1.01.7.4.1 Capillary approximation

Consider a spherical cavity of radius R with an inter­nal gas pressure of p.

To obtain the local equilibrium vacancy concen­tration, we take one atom from the cavity surface and place it in one of the vacancies in the crystal lattice next to the cavity surface. The cavity volume is thereby increased by one atomic volume, O, and its surface free energy is changed by

AFs(«) =Fs(« + 1) —Fs(«)

= (4я)1/3 (30п)2/3 [(1 + 1/п)2/3 — 1] g0 [105]

Here, g0 is the specific surface energy per unit area, assumed to be a constant, and n is the number of vacancies contained in the cavity. This number is related to the unrelaxed cavity volume by

4n 3

R3 = nfi [106]

3 1 J

When the cavity is large, that is, n ^ 1, we can use the approximation

1+n

and then obtain

AFs ffi 2у0(4я/3п)1/3(3П)2/3 ffi2R0O [107]

The growth of the cavity volume by O implies that the gas inside expands and thereby performs the work pO.

The overall change in Gibbs free energy is then

A G = -£Fv + TSfV + — O — pO — kT IndO R

Under local thermodynamic equilibrium, AG = 0, and we find that

= CV4exp{0R°—p)kT} [108]

The chemical potential ofvacancies at cavities is then

mV (r) = kT ‘n[cC/cVq] = —p) о [109]

This constitutes the well-known capillary approxi­mation, and it is often interpreted as the combined mechanical effect of surface tension and pressure. However, the above derivation shows that the surface tension 2g0/R is not a mechanical force, but a ther­modynamic or chemical force. The assumption that the surface tension acts like a mechanical pressure and generates a stress field whose radial component satisfies the boundary condition

Srr(R)=2R0 — p [110]

is incorrect. However, to clarify this point, we need to first introduce the correct definition of a surface stress.