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).

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.^[1]

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.^[2]

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.^[3]

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[link]

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^[4]:

"...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^[5] 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 amd this connection inspired him ^[6]:

"... 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.^[7]

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.^[6]

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.^[8] The analogy between reaction kinetics and diffusion leads to various nonlinear versions of Fik's law.^[9]

Diffusion in physics[link]

Elementary theory of diffusion coefficient[link]

The diffusion coefficient Failed to parse (Missing texvc executable; please see math/README to configure.): D

is the coefficient in the Fick's first law  Failed to parse (Missing texvc executable; please see math/README to configure.): 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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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, Failed to parse (Missing texvc executable; please see math/README to configure.): \ell

is the mean free path, and vT is the mean thermal speed:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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[link]

In Boltzman's kinetics of the mixture of gases, each gas has its own distribution function, Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): C_i(x,t)=\frac{1}{n_i}\int_c c f(x,c,t) \, dc . If the velocities Failed to parse (Missing texvc executable; please see math/README to configure.): C_i(x,t)

do not concide then there exists diffusion. 

In the Chapman-Enskog approximation, all the distribution functions are expressed through the dencities of the conserved quantities^[7]:

• individual concentrations of particles, Failed to parse (Missing texvc executable; please see math/README to configure.): n_i(x,t)=\int_c f_i(x,c,t)\, dc

(particles per volume),

• density of moment Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_i m_i n_i C_i(x,t)

(mi is the ith particle mass),

• density of kinetic energy Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): \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

Failed to parse (Missing texvc executable; please see math/README to configure.): P=k_{\rm B}nT

, where Failed to parse (Missing texvc executable; please see math/README to configure.): n=\sum_i n_i

is the total density.

For two gases, the difference between velocities, Failed to parse (Missing texvc executable; please see math/README to configure.): C_1-C_2

is given by the expression[7]:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): F_{i}

is the force applied to the molecules of the ith component and Failed to parse (Missing texvc executable; please see math/README to configure.): 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. Failed to parse (Missing texvc executable; please see math/README to configure.): \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);

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \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;

1. Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

1. Failed to parse (Missing texvc executable; please see math/README to configure.): 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,^[7]

• Failed to parse (Missing texvc executable; please see math/README to configure.): 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;

• Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \kappa_{12}r^{-\nu}

. The number Failed to parse (Missing texvc executable; please see math/README to configure.): A_1({\nu})

is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book[7])

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): V=\frac{\sum_i \rho_i C_i}{\rho}\, .

where Failed to parse (Missing texvc executable; please see math/README to configure.): \rho_i =m_i n_i

is the mass concentration of the ith species, Failed to parse (Missing texvc executable; please see math/README to configure.): \rho=\sum_i \rho_i
is the mass density.

By definition, the diffusion velocity of the ith component is Failed to parse (Missing texvc executable; please see math/README to configure.): v_i=C_i-V , Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_i \rho_i v_i=0 . The mass transfer of the ith component is described by the continuity equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \rho_i}{\partial t}+\nabla(\rho_i V) + \nabla (\rho_i v_i)=W_i \, ,

where Failed to parse (Missing texvc executable; please see math/README to configure.): W_i

is the net mass production rate in chemical reactions, Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_i W_i= 0

.

In these equations, the term Failed to parse (Missing texvc executable; please see math/README to configure.): \nabla(\rho_i V)

describes advection of the ith component and the term Failed to parse (Missing texvc executable; please see math/README to configure.): \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.^[10] For the diffusion velocities in multicomponent gases (N components) they used

Failed to parse (Missing texvc executable; please see math/README to configure.): v_i=-\left(\sum_{j=1}^N D_{ij}\mathbf{d}_j + D_i^{(T)} \nabla (\ln T) \right)\, ;

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{d}_j=\nabla X_j + (X_j-Y_j)\nabla (\ln P) + \mathbf{g}_j\, ;

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbf{g}_j=\frac{\rho}{P}\left( Y_j \sum_{k=1}^N Y_k (f_k-f_j) \right)\, .

Here, Failed to parse (Missing texvc executable; please see math/README to configure.): D_{ij}

is the diffusion coefficient matrix, Failed to parse (Missing texvc executable; please see math/README to configure.): D_i^{(T)}
is the thermal diffusion coefficient, Failed to parse (Missing texvc executable; please see math/README to configure.): f_i
is the body force per unite mass acting on the ith species, Failed to parse (Missing texvc executable; please see math/README to configure.): X_i=P_i/P
is the partial pressure fraction of the ith species (and Failed to parse (Missing texvc executable; please see math/README to configure.): P_i
is the partial pressure), Failed to parse (Missing texvc executable; please see math/README to configure.): Y_i=\rho_i/\rho
is the mass fraction of the ith species, and Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_i X_i=\sum_i Y_i=1

.

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[link]

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[link]

• Anisotropic diffusion, also known as the Perona-Malik equation, enhances high gradients
• Anomalous diffusion,^[11] 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,^[12] 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[link]

Underwater diving portal

• Advection
• Fick's laws of diffusion

References[link]

External links[link]

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