Editing Fick's laws of diffusion (section)

== Example solutions and generalization ==
Fick's second law is a special case of the [[convection–diffusion equation]] in which there is no [[advection|advective flux]] and no net volumetric source. It can be derived from the [[Continuity equation#Differential form|continuity equation]]:
: <math> \frac{\partial \varphi}{\partial t} + \nabla\cdot\mathbf{j} = R, </math>
where {{math|'''j'''}} is the total [[flux]] and {{mvar|R}} is a net volumetric source for {{math|'''φ'''}}. The only source of flux in this situation is assumed to be ''diffusive flux'':
: <math>\mathbf{j}_{\text{diffusion}} = -D \nabla \varphi . </math>

Plugging the definition of diffusive flux to the continuity equation and assuming there is no source ({{math|1=''R'' = 0}}), we arrive at Fick's second law:
: <math>\frac{\partial \varphi}{\partial t} = D\frac{\partial^2 \varphi}{\partial x^2} . </math>

If flux were the result of both diffusive flux and [[advection|advective flux]], the [[convection–diffusion equation]] is the result.

=== Example solution 1: constant concentration source and diffusion length ===
A simple case of diffusion with time {{mvar|t}} in one dimension (taken as the {{mvar|x}}-axis) from a boundary located at position {{math|1=''x'' = 0}}, where the concentration is maintained at a value {{math|''n''<sub>0</sub>}} is
: <math>n \left(x,t \right)=n_0 \operatorname{erfc} \left( \frac{x}{2\sqrt{Dt}}\right) ,</math>
where {{math|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 – that is, the 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 {{math|1=''n''(''x'', 0) = 0}}, {{math|''x'' > 0}} and that with {{math|1=''n''(''x'', 0) = ''n''<sub>0</sub>}}, {{math|''x'' ≤ 0}}), then the solution is amended only with coefficient {{sfrac|2}} in front of {{math|''n''<sub>0</sub>}} (as the diffusion now occurs in both directions). This case is valid when some solution with concentration {{math|''n''<sub>0</sub>}} is put in contact with a layer of pure solvent. (Bokstein, 2005) The length {{math|2{{sqrt|''Dt''}}}} is called the ''diffusion length'' and provides a measure of how far the concentration has propagated in the {{mvar|x}}-direction by diffusion in time {{mvar|t}} (Bird, 1976).

As a quick approximation of the error function, the first two terms of the [[Taylor series]] can be used:
: <math>n(x,t)=n_0 \left[ 1 - 2 \left(\frac{x}{2\sqrt{Dt\pi}}\right) \right] . </math>

If {{mvar|D}} is time-dependent, the diffusion length becomes
: <math> 2\sqrt{\int_0^t D( \tau ) \,d\tau}. </math>
This idea is useful for estimating a diffusion length over a heating and cooling cycle, where {{mvar|D}} varies with temperature.

=== Example solution 2: Brownian particle and mean squared displacement ===
Another simple case of diffusion is the [[Brownian motion]] of one particle. The particle's [[Mean squared displacement]] from its original position is:
<math display="block">\text{MSD} \equiv \left \langle (\mathbf{x}-\mathbf{x_0})^2 \right \rangle=2nDt , </math>
where <math>n</math> is the [[dimension]] of the particle's Brownian motion. For example, the diffusion of a molecule across a [[cell membrane]] 8&nbsp;nm thick is 1-D diffusion because of the spherical symmetry; However, the diffusion of a molecule from the membrane to the center of a [[Eukaryotic Cell|eukaryotic cell]] is a 3-D diffusion. For a cylindrical [[cactus]], the diffusion from photosynthetic cells on its surface to its center (the axis of its cylindrical symmetry) is a 2-D diffusion.

The square root of MSD, <math>\sqrt{2nDt}</math>, is often used as a characterization of how far the particle has moved after time <math>t</math> has elapsed. The MSD is symmetrically distributed over the 1D, 2D, and 3D space. Thus, the probability distribution of the magnitude of MSD in 1D is Gaussian and 3D is a Maxwell-Boltzmann distribution.

=== Generalizations ===
* In ''non-homogeneous media'', the diffusion coefficient varies in space, {{math|1=''D'' = ''D''(''x'')}}. This dependence does not affect Fick's first law but the second law changes: <math display="block">\frac{\partial \varphi(x,t)}{\partial t}=\nabla\cdot \bigl(D(x) \nabla \varphi(x,t)\bigr)=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}. </math>
* In ''[[anisotropic]] media'', the diffusion coefficient depends on the direction. It is a symmetric [[tensor]] {{math|1=''D<sub>ji</sub>'' = ''D<sub>ij</sub>''}}. Fick's first law changes to <math display="block">J=-D \nabla \varphi ,</math> it is the product of a tensor and a vector: <math display="block"> J_i=-\sum_{j=1}^3 D_{ij} \frac{\partial \varphi}{\partial x_j}.</math> For the diffusion equation this formula gives <math display="block">\frac{\partial \varphi(x,t)}{\partial t}=\nabla\cdot \bigl(D \nabla \varphi(x,t)\bigr)=\sum_{i=1}^3\sum_{j=1}^3D_{ij} \frac{\partial^2 \varphi(x,t)}{\partial x_i \partial x_j}. </math> The symmetric matrix of diffusion coefficients {{math|''D<sub>ij</sub>''}} should be [[Positive-definite matrix|positive definite]]. It is needed to make the right-hand side operator [[Elliptic operator|elliptic]].
* For ''inhomogeneous anisotropic media'' these two forms of the diffusion equation should be combined in <math display="block">\frac{\partial \varphi(x,t)}{\partial t}=\nabla\cdot \bigl(D(x) \nabla \varphi(x,t)\bigr)=\sum_{i,j=1}^3\left(D_{ij}(x) \frac{\partial^2 \varphi(x,t)}{\partial x_i \partial x_j}+ \frac{\partial D_{ij}(x)}{\partial x_i }  \frac{\partial \varphi(x,t)}{\partial x_i}\right). </math>
* The approach based on [[Diffusion#Einstein's mobility and Teorell formula|Einstein's mobility and Teorell formula]] gives the following generalization of Fick's equation for the ''multicomponent diffusion'' of the perfect components: <math display="block">\frac{\partial \varphi_i}{\partial t} = \sum_j \nabla\cdot\left(D_{ij} \frac{\varphi_i}{\varphi_j} \nabla \, \varphi_j\right) ,</math> where {{mvar|φ<sub>i</sub>}} are concentrations of the components and {{mvar|D<sub>ij</sub>}} is the matrix of coefficients. Here, indices {{mvar|i}} and {{mvar|j}} are related to the various components and not to the space coordinates.

The [[Diffusion#The theory of diffusion in gases based on Boltzmann's equation|Chapman–Enskog formulae for diffusion in gases]] include exactly the same terms. These physical models of diffusion are different from the test models {{math|1=∂<sub>''t''</sub>''φ<sub>i</sub>'' = Σ<sub>''j''</sub> ''D<sub>ij</sub>'' Δ''φ<sub>j</sub>''}} which are valid for very small deviations from the uniform equilibrium. Earlier, such terms were introduced in the [[Maxwell–Stefan diffusion]] equation.

For anisotropic multicomponent diffusion coefficients one needs a rank-four tensor, for example {{math|''D''<sub>''ij'',''αβ''</sub>}}, where {{math|''i'', ''j''}} refer to the components and {{math|1=''α'', ''β'' = 1, 2, 3}} correspond to the space coordinates.