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,. Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation.
Fick's first law: Movement of particles from high to low concentration is directly proportional to the particle's concentration gradient.
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

In 1855, physiologist Adolf Fick first reported 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, Ohm's law, and Fourier's law.
Fick's experiments 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. Today, Fick's laws form the core of our understanding of diffusion in solids, liquids, and gases. When a diffusion process does not follow Fick's laws, it is referred to as non-Fickian.

Fick's first law

Fick's first law relates the diffusive flux to the gradient of the concentration. It postulates that the flux goes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient, or in simplistic terms the concept that a solute will move from a region of high concentration to a region of low concentration across a concentration gradient. In one dimension, the law can be written in various forms, where the most common form is in a molar basis:
where
  • is the diffusion flux, of which the dimension is the amount of substance per unit area per unit time. measures the amount of substance that will flow through a unit area during a unit time interval,
  • is the diffusion coefficient or diffusivity. Its dimension is area per unit time,
  • is the concentration gradient,
  • is the concentration, with a dimension of amount of substance per unit volume,
  • is position, the dimension of which is length.
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. The modeling and prediction of Fick's diffusion coefficients is difficult. They can be estimated using the empirical Vignes correlation model or the physically motivated entropy scaling. In dilute aqueous solutions the diffusion coefficients of most ions are similar and have values that at room temperature are in the range of. For biological molecules the diffusion coefficients normally range from 10−10 to 10−11 m2/s.
In two or more dimensions we must use, the del or gradient operator, which generalises the first derivative, obtaining
where denotes the diffusion flux.
The driving force for the one-dimensional diffusion is the quantity, which for ideal mixtures is the concentration gradient.

Variations of the first law

Another form for the first law is to write it with the primary variable as mass fraction, then the equation changes to
where
  • the index denotes the th species,
  • is the diffusion flux of the th species,
  • is the molar mass of the th species,
  • is the mixture density.
The is outside the gradient operator. This is because
where is the partial density of the th species.
Beyond this, in chemical systems other than ideal solutions or mixtures, the driving force for the diffusion of each species is the gradient of chemical potential of this species. Then Fick's first law can be written
where
The driving force of Fick's law can be expressed as a fugacity difference:
where is the fugacity in Pa. is a partial pressure of component in a vapor or liquid phase. At vapor liquid equilibrium the evaporation flux is zero because.

Derivation of Fick's first law for gases

Four versions of Fick's law for binary gas mixtures are given below. These assume: thermal diffusion is negligible; the body force per unit mass is the same on both species; and either pressure is constant or both species have the same molar mass. Under these conditions, Ref. shows in detail how the diffusion equation from the kinetic theory of gases reduces to this version of Fick's law:
where is the diffusion velocity of species. In terms of species flux this is
If, additionally,, this reduces to the most common form of Fick's law,
If both species have the same molar mass, Fick's law becomes
where is the mole fraction of species.

Fick's second law

Fick's second law predicts how diffusion causes the concentration to change with respect to time. It is a partial differential equation which in one dimension reads
where
  • is the concentration in dimensions of, example mol/m3; is a function that depends on location and time,
  • is time, example s,
  • is the diffusion coefficient in dimensions of, example m2/s,
  • is the position, example m.
In two or more dimensions we must use the Laplacian, which generalises the second derivative, obtaining the equation
Fick's second law has the same mathematical form as the Heat equation and its fundamental solution is the same as the Heat kernel, except switching thermal conductivity with diffusion coefficient :

Derivation of Fick's second law

Fick's second law can be derived from Fick's first law and the mass conservation in absence of any chemical reactions:
Assuming the diffusion coefficient to be a constant, one can exchange the orders of the differentiation and multiply 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 Fick's second law becomes
which is analogous to the heat equation.
If the diffusion coefficient is not a constant, but depends upon the coordinate or concentration, Fick's second law yields
An important example is the case where 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, the solution for the concentration will be a linear change of concentrations along. In two or more dimensions we obtain
which is Laplace's equation, the solutions to which are referred to by mathematicians as harmonic functions.

Example solutions and generalization

Fick's second law is a special case of the convection–diffusion equation in which there is no advective flux and no net volumetric source. It can be derived from the continuity equation:
where is the total flux and is a net volumetric source for. The only source of flux in this situation is assumed to be diffusive flux:
Plugging the definition of diffusive flux to the continuity equation and assuming there is no source, we arrive at Fick's second law:
If flux were the result of both diffusive flux and advective flux, the convection–diffusion equation is the result.

Example solution 1: constant concentration source and diffusion length

A simple case of diffusion with time in one dimension from a boundary located at position, where the concentration is maintained at a value is
where is the complementary error function. This is the case when corrosive gases diffuse through the oxidative layer towards the metal surface. If, in its turn, the diffusion space is infinite, then the solution is amended only with coefficient in front of . This case is valid when some solution with concentration is put in contact with a layer of pure solvent. The length is called the diffusion length and provides a measure of how far the concentration has propagated in the -direction by diffusion in time .
As a quick approximation of the error function, the first two terms of the Taylor series can be used:
If is time-dependent, the diffusion length becomes
This idea is useful for estimating a diffusion length over a heating and cooling cycle, where varies with temperature.

Example solution 2: Brownian particle and mean squared displacement

Another simple case of diffusion is the Brownian motion of one particle. The particle's Mean squared displacement from its original position is:
where is the dimension of the particle's Brownian motion. For example, the diffusion of a molecule across a cell membrane 8 nm thick is 1-D diffusion because of the spherical symmetry; However, the diffusion of a molecule from the membrane to the center of a eukaryotic cell is a 3-D diffusion. For a cylindrical cactus, the diffusion from photosynthetic cells on its surface to its center is a 2-D diffusion.
The square root of MSD,, is often used as a characterization of how far the particle has moved after time has elapsed. The MSD is symmetrically distributed over the 1D, 2D, and 3D space. Thus, the probability distribution of the magnitude of MSD in 1D is Gaussian and 3D is a Maxwell-Boltzmann distribution.

Generalizations

  • In non-homogeneous media, the diffusion coefficient varies in space,. This dependence does not affect Fick's first law but the second law changes:
  • In anisotropic media, the diffusion coefficient depends on the direction. It is a symmetric tensor. Fick's first law changes to it is the product of a tensor and a vector: For the diffusion equation this formula gives The symmetric matrix of diffusion coefficients should be positive definite. It is needed to make the right-hand side operator elliptic.
  • For inhomogeneous anisotropic media these two forms of the diffusion equation should be combined in
  • The approach based on Einstein's mobility and Teorell formula gives the following generalization of Fick's equation for the multicomponent diffusion of the perfect components: where are concentrations of the components and is the matrix of coefficients. Here, indices and are related to the various components and not to the space coordinates.
The Chapman–Enskog formulae for diffusion in gases include exactly the same terms. These physical models of diffusion are different from the test models which are valid for very small deviations from the uniform equilibrium. Earlier, such terms were introduced in the Maxwell–Stefan diffusion equation.
For anisotropic multicomponent diffusion coefficients one needs a rank-four tensor, for example, where refer to the components and correspond to the space coordinates.