It is, in general, intuitive that diffusion will enhance with increasing temperature. Here, we attempt to derive diffusivity as a function of temperature from atomic theories as applicable to the vacancy mechanism of diffusion. For diffusion of atoms to occur, there should be availability of vacant neighboring sites around the atom. If Г (as defined before) is proportional to the number of nearest neighbors (b) and the probability of finding neighboring lattice sites vacant or vacancy concen­tration (Cv), the following mathematical relation can be written:

Г = bC, rn, (2.39)

where v is the atomic jump frequency. Hence, the number of successful jumps by an atom can be given by

v — nD exp (—AGm/RT), (2.40)

where nD is the lattice vibration frequency (same as the Debye frequency, typically -1013 s—1, which is defined as the theoretical maximum frequency of vibration that make up the diffusion medium crystal), DGm is the free energy maximum (in calo­ries per mole) along the diffusion path or activation barrier to vacancy migration (also known as free energy for migration), and T is the temperature (in K).

From Eq. (2.35), we know D — (1/6)12Г. By replacing the Г expression using Eqs. (2.39) and (2.40), we obtain

1 2

D — — l2bCv VDexp(—DGm/RT). (2.41)

6

We know from earlier chapters

Cv = exp(— AGf/RT), where AGf is the free energy of vacancy formation (see Section 2.2).

Using the general relation from the second law of thermodynamics, AG = AH — TAS, we can write

Rearranging Eq. (2.42), we get

The above equation can also be written as

D = Do exp (—Qsd/rt) ;

1 D 1 l2o (ASf + AS,

where Do — 61 bvD ex^l——- r—-

D0 is called the frequency factor and Qsd is called the activation energy for self­diffusion. D0 and QsD can be obtained by measuring D at different temperatures from experiments. A plot of ln (D) versus 1/T yields a straight line and the slope equals —QJR and the intercept on the y-axis is ln (D0). It is interesting to note that the final diffusivity term does not contain the defect concentration term, rather it has the activation enthalpy for vacancy formation. The derivation is equally applica­ble for substitutional vacancy diffusion and interstitial diffusion mechanisms. On assigning approximate values to the terms in Eq. (2.45) and considering a small positive value of the entropy term, D0 is generally found to be between 10-3 and 10 cm2 s-1.