In condensed matter physics, lattice diffusion (also called bulk or volume diffusion) refers to atomic diffusion within a crystalline lattice, which occurs by either interstitial or substitutional mechanisms. In interstitial lattice diffusion, a diffusant (such as carbon in an iron alloy), will diffuse in between the lattice structure of another crystalline element. In substitutional lattice diffusion (self-diffusion for example), the atom can only move by switching places with another atom. Substitutional lattice diffusion is often contingent upon the availability of point vacancies throughout the crystal lattice. Diffusing particles migrate from point vacancy to point vacancy by the rapid, essentially random jumping about (jump diffusion). Since the prevalence of point vacancies increases in accordance with the Arrhenius equation, the rate of crystal solid state diffusion increases with temperature. For a single atom in a defect-free crystal, the movement can be described by the "random walk" model.
Diffusion Coefficient for Interstitial Diffusion An atom diffuses in the interstitial mechanism by passing from one interstitial site to one of its nearest neighboring interstitial sites. The movement of atoms can be described as jumps, and the interstitial diffusion coefficient depends on the jump frequency. The jump frequency, Γ {\displaystyle \Gamma } , is given by:
Γ = z v exp ( − Δ G m R T ) {\displaystyle \Gamma =zv\exp \left({\frac {-\Delta G_{m}}{RT}}\right)}
where
z {\displaystyle z} is the number of nearest neighboring interstitial sites.
v {\displaystyle v} is vibration frequency of the interstitial atom due to thermal energy.
Δ G m {\displaystyle \Delta G_{m}} is the activation energy for the migration of the interstitial atom between sites.
R {\displaystyle R} is the gas constant.
T {\displaystyle T} is the thermodynamic temperature.
Δ G m {\displaystyle \Delta G_{m}} can be expressed as the sum of activation enthalpy term Δ H m {\displaystyle \Delta H_{m}} and the activation entropy term − T Δ S m {\displaystyle -T\Delta S_{m}} , which gives the diffusion coefficient as:
D = [ 1 z α 2 z v exp Δ S m R ] exp − Δ H m R T {\displaystyle D=\left[{\frac {1}{z}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}}{R}}\right]\exp {\frac {-\Delta H_{m}}{RT}}}
where
α {\displaystyle \alpha } is the jump distance. The diffusion coefficient can be simplified to an Arrhenius equation form:
D = D 0 exp − Q I R T {\displaystyle D=D_{0}\exp {\frac {-Q_{I}}{RT}}}
where
D 0 {\displaystyle D_{0}} is a temperature-independent material constant. D 0 = 1 z α 2 z v exp Δ S m R {\displaystyle D_{0}={\tfrac {1}{z}}\alpha ^{2}zv\exp {\tfrac {\Delta S_{m}}{R}}}
Q I {\displaystyle Q_{I}} is the activation enthalpy. Q I = Δ H m {\displaystyle Q_{I}=\Delta H_{m}}
… excerpt ends here. Continue reading the full article.
