Critical micelle concentration in three-dimensional lattice models of amphiphiles

Amphiphilic molecules with hydrophobic tails and hydrophilic heads form micelles of various shapes and sizes above a minimum threshold concentration known as the critical micelle concentration (CMC). The CMC as well as the size and the shape of the aggregates formed depend on various factors, e.g., the length of the amphiphiles, their internal rigidity, and temperature. In this letter we report the results of a detailed investigation of the dependence of the CMC on temperature for different lattice models of the amphiphilic selfassembly. Ensuring that the CMC can be unambiguously associated with a peak in the heat capacity as a function of the amphiphilic concentration, we show that for the amphiphiles of different lengths and head-to-tail ratios, the CMC decreases rapidly as a function of the chain length, consistently with the experimental results. However, for a given chain length, different lattice models predict that the CMC is always an increasing function of temperature. We point out that these lattice models, although widely used, are inadequate to explain the decrease of the CMC with temperature, seen experimentally for non-ionic surfactants. There is an increasing interest in understanding the self-assembly properties of amphiphilic molecules [1–4] due to their widespread application in fabricating various devices and moieties at the nanometre length scales. For example, micelle formation through the self-assembly of short hydrocarbon amphiphilic chains is used to prepare both ordered and disordered porous structures with pore sizes of the order of 40 Å [5]. Self-assembly of peptide ribbons or sheets has the potential to be used in drug delivery [6]. More recently, short amphiphilic chains have been used to create a medium with evenly distributed carbon nanotubes [7]. Pattern formations 3 Author to whom any correspondence should be addressed. 0953-8984/01/410861+09$30.00 © 2001 IOP Publishing Ltd Printed in the UK L861 L862 Letter to the Editor of di-block and tri-block co-polymers are also very well known [8]. Quite naturally, attention needs to be paid to understanding amphiphilic self-assembly in terms of different models with varying degree of coarse graining or complexity. In this letter, we report the results of a detailed investigation of the dependence of CMC on temperature and chain length carried out in various lattice models that are currently used to study the self-assembly of amphiphiles. We point out that additional features need to be incorporated to make these models more realistic. Traditionally, lattice models have been used to understand the micelle formation and phase separation processes of amphiphilic systems because their inherent simplicity is very appealing as compared to their off-lattice counterparts. These lattice models can be broadly divided into two categories. In the first category, the physical system is mapped onto extended Ising-like models with both bond and site variables [9]. These Ising-like models ignore the internal structure of the individual amphiphiles. However, they have successfully predicted both a closed-loop coexistence curve and the temperature dependence of the CMC observed experimentally for non-ionic surfactants (which are amphiphiles) [10]. It has been demonstrated from simple analytic arguments that the internal structure of the amphiphiles, e.g., head-to-tail ratios, the polarity and effective sizes of the amphiphilic head segments, etc, can dictate the final shapes of the micelles [2]. In a continuum description of the amphiphilic self-assembly, the geometrical features of amphiphiles are taken into account by modelling them as successive hydrophilic (head) and hydrophobic (tail) beads (monomers) connected by bonds and immersed in a solvent. Both the solvent molecules and the individual monomers of the amphiphiles are most often assumed to interact with a Lennard-Jones (LJ) potential. The successive monomers in a given chain interact with an anharmonic spring potential to inhibit disintegration of the chain. A specific choice of the LJ parameters then serves to model an amphiphile of a particular type. Unlike the Ising-type lattice models, the second category of lattice models, which we discuss here, partly take into account these geometrical structures of the amphiphiles and can be looked at as limiting cases of continuum models. Here an amphiphile made of m head and n tail segments connected by n + m− 1 bonds (denoted as HmTn) is restricted to moving on a lattice with suitably defined interactions between the amphiphiles and with the solvent particles in which they are immersed. The simplest of the lattice models of this type is the Larson model [11], where the interaction parameters are kept to a bare minimum by the special choice where the solvent particles are made of either the head or the tail particles as shown in figure 1(a). As a result, the phase digram associated with the self-assembly can be explored in terms of a single parameter, namely ht/(kBT ), where ht is a measure of the strength of the interaction between a head and a tail particle, kB is the Boltzmann constant, and T is the temperature. In this work we have used a more general model, where the solvent particles are in general different from the head and the tail segments. This model, which has a few additional parameters compared to the Larson model, has been studied extensively by Care and coworkers [12–15] and more recently by us [16]. Both the models capture many of the features of amphiphilic self-assembly. For example, Monte Carlo (MC) simulation showed the existence of lamellar and hexagonal phases and vesicular structures [11–15,17,18] in these models. The results obtained from these lattice models have also been compared with those from analytic mean-field theories [16, 19, 20]. It is worth mentioning that in both the Larson model and the Care model, the lattice is completely occupied either with an amphiphilic moiety or with a solvent particle. In this letter we have introduced a new lattice model where the solvent degrees of freedom are removed in favour of an effective interaction between the amphiphiles. Before we discuss Letter to the Editor L863 Figure 1. (a) The Larson model for H1T6. Each amphiphile is made of one head (•) and six tail (◦) monomers connected by six bonds; the solvent particles are chosen to be made of head particles. (b) In the Care model the solvent particles (filled squares) are chosen to be different. this effective-lattice model, we would like to make a few remarks about the continuum models. Most of the molecular dynamics (MD) simulation studies on continuum models have been carried out for a small number ( 100) of realistic amphiphilic chains [22–25]. Recently Maillet, Lachet, and Coveney have [26] reported a fully atomistic study of the structure and dynamics of micelles of approximately 50 realistic amphiphiles surrounded by 3000 water molecules, taking into account the relevant long-range interactions and stretching the MD run up to 3 ns. Because of the small concentration of the amphiphiles (relevant for micelle formation), monitoring the solvent degrees of freedom exhausts most of the computer time. This inhibits the study of aggregation properties for a large number of amphiphiles. Naturally, simpler effective-continuum models have also been studied through MC and MD simulation where the solvent degrees of freedom have been eliminated at the cost of effective interaction parameters among amphiphilic segments only. These off-lattice models without the solvent particles are very enticing for numerical expediency and many of the features are qualitatively similar [27–30]. The lattice versions of these continuum models without the solvent particles (we will refer to them as effective-lattice models (ELM)) have not been studied in the context of amphiphilic self-assembly so far. The purpose of this letter is twofold. First, we report the self-assembling properties of the amphiphiles in the ELM proposed here and compare the results with those obtained from other lattice models which include the solvent particles explicitly. Second, by carrying out extensive MC simulation for different chain lengths, concentrations, and temperatures on these lattice models, both with and without the solvent particles, we establish an important generic result: the CMC in the lattice models in this class always increases with increasing temperature. We point out that this is an obvious limitation of these lattice models because experimental investigations on various amphiphiles show [31] that the CMC for non-ionic surfactants decreases with increasing temperature. We discuss the limitations of these models and suggest further improvements. First we show the MC simulation results for the Care model where the amphiphiles are confined to a three-dimensional (3D) cubic lattice of size L. In this model an amphiphile HmTn of length (m + n − 1) consists of m hydrophilic heads (H) and n hydrophobic tails (T) connected by m + n − 1 bonds. We use the notation unimer to represent each isolated amphiphile while a monomer represents either a head or a tail particle. We consider NA of such amphiphiles which occupy (m+n)NA lattice sites. The remaining Nw = L3− (m+n)NA L864 Letter to the Editor sites are occupied by the solvent particles. The total energy of the system is given by H = T SnT S + HSnHS + HHnHH + ∑