Fick's laws of diffusion


Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. 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.
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 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. 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 10−11 to 10−10 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 vector.
The driving force for the one-dimensional diffusion is the quantity, which for ideal mixtures is the concentration gradient.

Alternative formulations of the first law

Another form for the first law is to write it with the primary variable as mass fraction, for when the total concentration of the mixture is approximately constant, then the equation changes to:
where
Note that the is outside the gradient operator. This is because:
where is the concentration of the substance for the th species.
Beyond this, in chemical systems other than ideal solutions or mixtures, the driving force for diffusion of each species is the gradient of chemical potential of this species. Then Fick's first law can be written
where
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
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 laws

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 2 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 has the particle 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 Maxwell-Boltzmann distribution.

Generalizations

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.

Applications

Equations based on Fick's law have been commonly used to model transport processes in foods, neurons, biopolymers, pharmaceuticals, porous soils, population dynamics, nuclear materials, plasma physics, and semiconductor doping processes. Theory of all voltammetric methods is based on solutions of Fick's equation. Much 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". It can be shown that the Fick's law can be obtained from the Maxwell–Stefan diffusion equations
of multi-component mass transfer. The Fick's law is limiting case of the Maxwell–Stefan equations, when the mixture is extremely dilute and every chemical species is interacting only with the bulk mixture and not with other species. To account for the presence of multiple species in a non-dilute mixture, several variations of the Maxwell–Stefan equations are used. See also non-diagonal coupled transport processes.

Fick's flow in liquids

When two miscible liquids are brought into contact, and diffusion takes place, the macroscopic concentration evolves following Fick's law. On a mesoscopic scale, that is, between the macroscopic scale described by Fick's law and molecular scale, where molecular random walks take place, fluctuations cannot be neglected. Such situations can be successfully modeled with Landau-Lifshitz fluctuating hydrodynamics. In this theoretical framework, diffusion is due to fluctuations whose dimensions range from the molecular scale to the macroscopic scale.
In particular, fluctuating hydrodynamic equations include a Fick's flow term, with a given diffusion coefficient, along with hydrodynamics equations and stochastic terms describing fluctuations. When calculating the fluctuations with a perturbative approach, the zero order approximation is Fick's law. The first order gives the fluctuations, and it comes out that fluctuations contribute to diffusion. This represents somehow a tautology, since the phenomena described by a lower order approximation is the result of a higher approximation: this problem is solved only by renormalizing the fluctuating hydrodynamics equations.

Sorption rate and collision frequency of diluted solute

The adsorption or absorption rate of a diluted solute to a surface or interface in a solution can be calculated using Fick's laws of diffusion, whose solution is typically a Gaussian function. Considering one dimension that is perpendicular to the surface, the probability of any given solute molecule in the solution hit the surface is the error function of its diffusive broadening over the time of interest. Thus integrate these error functions and integrate it with all solute molecules in the bulk gives the adsorption rate of the solute in unit s−1 to an area of interest:
where
The rate is proportional to t1/2 because diffusion is a self-mimetic fractal process. At a high observation resolution, i.e. a small t, the molecule collides with the surface many times which will be collapsed into only one collision count at longer observation time resolution. Due to this fractal nature, the analytical solution should be multiply by a factor of 2 according to a Monte Carlo simulation:
In the short time limit, in the order of the diffusion time a2/D, where a is the particle radius, the diffusion is described by the Langevin equation. At a longer time, the Langevin equation merges into the Stokes–Einstein equation. The latter is appropriate for the condition of the diluted solution, where long-range diffusion is considered. According to the fluctuation-dissipation theorem based on the Langevin equation in the long-time limit and when the particle is significantly denser than the surrounding fluid, the time-dependent diffusion constant is:
where
For a single molecule such as organic molecules or biomolecules in water, the exponential term is negligible due to the small product of mμ in the picosecond region.
When the area of interest is the size of a molecule, the adsorption rate equation represents the collision frequency of two molecules in a diluted solution, with one molecule a specific side and the other no steric dependence, i.e., a molecule hit one side of the other. The diffusion constant need to be updated to the relative diffusion constant between two diffusing molecules. This estimation is especially useful in studying the interaction between a small molecule and a larger molecule such as a protein. The effective diffusion constant is dominated by the smaller one whose diffusion constant can be used instead.
The above hitting rate equation is also useful to predict the kinetics of molecular self-assembly on a surface. Molecules are randomly oriented in the bulk solution. Assuming 1/6 of the molecules has the right orientation to the surface binding sites, i.e. 1/2 of the z-direction in x, y, z three dimensions, thus the concentration of interest is just 1/6 of the bulk concentration. Put this value into the equation one should be able to calculate the theoretical adsorption kinetic curve using the Langmuir adsorption model. In a more rigid picture, 1/6 can be replaced by the steric factor of the binding geometry.

Biological perspective

The first law gives rise to the following formula:
in which,
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.
Under the condition of a diluted solution when diffusion takes control, the membrane permeability mentioned in the above section can be theoretically calculated for the solute using the equation mentioned in the last section :
where
fabrication technologies, model processes like CVD, thermal oxidation, wet oxidation, doping, etc. use diffusion equations obtained from Fick's law.
In certain cases, the solutions are obtained for boundary conditions such as constant source concentration diffusion, limited source concentration, or moving boundary diffusion.