioned in the Introduction, fluorescence quenching provides a j ,N), which is straightforward method to determine P0(n simply the inverse of the grand partition function according to eq 12. Applicability of this method is limited j ,N) (measurable in a single to relatively large values of P0(n photon counting experiment) and, hence, to low quencher concentrations (phase separated mixtures are an exception). This particular case, i.e., xj , 1, is of our special concern here. As shown in Figure 4, the distribution is not changed much as a function of the interaction energy parameter, except for high positive values of w, where the phase separation takes place. The cumulant expansion converges rapidly; it nearly coincides with exact results already in the third order (not shown). The question is how to distinguish these changes experimentally. Figure 5 gives the answer. The dependence of ln Ξ/n j (or, equivalently, -ln P0/xjN) on xj proves to be quite sensitive to the interaction parameter. It is practically linear for small xj, with the slope bearing information on the value of w. The plot of ln Ξ/n j as a function of w for fixed xj (see Figure 6) is essentially asymmetric with respect to the sign of w giving clear evidence of the mean-field approximation breakdown. Although third-order cumulant expansion is already in excellent agreement with exact (albeit numerical) results, it is not quite handy. In an attempt to derive a more simple approximation, let us see what advantage can we take from that xj , 1. In this limit, we can assume that minority species form no more than one dimer. This assumption should work particularly well for negative w. The probability that 2 of n particles become nearest neighbors on a lattice with N . n sites is approximately n(n - 1)z/2N. Since n12 can take only two values in this model, namely, zn and zn - 2, the latter corresponding to configurations with one dimer, we can immediately evaluate the sum in eq 11 as
g(n12,n,N) exp(-n12w) ≈ ∑ n 12
[
1+
n(n - 1) N
]
ζ exp(-znw) (41)
Surfactant Distribution in Mixed Micelles
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Figure 7. Plot of y(xj) ) (1/z)((1/n) ln Ξ - 1 - (xj/2)) versus xj for a periodic 2D lattice as a function of the interaction energy parameter w ) -0.8, -0.6, -0.4, -0.2, 0, 0.2, 0.4, 0.6 (top to bottom). The results are compared for a square lattice with N ) 64 (O) and N)100 (+) sites and for a triangular lattice with N ) 64 sites (solid curves).
limit
κj ≈ (-2)j
Figure 6. Plot of ln Ξ/n j as a function of w for a periodic 2D square lattice with N ) 64 sites for xj ) 0.01 and xj ) 0.05. Mean-field approximation (dotted curves) and third-order truncated cumulant expansion (solid), as compared with numerical results (open circles) and dilute solution approximation (dashed curves).
where ζ ) (z/2)(e2w - 1). Now it is straightforward to derive an approximate expression for the grand partition function. After some algebra we arrive at
1 1 ln Ξ ) 1 + - ζ xj n 2
(
)
(42)
This can be regarded as the dilute solution approximation. It is directly related to the second-order virial expansion. In particular, 1/2 in the right-hand side of eq 42 is due to the excluded-volume effect. It is possible to develop a consistent dilute solution approximation on more general grounds than merely a lattice model. The results should be similar, as long as one assumes that the aggregation number is independent of the micelle composition. Importantly, eq 42 is expected to hold with ζ having a different, more general, interpretation. Dilute solution approximation compares well with numerical results29 for small xj, as illustrated in Figures 5 and 6, but worse when xj is increased. This disagreement is understandable. It is caused by formation of more than one bond between the minority species. For large positive values of w, cluster formation and, eventually, phase separation take place, that is what the dilute solution approximation simply cannot handle. But we always have the possibility to return to the third-order cumulant expansion in such a case. Now let us consider the limit of small xj starting from the cumulant expansion itself. We have for j > 1 in this (29) There is no need to evaluate g(n12,n,N) as long as only the grand partition function is of interest. The latter can be calculated by a very powerful matrix method which reduces to diagonalization of a 2N1/2 × 2N1/2matrix.26 This is much faster than to evaluate the whole g(n12,n,N) spectrum where all 2N configurations are involved.
zn(n - 1) 2N
(43)
Substituting eq 43 into eq 33 we immediately reproduce eq 41, as expected. Contribution of each κj to the final result can be recognized in the Taylor series expansion of 1 /2(e2w - 1). The mean-field approximation takes only the first term in this expansion, that restricts its applicability to |w| , 1. 1 1 ln Ξ - 1 According to eq 42, the function y(xj) ) z n xj should be universal, i.e., independent of the lattice 2 structure. The actual behavior of this function is compared in Figure 7 for a square lattice with N ) 64 and 100, and for a triangular lattice with N ) 64. The results for the square lattice with different number of sites match perfectly in the range of w considered. They also agree very well with those for the triangular lattice, except for large positive values of w. This is where the phase separation starts. It is easier for the molecules of the same kind to form clusters on a lattice with a larger coordination number as reflected in Table 1. The dilute solution approximation does not work here. Starting with eq 41 it is straightforward to find the activity factors in the limit of xj , 1. We have
(
)
ln f1 ) ζxj2
(44)
ln f2 ) R - 2ζxj
(45)
for the majority and the minority species, respectively. Thus, it should be possible to determine the parameter ζ from the cmc measurements and, independently, from the fluorescence quenching experiments. In addition, the interchange energy parameter R ) zw can be obtained in the former case and the aggregation number in the latter by extrapolating to infinite dilution. One can further estimate w by solving a transcendental equation e2w) 1 + 2wζ/R. If equal values are obtained for R and ζ, then w must be small. This is where the mean-field approximation comes into play. VIII. Concluding Remarks We have analyzed the composition distribution of mixed micelles, using basic principles of classical statistical
4680 Langmuir, Vol. 12, No. 20, 1996
mechanics, with special emphasis on the potential application to fluorescence quenching experiments in such systems. One of our original ideas was to go beyond the mean-field approximation in describing interactions between surfactant molecules which approach currently prevails in the literature. Thus we have made a few simplifying assumptions and reduced the problem to the lattice model with nearest-neighbor interactions, neglecting any possible composition-induced size or shape changes of the micelles. Being well aware of how serious these simplifications are, we have not concerned ourselves with making any quantitative comparisons to the available experimental data, although we expect the results to be
Barzykin and Almgren
applicable to mixtures of nonionic surfactants. But we have been able to explore explicitly the thermodynamic properties of a finite system of interacting molecules starting from microscopically defined interactions and we have shown how substantial the deviations from the meanfield predictions can be. More detailed molecular theory must account for the size and, possibly, shape distribution of the micelles as well as explicitly consider various conformations of the surfactant molecules.30 LA960107T (30) Szleifer, I.; Ben-Shaul, A.; Gelbart, W. M. J. Chem. Phys. 1987, 86, 709.
