Disorder, lattice model

After 1997, other pseudolattice approaches to equilibrium properties of electrolyte solutions deserve some comment. Moggia and Bianco (Moggia Bianco, 2007 Moggia, 2008) provided expressions for the activity and osmotic coefficients following a pseudolattice approach. The authors assumed that the solute ions evolve from a disordered lattice model within a continuous solvent at extremely dilute solutions to a disordered lattice of local arrangements of both solute ions and solvent dipoles at higher concentrations, and they were able to satisfactorily explain the thermodynamic properties of these systems. 

Micellar structure has been a subject of much discussion [104]. Early proposals for spherical [159] and lamellar [160] micelles may both have merit. A schematic of a spherical micelle and a unilamellar vesicle is shown in Fig. Xni-11. In addition to the most common spherical micelles, scattering and microscopy experiments have shown the existence of rodlike [161, 162], disklike [163], threadlike [132] and even quadmple-helix [164] structures. Lattice models (see Fig. XIII-12) by Leermakers and Scheutjens have confirmed and characterized the properties of spherical and membrane like micelles [165]. Similar analyses exist for micelles formed by diblock copolymers in a selective solvent [166]. Other shapes proposed include ellipsoidal [167] and a sphere-to-cylinder transition [168]. Fluorescence depolarization and NMR studies both point to a rather fluid micellar core consistent with the disorder implied by Fig. Xm-12. 

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. 

The example illustrates how Monte Carlo studies of lattice models can deal with questions which reach far beyond the sheer calculation of phase diagrams. The reason why our particular problem could be studied with such success Hes of course in the fact that it touches a rather fundamental aspect of the physics of amphiphilic systems—the interplay between structure and wetting behavior. In fact, the results should be universal and apply to all systems where structured, disordered phases coexist with non-struc-tured phases. It is this universal character of many issues in surfactant physics which makes these systems so attractive for theoretical physicists. 

FIG. 13 Phase diagram of a vector lattice model for a balanced ternary amphiphilic system in the temperature vs surfactant concentration plane. W -I- O denotes a region of coexistence between oil- and water-rich phases, D a disordered phase, Lj an ordered phase which consists of alternating oil, amphiphile, water, and again amphi-phile sheets, and L/r an incommensurate lamellar phase (not present in mean field calculations). The data points are based on simulations at various system sizes on an fee lattice. (From Matsen and Sullivan [182]. Copyright 1994 APS.)... 

Independently, Burger [231] develops analytical equations for lattice models without substitutional disorder. His results are special cases of the models presented by Ruland. 

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. 

Usually it is assumed that tc is the only temperature-dependent variable in Eq. 9. This might be the case for an order-disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square fluctuation amplitudes are reduced by the so-called depopulation factor (l - and become in this way temperature-dependent also. The temperature dependence of tc in this model is given by Eq. 8, i.e. r would be zero at Tc, proportional to (T - Tc) above Tc and proportional to (Tc - T) below Tc. 

Section IV is devoted to excitons in a disordered lattice. In the first subsection, restricted to the 2D radiant exciton, we study how the coherent emission is hampered by such disorder as thermal fluctuation, static disorder, or surface annihilation by surface-molecule photodimerization. A sharp transition is shown to take place between coherent emission at low temperature (or weak extended disorder) and incoherent emission of small excitonic coherence domains at high temperature (strong extended disorder). Whereas a mean-field theory correctly deals with the long-range forces involved in emission, these approximations are reviewed and tested on a simple model case the nondipolar triplet naphthalene exciton. The very strong disorder then makes the inclusion of aggregates in the theory compulsory. From all this study, our conclusion is that an effective-medium theory needs an effective interaction as well as an effective potential, as shown by the comparison of our theoretical results with exact numerical calculations, with very satisfactory agreement at all concentrations. Lastly, the 3D case of a dipolar exciton with disorder is discussed qualitatively. 

An intermediate case (Bazin et al., 2002) between bulk and surface diffraction is reached for nanoparticles when the contribution from surface atoms becomes significant and diffraction analysis in the limit of infinite periodic lattice models inadequately describes the diffraction data. A case study with diamond nanoparticles (Palosz et al., 2002) describes elegantly the possibilities and limitations of diffraction analysis of such samples there is a focus on the nonperiodic structure such as strain and disorder induced by the dominant presence of a nonideal surface termination. 

Ab initio density functional methods using pseudopotentials and plane wave basis set are naturally well-adapted to treat periodic lattice models of the zeolite catalyst. Although, in the light of the above discussion, it does not correspond entirely to the real situation of a zeolite framework with substitutional disorder, such a model still offers the most of the guarantee that none of the physico-chemically important interactions are missed by our calculations. 

In the current paper, we discuss some of the new approaches and results that have been developed and obtained recently within the context of such molecular modeling research, and in particular with the mean field and Monte Carlo studies of a lattice model. The next section describes the Gaussian random field method (Woo et al, 2001), which provides a computationally efficient route to generate realistic representations of the disordered mesoporous glasses. Application of the mean field theory, and Monte Carlo simulations are described in Secs. 3 and 4, respectively. 

As mentioned before, the disordered solids will be mostly modelled in this book using randomly diluted site or bond lattice models. A knowledge of percolation cluster statistics will therefore be necessary and widely employed. Although this lattice percolation kind of disorder will not be the only kind of disorder used to model such solids, as can be seen later in this book, the widely established results for percolation statistics have been employed successsfully to understand and formulate analytically various breakdown properties of disordered solids. We therefore give here a very brief introduction to the percolation theory. For details, see the book by Stauffer and Aharony (1992). 

The second thing we need to characterise is the type of disorder we shall consider. As mentioned before, a major part of this book will be concerned with the lattice models of disorder, with the statistics governed by that of random percolation. The simplest physical picture of this kind of disorder... 

In the discrete lattice model, discussed above, each bond is identical, having identical threshold values for its failure. In the laboratory simulation experiments (discussed in the previous section) on metal foils to model such systems, holes of fixed size are punched on lattice sites and the bonds between these hole sites are cut randomly. If, however, the holes are punched at arbitrary points (unlike at the lattice sites as discussed before), one gets a Swiss-cheese model of continuum percolation. For linear responses like the elastic modulus Y or the conductivity E of such continuum disordered systems, there are considerable differences (Halperin et al 1985) and the corresponding exponent values for continuum percolation are higher compared to those of discrete lattice systems (see Section 1.2.1 (g)). We discuss here the corresponding difference (Chakrabarti et al 1988) for the fracture exponent Tf. It is seen that the fracture exponent Tf for continuum percolation is considerably higher than that Tf for lattice percolation Tf = Tf 4- (1 -h x)/2, where x = 3/2 and 5/2 in d = 2 and 3 respectively. 

FIG. 3 Models for the structure of carbonaceous adsorbents, (a) "Disordered lattice. (From Ref. 71.) (b) Crumpled paper sheets. (From Refs. 72 and 73.)... 

Eustathopoulos has also summarized some theoretical calculations of a for binary alloys, which are based on lattice models in which the interface layers are treated in the Bragg-Williams approximation, assuming complete atomic disorder in the interface and in the crystal. The calculated surface free energies are related to the surface free energies of the pure components and to the activities in the bulk phases. 

Abstract The influence of randomly distributed impurities on liquid crystal (LC) orientational ordering is studied using a simple Lebwohl-Lasher t5q)e lattice model in two d=2) and three d=3) dimensions. The impurities of concentration p impose a random anisotropy field-type of disorder of strength w to the LC nematic phase. Orientational correlations can be well presented by a single coherence length for a weak enough w. We show that the Imry-Ma... 

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