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 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
where
- \, J is the diffusion flux in dimensions of , example \bigg(\frac{\mathrm{mol}}{ m^2\cdot s}\bigg)
- \, D is the diffusion coefficient or diffusivity in dimensions of , example \bigg(\frac{m^2}{s}\bigg)
- \, \phi is the concentration in dimensions of , example \bigg(\frac\mathrm{mol}{m^3}\bigg)
- \, x is the position , 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.
In two or more dimensions we must use \nabla, the del or gradient operator, which generalises the first derivative, obtaining

# 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.
Where
- \,\phi is the concentration in dimensions of , 
- \, t is time 
- \, D is the diffusion coefficient in dimensions of , 
- \, x is the position , 
It can be derived from the Fick's First law and the mass balance:
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:
\frac{\partial \phi}{\partial t} = 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:
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
which is Laplace's equation, the solutions to which are called harmonic functions by mathematicians.

# 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
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 
- \, T is the temperature in units of  (kelvins or degrees Rankine)
- \, R is the gas constant in dimensions of 
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 .

# Biological perspective
The first law gives rise to the following formula:
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,
-r moving boundary diffusion (where junction depth keeps moving into
the substrate).