Ising model approach, interaction

In the following sections, we shah demonstrate that the observed behavior of electro-optic activity with chromophore number density can be quantitatively explained in terms of intermolecular electrostatic interactions treated within a self-consistent framework. We shall consider such interactions at various levels to provide detailed insight into the role of both electronic and nuclear (molecular shape) interactions. Treatments at several levels of mathematical sophistication will be discussed and both analytical and numerical results will be presented. The theoretical approaches presented here also provide a bridge to the fast-developing area of ferro- and antiferroelectric liquid crystals [219-222]. Let us start with the simplest description of our system possible, namely, that of the Ising model [223,224]. This model is a simple two-state representation of the to-... 

For strongly asymmetric mixtures (e.g., mixtures where the A-chains are stiff while the B-chains are flexible) the semi-grandcanonical approach is clearly not feasible, and one must work in a canonical ensemble where both the number of A-chains nA and the number of B-chains nB are fixed. However, the finite size scaling ideas for PL(M) as exposed above still can be exploited if one considers the order parameter M in L x L subsystems of a much larger system [267]. The usefulness of this concept was demonstrated earlier for Ising models and Len-nard-Jones fluids [268-271]. Gauger and Pakula [267] find an entropy-driven phase separation without any intermolecular interactions. 

Discrete models. The most advanced are discrete models that explicitly take into account the interactions between nearest neighbour (NN) layers and even next to nearest neighbour (NNN) layers. Among those approaches the most successful are Ising models [24] and the XY models, particularly, the so-called clock model [6,26]. 

The Langmuir approach was a starting point for developing the more realistic formalism in the framework of the lattice gas theories based on the Ising model [24]. It seems intuitively obvious that the lattice gas model is well suited for representing localized adsorption. The adsorbed phase is considered a two-dimensional lattice gas. The most popular isotherm involving molecular interaction effects is the Fowler-Guggenheim equation [25]... 

The lattice models have been used frequently in theoretical studies of adsorption. The wide popularity of this approach results from its flexibility and simplicity. It may be applied for systems of various dimensionahties, for many lattice geometries, for different models of adsorbate-adsorbate molecular interactions, and many spatially varying external fields. Most studies have focused on a two-dimensional or three-dimensional cubic lattice, with only isotropic nearest-neighbor couplings. The isomorphism between the Ising model and the classical lattice gas or a coarse model of binary mixture is well known and very helpfijl for theoretical analysis. The lattice models can be also appHed to describe the systems involving polymers. 

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