Fourier analysis

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File:DiffusionMicroMacro.gif

Molecular diffusion from a microscopic and macroscopic point of view. Initially, there are solute molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container. Top: A single molecule moves around randomly. Middle: With more molecules, there is a clear trend where the solute fills the container more and more uniformly. Bottom: With an enormous number of solute molecules, randomness becomes undetectable: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. This smooth flow is described by Fick's laws.

Fick's laws of diffusion describe diffusion and can be used to solve for the diffusion coefficient, D. They were derived by Adolf Fick in 1855.

Fick's first law

Fick's first law relates the diffusive flux to the concentration under the assumption of steady state. 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 (spatial derivative). In one (spatial) dimension, the law is

$J=-D\frac{\partial \varphi }{\partial x}$

where

• $J$ is the "diffusion flux" [(amount of substance) per unit area per unit time], example $(\frac{\mathrm{m}\mathrm{o}\mathrm{l}}{{\mathrm{m}}^2\cdot \mathrm{s}})$. $J$ measures the amount of substance that will flow through a small area during a small time interval.

• $D$ is the diffusion coefficient or diffusivity in dimensions of [length² time⁻¹], example $(\frac{{\mathrm{m}}^2}{\mathrm{s}})$

• $\varphi$ (for ideal mixtures) is the concentration in dimensions of [amount of substance per unit volume], example $(\frac{\mathrm{m}\mathrm{o}\mathrm{l}}{{\mathrm{m}}^3})$

• $x$ is the position [length], example $\mathrm{m}$

$D$ 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 0.6x10⁻⁹ to 2x10⁻⁹ m²/s. For biological molecules the diffusion coefficients normally range from 10⁻¹¹ to 10⁻¹⁰ m²/s.

In two or more dimensions we must use $\mathrm{\nabla }$, the del or gradient operator, which generalises the first derivative, obtaining

$\mathbf{J}=-D\mathrm{\nabla }\varphi$.

The driving force for the one-dimensional diffusion is the quantity $-\frac{\partial \varphi }{\partial x}$ which for ideal mixtures is the concentration gradient. 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 as:

$J_i=-\frac{D{c}_i}{RT}\frac{\partial {\mu }_i}{\partial x}$

where the index i denotes the ith species, c is the concentration (mol/m^{3), R is the universal gas constant (J/(K mol)), T is the absolute temperature (K), and μ is the chemical potential (J/mol).}

If the primary variable is mass fraction ($y_i$, given, for example, in $\frac{\mathrm{k}g}{\mathrm{k}g}$), then the equation changes to:

$J_i=-\rho D\mathrm{\nabla }{y}_i$

where $\rho$ is the fluid density (for example, in $\frac{\mathrm{k}g}{{\mathrm{m}}^3}$). Note that the density is outside the gradient operator.

Fick's second law

Fick's second law predicts how diffusion causes the concentration to change with time:

$\frac{\partial \varphi }{\partial t}=D\frac{{\partial }^2\varphi }{\partial x^2}$

where

• $\varphi$ is the concentration in dimensions of [(amount of substance) length⁻³], example $(\frac{\mathrm{m}\mathrm{o}\mathrm{l}}{m^3})$
• $t$ is time [s]
• $D$ is the diffusion coefficient in dimensions of [length² time⁻¹], example $(\frac{m^2}{s})$
• $x$ is the position [length], example $m$

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

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

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

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

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

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

which is analogous to the heat equation.

If the diffusion coefficient is not a constant, but depends upon the coordinate and/or concentration, Fick's Second Law yields

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

An important example is the case where $\varphi$ 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

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

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

Example solution in one dimension: diffusion length

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 $n_0$ is

$n(x,t)=n_0\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}\left(\frac{x}{2\sqrt{Dt}}\right)$.

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 (i. e., 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 $n(x,0)=n_0,x\le 0$), then the solution is amended only with coefficient ½ in front of n₀ (this might seem obvious, 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. (Bokshtein, 2005) The length $2\sqrt{Dt}$ 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:

$n(x,t)=n_0[1-2\left(\frac{x}{2\sqrt{Dt\pi }}\right)]$

If $D$ is time-dependent, the diffusion length becomes $2\sqrt{\underset{0}{\overset{t^{\prime }}{\int }}D(t^{\prime })d{t}^{\prime }}$. This idea is useful for estimating a diffusion length over a heating and cooling cycle, where D varies with temperature.

Generalizations

1. In the inhomogeneous media, the diffusion coefficient varies in space, $D=D(x)$. This dependence does not affect Fick's first law but the second law changes:

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

2. In the anisotropic media, the diffusion coefficient depends on the direction. It is a symmetric tensor $D=D_{ij}$. Fick's first law changes to

$J=-D\mathrm{\nabla }\varphi ,\text{ it is the product of a tensor and a vector: }{J}_i=-\sum _{j=1}^3{D}_{ij}\frac{\partial \varphi }{\partial x_j}.$

For the diffusion equation this formula gives

$\frac{\partial \varphi (x,t)}{\partial t}=\mathrm{\nabla }\cdot (D\mathrm{\nabla }\varphi (x,t))=\sum _{j=1}^3{D}_{ij}\frac{{\partial }^2\varphi (x,t)}{\partial x_i\partial x_j}.$

The symmetric matrix of diffusion coefficients $D_{ij}$ should be positive definite. It is needed to make the right hand side operator elliptic.

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

$\frac{\partial \varphi (x,t)}{\partial t}=\mathrm{\nabla }\cdot (D(x)\mathrm{\nabla }\varphi (x,t))=\sum _{i,j=1}^3(D_{ij}(x)\frac{{\partial }^2\varphi (x,t)}{\partial x_i\partial x_j}+\frac{\partial D_{ij}(x)}{\partial x_i}\frac{\partial \varphi (x,t)}{\partial x_j}).$

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

$\frac{\partial {\varphi }_i}{\partial t}=\sum _j\mathrm{d}\mathrm{i}\mathrm{v}\left(D_{ij}\frac{{\varphi }_i}{{\varphi }_j}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}{\varphi }_j\right).$

where ${\varphi }_i$ are concentrations of the components and $D_{ij}$ is the matrix of coefficients. Here, indexes i,j are related to the various components and not to the space coordinates.

The Chapman-Enskog formulas for diffusion in gases include exactly the same terms. It should be stressed that these physical models of diffusion are different from the toy-models ${\partial }_t{\varphi }_i=\sum _j{D}_{ij}\mathrm{\Delta }{\varphi }_j$ 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 4-index quantities, for example, $D_{ij\alpha \beta }$, where i, j are related to the components and α, β=1,2,3 correspond to the space coordinates.

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. The 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. 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 does happen), we refer to such processes as non-Fickian, in that they are exceptions that "prove" the importance of the general rules that Fick outlined in 1855.

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, semiconductor doping process, etc. Theory of all voltammetric methods is based on solutions of Fick's equation. A large amount of 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 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 (Onsager relationship).

Biological perspective

The first law gives rise to the following formula:

$\text{Flux}=-P(c_2-c_1)$

in which,

• $P$ is the permeability, an experimentally determined membrane "conductance" for a given gas at a given temperature.
• $c_2-c_1$ is the difference in concentration of the gas across the membrane for the direction of flow (from $c_1$ to $c_2$).

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.

Fick's flow in liquids

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.

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 fluctuating hydrodynamics equations.

Semiconductor fabrication applications

Integrated circuit Fabrication technologies, model processes like CVD, Thermal Oxidation, and 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).

Derivation of Fick's 1st law in 1 dimension

The following derivation is based on a similar argument made in Berg 1977 (see references).

Consider a collection of particles performing a random walk in one dimension with length scale $\mathrm{\Delta }x$ and time scale $\mathrm{\Delta }t$. Let $N(x,t)$ be the number of particles at position $x$ at time $t$.

At a given time step, half of the particles would move left and half would move right. Since half of the particles at point $x$ move right and half of the particles at point $x+\mathrm{\Delta }x$ move left, the net movement to the right is:

$-\frac{1}{2}[N(x+\mathrm{\Delta }x,t)-N(x,t)]$

The flux, J, is this net movement of particles across some area element of area a, normal to the random walk during a time interval $\mathrm{\Delta }t$. Hence we may write:

$J=-\frac{1}{2}[\frac{N(x+\mathrm{\Delta }x,t)}{a\mathrm{\Delta }t}-\frac{N(x,t)}{a\mathrm{\Delta }t}]$

Multiplying the top and bottom of the righthand side by $(\mathrm{\Delta }x{)}^2$ and rewriting, we obtain:

$J=-\frac{{\left(\mathrm{\Delta }x\right)}^2}{2\mathrm{\Delta }t}[\frac{N(x+\mathrm{\Delta }x,t)}{a(\mathrm{\Delta }x{)}^2}-\frac{N(x,t)}{a(\mathrm{\Delta }x{)}^2}]$

We note that concentration is defined as particles per unit volume, and hence $\varphi (x,t)=\frac{N(x,t)}{a\mathrm{\Delta }x}$.

In addition, $\frac{{\left(\mathrm{\Delta }x\right)}^2}{2\mathrm{\Delta }t}$ is the definition of the diffusion constant in one dimension, $D$. Thus our expression simplifies to:

$J=-D[\frac{\varphi (x+\mathrm{\Delta }x,t)}{\mathrm{\Delta }x}-\frac{\varphi (x,t)}{\mathrm{\Delta }x}]$

In the limit where $\mathrm{\Delta }x$ is infinitesimal, the righthand side becomes a space derivative:

$J=-D\frac{\partial \varphi }{\partial x}$

See also

• Diffusion
• Osmosis
• Mass flux
• Maxwell-Stefan diffusion
• Churchill-Bernstein Equation
• Nernst-Planck equation
• Gas exchange
• False diffusion

Notes

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References

• W.F. Smith, Foundations of Materials Science and Engineering 3rd ed., McGraw-Hill (2004)
• H.C. Berg, Random Walks in Biology, Princeton (1977)
• R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, John Wiley & sons, (1976)
• J. Crank, The Mathematics of Diffusion, Oxford University Press (1980)
• Thermodynamics and Kinetics in Materials Science: A Short Course. Bokshtein, B. S. Mendelev, M. I. Srolovitz, D. J. Eds. Oxford University Press: Oxford (2005) – pp. 167–171.

External links

• Diffusion fundamentals
• Diffusion in Polymer based Materials
• Fick's equations, Boltzmann's transformation, etc. (with figures and animations)
• Wilson, Bill. Fick's Second Law. Connexions. 21 Aug. 2007
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