Fick's laws of diffusion describe diffusion, and define the diffusion coefficient D.

History

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

Fick's First Law

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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 (J_in=J_out).

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

Where

• $J$ is the diffusion flux in dimensions of href="http://articles.gourt.com/en/amount of substance">amount of substance) length^-2 time^-1, m^-2 s^-1
• $D$ is the diffusion coefficient or diffusivity in dimensions of time^-1, s^-1
• $\backslash Phi$ is the concentration in dimensions of of substance) length^-3, m^-3
• $x$ is the position [m

Fick's Second Law

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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.

$\backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; t\}\; =\; D\; \backslash frac\{\backslash partial^2\; \backslash phi\}\{\backslash partial\; x^2\}$

Where

• $\backslash Phi$ is the concentration in dimensions of of substance) length^-3, m^-3
• $t$ is time ^*
• $D$ is the diffusion coefficient in dimensions of time^-1, s^-1
• $x$ is the position [m

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

$\backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; t\}\; =-\backslash frac\; \{\backslash partial\}\; \{\backslash partial\; x\}\; J\; =\; \backslash frac\; \{\backslash partial\}\; \{\backslash partial\; x\}\; (D\; \backslash frac\; \{\backslash partial\}\; \{\backslash partial\; x\}\; \backslash phi)$

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

$\backslash frac\; \{\backslash partial\}\; \{\backslash partial\; x\}\; (D\; \backslash frac\; \{\backslash partial\}\; \{\backslash partial\; x\}\; \backslash phi)\; =\; D\; \backslash frac\; \{\backslash partial\}\; \{\backslash partial\; x\}\; \backslash frac\; \{\backslash partial\}\; \{\backslash partial\; x\}\; \backslash phi=\; D\; \backslash frac\{\backslash partial^2\; \backslash phi\}\{\backslash partial\; x^2\}$

and, thus, receive the form of the Fick's equations as was stated above.

For the case of 3-dimensional diffusion the Second Fick's Law looks like:

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

where $\backslash nabla$ is the usual del operator.

Finally if the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, the Second Fick's Law looks like:

$\backslash frac\{\backslash partial\; \backslash phi\}\{\backslash partial\; t\}\; =\; \backslash nabla\; (D\; \backslash nabla\; \backslash phi)$

Applicability

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Equations based on Fick's law have been commonly used to model transport processes in foods, biopolymers, pharmaceuticals, porous soils, 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

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

$D\; =\; D\_0\; e^\{-\backslash frac\{E\_\{A\}\}\{RT\}\}$

Where:

• D is the diffusion coefficient
• D₀ is the maximum diffusion coefficient (at infinite temperature)
• E_A is the activation energy for diffusion in dimensions of (amount of substance)^-1
• T is the temperature in units of temperature (kelvins or degrees Rankine)
• R is the gas constant in dimensions of temperature^-1 (amount of substance)^-1

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 ^*.

A Biological Perspective

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The first law gives rise to the formula

$\backslash mathrm\{Rate\backslash \; of\backslash \; diffusion\}\; =\; \backslash frac\{K\; A\; (P\_2\; -\; P\_1)\}\{D\}$

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

• Constant for a given gas at a given temperature by an experimentally determined factor, $K$
• Proportional to the surface area over which diffusion is taking place, $A$
• Proportional to the difference in partial pressures of the gas across the membrane, $P\_2\; -\; P\_1$
• Inversely proportional to the distance over which diffusion must take place, or in other words the thickness of the membrane, $D$.

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.

See also

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• Gas exchange
• Lung
• Alveoli

External links

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• Diffusion coefficient - diffusion-polymers.com

References

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• 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^rd ed., McGraw-Hill (2004)
• H.C. Berg, Random Walks in Biology, Princeton (1977)

Diffusion | Statistical mechanics | Physical chemistry | Eponymous laws

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