# A New Solution to the Grain Boundary Grooving Problem in Polycrystalline Thin Films When Evaporation and Diffusion Meet in Power Electronic Devices

## TL;DR

This paper presents a new mathematical solution for understanding how grain boundaries in thin films change shape when evaporation and diffusion happen together.

## Contribution

A novel analytical solution for the grain boundary grooving problem combining evaporation and diffusion effects is proposed.

## Key findings

- A fourth-order partial differential equation was solved to model the groove profile under combined evaporation and diffusion.
- The coupling of evaporation and diffusion significantly affects the geometric characteristics of the groove profile.
- Boundary conditions were shown to influence the groove profile's characteristics.

## Abstract

This paper constituted an extension of two previous studies concerning the mathematical development of the grain boundary grooving in polycrystalline thin films in the cases of evaporation/condensation and diffusion taken separately. The thermal grooving processes are deeply controlled by the various mass transfer mechanisms of evaporation–condensation, surface diffusion, lattice diffusion, and grain boundary diffusion. This study proposed a new original analytical solution to the mathematical problem governing the grain groove profile in the case of simultaneous effects of evaporation–condensation and diffusion in polycrystalline thin films by resolving the corresponding fourth-order partial differential equation ∂y∂t=C∂2y∂x2−B∂4y∂x4 obtained from the approximation ∂y∂x2≪1. The comparison of the new solution to that of diffusion alone proved an important effect of the coupling of evaporation and diffusion on the geometric characteristics of the groove profile. A second analytical solution based on the series development was also proposed. It was proved that changes in the boundary conditions of the grain grooving profile largely affected the different geometric characteristics of the groove profile.