Model lattice

The simplest model considers a regular lattice where each effective bead of the polymer takes a single lattice site, and a bond connecting two beads is just a nearest-neighbor link on the lattice (Fig. 1.4). ° Since each lattice site can at most be occupied by one bead, the walk cannot intersect itself (or other walks, respectively) and thus an excluded volume interaction is auto- 

For most static properties of single chains, the algorithm of choice is 

An algorithm that incorporates large nonlocal moves of bonds and works for dense polymer systems (even without any vacancies, / = 1) is the collective motion algorithm where one transports beads from kinks or chain ends along the chain contour to another position along the chain, for several chains simultaneously, so that in this way this rearrange- 

The lattice algorithm that is now most widely used for the simulation of many-chain systems is the bond fluctuation model (Fig. 1.5). It has been used to model the dynamics of both two-dimensional and three-dimensional polymer melts, the glass transition (see also Chapter 

To solve Eq. (4.86) we employ the Jacobi-Newton iteration technique, which proceeds iteratively in an alternating sequence of local and global minimization steps. Let be the local density at lattice site i in the A th local and the 1th global minimization step. A local estimate for the corresponding minimum value of fl is obtained via Newton s method [see Eq. (D.6)] that is, 

It is important to realize that throughout each local minimization cycle the densities at nearest-neighbor sites of site i represented by the set remain fixed at the initial values assigned to them at the beginning of the local cycle. The iterative solution of Eq. (D.14) is halted if [see Eq. (D.7)  

To initiate the iterative scheme, suitable starting solutions for Eq. (D.14) are obviously required. These solutions ai e provided by the morphologies M at T = 0 for which can be calculated analytically from the expressions for fl compiled in Table 4.1. 

Here wc work out cxpro.s.sions for the number A aa ( bb) of A-A (B-B) pairs. These pairs are directly connected sites, both of which are occupied by a molecule of species A (B). Likewise, expressions for Vab (s) and the total number of molecules of species A and B at the solid substrate, Aaw (s) and Abw (s), may be derived easily. The resulting expressions presented in Eqs. (4.118)-(4.121) contain terms that can be cast as 

20) as well as (D.22) permit us to write down the mean-field expressions 

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There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

In polymer solutions and blends, it becomes of interest to understand how the surface tension depends on the molecular weight (or number of repeat units, IV) of the macromolecule and on the polymer-solvent interactions through the interaction parameter, x- In terms of a Hory lattice model, x is given by the polymer and solvent interactions through... 

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. 

Of particular interest has been the study of the polymer configurations at the solid-liquid interface. Beginning with lattice theories, early models of polymer adsorption captured most of the features of adsorption such as the loop, train, and tail structures and the influence of the surface interaction parameter (see Refs. 57, 58, 62 for reviews of older theories). These lattice models have been expanded on in recent years using modem computational methods [63,64] and have allowed the calculation of equilibrium partitioning between a poly-... 

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. 

Fig. XIII-12. Lattice model of a spherical micelle. (From Ref. 160a.)...

An unexpected conclusion from this fonuulation, shown in various degrees of generality in 1970-71, is that for systems that lack tlie synunetry of simple lattice models the slope of the diameter of the coexistence curve... 

Harris J and Rice S A 1988 A lattice model of a supported monolayer of amphiphile molecules—Monte Carlo simulations J. Ohem. Phys. 88 1298-306... 

Fabbri U and Zannoni C 1986 A Monte Carlo investigation of the Lebwohl-Lasher lattice model in the vicinity of its orientational phase transition Mol. Phys. 58 763-88... 

A compact aud readable iutroductiou to Moute Carlo, with examples aud exercises, plus useful pointers to the literature on lattice models. 

Off-lattice models enjoy a growing popularity. Again, a particle corresponds to a small number of atomistic repeat units... 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

For structures with a high curvature (e.g., small micelles) or situations where orientational interactions become important (e.g., the gel phase of a membrane) lattice-based models might be inappropriate. Off-lattice models for amphiphiles, which are quite similar to their counterparts in polymeric systems, have been used to study the self-assembly into micelles [ ], or to explore the phase behaviour of Langmuir monolayers [ ] and bilayers. In those systems, various phases with a nematic ordering of the hydrophobic tails occur. 

As early as 1969, Wlieeler and Widom [73] fomuilated a simple lattice model to describe ternary mixtures. The bonds between lattice sites are conceived as particles. A bond between two positive spins corresponds to water, a bond between two negative spins corresponds to oil and a bond coimecting opposite spins is identified with an amphiphile. The contact between hydrophilic and hydrophobic units is made infinitely repulsive hence each lattice site is occupied by eitlier hydrophilic or hydrophobic units. These two states of a site are described by a spin variable s., which can take the values +1 and -1. Obviously, oil/water interfaces are always completely covered by amphiphilic molecules. The Hamiltonian of this Widom model takes the form... 

Lattice models have been studied in mean field approximation, by transfer matrix methods and Monte Carlo simulations. Much interest has focused on the occurrence of a microemulsion. Its location in the phase diagram between the oil-rich and the water-rich phases, its structure and its wetting properties have been explored [76]. Lattice models reproduce the reduction of the surface tension upon adsorption of the amphiphiles and the progression of phase equilibria upon increasmg the amphiphile concentration. Spatially periodic (lamellar) phases are also describable by lattice models. Flowever, the structure of the lattice can interfere with the properties of the periodic structures. 

Kremer K and Binder K 1988 Monte Carlo simulations of lattice models for macromolecules Comp. Phys. Rep. 7 259... 

Lattice models are particularly suited for answering the questions posed. We will show that the two physical restrictions are sufficient to rationalize the emergence of very limited (believed to be only of the order of a thousand... 

C2.5.3.4 EXPLORING THE PROTEIN FOLDING MECHANISM USING THE LATTICE MODEL... 

Using lattice models we have also established tliat folding rates correlate well witli Z = (E - / 5, where E- is... 

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