Growth Inside a Corner: The Limiting Interface Shape

TL;DR

This paper studies the long-term shape of a crystal growing inside a corner by depositing cubes, deriving a governing equation, solving it analytically, and confirming results with simulations, also extending to higher dimensions.

Contribution

It introduces a conjectured evolution equation for the 3D interface shape and provides an analytical solution validated by simulations, extending understanding of crystal growth in corners.

Findings

Analytical solution matches simulation results.

Derived a governing equation for 3D interface evolution.

Generalized the model to arbitrary dimensions.

Abstract

We investigate the growth of a crystal that is built by depositing cubes onto the inside of a corner. The interface of this crystal evolves into a limiting shape in the long-time limit. Building on known results for the corresponding two-dimensional system and accounting for the symmetries of the three-dimensional problem, we conjecture a governing equation for the evolution of the interface profile. We solve this equation analytically and find excellent agreement with simulations of the growth process. We also present a generalization to arbitrary spatial dimension.