22.1 Introduction

Here Q is the activation energy for creep (J mol^–1), R is the universal gas constant (8.31 J mol^–1 K^–1) and T is the absolute temperature (K). This is an example of Arrhenius’s law. Like all good laws, it has great generality. Thus it applies not only to the rate of creep, but also to the rate of oxidation (Chapter 25), the rate of corrosion (Chapter 27), the rate of diffusion (this chapter), even to the rate at which bacteria multiply and milk turns sour. It states that the rate of the process (creep, corrosion, diffusion, etc.) increases exponentially with temperature (or, equivalently, that the time for a given amount of creep, or of oxidation, decreases exponentially with temperature) in the way shown in Figure 22.1. If the rate of a process that follows Arrhenius’s law is plotted on a log_e scale against 1/T, a straight line with a slope of –Q/R is obtained (Figure 22.2). The value of Q characterizes the process—a feature we have already used in Chapter 21.

In this chapter, we discuss the origin of Arrhenius’s law and its application to diffusion. In the next, we examine how it is that the rate of diffusion determines that of creep.

22.2 Diffusion and Fick’s Law

First, what do we mean by diffusion? A good way of demonstrating it is to take a glass Petri dish, and run a very shallow layer of water into it. Wait until the water is totally still, and then drop a very small crystal of potassium permanganate into it. The edge of the crystal will immediately begin to dissolve in the water, producing a dark pink ring of solution centered on the crystal—see Figure 22.3(a). The ring then begins to spread radially outward into the layer of water—initially very quickly, but progressively more slowly as time passes. Figures 22.3(a) to 22.3(d) are time-lapse photographs showing the ring spreading. This is caused by the movement of potassium permanganate ions by random exchanges with the water molecules. The ions move from regions where they are concentrated to regions where they are less concentrated—the ions diffuse down the concentration gradient. This behavior is described by Fick’s first law of diffusion:

J=–Ddcdx

J is the number of ions diffusing down the concentration gradient per second per unit area; it is called the flux of ions (Figure 22.4). c is the concentration of ions in the water—the number of ions (in this case permanganate ions) per unit volume of solution. x is the distance measured along the flux lines, and D is the diffusion coefficient for permanganate ions in solution—it has units of m² s^–1.

This diffusive behavior is not just limited to ions in water—it occurs in all liquids, and more remarkably, in all solids as well. As an example, in the alloy brass—a mixture of zinc in copper—zinc atoms diffuse through the solid copper in just the way that ions diffuse through water. Because the materials of engineering are mostly solids, we shall now confine ourselves to talking about diffusion in the solid state.

Physically, diffusion occurs because atoms, even in a solid, are able to move—to jump from one atomic site to another. Figure 22.5 shows a solid in which there is a concentration gradient of black atoms: there are more to the left of the broken line than there are to the right. If atoms jump across the broken line at random, then there will be a net flux of black atoms to the right (simply because there are more on the left to jump), and, of course, a net flux of white atoms to the left. Fick’s law describes this. It is derived in the following way.

The atoms in a solid vibrate, or oscillate, about their mean positions, with a frequency υ (typically about 10¹³ s^–1). The crystal lattice defines these mean positions. At a temperature T, the average energy (kinetic plus potential) of a vibrating atom is 3kT where k is Boltzmann’s constant (1.38 × 10^–23 J atom^–1 K^–1). But this is only the average energy. As atoms (or molecules) vibrate, they collide, and energy is continually transferred from one to another. Although the average energy is 3kT, at any instant, there is a certain probability that an atom has more or less than this. A very small fraction of the atoms have, at a given instant, much more—enough, in fact, to jump to a neighboring atom site. It can be shown from statistical mechanical theory that the probability, p, that an atom will have, at any instant, an energy ≥q is

p=e–q/kT

Why is this relevant to the diffusion of zinc in copper? Imagine two adjacent lattice planes in the brass with two slightly different zinc concentrations, as shown in exaggerated form in Figure 22.6. Let us denote these two planes as A and B. Now for a zinc atom to diffuse from A to B, down the concentration gradient, it has to “squeeze” between the copper atoms (a simplified statement—but we shall elaborate on it in a moment). This is another way of saying: the zinc atom has to overcome an energy barrier of height q, as shown in Figure 22.6. Now, the number of zinc atoms in layer A is n_A, so the number of zinc atoms that have enough energy to climb over the barrier from A to B at any instant is

nAp=nAe–q/kT

For these atoms actually to climb over the barrier from A to B, they must of course be moving in the correct direction. The number of times each zinc atom oscillates toward B is ≈ υ/6 s^–1 (there are six possible directions in which the zinc atoms can move in three dimensions, only one of which is from A to B). Thus the number of atoms that actually jump from A to B per second is

υ6nAe–q/kT

where c_A and c_B are the zinc concentrations at A and B and r₀ is the atom size. Substituting for n_A and n_B in Equation (22.8) gives

J=υ6r0(cA–cB)e–q/kT

and this is just Fick’s law (Equation 22.2) with

D=D0e–Q/RT

This method of writing D emphasizes its exponential dependence on temperature, and gives a conveniently sized activation energy (expressed per mole of diffusing atoms rather than per atom). Thinking again of creep, the thing about Equation (22.12) is that the exponential dependence of D on temperature has exactly the same form as the dependence of ɛ˙ss on temperature that we seek to explain.

22.3 Data for Diffusion Coefficients

Materials handbooks list data for D₀ and Q for various atoms diffusing in metals and ceramics—Table 22.1 gives some useful values. Diffusion occurs in polymers, composites, and glasses, too, but the data are less reliable. The last column of the table shows the normalized activation energy, Q/RT_M, where T_M is the melting point (in degrees kelvin). It has been found that, for a given class of material (e.g., f.c.c. metals, or refractory oxides), the diffusion parameter D₀ for mass transport—the one that is important in creep—is roughly constant; and the activation energy is proportional to the melting temperature T_M (K) so Q/RT_M, too, is a constant (which is why creep is related to the melting point). This means that many diffusion problems can be solved approximately using the data given in Table 22.2, which shows the average values of D₀ and Q/RT_M, for material classes.

22.4 Mechanisms of Diffusion

Bulk diffusion: Interstitial and vacancy diffusion

Diffusion in the bulk of a crystal can occur by two mechanisms. The first is interstitial diffusion. Atoms in all crystals have spaces, or interstices, between them, and small atoms dissolved in the crystal can diffuse by squeezing between atoms, jumping—when they have enough energy—from one interstice to another (Figure 22.7). Carbon, a small atom, diffuses through steel in this way; in fact C, O, N, B, and H diffuse interstitially in most crystals. These small atoms diffuse very quickly. This is reflected in their very small values of Q/RT_M, seen in the last column of Table 22.1.

The second mechanism is that of vacancy diffusion. When zinc diffuses in brass, for example, the zinc atom (comparable in size to the copper atom) cannot fit into the interstices—the zinc atom has to wait until a vacancy, or missing atom, appears next to it before it can move. This is the mechanism by which most diffusion in crystals takes place (Figures 22.8 and 11.4).

Fast diffusion paths: Grain boundary and dislocation core diffusion

Diffusion in the bulk crystals may sometimes be short circuited by diffusion down grain boundaries or dislocation cores. The boundary acts as a planar channel, about two atoms wide, with a local diffusion rate which can be as much as 10⁶ times greater than in the bulk (Figures 22.9 and 11.4). The dislocation core, too, can act as a high conductivity “wire” of cross-section about (2b)², where b is the atom size (Figure 22.10). Of course, their contribution to the total diffusive flux depends also on how many grain boundaries or dislocations there are: when grains are small or dislocations numerous, their contribution becomes important.

Worked example

The demonstration of diffusion shown in Figure 22.3 can be used to verify Equation (22.13) and also give us an estimate of the diffusion coefficient. The Following Table gives the data.

Figure no.	Radius (mm)	Time (min)	Change in radius = x	Square root of time = t
(a) (b) (c) (d)	2.6 8.3 14.9 30.3	0 3 9 57	0 5.7 12.3 27.7	0 1.7 3.0 7.6

The plot of x against t is a pretty good straight line, consistent with Equation (22.13). The value of D can also be estimated because the slope of the plot xt=D≈27.7mm7.6min1/2,givingD≈27.72mm257×60s=0.22mm2s–1.

Examples

Answers