Mechanisms of diffusion

5.4.2 Mechanisms of diffusion

The transport of atoms through the lattice may conceivably occur in many ways. The term ‘interstitial diffusion’ describes the situation when the moving atom does not lie on the crystal lattice, but instead occupies an interstitial position. Such a process is likely in interstitial alloys where the migrating atom is very small (e.g. carbon, nitrogen or hydrogen in iron). In this case, the diffusion process for the atoms to move from one interstitial position to the next in a perfect lattice is not defect- controlled. A possible variant of this type of diffusion has been suggested for substitutional solutions in which the diffusing atoms are only temporarily interstitial and are in dynamic equilibrium with others in substitutional positions. However, the energy to form such an interstitial is many times that to produce a vacancy and, consequently, the most likely mechanism is that of the continual migration of vacancies. With vacancy diffusion, the probability that an atom may jump to the next site will depend on: (1) the probability that the site is vacant (which in turn is proportional to the fraction of vacancies in the crystal) and (2) the probability that it has the required activation energy to make the transition. For self-diffusion, where no complications exist, the diffusion coefficient is therefore given by

D = 1 6 a 2 fv exp[(S f +S m )/k] × exp[−E f /kT ] exp[ −E m /kT ] =D 0 exp[ −(E f +E m )/kT ].

The factor f appearing in D 0 is known as a correlation factor and arises from the fact that any particular diffusion jump is influenced by the direction of the previous jump. Thus, when an atom and a vacancy exchange places in the lattice there is a greater probability of the atom returning to its original site than moving to another site, because of the presence there of a vacancy; f is 0.80 and 0.78 for fcc

and bcc lattices, respectively. Values for E f and E m are discussed in Chapter 3: E f is the energy of formation of a vacancy, E m the energy of migration, and the sum of the two energies, Q =E f +E m ,

is the activation energy for self-diffusion 2 E d .

In alloys, the problem is not so simple and it is found that the self-diffusion energy is smaller than in pure metals. This observation has led to the suggestion that in alloys the vacancies associate preferentially with solute atoms in solution; the binding of vacancies to the impurity atoms increases the effective vacancy concentration near those atoms so that the mean jump rate of the solute atoms is much increased. This association helps the solute atom on its way through the lattice but, conversely, the speed of vacancy migration is reduced because it lingers in the neighborhood of the solute atoms, as shown in Figure 5.6. The phenomenon of association is of fundamental importance in all kinetic studies, since the mobility of a vacancy through the lattice to a vacancy sink will be governed by its ability to escape from the impurity atoms which trap it. This problem has been mentioned in Chapter 3.

When considering diffusion in alloys it is important to realize that in a binary solution of A and B the diffusion coefficients D A and D B are generally not equal. This inequality of diffusion was first demonstrated by Kirkendall using an α-brass/copper couple (Figure 5.7). He noted that if the position of the interfaces of the couple were marked (e.g. with fine W or Mo wires), during diffusion the markers move towards each other, showing that the zinc atoms diffuse out of the alloy more rapidly than copper atoms diffuse in. This being the case, it is not surprising that several workers have shown that porosity develops in such systems on that side of the interface from which there is a net loss of atoms.

The Kirkendall effect is of considerable theoretical importance, since it confirms the vacancy mechanism of diffusion. This is because the observations cannot easily be accounted for by any other

2 The entropy factor exp[(S f +S m )/k] is usually taken to be unity.

250 Physical Metallurgy and Advanced Materials

Vacancy atom

Figure 5.6 Solute atom–vacancy association during diffusion.

Copper

Molybdenum wire markers

α - Brass

Figure 5.7 α -Brass–copper couple for demonstrating the Kirkendall effect.

postulated mechanisms of diffusion, such as direct place exchange, i.e. where neighboring atoms merely change place with each other. The Kirkendall effect is readily explained in terms of vacancies, since the lattice defect may interchange places more frequently with one atom than the other. The effect is also of some practical importance, especially in the fields of metal-to-metal bonding, sintering and creep.