Results from Lattice Models

In view of these observations, one would like to establish the Ising-like nature of the critical point by an RG treatment. Unfortunately, lattice models, as successfully applied to describe the criticality of nonionic fluids, may be of little help in this regard, because predictions for the Coulomb gas have proved to be surprisingly different from those for the continuum RPM. Discretization effects—and, more generally, the relevance of the results of lattice models with respect to the fluid—still need to be explored in detail. On the other hand, an RG treatment of the RPM or UPM is still lacking and, as Fisher [278] notes, the way ahead remains misty. 

Several models have attributed the phenomenon to increased lattice distortion and defect density, surface passivation, or shortened electron life. In an earlier investigation, it was suggested that etch rate reduction is caused by the increased lattice distortion from high doping. Later studies indicated that the reduced etch rate of high boron doped material is not likely to result from lattice distortion or stress. 

The interpretation of the spectral details was improved by comparing results from two model calculations, one for an undoped cerium oxide lattice and the other for a Sc doped lattice. The main hydrogen site was found to be at the Ce, giving features about 650 cm. The minority population was next to the doped site, giving features about 850 cm [63]. The temperature dependence of the y(OH...O), 805 cm, in RbHS04 was also measured. 

To predict the experiments, we want to compute the mathematical form of m T) near the critical temperature T Tc, where m(T) 0. We could get this function from the lattice or van der Waals gas models, but instead we will find the same result from a model that is simpler and more general, called the Landau model. 

In the gas phase, physical and chemical properties of elementary species can be studied in the absence of external perturbations, while in condensed phases, solution or solid, the dense environments formed by the solvent or the lattice have important effects on chemical processes. Due to the relative simpUcity of gas-phase systems, it is possible to probe the relationships between electronic structure, reactivity, and energetics, and the information obtained can be compared directly with the results from theoretical models. 

Consequently, we focus here on computer simulations exclusively. The outline of the remainder of this chapter is as follows Section 1.2 presents on overview of polymer models (from lattice models to atomistic descriptions) and will also describe the most important aspects of Monte Carlo simulations of these models. As an example, recent work on simple short alkanes and solutions of alkanes in supercritical carbon dioxide [47,48] will be presented, to clarify to what extent a comparison of Monte Carlo results on phase behavior and experimental data is sensible, and which experimental input into the models is indispensable to make them predictive. 

Various functional forms for / have been proposed either as a result of empirical observation or in terms of specific models. A particularly important example of the latter is that known as the Langmuir adsorption equation [2]. By analogy with the derivation for gas adsorption (see Section XVII-3), the Langmuir model assumes the surface to consist of adsorption sites, each having an area a. All adsorbed species interact only with a site and not with each other, and adsorption is thus limited to a monolayer. Related lattice models reduce to the Langmuir model under these assumptions [3,4]. In the case of adsorption from solution, however, it seems more plausible to consider an alternative phrasing of the model. Adsorption is still limited to a monolayer, but this layer is now regarded as an ideal two-dimensional solution of equal-size solute and solvent molecules of area a. Thus lateral interactions, absent in the site picture, cancel out in the ideal solution however, in the first version is a properly of the solid lattice, while in the second it is a properly of the adsorbed species. Both models attribute differences in adsorption behavior entirely to differences in adsorbate-solid interactions. Both present adsorption as a competition between solute and solvent. 

The behavior of an adsorbate on a single patch of size L has been represented by the familiar two-dimensional lattice gas model Hamiltonian with the added term resulting from the presence of a boundary field ... 

The entropy of mixing disoriented polymer and solvent may be obtained, according to the original assumptions pertaining to the lattice model, by subtracting Eq. (9) from (8). The result reduces to ... 

Monte Carlo simulation shows [8] that at a given instance the interface is rough on a molecular scale (see Fig. 2) this agrees well with results from molecular-dynamics studies performed with more realistic models [2,3]. When the particle densities are averaged parallel to the interface, i.e., over n and m, and over time, one obtains one-dimensional particle profiles/](/) and/2(l) = 1 — /](/) for the two solvents Si and S2, which are conveniently normalized to unity for a lattice that is completely filled with one species. Figure 3 shows two examples for such profiles. Note that the two solvents are to some extent soluble in each other, so that there is always a finite concentration of solvent 1 in the phase... 

Fig. 5.8. Comparison of chemically realistic input for the optimization procedure and results for the corresponding lattice model, using the two-bond and four-bond optimization procedure a mean length (L) (in lattice units), b mean angle (0), and c the mean reduced barrier (W)/kBT. From [32]...

In order to understand the thermodynamic issues associated with the nanocomposite formation, Vaia et al. have applied the mean-field statistical lattice model and found that conclusions based on the mean field theory agreed nicely with the experimental results [12,13]. The entropy loss associated with confinement of a polymer melt is not prohibited to nanocomposite formation because an entropy gain associated with the layer separation balances the entropy loss of polymer intercalation, resulting in a net entropy change near to zero. Thus, from the theoretical model, the outcome of nanocomposite formation via polymer melt intercalation depends on energetic factors, which may be determined from the surface energies of the polymer and OMLF. 

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