Fick's laws of diffusion

Another form for the first law is to write it with the primary variable as mass fraction (y_i, given for example in kg/kg), for when the total concentration of the mixture is approximately constant, then the equation changes to:

${\displaystyle J_{i}=-\rho D\nabla y_{i}}$

where

• the index i denotes the ith species.
• J is still the same diffusion flux as in the common form, namely as amount of substance per unit area per unit time.
• ρ is the total concentration of the mixture, as in fluid density.
• y is the aforementioned mass fraction, which is dimensionless.

Note that the ${\displaystyle \rho }$ is outside the gradient operator. This is because:

${\displaystyle y_{i}={\frac {\rho _{si}}{\rho }}}$

where ρ_si is the concentration of the substance for the ith 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 (one-dimensional case) can be written

${\displaystyle J_{i}=-{\frac {Dc_{i}}{RT}}{\frac {\partial \mu _{i}}{\partial x}}}$

where

• the index i denotes the ith species.
• c is the concentration (mol/m³).
• R is the universal gas constant (J/K/mol).
• T is the absolute temperature (K).
• µ is the chemical potential (J/mol).

Fick's second law can be derived from Fick's first law and the mass conservation in absence of any chemical reactions:

${\displaystyle {\frac {\partial \varphi }{\partial t}}+{\frac {\partial }{\partial x}}J=0\Rightarrow {\frac {\partial \varphi }{\partial t}}-{\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)\,=0}$

Assuming the diffusion coefficient D to be a constant, one can exchange the orders of the differentiation and multiply by the constant:

${\displaystyle {\frac {\partial }{\partial x}}\left(D{\frac {\partial }{\partial x}}\varphi \right)=D{\frac {\partial }{\partial x}}{\frac {\partial }{\partial x}}\varphi =D{\frac {\partial ^{2}\varphi }{\partial x^{2}}}}$

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

${\displaystyle {\frac {\partial \varphi }{\partial t}}=D\nabla ^{2}\varphi ,}$

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

${\displaystyle {\frac {\partial \varphi }{\partial t}}=\nabla \cdot (D\nabla \varphi ).}$

An important example is the case where φ 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 D, the solution for the concentration will be a linear change of concentrations along x. In two or more dimensions we obtain

${\displaystyle \nabla ^{2}\varphi =0}$

which is Laplace's equation, the solutions to which are referred to by mathematicians as harmonic functions.

A simple case of diffusion with time t in one dimension (taken as the x-axis) from a boundary located at position x = 0, where the concentration is maintained at a value n₀ is

${\displaystyle n\left(x,t\right)=n_{0}\mathrm {erfc} \left({\frac {x}{2{\sqrt {Dt}}}}\right)}$.

where erfc is the complementary error function. This is the case when corrosive gases diffuse through the oxidative layer towards the metal surface (if we assume that concentration of gases in the environment is constant and the diffusion space – that is, the corrosion product layer – is semi-infinite, starting at 0 at the surface and spreading infinitely deep in the material). If, in its turn, the diffusion space is infinite (lasting both through the layer with n(x,0) = 0, x > 0 and that with n(x,0) = n₀, x ≤ 0), then the solution is amended only with coefficient 1/2 in front of n₀ (as the diffusion now occurs in both directions). This case is valid when some solution with concentration n₀ is put in contact with a layer of pure solvent. (Bokstein, 2005) The length 2√Dt is called the diffusion length and provides a measure of how far the concentration has propagated in the x-direction by diffusion in time t (Bird, 1976).

As a quick approximation of the error function, the first 2 terms of the Taylor series can be used:

${\displaystyle n\left(x,t\right)=n_{0}\left[1-2\left({\frac {x}{2{\sqrt {Dt\pi }}}}\right)\right]}$

If D is time-dependent, the diffusion length becomes

${\displaystyle 2{\sqrt {\int _{0}^{t}D\tau \,d\tau }}.}$.

This idea is useful for estimating a diffusion length over a heating and cooling cycle, where D varies with temperature.

Another simple case of diffusion is the Brownian motion of one particle. The particle's Mean squared displacement from its original position is:

${\displaystyle {\text{MSD}}\equiv \langle (\mathbf {x} -\mathbf {x_{0}} )^{2}\rangle =2nDt}$

where ${\displaystyle n}$ 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 (the axis of its cylindrical symmetry) is a 2-D diffusion.

The square root of MSD, ${\displaystyle {\sqrt {2nDt}}}$, is often used as a characterization of how far has the particle moved after time ${\displaystyle t}$ 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.

• 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:

${\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=D(x)\Delta \varphi (x,t)+\sum _{i=1}^{3}{\frac {\partial D(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}}$

• In anisotropic media, the diffusion coefficient depends on the direction. It is a symmetric tensor D = D_ij. Fick's first law changes to

${\displaystyle J=-D\nabla \varphi }$,

it is the product of a tensor and a vector:

${\displaystyle J_{i}=-\sum _{j=1}^{3}D_{ij}{\frac {\partial \varphi }{\partial x_{j}}}.}$

For the diffusion equation this formula gives

${\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D\nabla \varphi (x,t){\bigr )}=\sum _{i=1}^{3}\sum _{j=1}^{3}D_{ij}{\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}.}$

The symmetric matrix of diffusion coefficients D_ij should be positive definite. It is needed to make the right hand side operator elliptic.

• For inhomogeneous anisotropic media these two forms of the diffusion equation should be combined in

${\displaystyle {\frac {\partial \varphi (x,t)}{\partial t}}=\nabla \cdot {\bigl (}D(x)\nabla \varphi (x,t){\bigr )}=\sum _{i,j=1}^{3}\left(D_{ij}(x){\frac {\partial ^{2}\varphi (x,t)}{\partial x_{i}\partial x_{j}}}+{\frac {\partial D_{ij}(x)}{\partial x_{i}}}{\frac {\partial \varphi (x,t)}{\partial x_{i}}}\right).}$

• The approach based on Einstein's mobility and Teorell formula gives the following generalization of Fick's equation for the multicomponent diffusion of the perfect components:

${\displaystyle {\frac {\partial \varphi _{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {\varphi _{i}}{\varphi _{j}}}\nabla \,\varphi _{j}\right).}$

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.

The Chapman–Enskog formulae for diffusion in gases include exactly the same terms. These physical models of diffusion are different from the test models ∂_tφ_i = ∑_j D_ij Δφ_j 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 D_ij,αβ, where i, j refer to the components and α, β = 1, 2, 3 correspond to the space coordinates.

When two miscible liquids are brought into contact, and diffusion takes place, the macroscopic (or average) 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.[8]

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.

The adsorption or absorption rate of a diluted solute to a surface or interface in a (gas or liquid) 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:[9]

Scheme of molecular diffusion in the solution. Orange dots are solute molecules, solvent molecules are not drawn, black arrow is an example random walk trajectory, and the red curve is the diffusive Gaussian broadening probability function from the Fick's law of diffusion.[9] ^{Fig. 9}

${\displaystyle r={\frac {1}{t}}\left(\int _{z=0}^{\infty }AC{\int _{x=z}^{\infty }{\frac {1}{\sqrt {4\pi Dt}}}e^{-{\frac {x^{2}}{4Dt}}}dx}dz\right)=AC{\sqrt {\frac {D}{\pi t}}}}$

where

• x is the distance of the probability function from the original location of a solute molecule (time t location references to its location at time 0, ${\displaystyle \Delta t=t-0}$)
• z is the original distance of the molecule from the surface.
• A is the surface area of the surface of interest.
• C is the number concentration of the molecule in the bulk solution.
• D is the effective diffusion constant of the solute molecule measured at time resolution t.
• t is the time of interest. Because of the non-Gaussian tail of a typical diffusion system, t should be chosen to at a value such that r ≈ 1/t to reflect the major Gaussian peak effect, i.e. around the first passage time t ≈ 4/(π D A² C ²).
• Note, (1) D is dependent on t, and the probability function is commonly non-Gaussian. (2) The 3D solution of hitting a small (comparing to the diffusion length) area on the surface is the same as the 1D solution, just using relative argument to imagine the ball diffuses to half of the sphere following a radial probability density function of the Maxwell–Boltzmann distribution.[10]

The rate is proportional to t^1/2 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.[10] Due to this fractal nature, the analytical solution should be multiply by a factor of 2 according to a Monte Carlo simulation:[10]

${\displaystyle r=2AC{\sqrt {\frac {D}{\pi t}}}}$

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:[11]

${\displaystyle D(t)=\mu \,k_{\rm {B}}T(1-e^{-t/(m\mu )})}$

where

• k_B is Boltzmann's constant;
• T is the absolute temperature.
• μ is the mobility of the particle in the fluid or gas, which can be calculated using the Einstein relation (kinetic theory).
• m is the mass of the particle.
• t is time.

For a single molecule such as organic molecules or biomolecules (e.g proteins) 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 (specifically, a long cylindrical molecule such as DNA), 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 (random orientation) 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.

The first law gives rise to the following formula:[12]

${\displaystyle {\text{flux}}={-P\left(c_{2}-c_{1}\right)}}$

in which,

• P is the permeability, an experimentally determined membrane "conductance" for a given gas at a given temperature.
• c₂ − c₁ is the difference in concentration of the gas across the membrane for the direction of flow (from c₁ to c₂).

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 (use with particular care because the equation is derived for dense solutes, while biological molecules are not denser than water):[9]

${\displaystyle P=2A_{p}\eta _{tm}{\sqrt {D/(\pi t)}}}$

where

• ${\displaystyle A_{P}}$ is the total area of the pores on the membrane (unit m²).
• ${\displaystyle \eta _{tm}}$ transmembrane efficiency (unitless), which can be calculated from the stochastic theory of chromatography.
• D is the diffusion constant of the solute unit m²s⁻¹.
• t is time unit s.
• c₂, c₁ concentration should use unit mol m⁻³, so flux unit becomes mol s⁻¹.

Integrated circuit 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 (where junction depth keeps moving into the substrate).