d c d t = ∇ c ⋅ v → + ∂ c ∂ t {\displaystyle {\frac {dc}{dt}}=\nabla c\cdot {\overrightarrow {v}}+{\frac {\partial c}{\partial t}}} v → ( r → ) {\displaystyle {\overrightarrow {v}}({\overrightarrow {r}})} : Velocity c ( r → , t ) {\displaystyle c({\overrightarrow {r}},t)} : Time-Dependent Field Rate of accumulation is the negative of the divergence of the flux of the quantity plus the rate of production ∂ c i ∂ t = − ∇ ⋅ J i → + ρ i  ̇ {\displaystyle {\frac {\partial c_{i}}{\partial t}}=-\nabla \cdot {\overrightarrow {J_{i}}}+{\dot {\rho _{i}}}} ρ i  ̇ ( r → ) {\displaystyle {\dot {\rho _{i}}}({\overrightarrow {r}})} : Rate of production of the density of i {\displaystyle i} in Δ V {\displaystyle \Delta V} ∇ ⋅ J i → {\displaystyle \nabla \cdot {\overrightarrow {J_{i}}}} : The divergence of J i → {\displaystyle {\overrightarrow {J_{i}}}} M i  ̇ = ∫ Δ V − ∇ ⋅ J i → + ρ i  ̇ d x {\displaystyle {\dot {M_{i}}}=\int _{\Delta V}-\nabla \cdot {\overrightarrow {J_{i}}}+{\dot {\rho _{i}}}\,dx} M i  ̇ {\displaystyle {\dot {M_{i}}}} : Rate at which i {\displaystyle i} flows through area Δ A → {\displaystyle \Delta {\overrightarrow {A}}} ∫ B ( Δ V ) J → ⋅ n  ̇ d A = ∫ Δ V ∇ ⋅ J → d V {\displaystyle \int _{\mathrm {B} (\Delta V)}{\overrightarrow {J}}\cdot {\dot {n}}\,dA=\int _{\Delta V}\nabla \cdot {\overrightarrow {J}}\,dV} B ( Δ V ) {\displaystyle \mathrm {B} (\Delta V)} : Oriented surface around a volume M 11 x 1 + M 12 x 2 + ⋯ + M 1 n x n = y 1 M 21 x 1 + M 22 x 2 + ⋯ + M 2 n x n = y 2 ⋮ ⋮ ⋮ ⋮ M m 1 x 1 + M m 2 x 2 + ⋯ + M m n x n = y m {\displaystyle {\begin{alignedat}{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{alignedat}}} The vector equation is equivalent to a matrix equation of the form M x = y {\displaystyle 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 = [ M 11 M 12 ⋯ M 1 n M 21 M 22 ⋯ M 2 n ⋮ ⋮ ⋱ ⋮ M m 1 M m 2 ⋯ M m n ] , x = [ x 1 x 2 ⋮ x n ] , y = [ y 1 y 2 ⋮ y m ] {\displaystyle 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}}} M _ e → = λ e → {\displaystyle {\underline {M}}{\overrightarrow {e}}=\lambda {\overrightarrow {e}}} M _ {\displaystyle {\underline {M}}} : n x n {\displaystyle nxn} square matrix or tensor e → {\displaystyle {\overrightarrow {e}}} : eigenvector (special vector) λ {\displaystyle \lambda } : eigenvalue (special scalar multiplier) [ diagonalized matrix ] = [ eigenvector column matrix ] − 1 [ square matrix ] [ eigenvector column matrix ] {\displaystyle \left[{\mbox{diagonalized matrix}}\right]=\left[{\mbox{eigenvector column matrix}}\right]^{-1}\left[{\mbox{square matrix}}\right]\left[{\mbox{eigenvector column matrix}}\right]} T d S = d u − ∑ j Ψ j d ζ j {\displaystyle TdS=du-\sum _{j}\Psi _{j}d\zeta _{j}\;} ∑ j Ψ j d ζ j = − P d v + φ d q + γ d A + μ 1 d c 1 + ⋯ {\displaystyle \sum _{j}\Psi _{j}d\zeta _{j}=-Pdv+\phi dq+\gamma dA+\mu _{1}dc_{1}+\cdots \;} T σ  ̇ = − J Q → T ⋅ ∇ T − ∑ j J i → ⋅ ∇ Ψ j {\displaystyle T{\dot {\sigma }}=-{\frac {\overrightarrow {J_{Q}}}{T}}\cdot \nabla T-\sum _{j}{\overrightarrow {J_{i}}}\cdot \nabla \Psi _{j}} σ  ̇ {\displaystyle {\dot {\sigma }}} : Rate of entropy-density creation J Q → {\displaystyle {\overrightarrow {J_{Q}}}} : Flux of heat J i → {\displaystyle {\overrightarrow {J_{i}}}} : Conjugate force ∇ Ψ j {\displaystyle \nabla \Psi _{j}} : Conjugate flux J Q → = − K ∇ T {\displaystyle {\overrightarrow {J_{Q}}}=-K\nabla T} J i → = − M i c i ∇ μ i {\displaystyle {\overrightarrow {J_{i}}}=-M_{i}c_{i}\nabla \mu _{i}} J q → = − ρ ∇ φ {\displaystyle {\overrightarrow {J_{q}}}=-\rho \nabla \phi } The local generation of entropy, σ  ̇ {\displaystyle {\dot {\sigma }}} is nonnegative σ  ̇ = ∂ s ∂ t + ∇ ⋅ J Q → ≥ 0 {\displaystyle {\dot {\sigma }}={\frac {\partial s}{\partial t}}+\nabla \cdot {\overrightarrow {J_{Q}}}\geq 0} ∂ J Q ∂ F Q F Q + ∂ J Q ∂ F q F q + ⋯ + ∂ J Q ∂ F N c F N c = J Q ( F Q , F q , F 1 , F 2 , . . . , F N c ) ∂ J q ∂ F Q F Q + ∂ J q ∂ F q F q + ⋯ + ∂ J q ∂ F N c F N c = J q ( F Q , F q , F 1 , F 2 , . . . , F N c ) ⋮ ⋮ ⋮ ⋮ ∂ J N c ∂ F Q F Q + ∂ J N c ∂ F q F q + ⋯ + ∂ J N c ∂ F N c F N c = J N c ( F Q , F q , F 1 , F 2 , . . . , F N c ) {\displaystyle {\begin{alignedat}{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{alignedat}}} Abbreviated form: J α = ∑ β L α β F β {\displaystyle J_{\alpha }=\sum _{\beta }L_{\alpha \beta }F_{\beta }\;} L α β = ∂ J α ∂ F β {\displaystyle L_{\alpha \beta }={\frac {\partial J_{\alpha }}{\partial F_{\beta }}}} T d S = d u + d w − ∑ i = 1 N c − 1 ( μ i − μ N c ) d c i {\displaystyle TdS=du+dw-\sum _{i=1}^{N_{c}-1}\left(\mu _{i}-\mu _{N_{c}}\right)dc_{i}} F i → = − ∇ ( μ i − μ N c ) {\displaystyle {\overrightarrow {F_{i}}}=-\nabla \left(\mu _{i}-\mu _{N_{c}}\right)} F 1 → = − ∇ Φ 1 {\displaystyle {\overrightarrow {F_{1}}}=-\nabla \Phi _{1}} L α β = L β α {\displaystyle L_{\alpha \beta }=L_{\beta \alpha }\;} ∂ J α ∂ F β = ∂ J β ∂ F α {\displaystyle {\frac {\partial J_{\alpha }}{\partial F_{\beta }}}={\frac {\partial J_{\beta }}{\partial F_{\alpha }}}} 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 = − k T [ L 11 c 1 − L 1 ∗ 1 c ∗ 1 ] ∂ c ∗ 1 ∂ x {\displaystyle 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 ∂ c ∗ 1 ∂ x {\displaystyle J_{^{*}1}^{C}=-^{*}D{\frac {\partial c_{^{*}1}}{\partial x}}} J ∗ 1 C = − k T [ L 11 c 1 − L 1 ∗ 1 c ∗ 1 ] ∂ c ∗ 1 ∂ x {\displaystyle 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 ∂ c ∗ 1 ∂ x {\displaystyle J_{^{*}1}^{C}=-^{*}D_{1}{\frac {\partial c_{^{*}1}}{\partial x}}} Constraint associated with vacancy mechanism: J 1 c → + J 2 c → + J v c → = 0 {\displaystyle {\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 ∂ μ 1 ∂ x + c 2 ∂ μ 2 ∂ x + c v ∂ μ v ∂ x {\displaystyle 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: μ i = μ i ∘ + k T ln ⁡ ( γ i < Ω > c i ) {\displaystyle \mu _{i}=\mu _{i}^{\circ }+kT\ln(\gamma _{i}<\Omega >c_{i})} Flux is proportional to the concentration gradient J 1 c = − k T [ L 11 c 1 − L 12 c 2 ] [ 1 + ∂ ln ⁡ γ 1 ∂ ln ⁡ c 1 + ∂ ln < Ω > ∂ ln ⁡ c 1 ] ∂ c 1 ∂ x {\displaystyle J_{1}^{c}=-kT\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 ∂ c 1 ∂ x {\displaystyle J_{1}^{c}=-D_{1}{\frac {\partial c_{1}}{\partial x}}} Assumptions that simplify D 1 {\displaystyle D_{1}\;} Concentration-independent average site volume < Ω > {\displaystyle <\Omega >\;} The coupling (off-diagonal) terms, L 12 / c 2 {\displaystyle L_{12}/c_{2}\;} and L 1 ∗ 1 / c ∗ 1 {\displaystyle L_{1^{*}1}/c_{^{*}1}\;} , are small compared with the direct term L 11 / c 2 {\displaystyle L_{11}/c_{2}\;} D 1 ≈ [ 1 + ∂ ln ⁡ γ 1 ∂ ln ⁡ c 1 ] ∗ D 1 {\displaystyle D_{1}\approx \left[1+{\frac {\partial \ln \gamma _{1}}{\partial \ln c_{1}}}\right]{}^{*}D_{1}} 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 Ω 1 D 2 + c 2 + Ω 2 D 1 ] ∂ c 1 ∂ x {\displaystyle 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 Ω 1 D 2 + c 2 + Ω 2 D 1 {\displaystyle c_{1}\Omega _{1}D_{2}+c_{2}+\Omega _{2}D_{1}\;} , can be simplified through Ω 1 = Ω 2 =< Ω > {\displaystyle \Omega _{1}=\Omega _{2}=<\Omega >\;} The L-frame and the V-frame are the same J 1 c → = L 11 F 1 → {\displaystyle {\overrightarrow {J_{1}^{c}}}=L_{11}{\overrightarrow {F_{1}}}} J 1 c → = − L 11 ∇ Φ 1 {\displaystyle {\overrightarrow {J_{1}^{c}}}=-L_{11}\nabla \Phi _{1}} J 1 c → = − L 11 ∇ μ 1 {\displaystyle {\overrightarrow {J_{1}^{c}}}=-L_{11}\nabla \mu _{1}} μ 1 = μ 1 ∘ + k T ln ⁡ ( K 1 c 1 ) {\displaystyle \mu _{1}=\mu _{1}^{\circ }+kT\ln(K_{1}c_{1})} ∇ μ 1 = k T c 1 ∇ c 1 {\displaystyle \nabla \mu _{1}={\frac {kT}{c_{1}}}\nabla c_{1}} J 1 c → = − L 11 k T c 1 ∇ c 1 {\displaystyle {\overrightarrow {J_{1}^{c}}}=-L_{11}{\frac {kT}{c_{1}}}\nabla c_{1}} Evaluate L 11 {\displaystyle L_{11}} by substitution of interstitial mobility, M 1 {\displaystyle M_{1}} v 1 c → = − M 1 ∇ μ 1 {\displaystyle {\overrightarrow {v_{1}^{c}}}=-M_{1}\nabla \mu _{1}} v 1 c → = − M 1 k T c 1 ∇ c 1 {\displaystyle {\overrightarrow {v_{1}^{c}}}={\frac {-M_{1}kT}{c_{1}}}\nabla c_{1}} J 1 c → = v 1 c → c 1 {\displaystyle {\overrightarrow {J_{1}^{c}}}={\overrightarrow {v_{1}^{c}}}c_{1}} J 1 c → = − M 1 k T ∇ c 1 {\displaystyle {\overrightarrow {J_{1}^{c}}}=-M_{1}kT\nabla c_{1}} J 1 → = L 11 F 1 → {\displaystyle {\overrightarrow {J_{1}}}=L_{11}{\overrightarrow {F_{1}}}} J 1 → = − L 11 ∇ Φ 1 {\displaystyle {\overrightarrow {J_{1}}}=-L_{11}\nabla \Phi _{1}} J 1 → = − L 11 ∇ ( μ 1 + q 1 φ ) {\displaystyle {\overrightarrow {J_{1}}}=-L_{11}\nabla (\mu _{1}+q_{1}\phi )} J 1 → = − D 1 ∇ c 1 − D 1 c 1 q 1 k T ∇ ( φ ) {\displaystyle {\overrightarrow {J_{1}}}=-D_{1}\nabla c_{1}-{\frac {D_{1}c_{1}q_{1}}{kT}}\nabla (\phi )} E → = − ∇ φ {\displaystyle {\overrightarrow {E}}=-\nabla \phi \;} : Electric field Absence of concentration gradient: J q → = q 1 J 1 → {\displaystyle {\overrightarrow {J_{q}}}=q_{1}{\overrightarrow {J_{1}}}} J q → = − D 1 c 1 q 1 2 k T ∇ ( φ ) {\displaystyle {\overrightarrow {J_{q}}}=-{\frac {D_{1}c_{1}q_{1}^{2}}{kT}}\nabla (\phi )} Electrical conductivity: ρ = D 1 c 1 q 1 2 k T {\displaystyle \rho ={\frac {D_{1}c_{1}q_{1}^{2}}{kT}}} Two fluxes when electric field is applied to a dilute solution of interstitial atoms in metal J q {\displaystyle J_{q}\;} : Flux of conjuction electrons J 1 {\displaystyle J_{1}\;} : Flux of interstitials F q = E {\displaystyle F_{q}=E\;} F q = − ∇ φ {\displaystyle F_{q}=-\nabla \phi } J 1 → = − L 11 ∇ μ 1 + L 1 q E → {\displaystyle {\overrightarrow {J_{1}}}=-L_{11}\nabla \mu _{1}+L_{1q}{\overrightarrow {E}}} J 1 → = − D 1 ( ∇ c 1 − c 1 β k T E → ) {\displaystyle {\overrightarrow {J_{1}}}=-D_{1}\left(\nabla c_{1}-{\frac {c_{1}\beta }{kT}}{\overrightarrow {E}}\right)} Interstitial flux with thermal gradient where both heat flow and mass diffusion of interstitial component occurs: J 1 → = − L 11 ∇ μ 1 − L 1 q T ∇ T {\displaystyle {\overrightarrow {J_{1}}}=-L_{11}\nabla \mu _{1}-{\frac {L_{1q}}{T}}\nabla T} J 1 → = − D 1 ∇ c 1 − D 1 c 1 Q 1 trans k T 2 ∇ T {\displaystyle {\overrightarrow {J_{1}}}=-D_{1}\nabla c_{1}-{\frac {D_{1}c_{1}Q_{1}^{\mbox{trans}}}{kT^{2}}}\nabla T} 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 J A → = L A A F A → {\displaystyle {\overrightarrow {J_{A}}}=L_{AA}{\overrightarrow {F_{A}}}} J A → = − L A A ∇ Φ A {\displaystyle {\overrightarrow {J_{A}}}=-L_{AA}\nabla \Phi _{A}} J A → = − L A A ∇ ( μ A − μ v ) {\displaystyle {\overrightarrow {J_{A}}}=-L_{AA}\nabla (\mu _{A}-\mu _{v})} J V → = − J A → {\displaystyle {\overrightarrow {J_{V}}}=-{\overrightarrow {J_{A}}}} ∂ c ∂ t = n  ̇ − ∇ ⋅ J → {\displaystyle {\frac {\partial c}{\partial t}}={\dot {n}}-\nabla \cdot {\overrightarrow {J}}} n  ̇ {\displaystyle {\dot {n}}} : source or sink term J → {\displaystyle {\overrightarrow {J}}} : any flux in a V-frame ∂ c ∂ t = ∇ ⋅ J → {\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot {\overrightarrow {J}}} ∂ c ∂ t = ∇ ⋅ ( D ∇ c ) {\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot (D\nabla c)} ∂ c ∂ t = D o ∇ 2 c {\displaystyle {\frac {\partial c}{\partial t}}=D_{o}\nabla ^{2}c} ∂ h ∂ t = − ∇ ⋅ J Q → {\displaystyle {\frac {\partial h}{\partial t}}=-\nabla \cdot {\overrightarrow {J_{Q}}}} c P ∂ T ∂ t = − ∇ ⋅ ( − K ∇ T ) {\displaystyle c_{P}{\frac {\partial T}{\partial t}}=-\nabla \cdot (-K\nabla T)} ∂ T ∂ t = ∇ ⋅ ( K c P ∇ T ) {\displaystyle {\frac {\partial T}{\partial t}}=\nabla \cdot \left({\frac {K}{c_{P}}}\nabla T\right)} ∇ ⋅ ( κ ∇ T ) {\displaystyle \nabla \cdot (\kappa \nabla T)} h {\displaystyle h\;} : enthalpy density c P {\displaystyle c_{P}\;} : heat capacity K / c p = κ {\displaystyle K/c_{p}=\kappa \;} : thermal diffusivity ∂ c ∂ t = D ∇ 2 c {\displaystyle {\frac {\partial c}{\partial t}}=D\nabla ^{2}c} c ( x , t ) = c  ̄ + Δ c 2 erf ( x 4 D t ) {\displaystyle c(x,t)={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4Dt}}}\right)} c ( x , t ) = c o Δ x 4 π D t e − x 2 / ( 4 D t ) {\displaystyle c(x,t)={\frac {c_{o}\Delta x}{\sqrt {4\pi Dt}}}e^{-x^{2}/(4Dt)}} c ( x , t ) = n d 4 π D t e − x 2 / ( 4 D t ) {\displaystyle c(x,t)={\frac {n_{d}}{\sqrt {4\pi Dt}}}e^{-x^{2}/(4Dt)}} Source strength, n d = ∫ − ∞ ∞ c ( x ) d x {\displaystyle n_{d}=\int _{-\infty }^{\infty }c(x)dx} ∂ c ∂ t = ∇ ⋅ [ D ( c ) ∇ c ] {\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot [D(c)\nabla c]} Interdiffusivity: D ( c 1 ) = − 1 2 τ d x d c 1 ∫ c 1 R c 1 x ( c ′ ) d c ′ {\displaystyle D(c_{1})=-{\frac {1}{2\tau }}{\frac {dx}{dc_{1}}}\int _{c_{1}^{R}}^{c_{1}}x(c')dc'} ∂ c ∂ t = ∇ ⋅ [ D ( t ) ∇ c ] {\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot [D(t)\nabla c]} ∂ c ∂ t = D ( t ) ∇ 2 c ] {\displaystyle {\frac {\partial c}{\partial t}}=D(t)\nabla ^{2}c]} Change of variable: τ D = ∫ 0 t D ( t ′ ) d t ′ {\displaystyle \tau _{D}=\int _{0}^{t}D(t')dt'} Transformed equation: ∂ c ∂ τ D = ∇ 2 c {\displaystyle {\frac {\partial c}{\partial \tau _{D}}}=\nabla ^{2}c} Solution: c ( x , τ D ) = c  ̄ + Δ c 2 erf ( x 4 τ D ) {\displaystyle c(x,\tau _{D})={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4\tau _{D}}}}\right)} c ( x , t ) = c  ̄ + Δ c 2 erf ( x 4 ∫ 0 t D ( t ′ ) d t ′ ) {\displaystyle c(x,t)={\bar {c}}+{\frac {\Delta c}{2}}{\mbox{erf}}\left({\frac {x}{\sqrt {4\int _{0}^{t}D(t')dt'}}}\right)} J → = − D ∇ c {\displaystyle {\overrightarrow {J}}=-\mathbf {D} \nabla c} The diagonal elements of D ^ {\displaystyle {\hat {\mathbf {D} }}} are the eigenvalues of D {\displaystyle \mathbf {D} } , and the coordinate system of D ^ {\displaystyle {\hat {\mathbf {D} }}} defines the principal axes. ∂ c ∂ t = − ∇ ⋅ J → {\displaystyle {\frac {\partial c}{\partial t}}=-\nabla \cdot {\overrightarrow {J}}} ∂ c ∂ t = ∇ ⋅ D ^ ∇ c {\displaystyle {\frac {\partial c}{\partial t}}=\nabla \cdot {\hat {\mathbf {D} }}\nabla c} Relation of D {\displaystyle \mathbf {D} } and D ^ {\displaystyle {\hat {\mathbf {D} }}} : D ^ = R _ D R _ − 1 {\displaystyle {\hat {\mathbf {D} }}={\underline {R}}\mathbf {D} {\underline {R}}^{-1}} ∇ 2 c = 0 {\displaystyle \nabla ^{2}c=0} J = − D d c d x {\displaystyle J=-D{\frac {dc}{dx}}} J = D c 0 − c L L {\displaystyle J=D{\frac {c^{0}-c^{L}}{L}}} Laplacian Operator: ∇ 2 c = 1 r ∂ ∂ r ( r ∂ c ∂ r ) + 1 r 2 ∂ 2 c ∂ θ 2 + ∂ 2 c ∂ x 2 {\displaystyle \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 in − c in − c out ln ⁡ ( r out / r in ) ln ⁡ ( r r in ) {\displaystyle 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)} Laplacian operator in spherical coordinates ∇ 2 c = 1 r 2 [ ∂ ∂ r ( r 2 ∂ c ∂ r ) + 1 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ c ∂ θ ) + 1 sin 2 ⁡ θ ∂ 2 c ∂ φ 2 ] {\displaystyle \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]} Steady-state conditions D {\displaystyle D} varies with position ∂ ⋅ ( D ∇ c ) = 0 {\displaystyle \partial \cdot (D\nabla c)=0} Solution is obtained by integration: c ( x ) = c ( x 1 ) + a 1 ∫ x 1 x d ζ D ( ζ ) {\displaystyle c(x)=c(x_{1})+a_{1}\int _{x_{1}}^{x}{\frac {d\zeta }{D(\zeta )}}} c ( x , y , z , t ) = n d x 4 π D t e − x 2 / ( 4 D t ) × n d y 4 π D t e − y 2 / ( 4 D t ) × n d z 4 π D t e − z 2 / ( 4 D t ) {\displaystyle 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)}} Uniform distribution of point, line, or plana source placed along x > 0 {\displaystyle x>0} Contribution at a general position x {\displaystyle x} from the source: c ζ ( x , t ) = c o d ζ 4 π D t e − ( ζ − x ) 2 / ( 4 D t ) {\displaystyle 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 ) = c o 4 π D t ∫ 0 ∞ e − ( ζ − x ) 2 / ( 4 D t ) d ζ {\displaystyle c(x,t)={\frac {c_{o}}{\sqrt {4\pi Dt}}}\int _{0}^{\infty }e^{-(\zeta -x)^{2}/(4Dt)}d\zeta } c ( x , t ) = c o 2 + c o 2 erf ( x 2 D t ) {\displaystyle c(x,t)={\frac {c_{o}}{2}}+{\frac {c_{o}}{2}}{\mbox{erf}}\left({\frac {x}{2{\sqrt {Dt}}}}\right)} System : Three Dimensions, ( x , y , z ) {\displaystyle (x,y,z)} Equation : d c d t = D ∇ 2 c {\displaystyle {\frac {dc}{dt}}=D\nabla ^{2}c} Solution : c ( r , θ , z , t ) = R ( r ) Θ ( θ ) Z ( z ) T ( t ) {\displaystyle c(r,\theta ,z,t)=R(r)\Theta (\theta )Z(z)T(t)\;} Laplace transform of a function f ( x , t ) {\displaystyle f(x,t)\;} L { f ( x , t ) } = f ^ ( x , p ) {\displaystyle L\{f(x,t)\}={\hat {f}}(x,p)} L { f ( x , t ) } = ∫ 0 ∞ e − p t f ( x , t ) d t {\displaystyle L\{f(x,t)\}=\int _{0}^{\infty }e^{-pt}f(x,t)dt} T ′ = k T 2 π m 1 L well e − ( E A − E well ) / ( k T ) {\displaystyle \mathrm {T} '={\sqrt {\frac {kT}{2\pi m}}}{\frac {1}{L^{\mbox{well}}}}e^{-(E^{A}-E^{\mbox{well}})/(kT)}} T ′ = k T 2 π m 1 L well e − ( E m ) / ( k T ) {\displaystyle \mathrm {T} '={\sqrt {\frac {kT}{2\pi m}}}{\frac {1}{L^{\mbox{well}}}}e^{-(E^{m})/(kT)}} T ′ = 1 2 π β m e − ( E A − E well ) / ( k T ) {\displaystyle \mathrm {T} '={\frac {1}{2\pi }}{\sqrt {\frac {\beta }{m}}}e^{-(E^{A}-E^{\mbox{well}})/(kT)}} T ′ = ν e − E m / ( k T ) {\displaystyle \mathrm {T} '=\nu e^{-E^{m}/(kT)}} T ′ = ν e − G m / ( k T ) {\displaystyle \mathrm {T} '=\nu e^{-G^{m}/(kT)}} < R 2 ( N τ ) >= N τ < r 2 > + 2 < ( ∑ j = 1 N τ − 1 ∑ i = 1 N τ − j | r i → | | r → i + j | cos ⁡ θ i , i + j > {\displaystyle =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}>} < R 2 ( τ ) >= 6 D τ {\displaystyle =6D\tau \;} D = T < r 2 > 2 {\displaystyle D={\frac {\mathrm {T} }{2}}} Correlation factor: f = 1 + 2 N τ < r 2 > < ( ∑ j = 1 N τ − 1 ∑ i = 1 N τ − j | r i → | | r → i + j | cos ⁡ θ i , i + j > {\displaystyle \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 = < r 2 > N τ 6 τ f {\displaystyle D={\frac {N_{\tau }}{6\tau }}\mathbf {f} } D = T τ < r 2 > 6 τ f {\displaystyle D={\frac {\mathrm {T} \tau }{6\tau }}\mathbf {f} } D = T < r 2 > 6 f {\displaystyle D={\frac {\mathrm {T} }{6}}\mathbf {f} } f = 1 + < cos ⁡ θ > 1 − < cos ⁡ θ > {\displaystyle \mathbf {f} ={\frac {1+<\cos \theta >}{1-<\cos \theta >}}} f ≈ z − 1 z + 1 {\displaystyle \mathbf {f} \approx {\frac {z-1}{z+1}}} ∗ D ( m a s s 1 ) ∗ D ( m a s s 2 ) = T 1 T 2 = T 1 ′ T 2 ′ = ν 1 ν 2 = m 1 m 2 {\displaystyle {\frac {^{*}D(mass1)}{^{*}D(mass2)}}={\frac {\mathrm {T} _{1}}{\mathrm {T} _{2}}}={\frac {\mathrm {T} '_{1}}{\mathrm {T} '_{2}}}={\frac {\nu _{1}}{\nu _{2}}}={\frac {m_{1}}{m_{2}}}}
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