Grain-boundary grooving by surface diffusion with asymmetric and strongly anisotropic surface energies

Abstract

Dirichlet extremum problem associated with the asymmetric grain-boundary thermal grooving under the Dirac δtype anisotropic surface stiffness in bicrystal thin solid films Grain-boundary migration controls grain growth, which is important in material processing and synthesis. When a vertical grain boundary ends at a horizontal free surface, a groove forms at the tip to reduce the combined grain-boundary and surface energies. The groove affects the migration of the grain boundary, and its effect must be understood. This work studies grain-boundary grooving by capillarity-driven surface diffusion with asymmetric and strongly anisotropic surface energies. The surface energies are described by the delta-function facet model that holds for temperatures above the roughening temperature of the bicrystal. Since the asymmetric anisotropy does not introduce a length scale, the asymmetric groove still grows with time t as t 1/4 . Thus, the nonlinear partial differential equation that governs grooving is reduced by a self-similar transformation to an ordinary differential equation, which is then solved numerically by shooting methods. We vary systematically the crystallographic orientations of the bicrystal and the normalized grain-boundary energy. We find that the self-similar groove profile is faceted and asymmetric for most bicrystals. However, a few asymmetric bicrystals can also yield smooth and symmetric grooves. Some bicrystals may even form ridges instead of grooves. We also find that the asymmetric surface energies tilt the grain-boundary tip sideways, which induces the grain boundary to migrate. We show that anisotropic groove profiles measured in SrTiO 3 and Ni can be well fitted by our model.

Cathode edge displacement by voiding coupled with grain boundary grooving in bamboo like metallic interconnects by surface drift-diffusion under the capillary and electromigration forces

“…Here, we should mentioned that we have also developed a wetting boundary condition (or interactive-BC) for the electrode-interconnect contact area using the irreversible thermodynamic similar to the GB-TJ formulation. Where, we have found that the cathode area shrinkage or the ridge formation is solely controlled by the surface Gibbs free energies at triple junction, exactly analog to the classical wetting theory, unless one has strong anisotropy, where the torque term might dominate the surface free energies and causes the ridge formation (Min and Wong, 2006).…”

Thermal grain-boundary grooving in bicrystal thin solid films having strong anisotropic surface Gibbs free energy represented by the modified cycloid-curtate function

“…Then, one may combine Eqs. (8) and (9) to obtain the following explicit exact expressions using the definition of the normalized time presented previously…”

“…Recently, Zhang et al [7] and Xin and Wong [8], prescribed an orientation dependent surface stiffness instead of the surface free energy explicitly in their treatments. Xin and Wong [8] and Min and Wong [9] while utilizing spike function for the faceting in their self-similar transformation theory, discovered very interesting effect of the anisotropy on the Mullins rate constant B measured directly from the Mullins profile function, which shows substantial amount of reduction, where the surface stiffness function shows quasiisotropic behavior, and the surface topography is very smooth (no faceting formation). Recently, Ogurtani [10] has reached exactly the same conclusion for the four-fold symmetry by applying special analysis on the surface Gibbs free energy function adapted from Ramasubramaniam and Shenoy [11] in addition to the verifications by the computer simulation experiments.…”

A tangent-plane marker-particle method for the computation of three-dimensional solid surfaces evolving by surface diffusion on a substrate

“…(2) is subject to the following boundary conditions. At the contact line x ¼ x c ðyÞ, the height h of the wire surface is zero, the contact angle is in equilibrium, and the mass flux tangent to the wire surface and into the substrate is zero [16,24,23]:…”