Lattice diffusion coefficient

In condensed matter physics, lattice diffusion (also called bulk or volume diffusion) refers to atomic diffusion within a crystalline lattice,^[1] 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.

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}}

In the case of interstitial diffusion, the activation enthalpy Q I {\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 increases exponentially with temperature at a rate determined by the activation enthalpy Q I {\displaystyle Q_{I}} .

The rate of self-diffusion can be measured experimentally by introducing radioactive 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:

D A ∗ = D A = 1 6 α 2 Γ {\displaystyle D_{A}^{*}=D_{A}={\frac {1}{6}}\alpha ^{2}\Gamma }

where

• D A ∗ {\displaystyle D_{A}^{*}} is the diffusion coefficient of radioactive A atoms in pure A.
• D A {\displaystyle D_{A}} is the diffusion coefficient of A atoms in pure A.
• Γ {\displaystyle \Gamma } is the jump frequency for both the A* and A atoms.
• α {\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:

Γ = z v X v exp ⁡ − Δ G m R T {\displaystyle \Gamma =zvX_{v}\exp {\frac {-\Delta G_{m}}{RT}}}

where

• z {\displaystyle z} is the number of nearest neighbors.
• v {\displaystyle v} is the frequency of temperature-independent atomic vibration.
• X v {\displaystyle X_{v}} is the vacancy fraction of the lattice.
• Δ G m {\displaystyle \Delta G_{m}} is the activation energy barrier to atomic migration.

In thermodynamic equilibrium,

X v = X v e = exp ⁡ − Δ G v R T {\displaystyle X_{v}=X_{v}^{e}=\exp {\frac {-\Delta G_{v}}{RT}}}

where Δ G v {\displaystyle \Delta G_{v}} is the free energy of vacancy formation for a single vacancy.

The diffusion coefficient in thermodynamic equilibrium can be expressed with Δ G m {\displaystyle \Delta G_{m}} and Δ G v {\displaystyle \Delta G_{v}} , giving:

D A = 1 6 α 2 z v exp ⁡ − ( Δ G m + Δ G v ) R T {\displaystyle D_{A}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {-(\Delta G_{m}+\Delta G_{v})}{RT}}}

Substituting ΔG = ΔH – TΔS gives:

D A = 1 6 α 2 z v exp ⁡ Δ S m + Δ S v R exp ⁡ − ( Δ H m + Δ H v ) R T {\displaystyle D_{A}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}+\Delta S_{v}}{R}}\exp {\frac {-(\Delta H_{m}+\Delta H_{v})}{RT}}}

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

D A = D 0 exp ⁡ − Q S R T {\displaystyle D_{A}=D_{0}\exp {\frac {-Q_{S}}{RT}}}

where

• D 0 {\displaystyle D_{0}} is approximately a constant. D 0 = 1 6 α 2 z v exp ⁡ Δ S m + Δ S v R {\displaystyle D_{0}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}+\Delta S_{v}}{R}}}
• Q S {\displaystyle Q_{S}} is the activation enthalpy. Q S = Δ H m + Δ H v {\displaystyle Q_{S}=\Delta H_{m}+\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 a vacancy 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:

D v = 1 6 α 2 Γ v {\displaystyle D_{v}={\frac {1}{6}}\alpha ^{2}\Gamma _{v}}

where Γ v {\displaystyle \Gamma _{v}} is the jump frequency of a vacancy.

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

D v = 1 6 α 2 z v exp ⁡ Δ S m R exp ⁡ − Δ H m R T {\displaystyle D_{v}={\frac {1}{6}}\alpha ^{2}zv\exp {\frac {\Delta S_{m}}{R}}\exp {\frac {-\Delta H_{m}}{RT}}}

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

D v = D A X v e {\displaystyle D_{v}={\frac {D_{A}}{X_{v}^{e}}}}

where the equilibrium vacancy fraction X v e = exp ⁡ − Δ G v R T {\displaystyle X_{v}^{e}=\exp {\frac {-\Delta G_{v}}{RT}}}

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

D ~ = D ~ A X B + D ~ B X A {\displaystyle {\tilde {D}}={\tilde {D}}_{A}X_{B}+{\tilde {D}}_{B}X_{A}}

where X A {\displaystyle X_{A}} and X B {\displaystyle X_{B}} are the amount fractions of species A and B, respectively.

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