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Fick's First law

Fick's first law relates the diffusive flux to the concentration field, by postulating 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, this is

$J=-D{\frac {\partial \phi }{\partial x}}\,\!$

where

$J\,\!$ is the diffusion flux in dimensions of [(amount of substance) length⁻² time^-1], example $({\tfrac {\mathrm {mol} }{m^{2}\cdot s}})$. $J$ 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 $({\tfrac {m^{2}}{s}})$

$\phi \,\!$ (for ideal mixtures) is the concentration in dimensions of [(amount of substance) length⁻³], example $({\tfrac {\mathrm {mol} }{m^{3}}})$

$x\,\!$ is the position [length], example $\,m$

$\,D$ 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

$J=-D\nabla \phi \,\!$.

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:

$J_{i}=-{\frac {Dc_{i}}{RT}}{\frac {\partial \mu }{\partial x_{i}}}\,\!$

where the index i denotes the ith species, c is the concentration (mol/m^{3), R is the universal gas constant (J/(K mol)), T is the absolute temperature (K), and μ is the chemical potential (J/mol).}

Second law

Fick's second law predicts how diffusion causes the concentration field 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], [mol m^-3]
• $\,t$ is time [s]
• $\,D$ is the diffusion coefficient in dimensions of [length² time^-1], [m² s^-1]
• $\,x$ is the position [length], [m]

It can be derived from Fick's First law and the mass balance:

${\frac {\partial \phi }{\partial t}}=-\,{\frac {\partial }{\partial x}}\,J={\frac {\partial }{\partial x}}{\bigg (}\,D\,{\frac {\partial }{\partial x}}\phi \,{\bigg )}\,\!$

Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiating and multiplying 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 the Second Fick's Law is:

${\frac {\partial \phi }{\partial t}}=D\,\nabla ^{2}\,\phi \,\!$,

which is analogous to the heat equation.

If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, the Second Fick's Law becomes:

${\frac {\partial \phi }{\partial t}}=\nabla \cdot (\,D\,\nabla \,\phi \,)\,\!$

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.