Fractal dimension and degree of order in sequential deposition of mixture

TL;DR

This paper models the sequential deposition of particle mixtures with power-law size distribution, analyzing the resulting patterns' fractal dimensions and order, including effects of correlation introduced by a tuning parameter.

Contribution

It introduces models for correlated and uncorrelated sequential deposition of particle mixtures, deriving the fractal dimension and pattern scale invariance in the long-term limit.

Findings

Pattern becomes scale invariant over time.

Fractal dimension increases with correlation parameter beta.

Pattern reaches a non-zero fractal dimension at high correlation.

Abstract

We present a number models describing the sequential deposition of a mixture of particles whose size distribution is determined by the power-law $p(x) \sim \alpha x^{\alpha-1}$, $x\leq l$ . We explicitly obtain the scaling function in the case of random sequential adsorption (RSA) and show that the pattern created in the long time limit becomes scale invariant. This pattern can be described by an unique exponent, the fractal dimension. In addition, we introduce an external tuning parameter beta to describe the correlated sequential deposition of a mixture of particles where the degree of correlation is determined by beta, while beta=0 corresponds to random sequential deposition of mixture. We show that the fractal dimension of the resulting pattern increases as beta increases and reaches a constant non-zero value in the limit $\beta \to \infty$ when the pattern becomes perfectly ordered or non-random fractals.