# Materials Science and Engineering/Equations/Kinetics
## Mathematical Foundation
### Time-Dependent Field

    ${\overrightarrow {v}}({\overrightarrow {r}})$: 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
    ${\dot {\rho _{i}}}({\overrightarrow {r}})$: Rate of production of the density of $i$ in $\Delta V$
    $\nabla \cdot {\overrightarrow {J_{i}}}$: The divergence of ${\overrightarrow {J_{i}}}$
    ${\dot {M_{i}}}$: Rate at which $i$ flows through area $\Delta {\overrightarrow {A}}$

## Divergence Theorem

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

## General Set of Linear Equations

The vector equation is equivalent to a matrix equation of the form
where M is an m×n matrix, x is a column vector with n entries, and y is a column vector with m entries.
    $M={\begin{bmatrix}M_{11}&M_{12}&\cdots &M_{1n}\\M_{21}&M_{22}&\cdots &M_{2n}\\\vdots &\vdots &\ddots &\vdots \\M_{m1}&M_{m2}&\cdots &M_{mn}\end{bmatrix}},\quad \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}},\quad \mathbf {y} ={\begin{bmatrix}y_{1}\\y_{2}\\\vdots \\y_{m}\end{bmatrix}}$

### Eigenvalue Equation

    ${\underline {M}}$: $nxn$ square matrix or tensor
    ${\overrightarrow {e}}$: eigenvector (special vector)
    $\lambda$: eigenvalue (special scalar multiplier)

### Transformation of Rank-Two Tensor
## Irreversible Thermodynamics
### Differential Change in Entropy

    $\sum _{j}\Psi _{j}d\zeta _{j}=-Pdv+\phi dq+\gamma dA+\mu _{1}dc_{1}+\cdots \;$

### Entropy Production

    ${\dot {\sigma }}$: Rate of entropy-density creation
    ${\overrightarrow {J_{Q}}}$: Flux of heat
    ${\overrightarrow {J_{i}}}$: Conjugate force
    $\nabla \Psi _{j}$: Conjugate flux

### Basic Postulate of Irreversible Thermodynamics

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

### Coupling Between Forces and Fluxes

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

### Force-Flux Relations with Constrained Extensive Quantities

    ${\overrightarrow {F_{i}}}=-\nabla \left(\mu _{i}-\mu _{N_{c}}\right)$

### Onsager Symmetry Principle
## 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

### 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: $\mu _{i}=\mu _{i}^{\circ }+kT\ln(\gamma _{i}<\Omega >c_{i})$
Flux is proportional to the concentration gradient
Assumptions that simplify $D_{1}\;$
* Concentration-independent average site volume $<\Omega >\;$
* The coupling (off-diagonal) terms, $L_{12}/c_{2}\;$ and $L_{1^{*}1}/c_{^{*}1}\;$, are small compared with the direct term $L_{11}/c_{2}\;$

#### 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:
* The interdiffusivity, $c_{1}\Omega _{1}D_{2}+c_{2}+\Omega _{2}D_{1}\;$, can be simplified through $\Omega _{1}=\Omega _{2}=<\Omega >\;$
* The L-frame and the V-frame are the same

### Diffusion of Interstitial Particles in Concentration Gradient

* ${\overrightarrow {J_{1}^{c}}}=L_{11}{\overrightarrow {F_{1}}}$
* ${\overrightarrow {J_{1}^{c}}}=-L_{11}\nabla \Phi _{1}$
* ${\overrightarrow {J_{1}^{c}}}=-L_{11}\nabla \mu _{1}$
* $\mu _{1}=\mu _{1}^{\circ }+kT\ln(K_{1}c_{1})$
* $\nabla \mu _{1}={\frac {kT}{c_{1}}}\nabla c_{1}$
* Evaluate $L_{11}$ by substitution of interstitial mobility, $M_{1}$
** ${\overrightarrow {v_{1}^{c}}}=-M_{1}\nabla \mu _{1}$
** ${\overrightarrow {v_{1}^{c}}}={\frac {-M_{1}kT}{c_{1}}}\nabla c_{1}$
** ${\overrightarrow {J_{1}^{c}}}={\overrightarrow {v_{1}^{c}}}c_{1}$

### Diffusion of Charged Ions in Ionic Conductors

* ${\overrightarrow {J_{1}}}=L_{11}{\overrightarrow {F_{1}}}$
* ${\overrightarrow {J_{1}}}=-L_{11}\nabla \Phi _{1}$
* ${\overrightarrow {J_{1}}}=-L_{11}\nabla (\mu _{1}+q_{1}\phi )$
* ${\overrightarrow {E}}=-\nabla \phi \;$: Electric field
* Absence of concentration gradient:
** ${\overrightarrow {J_{q}}}=q_{1}{\overrightarrow {J_{1}}}$
** ${\overrightarrow {J_{q}}}=-{\frac {D_{1}c_{1}q_{1}^{2}}{kT}}\nabla (\phi )$
* 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
* $F_{q}=E\;$
* $F_{q}=-\nabla \phi$

### Mass Diffusion in Thermal Gradient

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

### 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
* ${\overrightarrow {J_{A}}}=L_{AA}{\overrightarrow {F_{A}}}$
* ${\overrightarrow {J_{A}}}=-L_{AA}\nabla \Phi _{A}$
* ${\overrightarrow {J_{A}}}=-L_{AA}\nabla (\mu _{A}-\mu _{v})$
* ${\overrightarrow {J_{V}}}=-{\overrightarrow {J_{A}}}$

## Fick's Second Law
### Diffusion Equation in the General Form

    ${\dot {n}}$: source or sink term
    ${\overrightarrow {J}}$: any flux in a V-frame

### Heat Equation

    $h\;$: enthalpy density
    $c_{P}\;$: heat capacity
    $K/c_{p}=\kappa \;$: thermal diffusivity

### Constant Diffusivity

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

### One-Dimensional Diffusion Along x from an Initial Step Function

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

### Localized Source

* Source strength, $n_{d}=\int _{-\infty }^{\infty }c(x)dx$

### Diffusivity as a Function of Concentration

* Interdiffusivity: $D(c_{1})=-{\frac {1}{2\tau }}{\frac {dx}{dc_{1}}}\int _{c_{1}^{R}}^{c_{1}}x(c')dc'$

### Diffusivity as a Function of Time

* Change of variable: $\tau _{D}=\int _{0}^{t}D(t')dt'$
* Transformed equation: ${\frac {\partial c}{\partial \tau _{D}}}=\nabla ^{2}c$
* Solution:
    $c(x,\tau _{D})={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4\tau _{D}}}}\right)$
    $c(x,t)={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4\int _{0}^{t}D(t')dt'}}}\right)$

### Diffusivity as a Function of Direction

* The diagonal elements of ${\hat {\mathbf {D} }}$ are the eigenvalues of $\mathbf {D}$, and the coordinate system of ${\hat {\mathbf {D} }}$ defines the principal axes.
** ${\frac {\partial c}{\partial t}}=-\nabla \cdot {\overrightarrow {J}}$
** ${\frac {\partial c}{\partial t}}=\nabla \cdot {\hat {\mathbf {D} }}\nabla c$
* Relation of $\mathbf {D}$ and ${\hat {\mathbf {D} }}$:

## Steady-State Solutions
### Cylindrical Shell

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

### Spherical Shell

* Laplacian operator in spherical coordinates
$\nabla ^{2}c={\frac {1}{r^{2}}}\left[{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial c}{\partial r}}\right)+{\frac {1}{\sin {\theta }}}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial c}{\partial \theta }}\right)+{\frac {1}{\sin ^{2}\theta }}{\frac {\partial ^{2}c}{\partial \phi ^{2}}}\right]$

### Variable Diffusivity

* Steady-state conditions
* $D$ varies with position
* Solution is obtained by integration:

### 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:
    $c(x,t)={\frac {c_{o}}{\sqrt {4\pi Dt}}}\int _{0}^{\infty }e^{-(\zeta -x)^{2}/(4Dt)}d\zeta$

### Method of Separation of Variables

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

### Method of Laplace Transforms

* Laplace transform of a function $f(x,t)\;$

## Atomic Models of Diffusion
### Diffusion and Correlated Jumps

* Correlation factor:
* Macroscopic Diffusivity and Microscopic Parameters: