Fick's Law

'''Fick's laws of diffusion''' describes the diffusion process in an environement.

## History

Fick's laws of diffusion were derived by Adolf Fick in the year 1855.

## Fick's first law

Fick's first law is used in steady-state diffusion, i.e., when the concentration within the diffusion volume does notchange with respect to time $(J_{in} = J_{out})$. In one (spatial) dimension, this is:

(1)
\begin{align} J = D \frac{\partial \phi}{\partial x} \end{align}

where

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

$D$ 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. For the biological molecules the diffusion coefficients normally range from $10^{-11}$ to $10^{-10}$ $\frac{m^{2}}{s}$.

In two or more dimensions we must use $\nabla$, the del or gradient operator, which generalises the first derivative, obtaining

(2)
\begin{align} J=- D\nabla \phi \end{align}

## Fick's 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.

(3)
\begin{align} \frac{\partial \phi}{\partial t} = D\,\frac{\partial^2 \phi}{\partial x^2} \end{align}

where

• $\phi$ is the concentration in dimensions of [(amount of substance) $length^{-2}$], [mol $m^{-3}$]
• $t$ is time [s]
• $D$ is the diffusion coefficient in dimensions of [$length^{-3}$ $time^{-1}$], [$m^{2}$ $s^{-1}$]
• $x$ is the position [length], [m]

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

(4)
\begin{align} \frac{\partial \phi}{\partial t} =-\frac{\partial}{\partial x}J = \frac{\partial}{\partial x}D\frac{\partial}{\partial x}\phi \end{align}

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

(5)
\begin{align} \frac{\partial}{\partial x}D\frac{\partial}{\partial x} \phi = D\frac{\partial}{\partial x} \frac{\partial}{\partial x} \phi = D\frac{\partial^2\phi}{\partial x^2} \end{align}

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:

(6)
\begin{align} \frac{\partial \phi}{\partial t} = D\nabla^{2}\phi \end{align}

also called 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:

(7)
\begin{align} \frac{\partial \phi}{\partial t} = \nabla . (D\nabla\phi) \end{align}

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

(8)
\begin{align} \nabla^{2} \phi =0 \end{align}

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

## Temperature dependence of the diffusion coefficient

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

(9)
\begin{align} D = D_{0}. e^{-\frac{E_{A}}{R.T}} \end{align}

where

• $D$ is the diffusion coefficient
• $D_{0}$ is the maximum diffusion coefficient (at infinite temperature)
• $E_{A}$ is the activation energy for diffusion in dimensions of [energy $(amount of substance)^{-1}$]
• $T$ is the temperature in units of [absolute temperature] (kelvin s or degrees Rankine)
• $R$ 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 (http://www.cco.caltech.edu/~brokawc/Bi145/Diffusion.html).

## Biological perspective

The first law gives rise to the following formula:<ref>GeorgiaPhysiology|3/3ch9/s3ch9_2</ref>

(10)
\begin{align} Rate\ of\ diffusion\ = \frac{{K A (P_{2} - P_{1})}}{d} \end{align}

It states that the rate of diffusion of a gas across a membrane is

*$K$ is experimentally determined constant for a given gas at a given temperature.
*$A$ is proportional to the surface area over which diffusion is taking place.
*$P_{2} - P_{1}$ is proportional to the difference in partial pressures of the gas across the Artificial membrane.
*$d$ is inversely proportional to the distance over which diffusion must take place, or in other words the thickness of the membrane.

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.

## References

• A. Fick, ''Phil. Mag.'' (1855), '''10''', 30.
• A. Fick, ''Poggendorff's Annel. Physik.'' (1855), '''94''', 59.
• W.F. Smith, ''Foundations of Materials Science and Engineering 3<sup>rd</sup> ed.'', McGraw-Hill (2004)
• H.C. Berg, ''Random Walks in Biology'', Princeton (1977)