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:

 $D_{I}={\frac {\mathrm {T} r^{2}}{6}}$

• 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, $\mathrm {T} =4\mathrm {T} '$:

$D_{I}={\frac {a^{2}\mathrm {T} '}{6}}$

$D_{I}={\frac {a^{2}\mathrm {T} }{24}}$

 $D_{I}={\frac {\mathrm {T} r^{2}}{6}}$

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

 ${}^{*}D=\mathbf {f} a^{2}\nu e^{(S_{v}^{m}+S_{v}^{f})/k_{e}}e^{-(H_{v}^{m}+H_{v}^{f})/(kT)}$
 ${}^{*}D=^{*}D^{\circ }e^{-E/(kT)}$

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

$E={\frac {\Delta H}{3}}+H_{CV}^{m}$