Several predictions stem from this model.

· D_b8 should increase linearly with misorientation, at least in the low angle region where the dislocation cores don't overlap. This is borne out by the data in Fig. 6-6.

· The activation energy for grain boundary diffusion Qb = Qp should be independent of 0, at least in the low angle region. (See Fig. 6-6.)

· D_b8 should not be isotropic in the boundary, but should be appre­ciably larger in the direction of the pipes than normal to them. As 0 increases, and the cores get closer together, the anisotropy should decrease. The data in Fig. 6-7 shows this. Also, the anisotropy per­sists even at very high angles where the dislocation model is no longer valid. Similar results have been demonstrated for other metals.

The data for silver in Fig. 6-6 and Eq. (6-9) lead to the equation D p 0.1 exp(-82,500/RT) cm²/s (6-10)

For the pipe diffusivity. This is consistent with the data for high angle

500°C

400*C

I I — -- —0— -- 0= 20° , a.. ---o--- 0=13°

—...9 ...CZ..... ......... "..........._b...... ·......."1_...... -..,,...■..... _

B= go

1.25

1.55

—13.5 —14.0 —14.5 —15.0 —15.5

1.35 1.45 1000/T

" '⁴'*......

. "■0......

'....._o. _ -......

"..Z.,■:_{. 50.}_ II_%

⁴⁴%^-:-.⁴.••• '⁴%

Fig. 6-6—Dependence of log(D_b5) on temperature and 6 for [100] tilt boundaries in Ag. [D. Turnbull, R. Hoffman, Acta Met., 2 (1954) 419.]

20—

0

O 10 20 30 40

8

Fig. 6-7--Anisotropy of DO on 6 for [100] tilt boundaries in Ag. [R. Hoffman, Acta Met., 4 (1956) 98.]

boundaries in Table 6-1. Using Eq. (6-10) and the data in Table 6-1 the ratio D_p/D₁ at T /T,, = 0.7, 0.5, and 0.3 are, 5 x 10⁵, 6 x 10⁷, and 6 x 10¹², respectively. These are typical of other metals.

At larger misorientations the dislocation model is no longer valid, and as 0 keeps increasing one ultimately rotates the two crystals through an angle corresponding to a symmetry operation and the boundary dis­appears. In between, the boundary will pass through coincidence ori­entations where the energy is lower, the lowest order coincidence being a coherent twin boundary in fcc metals. Fig. 6-8 shows such an ex­ample for diffusion parallel and perpendicular to the axis of rotation in [011] tilt boundaries in aluminum.

Table 6-1. High Angle Grain Boundary & Lattice D in Metals

		Doi	Dob	Qi	Qb
Metal	Struc	(cm^2/s)	(cm^2/s)	(Id/mol)	(kJ/mol)	Ref.
Ag	fcc	0.04	0.03
Au	fcc	0.04	0.03
Ni	fcc	0.92	0.07

Lattice data from Table 2-5 and 2-6. 8 assumed 3 x 10^-8 cm. 'D. Turnbull, R. Hoffman, Acta Met., 2 (1954) 419.

²D. Gupta, K. W. Asai, Thin Solid Films, 22 (1974) 121. ³A. R. Wazan, J. Appl. Phys., 36 (1965) 3596.

High Diffusivity Paths 199

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