JOURNAL OF CHEMICAL PHYSICS

VOLUME 112, NUMBER 10

8 MARCH 2000

Transition in three-dimensional micellar systems M. Girardi and W. Figueiredoa) Departamento de Fı´sica—Universidade Federal de Santa Catarina 88040-900, Floriano´polis, SC, Brazil

共Received 16 November 1999; accepted 8 December 1999兲 We have studied the thermal behavior of aggregates of amphiphilic molecules in water for a three-dimensional lattice model. We have employed extensive Monte Carlo simulations to find the distribution curve for aggregate size as a function of temperature. Our model consists of amphiphile molecules with a single polar head and three monomers in its hydrophobic tail. In this lattice model each amphiphile occupies four sites of the lattice, and the water molecules occupy only one site. We determine the free amphiphile concentration and the aggregate-size distribution P(n) as functions of total surfactant concentration for different temperatures. As for the two-dimensional case, the degree of micellar organization is characterized by ⌬⬅ P(n max)⫺P(nmin), where n min and n max ⬎nmin are the aggregate sizes at which P exhibits its local minimum and maximum. For this three-dimensional model we also show that ⌬ vanishes continuously as we increase the temperature. However, the exponent associated with the micellar to nonmicellar transition is twice that found in the two-dimensional simulations. © 2000 American Institute of Physics. 关S0021-9606共00兲51009-5兴

I. INTRODUCTION

model is very simple, it contains the essential aspects of the real micelle formation, except that it gives a high polydispersity due to the small size of the molecules.2 Many simulations have been performed in past years for the threedimensional assembly of amphiphiles. Larson,6–8 for instance, considered the self-assembly of surfactants at very high concentrations and found different crystalline phases. Brindle and Care9 found the phase diagram of the mixture of amphiphile and water for chains with a length of 4. They showed that, at low concentrations, the model exhibited a clear micellar behavior despite the high dispersivity. Also, Desplate and Care10 determined an analytical expression for the cluster-size distribution. Nelson et al.11 investigated the size and shape of self-assembled micelles, and showed that the distribution curve exhibits a Gaussian peak at spherical micelle shape. In order to model our system, we take five nearestneighbor energies. The interactions between an amphiphile tail monomer and water, and between tails and heads, ⑀ TW and ⑀ TH , respectively, are assumed to be repulsive. The three remaining interactions—⑀ TT 共between tail monomers兲, ⑀ HH 共between heads兲, and ⑀ HW 共between a head and water兲— are attractive. We also associate an energy ⑀ b ⬎0 to each bent bond in the amphiphile, thereby favoring a linear conformation. For simplicity, we take the same magnitude for all the interactions, i.e., ⑀ TW ⫽ ⑀ TH ⫽ ⑀ b ⫽ ⑀ , and ⑀ TT ⫽ ⑀ HH ⫽ ⑀ HW ⫽⫺ ⑀ . We set ⑀ ⫽1 and use the scaled temperature t ⫽ k B T/ ⑀ , where k B is the Boltzmann constant and T is the temperature. The simulation was performed along the following steps: First, we randomly distribute the amphiphile molecules on the lattice, forbidding double occupancy. The remaining sites are regarded as harboring water molecules. As our amphiphile molecules are small, we only use chain reptation movements to generate trial configurations that guarantee the ergodicity of the algorithm. The latter is accepted according

1

In a recent paper, we have studied the self-assembly in a two-dimensional lattice model of amphiphilic molecules in an aqueous solution via Monte Carlo simulation. We have shown that the formation of micelles2–4 depends on the temperature of the heat bath for a given concentration of surfactants. We have defined a parameter which characterizes the transition from a micellar to a nonmicellar state. The definition was based on the fingerprint5 of what is called micelle: the appearance of a minimum and a maximum in the aggregate-size distribution curve P(n). In this work we have extended our simulations to three dimensions and found the corresponding curve for P(n) for different values of temperature. We have calculated ⌬⬅ P(n max)⫺P(nmin), where n min and n max are the aggregate sizes at which P exhibits its local minimum and maximum. The parameter ⌬ decreases quadratically with temperature and vanishes at a given temperature. For the two-dimensional case, the decreases were considered linear. In the next section we introduce the model, and in Sec. III we present our results and conclusions. II. MODEL

The model we consider here is a three-dimensional version which we have studied previously in two dimensions.1 The amphiphile molecules are represented by monomers occupying four adjacent sites in the cubic lattice 共coordination number 6兲, and the remaining sites represent water molecules. One of the monomers represents the polar head of the molecule; the other three are identified with its hydrophobic hydrocarbon part. Beyond the usual prohibition against two monomers occupying the same site, the model includes interactions between nearest neighbors only. Although this a兲

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

M. Girardi and W. Figueiredo

FIG. 1. Total energy as a function of the number of Monte Carlo steps 共MCs兲. The temperature is t⫽1.50, the lattice size is L⫽40, total concentration of amphiphiles X t ⫽1.25%. Curve 共a兲 means a random initial configuration, and 共b兲 is a lamellar initial low-energy configuration.

to the Metropolis prescription. We used cubic lattices of size L⫽40, with periodic boundary conditions. Simulations performed with L⫽20 gave essentially identical results. We take as a Monte Carlo step 共MCs兲 one complete run over the surfactant molecules. After typically 5⫻104 MCs, the system reaches equilibrium, and a large number of Boltzmannweighted configurations are generated. The optimal number of MCs depends on the concentration of surfactant molecules and the temperature. At very low temperatures, convergence to equilibrium is very slow, and it is necessary to consider many more than 106 MCs to bypass metastable states and attain equilibrium.12 In order to avoid these very slow convergence processes, we restricted our study to t⬎1.3. Following equilibration we generated more 105 MCs in order to calculate the thermal properties and their respective fluctuations.

FIG. 2. Free surfactant concentration as a function of total concentration for two lattice sizes. The temperature is 1.50. The lines serve as a guide to the eye, the dotted line corresponds to (L⫽40) and the full line corresponds to (L⫽20).

where N n is the number of aggregates containing n amphiphile molecules. In this way, the total concentration X t ⬁ can be written as X t ⫽ 兺 n⫽1 nX n . We show in Fig. 2 a typical curve for the equilibrium density of free molecules in solution for two different lattice sizes. This curve was obtained at t⫽1.48. As explained before, there is no difference between the results for the lattice sizes L⫽20 and L⫽40. This indicates that the finite-size effects are not important here because the size of the amphiphiles is small compared to the

III. RESULTS

We show in Fig. 1 a typical curve for the total energy of the system as a function of the number of Monte Carlo steps, for t⫽1.50, total concentration of surfactants X t ⫽1.25%, and a lattice of size L⫽40. In this figure we have used two different initial configurations, and we see that after 5⫻104 MCs the thermalization is easily found. As in the two-dimensional case, we obtained aggregates of various sizes n⭓2. We consider a cluster to be an aggregate only when a hydrocarbon chain has at least one firstneighbor contact with another hydrocarbon chain. We define X n , the concentration 共number density兲 of aggregates of size n, by X n⫽

Nn L3

,

共1兲

FIG. 3. Distribution function P(n) as a function of the size of the aggregates n for three temperatures, as indicated. The total surfactant concentration is X t⫽1.25%. ⌬ is the difference in height between the maximum and minimum of the distribution at t⫽1.44. The lines serve to guide the eye.

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J. Chem. Phys., Vol. 112, No. 10, 8 March 2000

FIG. 4. Free amphiphile concentration X 1 as a function of total surfactant concentration X t for three different temperatures, as indicated. The lines serve as a guide to the eye.

linear size L. We clearly show that the free surfactant concentration decreases just above the critical micellar concentration 共CMC兲. This result was also seen through stochastic dynamic simulations performed by Gottberg et al.13 for molecules of size 4. In our simulations we have taken total concentrations of up to 2.50% in order to get very dilute solutions. We show in Fig. 3 the aggregate-size distribution function, P(n)⬅nX n , for three different temperatures. For t⫽1.44, the local minimum and maximum characteristics of micelle formation are clearly evident. On the other hand, for t⫽1.53 and t⫽1.60, the distribution function decreases monotonically with n. Although the free surfactant curves, exhibited in Fig. 4, are very similar, those for t⫽1.53 and t⫽1.60 do not exhibit the signature of micellar organization. That is, the full signature of micelle formation implies a well-defined micelle concentration and a distribution function showing a local minimum and maximum.5 On the basis of Figs. 3 and 4, then, it is only for t⫽1.44 that we can speak of a true micelle formation. In this case we can define the parameter ⌬ as the difference in height between the maximum and the minimum of P(n). We performed detailed simulations (105 Monte Carlo steps兲, and considered 40 samples for each histogram to study the transition from the micellar state to the nonmicellar one as a function of temperature. We obtain a transition temperature t M ⫽1.53⫾0.01 for an amphiphile concentration of 1.25%. In Fig. 5 we show a plot of the parameter ⌬ as a function of temperature in the vicinity of the transition. The exponent ␤, which characterizes the micellar to nonmicellar behavior, is given by ␤⫽2.1⫾0.4. This value is almost twice the value found in the two dimensions. It is worth mentioning that this kind of transition is different from the macroscopic phase separation observed in the lattice gases. A detailed discussion of this point can be found in the work of Floriano et al.14 In summary, we have performed Monte Carlo simula-

Transition in three-dimensional micellar systems

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FIG. 5. Plot of the parameter ⌬ as a function of t for the total surfactant concentration X t ⫽1.25%. The error bars are indicated in the figure. The line is the best fit to the simulation data. For t very near t M ⫽1.53, we have the following power law: ⌬⬀(t⫺t M ) ␤ , and ␤ ⫽2.1⫾0.4.

tions on a cubic lattice, for a model of amphiphile molecules in water. We determined the free amphiphile concentration as a function of total surfactant concentration for various temperatures. Although the free amphiphile concentration exhibits the same behavior for the temperatures considered, there is a transition temperature above which we do not have true micelle formation. The exponent associated with the continuous transition from the micellar to the nonmicellar state is near 2.

ACKNOWLEDGMENTS

This work was supported by the Brazilian agencies CNPq and FINEP.

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