When the crystal lattice is under pressure p, the self­diffusion coefficient changes and is then given by

DSD(T, p) —DSDexp(_QSD/kT) exp(_pVSD/kT) [24]

The activation volume VsD can be obtained experi­mentally by measuring the self-diffusion coefficient as a function of an externally applied pressure. Such measurements have been carried out only for a few

 

0 0 0.5 1 1.5 2 Experimental vacancy migration energy (eV) Figure 10 Comparison of computed vacancy migration energies according to models by Flynn and Kornblit with measured values.

 

Table 5 Preexponentials for tracer self-diffusion Metal M a (nm) &d(K) E m(eV Experimental value S&M Flynn (m2 s 1) Ag 107.9 0.409 229 0.66 4.5e-6 3.00e-6 5.90e-6 Al 26.98 0.405 430.6 0.61 4.7e-6 5.69e-6 1.08e-6 Au 197 0.408 162.7 0.71 3.5e-6 2.29e-6 4.16e-6 Cu 63.54 0.361 349.6 0.70 1.6e-5 3.55e-6 7.02e-6 Ni 58.71 0.352 481.4 1.04 9.2e-5 4.39e-6 9.18e-6 Pb 207.2 0.495 106.6 0.43 6.65e-5 2.11e-6 4.01e-6 Pd 106.4 0.389 278 1.0 2.1e-5 3.53e-6 6.46e-6 Pt 195.1 0.392 240 1.4 6.0e-6 3.11e-6 5.66e-6

СЛ

10-5

c

0

C

О

CL

X

0

0

Q.

13

о

0

10-6

10-6 10-5 0.0001

Experimental preexponential D0 (m2s-1)

Figure 11 Comparison of preexponential factors for tracer self-diffusion as computed with two models and as measured.

metals, and it has been found that the activation volumes have positive values. Therefore, self-diffusion decreases with applied pressure. However, it has been noticed that the self-diffusion coefficient at melting appears to be constant, and this can be explained by the fact that the melting temperature increases in general with pressure. It follows then from the condition

d [ln DSD (p, Tm (p))] /dp|p=0 = 0

that

Q

Tm dp p=0

where T0 is the melting temperature under ambient conditions.

Brown and Ashby16 have used this relationship to evaluate the activation volumes for self-diffusion

for a variety of metals. Using more recent values for the pressure derivative of the melting tempera­ture by Wallace17 and Wang et a/.,18 one obtains activation volumes as shown in Figure 12. They are in reasonably good agreement with the experi­mental values where they exist. With the exception of Pt, the predicted values are also similar, giving an activation volume of about 0.85O for fcc metals, 0.65O for hcp metals, and around 0.4O for bcc metals.

The equilibrium vacancy concentration in a solid under pressure p is given by

-ref

bT where VV is the vacancy formation volume. Since the self-diffusion coefficient is the product of the thermal vacancy concentration and the vacancy migra­tion coefficient, the activation volume for self-diffusion is the sum of two contributions, namely

Vsd = vf + vm = O + V^el + vm [27]

with V™ being the activation volume for vacancy migration.

If one takes the average of the predicted activation volumes shown in Figure 12, and the vacancy relax­ation volumes from Table 3, one obtains values for V™ listed in Table 6 and also shown in Figure 12.