Definitions

# 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.

## First law

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

$bigg. J = - D frac\left\{partial phi\right\}\left\{partial x\right\} bigg.$
where

• $J$ is the diffusion flux in dimensions of [(amount of substance) length−2 time-1], example $bigg\left(frac\left\{mathrm\left\{mol\right\}\right\}\left\{ m^2cdot s\right\}bigg\right)$. $J$ measures the amount of substance that will flow through a small area during a small time interval.
• $, D$ is the diffusion coefficient or diffusivity in dimensions of [length2 time−1], example $bigg\left(frac\left\{m^2\right\}\left\{s\right\}bigg\right)$
• $, phi$ (for ideal mixtures) is the concentration in dimensions of [(amount of substance) length−3], example $bigg\left(fracmathrm\left\{mol\right\}\left\{m^3\right\}bigg\right)$
• $, x$ is the position [length], example $,m$

$, D$ is proportional to the squared 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.

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

$J=- Dnabla phi$.

The driving force for the one-dimensional diffusion is the quantity $- frac\left\{partial phi\right\}\left\{partial x\right\}$

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 potential of this species. Then Fick's first law (one-dimensional case) can be written as:

$J_i = - frac\left\{D c_i\right\}\left\{RT\right\} frac\left\{partial mu\right\}\left\{partial x_i\right\}$

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).

## Second law

Fick's second law predicts how diffusion causes the concentration field to change with time:

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

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 Fick's First law and the mass balance:

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

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

$frac\left\{partial\right\}\left\{partial x\right\}bigg\left(,D,frac\left\{partial\right\}\left\{partial x\right\} phi,bigg\right) = D,frac\left\{partial\right\}\left\{partial x\right\} frac\left\{partial\right\}\left\{partial x\right\} ,phi = D,frac\left\{partial^2phi\right\}\left\{partial x^2\right\}$
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:

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

which is analogous to 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:

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

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

$nabla^2,phi =0!$

which is Laplace's equation, the solutions to which are called harmonic functions by 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 $n\left(x,t\right)$ from a boundary located at position $x=0$ where the density is maintained at a value $n\left(0\right)$ is

$n left\left(x,t right\right)=n\left(0\right)$ $mathrm\left\{erfc\right\} left\left(begin\left\{matrix\right\}frac\left\{x\right\}\left\{sqrt\left\{4Dt\right\}\right\}end\left\{matrix\right\}right\right)$.

where erfc is the complementary error function. The length $sqrt\left\{4Dt\right\}$ 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 examples

## 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).

## Temperature dependence of the diffusion coefficient

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

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

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.

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.

An approximate dependence of the diffusion coefficient on temperature in liquids can often be found using Stokes-Einstein equation, which predicts that:

$frac \left\{D_\left\{T1\right\}\right\} \left\{D_\left\{T2\right\}\right\} = frac \left\{T_1\right\} \left\{T_2\right\} frac \left\{mu_\left\{T2\right\}\right\} \left\{mu_\left\{T1\right\}\right\}$

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 viscosity of 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:
$frac \left\{D_\left\{P1\right\}\right\} \left\{D_\left\{P2\right\}\right\} = frac \left\{rho_\left\{P2\right\}\right\} \left\{rho_\left\{P1\right\}\right\}$
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)

## Biological perspective

The first law gives rise to the following formula:

$Flux = \left\{-Pcdot Acdot \left(c_2 - c_1\right)\right\},!$

in which,

• $, P$ is the permeability, an experimentally determined membrane "conductance" for a given gas at a given temperature.
• $, A$ is the surface area over which diffusion is taking place.
• $, c_2 - c_1$ is the difference in concentration of the gas across the membrane for the direction of flow (from $c_1$ to $c_2$).

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.

## Semiconductor fabrication applications

IC Fabrication technologies, model processes like CVD, Thermal Oxidation, and Wet Oxidation, Doping etc using Diffusion equations obtained from Ficks law.

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

### Notations

• 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)