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

$\mathrm{T}=4{\mathrm{T}}^{\prime }$

${\mathrm{T}}^{\prime }=\nu e^{S^m/(kT)}{e}^{-H^m/(kT)}$

• 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
    • $c^{\prime }$: number of atoms in the $\alpha$ plane per unit area
    • Carbon concentration on each of the three sites: $c^{\prime }/3$
    • Jump rate of atoms from the type 1 and 3 sites between plan $\alpha$ and $\beta$: $(c^{\prime }/3){\mathrm{T}}^{\prime }$
    • Contribution to the flux from the three sites: $J^{\alpha \to \beta }=\frac{2{\mathrm{T}}^{\prime }{c}^{\prime }}{3}$
  • Convert to the number of atoms per unit volume: $c=2c^{\prime }/a$

$J^{\alpha \to \beta }=\frac{a{\mathrm{T}}^{\prime }c}{3}$

• • Find the reverse flux by using a first-order expansion

$J^{\beta \to \alpha }=\frac{a{\mathrm{T}}^{\prime }}{3}(c+\frac{a}{2}\frac{\partial c}{\partial y})$

• • Find the net flux

$J^{\text{net}}=J^{\alpha \to \beta }-J^{\beta \to \alpha }$

$J^{\text{net}}=-\frac{a^2{\mathrm{T}}^{\prime }}{6}\frac{\partial c}{\partial y}$

• • Compare with Fick's law expression, $J^{\text{net}}=-D_I\frac{\partial c}{\partial y}$, and total jump frequency, $\mathrm{T}=4{\mathrm{T}}^{\prime }$:

$D_I=\frac{a^2{\mathrm{T}}^{\prime }}{6}$

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

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

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, ${\mathrm{T}}_V$

${\mathrm{T}}_V=12{{\mathrm{T}}_V}^{\prime }$

${{\mathrm{T}}_V}^{\prime }=\nu e^{S_V^m/k}{e}^{-H_V^m/(kT)}$

• $X_V$: fraction of sites randomly occupied by vacancies
• Jump rate of host atoms:

${\mathrm{T}}_A=X_V{\mathrm{T}}_V$

• Self-diffusivity with $r=a/\sqrt{2}$:

${}^{*}D=\frac{{\mathrm{T}}_A{r}^2𝐟}{6}$

${}^{*}D=X_V{{\mathrm{T}}_V}^{\prime }{a}^2𝐟$

• With uncorrelated vacancy diffusion, the vacancy diffusivity is

$D_V=\frac{{\mathrm{T}}_V{r}^2𝐟}{6}={{\mathrm{T}}_V}^{\prime }{a}^2$

• The vacancy diffusivity is related to the self-diffusivity

${}^{*}D=X_V{D}_V𝐟$

• $X_V=X_V^{\text{eq}}$ when the vacancies are in thermal equilibrium

$X_V^{\text{eq}}=e^{-G_v^f/(kT)}=e^{-S_v^f/(k)}{e}^{-H_v^f/(kT)}$

• • $S_v^f$: vacancy vibrational entropy
  • $H_v^f$: enthalpy of formation

 ${}^{*}D=𝐟a^2\nu e^{(S_v^m+S_v^f)/k_e}{e}^{-(H_v^m+H_v^f)/(kT)}$
 ${}^{*}D{=}^{*}{D}^{\circ }{e}^{-E/(kT)}$

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)

$K_K^{\times }+C{l}_{Cl}^{\times }={V_K}^{\prime }+V_{Cl}^{\circ }+K_K^{\times }+C{l}_{Cl}^{\times }$

$\text{null}={V_K}^{\prime }+V_{Cl}^{\circ }$

• • Relation between free energy of formation, $G_S^f$, and the equilibrium constant, $K^{\text{eq}}$

$G_s^f=-kT\ln K^{\text{eq}}$

$K^{\text{eq}}=e^{-G_s^f/(kT)}$

$K^{\text{eq}}=a_{AV}{a}_{CV}$

• • The activities correspond to anion and cation vacancies
  • With dilute concentrations of vacancies, Raoult's law applies, and activities are equal to site fractions

$\left[{V_K}^{\prime }\right]\left[V_{Cl}^{\circ }\right]=K^{\text{eq}}=e^{-G_s^f/(kT)}$

• • A requirement of electrical neutrality is that the number of potassium vacancies is equal to the number of chlorine vacancies

$\left[{V_K}^{\prime }\right]=\left[V_{Cl}^{\circ }\right]=e^{-G_s^f/(2kT)}$

• • Vacancy self-diffusion in a metal

${}^{*}{D}^K=g{a}^2𝐟\nu e^{(S_{CV}^m+S_S^f/2)/k}{e}^{-(H_{CV}^m+H_S^f/2)/(kT)}$

• • $g$: geometric factor
  • $𝐟$: 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

$A{g}_{Ag}^{\times }=A{g}_i^{\circ }+V_{A^{\prime }g}$

• $a_{A{g}_{Ag}^{\times }}=1$

$K^{\text{eq}}=\left[A{g}_i^{\circ }\right]\left[V_{A^{\prime }g}\right]=e^{-G_F^f/(kT)}$

• Elecrical neutrality condition:

$\left[A{g}_i^{\circ }\right]\left[V_{A^{\prime }g}\right]=e^{-G_F^f/(2kT)}$

• 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 $C{a}^{++}$ through doping $KCl$ with $CaC{l}_2$
  • Step 1: Two cation and two anion vacancies form
  • Step 2: Single $C{a}^{++}$ cation and two $Cl$ anions incorporated
  • Cation and anionic vacancy populations relate to the site fraction of extrinsic Ca^{++} impurity

$\left[C{a}_K^{\circ }\right]+\left[V_{Cl}^{\circ }\right]=\left[{V_K}^{\prime }\right]$

$\left[{V_K}^{\prime }\right](\left[{V_K}^{\prime }\right]-\left[C{a}_K^{\circ }\right])=e^{-G_S^f/(kT)}={\left[{V_K}^{\prime }\right]}_{\text{pure}}^2$

• • The equation can be solved to find the vacancy site fraction
  • Two limiting cases of the behavior of $\left[{V_K}^{\prime }\right]$
    • Intrinsic: ${\left[{V_K}^{\prime }\right]}_{\text{pure}}\gg \left[C{a}_K^{\circ }\right]$, then $\left[{V_K}^{\prime }\right]={\left[{V_K}^{\prime }\right]}_{\text{pure}}$
    • Extrinsic: ${\left[{V_K}^{\prime }\right]}_{\text{pure}}\ll \left[C{a}_K^{\circ }\right]$, then $\left[{V_K}^{\prime }\right]={\left[C{a}_K^{\circ }\right]}_{\text{pure}}$

Self-Diffusion in Nonstochiometric Crystals

• • Oxidation of $FeO$

$FeO+\frac{x}{2}{O}_2=Fe{O}_{1+x}$

• • Consider the sum of two reactions

$2F{e}^{++}=2F{e}^{+++}+2e^{-}$

$\frac{1}{2}{O}_2+2e^{-}=O^{--}$

$2F{e}^{++}+\frac{1}{2}{O}_2=2F{e}^{+++}+O^{--}$

• • A cation vacancy must be created with regard to every O atom added

$2F{e}_{Fe}^{\times }+\frac{1}{2}{O}_2=2F{e}_{Fe}^{\circ }+O_O^{\times }+V_{F^{″}e}$

• • Relationship between cation vacancy site fraction and oxygen gas pressure

$\frac{1}{2}=O_O^{\times }+V_{F^{″}e}+2h_{Fe}^{\circ }$

$h_{Fe}^{\circ }=F{e}_{Fe}^{\circ }-F{e}_{Fe}^{\times }$

• • Equilibrium constant of the reaction:

$K^{\text{eq}}=\frac{\left[V_{F^{″}e}\right]{\left[h_{Fe}^{\circ }\right]}^2}{P_{O_2^{1/2}}}=e^{-\mathrm{\Delta }G/(kT)}$

• • Electrical neutrality condition with oxidation-induced cation vacancies as dominant charged defects

$\left[h_{Fe}^{\circ }\right]=2\left[V_{F^{″}e}\right]$

• • Solve to find $\left[V_{F^{″}e}\right]$

$\left[V_{F^{″}e}\right]={\left(\frac{1}{4}\right)}^{1/3}{e}^{-\mathrm{\Delta }G/(3kT)}(P_{O_2}{)}^{1/6}$

• • Activation energy

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