Materials Science and Engineering/Derivations/Kinetics

Diffusion of Solute Atoms in BCC Crystal by the Interstitial Mechanism

 * Connection between jump rate, $$\Tau$$, and intersite jump distance, $$r$$, and the correlation factor:

$$D_I = \frac{\Tau r^2}{6}$$


 * Each interstitial site is associated with four nearest-neighbors
 * $$\Tau = 4 \Tau'\;$$
 * $$\Tau' = \nu e^{S^m/(kT)} e^{-H^m/(kT)}$$


 * 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 $$\beta\;$$: $$(c'/3)\Tau'\;$$
 * Contribution to the flux from the three sites: $$J^{\alpha \rightarrow \beta} = \frac{2 \Tau' c'}{3}$$
 * Convert to the number of atoms per unit volume: $$c = 2c' / a\;$$
 * $$J^{\alpha \rightarrow \beta} = \frac{a \Tau' c}{3}$$


 * Find the reverse flux by using a first-order expansion
 * $$J^{\beta \rightarrow \alpha} = \frac{a \Tau'}{3} \left ( c + \frac{a}{2} \frac{\partial c}{\partial y} \right )$$


 * Find the net flux
 * $$J^{\mbox{net}} = J^{\alpha \rightarrow \beta} - J^{\beta \rightarrow \alpha} $$
 * $$J^{\mbox{net}} = - \frac{a^2 \Tau'}{6} \frac{\partial c}{\partial y} $$


 * Compare with Fick's law expression, $$J^{\mbox{net}} = -D_I \frac{\partial c}{\partial y}$$, and total jump frequency, $$\Tau = 4 \Tau'$$:


 * $$D_I = \frac{a^2 \Tau'}{6}$$
 * $$D_I = \frac{a^2 \Tau}{24}$$

$$D_I = \frac{\Tau r^2}{6}$$

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, $$\Tau_V$$
 * $$\Tau_V = 12 \Tau_V'\;$$
 * $$\Tau_V' = \nu e^{S_V^m/k} e^{-H_V^m/(kT)}$$


 * $$X_V\;$$: fraction of sites randomly occupied by vacancies
 * Jump rate of host atoms:
 * $$\Tau_A = X_V \Tau_V\;$$


 * Self-diffusivity with $$r = a/\sqrt{2}$$:
 * $${}^*D = \frac{\Tau_A r^2 \mathbf{f}}{6}$$
 * $${}^*D = X_V \Tau_V' a^2 \mathbf{f}$$


 * With uncorrelated vacancy diffusion, the vacancy diffusivity is
 * $$D_V = \frac{\Tau_V r^2 \mathbf f}{6} = \Tau_V' a^2$$


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

$${}^*D = \mathbf{f} a^2 \nu e^{(S_v^m + S_v^f)/k_e} e^{-(H_v^m+H_v^f)/(kT)}$$ $${}^*D = ^*D^{\circ} e^{-E/(kT)}$$
 * $$S_v^f\;$$: vacancy vibrational entropy
 * $$H_v^f\;$$: enthalpy of formation

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
 * $$2Fe_{Fe}^{\times} + \frac{1}{2} O_2 = 2 Fe_{Fe}^{\circ} + O_O^{\times} + V_{Fe}''$$


 * 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


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