Fick's Law Of Diffusion

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 not change with respect to time (J_in=J_out).

J = - D rac{\partial \phi}{\partial x}

Where

$J$ is the diffusion flux in dimensions of 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

$\Phi$ is the concentration in dimensions of of substance) length^-3 , m^-3

$x$ is the position [m

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.

:

rac{\partial \phi}{\partial t} = D rac{\partial^2 \phi}{\partial x^2}

Where

$\Phi$ is the concentration in dimensions of of substance) length^-3 , dm^-3

$t$ is time {Link without Title}

$D$ 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:

:

rac{\partial \phi}{\partial t} =-rac {\partial} {\partial x} J = rac {\partial} {\partial x} (D rac {\partial} {\partial x} \phi)

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

rac {\partial} {\partial x} (D rac {\partial} {\partial x} \phi) = D rac {\partial} {\partial x}  rac {\partial} {\partial x} \phi= D rac{\partial^2 \phi}{\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:
:

rac{\partial \phi}{\partial t} = D 
abla^2 \phi

,
where

abla

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:
:

rac{\partial \phi}{\partial t} =  
abla (D 
abla \phi)

APPLICABILITY

Equations based on Fick's law have been commonly used to model Transport Processes in Food s, Biopolymer s, Pharmaceuticals , Porous Soil s, Semiconductor Doping process, etc. A large amount of Experiment al 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

D = D_0 e^{-rac{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 ( Kelvin s 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 mm&²/s, and in water, its coefficient is 0.0016 mm&²/s {Link without Title} .

A BIOLOGICAL PERSPECTIVE

The first law gives rise to the formula

:

\mathrm{Rate\ of\ diffusion} = rac{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 Pressure s 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

Gas Exchange

Lung

Alveoli

EXTERNAL LINKS

Diffusion coefficient - diffusion-polymers.com

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^rd ed.'', McGraw-Hill (2004)

H.C. Berg, ''Random Walks in Biology'', Princeton (1977)

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Information About

Fick's Law Of Diffusion