Self-avoiding Lévy walk: A model for very stiff polymers

Non-Gaussian polymers described by alpha-stable chain statistics: Model, effective interactions in binary mixtures, and application to on-surface separation

The Gaussian chain model is the classical description of a polymeric chain, which provides analytical results regarding end-to-end distance, the distribution of segments around the mass center of a chain, coarse-grained interactions between two chains and effective interactions in binary mixtures. This hierarchy of results can be calculated thanks to the α stability of the Gaussian distribution. In this paper we show that it is possible to generalize the model of Gaussian chain to the entire class of α-stable distributions, obtaining the analogous hierarchy of results expressed by the analytical closed-form formulas in the Fourier space. This allows us to establish the α-stable chain model. We begin with reviewing the applications of Levy flights in the context of polymer sciences, which include: chains described by the heavy-tailed distributions of persistence length; polymers adsorbed to the surface; and the chains driven by a noise with power-law spatial correlations. Further, we derive the distribution of segments around the mass center of the α-stable chain and construct the coarse-grained interaction potential between two chains. These results are employed to discuss the model of binary mixture consisting of the α-stable chains. In what follows, we establish the spinodal decomposition condition generalized to the mixtures of the α-stable polymers. This condition is further applied to compare the on-surface phase separation of adsorbed polymers (which are known to be described with heavy-tailed statistics) with the phase separation condition in the bulk. Finally, we predict the four different scenarios of simultaneous mixing and demixing in the two- and three-dimensional systems.

Position and Orientation Distributions for Locally Self-Avoiding Walks in the Presence of Obstacles

This paper presents a new approach to study the statistics of lattice random walks in the presence of obstacles and local self-avoidance constraints (excluded volume). By excluding sequentially local interactions within a window that slides along the chain, we obtain an upper bound on the number of self-avoiding walks (SAWs) that terminate at each possible position and orientation. Furthermore we develop a technique to include the effects of obstacles. Thus our model is a more realistic approximation of a polymer chain than that of a simple lattice random walk, and it is more computationally tractable than enumeration of obstacle-avoiding SAWs. Our approach is based on the method of the lattice-motion-group convolution. We develop these techniques theoretically and present numerical results for 2-D and 3-D lattices (square, hexagonal, cubic and tetrahedral/diamond). We present numerical results that show how the connectivity constant mu changes with the length of each self-avoiding window and the total length of the chain. Quantities such as R and others such as the probability of ring closure are calculated and compared with results obtained in the literature for the simple random walk case.