Lattice models, of microemulsions

B. Widom. Lattice model of microemulsions. J Chem Phys 54 6943-6954, 1986. 

Probably the simplest of all lattice models of microemulsions is the Widom-Wheeler model [10,11]. In addition to its simplicity, its appeal lies in the fact that it is isomorphic to the Ising model of magnetism, a model well studied and readily simulated. 

Ginzburg-Landau models can be derived in a straightforward way from all microscopic lattice models of microemulsions. This has been done explicitly for the Widom model [43], for the three-component model [44], for vector models [45], and for the charge-frustrated Ising model [37]. In the case of the three-component model of Eqs. (2) and (3), the derivation shows, for example, that... 

Finally, one word about the lattice theories of microemulsions [30 36]. In these models the space is divided into cells in which either water or oil can be found. This reduces the problepi to a kind of lattice gas, for which there is a rich literature in statistical mechanics that could be extended to microemulsions. A predictive treatment of both droplet and bicontinuous microemulsions was developed recently by Nagarajan and Ruckenstein [37], which, in contrast to the previous theoretical approaches, takes into account the molecular structures of the surfactant, cosurfactant, and hydrocarbon molecules. The treatment is similar to that employed by Nagarajan and Ruckenstein for solubilization [38]. 

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

Lattice models for bulk mixtures have mostly been designed to describe features which are characteristic of systems with low amphiphile content. In particular, models for ternary oil/water/amphiphile systems are challenged to reproduce the reduction of the interfacial tension between water and oil in the presence of amphiphiles, and the existence of a structured disordered phase (a microemulsion) which coexists with an oil-rich and a water-rich phase. We recall that a structured phase is one in which correlation functions show oscillating behavior. Ordered lamellar phases have also been studied, but they are much more influenced by lattice artefacts here than in the case of the chain models. 

A. Ciach, J. S. Hoye, G. Stell. Microscopic model for microemulsion. II. Behavior at low temperatures and critical point. J Chem Phys 90 1222-1228, 1989. A. Ciach. Phase diagram and structure of the bicontinuous phase in a three dimensional lattice model for oil-water-surfactant mixtures. J Chem Phys 95 1399-1408, 1992. 

Whereas in approach 1 lattice models are used, we will work in the continuum, making extensive use of interface thermodynamics. The advantage of such an approach, as it turns out, is that detailed properties such as the size distribution of microemulsion droplets and the interfacial tension of a flat monolayer separating a microemulsion and an excess phase can be predicted. On the other hand, the lattice approaches as summarized in item 1 predict global phase behavior, which is not (yet) possible with the thermodynamic formalism reviewed in the following section. The reason is that currently a realistic model for the middle phase is lacking. A more detailed discussion regarding this issue is presented in Sec. VIII. 

A further step in the direction of a more detailed and realistic description of the amphiphile is taken by models in which the amphiphile is modeled by a polymer-like chain of subunits that are water-like at one end and oil-like at the other end [23]. Such models have been studied both in the continuum [24-26] and on a lattice [27 -31]. The advantage of these models is that they allow a calculation of the effect of the amphiphile chain length on the properties of microemulsions. These models have been studied mostly by computer simulations. 

As to the first question, lattice models do exhibit oil/water inlerfacial tensions that are reduced to various degrees from the value in the absence of amphiphile. For example, in the three-component model solved within mean-field theory, a reduction on the order of 30 was found in the oil/water interfacial tension at three-phase coexistence with the microemulsion [101]. When simulated so that fluctuations were included [102], the reduction increased to about a factor of 100, which is characteristic of a weak amphiphile. Other lattice models [103] have obtained reductions as large as a factor of 800, larger than that provided by even the strong amphiphile C6E3 [104]. 

The complexity of the equilibrium phases and nonequilibrium phenomena exhibited by multicomponent oil-water-surfactant systems is amply demonstrated in numerous contributions in this volume. Therefore, the need for theoretical (and computational) methods that make the interpretation of experimental observations easier and serve as predictive tools is readily apparent. Excellent treatments of the current status of theoretical advances in dealing with microemulsions are available in recent monographs and compendia (see, e.g., Refs. 1-3 and references therein). These references deal with systems consisting of significant fractions of oil and water and focus on the different phases and intricate microstructures that develop in such systems as the surfactant and salt concentrations are varied. In contrast, the present chapter focuses exclusively on simulations, particularly on a first level introduction to the use of lattice Monte Carlo methods for modeling self-association and phase equilibria in surfactant solutions with and without an oil phase. Although results on phase equilibria are presented, we spend a substantial portion of the review on micellization in surfactant-water mixtures, as this forms the necessary first step in the eventual identification of the most essential parameters needed in computer models of surfactant-water-oil systems. 

The model based on the lattice fluid SCF theory offers a means to calculate fundamental interfacial properties of microemulsions from pure component properties [25]. Because all of the relevant interfacial thermodynamic properties are calculated explicitly and the surfactant and oil molecular architectures are considered, the model is applicable to a wide range of microemulsion systems. The interfacial tension, bending moment, and interaction strength between the droplets can be calculated in a consistent manner and analyzed in terms of the detailed interfacial composition. The mechanism of the density effect on the natural curvature includes both an enthalpic and an entropic component. As density is decreased, the solvation of the surfactant tails is less favorable enthalpically, and the solvent is expelled from the interfacial region. Entropy also contributes to this oil expulsion due to the density difference between the interfacial region and the bulk. The oil expulsion and increased tail-tail interactions decrease the natural curvature. 

In another model, de Gennes and Taupin [114] proposed the use of a simple cubic lattice the size of the cube was taken as the same as the droplet diameter or the persistence length The latter has been related to the interfaces in the following way the interface is flat at scales less than and (ii) interfaces with an area of ( k) have independent orientations. The cubes are, of course, composed of oil or water. When a series of adjacent cubes are made of either oil or water, there is no interface in between. Therefore, progressive increase in the number of adjacent cubes of the same type will lead to the conversion of one type of microemulsion to another via a bicontinuous structure. 

In terms of interaction parameters J and L within the spin-lattice model, Gompper and Schick have calculated the structure function of bicontinuous microemulsion systems. Comparing the structure functions obtainable from SANS with those calculated ones by Gompper and Schick, we could determine the interaction parameters in right order of magnitude. 

Our main purpose in this paper is to clarify the essential differences of structures and interaction parameters among middle-temperature lamellar (MTL) phase and the two microemulsion, LTM and HTM, phases. We have determined the structural parameters by phenomenological model analysis and the microscopic interaction parameters by microscopic spin-lattice model analysis, respectively. 

A microscopic lattice model is proposed for microemulsion system by Gompper and Schick [6,7]. They choose a Hamiltonian H, which incorporates a statistical variable P on each site of the lattice. The variable has a value, P = 1, if there is a a molecule at site i, otherwise P = 0. Starting from such a Hamiltonian, Gompper and Schick calculated scattering function for a particular amphiphilic system in which the role of water and oil is symmetric or balanced . In a balanced system, the volume fractions of water and oil are equal, and water and oil are equally favored by the amphiphiles, i.e., hydro- and lipophilicity of the amphiphiles are equal. 

A variety of thermodynamic models qualitatively capture the central equilibrium and dynamic features of microemulsion behavior (41 5). These range from phenomenological models, which treat the oil-water interface as fluctuating membranes to lattice models which describes discrete surfactants interacting with oil and water. Membrane models range from simple cubic lattice descriptions to fluctuating film models that include curvature-dependent bending elasticity to prescribe lower limits on domain size. 

Lattice models seek to describe microemulsion structural domains down to the near-molecular level. These include models of amphiphilicity, imposed by mean field attractive and repulsive interactions applied to simplified diatomic or oligomeric amphiphiles, oil and water. Recent versions of these models include bending energy contributions for surfactants meeting at angles to approximate realistic molecular features and accurately capture microemulsion phase diagram. In all cases, microemulsion phase behavior is most accurately captured when the models consider key physical attributes (either exphc-itly or implicitly), including the balance between entropic (which tend to disperse oil-water into ever finer domains) and interfacial (which drive phase separation and place limits on domain curvature) contributions. 

K. Chen, C. Ebner, C. Jayaprakash, R. Pandit. Microemulsions in oil-water-surfactant mixtures Systematics of a lattice-gas model. Phys Rev A 55 6240, 1988. 

A description of the percolation phenomenon in ionic microemulsions in terms of the macroscopic DCF will be carried out based on the static lattice site percolation (SLSP) model [152]. In this model the statistical ensemble of various... 

This chapter will focus on a simpler version of such a spatially coarse-grained model applied to micellization in binary (surfactant-solvent) systems and to phase behavior in three-component solutions containing an oil phase. The use of simulations for studying solubilization and phase separation in surfactant-oil-water systems is relatively recent, and only limited results are available in the literature. We consider a few major studies from among those available. Although the bulk of this chapter focuses on lattice Monte Carlo (MC) simulations, we begin with some observations based on molecular dynamics (MD) simulations of micellization. In the case of MC simulations, studies of both micellization and microemulsion phase behavior are presented. (Readers unfamiliar with details of Monte Carlo and molecular dynamics methods may consult standard references such as Refs. 5-8 for background.)... 

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