A dynamical model for fractal and compact growth in supercooled systems

Ho-Kei Chan,Ingo Dierking

doi:10.1088/2399-6528/ab8b33

Ho-Kei Chan, Ingo Dierking

Open Access

https://doi.org/10.1088/2399-6528/ab8b33

Copy DOI

Abstract

A dynamical model that can exhibit both fractal percolation growth and compact circular growth is presented. At any given cluster size, the dimension of a cluster growing on a two-dimensional square lattice depends on the ratio between the rates of two probabilistic processes, namely (i) the aggregation of lattice sites into the growing cluster and (ii) the relaxation of lattice sites into those available for potential aggregation. The proposed model approaches the limit of two-dimensional invasion percolation if the aggregation process is much faster than the relaxation process, and it approaches Eden’s model for compact circular growth if the relaxation process is much faster than the aggregation process. Experimental examples of the fractal-growth regime include the percolation-like growth of bent-core smectics and calamitic smectics, where such fractal growth is attributed to the slow relaxation of molecules in a viscous supercooled medium.

Highlights

Fractal geometry has been highly successful in describing complex patterns formed in a wealth of growth processes, with examples ranging from dielectric breakdown, colloidal aggregations, biological systems, and geology [1,2,3,4,5,6,7]
Experimental examples of the fractal-growth regime include the percolation-like growth of bent-core smectics and calamitic smectics, where such fractal growth is attributed to the slow relaxation of molecules in a viscous supercooled medium
The model approaches the limit of two-dimensional invasion percolation if the aggregation rate fa of the growing cluster is much higher than the relaxation rate fr of the medium, and it approaches Eden’s model for compact circular growth if the relaxation rate fr is much higher than the aggregation rate fa

Summary

April 2020

A dynamical model that can exhibit both fractal percolation growth and compact circular growth is. At any given cluster size, the dimension of a cluster growing on a two-dimensional square this work must maintain attribution to the lattice depends on the ratio between the rates of two probabilistic processes, namely (i) the aggregation author(s) and the title of the work, journal citation of lattice sites into the growing cluster and (ii) the relaxation of lattice sites into those available for and DOI. The proposed model approaches the limit of two-dimensional invasion percolation if the aggregation process is much faster than the relaxation process, and it approaches. Eden’s model for compact circular growth if the relaxation process is much faster than the aggregation process. Experimental examples of the fractal-growth regime include the percolation-like growth of bent-core smectics and calamitic smectics, where such fractal growth is attributed to the slow relaxation of molecules in a viscous supercooled medium

Introduction

Theoretical model and computational method

Simulation results

Experimental examples

Conclusions and future work