A Simplified Lattice Model

Introduction 53. 2 Strictly Regular Solutions 55 3. Quasi-Chemical Approximation 59. 4 General Remarks concerning the Order-Disorder Problem 64 5 Mole- 

It is assumed in this model that only depends on the composition of the system but that is unaffected by the mixing. Later considerations will however invalidate this assumption. 

Neither shall we take account of the thermal expansion of the lattice. Here again we shall see later that this assumption is not correct except at very low temperatures and that the changes in lattice distances due to changes in composition give important contributions to the excess functions. 

Assumptions (a)-(d) are the basic assumptions of what is called by Guggenheim a strictly regular solution ). 

We shall first consider molecules of the same size ( 2-4) and then the statistical theory of pol miers ( 5-7) 

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The interface between two immiscible liquids may be considered to be a surface solution of surfactant in a special kind of solvent. In order to calculate the entropy of such a solution, we will adopt a simplified lattice model and use lattice statistics, a widely used method for describing surface solutions. The transition from three-dimensional (3-d) to two-dimensional (2-d) geometry may cause errors in statistical formulas, if some peculiarities of 2-d solutions are overlooked. 

The current understanding of the protein folding process has benefited much from studies that focus on computer simulations of simplified lattice models. These studies try to construct as simple a model as possible that will capture some of the more important properties of the real polypeptide chain. Once such a model is defined it can be explored and studied at a level of detail that is hard to achieve with more realistic (and thus more complex) atomistic models. 

Fig 5.6 gives a simplified idea on the geometry of a square lattice model reflecting an octahedral coordination. 

On a lattice, so-called crankshaft moves are trivial implementations of concerted rotations [77]. They have been generalized to the off-lattice case [78] for a simplified protein model. For concerted rotation algorithms that allow conformational changes in the entire stretch, a discrete space of solutions arises when the number of constraints is exactly matched to the available degrees of freedom. The much-cited work by Go and Scheraga [79] formulates the loop-closure problem as a set of algebraic equations for six unknowns reducible... 

In theory, it would also be necessary to take into accormt the movement of all the other electrons. However, for the sake of simplicity, we will adopt a ID lattice model, and we will suppose that the movement of the electron is not affected by the presence of other electrorrs. This simplified model, though, is a good means of describing the electronic properties that interest us in this book. This model will particrrlarly allow us to demonstrate the concept of the electron s effective mass. 

Kronig Penny Model. A simplified, onedimensional model of the crystal lattice, giving the essentials of the behaviour of electrons in a periodic potential. See also... 

A) Simplified microscopic model of a quartz crystal lattice B) Longitudinal effect C) Transverse effect Si " and 2 O refer to centers of gravity (circles) for charges associated with the two types of atoms, where the tetrahedral "Si04 structure has been projected onto a plane (as a hexagon)... 

Another advantage of MC is that it may be applied to systems for which F, = -V U cannot be calculated or is not meaningful. This is of particular importance for modeling self-assembling systems on a lattice. Attaining chemical equilibrium everywhere in the system requires frequent exchange of monomers between aggregates, which is possible for extremely simplified lattice models only. 

The main ideas and results of scaling theory can be demonstrated through a simplified mesoscopic model. Let us assume that the near-critical state (in the one-phase region) is a t/-dimensional ideal lattice gas of fluctuation clusters with a mesoscopic lattice spacing 2 and size L. Then the excess pressure, reads... 

Highly simplified models of protein structure embedded into low coordination lattices have been used for tertiary structure prediction for almost 20 years [65, 66, 75]. For example, Covell and Jemigan [64] enumerated all possible conformations of five small proteins restricted to fee and bcc lattices. They found that the nativelike conformation always has an energy within 2% of the lowest energy. Virtually simultaneously. Hinds and Levitt [28] used a diamond lattice model where a single lattice unit represents several residues. While such a representation cannot reproduce the geometric details of helices or P-sheets, the topology of native folds could be recovered with moderate accuracy. 

Methods treating a lipid as a particle are also applied in studies of the bending rigidity of membranes. These simplified methods constitute an intermediate bridging the gap between the simulation methods at the atomic level and the more simplified lattice model. In the lattice model it is assumed that atomic positions of lipids are restricted to discrete lattice points. Of course, this simplificatipn results in loss of realistic images for individual lipid mol ules, but it makes application of the formulae of the statistical mechanics more practical. Thus, the lattice model is important in the field of computational physics to calculate thermodynamic and physical properties. 

Before we leave this subject we wish to mention a suggestion made by Guggenheim [1948, 1949, 1952]. In. order to force agreement between the simplified lattice model and experimental data, he proposes to regard the quasi-chemical equation (3.3.14) as defining w. Then w may be considered as a temperature dependent parameter which Guggenheim calls the cooperative free energy. This parameter is then adjusted to the experimental data. 

This ows that the assumptions used in the simplified lattice model of Ch. Ill cannot be consistent. Indeed this model predicts in its zeroth approximation ( 2) a vanishing excess entropy (cf. 3.2.15), instead of an exce.ss entropy proportional to the excess free energy. This is clearly due to the assumption that the excess free energy is only related to changes in the lattice partition functions and that in spite of changes in the interaction, the excess volume is zero. 

The structural effects depend primarily on the configurational specific heat and its derivative with respect to temperature. This shows quite dearly why in a rigid lattice approach like that used in the simplified lattice model the structural effect vanishes. 

Fig. 5 shows data from a simulation of TIP4P water that is confined on both sides by a rhombohedral mercury crystal with (111) surface structure. Bosio et al. [135] deduce from their X-ray studies that a solid o-mercury lattice with a larger lattice constant in the z direction may be used as a good structural model for liquid mercury. Thus, the mercury phase was modeled as a rigid crystal in order to simplify the simulations. The surface of such a crystal shows rather low corrugation. 

While the locations of the spins are not random - indeed, the spins populate sites of a regular lattice - the interactions themselves are completely random. Frustration, too, has been retained. Thus, arguably, two of the three fundamental properties of real spin glass systems are satisfied. What remains to be seen, of course, is the extent to which this simplified model retains the overall physics. 

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