Mixtures lattice model

As early as 1969, Wlieeler and Widom [73] fomuilated a simple lattice model to describe ternary mixtures. The bonds between lattice sites are conceived as particles. A bond between two positive spins corresponds to water, a bond between two negative spins corresponds to oil and a bond coimecting opposite spins is identified with an amphiphile. The contact between hydrophilic and hydrophobic units is made infinitely repulsive hence each lattice site is occupied by eitlier hydrophilic or hydrophobic units. These two states of a site are described by a spin variable s., which can take the values +1 and -1. Obviously, oil/water interfaces are always completely covered by amphiphilic molecules. The Hamiltonian of this Widom model takes the form... 

In this section and the last, we have examined the lattice model of the Flory-Huggins theory for general expressions relating AHj and ASj to the composition of the mixture. The separate components can therefore be put together to give an expression for AGj as a function of temperature and composition ... 

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. 

Note that large density fluctuations are suppressed by construction in a random lattice model. In order to include them, one could simply simulate a mixture of hard disks with internal conformational degrees of freedom. Very simple models of this kind, where the conformational degrees of freedom affect only the size or the shape of the disks, have been studied by Fraser et al. [206]. They are found to exhibit a broad spectrum of possible phase transitions. 

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. 

The structure of a simple mixture is dominated by the repulsive forces between the molecules [15]. Any model of a liquid mixture and, a fortiori of a polymer solution, should therefore take proper account of the configurational entropy of the mixture [16-18]. In the standard lattice model of a polymer solution, it is assumed that polymers live on a regular lattice of n sites with coordination number q. If there are n2 polymer chains, each occupying r consecutive sites, then the remaining m single sites are occupied by the solvent. The total volume of the incompressible solution is n = m + m2. In the case r = 1, the combinatorial contribution of two kinds of molecules to the partition function is... 

Monte Carlo may be used to study the lateral distribution of lipid molecules in mixed bilayers. This of course is a very challenging problem, and, to date, the only way to obtain relevant information for this is to reduce the problem to a very simplistic two-dimensional lattice model. In this case, the lipid molecules occupy a given site and can be in one of the predefined number of different states. These pre-assigned states (usually about 10 are taken), are representative conformations of lipids in the gel or in the liquid state. Each state interacts in its own way with the neighbouring molecules (sitting on neighbouring sites). Typically, one is interested in the lateral phase behaviour near the gel-to-liquid phase transition of the bilayer [69,70]. For some recent simulations of mixtures of DMPC and DSPC, see the work of Sugar [71]. 

The addition of water to solutions of PBT dissolved in a strong acid (MSA) causes phase separation in qualitative accord with that predicted by the lattice model of Flory (17). In particular, with the addition of a sufficient amount of water the phase separation produces a state that appears to be a mixture of a concentrated ordered phase and a dilute disordered phase. If the amount of water has not led to deprotonation (marked by a color change) then the birefringent ordered phase may be reversibly transformed to an isotropic disordered phase by increased temperature. This behavior is in accord with phase separation in the wide biphasic gap predicted theoretically (e.g., see Figure 8). The phase separation appears to occur spinodally, with the formation of an ordered, concentrated phase that would exist with a fibrillar morphology. This tendency may be related to the appearance of fibrillar morphology in fibers and films of such polymers prepared by solution processing. 

Solution of long-chain molecules When two liquids mix to form a mixture, the entropy change is similar to that of the volume expansion, as long as the solute molecules have the same size as the solvent molecules and are randomly distributed. But when the solute forms long-chain molecules, the correct method of calculating the entropy was given by Flory. First consider a lattice model where the solvent and the solute molecules have the same volume. Let i and 2 be the number of solvent and... 

Mixture Models Broken-Down Ice Structures. Historically, the mixture models have received considerably more attention than the uniformist, average models. Somewhat arbitrarily, we divide these as follows (1) broken-down ice lattice models (i.e., ice-like structural units in equilibrium with monomers) (2) cluster models (clusters in equilibrium with monomers) (3) models based on clathrate-like cages (again in equilibrium with monomers). In each case, it is understood that at least two species of water exist—namely, a bulky species representing some... 

Wu et al., [88] compared several local composition models with LMC simulations for lattice mixtures. The models tested included UNIQUAC, the AD model for lattice fluids of Aranovich and Donohue [89], and the Born-Green-Yvon (BGY) model of Lipson [90]. 

The necessity of introducing a combinatorial contribution to the chemical potential is a result of the neglect of size effects in the thermodynamics of pairwise interacting surface models. It also appears in lattice models that do not allow for a realistic representation of molecular sizes and are often simplified to models of equally sized lattice objects. The task of the combinatorial contribution is to represent the chemical potential of virtually homogeneous interacting objects of different size in 1 mol of a liquid mixture of a given composition with respect to the size and shape of the molecules. 

To extend a close-packed lattice model Equation (15) to a lattice fluid model, we adopt a two-step process as shown in Figure 15 to establish an EOS (Hu et al., 1992). In the first step, pure chain molecules at close-packed lattice are mixed to form a close-packed mixture. In the second step, the close-packed mixture is mixed with N0 holes to form an expanded realistic system with volume V at temperature T and pressure p. According the two-step process, the Helmholtz energy of mixing can be expressed as... 

K number of component in mixture or the curvature of cylindrical pore defined by K = 1 /Rex L the length of cylindrical pore m the number of periods in the lamellar N the number of molecule in the fluid mixture Nh the number of head units of a polymer chain Nr total number of sites in lattice model... 

A Statistical-Mechanics based Lattice-Model Equation of state (EOS) for modelling the phase behaviour of polymer-supercritical fluid mixtures is presented. The EOS can reproduce qualitatively all experimental trends observed, using a single, adjustable mixture parameter and in this aspect is better than classical cubic EOS. Simple mixtures of small molecules can also be quantitatively modelled, in most cases, with the use of a single, temperature independent adjustable parameter. 

The lattice model thus provides the capability to obtain good, quantitative fits to experimental VLE data for binary mixtures of molecules below their critical point. Its value lies in the fact that it performs equally well regardless of the size difference between the component molecules. 

The UNIFAC (Unified quasi chemical theory of liquid mixtures Functional-group Activity Coefficients) group-contribution method for the prediction of activity coefficients in non-electrolyte liquid mixtures was first introduced by Fredenslund et al. (1975). It is based on the Unified Quasi Chemical theory of liquid mixtures (UNIQUAC) (Abrams and Prausnitz, 1975), which is a statistical mechanical treatment derived from the quasi chemical lattice model (Guggenheim, 1952). UNIFAC has been extended to polymer solutions by Oishi and Prausnitz (1978) who added a free volume contribution term (UNIFAC-FV) taken from the polymer equation-of-state of Flory (1970). 

We consider a two-dimensional mixture which contains N end-attached chains and N0 solvent molecules and employ the simple cubic lattice model for its representation. The lat-... 

We consider a system consisting of two identical polymer brushes (grafted to two parallel plates) and solvent molecules. The cubic lattice model is employed to represent the mixture of polymer chains and solvent (schematically shown in Fig. 1). Each polymer segment and each solvent molecule occupies one lattice site. The lattice is divided into 2m layers parallel to the plates, numbered from any of the two plates as 1, 2,..., m+ 1,...,2m+1 i,...,2m. Each lattice layer... 

At its most satisfying level, a statistical thermodynamic theory would begin by specifying realistic interaction potentials for the molecular components of a complex mixture and from these potentials the thermodynamic functions and phase behavior would be predicted without further approximation. For the next decade or so, there is little hope to accomplish such a theory for microstructured fluids. However, predictive theories can be obtained with the aid of elemental structures models. Also, lattice models... 

Of course, some general aspects of our treatment could be easily extended to a general form of f b ireJ as in the semi-infinite case [226],but for explicit numerical work a specific form of fs(b ire) ((()) is needed. Equation (10) can be justified for Ising-type lattice models near the critical point [216,220], i.e. when ( ) is near ( >crit=l/2, as well as in the limits f]>—>0 or <()—>1 [11]. The linear term —pj( ) is expected due to the preferential attraction of component B to the walls, and to missing neighbors for the pairwise interactions near the walls while the quadratic term can be attributed to changes in the pairwise interactions near the walls [144,216,227]. We consider Eq. (10) only as a convenient model assumption to illustrate the general theoretical procedures - there is clear evidence that Eq. (10) is not accurate for real polymer mixtures [74,81,82,85]. 

Sanchez and Lacombe (1976) developed an equation of state for pure fluids that was later extended to mixtures (Lacombe and Sanchez, 1976). The Sanchez-Lacombe equation of state is based on hole theory and uses a random mixing expression for the attractive energy term. Random mixing means that the composition everywhere in the solution is equal to the overall composition, i.e., there are no local composition effects. Hole theory differs from the lattice model used in the Flory-Huggins theory because here the density of the mixture is allowed to vary by increasing the fraction of holes in the lattice. In the Flory-Huggins treatment every site is occupied by a solvent molecule or polymer segment. The Sanchez-Lacombe equation of state takes the form... 

The Kumar equation of state (Kumar, 1986 Kumar et al., 1987) is a modification of the Panayiotou-Vera model that was developed to simplify the calculations for multicomponent mixtures. Since the Panayiotou-Vera equation is based on the lattice model with the quasichemical approach for the nonrandomness of the molecules in the mixture, the quasichemical expressions must be solved. For a binary system the quasichemical expressions reduce to one quadratic expression with one unknown, but the number of coupled... 

We shall not attempt to review and compare critically various theories of liquid crystallinity in this chapter. Inasmuch as theory based on a lattice model has proved most successful in the treatment of liquid crystallinity in polymeric systems, we shall present an abbreviated account of that theory confined to its essential aspects. The versatility of this theory has permitted its extension to polydisperse systems, to mixtures of rodlike polymers with random coils and to some of the many kinds of semirigid chains. These ramifications of the theory will be discussed in this chapter... 

For polymers, x is usually defined on a per monomer basis or on the basis of a reference volume of order one monomer in size. However, x is usually not computed from formulas for van der Waals interactions, but is adjusted to obtain the best agreement between the Flory-Huggins theory and experimental data on the scattering or phase behavior of mixtures (Balsara 1996). In this fitting process, inaccuracies and ambiguities in the lattice model, as well as in the mean-field approximations used to obtain Eq. (2-28), are papered over, and contributions to the free energy from sources other than simple van der Waals interactions get lumped into the x parameter. The temperature dependences of x for polymeric mixtures are often fit to... 

Highly ordered lamellar gel microstructures are formed by certain surfactants and mixtures of a surfactant and long-chain fatty alcohols in water. Using small angle X-ray scattering (SAXS), an ordered lamellar stack lattice model was proposed for the gel formed by 10% w/w cetostearyl alcohol containing 0.5% cetri-mide surfactant. In contrast, the microstructure of a Brij 96 gel depends on the surfactants concentration. A hexagonal liquid-crystalline gel structure was... 

The Blume-Emery-Griffiths (BEG) model is one of the well-known spin lattice models in equilibrium statistical mechanics. It was originally introduced with the aim to account for phase separation in helium mixtures [30]. Besides various thermodynamic properties, the model has been extended to study the structural phase transitions in many bulk systems. By... 

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