Self-avoiding chains

A self-avoiding chain (self-avoiding walk) drawn on a triangular lattice (from J.F, Renardy, 1971). The starting point of this chain is at O. 

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. 

Polymer chains at low concentrations in good solvents adopt more expanded confonnations tlian ideal Gaussian chains because of tire excluded-volume effects. A suitable description of expanded chains in a good solvent is provided by tire self-avoiding random walk model. Flory 1151 showed, using a mean field approximation, that tire root mean square of tire end-to-end distance of an expanded chain scales as... 

The basic features of folding can be understood in tenns of two fundamental equilibrium temperatures that detennine tire phases of tire system [7]. At sufficiently high temperatures (JcT greater tlian all tire attractive interactions) tire shape of tire polypeptide chain can be described as a random coil and hence its behaviour is tire same as a self-avoiding walk. As tire temperature is lowered one expects a transition at7 = Tq to a compact phase. This transition is very much in tire spirit of tire collapse transition familiar in tire theory of homopolymers [10]. The number of compact... 

Simplified models for proteins are being used to predict their stmcture and the folding process. One is the lattice model where proteins are represented as self-avoiding flexible chains on lattices, and the lattice sites are occupied by the different residues (29). When only hydrophobic interactions are considered and the residues are either hydrophobic or hydrophilic, simulations have shown that, as in proteins, the stmctures with optimum energy are compact and few in number. An additional component, hydrogen bonding, has to be invoked to obtain stmctures similar to the secondary stmctures observed in nature (30). 

Let us consider a simple self-avoiding walk (SAW) on a lattice. The net interaction of solvent-solvent, chain-solvent and chain-chain is summarized in the excluded volume between the monomers. The empty lattice sites then represent the solvent. In order to fulfill the excluded volume requirement each lattice site can be occupied only once. Since this is the only requirement, each available conformation of an A-step walk has the same probability. If we fix the first step, then each new step is taken with probability q— 1), where q is the coordination number of the lattice ( = 4 for a square lattice, = 6 for a simple cubic lattice, etc.). 

The large-scale structure of polymer chains in a good solvent is that of a self-avoiding random walk (SAW), but in melts it is that of a random walk (RW).11 The large-scale structure of these mathematical models, however, is... 

Self-avoiding random walks (SARW) statistics has been proposed [1] for single that is for non-interacting between themselves ideal polymeric chains (free-articulated Kuhn s chains [2]) into ideal solvents, in which the all-possible configurations of the polymeric chain are energetically equal. From this statistics follows, that under the absence of external forces the conformation of a polymeric chain takes the shape of the Flory ball, the most verisimilar radius Rf of which is described by known expression [3, 4]... 

Condition of the self-avoiding RW trajectories absence on the d-dimensional lattice demands the circumstance at which more than one link of the chain can not be stood in every cell. Links of the chain are inseparable they cannot be divided one from another and located into the cells in random order. Thereby, number of different methods of mN differing links location per Z identical cells under condition that in every cell more than one link of the chain cannot be stood is equal to Z / (Z-mN) . 

Thereby, F represents by itself a free energy of random walks independent on the conformational state of a chain F(x) brings a positive contribution into F and the sense of this consists in a fact that the terms F(x) and S(x) represent the limitations imposed on the trajectories of random walk by request of the self-avoiding absence. These limitations form the self-organization effect of the polymeric chain the conformation of polymeric chain is the statistical form of its self-organization. 

Self-avoiding random walks statistics for intertwining polymeric chains and based on it thermodynamics of their conformational state in m-ball permitted to obtain the theoretical expressions for elasticity modules and main tensions appearing at the equilibrium deformation of /n-ball. Calculations on the basis of these theoretical expressions without empirical adjusting parameters are in good agreement with the experimental data. 

A self-avoiding walk on a lattice is a random walk subject to the condition that no lattice site may be visited more than once in the walk. Self-avoiding walks were first introduced as models of polymer chains which took into account in a realistic manner the excluded volume effect1 (i.e., the fact that no element of space can be occupied more than once by the polymer chain). Although the mathematical problem of... 

The graphs in Eq. (45) are the same as occur in the susceptibility coefficients dn, but each graph is now multiplied by a mean-square length factor. Taking for the self-avoiding walk approximation simple chains C , and using the results of Section IV-C we find... 

C. Domb and F. T. Hioe, to be published J. Chem. Phys. (in press) F. T. Hioe Self-Avoiding Walk Model of a Polymer Chain, Thesis, University of London,... 

III, Stochastic Model for Restricted Self-Avoiding Chains with Nearest-Neighbor... 

In his paper Domb presents a detailed analysis of the statistical properties of self-avoiding walks on lattices.1 These walks serve as models for linear polymer chains with hard-core intramolecular interactions associated with the exclusion of multiple occupancies of the lattice sites by the chain so-called chains with excluded volume. 

Self-avoiding walks with nearest-neighbor interactions of attraction (corresponding to negative values of e), are of particular interest to us. As a consequence of the presence of the forces of attraction there is a possibility of configurational transition in the chain of the kind encountered... 

HI. STOCHASTIC MODEL FOR RESTRICTED SELF-AVOIDING CHAINS WITH NEAREST-NEIGHBOR INTERACTIONS... 

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