Space-lattice 935 -models

DA Elmds, M Levitt. Exploring conformational space with a simple lattice model for protein structure. J Mol Biol 243 668-682, 1994. 

Lattice models have the advantage that a number of very clever Monte Carlo moves have been developed for lattice polymers, which do not always carry over to continuum models very easily. For example, Nelson et al. use an algorithm which attempts to move vacancies rather than monomers [120], and thus allows one to simulate the dense cores of micelles very efficiently. This concept cannot be applied to off-lattice models in a straightforward way. On the other hand, a number of problems cannot be treated adequately on a lattice, especially those related to molecular orientations and nematic order. For this reason, chain models in continuous space are attracting growing interest. 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

For more precise work, micro hot stage methods under a microscope are used. For all compds, except those which are isotropic or become so on heating, the mp can best be observed by means of a polarizing microscope, since the temp at which color disappears arid the space lattice is ruptured is the true mp. Among numerous models of micro-hot stages, the Kofler micro-hot stage has attained widespread use and is commercially available (Refs 3 4)... 

Kinetic gelation simulations seek to follow the reaction kinetics of monomers and growing chains in space and time using lattice models [43]. In one example, Bowen and Peppas [155] considered homopolymerization of tetrafunctional monomers, decay of initiator molecules, and motion of monomers in the lattice network. Extensive kinetic simulations such as this can provide information on how the structure of the gel and the conversion of monomer change during the course of gelation. Application of this type of model to polyacrylamide gels and comparison to experimental data has not been reported. 

In contrast to the lattice models discussed below, off-lattice models allow the chemical species under consideration to occupy in principle any position in space, so that important information concerning the relaxation and space distribution of the constituents of the system can be obtained. We discuss next some applications of these models to electrochemical problems. 

The main idea of a lattice model is to assume that atomic or molecular entities constituting the system occupy well-defined lattice sites in space. This method is sometimes employed in simulations with the grand canonical ensemble for the simulation of surface electrochemical proceses. The Hamiltonians H of the lattice gas for one and two adsorbed species from which the ttansition probabilities 11 can be calculated have been discussed by Brown et al. (1999). We discuss in some detail MC lattice model simulations applied to the electrochemical double layer and electrochemical formation and growth two-dimensional phases not addressed in the latter review. MC lattice models have also been applied recently to the study the electrox-idation of CO on metals and alloys (Koper et al., 1999), but for reasons of space we do not discuss this topic here. 

This chapter is concerned with the application of liquid state methods to the behavior of polymers at surfaces. The focus is on computer simulation and liquid state theories for the structure of continuous-space or off-lattice models of polymers near surfaces. The first computer simulations of off-lattice models of polymers at surfaces appeared in the late 1980s, and the first theory was reported in 1991. Since then there have been many theoretical and simulation studies on a number of polymer models using a variety of techniques. This chapter does not address or discuss the considerable body of literature on the adsorption of a single chain to a surface, the scaling behavior of polymers confined to narrow spaces, or self-consistent field theories and simulations of lattice models of polymers. The interested reader is instead guided to review articles [9-11] and books [12-15] that cover these topics. 

Unless one is willing to become involved in many intricacies, a lattice model with united atoms (segments) features segments which are all of equal size. The price we have to pay for this is that there is no unique way to convert from lattice units to real space coordinates. We will discuss this point in the Result sections in more detail. 

Figure 11.3 Arrangement of atoms in an ionic solid such as NaCl. (a) shows a cubic lattice with alternating Na+ and Cl- ions, (b) is a space-filling model of the same structure, in which the small spheres are Na+ ions, the larger Cl-. The structure is described as two interlocking face-centred cubic lattices of sodium and chlorine ions. 

Off-lattice models consider chains composed of interacting units in the free space. Single chains or simulation boxes containing many-chain systems can be investigated. Usually the solvent is only considered according to its quality effects in thermal systems. Therefore it is assumed to fill the remaining space act-... 

The changes in reorientation of surface atoms were explained using the dynamic model of the crystal space lattice. It was assumed that during anodic polarization, when the oxidation of adsorbed water is taking place, atoms oscillate mainly in a direction perpendicular to the electrode surface. This process leads to periodic separation of atoms in the first surface layer. Thus, the location of atoms in different orientations is possible. It was stated that various techniques of electrode pretreatment used for... 

As an example of the usefulness of molecular visualization, Figure 1 shows a 128-molecule sample of ordinary hexagonal ice. The molecules are drawn in the Space Filling model style, commonly known to provide a reasonable representation of the effective size of most molecules. Figure la shows the ideal lattice structure (T = O K). It can be seen that the structure is exceptionally open, with channels that permeate the entire lattice. Essentially, the picture provides a hands-on molecular illustration of the uniqueness of water (the density of the solid is so low that it actually floats on the liquid). 

Applying this criterion to. a lattice model, v corresponds roughly to the volume of a unit cell, and / to a lattice spacing. Hence the values of n 10 for exact enumeration quoted in Section IV seem quite reasonable. Certainly there is no support for the claim by Flory and Fisk31 that the 6/5 power law is attained only for n > 10.6... 

It is convenient for many purposes to have models available for inspection in order to realize fully the three-dimensional aspect of molecular and lattice structures. "Bafl-and-stick" models of various stages of sophistication are useful when it is necessary to be able to see through the structure under consideration. Space-filling models of atoms with both covalent and van der Waals radii are particularly helpful when steric effects are important. The space-filling models and the more sophisticaied stick models tend to be rather expensive, but there are several inexpensive modifications of the "ball-and-stick type available. It is extremely useful to have such a set at hand when considering molecular structures. 

In studying processes of accumulation of the Frenkel defects, one uses three different types of simple models the box, continuum, and discrete (lattice) models. In the simplest, box model, which was proposed first in [22], one studies the accumulation of complementary particles in boxes having a certain capacity, with walls impenetrable for diffusion of particles among the boxes. The continuum model treats respectively a continuous medium the intrinsic volume of similar defects at any point of the space is not bounded. In the model of a discrete medium a single cell (e.g., crystalline lattice site) cannot contain more than one defect (v or i). 

However, these results do not immediately carry over to the problems of interest here where (while PBCs are the norm) the ensembles are frequently open or constant pressure, and the systems do not fit in to the lattice model framework. Even in the apparently simple case of crystalline solids in NVT, the free translation of the center of mass introduces /-dependent phase space factors in the configurational integral which manifest themselves as additional finite-size corrections to the free energy these may not yet be fully understood [58, 97]. If one adopts the traditional stance, then, one is typically faced with having to make extrapolations of the free-energy densities in each of the two phases, without a secure understanding of the underlying form (jf . ..) of the corrections involved. 

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