Migration of a material interface

The total free energy of the system in this case is 

It follows immediately that the chemical potential for migration of the interface is 

If there is no internal or chemical energy change associated with the motion of the interface, it then tends to advance locally in the nj direction if X 0, thereby reducing system free energy the interface, however, tends to move in the opposite direction if y 0. It follows that y = 0 is a local equilibrium condition for the surface. Stable equilibrium is expected if x increases from zero with a perturbation in position in the nJ direction or if X decreases from zero with a perturbation in position in the —nJ direction. The equilibrium condition relates the local mean curvature k of the surface to energy densities according to 

Many conditions of this general type, relating surface curvature to energy densities, are commonly identified as a Gibbs-Thomson relationship. 

The spherically symmetric elasticity problem is readily analyzed to determine the elastic fields throughout the materials involved (Timoshenko and Goodier 1987) the details are left as an exercise. The value of k. on the spherical interface is —2/R. Substitution of the limiting values of the elastic fields on the interface into (8.22) yields 

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