Outside the grain boundary, diffusion would obey the equation

ac_t/at = D₁V²c₁ (6-4)

The problem thus becomes one of determining the solution c(x,y,t) which will simultaneously satisfy these differential equations in the respective regions and be continuous across the boundary between the slab and the grain, that is

Solutions. Experimentally, solute is applied to the free surface at y = 0 and allowed to diffuse into the sample. Solutions have been ob­tained for the boundary conditions of either constant surface compo­sition, or the application of a thin film to the surface. If Db is much greater than D₁ the solute penetration around the grain boundaries will be deeper than through the lattice. It is the distribution of this material which diffused along the boundary to a depth well beneath the layer entering through the lattice that allows the determination of Db. Work­ing with the case of constant surface composition, Fisher found that the concentration in the grain boundary rises quickly at first but then at an ever-decreasing rate. Thus the grain boundary concentration at any point on the boundary will be near its final value during much of the anneal. To simplify the analysis he assumed that the boundary composition at each point stays at its final value throughout the ex­periment and that the flux in the lattice is perpendicular to the bound­ary. This yields a simple, approximate solution for the solute that en­ters the lattice via the boundary. The same type of assumptions have been used by others in considering other geometries.

The first exact solution, and the one most frequently used, is due to Whipple. He also assumed a constant surface composition, and used a Fourier-Laplace transform to obtain a solution in integral form.' Nu­merical analysis indicated' that beneath the surface layer due to lattice diffusion where dln(C)/ d(y²) is constant, Whipple's solution gives a region where the slope din(c)/d(y⁶¹⁵) is constant and D_b can be ob­tained from the equation⁵

fdln(C)/ cl(y⁶¹⁵)1⁵¹³ = 0.66(D₁/0^1/2 (1/8D_b) (6-6)

Note that one can determine only the product DA not Db alone. This equation is valid only in the region of y greater than, say, 4\D _It, where the tracer present has entered the crystal by diffusing in along the boundary and then out into the crystal from the grain boundary. If such a region is to develop, Db must be much greater than D₁. Whether or not it develops is indicated by the parameter

p = (D_b/D,)(5/21/D,t) (6-7)

which should be 10 or greater for Eq. (6-6) to be used. Figure 6-4 is adapted from Whipple and shows contours for a concentration 0.2 times the surface concentration, for = 0.1, 1.0, and 10. Note that there is no significant extra penetration along the grain boundary until p

³R. T. Whipple, Phil. Mag., 45 (1954) 1225.

'H. Levine, C. MacCallum, J. Appl. Phys., 31 (1960) 595.

⁵See N. L. Peterson, in Grain Boundary Structure and Kinetics, ASM, Metals Park, OH (1980), pp. 209-37.

X

y			p = 0.1 i3 = to 0 =10.0

Fig. 6-4 — Iso-concentration contours showing the degree of extra penetration near grain boundary for various values of /3. y = 1 is the penetration in the absence of a grain boundary. (R. T. Whipple)

1. If 8 = 4 x 10^-8 cm, and D₁ = 10^-" cm²/s, then t = 10⁵ s (------28 hr). This means that D_b/D_i must be greater than 5 x 10⁴ before there is appreciable penetration at the boundaries. Physically the reason for this is that the grain boundary slab is so thin that the grain boundary flux is not sufficient to bring in enough material to distort the contours until D_b/D_i > 5 x 10⁴.

Measurement of Db. If Db is much greater than D_I, solute will dif­fuse along the boundary to a greater depth than in the lattice before being drained off into the grains by lattice diffusion. Two methods for determining Db have been used extensively:

· measure the distribution of in-diffused solute in a series of thin slices cut parallel to the sample surface, c(y), and use Eq. (6-6) to give D_b8.

· measure the depth of penetration of a given concentration at the boundary Ay compared to the lattice penetration from the surface far from the boundary.

The second technique is useful for comparing the relative DO of different boundaries. The first is used for accurate determination of

Experimentally one invariably applies a thin film of tracer to the surface rather than maintaining a constant surface concentration as as­sumed in Whipple's solution. Suzuoka found the solution for the thin film case.⁶ The solution differs from that of Whipple, but the slope of clinC / dy^{6 I 5} is nearly the same. If /3 10 the equation for determining DO given by Suzuoka's solution results only in the constant 0.66 in Eq. (6-6) being decreased by a few percent.

A thin-film tracer experiment in a crystal free of grain boundary effects produces a penetration curve with In (c) vs y² being a straight line. If grain boundary diffusion makes a significant contribution, the tracer penetrates much deeper, the penetration curve drops more slowly and gives a line for in(C) vs ^y6"5. Thus the penetration curve consists of a sum of two terms and can be described by the equation

c(y,t) = A₁ exp(—y²/4Dt) + A_ll exp(—y^6/5/b) (6-8)

where A₁, A_ll, and b are known constants. If ln(C) drops more slowly than y², this is a clear indication that some high diffusivity path is operating in addition to lattice diffusion. Fig. 6-5 shows a penetration curve for a gold tracer diffused into a fine grained thin film of gold

⁶T. Suzuoka, J. Phys. Soc. Japan, 19 (1964) 839.

PENETRATION DISTANCE Y(I0^-4Cm)

1.0 2.0 3.0 4.0 5.0 6.0

T ^-r i I ^---1^{- -}r^{- —}I ^-1- 1:

Au¹⁹⁵IN Au-I.2 Ta = ALLOY -

TEMP 253.0°C -

TIME 7.776x10⁵sec

ID

rt (ID^-ccret)

D

80₁₃-5.10 10⁹cm³/sec

1

20 4.0 &0 8.0 10.0 12.0 140

Y^6/5 00^-5CM^6/5)

Fig. 6-5 — Penetration plot for Au* into polycrystalline Au. A three term exponential fitting procedure yields three lines, and values of D for diffusion in the lattice (inset, I), sub-grains (II) and grain boundaries (III). [D. Gupta, J. Appl. Phys., 44 (1973) 4455.]

at 253° C (0.4 T_m). The data has been fit to an equation of the form of Eq. (6-8) but with an additional y⁶¹⁵ term. The small inset shows that the data points closest to the surface fit ln(C) vs y². The main graph is a plot of 11'0) vs. y⁶/⁵ with two straight lines, labeled II and III. The one main line (III) is due to diffusion along grain boundaries, and clearly penetrates quite deeply.

Eqs. (6-5) and (6-6) are only valid for bicrystals, or polycrystal-line samples for which the grain diameter 2R is much greater than the mean diffusion distance in the lattice 2N/D_it. If R is less than or equal to Vti_lt, then different effects are seen. These are discussed in Sec. 6­3 along with the effect of arrays of dislocations.

Before leaving this analysis it should be pointed out that the math­ematical analysis given above is also applicable to the case of surface diffusion. The slab in Fig. 6-2 is a plane of symmetry so there will be no net flux across it. Thus if the half of the bicrystal to the left of the slab is removed, the high-diffusivity slab remains, but now it cor­responds to a solid-vapor interface. In the derivation of Eq. (6-3), the only change required is to remove the factor of 2 since the volume element in Fig. 6-3 now loses material to the lattice on only one side.

6.2 EXPERIMENTAL OBSERVATIONS ON GRAIN
BOUNDARY DIFFUSION

As a result of transmission electron microscopy and computer mod­eling, there are now quite detailed models for the low temperature structure of grain boundaries, and quite reasonable models for their high temperature kinetic behavior.' There has also been a substantial number of measurements of DO as a function of temperature and boundary structure. ⁸

Grain Boundary Misorientation. The dislocation model for a low angle grain boundary predicts that a low angle [001] tilt boundary con­sists of edge dislocations parallel to the [001] direction and a separation distance of s = a_c,/[2sin(0/2)].⁹ The lattice between the dislocation cores is elastically strained but relatively perfect. Turnbull and Hoff-

'See R. W. Balluffi, in Diffusion in Crystalline Solids, ed. G. E. Murch, A. S. Nowick, Academic Press, 1984, p. 320-78.

N. L. Peterson, in Grain Boundary Structure and Kinetics, ASM Metals Park, OH (1980) 209-37. And, G. Martin, B. Perraillon, in Grain Boundary Structure and Ki­netics, ASM Metals Park, OH (1980) 239-95.

⁹The two halves of a bicrystal containing a (001) tilt boundary have a common [100] direction in the plane of the boundary and can be brought into coincidence by rotation about the [100]. If the grain boundary also is a plane of symmetry, it is said to be a symmetric tilt boundary. If the common [100] direction is normal to the boundary, the boundary is called a (100) twist boundary.

man postulated that in the core of these dislocations the diffusion coef­ficient D _p is much greater than D_i. Thus instead of replacing the grain boundary by a slab of uniform thickness 5 and diffusivity Db, the boundary is described as a planar array of 'pipes' of radius 'a' and spacing s. For diffusion in the direction of the dislocation cores or pipes this gives the equation

p = = D_p(77U² S) D_po-a²[2sin(0/2)/a₀] D_pira²0/a_o (6-9)

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