# Grooving of a grain boundary by evaporation-condensation processes: Surface evolution below the roughening transition.

Original Entry by Michelle Borkin, AP225 Fall 2009

## Contents

## Overview

H.A. Stone, M. Aziz and D. Margetis, J. Appl. Phys. 97, 113535-1-6, 2005

## Keywords

grain boundary, solid–vapor interface, evaporation–condensation, thermodynamic roughening

## Summary

This paper's focus is understanding and mathematically defining how grooves form at grain boundaries at the intersection of a planar surface. This is important in describing grain growth of thin films since the deepening of grain boundary grooves along the surface will set the boundaries resulting in slower grain growth. The specific condition studied here is how these grooves evolve when below the thermodynamic roughening transition by evaporation–condensation processes. Also, the case examined is a solid–vapor interface. It turns out the dynamics are governed by a nonlinear partial differential equation and the groove profile is defined by a nonlinear ordinary differential equation. This groove profile is important since the slope determines the driving force for how much grain growth is required for the boundary to break free of the groove. Both numerical and approximate analytical solutions are presented and agree with each other: the grooves' width and depth vary as <math>t^{1/2}</math> where <math>t</math> is time. Having such mathematical descriptions are useful for comparing to experiments in order to extract estimates for surface diffusion coefficients and determining features of the surface energy.

## Soft Matter

The theoretical description of this grain boundary evolution focuses on studying the surface free energy. (This has traditionally been tricky since there is a singularity in the surface free energy at the edge of the facet.) Defining the surface free energy of the solid–vapor interface as <math>\gamma</math> and the grain boundary free energy as <math>\gamma_{b}</math>, then the most energetically favorable position is for the groove to form at the grain boundary when <math>\gamma_{b} < 2\gamma</math> thus the angle <math>2\theta</math> is dictated by <math>\gamma_{b} / 2\gamma = cos(\theta)</math>.

The evolution of the groove over time below the thermodynamic roughening limit, where <math>h(t)</math> is the one-dimensional groove profile, is defined by first looking at the surface free energy (<math>G</math>) per unit area:

<math>G\left(\frac{\partial h}{\partial x}\right) = g_{0} + g_{1}\left|\frac{\partial h}{\partial x}\right| + \frac{1}{3}g_{3}\left|\frac{\partial h}{\partial x}\right|</math>

where <math>g_{0}</math> is the free energy of the terrace, <math>g_{1}</math> is the step line tension, and <math>g_{3}</math> is the strength of step–step interactions entropic repulsive interactions. Next, the chemical potential <math>\mu</math> is defined by:

<math>\mu - \mu_{0} = -2g_{3}\Omega\left|\frac{\partial h}{\partial x}\right|\frac{\partial^{2} h}{\partial x^{2}}</math>

where <math>\mu_{0}</math> is the chemical potential of the vapor and <math>\Omega</math> is the atomic volume. And for small deviations from equilibrium, the surface evolution is described by:

<math>\frac{\partial h}{\partial t} = -k\left|\frac{\partial h}{\partial x}\right|^{b}(\mu - \mu_{0})</math>

where k is a constant and b=1. Combining the above equations produces the nonlinear PDE that governs the grain growth boundary:

<math>\frac{\partial h}{\partial t} = \beta\left|\frac{\partial h}{\partial x}\right|^{\alpha}\frac{\partial^{2} h}{\partial x^{2}}</math>

where <math>\beta =2kg_{3}\Omega</math> and <math>\alpha = b + 1</math>.

The paper then goes onto examine in more detail the cases of <math>\alpha = 1</math> and <math>\alpha = 2</math>. A summary of the analytical results can be found in **Table 1**. Of important note is that in all cases the surface profile has the same scaling form proportional to
<math>t^{1/2}</math>.