Growing interfaces in quenched disordered media

TL;DR

This paper derives microscopic and macroscopic equations for the growth of interfaces in quenched disordered media, specifically analyzing the Tang and Leschhorn model, and confirms the results with Monte Carlo simulations.

Contribution

It introduces a simple derivation of evolution equations for interface height, roughness, and activity density, highlighting the interplay of diffusion and substratum effects.

Findings

Analytical expression for the dynamical exponent β=0.629

Macroscopic roughness evolution equation at critical pressure p=0.461

Excellent agreement between theoretical results and Monte Carlo simulations

Abstract

We present the microscopic equation of growing interface with quenched noise for the Tang and Leschhorn model [{\em Phys. Rev.} {\bf A 45}, R8309 (1992)]. The evolution equations for the mean heigth and the roughness are reached in a simple way. Also, an equation for the interface activity density (i.e. interface density of free sites) as function of time is obtained. The microscopic equation allows us to express these equations in two contributions: the diffusion and the substratum one. All the equation shows the strong interplay between both contributions in the dynamics. A macroscopic evolution equation for the roughness is presented for this model for the critical pressure $p=0.461$. The dynamical exponent $\beta=0.629$ is analitically obtained in a simple way. Theoretical results are in excellent agreement with the Monte Carlo simulation.