Fick's law of diffusion
Fick's laws of diffusion describe
diffusionand can be used to solve for the diffusion coefficient "D". They were derived by Adolf Fickin the year 1855.
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
* is the diffusion flux in dimensions of [(
amount of substance) length−2 time-1] , example . measures the amount of substance that will flow through a small area during a small time interval.
* is the diffusion coefficient or diffusivity in dimensions of [length2 time−1] , example
* (for ideal mixtures) is the concentration in dimensions of [(amount of substance) length−3] , example
* is the position [length] , example
is proportional to the squared velocity of the diffusing particles, which depends on the temperature,
viscosityof 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 , the
delor gradientoperator, which generalises the first derivative, obtaining
The driving force for the one-dimensional diffusion is the quantity
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 potentialof this species. Then Fick's first law (one-dimensional case) can be written as:
where the index i denotes the ith species, c is the concentration (mol/m3), R is the
universal gas constant(J/(K mol)), T is the absolute temperature (K), and μ is the chemical potential (J/mol).
Fick's second law predicts how diffusion causes the concentration field to change with time:
* is the concentration in dimensions of [(amount of substance) length-3] , [mol m-3]
* is time [s]
* is the diffusion coefficient in dimensions of [length2 time-1] , [m2 s-1]
* is the position [length] , [m]
It can be derived from Fick's First law and the
Assuming the diffusion coefficient "D" to be a constant we can exchange the orders of the differentiating and multiplying by the constant:
: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:
which is analogous to the
If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, the Second Fick's Law becomes:
An important example is the case where 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 , the solution for the concentration will be a linear change of concentrations along . In two or more dimensions we obtain
Laplace's equation, the solutions to which are called harmonic functionsby mathematicians.
Example solution in one dimension: diffusion length
A simple case of diffusion with time "t" in one dimension (taken as the "x"-axis) of a density from a boundary located at position where the density is maintained at a value is
where "erfc" is the complementary
error function. The length is called the diffusion length and provides a measure of how far the density has propagated in the "x-"direction by diffusion in time "t".
For more detail on diffusion length, see these [http://www.timedomaincvd.com/CVD_Fundamentals/xprt/diffusion_length.html examples] .
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 polymerscience 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 ( Onsagerrelationship).
Temperature dependence of the diffusion coefficient
The diffusion coefficient at different temperatures is often found to be well predicted by
* is the diffusion coefficient
* is the maximum diffusion coefficient (at infinite temperature)
* is the
activation energyfor diffusion in dimensions of [energy (amount of substance)−1]
* is the temperature in units of [absolute temperature] (
kelvins or degrees Rankine)
* is the
gas constantin dimensions of [energy temperature−1 (amount of substance)−1]
An equation of this form is known as the
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 Diffusion ] ] .
An approximate dependence of the diffusion coefficient on temperature in liquids can often be found using
Stokes-Einstein equation, which predicts that:
where:: T1 and T2 denote temperatures 1 and 2, respectively: D is the diffusion coefficient (m²/s): T is the absolute temperature (K),: μ is the
dynamic viscosityof the solvent (Pa·s)
Pressure dependence of the diffusion coefficient
For self-diffusion in gases at two different pressures (but the same temperature), the following empirical equation has been suggested: [E.L. Cussler, "Diffusion. Mass Transfer in Fluid Systems", 2nd edition, Cambridge University Press, 1997.] :where:: P1 and P2 denote pressures 1 and 2, respectively: D is the diffusion coefficient (m²/s): ρ is the gas mass density (kg/m3)
The first law gives rise to the following formula: [GeorgiaPhysiology|3/3ch9/s3ch9_2]
* is the permeability, an experimentally determined membrane "conductance" for a given gas at a given temperature.
* is the surface area over which diffusion is taking place.
* is the difference in
concentrationof the gas across the membrane for the direction of flow (from to ).
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
The exchange rate of a gas across a fluid membrane can be determined by using this law together with
Semiconductor fabrication applications
IC Fabrication technologies, model processes like CVD, Thermal Oxidation,and Wet Oxidation, Doping etc using Diffusion equations obtainedfrom Ficks law.
In certain cases, the solutions are obtained for boundary conditions such asconstant source concentration diffusion, limited source concentration,or moving boundary diffusion (where junction depth keeps moving intothe substrate).
* 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 3rd ed.", McGraw-Hill (2004)
* H.C. Berg, "Random Walks in Biology", Princeton (1977)
* [http://www.timedomaincvd.com/CVD_Fundamentals/xprt/intro_diffusion.html Diffusion fundamentals]
* [http://composite-agency.com/messages/3875.html Diffusion coefficient (of gases & liquids in polymer and composite materials)]
* [http://www.stthomas.edu/biol/sciencebear2/respiratory/respiratorycalc/respiratorycalc.html Ficks Law Calculator and a host of other science tools] - Rex Njoku & Dr.Anthony Steyermark
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