Principles of Diffusion

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Fick's laws are the basis for describing diffusion. They are mostly expressed as, in one-dimensional and three-dimensional form respectively:

First law:   J = − D ∂c   ╍╍  J = − D ∇c
∂x
Second law:   ∂c = D ∂2c   ╍╍   ∂c = D ∇2c
∂t ∂x2 ∂t
where J is flux – amount per area and per time –, ∂ partial differential operator, D diffusion coefficient, c concentration, x distance, and ∇ gradient operator: in rectangular coordinates x, y, z:
∇ = ( ∂, ∂, ∂)
∂x ∂y ∂z
and the underline in J is to indicate that it is 3-dimensional: J = (Jx, Jy, Jz) – this will be the standard method here to indicate that something is a vector, a 3-dimensional entity.

There is a lot to say about these forms of Fick's laws. The first law is only valid in homogeneous media, at least, homogeneous in the direction of the respective gradient. For gases, the form:

J = − ℘∇P
is better since this is valid also for permeabilities ℘ (1) that vary from location to location. Consequently, the same problem is with the second law which is actually a conservation-of-mass equation based on the first law:
∂c = −  ∂J   ╍╍   ∂c = − ∇⋅J
∂t ∂x ∂t
which is easily seen since it states that the change in concentration equals the difference in what comes in (Jin) and goes out (Jout). Consequently, this form is universally valid. The ⋅ indicates a vector product (2).

In biology, homogeneous media are rare, and often is worked with 'apparent' diffusion coefficients. It will be obvious that is not without pitfalls.

Furthermore, biological molecules often have electrical charge and that also can and will be a driving force for movement. Even for neutral gases like CO2 this has to be considered since it reacts with water forming the charged bicarbonate ion, HCO3−. For ions, Fick's first law has to be replaced by the Nernst-Planck equation:

J = − D{∇c + Z c F ∇ V}
RT
where Z is the electrical charge of the ion, F is the Faraday constant (3), R is the universal gas constant (3), T is absolute temperature (K), and V is the electrical potential. Consequently, ion movements will be accompanied by voltage gradients – the basis for cell membrane potentials. Note, that again this is for homogeneous media; the gas form will be universally valid:
J = − ℘{∇P + Z P F ∇ V}
RT
(1) Caveat! This conforms the original definition of permeability (4), but since, several other types have been defined too!
(2) The vector product between two vectors a and b is defined as a⋅b = (axbx+ayby+azbz)
(3) F = 96485 C·mol−1   R = 8,3 J·K−1·mol−1
(4) Jost W: Diffusion in Solids, Liquids, Gases. Acad Press Inc New York XI (1952). https://doi.org/10.1002/ange.19530651912

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