Diffusion laws

5.4.1 Diffusion laws

Some knowledge of diffusion is essential in understanding the behavior of materials, particularly at elevated temperatures. A few examples include such commercially important processes as annealing, heat treatment, the age hardening of alloys, sintering, surface hardening, oxidation and creep. Apart

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Initial distribution

Final distribution

Solute concentration

Distance

Figure 5.4 Effect of diffusion on the distribution of solute in an alloy.

from the specialized diffusion processes, such as grain boundary diffusion and diffusion down disloca- tion channels, a distinction is frequently drawn between diffusion in pure metals, homogeneous alloys and inhomogeneous alloys. In a pure material self-diffusion can be observed by using radioactive tracer atoms. In a homogeneous alloy diffusion of each component can also be measured by a tracer method, but in an inhomogeneous alloy diffusion can be determined by chemical analysis merely from the broadening of the interface between the two metals as a function of time. Inhomogeneous alloys are common in metallurgical practice (e.g. cored solid solutions) and in such cases diffusion always occurs in such a way as to produce a macroscopic flow of solute atoms down the concentration gradient. Thus, if a bar of an alloy, along which there is a concentration gradient (Figure 5.4), is heated for a few hours at a temperature where atomic migration is fast, i.e. near the melting point, the solute atoms are redis- tributed until the bar becomes uniform in composition. This occurs even though the individual atomic movements are random, simply because there are more solute atoms to move down the concentration gradient than there are to move up. This fact forms the basis of Fick’s law of diffusion, which is

dn/dt = −Ddc/dx. (5.2) Here the number of atoms diffusing in unit time across unit area through a unit concentration gradient

is known as the diffusivity or diffusion coefficient, 1 D. It is usually expressed as units of cm 2 s −1 or m 2 s −1 and depends on the concentration and temperature of the alloy. To illustrate, we may consider the flow of atoms in one direction x, by taking two atomic planes A and

B of unit area separated by a distance b, as shown in Figure 5.5. If c 1 and c 2 are the concentrations of diffusing atoms in these two planes (c 1 > c 2 ), the corresponding number of such atoms in the respective planes is n 1 =c 1 b and n 2 =c 2 b. If the probability that any one jump in the +x direction is p x , then the number of jumps per unit time made by one atom is p x ν , where ν is the mean frequency with which an atom leaves a site irrespective of directions. The number of diffusing atoms leaving

A and arriving at B in unit time is ( p x ν c 1 b) and the number making the reverse transition is ( p x ν c 2 b) so that the net gain of atoms at B is

p x ν b(c 1 −c 2 ) =J x ,

with J x the flux of diffusing atoms. Setting c 1 −c 2 = −b(dc/dx), this flux becomes:

J x = −p x v 2 1 v 2 b (dc/dx) =− 2 vb (dc/dx)

= −D(dc/dx). (5.3)

1 The conduction of heat in a still medium also follows the same laws as diffusion.

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c Conc

n 1 atoms diffusing

n 2 atoms

Unit area

Figure 5.5 Diffusion of atoms down a concentration gradient.

In cubic lattices, diffusion is isotropic and hence all six orthogonal directions are equally likely, so that p x 1 = 6 . For simple cubic structures b = a and thus

D x 2 =D y =D z 1 = 6 va = D, (5.4)

whereas in fcc structures b

12 va 2 , and in bcc structures D = 24 va 2 . Fick’s first law only applies if a steady state exists in which the concentration at every point is

1 = a/ 1 2 and D =

invariant, i.e. (dc/dt) = 0 for all x. To deal with non-stationary flow in which the concentration at

a point changes with time, we take two planes A and B, as before, separated by unit distance and consider the rate of increase of the number of atoms (dc/dt) in a unit volume of the specimen; this is equal to the difference between the flux into and that out of the volume element. The flux across one plane is J x and across the other J x + dJ x , the difference being −dJ x . We thus obtain Fick’s second law of diffusion:

When D is independent of concentration this reduces to

dc d 2 c =D dt x dx 2

(5.6) and in three dimensions becomes

An illustration of the use of the diffusion equations is the behavior of a diffusion couple, where there is a sharp interface between pure metal and an alloy. Figure 5.4 can be used for this example and as

248 Physical Metallurgy and Advanced Materials the solute moves from alloy to the pure metal, the way in which the concentration varies is shown by

the dotted lines. The solution to Fick’s second law is given by x/[2 c √

where c 0 is the initial solute concentration in the alloy and c is the concentration at a time t at a distance x from the interface. The integral term is known as the Gauss error function (erf ( y)) and as y → ∞, erf ( y) → 1. It will be noted that at the interface where x = 0, then c = c 0 / 2, and in those regions where the curvature ∂ 2 c/∂x 2 is positive the concentration rises, in those regions where the curvature is negative the concentration falls, and where the curvature is zero the concentration remains constant.

This particular example is important because it can be used to model the depth of diffusion after time t, e.g. in the case hardening of steel, providing the concentration profile of the carbon after

a carburizing time t, or dopant in silicon. Starting with a constant composition at the surface, the value of x where the concentration falls to half the initial value, i.e. 1

2 , is given by x = (Dt). Thus, knowing D at a given temperature, the time to produce a given depth of diffusion can be estimated.

− erf(y) = 1

The diffusion equations developed above can also be transformed to apply to particular diffusion geometries. If the concentration gradient has spherical symmetry about a point, c varies with the radial distance r and, for constant D,

When the diffusion field has radial symmetry about a cylindrical axis, the equation becomes

and the steady-state condition (dc/dt) = 0 is given by

d 2 c 1 dc dr 2 + r dr = 0,

(5.10) which has a solution c = A ln(r) + B. The constants A and B may be found by introducing the

appropriate boundary conditions and for c =c 0 at r =r 0 and c =c 1 at r =r 1 the solution becomes

c = +c

c 0 ln(r 1 / r)

1 ln(r/r 0 )

ln(r 1 / r 0 )

The flux through any shell of radius r is −2πrD(dc/dr) or 2πD

ln(r / r 0 )

Diffusion equations are of importance in many diverse problems and in Chapter 3 are applied to the diffusion of vacancies from dislocation loops and the sintering of voids.

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