For those that are unfamiliar with Flicks Law.
Fick's first law relates the diffusive flux to the concentration under the assumption of steady state. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient (spatial derivative). In one (spatial) dimension, the law is
\bigg. J = - D \frac{\partial \phi}{\partial x} \bigg.
where
J is the "diffusion flux" [(amount of substance) per unit area per unit time], example (\tfrac{\mathrm{mol}}{ \mathrm m^2\cdot \mathrm s}). J measures the amount of substance that will flow through a small area during a small time interval.
\, D is the diffusion coefficient or diffusivity in dimensions of [length2 time−1], example (\tfrac{\mathrm m^2}{\mathrm s})
\, \phi (for ideal mixtures) is the concentration in dimensions of [amount of substance per unit volume], example (\tfrac{\mathrm {mol}}{\mathrm m^3})
\, x is the position [length], example \,\mathrm m
\, D is proportional to the squared velocity of the diffusing particles, which depends on the temperature, viscosity of the fluid and the size of the particles according to the Stokes-Einstein relation. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of 0.6x10−9 to 2x10−9 m2/s. For biological molecules the diffusion coefficients normally range from 10−11 to 10−10 m2/s.
In two or more dimensions we must use \nabla, the del or gradient operator, which generalises the first derivative, obtaining
\mathbf{J}=- D\nabla \phi .
The driving force for the one-dimensional diffusion is the quantity - \frac{\partial \phi}{\partial x} which for ideal mixtures is the concentration gradient. In chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of chemical potential of this species. Then Fick's first law (one-dimensional case) can be written as:
J_i = - \frac{D c_i}{RT} \frac{\partial \mu_i}{\partial x}
where the index i denotes the ith species, c is the concentration (mol/m3), R is the universal gas constant (J/(K mol)), T is the absolute temperature (K), and μ is the chemical potential (J/mol).
If the primary variable is mass fraction (y_i, given, for example, in \tfrac{\mathrm kg}{\mathrm kg}), then the equation changes to:
J_i=- \rho D\nabla y_i
where \rho is the fluid density (for example, in \tfrac{\mathrm kg}{\mathrm m^3}). Note that the density is outside the gradient operator.
Fick's second law[edit]
Fick's second law predicts how diffusion causes the concentration to change with time:
\frac{\partial \phi}{\partial t} = D\,\frac{\partial^2 \phi}{\partial x^2}\,\!
where
\,\phi is the concentration in dimensions of [(amount of substance) length−3], example (\tfrac{\mathrm{mol}}{m^3})
\, t is time [s]
\, D is the diffusion coefficient in dimensions of [length2 time−1], example (\tfrac{m^2}{s})
\, x is the position [length], example \,m
It can be derived from Fick's First law and the mass conservation in absence of any chemical reactions:
\frac{\partial \phi}{\partial t} +\,\frac{\partial}{\partial x}\,J = 0\Rightarrow\frac{\partial \phi}{\partial t} -\frac{\partial}{\partial x}\bigg(\,D\,\frac{\partial}{\partial x}\phi\,\bigg)\,=0\!
Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiation and multiply by the constant:
\frac{\partial}{\partial x}\bigg(\,D\,\frac{\partial}{\partial x} \phi\,\bigg) = D\,\frac{\partial}{\partial x} \frac{\partial}{\partial x} \,\phi = D\,\frac{\partial^2\phi}{\partial x^2}
and, thus, receive the form of the Fick's equations as was stated above.
For the case of diffusion in two or more dimensions Fick's Second Law becomes
\frac{\partial \phi}{\partial t} = D\,\nabla^2\,\phi\,\!,
which is analogous to the heat equation.
If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, Fick's Second Law yields
\frac{\partial \phi}{\partial t} = \nabla \cdot (\,D\,\nabla\,\phi\,)\,\!
An important example is the case where \,\phi is at a steady state, i.e. the concentration does not change by time, so that the left part of the above equation is identically zero. In one dimension with constant \, D, the solution for the concentration will be a linear change of concentrations along \, x. In two or more dimensions we obtain
\nabla^2\,\phi =0\!
which is Laplace's equation, the solutions to which are called harmonic functions by mathematicians.
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 n_0 is
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\left(x,0\right) = 0, x >0 and that with n\left(x,0\right) = n_0, x \le 0 ), then the solution is amended only with coefficient ½ 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. (Bokshtein, 2005) The length 2\sqrt{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 \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\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 the 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:
\frac{\partial \phi(x,t)}{\partial t}=\nabla\cdot (D(x) \nabla \phi(x,t))=D(x) \Delta \phi(x,t)+\sum_{i=1}^3 \frac{\partial D(x)}{\partial x_i} \frac{\partial \phi(x,t)}{\partial x_i}\
2. In the anisotropic media, the diffusion coefficient depends on the direction. It is a symmetric tensor D=D_{ij}. Fick's first law changes to
J=-D \nabla \phi \ , \mbox{ it is the product of a tensor and a vector: } \;\; J_i=-\sum_{j=1}^3D_{ij} \frac{\partial \phi}{\partial x_j} \ .
For the diffusion equation this formula gives
\frac{\partial \phi(x,t)}{\partial t}=\nabla\cdot (D \nabla \phi(x,t))=\sum_{j=1}^3D_{ij} \frac{\partial^2 \phi(x,t)}{\partial x_i \partial x_j}\ .
The symmetric matrix of diffusion coefficients D_{ij} should be positive definite. It is needed to make the right hand side operator elliptic.
3. For the inhomogeneous anisotropic media these two forms of the diffusion equation should be combined in
\frac{\partial \phi(x,t)}{\partial t}=\nabla\cdot (D(x) \nabla \phi(x,t))=\sum_{i,j=1}^3\left(D_{ij}(x) \frac{\partial^2 \phi(x,t)}{\partial x_i \partial x_j}+ \frac{\partial D_{ij}(x)}{\partial x_i } \frac{\partial \phi(x,t)}{\partial x_j}\right)\ .
4. The approach based on the Einstein's mobility and Teorell formula gives the following generalization of Fick's equation for the multicomponent diffusion of the perfect components:
\frac{\partial \phi_i}{\partial t} =\sum_j {\rm div}\left(D_{ij} \frac{\phi_i}{\phi_j} {\rm grad} \, \phi_j\right) \, .
where \phi_i are concentrations of the components and D_{ij} is the matrix of coefficients. Here, indexes i,j are related to the various components and not to the space coordinates.
The Chapman-Enskog formulas for diffusion in gases include exactly the same terms. It should be stressed that these physical models of diffusion are different from the toy-models \partial_t \phi_i = \sum_j D_{ij} \Delta \phi_j 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 4-index quantities, for example, D_{ij\, \alpha \beta}, where i, j are related to the components and α, β=1,2,3 correspond to the space coordinates.
Sorry I could not help myself.
Serious question.
Reg/ Neil.
You are both right to state that mine water has caused saline contamination to our aquifers but is there any evidence that it has been harmful to humans?