Diffusion-limited polymerization and surface growth.

Abstract

Simulations have been carried out to investigate the structures generated by a nonequilibrium Laplacian-random-walk model for the growth of polymer chains which was recently developed by Lyklema et al. Instead of solving the Laplace equation numerically to obtain the growth probabilities at the active end of the growing polymer chain, random walkers are used to simulate a random growth process controlled by a harmonic field. The results from two-dimensional simulations are in good agreement with those of Lyklema et al., Bradley and Kung, and Debierre and Turban. The growth of single chains on square lattices or cubic lattices leads to structures which can be described in terms of a single geometric scaling exponent (fractal dimensionality) for both absorbing and reflecting boundary conditions at the inactive (nongrowing chain sites). The main objective of this work was the investigation of models in which many chains grow from a line or plane of active sites to represent growth from a surface. For these models at least two effective exponents are needed to describe the structure of the system. The distribution of chain length can be described in terms of a power law (with an exponential cutoff) and the total density profile can also be described by a power law. The two-point density-density correlation function ${C}^{h}$(r) for those sites located at a distance h from the original surface of active sites can be described in terms of the scaling form ${C}^{h}$(r)\ensuremath{\sim}${h}^{\mathrm{\ensuremath{-}}\ensuremath{\mu}}$f(r/${h}^{\ensuremath{\omega}}$), where f(x) is a scaling function. These results indicate that the structure of the surface deposits grown by the models should be described in terms of a non-self-similar (possibly self-affine) fractal geometry.