Diffusion: Fick’s laws

 

Principle

 

Fick's first law

Fick's first law is used in steady-state diffusion (see Diffusion: general), i.e., when the concentration within the diffusion volume does not change with respect to time (Jin = Jout).

,                      (1)

where

  J is the diffusion flux in dimensions of  (mol m-2 s-1);

  D is the diffusion coefficient or diffusivity, (m2 s-1);

  φ is the concentration (mol m-3);

  x is the position (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. For the biological molecules the diffusion coefficient normally ranges from 10-11 to 10-10 m2 s-1.

 

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.

 

,                                (2)

Where:

φ is the concentration (mol m-3);

 t is time (s);

 D is the constant diffusion coefficient (m2 s-1);

 x is the position (m).

With some calculation, it can be derived from the First Fick's law and the mass balance.

 

 

Application

 

Equations based on Fick's law have been commonly used to model transport processes in foods, porous soils, semiconductor doping process, etc.. In biomedicine, transport of biopolymers, pharmaceuticals, in neurons, etc. are modeled with the Fick equations.

Typically, a compound's D is ~10,000x greater in air than in water. CO2 in air has a diffusion coefficient of 16 mm˛/s, and in water, its coefficient is 0.0016 mm˛/s.

 

Physiological examples

For example for the steady states, when concentration does not change by time, the left part of the above equation will be zero and therefore in one dimension and when D is constant, the equation (2) becomes 0 = D dφ/dx2. The solution for the concentration φ will be the linear change of concentrations along x. This is what by approximation happens in the alveolar-capillary membrane and with diffusion of particles from the liquid to liquid phase in the wall of capillaries in the tissues.

 

Biological perspective

The first law gives rise to the following formula:

  Rate of diffusion = KA(P2 – P1)

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 pressures of the gas across the membrane, P2 − P1

It is also inversely proportional to the distance over which diffusion must take place, or in other words the thickness of the membrane. The factor K comprises this thickness.

The exchange rate of a gas across a fluid membrane can be determined by using this law together with Graham's law.

 

 

More Info

 

Temperature dependence of the diffusion coefficient

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

 

  ,                   (5)

where:

 D0 is the maximum diffusion coefficient at infinite temperature,

 EA is the activation energy for diffusion (J/mol),

 T is absolute temperature (K),

 R is the gas constant in dimensions of (J/(K∙mol)).


3D diffusion

For the case of 3-dimensional diffusion the Second Fick's Law looks like:

                     (3)

where is the del operator. In a 3D system with perpendicular coordinates (x, y, z), this is a Cartesian coordinate system R3, del is defined as:

,

where i, j and k are the unit-vectors in the direction of the respective coordinate (the standard basis in R3).

 

Finally if the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, the Second Fick's Law looks like:

.           (4)