Materials Science and Engineering/Equations/Kinetics

Mathematical Foundation

Time-Dependent Field

 $\frac{dc}{dt}=\mathrm{\nabla }c\cdot \overrightarrow{v}+\frac{\partial c}{\partial t}$

$\overrightarrow{v}(\overrightarrow{r})$: Velocity

$c(\overrightarrow{r},t)$: Time-Dependent Field

Accumulation

Rate of accumulation is the negative of the divergence of the flux of the quantity plus the rate of production

 $\frac{\partial c_i}{\partial t}=-\mathrm{\nabla }\cdot \overrightarrow{J_i}+\dot{{\rho }_i}$

$\dot{{\rho }_i}(\overrightarrow{r})$: Rate of production of the density of $i$ in $\mathrm{\Delta }V$

$\mathrm{\nabla }\cdot \overrightarrow{J_i}$: The divergence of $\overrightarrow{J_i}$

 $\dot{M_i}=\underset{\mathrm{\Delta }V}{\int }-\mathrm{\nabla }\cdot \overrightarrow{J_i}+\dot{{\rho }_i}dx$

$\dot{M_i}$: Rate at which $i$ flows through area $\mathrm{\Delta }\overrightarrow{A}$

Divergence Theorem

 $\underset{\mathrm{B}(\mathrm{\Delta }V)}{\int }\overrightarrow{J}\cdot \dot{n}dA=\underset{\mathrm{\Delta }V}{\int }\mathrm{\nabla }\cdot \overrightarrow{J}dV$

$\mathrm{B}(\mathrm{\Delta }V)$: Oriented surface around a volume

General Set of Linear Equations

 $\begin{array}{cccccccccccccc}{M}_{11}{x}_1& & +& & M_{12}{x}_2& & +\cdots +& & M_{1n}{x}_n& & =& & & y_1\\ M_{21}{x}_1& & +& & M_{22}{x}_2& & +\cdots +& & M_{2n}{x}_n& & =& & & y_2\\ ⋮& & & & ⋮& & & & ⋮& & & & & ⋮\\ M_{m1}{x}_1& & +& & M_{m2}{x}_2& & +\cdots +& & M_{mn}{x}_n& & =& & & y_m\\ \end{array}$

The vector equation is equivalent to a matrix equation of the form

 $M𝐱=𝐲$

where M is an m×n matrix, x is a column vector with n entries, and y is a column vector with m entries.

$M=\left[\begin{array}{cccc}{M}_{11}& M_{12}& \cdots & M_{1n}\\ M_{21}& M_{22}& \cdots & M_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ M_{m1}& M_{m2}& \cdots & M_{mn}\end{array}\right],𝐱=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right],𝐲=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_m\end{array}\right]$

Eigenvalue Equation

 $\underset{\_}{M}\overrightarrow{e}=\lambda \overrightarrow{e}$

$\underset{\_}{M}$: $nxn$ square matrix or tensor

$\overrightarrow{e}$: eigenvector (special vector)

$\lambda$: eigenvalue (special scalar multiplier)

Transformation of Rank-Two Tensor

 $\left[\text{diagonalized matrix}\right]={\left[\text{eigenvector column matrix}\right]}^{-1}\left[\text{square matrix}\right]\left[\text{eigenvector column matrix}\right]$

Irreversible Thermodynamics

Differential Change in Entropy

 $TdS=du-\sum _j{\mathrm{\Psi }}_{j}d{\zeta }_j$

$\sum _j{\mathrm{\Psi }}_{j}d{\zeta }_j=-Pdv+\varphi dq+\gamma dA+{\mu }_{1}d{c}_1+\cdots$

Entropy Production

 $T\dot{\sigma }=-\frac{\overrightarrow{J_Q}}{T}\cdot \mathrm{\nabla }T-\sum _j\overrightarrow{J_i}\cdot \mathrm{\nabla }{\mathrm{\Psi }}_j$

$\dot{\sigma }$: Rate of entropy-density creation

$\overrightarrow{J_Q}$: Flux of heat

$\overrightarrow{J_i}$: Conjugate force

$\mathrm{\nabla }{\mathrm{\Psi }}_j$: Conjugate flux

Empirical Force-Flux Law

Fourier's

 $\overrightarrow{J_Q}=-K\mathrm{\nabla }T$

Modified Fick's

 $\overrightarrow{J_i}=-M_i{c}_i\mathrm{\nabla }{\mu }_i$

Ohm's

 $\overrightarrow{J_q}=-\rho \mathrm{\nabla }\varphi$

Basic Postulate of Irreversible Thermodynamics

The local generation of entropy, $\dot{\sigma }$ is nonnegative

 $\dot{\sigma }=\frac{\partial s}{\partial t}+\mathrm{\nabla }\cdot \overrightarrow{J_Q}\ge 0$

Coupling Between Forces and Fluxes

 $\begin{array}{cccccccccccccc}\frac{\partial J_Q}{\partial F_Q}{F}_Q& & +& & \frac{\partial J_Q}{\partial F_q}{F}_q& & +\cdots +& & \frac{\partial J_Q}{\partial F_{N_c}}{F}_{N_c}& & =& & & J_Q(F_Q,F_q,F_1,F_2,...,F_{N_c})\\ \frac{\partial J_q}{\partial F_Q}{F}_Q& & +& & \frac{\partial J_q}{\partial F_q}{F}_q& & +\cdots +& & \frac{\partial J_q}{\partial F_{N_c}}{F}_{N_c}& & =& & & J_q(F_Q,F_q,F_1,F_2,...,F_{N_c})\\ ⋮& & & & ⋮& & & & ⋮& & & & & ⋮\\ \frac{\partial J_{N_c}}{\partial F_Q}{F}_Q& & +& & \frac{\partial J_{N_c}}{\partial F_q}{F}_q& & +\cdots +& & \frac{\partial J_{N_c}}{\partial F_{N_c}}{F}_{N_c}& & =& & & J_{N_c}(F_Q,F_q,F_1,F_2,...,F_{N_c})\\ \end{array}$

Abbreviated form:

 $J_{\alpha }=\sum _{\beta }{L}_{\alpha \beta }{F}_{\beta }$

$L_{\alpha \beta }=\frac{\partial J_{\alpha }}{\partial F_{\beta }}$

Force-Flux Relations with Constrained Extensive Quantities

 $TdS=du+dw-\sum _{i=1}^{N_c-1}({\mu }_i-{\mu }_{N_c})d{c}_i$

$\overrightarrow{F_i}=-\mathrm{\nabla }({\mu }_i-{\mu }_{N_c})$

Diffusion Potential

 $\overrightarrow{F_1}=-\mathrm{\nabla }{\mathrm{\Phi }}_1$

Onsager Symmetry Principle

 $L_{\alpha \beta }=L_{\beta \alpha }$

 $\frac{\partial J_{\alpha }}{\partial F_{\beta }}=\frac{\partial J_{\beta }}{\partial F_{\alpha }}$

Driving Forces and Fluxes

Diffusion in Absence of Chemical Effects

• Components diffuse in chemically homogeneous material
• Diffusion measured with radioactive tracer
• Fick's law flux equation derived when self-diffusion occurs by the vacancy-exchange mechanism.
• The crystal is network-constrained
• There are three components:
  • Inert atoms
  • Radioactive atoms
  • Vacancies
• C-frame: single reference frame
• Vacancies assumed to be in equilibrium throughout
• Raoultian behavior

 $J_{{}^{*}1}^C=-kT[\frac{L_{11}}{c_1}-\frac{L_{1^{*}1}}{c_{{}^{*}1}}]\frac{\partial c_{{}^{*}1}}{\partial x}$

 $J_{{}^{*}1}^C={-}^{*}D\frac{\partial c_{{}^{*}1}}{\partial x}$

Diffusion of i in Chemically Homogeneous Binary Solution

 $J_{{}^{*}1}^C=-kT[\frac{L_{11}}{c_1}-\frac{L_{1^{*}1}}{c_{{}^{*}1}}]\frac{\partial c_{{}^{*}1}}{\partial x}$
 $J_{{}^{*}1}^C={-}^{*}{D}_1\frac{\partial c_{{}^{*}1}}{\partial x}$

Diffusion of Substitutional Particles in Concentration Gradient

• Constraint associated with vacancy mechanism: $\overrightarrow{J_1^c}+\overrightarrow{J_2^c}+\overrightarrow{J_v^c}=0$
  • Difference in fluxes of the two substitutional species requires net flux of vacancies.
• Gibbs-Duhem relation: $c_1\frac{\partial {\mu }_1}{\partial x}+c_2\frac{\partial {\mu }_2}{\partial x}+c_v\frac{\partial {\mu }_v}{\partial x}$
• Chemical potential gradients related to concentration gradients: ${\mu }_i={\mu }_i^{\circ }+kT\ln ({\gamma }_i<\mathrm{\Omega }>c_i)$

Flux is proportional to the concentration gradient

 $J_1^c=-kT[\frac{L_{11}}{c_1}-\frac{L_{12}}{c_2}][1+\frac{\partial \ln {\gamma }_1}{\partial \ln c_1}+\frac{\partial \ln <\mathrm{\Omega }>}{\partial \ln c_1}]\frac{\partial c_1}{\partial x}$
 $J_1^c=-D_1\frac{\partial c_1}{\partial x}$

Assumptions that simplify $D_1$

• Concentration-independent average site volume $<\mathrm{\Omega }>$
• The coupling (off-diagonal) terms, $L_{12}/c_2$ and $L_{1^{*}1}/c_{{}^{*}1}$, are small compared with the direct term $L_{11}/c_2$

 $D_1\approx [1+\frac{\partial \ln {\gamma }_1}{\partial \ln c_1}]{}^{*}{D}_1$

Diffusion in a Volume-Fixed (V-Frame)

• Velocity of a local C-frame with respect to the V-frame: velocity of any inert marker with respect to the V-frame
• Flux of 1 in the V-frame:

 $J_1^v=-[c_1{\mathrm{\Omega }}_1{D}_2+c_2+{\mathrm{\Omega }}_2{D}_1]\frac{\partial c_1}{\partial x}$

• The interdiffusivity, $c_1{\mathrm{\Omega }}_1{D}_2+c_2+{\mathrm{\Omega }}_2{D}_1$, can be simplified through ${\mathrm{\Omega }}_1={\mathrm{\Omega }}_2=<\mathrm{\Omega }>$
• The L-frame and the V-frame are the same

Diffusion of Interstitial Particles in Concentration Gradient

• $\overrightarrow{J_1^c}=L_{11}\overrightarrow{F_1}$
• $\overrightarrow{J_1^c}=-L_{11}\mathrm{\nabla }{\mathrm{\Phi }}_1$
• $\overrightarrow{J_1^c}=-L_{11}\mathrm{\nabla }{\mu }_1$
• ${\mu }_1={\mu }_1^{\circ }+kT\ln (K_1{c}_1)$
• $\mathrm{\nabla }{\mu }_1=\frac{kT}{c_1}\mathrm{\nabla }{c}_1$

 $\overrightarrow{J_1^c}=-L_{11}\frac{kT}{c_1}\mathrm{\nabla }{c}_1$

• Evaluate $L_{11}$ by substitution of interstitial mobility, $M_1$
  • $\overrightarrow{v_1^c}=-M_1\mathrm{\nabla }{\mu }_1$
  • $\overrightarrow{v_1^c}=\frac{-M_{1}kT}{c_1}\mathrm{\nabla }{c}_1$
  • $\overrightarrow{J_1^c}=\overrightarrow{v_1^c}{c}_1$

 $\overrightarrow{J_1^c}=-M_{1}kT\mathrm{\nabla }{c}_1$

Diffusion of Charged Ions in Ionic Conductors

• $\overrightarrow{J_1}=L_{11}\overrightarrow{F_1}$
• $\overrightarrow{J_1}=-L_{11}\mathrm{\nabla }{\mathrm{\Phi }}_1$
• $\overrightarrow{J_1}=-L_{11}\mathrm{\nabla }({\mu }_1+q_1\varphi )$

 $\overrightarrow{J_1}=-D_1\mathrm{\nabla }{c}_1-\frac{D_1{c}_1{q}_1}{kT}\mathrm{\nabla }(\varphi )$

• $\overrightarrow{E}=-\mathrm{\nabla }\varphi$: Electric field
• Absence of concentration gradient:
  • $\overrightarrow{J_q}=q_1\overrightarrow{J_1}$
  • $\overrightarrow{J_q}=-\frac{D_1{c}_1{q}_1^2}{kT}\mathrm{\nabla }(\varphi )$

• Electrical conductivity:
  • $\rho =\frac{D_1{c}_1{q}_1^2}{kT}$

Electromigration in Metals

• Two fluxes when electric field is applied to a dilute solution of interstitial atoms in metal
  • $J_q$: Flux of conjuction electrons
  • $J_1$: Flux of interstitials
• $F_q=E$
• $F_q=-\mathrm{\nabla }\varphi$

 $\overrightarrow{J_1}=-L_{11}\mathrm{\nabla }{\mu }_1+L_{1q}\overrightarrow{E}$
 $\overrightarrow{J_1}=-D_1(\mathrm{\nabla }{c}_1-\frac{c_1\beta }{kT}\overrightarrow{E})$

Mass Diffusion in Thermal Gradient

• Interstitial flux with thermal gradient where both heat flow and mass diffusion of interstitial component occurs:

 $\overrightarrow{J_1}=-L_{11}\mathrm{\nabla }{\mu }_1-\frac{L_{1q}}{T}\mathrm{\nabla }T$
 $\overrightarrow{J_1}=-D_1\mathrm{\nabla }{c}_1-\frac{D_1{c}_1{Q}_1^{\text{trans}}}{k{T}^2}\mathrm{\nabla }T$

Mass Diffusion Driven by Capillarity

• The system consists of two network-constrained components:
  • Host atoms
  • Vacancies
• No mass flow within the crystal (the crystal C-frame is also the V-frame)
• Constant temperature and no electric field
• $\overrightarrow{J_A}=L_{AA}\overrightarrow{F_A}$
• $\overrightarrow{J_A}=-L_{AA}\mathrm{\nabla }{\mathrm{\Phi }}_A$
• $\overrightarrow{J_A}=-L_{AA}\mathrm{\nabla }({\mu }_A-{\mu }_v)$
• $\overrightarrow{J_V}=-\overrightarrow{J_A}$

Fick's Second Law

Diffusion Equation in the General Form

 $\frac{\partial c}{\partial t}=\dot{n}-\mathrm{\nabla }\cdot \overrightarrow{J}$

$\dot{n}$: source or sink term

$\overrightarrow{J}$: any flux in a V-frame

Fick's Second Law

 $\frac{\partial c}{\partial t}=\mathrm{\nabla }\cdot \overrightarrow{J}$
 $\frac{\partial c}{\partial t}=\mathrm{\nabla }\cdot (D\mathrm{\nabla }c)$

Linearization of Diffusion Equation

 $\frac{\partial c}{\partial t}=D_o{\mathrm{\nabla }}^{2}c$

Heat Equation

 $\frac{\partial h}{\partial t}=-\mathrm{\nabla }\cdot \overrightarrow{J_Q}$
 $c_P\frac{\partial T}{\partial t}=-\mathrm{\nabla }\cdot (-K\mathrm{\nabla }T)$
 $\frac{\partial T}{\partial t}=\mathrm{\nabla }\cdot \left(\frac{K}{c_P}\mathrm{\nabla }T\right)$
 $\mathrm{\nabla }\cdot (\kappa \mathrm{\nabla }T)$

$h$: enthalpy density

$c_P$: heat capacity

$K/c_p=\kappa$: thermal diffusivity

Constant Diffusivity

$\frac{\partial c}{\partial t}=D{\mathrm{\nabla }}^{2}c$

One-Dimensional Diffusion Along x from an Initial Step Function

$c(x,t)=\overline{c}+\frac{\mathrm{\Delta }c}{2}\text{erf}\left(\frac{x}{\sqrt{4Dt}}\right)$

Localized Source

 $c(x,t)=\frac{c_o\mathrm{\Delta }x}{\sqrt{4\pi Dt}}{e}^{-x^2/(4Dt)}$
 $c(x,t)=\frac{n_d}{\sqrt{4\pi Dt}}{e}^{-x^2/(4Dt)}$

• Source strength, $n_d=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}c(x)dx$

Diffusivity as a Function of Concentration

 $\frac{\partial c}{\partial t}=\mathrm{\nabla }\cdot [D(c)\mathrm{\nabla }c]$

• Interdiffusivity: $D(c_1)=-\frac{1}{2\tau }\frac{dx}{d{c}_1}\underset{c_1^R}{\overset{c_1}{\int }}x(c^{\prime })d{c}^{\prime }$

Diffusivity as a Function of Time

 $\frac{\partial c}{\partial t}=\mathrm{\nabla }\cdot [D(t)\mathrm{\nabla }c]$
 $\frac{\partial c}{\partial t}=D(t){\mathrm{\nabla }}^{2}c]$

• Change of variable: ${\tau }_D=\underset{0}{\overset{t}{\int }}D(t^{\prime })d{t}^{\prime }$
• Transformed equation: $\frac{\partial c}{\partial {\tau }_D}={\mathrm{\nabla }}^{2}c$
• Solution:

$c(x,{\tau }_D)=\overline{c}+\frac{\mathrm{\Delta }c}{2}\text{erf}\left(\frac{x}{\sqrt{4{\tau }_D}}\right)$

$c(x,t)=\overline{c}+\frac{\mathrm{\Delta }c}{2}\text{erf}\left(\frac{x}{\sqrt{4\underset{0}{\overset{t}{\int }}D(t^{\prime })d{t}^{\prime }}}\right)$

Diffusivity as a Function of Direction

 $\overrightarrow{J}=-𝐃\mathrm{\nabla }c$

• The diagonal elements of $\widehat{𝐃}$ are the eigenvalues of $𝐃$, and the coordinate system of $\widehat{𝐃}$ defines the principal axes.

• • $\frac{\partial c}{\partial t}=-\mathrm{\nabla }\cdot \overrightarrow{J}$
  • $\frac{\partial c}{\partial t}=\mathrm{\nabla }\cdot \widehat{𝐃}\mathrm{\nabla }c$

• Relation of $𝐃$ and $\widehat{𝐃}$:

 $\widehat{𝐃}=\underset{\_}{R}𝐃{\underset{\_}{R}}^{-1}$

Steady-State Solutions

Harmonic Functions

 ${\mathrm{\nabla }}^{2}c=0$

One Dimension

 $J=-D\frac{dc}{dx}$
 $J=D\frac{c^0-c^L}{L}$

Cylindrical Shell

• Laplacian Operator: ${\mathrm{\nabla }}^{2}c=\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial c}{\partial r}\right)+\frac{1}{r^2}\frac{{\partial }^{2}c}{\partial {\theta }^2}+\frac{{\partial }^{2}c}{\partial x^2}$
• Integrate Twice and Apply the Boundary Conditions:

 $c(r)=c^{\text{in}}-\frac{c^{\text{in}}-c^{\text{out}}}{\ln (r^{\text{out}}/r^{\text{in}})}\ln \left(\frac{r}{r^{\text{in}}}\right)$

Spherical Shell

• Laplacian operator in spherical coordinates

${\mathrm{\nabla }}^{2}c=\frac{1}{r^2}[\frac{\partial }{\partial r}\left(r^2\frac{\partial c}{\partial r}\right)+\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }(\sin \theta \frac{\partial c}{\partial \theta })+\frac{1}{{\sin }^2\theta }\frac{{\partial }^{2}c}{\partial {\varphi }^2}]$

Variable Diffusivity

• Steady-state conditions
• $D$ varies with position

 $\partial \cdot (D\mathrm{\nabla }c)=0$

• Solution is obtained by integration:

 $c(x)=c(x_1)+a_1\underset{x_1}{\overset{x}{\int }}\frac{d\zeta }{D(\zeta )}$

Infinite Media with Instantaneous Localized Source

 $c(x,y,z,t)=\frac{n_{d_x}}{\sqrt{4\pi Dt}}{e}^{-x^2/(4Dt)}\times \frac{n_{d_y}}{\sqrt{4\pi Dt}}{e}^{-y^2/(4Dt)}\times \frac{n_{d_z}}{\sqrt{4\pi Dt}}{e}^{-z^2/(4Dt)}$

Solutions with the Error Function

• Uniform distribution of point, line, or plana source placed along $x>0$
• Contribution at a general position $x$ from the source:

$c_{\zeta }(x,t)=\frac{c_{o}d\zeta }{\sqrt{4\pi Dt}}{e}^{-(\zeta -x{)}^2/(4Dt)}$

• Integral over all sources:

$c(x,t)=\frac{c_o}{\sqrt{4\pi Dt}}\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-(\zeta -x{)}^2/(4Dt)}d\zeta$

 $c(x,t)=\frac{c_o}{2}+\frac{c_o}{2}\text{erf}\left(\frac{x}{2\sqrt{Dt}}\right)$

Method of Separation of Variables

• System : Three Dimensions, $(x,y,z)$
• Equation : $\frac{dc}{dt}=D{\mathrm{\nabla }}^{2}c$
• Solution : $c(r,\theta ,z,t)=R(r)\mathrm{\Theta }(\theta )Z(z)T(t)$

Method of Laplace Transforms

• Laplace transform of a function $f(x,t)$

 $L\{f(x,t)\}=\hat{f}(x,p)$
 $L\{f(x,t)\}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{e}^{-pt}f(x,t)dt$

Atomic Models of Diffusion

Model of One-Particle with Step Potential-Energy Wells

 ${\mathrm{T}}^{\prime }=\sqrt{\frac{kT}{2\pi m}}\frac{1}{L^{\text{well}}}{e}^{-(E^A-E^{\text{well}})/(kT)}$
 ${\mathrm{T}}^{\prime }=\sqrt{\frac{kT}{2\pi m}}\frac{1}{L^{\text{well}}}{e}^{-(E^m)/(kT)}$

Model of One-Particle with Step Potential-Energy Wells

 ${\mathrm{T}}^{\prime }=\frac{1}{2\pi }\sqrt{\frac{\beta }{m}}{e}^{-(E^A-E^{\text{well}})/(kT)}$
 ${\mathrm{T}}^{\prime }=\nu e^{-E^m/(kT)}$

Many-Body Model

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

Diffusion as Series of Discrete Jumps

 $<R^2(N_{\tau })>=N_{\tau }<r^2>+2<(\sum _{j=1}^{N_{\tau }-1}\sum _{i=1}^{N_{\tau -j}}|\overrightarrow{r_i}||{\overrightarrow{r}}_{i+j}|\cos {\theta }_{i,i+j}>$

Diffusivity and Mean-Square Particle Displacement

 $<R^2(\tau )>=6D\tau$

Relation of Macroscopic Diffusivity and Microscopic Jump Parameters

 $D=\frac{\mathrm{T}<r^2>}{2}$

Diffusion and Correlated Jumps

• Correlation factor:

 $𝐟=1+\frac{2}{N_{\tau }<r^2>}<(\sum _{j=1}^{N_{\tau }-1}\sum _{i=1}^{N_{\tau -j}}|\overrightarrow{r_i}||{\overrightarrow{r}}_{i+j}|\cos {\theta }_{i,i+j}>$

• Macroscopic Diffusivity and Microscopic Parameters:

 $D=\frac{<r^2>N_{\tau }}{6\tau }𝐟$
 $D=\frac{\mathrm{T}\tau <r^2>}{6\tau }𝐟$
 $D=\frac{\mathrm{T}<r^2>}{6}𝐟$

Atomic Models of Diffusivity

Metals

Correlation Factor

 $𝐟=\frac{1+<\cos \theta >}{1-<\cos \theta >}$
 $𝐟\approx \frac{z-1}{z+1}$

Isotope Effect

 $\frac{{}^{*}D(mass1)}{{}^{*}D(mass2)}=\frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}=\frac{\mathrm{T}{\text{'}}_1}{\mathrm{T}{\text{'}}_2}=\frac{{\nu }_1}{{\nu }_2}=\frac{m_1}{m_2}$