Incondensed matter physics,lattice diffusion (also calledbulk orvolume diffusion) refers toatomic diffusion within acrystalline lattice,^[1] which occurs by eitherinterstitial orsubstitutional mechanisms. In interstitial lattice diffusion, a diffusant (such ascarbon in aniron 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 ofpoint vacancies throughout the crystal lattice. Diffusing particles migrate from point vacancy to point vacancy by the rapid, essentiallyrandom jumping about (jump diffusion). Since the prevalence of point vacancies increases in accordance with theArrhenius equation, the rate of crystal solid state diffusion increases withtemperature. For a single atom in adefect-free crystal, the movement can be described by the "random walk" model.

An atom diffuses in theinterstitial 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 interstitialdiffusion coefficient depends on the jump frequency. The jumpfrequency,${\displaystyle \mathrm{\Gamma }}$, is given by:

${\displaystyle \mathrm{\Gamma }=zv\exp \left(\frac{-\mathrm{\Delta }{G}_m}{RT}\right)}$

where

• ${\displaystyle z}$ is the number of nearest neighboring interstitial sites.
• ${\displaystyle v}$ is vibration frequency of the interstitial atom due tothermal energy.
• ${\displaystyle \mathrm{\Delta }{G}_m}$ is theactivation energy for the migration of the interstitial atom between sites.

${\displaystyle \mathrm{\Delta }{G}_m}$ can be expressed as the sum of activationenthalpy term${\displaystyle \mathrm{\Delta }{H}_m}$ and the activationentropy term${\displaystyle -T\mathrm{\Delta }{S}_m}$, which gives the diffusion coefficient as:

${\displaystyle D=\left[\frac{1}{z}{\alpha }^{2}zv\exp \frac{\mathrm{\Delta }{S}_m}{R}\right]\exp \frac{-\mathrm{\Delta }{H}_m}{RT}}$

where

• ${\displaystyle \alpha }$ is the jump distance.

The diffusion coefficient can be simplified to anArrhenius equation form:

${\displaystyle D=D_0\exp \frac{-Q_I}{RT}}$

where

• ${\displaystyle D_0}$ is a temperature-independent material constant.${\displaystyle D_0=\frac{1}{z}{\alpha }^{2}zv\exp \frac{\mathrm{\Delta }{S}_m}{R}}$
• ${\displaystyle Q_I}$ is the activation enthalpy.${\displaystyle Q_I=\mathrm{\Delta }{H}_m}$

In the case of interstitial diffusion, the activation enthalpy${\displaystyle Q_I}$ is only dependent on the activation energy barrier to the movement of interstitial atoms from one site to another. The diffusion coefficient increasesexponentially with temperature at a rate determined by the activation enthalpy${\displaystyle Q_I}$.

The rate ofself-diffusion can be measured experimentally by introducingradioactive A atoms (A*) into pure A and measuring the rate at which penetration occurs at various temperatures. A* and A atoms have approximately identical jump frequencies since they are chemically identical. The diffusion coefficient of A* and A can be related to the jump frequency and expressed as:

${\displaystyle D_A^{\ast }=D_A=\frac{1}{6}{\alpha }^2\mathrm{\Gamma }}$

where

• ${\displaystyle D_A^{\ast }}$ is the diffusion coefficient of radioactive A atoms in pure A.
• ${\displaystyle D_A}$ is the diffusion coefficient of A atoms in pure A.
• ${\displaystyle \mathrm{\Gamma }}$ is the jump frequency for both the A* and A atoms.
• ${\displaystyle \alpha }$ is the jump distance.

An atom can make a successful jump when there are vacancies nearby and when it has enough thermal energy to overcome the energy barrier to migration. The number of successful jumps an atom will make in one second, or the jump frequency, can be expressed as:

${\displaystyle \mathrm{\Gamma }=zv{X}_v\exp \frac{-\mathrm{\Delta }{G}_m}{RT}}$

where

• ${\displaystyle z}$ is the number of nearest neighbors.
• ${\displaystyle v}$ is the frequency of temperature-independent atomic vibration.
• ${\displaystyle X_v}$ is the vacancy fraction of the lattice.
• ${\displaystyle \mathrm{\Delta }{G}_m}$ is the activation energy barrier to atomic migration.

Inthermodynamic equilibrium,

${\displaystyle X_v=X_v^e=\exp \frac{-\mathrm{\Delta }{G}_v}{RT}}$

where${\displaystyle \mathrm{\Delta }{G}_v}$ is the free energy of vacancy formation for a single vacancy.

The diffusion coefficient in thermodynamic equilibrium can be expressed with${\displaystyle \mathrm{\Delta }{G}_m}$ and${\displaystyle \mathrm{\Delta }{G}_v}$, giving:

${\displaystyle D_A=\frac{1}{6}{\alpha }^{2}zv\exp \frac{-(\mathrm{\Delta }{G}_m+\mathrm{\Delta }{G}_v)}{RT}}$

Substituting ΔG = ΔH – TΔS gives:

${\displaystyle D_A=\frac{1}{6}{\alpha }^{2}zv\exp \frac{\mathrm{\Delta }{S}_m+\mathrm{\Delta }{S}_v}{R}\exp \frac{-(\mathrm{\Delta }{H}_m+\mathrm{\Delta }{H}_v)}{RT}}$

The diffusion coefficient can be simplified to an Arrhenius equation form:

${\displaystyle D_A=D_0\exp \frac{-Q_S}{RT}}$

where

• ${\displaystyle D_0}$ is approximately a constant.${\displaystyle D_0=\frac{1}{6}{\alpha }^{2}zv\exp \frac{\mathrm{\Delta }{S}_m+\mathrm{\Delta }{S}_v}{R}}$
• ${\displaystyle Q_S}$ is the activation enthalpy.${\displaystyle Q_S=\mathrm{\Delta }{H}_m+\mathrm{\Delta }{H}_v}$

Compared to that of interstitial diffusion, the activation energy for self-diffusion has an extra term (ΔH_v). Since self-diffusion requires the presence of vacancies whose concentration depends on ΔH_v.

Diffusion of avacancy can be viewed as the jumping of a vacancy onto an atom site. It is the same process as the jumping of an atom into a vacant site but without the need to consider the probability of vacancy presence, since a vacancy is usually always surrounded by atom sites to which it can jump. A vacancy can have its own diffusion coefficient that is expressed as:

${\displaystyle D_v=\frac{1}{6}{\alpha }^2{\mathrm{\Gamma }}_v}$

where${\displaystyle {\mathrm{\Gamma }}_v}$ is the jump frequency of a vacancy.

The diffusion coefficient can also be expressed in terms of enthalpy of migration (${\displaystyle \mathrm{\Delta }{H}_m}$) and entropy of migration (${\displaystyle \mathrm{\Delta }{S}_m}$) of a vacancy, which are the same as for the migration of a substitutional atom:

${\displaystyle D_v=\frac{1}{6}{\alpha }^{2}zv\exp \frac{\mathrm{\Delta }{S}_m}{R}\exp \frac{-\mathrm{\Delta }{H}_m}{RT}}$

Comparing the diffusion coefficient between self-diffusion and vacancy diffusion gives:

${\displaystyle D_v=\frac{D_A}{X_v^e}}$

where the equilibrium vacancy fraction${\displaystyle X_v^e=\exp \frac{-\mathrm{\Delta }{G}_v}{RT}}$

In a system with multiple components (e.g. abinary alloy), thesolvent (A) and thesolute atoms (B) will not move in an equal rate. Each atomic species can be given its own intrinsic diffusion coefficient${\displaystyle {\tilde{D}}_A}$ and${\displaystyle {\tilde{D}}_B}$, expressing the diffusion of a certain species in the whole system. The interdiffusion coefficient${\displaystyle \tilde{D}}$ is defined by theDarken's equation as:

${\displaystyle \tilde{D}={\tilde{D}}_A{X}_B+{\tilde{D}}_B{X}_A}$

where${\displaystyle X_A}$ and${\displaystyle X_B}$ are theamount fractions of species A and B, respectively.

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• Diffusion

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