Lattices models of polymers

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. 

Certain difficulties remain, however, with this approach. First, such an important feature as a secondary structure did not find its place in this theory. Second, the techniques of sequence design ensuring exact reproduction of the given conformation are well developed only for lattice models of polymers. The existing techniques for continuum models are complex, intricate, and inefficient. Yet another aspect of the problem is the necessity of reaching in some cases beyond the mean field approximation. The first steps in this direction were made in paper [84], where an analog of the Ginzburg number for the theory of heteropolymers was established. 

These two methods are different and are usually employed to calculate different properties. Molecular dynamics has a time-dependent component, and is better at calculating transport properties, such as viscosity, heat conductivity, and difftisivity. Monte Carlo methods do not contain information on kinetic energy. It is used more in the lattice model of polymers, protein stmcture conformation, and in the Gibbs ensemble for phase equilibrium. 

The relation of the two-parameter limit to lattice models of polymer configurations has been discussed in [BD79]. 

The computer simulations are likely to be useful in two distinct situations— the first in which numerical data of a specified accuracy are required, possibly for some utilitarian purpose the second, perhaps more fundamental, in providing guidance to the theoretician s intuition, e.g., by comparing numerical results with those from approximate analytical approaches. As a consequence, the physical content of the model will depend upon the purpose of the calculation. Our attention here will be focused largely on the coarse-grained (lattice and off-lattice) models of polymers. Naturally, these models should reflect those generic properties of polymers that are the result of the chainlike structure of macromolecules. 

R. A. Vaia and E. P. Giannelis, Lattice model of polymer melt intercalation in organically-modified layered silicates, Macromolecules 30, 7990-7999 (1997). 

A. Sariban and K. Binder (1988) Phase-Separation of polymer mixtures in the presence of solvent. Macromolecules 21, pp. 711-726 ibid. (1991) Spinodal decomposition of polymer mixtures - a Monte-Carlo simulation. 24, pp. 578-592 ibid. (1987) Critical properties of the Flory-Huggins lattice model of polymer mixtures. J. Chem. Phys. 86, pp. 5859-5873 ibid. (1988) Interaction effects on linear dimensions of polymer-chains in polymer mixtures. Makromol. Chem. 189, pp. 2357-2365... 

A qualitatively similar behavior is seen in the simulations (Fig. 33b, c). It is rather clear, that neither for the experiments nor for the simulation an analysis of the data in terms of a model of a strictly incompressible binary mixture is adequate [63, 80, 221, 281, 282]. In this context, we would draw attention to a recent simulation of an off-lattice model of polymer blends in the isothermal-isobaric ensemble [283] relying on the incremental chemical potential method [284 286]. 

A. Sariban, K. Binder, Critical properties of the Flory-Huggins lattice model of polymer mixtures. J. Chem. Phys. 86, 5859-5873 (1987)... 

Vaia, R. A. Giannelis, E. P., Lattice Model of Polymer Melt Intercalation in Organically-Modified Layered Silicates. Macromolecules 1997, 30, 7990-7999. Vaia, R. A. Giannelis, E. P., Polymer Melt Intercalation in Organically-Modified Layered Silicates Model Predictions and Experiment. Macromolecules 1997, 30, 8000-8009. 

The lattice models of polymers reach their limits when one wants to study phenomena related to hydrodynamic flow. Although study of how chains in polymer brushes are deformed by shear flow has been attempted, by modeling the effect of this simply by assuming a smaller monomer jump rate against the velocity field rather than along it [61], the validity of such nonequilibrium Monte Carlo procedures is still uncertain. However, for problems regarding chain configurations in equilibrium, thermodynamics of polymers with various chemical architectures, and even the diffusive relaxation in melts, lattice models still find useful applications. 

Statistical thermodynamic theories provide a powerful tool to bridge between the microscopic chemical structures and the macroscopic properties. Lattice models have been widely used to describe the solution systems (Prigogine 1957). Chang (1939) and Meyer (1939) reported the earliest work related with the lattice model of polymer solution. The lattice model was then successfully established by Flory (1941, 1942) and Huggins (1942) to deal with the solutions of flexible polymers by using a mean-field approximation, and to derive the well-known Flory-Huggins equation. 

In the lattice model of polymer solutions, polymer chain is simply represented by a number of consecutively occupied lattice sites, each site corresponding to one chain unit. The rest single sites are assigned to solvents. This simple lattice treatment of polymer solutions allows a very convenient way to calculate thermodynamic properties of flexible and semiflexible polymer solutions from the statistical thermodynamic approach. By the mean-field assumption, the entropy part and the enthalpy part of partition function can be separately calculated. 

Monte Carlo simulations can calculate the phase diagrams of polymer solutions in a different way. In the lattice model of polymer solutions, each step of micro-... 

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