Diffusion

Basics of Diffusion

Fick’s Law

Diffusion flux $J = m / (tS)$ (unit: atoms cm^-2 s^-1)
- Mass diffusivity, or diffusion coefficient $D$ (unit: cm² s^-1)

First law

Also compare electricity: $I = -R dU / dx$

$J = - D \frac{dC}{dx}$

Second law

Because $dC / dt = - [J(x+dx) - J(x)] / dx = - dJ / dx$ , we have

$\frac{dC}{dt} = \frac{dD}{dC} \left(\frac{dC}{dx}\right)^2 + D \frac{d^2 C}{dx^2}$

and when $D \neq D(c)$

$\frac{dC}{dt} = D \frac{d^2 C}{dx^2}$

which means

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Diffusion and thermodynamics

Diffusion & chemical potential

The thermodynamic correction term: $1 - M_i RT \frac{\partial \ln \gamma_i}{\partial \ln x_i}$ .

Proof: […]

Mechanisms

Atomic mechanisms

The random walk derivation of Fick’s first law

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Therefore we have the atomic diffusion factor

$D = \frac{1}{6}\,\Gamma\lambda^2$

And the following discussions of mechanisms will focus on the expression of jump frequency $\Gamma$ .

Examples: diffusion in different systems

Diffusion in elemental crystals

Also see the point defect chapter. The interstitial and vacancy are two main defects in elemental crystals (especially pure metal).

Interstitial mechanism

This could also be, carbon particles move in iron.

Jump frequency $\Gamma = Z f_0 \exp\left(-\Delta G^{\#} / RT\right)$
- $Z$ : jumpable sites
- $f_0$ : intrinsic frequency
Diffusivity $D = D_0 \exp (-Q / RT)$

Vacancy mechanism

The vacancy mechanism is also called self-diffusion in pure metal.
Jump frequency $\Gamma = ZX_v^ef_0\exp(-\Delta G / RT)$ .
- $X_\text{v}^e$ - equilibrium vacancy concentration
Diffusivity $D = \frac{1}{6} Z X_\text{v} f_0 \lambda^2 \exp{-\frac{\Delta G}{RT}}$
For the equilibrium concentration of vacancies $X_{\mathrm{v}}=e^{\bar S_\mathrm{v}^{\mathrm{xs}} / k} e^{-\bar{H}_\mathrm{v} / k T}$ , derivation see the point defect chapter. Also can be derived from configruational entropy.
The self-diffusion can be see as the reverse process of vacancy diffusion $D_\text{self} = X_\text{v} D_\text{v}$

Biased-random walk, drift velocity: Fick’s first law with chemical potential

Diffusion in substitution alloys

Kirkendall effect

The move of boundaries (marked by Mo markers) is the result of the creation and destruction of atomic planes due to inequalities in diffusivities ( $D_\ce{Cu} \neq D_\ce{Ni}$ ).

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Darken analysis

Interdiffusion coefficient $\tilde D$

$\tilde D = X_BD_A + X_AD_B$

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Extended darken analysis

Include thermodynamic correction into the darken analysis.

$\tilde D = (X_AD^*_B + X_BD^*_A) \frac{X_AX_B}{RT}\frac{d^2 G}{dX_B^2}$

This is also the source of up-hill diffusion, which causes spinodal decomposition (automatic separation of phases).

Proof: […]

Diffusion in polycrystalline materials: short circuit diffusion

Grain boundary and dislocations have a lower activation energy.

Temperature effect

Easy to see that jump frequency $\Gamma \propto \exp(-Q/K_BT)$ , and $D = \frac{1}{6}\Gamma\lambda^2$ , $\ln D = -Q(1/K_BT)$

Other process Relates to Diffusion

Nucleation