# Temperature Effects

## Enthalpy of migration

In order for atoms to diffuse they must overcome the energy barrier associated with changing their position. The more kinetic energy the atoms have, the more likely it is that the energy barrier will be overcome. The greater the temperature of the system, the greater the kinetic energy of the atoms, therefore temperature has a significant effect on the diffusivity of the species. The energy needed to overcome the barrier is the enthalpy of migration.

For a system with attempt frequency v0 the frequency of successful jumps is given by an Arrhenius dependence:

\[v = {v_0}\exp \left( {\frac{{ - G*}}{{{k_B}T}}} \right)\]

Where ν0 is the pre-exponential, G* is the magnitude of the energy barrier (or the free energy associated with diffusion), KB is Boltzmann’s constant and *T* is the temperature.

Since

G* = H* - TS*

Where H* = enthalpy of migration and S* = entropy of migration

\[v = {v_0}\exp \left( {\frac{{S*}}{{{k_B}}}} \right)\exp \left( {\frac{{ - H*}}{{{k_B}T}}} \right)\]

Or

\[D = {D_0}\exp \left( {\frac{{ - H*}}{{{k_B}T}}} \right)\]

where \({D_0} = \frac{1}{6}{\lambda ^2}{v_0}\exp \left( {\frac{{S*}}{{{k_{\rm{B}}}}}} \right)\)

which can be assumed to remain constant with varying temperature

H* is often denoted as Q, the activation energy for diffusion.

## Enthalpy of vacancy formation

As we have discussed before, the diffusion rate in a substitutional lattice is dependent upon the number of vacancies present, which is also temperature dependent. Therefore, in the case of substitutional diffusion there is a further temperature effect to consider. The equilibrium number of vacancies, Xve, also shows an Arrhenius dependence:

\[{{\rm{X}}_{\rm{v}}}^e = \exp \left( {\frac{{ - \Delta {G_v}}}{{{k_B}T}}} \right)\] \[{{\rm{X}}_{\rm{v}}}^e = \exp \left( {\frac{{\Delta {S_v}}}{{{k_B}}}} \right)\exp \left( {\frac{{ - \Delta {H_v}}}{{{k_B}T}}} \right)\]

Where ΔGv = free energy change on formation of vacancies, ΔSV= entropy change on formation of vacancies and ΔHV= enthalpy change on formation of vacancies.

In the equation to calculate diffusivity, Q can be separated into two components;

- Qm – the enthalpy of migration due to lattice distortions
- Qf – the enthalpy of formation of a vacancy in an adjacent site

Hence

\[D = {D_0}\exp \left( {\frac{{ - {Q_{\rm{m}}}}}{{{k_B}T}}} \right)\exp \left( {\frac{{ - {Q_f}}}{{{k_B}T}}} \right)\]

Since the formation of a vacant site is not needed for interstitial atoms, Qinterstitial << Qsubstitional, and hence Dinterstitial >> Dsubstitional.

Temperature plays a significant role in the rate of diffusion, as it alters the equilibrium concentration of vacancies, and probability of a successful jump into a neighbouring site.

The animation below shows the effect of temperature on both substitutional and interstitial diffusion.