# Materials Science and Engineering/Derivations/Kinetics
## Atomic Models of Diffusivity
### Metals
#### Diffusion of Solute Atoms in BCC Crystal by the Interstitial Mechanism

* Connection between jump rate, $\mathrm {T}$, and intersite jump distance, $r$, and the correlation factor:
* Each interstitial site is associated with four nearest-neighbors
    $\mathrm {T} =4\mathrm {T} '\;$
    $\mathrm {T} '=\nu e^{S^{m}/(kT)}e^{-H^{m}/(kT)}$
* Lattice constant: $a\;$
** $r=a/2\;$
    $D_{I}={\frac {a^{2}}{6}}\nu e^{S^{m}/k}e^{-H^{m}/(kT)}$
    $D_{I}=D_{I}^{\circ }e^{-E/(kT)}$
* Consider concentration gradient and number of site-pairs that can contribute to flux across crystal plane
** Concentration gradient results in flux of atoms from three types of interstitial sites in $\alpha \;$ plane
*** $c'\;$: number of atoms in the $\alpha \;$ plane per unit area
*** Carbon concentration on each of the three sites: $c'/3\;$
*** Jump rate of atoms from the type 1 and 3 sites between plan $\alpha \;$ and $\beta \;$: $(c'/3)\mathrm {T} '\;$
*** Contribution to the flux from the three sites: $J^{\alpha \rightarrow \beta }={\frac {2\mathrm {T} 'c'}{3}}$
** Convert to the number of atoms per unit volume: $c=2c'/a\;$
    $J^{\alpha \rightarrow \beta }={\frac {a\mathrm {T} 'c}{3}}$
** Find the reverse flux by using a first-order expansion
    $J^{\beta \rightarrow \alpha }={\frac {a\mathrm {T} '}{3}}\left(c+{\frac {a}{2}}{\frac {\partial c}{\partial y}}\right)$
** Find the net flux
    $J^{\mbox{net}}=J^{\alpha \rightarrow \beta }-J^{\beta \rightarrow \alpha }$
    $J^{\mbox{net}}=-{\frac {a^{2}\mathrm {T} '}{6}}{\frac {\partial c}{\partial y}}$
** Compare with Fick's law expression, $J^{\mbox{net}}=-D_{I}{\frac {\partial c}{\partial y}}$, and total jump frequency, $\mathrm {T} =4\mathrm {T} '$:
    $D_{I}={\frac {a^{2}\mathrm {T} '}{6}}$
    $D_{I}={\frac {a^{2}\mathrm {T} }{24}}$

#### Self-Diffusion in FCC Structure by Vacancy Mechanism

* There are twelve nearest neighbors on an fcc lattice
* Vacancies randomly occupy sites and are associated with jump frequency, $\mathrm {T} _{V}$
    $\mathrm {T} _{V}=12\mathrm {T} _{V}'\;$
    $\mathrm {T} _{V}'=\nu e^{S_{V}^{m}/k}e^{-H_{V}^{m}/(kT)}$
* $X_{V}\;$: fraction of sites randomly occupied by vacancies
* Jump rate of host atoms:
    $\mathrm {T} _{A}=X_{V}\mathrm {T} _{V}\;$
* Self-diffusivity with $r=a/{\sqrt {2}}$:
    $^{*}D={\frac {\mathrm {T} _{A}r^{2}\mathbf {f} }{6}}$
    $^{*}D=X_{V}\mathrm {T} _{V}'a^{2}\mathbf {f}$
* With uncorrelated vacancy diffusion, the vacancy diffusivity is
    $D_{V}={\frac {\mathrm {T} _{V}r^{2}\mathbf {f} }{6}}=\mathrm {T} _{V}'a^{2}$
* The vacancy diffusivity is related to the self-diffusivity
    $^{*}D=X_{V}D_{V}\mathbf {f}$
* $X_{V}=X_{V}^{\mbox{eq}}\;$ when the vacancies are in thermal equilibrium
    $X_{V}^{\mbox{eq}}=e^{-G_{v}^{f}/(kT)}=e^{-S_{v}^{f}/(k)}e^{-H_{v}^{f}/(kT)}$
** $S_{v}^{f}\;$: vacancy vibrational entropy
** $H_{v}^{f}\;$: enthalpy of formation

### Ionic Solids
#### Intrinsic Crystal Self-Diffusion with Schottky Defects

* Predominant point defects are cation and anion vacancy complexes
* Self-diffusion occurs by a vacancy mechanism
** Defect-creation (Kroger-Vink notation)
    $K_{K}^{\times }+Cl_{Cl}^{\times }=V_{K}'+V_{Cl}^{\circ }+K_{K}^{\times }+Cl_{Cl}^{\times }$
    ${\mbox{null}}=V_{K}'+V_{Cl}^{\circ }$
** Relation between free energy of formation, $G_{S}^{f}$, and the equilibrium constant, $K^{\mbox{eq}}\;$
    $G_{s}^{f}=-kT\ln K^{\mbox{eq}}$
    $K^{\mbox{eq}}=e^{-G_{s}^{f}/(kT)}$
    $K^{\mbox{eq}}=a_{AV}a_{CV}\;$
** The activities correspond to anion and cation vacancies
** With dilute concentrations of vacancies, Raoult's law applies, and activities are equal to site fractions
    $\left[V_{K}'\right]\left[V_{Cl}^{\circ }\right]=K^{\mbox{eq}}=e^{-G_{s}^{f}/(kT)}$
** A requirement of electrical neutrality is that the number of potassium vacancies is equal to the number of chlorine vacancies
    $\left[V_{K}'\right]=\left[V_{Cl}^{\circ }\right]=e^{-G_{s}^{f}/(2kT)}$
** Vacancy self-diffusion in a metal
    $^{*}D^{K}=ga^{2}\mathbf {f} \nu e^{(S_{CV}^{m}+S_{S}^{f}/2)/k}e^{-(H_{CV}^{m}+H_{S}^{f}/2)/(kT)}$
** $g$: geometric factor
** $\mathbf {f}$: correlation factor
** Activation energy of self-diffusion
    $E=H_{CV}^{m}+H_{S}^{f}/2$

#### Intrinsic Crystal Self-Diffusion with Frenkel Defects

* Frenkel pair formation
    $Ag_{Ag}^{\times }=Ag_{i}^{\circ }+V_{Ag}'$
* $a_{Ag_{Ag}^{\times }}=1$
    $K^{\mbox{eq}}=\left[Ag_{i}^{\circ }\right]\left[V_{Ag}'\right]=e^{-G_{F}^{f}/(kT)}$
* Elecrical neutrality condition:
    $\left[Ag_{i}^{\circ }\right]\left[V_{Ag}'\right]=e^{-G_{F}^{f}/(2kT)}$
* Activation energy of self-diffusivity of cations
    $E=H_{I}^{m}+{\frac {H_{F}^{f}}{2}}$

#### Extrinsic Crystal Self-Diffusion with Frenkel Defects

* Extrinsic defects result from the addition of aliovalent solute
* Extrinsic cation-site vacancies are created by incorporation of $Ca^{++}$ through doping $KCl$ with $CaCl_{2}$
** Step 1: Two cation and two anion vacancies form
** Step 2: Single $Ca^{++}$ cation and two $Cl$ anions incorporated
** Cation and anionic vacancy populations relate to the site fraction of extrinsic Ca^{++} impurity
    $\left[Ca_{K}^{\circ }\right]+\left[V_{Cl}^{\circ }\right]=\left[V_{K}'\right]$
    $\left[V_{K}'\right]\left(\left[V_{K}'\right]-\left[Ca_{K}^{\circ }\right]\right)=e^{-G_{S}^{f}/(kT)}=\left[V_{K}'\right]_{\mbox{pure}}^{2}$
** The equation can be solved to find the vacancy site fraction
** Two limiting cases of the behavior of $\left[V_{K}'\right]$
*** Intrinsic: $\left[V_{K}'\right]_{\mbox{pure}}\gg \left[Ca_{K}^{\circ }\right]$, then $\left[V_{K}'\right]=\left[V_{K}'\right]_{\mbox{pure}}$
*** Extrinsic: $\left[V_{K}'\right]_{\mbox{pure}}\ll \left[Ca_{K}^{\circ }\right]$, then $\left[V_{K}'\right]=\left[Ca_{K}^{\circ }\right]_{\mbox{pure}}$

#### Self-Diffusion in Nonstochiometric Crystals

** Oxidation of $FeO$
    $FeO+{\frac {x}{2}}O_{2}=FeO_{1+x}$
** Consider the sum of two reactions
    $2Fe^{++}=2Fe^{+++}+2e^{-}$
    ${\frac {1}{2}}O_{2}+2e^{-}=O^{--}$
    $2Fe^{++}+{\frac {1}{2}}O_{2}=2Fe^{+++}+O^{--}$
** A cation vacancy must be created with regard to every O atom added
    $2Fe_{Fe}^{\times }+{\frac {1}{2}}O_{2}=2Fe_{Fe}^{\circ }+O_{O}^{\times }+V_{Fe}''$
** Relationship between cation vacancy site fraction and oxygen gas pressure
    ${\frac {1}{2}}=O_{O}^{\times }+V_{Fe}''+2h_{Fe}^{\circ }$
    $h_{Fe}^{\circ }=Fe_{Fe}^{\circ }-Fe_{Fe}^{\times }$
** Equilibrium constant of the reaction:
    $K^{\mbox{eq}}={\frac {\left[V_{Fe}''\right]\left[h_{Fe}^{\circ }\right]^{2}}{P_{O_{2}^{1/2}}}}=e^{-\Delta G/(kT)}$
** Electrical neutrality condition with oxidation-induced cation vacancies as dominant charged defects
    $\left[h_{Fe}^{\circ }\right]=2\left[V_{Fe}''\right]$
** Solve to find $\left[V_{Fe}''\right]$
    $\left[V_{Fe}''\right]=\left({\frac {1}{4}}\right)^{1/3}e^{-\Delta G/(3kT)}(P_{O_{2}})^{1/6}$
** Activation energy
    $E={\frac {\Delta H}{3}}+H_{CV}^{m}$