Analytical Solutions to the Problem of the Grain Groove Profile

Abstract

During the last sixty years, the problem of the formation of grain boundary grooving in polycrystalline thin films, was largely studied, analyzed and commented. The thermal effect on the properties of the grain boundary grooving was first studied by Mullins in his famous paper published in 1957 and then by other authors. This paper constitutes a new contribution on the correction of Mullins problem in the case of the evaporation-condensation and proposes a more accurate solution of the partial differential equation governing the geometric profile of the grain boundary grooving. The Mullins hypothesis neglecting the first derivative (|y'|≪1) in the main equation was defeated by our new solution. In this paper, we proved that the new proposed mathematical solution giving the solution y(x, t) is valid for all x values without any approximation on the first derivative y'.

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

“…𝑦(𝑥, 𝑡) = 𝑚 (𝐵𝑡) / 𝑔 𝑥 (𝐵𝑡) / 𝑦(𝑢, 𝑡) = 𝑚 (𝐵𝑡) / 𝑔(𝑢) (7) The different derivatives of 𝑦(𝑥, 𝑡) and 𝑢(𝑥, 𝑡) are given by Equation (8): The above partial differential equation cannot analytically be resolved without approximation. In the following sections, we resumed the essential results obtained in previous studies [61][62][63].…”

“…We proposed, in previous works [ 61 , 62 , 63 ], analytical solutions to the mathematical problem in the case of the evaporation–condensation in polycrystalline thin films by resolving the corresponding second non-linear partial differential equation [ 61 , 62 ] without any approximation when materials are submitted to thermal and mechanical stress and fatigue effects. One proved the non-validity of Mullins approximation that neglected the first derivative in the mathematical equation associated with the evaporation case.…”

New Solution of the Grain Boundary Grooving Problem in Polycrystalline Thin Films when Evaporation and Diffusion meet in Power Electronic Devices

“…The above partial differential equation cannot analytically be resolved without approximation in the case of both evaporation and diffusion processes. In the following sections, we resumed the essential results obtained in previous studies [51][52][53].…”

“…We proposed in previous works [51][52][53] analytical solutions to the mathematical problem in the case of the evaporation-condensation in polycrystalline thin films by resolving the corresponding second non-linear partial differential equation [51,52] without any approximation, when materials are submitted to thermal and mechanical stress, and fatigue effects. One proved the non-validity of Mullins approximation that neglected the first derivative in the mathematical equation associated to the evaporation case.…”

Determination of the time-dependent thermal grooving coefficient

“…As a practical application important in characterising the strength and stability of polycrystalline materials, we consider the study of a groove that forms when a vertical grain boundary meets a horizontal free surface. This particularly occurs in the thermal treatment and metallization of electronic components of power modules, [16]. As a mathematical model, we consider the fourth-order time-dependent Mullins partial differential equation governing the thermal grooving by surface diffusion [29,27], which is the characteristic mechanism for mass transport at special metal surfaces such as gold.…”