Diffusion

This article is about the generic concept of the time-dependent random process. For other uses, see Diffusion (disambiguation).

File:Diffusion.svg

A diffusion process in science. Some particles are dissolved in a glass of water. Initially, the particles are all near one corner of the glass. If the particles all randomly move around ("diffuse") in the water, then the particles will eventually become distributed randomly and uniformly (but diffusion will still continue to occur, just that there will be no net flux).

File:Diffusion v2 20101120.ogv

Time lapse video of diffusion of a dye dissolved in water into a gel.

Diffusion is one of several transport phenomena that occur in nature. A distinguishing feature of diffusion is that it results in mixing or mass transport without requiring bulk motion. Thus, diffusion should not be confused with convection or advection, which are other transport mechanisms that use bulk motion to move particles from one place to another. In Latin word "diffundere" means "to spread out".

There are two ways to introduce the notion of diffusion: either a phenomenological approach starting with Fick’s laws and their mathematical consequences, or a physical and atomistic one, by considering the random walk of the diffusing particles.

In the phenomenological approach, according to Fick's laws, the diffusion flux is proportional to the minus gradient of concentrations. It goes from regions of higher concentration to regions of lower concentration. Later on, various generalizations of the Fick's laws were developed in the frame of thermodynamics and non-equilibrium thermodynamics.

From the atomistic point of view, diffusion is considered as a result of the random walk of the diffusing particles. In molecular diffusion, the moving molecules are self propelled by thermal energy. Random walk of small particles in suspension in a fluid was discovered in 1827 by Robert Brown. The theory of the Brownian motion and the atomistic backgrounds of diffusion were developed by Albert Einstein.

Now, the concept of diffusion is widely used in science: in physics (particle diffusion), chemistry and biology, in sociology, economics and finance (diffusion of people, ideas and of price values). It appears every time, when the concept of random walk in ensembles of individuals is applicable.

History of diffusion in physics

In technology, diffusion in solids was used long before the theory of diffusion was created. For example, the cementation process that produces steel from the iron includes carbon diffusion and was described already by Pliny the Elder, the diffusion of colours of stained glasses or earthenwares and Chinas was well known for many centuries.

In modern science, the first systematic experimental study of diffusion was performed by Thomas Graham. He studied diffusion in gases and the main phenomenon was described by him in 1831-1833:

"...gases of different nature, when brought into contact, do not arrange themselves according to their density, the heaviest undermost, and the lighter uppermost, but they spontaneously diffuse, mutually and equally, through each other, and so remain in the intimate state of mixture for any length of time.”

The measuments of Graham allowed James Clerk Maxwell to derive in 1867 the coefficient of diffusion of CO₂ in air. The error is less than 5%.

In 1855, Adolf Fick, the 26-year old anatomy demonstrator from Zürich proposed his law of diffusion. He used Graham's research and his goal was "the development of a fundamental law, for the operation of diffusion in a single element of space". He declared a deep analogy between diffusion and conduction of heat or electricity and created the formalism that is similar to Fourier's law for heat conduction (1822) and Ohm's law for electrical current (1827).

Robert Boyle demonstrated diffusion in solids in 17th century by penetration of Zinc into a copper coin. Nevertheless, diffusion in solids was not systematically studied till the second part of the 19th century. William Chandler Roberts-Austen, the well known British metallurgist, studied systematically solid state diffusion on the example of gold in lead in 1896. He has been the former assistant of Thomas Graham and this connection inspired him:

"... My long connection with Graham's researches made it almost a duty to attempt to extend his work on liquid diffusion to metals."

In 1858, Rudolf Clausius introduced the concept of the mean free path. In the same year, James Clerk Maxwell developed the first atomistic theory of transport processes in gases. The modern atomistic theory of diffusion and Brownian motion was developed by Albert Einstein, Marian Smoluchowski and Jean-Baptiste Perrin. The role of Ludwig Boltzmann in the development of the atomistic backgrounds of the macroscopic transport processes was great. His Boltzmann equation serves more than 140 years as a source of ideas and problems in mathematics and physics of transport processes.

In 1920-1921 George de Hevesy measured self-diffusion using radioisotopes. He studied self-diffusion of radioactive isotopes of lead in liquid and solid lead.

Yakov Frenkel (or, sometimes, Jakov or Jacov) proposed in 1926 and then elaborated the idea of diffusion in crystals through local defects (vacancies and interstitial atoms). He introduced several mechanisms of diffusion and found rate constants from experimental data. Later, this idea was developed further by Carl Wagner and Walter H. Schottky. Nowadays, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals.

The ideas of Frenkel represent diffusion process in condensed matter as an ensemble of elementary jumps and quasichemical interactions of particles and defects. Henry Eyring with co-authors applied his theory of absolute reaction rates to this quasichemical representation of diffusion. The analogy between reaction kinetics and diffusion leads to various nonlinear versions of Fick's law.

Basic models of diffusion

Diffusion flux

Each model of diffusion expresses the diffusion flux through concentrations, densities and their derivatives. Flux is a vector \(J\). The transfer of a physical quantity \(N\) through a small area \(\Delta S\) with normal \(\nu\) per time \(\Delta t\) is \[\Delta N = (J,\nu) \Delta S \Delta t +o(\Delta S \Delta t)\, ,\] where \((J,\nu)\) is the inner product and \(o(...)\) is the little-o notation. If we use the notation of vector area \(\Delta \mathbf{S}=\nu \Delta S\) then \[\Delta N = (J, \Delta \mathbf{S}) \Delta t +o(\Delta \mathbf{S} \Delta t)\, . \] The dimension of the diffusion flux is [flux]=[quantity]/([time]·[area]). The diffusing physical quantity \(N\) may be the number of particles, mass, energy, electric charge, or any other scalar extensive quantity. For its density, \(n\), the diffusion equation has the form \[\frac{\partial n}{\partial t}= - \nabla J +W \, ,\] where \(W\) is intensity of any local sourse of this quantity (the rate of a chemical reaction, for example). For the diffusion equation, the no-flux boundary conditions can be formulated as \((J(x),\nu(x))=0\) on the boundary, where \(\nu\) is the normal to the boundary at point \(x\).

Fick's law and equations

Main article: Fick's laws of diffusion

Fick's first law: the diffusion flux is proportional to the minus concentration gradient: \[J=-D \nabla n \ , \;\; J_i=-D \frac{\partial n}{\partial x_i} \ .\] The corresponding diffusion equation (Fick's second law) is \[\frac{\partial n(x,t)}{\partial t}=\nabla( D \nabla n(x,t))=D \Delta n(x,t)\ , \] where \(\Delta\) is the Laplace operator, \[\Delta n(x,t) = \sum_i \frac{\partial^2 n(x,t)}{\partial x_i^2} \ .\]

Generalizations.

1. In the inhomogeneous media, the diffusion coefficient varies in space, \(D=D(x)\). This depencdence does not affect Fick's first law but the second law changes: \[\frac{\partial n(x,t)}{\partial t}=\nabla (D(x) \nabla n(x,t))=D(x) \Delta n(x,t)+\sum_{i=1}^3 \frac{\partial D(x)}{\partial x_i} \frac{\partial n(x,t)}{\partial x_i}\ \] 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 \[J=-D \nabla n \ , \mbox{ it is the product of a tensor and a vector: } \;\; J_i=-\sum_{j=1}^3D_{ij} \frac{\partial n}{\partial x_j} \ .\] For the diffusion equation this formula gives \[\frac{\partial n(x,t)}{\partial t}=\nabla( D \nabla n(x,t))=\sum_{j=1}^3D_{ij} \frac{\partial^2 n(x,t)}{\partial x_i \partial x_j}\ . \] 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 \[\frac{\partial n(x,t)}{\partial t}=\nabla( D(x) \nabla n(x,t))=\sum_{i,j=1}^3\left(D_{ij}(x) \frac{\partial^2 n(x,t)}{\partial x_i \partial x_j}+ \frac{\partial D_{ij}(x)}{\partial x_i } \frac{\partial n(x,t)}{\partial x_j}\right)\ . \]

Onsager's equations for multicomponent diffusion and thermodiffusion

Fick's law describes diffusion of an admixture in a media. The concentration of this admixture should be small and the gradient of this concentration should be also small. The driving force of diffusion in Fick's law is the antigradient of concentration, \(-\nabla n\).

In 1931, Lars Onsager included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For multi-component transport, \[J_i=\sum_j L_{ij} X_j \, ,\] where \(J_i\) is the flux of the ith physical quantity (component) and \(X_j\) is the jth thermodynamic force.

The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the entropy density s (he used the term "force" in quotation marks or "driving force"): \[X_i= {\rm grad} \frac {\partial s(n)}{\partial n_i}\ ,\] where \(n_i\) are the "thermodynamic coordinates". For the heat and mass transfer one can take \(n_0=u\) (the density of internal energy) and \(n_i\) is the concentration of the ith component. The corresponding driving forces are the space vectors \[X_0= {\rm grad} \frac{1}{T}\ , \;\;\; X_i= - {\rm grad} \frac{\mu_i}{T}\; (i >0) ,\] because \({\rm d}s=\frac{1}{T}{\rm d}u-\sum_{i \geq 1}\frac{\mu_i}{T} {\rm d} n_i\) where T is the absolute temperature and \(\mu_i\) is the chemical potential of the ith component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore, the terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients.

For the linear Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium: \[X_i= \sum_{k \geq 0} \left.\frac{\partial^2 s(n)}{\partial n_i \partial n_k}\right|_{n=n^*} {\rm grad} n_k \ ,\] where the derivatives of s are calculated at equilibrium n^*. The matrix of the kinetic coefficients \(L_{ij}\) should be symmetric (Onsager reciprocal relations) and positive definite (for the entropy growth).

The transport equations are \[\frac{\partial n_i}{\partial t}= - {\rm div} J_i =- \sum_{j\geq 0} L_{ij}{\rm div} X_j = \sum_{k\geq 0} \left[-\sum_{j\geq 0} L_{ij} \left.\frac{\partial^2 s(n)}{\partial n_j \partial n_k}\right|_{n=n^*}\right] \Delta n_k\ .\] Here, all the indexes i, j, k=0,1,2,... are related to the internal energy (0) and various components. The expression in the square brackets is the matrix \(D_{ik}\)of the diffusion (i,k>0), thermodiffusion (i>0, k=0 or k>0, i=0) and thermal conductivity (i=k=0) coefficients.

Under isothermal conditions T=const. The relevant thermodynamic potential is the free energy (or the free entropy). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, \(-(1/T)\nabla\mu_j\), and the matrix of diffusion coefficients is \[D_{ik}=\frac{1}{T}\sum_{j\geq 1} L_{ij} \left.\frac{\partial \mu_j(n,T)}{ \partial n_k}\right|_{n=n^*}\] (i,k>0).

There is intrinsic arbitrariness in the definition of the thermodynamic forces and kinetic coefficients because they are not measurable separately and only their combinations \(\sum_j L_{ij}X_j\) can be measured. For example, in the original work of Onsager the thermodynamic forces include additional multiplier T, whereas in the Course of Theoretical Physics this multiplier is omitted but the sign of the thermodynamic forces is opposite. All these changes are supplemented by the corresponding changes in the coefficients and do not effect the measurable quantities.

Nondiagonal diffusion must be nonlinear

The formalism of linear irreversible thermodynamics (Onsager) generates the systems of linear diffusion equations in the form \[\frac{\partial n_i}{\partial t} =\sum_j D_{ij} \Delta c_j \, .\] If the matrix of diffusion coefficients is diagonal then this system of equations is just a collection of decoupled Fick's equations for various components. Assume that diffusion is non-diagonal, for example, \(D_{12}\neq 0\), and consider the state with \(c_2= \ldots = c_n=0\). At this state, \(\partial n_2/ \partial t = D_{12} \Delta n_1\). If \(D_{12} \Delta n_1(x) < 0\) at some points then \(n_2(x)\) becomes negative at these points in a short time. Therefore, linear non-diagonal diffusion does not preserve positivity of concentrations. Non-diagonal equations of multicomponent diffusion must be non-linear.

Diffusion in porous media

Diffusion of Gasses in porous media As we know that in porous media there are pores as well as solid material existing to gather all the way along the spacial dimensions of the porous media. So, there exist a combination of diffusion mechanisms in poroous media. To understand this, lets consider the example of ceramic membranes. when a gas enters into a ceramic membrane it undergoes the molecular diffusion when moving through the material of the porous media and knudsen diffusion ( if the diameter of pore is less than the length of the pore) and molecular difussion ( can be self diffusion or mutual diffusion depending on the number of species being diffused) in series when moving in the pores. while moving in the pore the poiseuille diffusion also happens in parrllel with knudsen and molecular diffusion. Depending on the nature of material and operating conditions one or more types of diffusion may be small and can be neglected but basic phenomenon is the same. for more information see <M. Coroneo,G.Montante,M.GiacintiBaschetti,A.Paglianti,CFD modelling of inorganic membrane modules for gas mixture separation,Chemical EngineeringScience64(2009)1085—1094>

Diffusion in physics

Elementary theory of diffusion coefficient

File:Translational motion.gif

Random collisions of particles in a gas.

The diffusion coefficient \(D\) is the coefficient in the Fick's first law \(J=- D {\partial n}/{\partial x}\), where J is the diffusion flux (amount of substance) per unit area per unit time, n (for ideal mixtures) is the concentration, x is the position [length].

Let us consider two gases with molecules of the same diameter d and mass m (self-diffusion). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient

\[D=\frac{1}{3} \ell v_T = \frac{2}{3}\sqrt{\frac{k_{\rm B}^3}{\pi^3 m}}\frac{T^{3/2}}{Pd^2}\, ,\] where k_B is the Boltzmann constant, T is the temperature, P is the pressure, \(\ell\) is the mean free path, and v_T is the mean thermal speed: \[\ell = \frac{k_{\rm B}T}{\sqrt 2 \pi d^2 P}\, , \;\;\; v_T=\sqrt{\frac{8k_{\rm B}T}{\pi m}}\, .\] We can see that the diffusion coefficient in the mean free path approximation grows with T as T^3/2 and decreases with P as 1/P. If we use for P the ideal gas law P=RnT with the total concentration n, then we can see that for given concentration n the diffusion coefficient grows with T as T^1/2 and for given temperature it decreases with the total concentration as 1/n.

For two different gases, A and B, with molecular masses m_A, m_B and molecular diameters d_A, d_B, the mean free path estimate of the diffusion coefficient of A in B and B in A is: \[D_{\rm AB}=\frac{1}{3}\sqrt{\frac{k_{\rm B}^3}{\pi^3}}\sqrt{\frac{1}{2m_{\rm A}}+\frac{1}{2m_{\rm B}}}\frac{4T^{3/2}}{P(d_{\rm A}+d_{\rm B})^2}\, ,\]

The theory of diffusion in gases based on Boltzmann's equation

In Boltzman's kinetics of the mixture of gases, each gas has its own distribution function, \(f_i(x,c,t)\), where t is the time moment, x is position and c is velocity of molecule of the of the ith component of the mixture. Each component has its mean velocity \(C_i(x,t)=\frac{1}{n_i}\int_c c f(x,c,t) \, dc\). If the velocities \(C_i(x,t)\) do not concide then there exists diffusion.

In the Chapman-Enskog approximation, all the distribution functions are expressed through the densities of the conserved quantities:

• individual concentrations of particles, \(n_i(x,t)=\int_c f_i(x,c,t)\, dc\) (particles per volume),
• density of moment \(\sum_i m_i n_i C_i(x,t)\) (m_i is the ith particle mass),
• density of kinetic energy \(\sum_i \left( n_i\frac{m_i C^2_i(x,t)}{2} + \int_c \frac{m_i (c_i-C_i(x,t))^2}{2} f_i(x,c,t)\, dc \right)\).

The kinetic temperature T and pressure P are defined in 3D space as \[\frac{3}{2}k_{\rm B}T=\frac{1}{n} \int_c \frac{m_i (c_i-C_i(x,t))^2}{2} f_i(x,c,t)\, dc\]; \(P=k_{\rm B}nT\), where \(n=\sum_i n_i\) is the total density.

For two gases, the difference between velocities, \(C_1-C_2\) is given by the expression: \[C_1-C_2=-\frac{n^2}{n_1n_2}D_{12}\left\{ \nabla \left(\frac{n_1}{n}\right)+ \frac{n_1n_2 (m_2-m_1)}{n (m_1n_1+m_2n_2)}\nabla P- \frac{m_1n_1m_2n_2}{P(m_1n_1+m_2n_2)}(F_1-F_2)+k_T \frac{1}{T}\nabla T\right\}\], where \(F_{i}\) is the force applied to the molecules of the ith component and \(k_T\) is the thermodiffusion ratio.

The coefficient D₁₂ is positive. This is the diffusion coefficient. Four terms in the formula for C₁-C₂ describe four main effects in the diffusion of gases:

1. \(\nabla \left(\frac{n_1}{n}\right)\) describes the flux of the first component from the areas with the high ratio n₁/n to the areas with lower values of this ratio (and, analogously the flux of the second component from high n₂/n to low n₂/n because n₂/n=1-n₁/n);
2. \(\frac{n_1n_2 (m_2-m_1)}{n (m_1n_1+m_2n_2)}\nabla P\) describes the flux of the heavier molecules to the areas with higher pressure and the lighter molecules to the areas with lower pressure, this is barodiffusion;
3. \(\frac{m_1n_1m_2n_2}{P(m_1n_1+m_2n_2)}(F_1-F_2)\) describes diffusion caused by the difference of the forces applied to molecules of different types. For example, in the Earth's gravitational field, the heavier molecules should go down, or in electric field the charged molecules should move, until this effect is not equilibrated by the sum of other terms. This effect should not be confused with barodiffusion caused by the pressure gradient.
4. \(k_T \frac{1}{T}\nabla T\) describes thermodiffusion, the diffusion flux caused by the temperature gradient.

All these effects are called diffusion because they describe the differences between velocities of different components in the mixture. Therefore, these effects cannot be described as a bulk transport and differ from advection or convection.

In the first approximation,

• \(D_{12}=\frac{3}{2n(d_1+d_2)^2}\left[\frac{kT(m_1+m_2)}{2\pi m_1m_2}\right]^{1/2}\) for rigid spheres;
• \(D_{12}=\frac{3}{8nA_1({\nu})\Gamma(3-\frac{2}{\nu-1})}\left[\frac{kT(m_1+m_2)}{2\pi m_1m_2}\right]^{1/2} \left(\frac{2kT}{\kappa_{12}}\right)^{\frac{2}{\nu-1}}\) for repulsing force \(\kappa_{12}r^{-\nu}\).

The number \(A_1({\nu})\) is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book)

We can see that the dependence on T for the rigid spheres is the same as for the simple mean free path theory but for the power repulsion laws the exponent is different. Dependence on a total concentration n for a given temperature has always the same character, 1/n.

In applications to gas dynamics, the diffusion flux and the bulk flow should be joined in one system of transport equations. The bulk flow describes the mass transfer. Its velocity V is the mass average velocity. It is defined through the momentum density and the mass concentrations: \[V=\frac{\sum_i \rho_i C_i}{\rho}\, .\] where \(\rho_i =m_i n_i\) is the mass concentration of the ith species, \(\rho=\sum_i \rho_i\) is the mass density.

By definition, the diffusion velocity of the ith component is \(v_i=C_i-V\), \(\sum_i \rho_i v_i=0\). The mass transfer of the ith component is described by the continuity equation \[\frac{\partial \rho_i}{\partial t}+\nabla(\rho_i V) + \nabla (\rho_i v_i)=W_i \, ,\] where \(W_i\) is the net mass production rate in chemical reactions, \(\sum_i W_i= 0\).

In these equations, the term \(\nabla(\rho_i V)\) describes advection of the ith component and the term \(\nabla (\rho_i v_i)\) represents diffusion of this component.

In 1948, Wendell H. Furry proposed to use the form of the diffusion rates found in kinetic theory as a framework for the new phenomenological approach to diffusion in gases. This approach was developed further by F.A. Williams and S.H. Lam. For the diffusion velocities in multicomponent gases (N components) they used \[v_i=-\left(\sum_{j=1}^N D_{ij}\mathbf{d}_j + D_i^{(T)} \nabla (\ln T) \right)\, ;\] \[\mathbf{d}_j=\nabla X_j + (X_j-Y_j)\nabla (\ln P) + \mathbf{g}_j\, ;\] \[\mathbf{g}_j=\frac{\rho}{P}\left( Y_j \sum_{k=1}^N Y_k (f_k-f_j) \right)\, .\] Here, \(D_{ij}\) is the diffusion coefficient matrix, \(D_i^{(T)}\) is the thermal diffusion coefficient, \(f_i\) is the body force per unite mass acting on the ith species, \(X_i=P_i/P\) is the partial pressure fraction of the ith species (and \(P_i\) is the partial pressure), \(Y_i=\rho_i/\rho\) is the mass fraction of the ith species, and \(\sum_i X_i=\sum_i Y_i=1\).

File:Diffusion center.gif

The above palette shows change in excess carriers being generated (green:electrons and purple:holes) with increasing light intensity (Generation rate /cm3) at the center of an intrinsic semiconductor bar. Electrons have a higher diffusion constant than holes, leading to fewer excess electrons at the center as compared to holes.

Separation diffusion from convection in gases

While Brownian motion of multi-molecular mesoscopic particles (like pollen grains studied by Brown) is observable under an optical microscope, molecular diffusion can only be probed in carefully controlled experimental conditions. Since Graham experiments, it is well known that avoiding of convection is necessary and this may be a non-trivial task.

Under normal conditions, molecular diffusion dominates only on length scales between nanometer and millimeter. On larger length scales, transport in liquids and gases is normally due to another transport phenomenon, convection, and to study diffusion on the larger scale, special efforts are needed.

Therefore, some often cited examples of diffusion are wrong: If cologne is sprayed in one place, it will soon be smelled in the entire room, but a simple calculation shows that this can't be due to diffusion. Convective motion persists in the room because the temperature inhomogeneity. If ink is dropped in water, one usually observes an inhomogeneous evolution of the spatial distribution, which clearly indicates convection (caused, in particular, by this dropping).^{[citation needed]}

In contrast, heat conduction through solid media is an everyday occurrence (e.g. a metal spoon partly immersed in a hot liquid). This explains why the diffusion of heat was explained mathematically before the diffusion of mass.

Other types of diffusion

• Anisotropic diffusion, also known as the Perona-Malik equation, enhances high gradients
• Anomalous diffusion, in porous medium
• Atomic diffusion, in solids
• Eddy diffusion, in coarse-grained description of turbulent flow
• Effusion of a gas through small holes
• Electronic diffusion, resulting in an electric current called the diffusion current
• Facilitated diffusion, present in some organisms
• Gaseous diffusion, used for isotope separation
• Heat equation, diffusion of thermal energy
• Itō diffusion, mathematisation of Brownian motion, continuous stochastic process.
• Knudsen diffusion of gas in long pores with frequent wall collisions
• Momentum diffusion ex. the diffusion of the hydrodynamic velocity field
• Photon diffusion
• Random walk, model for diffusion
• Reverse diffusion, against the concentration gradient, in phase separation
• Rotational diffusion, random reorientations of molecules
• Surface diffusion, diffusion of adparticles on a surface
• Turbulent diffusion, transport of mass, heat, or momentum within a turbulent fluid

See also

• Advection
• Fick's laws of diffusion

External links

• Diffusion in a Bipolar Junction Transistor Demo
• Diffusion Furnace for doping of semiconductor wafers. POCl3 doping of Silicon.
• A Java applet implementing Diffusion

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