# Fick's law of diffusion

For the technique of measuring cardiac output, see Fick principle.

Fick's laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient D. They were derived by Adolf Fick in the year 1855.

## First law

Fick's first law is used in steady-state diffusion, i.e., when the concentration within the diffusion volume does not change with respect to time ${\displaystyle (\,J_{\mathrm {in} }=J_{\mathrm {out} })}$. In one (spatial) dimension, this is

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

where

• ${\displaystyle \,J}$ is the diffusion flux in dimensions of [(amount of substance) length−2 time-1], example ${\displaystyle {\bigg (}{\frac {\mathrm {mol} }{m^{2}\cdot s}}{\bigg )}}$
• ${\displaystyle \,D}$ is the diffusion coefficient or diffusivity in dimensions of [length2 time−1], example ${\displaystyle {\bigg (}{\frac {m^{2}}{s}}{\bigg )}}$
• ${\displaystyle \,\phi }$ is the concentration in dimensions of [(amount of substance) length−3], example ${\displaystyle {\bigg (}{\frac {\mathrm {mol} }{m^{3}}}{\bigg )}}$
• ${\displaystyle \,x}$ is the position [length], example ${\displaystyle \,m}$

${\displaystyle \,D}$ is proportional to the 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 ${\displaystyle \nabla }$, the del or gradient operator, which generalises the first derivative, obtaining

${\displaystyle J=-D\nabla \phi }$.

## Second law

Fick's second law is used in non-steady or continually changing state diffusion, i.e., when the concentration within the diffusion volume changes with respect to time.

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

Where

• ${\displaystyle \,\phi }$ is the concentration in dimensions of [(amount of substance) length-3], [mol m-3]
• ${\displaystyle \,t}$ is time [s]
• ${\displaystyle \,D}$ is the diffusion coefficient in dimensions of [length2 time-1], [m2 s-1]
• ${\displaystyle \,x}$ is the position [length], [m]

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

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

Assuming the diffusion coefficient D to be a constant we can exchange the orders of the differentiating and multiplying by the constant:

${\displaystyle {\frac {\partial }{\partial x}}{\bigg (}\,D\,{\frac {\partial }{\partial x}}\phi \,{\bigg )}=D\,{\frac {\partial }{\partial x}}{\frac {\partial }{\partial x}}\,\phi =D\,{\frac {\partial ^{2}\phi }{\partial x^{2}}}}$

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:

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

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:

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

An important example is the case where ${\displaystyle \phi }$ 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 ${\displaystyle \,D}$, the solution for the concentration will be a linear change of concentrations along ${\displaystyle \,x}$. In two or more dimensions we obtain

${\displaystyle \nabla ^{2}\,\phi =0\!}$

which is Laplace's equation, the solutions to which are called harmonic functions by mathematicians.

## Applicability

Equations based on Fick's law have been commonly used to model transport processes in foods, neurons, biopolymers, pharmaceuticals, porous soils, population dynamics, semiconductor doping process, etc. A large amount of experimental research in polymer science and food science has shown that a more general approach is required to describe transport of components in materials undergoing glass transition. In the vicinity of glass transition the flow behavior becomes "non-Fickian". See also non-diagonal coupled transport processes (Onsager relationship).

## Temperature dependence of the diffusion coefficient

The diffusion coefficient at different temperatures is often found to be well predicted by

${\displaystyle D=D_{0}\cdot e^{-{\frac {E_{A}}{R\cdot T}}},}$

where

• ${\displaystyle \,D}$ is the diffusion coefficient
• ${\displaystyle \,D_{0}}$ is the maximum diffusion coefficient (at infinite temperature)
• ${\displaystyle \,E_{A}}$ is the activation energy for diffusion in dimensions of [energy (amount of substance)−1]
• ${\displaystyle \,T}$ is the temperature in units of [absolute temperature] (kelvins or degrees Rankine)
• ${\displaystyle \,R}$ is the gas constant in dimensions of [energy temperature−1 (amount of substance)−1]

An equation of this form is known as the Arrhenius equation.

Typically, a compound's diffusion coefficient is ~10,000x greater in air than in water. Carbon dioxide in air has a diffusion coefficient of 16 mm²/s, and in water, its coefficient is 0.0016 mm²/s [1].

## Biological perspective

The first law gives rise to the following formula:[1]

${\displaystyle Flux={-P\cdot A\cdot (c_{2}-c_{1})}\,\!}$

in which,

• ${\displaystyle \,P}$ is the permeability, an experimentally determined membrane "conductance" for a given gas at a given temperature.
• ${\displaystyle \,A}$ is the surface area over which diffusion is taking place.
• ${\displaystyle \,c_{2}-c_{1}}$ is the difference in concentration of the gas across the membrane for the direction of flow (from ${\displaystyle c_{1}}$ to ${\displaystyle c_{2}}$).

Fick's first law is also important in radiation transfer equations. However, in this context it becomes inaccurate when the diffusion constant is low and the radiation becomes limited by the speed of light rather than by the resistance of the material the radiation is flowing through. In this situation, one can use a flux limiter.

The exchange rate of a gas across a fluid membrane can be determined by using this law together with Graham's law.

## Semiconductor fabrication applications

IC Fabrication technologies, model processes like CVD, Thermal Oxidation, and Wet Oxidation, Doping etc using Diffusion equations obtained from Ficks law.

In certain cases, the solutions are obtained for boundary conditions such as constant source concentration diffusion, limited source concentration, or moving boundary diffusion (where junction depth keeps moving into the substrate).