Grain growth in the initial deposit

This equation is derived by considering the transfer of material from a flat surface to a droplet. For the transfer of a small mass Sm from the flat surface of vapour pressure p° to the droplet of vapour pressure p, the Gibbs energy of transfer is 

The addition of this small mass increases the radius of the droplet, r, to r + dr, and the surface area by 8jrr dr. If the fluid density is p, then the 

The term f rin p/p° is clearly the chemical potential of a surface of radius r with respect to a flat surface of the same material as standard state. It follows that the difference in chemical potential between two surfaces, p1 — p11, where surface I is convex of radius n, and the other surface II is concave of radius r2 is given by 

The Einstein equation for the flux of atoms across the interface is 

The mobility, M, is a function of the substance, and obeys an Arrhenius expression M = Mo exp — (AE/RT) where AE is the energy of transfer across the grain boundary. 

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