NationMaster - Encyclopedia: Fick's law of diffusion

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. diffusion (disambiguation). ... Adolf Eugen Fick (1829-1901) was a German physiologist and inventor. ... Year 1855 (MDCCCLV) was a common year starting on Monday (link will display the full calendar) of the Gregorian Calendar (or a common year starting on Saturday of the 12-day slower Julian calendar). ...

1 First law
2 Second law
3 Applicability
4 Temperature dependence of the diffusion coefficient
5 Biological perspective
6 Semiconductor fabrication applications
7 See also
8 References
9 External links

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 $(, J_mathrm{in} = J_mathrm{out})$ . In one (spatial) dimension, this is diffusion (disambiguation). ...

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

where

$, J$ is the diffusion flux in dimensions of [(amount of substance) length⁻² time^-1], example $bigg(frac{mathrm{mol}}{ m^2cdot s}bigg)$

$, D$ is the diffusion coefficient or diffusivity in dimensions of [length² time⁻¹], example $bigg(frac{m^2}{s}bigg)$

$, phi$ is the concentration in dimensions of [(amount of substance) length⁻³], example $bigg(fracmathrm{mol}{m^3}bigg)$

$, x$ 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. 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. The amount of substance, n, of a sample or system is a physical quantity which is proportional to the number of elementary entities present. ... This is the diffusion coefficient in Ficks Law. ... Viscosity is a measure of the resistance of a fluid to deform under shear stress. ... In physics, in kinetic theory the Einstein relation is a previously unexpected connection revealed by Einstein in his 1905 paper on Brownian motion: linking D, the Diffusion constant, and Î¼, the mobility of the particles; where k is Boltzmanns constant, and T is the absolute temperature. ...

In two or more dimensions we must use $nabla$ , the del or gradient operator, which generalises the first derivative, obtaining In vector calculus, del is a vector differential operator represented by the nabla symbol: âˆ‡. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. ... For other uses, see Gradient (disambiguation). ...

$J=- Dnabla phi$ .

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.

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

Where

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

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

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

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

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

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:

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

also called the heat equation. The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ...

If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, the Second Fick's Law becomes:

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

An important example is the case where $φ$ 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

$nabla^2,phi =0!$

which is Laplace's equation, the solutions to which are called harmonic functions by mathematicians. In mathematics, Laplaces equation is a partial differential equation named after its discoverer, Pierre-Simon Laplace. ... In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U â†’ R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ...

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). Passive transport is a means of moving biochemicals, and other atomic or molecular substances, across membranes. ... Drawing by Santiago RamÃ³n y Cajal of neurons in the pigeon cerebellum. ... This article does not cite any references or sources. ... Pharmacology (in Greek: pharmakon (Ï†Î¬ÏÎ¼Î±ÎºÎ¿Î½) meaning drug, and lego (Î»ÎÎ³Ï‰) to tell (about)) is the study of how drugs interact with living organisms to produce a change in function. ... A pore, in general, is some form of opening, usually very small. ... Loess field in Germany Surface-water-gley developed in glacial till, Northern Ireland Technically, soil forms the pedosphere: the interface between the lithosphere (rocky part of the planet) and the biosphere, atmosphere, and hydrosphere. ... Population dynamics is the study of marginal and long-term changes in the numbers, individual weights and age composition of individuals in one or several populations, and biological and environmental processes influencing those changes. ... In semiconductor production, doping refers to the process of intentionally introducing impurities into an extremely pure (also referred to as intrinsic) semiconductor in order to change its electrical properties. ... In the scientific method, an experiment (Latin: ex- periri, of (or from) trying) is a set of observations performed in the context of solving a particular problem or question, to support or falsify a hypothesis or research concerning phenomena. ... A polymer (from Greek: Ï€Î¿Î»Ï…, polu, many; and Î¼ÎÏÎ¿Ï‚, meros, part) is a substance composed of molecules with large molecular mass composed of repeating structural units, or monomers, connected by covalent chemical bonds. ... A simplistic view of a materials glass transition temperature (Tg) is the temperature below which molecules have very little mobility. ... Lars Onsager (November 27, 1903-October 5, 1976) was a Norwegian physical chemist, winner of the 1968 Nobel Prize in Chemistry. ...

Temperature dependence of the diffusion coefficient

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

$D = D_0cdot e^{-frac{E_{A}}{Rcdot T}},$

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] (kelvins 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. The sparks generated by striking steel against a flint provide the activation energy to initiate combustion in this Bunsen burner. ... The kelvin (symbol: K) is a unit increment of temperature and is one of the seven SI base units. ... For the idealized thermodynamic cycle for a steam engine, see Rankine cycle. ... The gas constant (also known as the universal or ideal gas constant, usually denoted by symbol R) is a physical constant used in equations of state to relate various groups of state functions to one another. ... The Arrhenius equation is a simple, but remarkably accurate, formula for the temperature dependence of a chemical reaction rate, more correctly, of a rate coefficient, as this coefficient includes all magnitudes that affect reaction rate except for concentration. ...

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]

$mathrm{Rate of diffusion} = frac{{Kcdot Acdot (P_2 - P_1)}}{d} ,!$

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 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. In a mixture of ideal gases, each gas has a partial pressure which is the pressure which the gas would have if it alone occupied the volume. ... An artificial membrane, also called a synthetic membrane, is a membrane prepared for separation tasks in laboratory and industry. ... Flux limiters are used in high resolution schemes â€” numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by partial differential equations (PDEs). ...

The exchange rate of a gas across a fluid membrane can be determined by using this law together with Graham's law. Grahams law, also known as Grahams law of effusion, was formulated by Scottish physical chemist, Thomluyfkuyfj,gfhuas Graham. ...

Semiconductor fabrication applications

IC Fabrication technologies, model processes like CVD, Thermal Oxidation, and Wet Oxidation, Doping etc using Diffusion equations obtained from Ficks law. Integrated circuit of Atmel Diopsis 740 System on Chip showing memory blocks, logic and input/output pads around the periphery Microchips with a transparent window, showing the integrated circuit inside. ...

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).

References

^ Physiology at MCG 3/3ch9/s3ch9_2

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)

In 1828 the Medical Academy of Georgia was chartered by the state of Georgia with plans to offer a single course of lectures leading to a bachelors degree. ...

External links

Diffusion coefficient
Ficks Law Calculator and a host of other science tools - Rex Njoku & Dr.Anthony Steyermark

Categories: Diffusion | Statistical mechanics | Physical chemistry | Calques from German

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Fick's law of diffusion - Wikipedia, the free encyclopedia (589 words)
	Fick's laws of diffusion describe diffusion, and define the diffusion coefficient D.
	Fick's laws of diffusion were derived by Adolf Fick in the year 1855.
	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.

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