Nonsteady- Nonsteady-State

FIGURE6.5.(a) Steady-state diffusion through a slab. (b) Linear con-centration gradient which stays constant with time by constantly supplying solute atoms on the left and remov-ing the same number of atoms on the right side of the slab. Cand C_ are two assumed sur-face concentrations, where C! C.

which can be solved whenever a specific set of boundary condi-tions is known. It should be noted that the diffusion coefficient in Eq. (6.10) has only a constant value if one considers self-diffusion or possibly for very dilute systems. In general, however, D varies with concentration so that Eq. (6.10) must be written in more general terms as:

 C


t 

x

冢

冣

The solution of Eq. (6.10), assuming D to be constant with con-centration and for “infinitely long” samples (i.e., rods which are long enough so that the composition does not change at the outer ends), is:

erf

冢 冣

In other words the solution (6.11) is only valid when the length of a sample is larger than 10兹^D苶^t苶. The parameters in Eq. (6.11) are as follows: Cx is the concentration of the solute at the dis-tance x; Ciis the constant concentration of the solute at the in-terface dividing materials A and B after some time, t; and C0 is the initial solute concentration in material B. The initial solute

x

2兹^D苶^t苶 Ci Cx

Ci C⁰

X C_i

C_x C₀

Concentration of solute A

(2C_i– C₀)

0 x

t₂

t₀

t₁

Material A (Solute)

Material B (Host material) FIGURE6.6.Concentration profiles (called also penetration curves) for nonsteady-state diffusion of material A into material B for three differ-ent times. The concdiffer-entration of the solute in material B at the distance x is named Cx. The solute concentration at X
0, that is, at the interface between materials A and B, is C_i. The original solute concentration in material B (or at X
") is C⁰. A mirror image of the diffusion of B into A can be drawn if the mutual diffusivities are identical. This is omitted for clarity.

concentration in material A is 2Ci–C0; see Figure 6.6. The right-hand side in Eq. (6.11) is called the probability integral or Gauss-ian error function, which is defined by:

erf( y)
 兹 2

苶



冕

It is tabulated in handbooks similarly as trigonometric or other functions and is depicted in Figure 6.7. It can be observed that the error function (Figure 6.7) reproduces the concentration pro-file of Figure 6.6 quite well. (See also Problem 6.1.) A reasonable estimate for the distance, X, stating how far solute atoms may diffuse into a matrix during a time interval, t, can be obtained from:

X
2兹^D苶^t苶^{. (6.13)}

The boundary condition for which Eq. (6.10) was solved stipu-lates, as already mentioned, that the initial compositions of ma-terials A and B do not change at the free ends. Now, if the dif-fusion process is allowed to take place for long times, and if it is additionally conducted at high temperatures, a complete mixing of the two materials A and B will eventually take place. In other words, a uniform concentration of both components is then found along the entire couple, as indicated in Figure 6.6 by the horizontal line marked Ci. In this case, which is called complete interdiffusion, Eq. (6.11) is no longer a valid solution of Fick’s second law. Moreover, Eq. (6.11) loses its validity long before a uniform composition is attained.

FIGURE6.7.The Gaussian er-ror function erf(y) as a func-tion of y. Compare to Figure 6.6.

6.1.8

Interdiffusion

So far, we have considered only the diffusion of one component (say, material A) into another material (say, B). If both elements have the same diffusivity in each other (that is, if A diffuses with the same velocity into B as B diffuses into A), then a mirror im-age of the diffusion profiles shown in Figure 6.6 can be drawn for the other component. In many cases, however, this condition is not fulfilled. For example, copper diffuses with different velocity into nickel than vice versa (Table 6.1). Let us assume that element A diffuses faster into B than B into A. As a consequence, more A atoms cross the initial interface between the two elements in one direction than B atoms into the opposite direction. The result is that the initial interface shifts in the direction of element A.

This can be best observed, as done by Kirkendall, by inserting inert markers (usually fine wires of metals having a high melt-ing point) between the two bars of metals A and B before join-ing them. It is then observed that these markers move into the direction of the A metal upon prolonged heating of this diffusion couple, as schematically depicted in Figure 6.8. The shift is not very large, so that annealings close to the melting point of the el-ements lasting for many days are necessary.

After cooling, small sections parallel to the interface may be re-moved in a lathe which are then chemically analyzed for their com-positions. This latter procedure is, incidentally, common for many diffusion experiments. As an alternative, radioactive “tracer ele-ments” are utilized for determining the amount of diffused species rather than the less accurate chemical analysis. An even more ad-vanced technique utilizes the microprobe, which scans an X-ray beam along the specimen whose response signal eventually yields the composition. It might be noted in passing that in cases for which the atom flow of the species under consideration is con-siderably unbalanced, some porosity in the diffusion zone might develop.

Kirkendall markers after annealing

Initial location of interface

A B

FIGURE6.8.Schematic representa-tion of the movement of inert markers (inserted at the initial interface) due to faster movement of A atoms into B than vice versa (Kirkendall shift).

6.1.9

Kirkendall