Grooving of a Bicrystal by Surface Diffusion

Derivation of the Evolution Equation

We consider a surface with only one non-zero principal curvature, given by Kappa. The chemical potential of the surface is given by:

chemical potential equation (1)

(Herring, 1951) where gamma(theta) is the surface tension along the crystalline orientation theta and omega is the atomic volume. The chemical potential at zero curvature is given by mu0 and that lattice vacancies are in equilibrium and thus contribute no chemical potential.

The flux, J, due to surface diffusion is given by the Nerst-Einstein relation

Nerst - Einstein relation (2)

which, after substituting (1) becomes:

after substitution equation. (3)

Here, Ds(theta) is the surface diffusion and s is the arc length. The first set of symbols are c0, the number of atoms per unit area in one monolayer, k, the Boltzmann constant and T the temperature, are all considered constant.

Taking the negative divergence of J and multiplying by the atomic volume the normal velocity due to surface diffusion is obtained.

normal velocity equation (4)

If the surface profile is denoted by y(x,t) then equation (4) in Cartesian coordinates becomes

surface profile equation (5)

Here theta is chosen to be the angle from the y axis to the normal of the surface and can thus be given as theta equation.





Derivation of the Evolution Equation

Isotropic Materials

New Constitutive Functions

Symmetry Reductions

The Symmetric Grain Boundary Groove Problem