Why chloride corrosion is modelled by ficks second law?
Answers
Fick's second law predicts how diffusion causes the concentration to change with time. It is a partial differential equation which in one dimension reads:
∂ φ ∂ t = D ∂ 2 φ ∂ x 2 {\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,{\frac {\partial ^{2}\varphi }{\partial x^{2}}}\,\!}where
φ is the concentration in dimensions of [(amount of substance) length−3], example mol/m3; φ = φ(x,t) is a function that depends on location x and time tt is time [s]D is the diffusion coefficient in dimensions of [length2 time−1], example m2/sx is the position [length], example mIn two or more dimensions we must use the Laplacian Δ = ∇2, which generalises the second derivative, obtaining the equation
∂ φ ∂ t = D Δ φ {\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,\Delta \varphi } Derivation[edit]Fick's second law is a special case of the convection–diffusion equation in which there is no advective flux and no net volumetric source. It can be derived from the continuity equation:
∂ φ ∂ t + ∇ ⋅ j → = R , {\displaystyle {\frac {\partial \varphi }{\partial t}}+\nabla \cdot {\vec {j}}=R,}where j → {\displaystyle {\vec {j}}} is the total flux and R is a net volumetric source for φ {\displaystyle \varphi } . The only source of flux in this situation is assumed to be diffusive flux:
j → diffusion = − D ∇ φ {\displaystyle {\vec {j}}_{\text{diffusion}}=-D\,\nabla \varphi }Plugging the definition of diffusive flux to the continuity equation and assuming there is no source (R = 0), we arrive at Fick's second law:
∂ φ ∂ t = D ∂ 2 φ ∂ x 2 {\displaystyle {\frac {\partial \varphi }{\partial t}}=D\,{\frac {\partial ^{2}\varphi }{\partial x^{2}}}\,\!}If flux were the result of both diffusive flux and advective flux, the convection–diffusion equation is the result.
Example solution in one dimension: diffusion length[edit]A simple case of diffusion with time t in one dimension (taken as the x-axis) from a boundary located at position x = 0, where the concentration is maintained at a value n0 is
n ( x , t ) = n 0 e r f c ( x 2 D t ) {\displaystyle n\left(x,t\right)=n_{0}\mathrm {erfc} \left({\frac {x}{2{\sqrt {Dt}}}}\right)} .where erfc is the complementary error function. This is the case when corrosive gases diffuse through the oxidative layer towards the metal surface (if we assume that concentration of gases in the environment is constant and the diffusion space (i. e., corrosion product layer) is semi-infinite – starting at 0 at the surface and spreading infinitely deep in the material). If, in its turn, the diffusion space is infinite (lasting both through the layer with n(x,0) = 0, x > 0 and that with n(x,0) = n0, x ≤ 0), then the solution is amended only with coefficient 1⁄2 in front of n0 (this might seem obvious, as the diffusion now occurs in both directions). This case is valid when some solution with concentration n0 is put in contact with a layer of pure solvent. (Bokstein, 2005) The length 2√Dt is called the diffusion length and provides a measure of how far the concentration has propagated in the x-direction by diffusion in time t (Bird, 1976).
As a quick approximation of the error function, the first 2 terms of the Taylor series can be used:
n ( x , t ) = n 0 [ 1 − 2 ( x 2 D t π ) ] {\displaystyle n\left(x,t\right)=n_{0}\left[1-2\left({\frac {x}{2{\sqrt {Dt\pi }}}}\right)\right]}If D is time-dependent, the diffusion length becomes
2 ∫ 0 t D ( t ′ ) d t ′ {\displaystyle 2{\sqrt {\int _{0}^{t}D(t')dt'}}} .This idea is useful for estimating a diffusion length over a heating and cooling cycle, where D varies with temperature.
Generalizations[edit]1. In inhomogeneous media, the diffusion coefficient varies in space, D = D(x). This dependence does not affect Fick's first law but the second law changes:
∂ φ ( x , t ) ∂ t = ∇ ⋅ ( D ( x ) ∇ φ ( x , t ) ) = D ( x ) Δ φ ( x , t ) + ∑ i = 1 3 ∂ D ( x ) ∂ x i ∂ φ ( x , t ) ∂ x i {\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot (D(x)\nabla \varphi (x,t))=D(x)\Delta \varphi (x,t)+\sum _{i=1}^{3}{\frac {\partial D(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}\ }2. In anisotropic media, the diffusion coefficient depends on the direction. It is a symmetric tensor D = Dij. Fick's first law changes to
J = − D ∇ φ {\displaystyle J=-D\nabla \varphi \ } ,it is the product of a tensor and a vector:
J i = − ∑ j = 1 3 D i j ∂ φ ∂ x j . {\displaystyle \;\;J_{i}=-\sum _{j=1}^{3}D_{ij}{\frac {\partial \varphi }{\partial x_{j}}}\ .}For the diffusion equation this formula gives
∂ φ ( x , t ) ∂ t = ∇ ⋅ ( D ∇ φ ( x , t ) ) = ∑ i = 1 3 ∑ j = 1 3 D i j ∂ 2 φ ( x , t ) ∂ x i ∂ x j . {\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot (D\nabla \varphi (x,t))=\sum _{i=1}^{3}\sum _{j=1}^{3}D_{ij}{\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}\ .}The symmetric matrix of diffusion coefficients Dij should be positive definite. It is needed to make the right hand side operator elliptic.