The heart of crystalline materials

Diffusion occurs by stepwise migration of atoms from lattice site to lattice site across the crystal. It occurs in interstices and vacancies but more readily occurs in regions where the crystal presents linear defects (dislocations) or bidimensional defects (grain boundaries and sub-boundaries, sample surface).

In interstitial solid solutions, solute atoms (small dimension) migrate across interstitial sites in the lattice: we refer to an interstitial mechanism (see illustration). In substitutional solid solutions, the diffusion mechanisms use vacancies present in the material: an atom adjacent a vacancy can move to this unoccupied site, provoking a displacement of the vacancy in the reverse direction: we call this a vacancy mechanism (see illustration).

The number of vacancies \(N_v \)in a metal can be expressed by the relationship:

\(N_v = N \exp \left( - \frac{\Delta G_f}{kT} \right)\)

where \(N\) is the number of sites in the lattice, \(\Delta G_f\) is the free enthalpy for vacancy formation, \(k = {1,38062}.10^{-23}{\rm \, J/°C}\) is the Boltzman constant.

For an atom located close to a vacancy (or an atom in an interstitial position) to be able to move, it has to be given sufficient energy to separate adjacent atoms from their equilibrium position. The diffusion phenomenon therefore implies passing the potential energy barrier \(\Delta G_t\), as shown in the following diagram. The number of atoms with higher energy than this barrier increases with temperature, making diffusion mechanisms thermally activated processes (see diagram (cf. Thermal activation of diffusion mechanism)). Note also that diffusion mechanisms are favoured by low interatomic bond energy.

In the case of substitutional solid solutions characterized by vacancy diffusion mechanisms, diffusion of atoms can be described by a jump frequency of \(\Gamma_D\) from one equilibrium position to another. The jump frequency depends on the conjunction of two events: the proximity of a vacancy and the availability of sufficient energy to overcome the potential barrier between the occupied site and the vacancy. We can therefore write that the jump frequency is equal to:

\(\Gamma _D = \Gamma \Pi\)

where \(\Gamma\) is the efficient frequency of attack for the barrier and \(\Pi\) is the probability of the presence of a vacancy at the site adjacent to the diffusing atom.

We have

\(\Gamma = \nu \exp \left(-\Delta G_t/kT\right)\)

with \(\nu\) the frequency of attack for the potential barrier and \(\exp \left(-\Delta G_t/kT\right)\) the probability of passing the potential barrier

and \(\Pi = \exp \left(-\Delta G_f/kT\right)\).

Hence: \(\Gamma_D = \nu \exp \left[-\left(\Delta G_t + \Delta G_f \right)/kT \right]\)

However, \(\Delta G_{(t \textrm{ or } f)} = \Delta H_{(t \textrm{ or } f)} - T\Delta S_{(t \textrm{ or } f) }\)

Therefore, \(\Gamma_D = \left(\Gamma_D\right)_0 \exp \left[-\left(\Delta H_t+\Delta H_f\right)/kT\right]\)

with \(\Delta H_t\) the vacancy migration activation enthalpy (in the order of \({1}{\rm \, eV}\)), \(\Delta H_f\) the vacancy formation enthalpy (in the order of \({1}{\rm \, eV}\)) and \(\left(\Gamma_D\right)_0 = \nu \exp \left[\left(\Delta St +\Delta Sf\right)/k\right]\) independent of temperature.

In the case of a substitutional solid solution, the variation with temperature of the diffusion rate results in two phenomena: a variation in the number of vacancies and a change in mobility of vacancies. In the case of an interstitial diffusion, the diffusion sites are already formed by the lattice interstices and only the passage of the potential barrier characterized by \(\Delta H_t\) needs be considered.