Materials Science and Engineering/Equations/Kinetics

Time-Dependent Field
$$\frac{dc}{dt} = \nabla c \cdot \overrightarrow{v} + \frac{\partial c}{\partial t}$$


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

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


 * $$\dot{\rho_i}(\overrightarrow{r})$$: Rate of production of the density of $$i$$ in $$\Delta V$$
 * $$\nabla \cdot \overrightarrow{J_i}$$: The divergence of $$\overrightarrow{J_i}$$

$$\dot{M_i} = \int_{\Delta V} - \nabla \cdot \overrightarrow{J_i} + \dot{\rho_i}\,dx$$


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

Divergence Theorem
$$\int_{\Beta (\Delta V)} \overrightarrow{J} \cdot \dot{n} \,dA = \int_{\Delta V} \nabla \cdot \overrightarrow{J} \,dV$$


 * $$\Beta (\Delta V)$$: Oriented surface around a volume

General Set of Linear Equations
$$\begin{alignat}{7} M_{11} x_1 &&\; + \;&& M_{12} x_2 &&\; + \cdots + \;&& M_{1n} x_n &&\; = \;&&& y_1     \\ M_{21} x_1 &&\; + \;&& M_{22} x_2 &&\; + \cdots + \;&& M_{2n} x_n &&\; = \;&&& y_2     \\ \vdots\;\;\; &&    && \vdots\;\;\; &&              && \vdots\;\;\; &&     &&& \;\vdots \\ M_{m1} x_1 &&\; + \;&& M_{m2} x_2 &&\; + \cdots + \;&& M_{mn} x_n &&\; = \;&&& y_m     \\ \end{alignat}$$

The vector equation is equivalent to a matrix equation of the form $$M\mathbf{x}=\mathbf{y}$$ 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= \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
$$\underline{M} \overrightarrow{e} = \lambda \overrightarrow{e}$$


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

Transformation of Rank-Two Tensor
$$\left [ \mbox{diagonalized matrix} \right ] = \left [ \mbox{eigenvector column matrix} \right ]^{-1} \left [ \mbox{square matrix} \right ] \left [ \mbox{eigenvector column matrix} \right ]$$

Differential Change in Entropy
$$T dS = du - \sum_j \Psi_j d \zeta_j\;$$


 * $$\sum_j \Psi_j d \zeta_j = -Pdv + \phi dq + \gamma dA + \mu_1 dc_1+\cdots\;$$

Entropy Production
$$T \dot{\sigma} = - \frac{\overrightarrow{J_Q}}{T} \cdot \nabla T - \sum_j \overrightarrow{J_i} \cdot \nabla \Psi_j $$


 * $$\dot{\sigma}$$: Rate of entropy-density creation
 * $$\overrightarrow{J_Q}$$: Flux of heat
 * $$\overrightarrow{J_i}$$: Conjugate force
 * $$\nabla \Psi_j$$: Conjugate flux

Fourier's
$$\overrightarrow{J_Q} = -K \nabla T$$

Modified Fick's
$$\overrightarrow{J_i} = -M_i c_i \nabla \mu_i$$

Ohm's
$$\overrightarrow{J_q} = - \rho \nabla \phi$$

Basic Postulate of Irreversible Thermodynamics
The local generation of entropy, $$\dot{\sigma}$$ is nonnegative

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

Coupling Between Forces and Fluxes
$$\begin{alignat}{7} \frac{\partial J_Q}{\partial F_Q} F_Q &&\; + \;&& \frac{\partial J_Q}{\partial F_q} F_q &&\; + \cdots + \;&& \frac{\partial J_Q}{\partial F_{N_c}} F_{N_c} &&\; = \;&&& J_Q (F_Q, F_q, F_1, F_2, ..., F_{N_c})      \\ \frac{\partial J_q}{\partial F_Q} F_Q &&\; + \;&& \frac{\partial J_q}{\partial F_q} F_q &&\; + \cdots + \;&& \frac{\partial J_q}{\partial F_{N_c}} F_{N_c} &&\; = \;&&& J_q (F_Q, F_q, F_1, F_2, ..., F_{N_c})      \\ \vdots\;\;\; &&    && \vdots\;\;\; &&              && \vdots\;\;\; &&     &&& \;\vdots \\ \frac{\partial J_{N_c}}{\partial F_Q} F_Q &&\; + \;&& \frac{\partial J_{N_c}}{\partial F_q} F_q &&\; + \cdots + \;&& \frac{\partial J_{N_c}}{\partial F_{N_c}} F_{N_c} &&\; = \;&&& J_{N_c} (F_Q, F_q, F_1, F_2, ..., F_{N_c})      \\ \end{alignat}$$

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
$$TdS = du + dw - \sum_{i=1}^{N_c - 1} \left ( \mu_i - \mu_{N_c} \right ) dc_i$$


 * $$\overrightarrow{F_i} = - \nabla \left ( \mu_i - \mu_{N_c} \right )$$

Diffusion Potential
$$\overrightarrow{F_1} = - \nabla \Phi_1$$

Onsager Symmetry Principle
$$L_{\alpha \beta} = L_{\beta \alpha}\;$$

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

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

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

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

Diffusion of i in Chemically Homogeneous Binary Solution
$$J_{^*1}^C = -kT \left [ \frac{L_{11}}{c_1} - \frac{L_{1^*1}}{c_{^*1}} \right ] \frac{\partial c_{^*1}}{\partial x}$$ $$J_{^*1}^C = -^*D_1 \frac{\partial c_{^*1}}{\partial x}$$

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: $$\mu_i = \mu_i^{\circ} + kT \ln ( \gamma_i < \Omega > c_i )$$

Flux is proportional to the concentration gradient

$$J_1^c = -k T \left[ \frac{L_{11}}{c_1} - \frac{L_{12}}{c_2} \right ] \left [ 1+\frac{\partial \ln \gamma_1}{\partial \ln c_1} + \frac{\partial \ln < \Omega >}{\partial \ln c_1} \right ] \frac{\partial c_1}{\partial x}$$ $$J_1^c = -D_1 \frac{\partial c_1}{\partial x}$$

Assumptions that simplify $$D_1\;$$


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

$$D_1 \approx \left [ 1 + \frac{\partial \ln \gamma_1}{\partial \ln c_1} \right ] {}^*D_1$$

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:

$$J_1^v = - [c_1 \Omega_1 D_2 + c_2 + \Omega_2 D_1 ] \frac{\partial c_1}{\partial x}$$


 * 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

 * $$\overrightarrow{J_1^c} = L_{11} \overrightarrow{F_1}$$
 * $$\overrightarrow{J_1^c} = -L_{11} \nabla \Phi_1$$
 * $$\overrightarrow{J_1^c} = -L_{11} \nabla \mu_1$$
 * $$\mu_1 = \mu_1^{\circ} + kT \ln (K_1 c_1)$$
 * $$\nabla \mu_1 = \frac{kT}{c_1} \nabla c_1$$

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


 * Evaluate $$L_{11}$$ by substitution of interstitial mobility, $$M_1$$
 * $$\overrightarrow{v_1^c} = -M_1 \nabla \mu_1$$
 * $$\overrightarrow{v_1^c} = \frac{-M_1 k T}{c_1} \nabla c_1$$
 * $$\overrightarrow{J_1^c} = \overrightarrow{v_1^c} c_1$$

$$\overrightarrow{J_1^c} = -M_{1} kT \nabla c_1$$

Diffusion of Charged Ions in Ionic Conductors
$$\overrightarrow{J_1} = -D_1 \nabla c_1 - \frac{D_1 c_1 q_1}{kT} \nabla (\phi)$$
 * $$\overrightarrow{J_1} = L_{11} \overrightarrow{F_1}$$
 * $$\overrightarrow{J_1} = -L_{11} \nabla \Phi_1$$
 * $$\overrightarrow{J_1} = -L_{11} \nabla (\mu_1 + q_1 \phi)$$


 * $$\overrightarrow{E} = - \nabla \phi \;$$: 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
 * $$F_q = E\;$$
 * $$F_q = - \nabla \phi$$

$$\overrightarrow{J_1} = -L_{11} \nabla \mu_1 + L_{1q} \overrightarrow{E}$$ $$\overrightarrow{J_1} = -D_1 \left ( \nabla c_1 - \frac{c_1 \beta}{k T} \overrightarrow{E} \right )$$

Mass Diffusion in Thermal Gradient

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

$$\overrightarrow{J_1} = -L_{11} \nabla \mu_1 - \frac{L_{1q}}{T} \nabla T$$ $$\overrightarrow{J_1} = -D_1 \nabla c_1 - \frac{D_1 c_1 Q_1^{\mbox{trans}}}{k T^2} \nabla T$$

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
 * $$\overrightarrow{J_A} = L_{AA} \overrightarrow{F_A}$$
 * $$\overrightarrow{J_A} = - L_{AA} \nabla \Phi_A$$
 * $$\overrightarrow{J_A} = - L_{AA} \nabla ( \mu_A - \mu_v )$$
 * $$\overrightarrow{J_V} = - \overrightarrow{J_A}$$

Diffusion Equation in the General Form
$$\frac{\partial c}{\partial t} = \dot n - \nabla \cdot \overrightarrow{J}$$


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

Fick's Second Law
$$\frac{\partial c}{\partial t} = \nabla \cdot \overrightarrow{J}$$ $$\frac{\partial c}{\partial t} = \nabla \cdot (D \nabla c)$$

Linearization of Diffusion Equation
$$\frac{\partial c}{\partial t} = D_o \nabla^2 c$$

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


 * $$h\;$$: 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
$$c(x,t) = \bar {c} + \frac{\Delta c}{2} \mbox{erf} \left ( \frac{x}{\sqrt{4 D t}} \right )$$

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


 * Source strength, $$n_d = \int_{-\infty}^{\infty} c(x) dx$$

Diffusivity as a Function of Concentration
$$\frac{\partial c}{\partial t} = \nabla \cdot [D(c) \nabla c]$$


 * Interdiffusivity: $$D(c_1) = - \frac{1}{2 \tau} \frac{d x}{d c_1} \int_{c_1^R}^{c_1} x(c') dc'$$

Diffusivity as a Function of Time
$$\frac{\partial c}{\partial t} = \nabla \cdot [D(t) \nabla c]$$ $$\frac{\partial c}{\partial t} = D(t) \nabla^2 c]$$


 * 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
$$\overrightarrow{J} = - \mathbf{D} \nabla c$$


 * 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 $$\hat{\mathbf{D}}$$:

$$\hat{\mathbf{D}} = \underline{R} \mathbf D \underline{R}^{-1}$$

Harmonic Functions
$$\nabla^2 c = 0$$

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

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:

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

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

$$\partial \cdot (D \nabla c) = 0$$


 * Solution is obtained by integration:

$$c(x) = c(x_1) + a_1 \int_{x_1}^{x} \frac{d \zeta}{D( \zeta )}$$

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

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:


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


 * Integral over all sources:


 * $$c(x,t) = \frac{c_o}{\sqrt{4 \pi D t}} \int_0^{\infty} e^{-(\zeta - x)^2 / (4 D t)} d \zeta$$

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

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 $$f(x,t)\;$$

$$L\{f(x,t)\} = \hat{f} (x,p)$$ $$L\{f(x,t)\} = \int_0^{\infty} e^{-pt} f(x,t) dt$$

Model of One-Particle with Step Potential-Energy Wells
$$\Tau' = \sqrt{\frac{kT}{2 \pi m}} \frac{1}{L^{\mbox{well}}} e^{-(E^A - E^{\mbox{well}})/(kT)}$$ $$\Tau' = \sqrt{\frac{kT}{2 \pi m}} \frac{1}{L^{\mbox{well}}} e^{-(E^m)/(kT)} $$

Model of One-Particle with Step Potential-Energy Wells
$$\Tau' = \frac{1}{2 \pi} \sqrt{\frac{\beta}{m}} e^{-(E^A - E^{\mbox{well}})/(kT)}$$ $$\Tau' = \nu e^{-E^m/(kT)}$$

Many-Body Model
$$\Tau' = \nu e^{-G^m/(kT)}$$

Diffusion as Series of Discrete Jumps
$$ = N_{\tau} + 2 < ( \sum_{j=1}^{N_{\tau} - 1} \sum_{i=1}^{N_{\tau - j}} | \overrightarrow{r_i} || \overrightarrow{r}_{i+j}| \cos \theta_{i, i+j} >$$

Diffusivity and Mean-Square Particle Displacement
$$ = 6D \tau\;$$

Relation of Macroscopic Diffusivity and Microscopic Jump Parameters
$$D = \frac{\Tau }{2}$$

Diffusion and Correlated Jumps

 * Correlation factor:

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


 * Macroscopic Diffusivity and Microscopic Parameters:

$$D = \frac{ N_{\tau}}{6 \tau} \mathbf f$$ $$D = \frac{\Tau \tau}{6 \tau} \mathbf f$$ $$D = \frac{\Tau }{6} \mathbf f$$

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

Isotope Effect
$$\frac{^*D(mass 1)}{^*D(mass 2)} = \frac{\Tau_1}{\Tau_2} = \frac{\Tau'_1}{\Tau'_2} = \frac{\nu_1}{\nu_2} = \frac{m_1}{m_2}$$