Fick's laws of diffusion

Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, $D$ . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation.

For the technique of measuring cardiac output, see Fick principle.

Fick's first law: Movement of particles from high to low concentration (diffusive flux) is directly proportional to the particle's concentration gradient.^[1]

Fick's second law: Prediction of change in concentration gradient with time due to diffusion.

A diffusion process that obeys Fick's laws is called normal or Fickian diffusion; otherwise, it is called anomalous diffusion or non-Fickian diffusion.

History[edit]

In 1855, physiologist Adolf Fick first reported^[2] his now well-known laws governing the transport of mass through diffusive means. Fick's work was inspired by the earlier experiments of Thomas Graham, which fell short of proposing the fundamental laws for which Fick would become famous. Fick's law is analogous to the relationships discovered at the same epoch by other eminent scientists: Darcy's law (hydraulic flow), Ohm's law (charge transport), and Fourier's Law (heat transport).

Fick's experiments (modeled on Graham's) dealt with measuring the concentrations and fluxes of salt, diffusing between two reservoirs through tubes of water. It is notable that Fick's work primarily concerned diffusion in fluids, because at the time, diffusion in solids was not considered generally possible.^[3] Today, Fick's Laws form the core of our understanding of diffusion in solids, liquids, and gases (in the absence of bulk fluid motion in the latter two cases). When a diffusion process does not follow Fick's laws (which happens in cases of diffusion through porous media and diffusion of swelling penetrants, among others),^[4]^[5] it is referred to as non-Fickian.

$J$ is the diffusion flux, of which the is the amount of substance per unit area per unit time. $J$ measures the amount of substance that will flow through a unit area during a unit time interval.

dimension

$D$ is the diffusion coefficient or . Its dimension is area per unit time.

diffusivity

${\frac {d\varphi }{dx}}$ is the concentration gradient

$φ$ (for ideal mixtures) is the concentration, with a dimension of amount of substance per unit volume.

$x$ is position, the dimension of which is length.

$φ$ is the concentration in dimensions of $[{\mathsf {N}}{\mathsf {L}}^{-3}]$ , example mol/m³; $φ = φ (x, t)$ is a function that depends on location $x$ and time $t$

$t$ is time, example s

$D$ is the diffusion coefficient in dimensions of $[{\mathsf {L}}^{2}{\mathsf {T}}^{-1}]$ , example m²/s

$x$ is the position, example m

In non-homogeneous media, the diffusion coefficient varies in space, $D = D (x)$ . This dependence does not affect Fick's first law but the second law changes: ${\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=D(x)\Delta \varphi (x,t)+\sum _{i=1}^{3}{\frac {\partial D(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}$

In media, the diffusion coefficient depends on the direction. It is a symmetric tensor $D ji = D ij$ . Fick's first law changes to $J=-D\nabla \varphi ,$ it is the product of a tensor and a vector: $J_{i}=-\sum _{j=1}^{3}D_{ij}{\frac {\partial \varphi }{\partial x_{j}}}.$ For the diffusion equation this formula gives ${\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D\nabla \varphi (x,t){\bigr )}=\sum _{i=1}^{3}\sum _{j=1}^{3}D_{ij}{\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}.$ The symmetric matrix of diffusion coefficients $D ij$ should be positive definite. It is needed to make the right hand side operator elliptic.

anisotropic

For inhomogeneous anisotropic media these two forms of the diffusion equation should be combined in ${\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=\sum _{i,j=1}^{3}\left(D_{ij}(x){\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}+{\frac {\partial D_{ij}(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}\right).$

The approach based on gives the following generalization of Fick's equation for the multicomponent diffusion of the perfect components: ${\frac {\partial \varphi _{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {\varphi _{i}}{\varphi _{j}}}\nabla \,\varphi _{j}\right).$ where $φ i$ are concentrations of the components and $D ij$ is the matrix of coefficients. Here, indices $i$ and $j$ are related to the various components and not to the space coordinates.

Einstein's mobility and Teorell formula

k_B is the

Boltzmann constant

T is the

absolute temperature

μ is the mobility of the particle in the fluid or gas, which can be calculated using the

Fick's laws of diffusion

History[edit]

dimension

diffusivity

anisotropic

Einstein's mobility and Teorell formula

Boltzmann constant

absolute temperature

Einstein relation (kinetic theory)

Advection

Churchill–Bernstein equation

Diffusion

False diffusion

Gas exchange

Mass flux

Maxwell–Stefan diffusion

Nernst–Planck equation

Osmosis

Fick's equations, Boltzmann's transformation, etc.

Fick's Second Law