Self-avoiding random walks

The simplest model of polymers comprises random and self-avoiding walks on lattices [11,45,46]. These models are used in analytical studies [2,4], in particular in the numerical implementation of the self-consistent field theory [4] and in studies of adsorption of polymers [35,47-50] and melts confined between walls [24,51,52]. 

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 generalization of this scaling approach to the case of the good solvent is straightforward by replacing the relation between the blob size and the number of monomers in it ge from random to self-avoiding random walk statistics [26-31]. 

Eq. (3.1) describes the diffusive behavior of a chain (i.e., the movement of the center of mass) as well as its conformational rearrangements as a function of time. The equation is stochastic because the chain performs Brownian motion, and it has many different conformations which all have the same probability. The monomer-monomer interactions are described by the Fj. We will assume here that there are no long-range interactions present (in marked contrast to the case of polyelectrolytes ) and that hence the chain s structure is a random or self-avoiding walk. Motion in three-dimensional space is assumed throughout. The diffusion tensor Dy specifies the dynamics. Mathematical consistency of eq. (3.1) requires that Dy is symmetric and positive-definite for all possible golymer conformations (no other property is necessary). In the Rouse case, Dy is simply diagonal. 

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... 

A. Static Methods self-avoiding random walks... 

A. Static Methods Self-avoiding Random Walks... 

A. B. Harris. Self-avoiding walks on random lattices. Z Phys B 49 347-349, 1983. 

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... 

Conformation and Deformation of Linear Macromolecules in Concentrated Solutions and Melts in the Self-Avoiding Random Walks Statistics... 

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]... 

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. 

In presented work the analysis of osmotic pressure of the polymeric solutions has been done with taken into account the thermodynamics of conformation state of macromolecules following from the self-avoiding random walks statistics [13, 14],... 

Medvedevskikh Yu. G. Conformation and deformation of linear macromolecules in concentrated solutions and melts in the self-avoiding random walks statistics (see paper in presented book)... 

For a semi-flexible tube in a dilute environment, local repulsive potentials among parts of the fiber induce a self-avoiding random walk configuration (swollen coil [151]). In a crowded environment, the depletive action may dominate and the fiber will tend to collapse on itself, forming a globular phase. We know from standard statistical physics of polymers that this latter phase... 

II. Random Walks, Restricted Walks, and Self-Avoiding Walks.230... 

H. RANDOM WALKS, RESTRICTED WALKS, AND SELF-AVOIDING WALKS... 

Monte Carlo evidence confirming the power was provided subsequently by Gans,9 who introduced a new technique for overcoming attrition and generating long walks. As a result of a statistical analysis of a quarter of a million self-avoiding random walks on the diamond lattice,... 

Passage from a random to a self-avoiding walk and Ising analogs... 

Normal random walks Restricted walks (order r) Self-avoiding walks Ising analog... 

The whole construction procedure resembles a non-lattice-like self-avoiding random walk used in many MC- and MD-simulations. 

The fractal nature of monolayer diffusion fronts has been investigated both computationally [161] and analytically [164,165]. Findings for the case of a monolayer diffusing from a constant source row indicate a Df of exactly 1.75 for a self-avoiding random walk (H = 0) [165], Sapoval et al. [161] found that Df decreases below 1.75 when the equivalent of J/kBT in Eq. (1.26) exceeds a critical value of 1.76. Surely the minimum for Df is 1.0 as J/kBT —> oo. 

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