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Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature an ...
or
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...
. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, like in
spinodal decomposition Spinodal decomposition is a mechanism by which a single thermodynamic phase spontaneously separates into two phases (without nucleation). Decomposition occurs when there is no thermodynamic barrier to phase separation. As a result, phase separation ...
. The concept of diffusion is widely used in many fields, including
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
( particle diffusion),
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
, biology, sociology,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
, and finance (diffusion of people, ideas, and price values). The central idea of diffusion, however, is common to all of these: a substance or collection undergoing diffusion spreads out from a point or location at which there is a higher concentration of that substance or collection. A
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
is the change in the value of a quantity, for example, concentration,
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
, or temperature with the change in another variable, usually distance. A change in concentration over a distance is called a
concentration gradient Molecular diffusion, often simply called diffusion, is the thermal motion of all (liquid or gas) particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size (mass) ...
, a change in pressure over a distance is called a pressure gradient, and a change in temperature over a distance is called a temperature gradient. The word ''diffusion'' derives from the
Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...
word, ''diffundere'', which means "to spread out." A distinguishing feature of diffusion is that it depends on particle random walk, and results in mixing or mass transport without requiring directed bulk motion. Bulk motion, or bulk flow, is the characteristic of advection. The term convection is used to describe the combination of both transport phenomena. If a diffusion process can be described by Fick's laws, it's called a normal diffusion (or Fickian diffusion); Otherwise, it's called an anomalous diffusion (or non-Fickian diffusion). When talking about the extent of diffusion, two length scales are used in two different scenarios: # Brownian motion of an impulsive point source (for example, one single spray of perfume)—the square root of the
mean squared displacement In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference positio ...
from this point. In Fickian diffusion, this is \sqrt, where n is the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of this Brownian motion; # Constant concentration source in one dimension—the diffusion length. In Fickian diffusion, this is 2\sqrt.


Diffusion vs. bulk flow

"Bulk flow" is the movement/flow of an entire body due to a pressure gradient (for example, water coming out of a tap). "Diffusion" is the gradual movement/dispersion of concentration within a body, due to a concentration gradient, with no net movement of matter. An example of a process where both bulk motion and diffusion occur is human breathing. First, there is a "bulk flow" process. The
lungs The lungs are the primary organs of the respiratory system in humans and most other animals, including some snails and a small number of fish. In mammals and most other vertebrates, two lungs are located near the backbone on either si ...
are located in the thoracic cavity, which expands as the first step in external respiration. This expansion leads to an increase in volume of the
alveoli Alveolus (; pl. alveoli, adj. alveolar) is a general anatomical term for a concave cavity or pit. Uses in anatomy and zoology * Pulmonary alveolus, an air sac in the lungs ** Alveolar cell or pneumocyte ** Alveolar duct ** Alveolar macrophage * M ...
in the lungs, which causes a decrease in pressure in the alveoli. This creates a pressure gradient between the air outside the body at relatively high pressure and the alveoli at relatively low pressure. The air moves down the pressure gradient through the airways of the lungs and into the alveoli until the pressure of the air and that in the alveoli are equal, that is, the movement of air by bulk flow stops once there is no longer a pressure gradient. Second, there is a "diffusion" process. The air arriving in the alveoli has a higher concentration of oxygen than the "stale" air in the alveoli. The increase in oxygen concentration creates a concentration gradient for oxygen between the air in the alveoli and the blood in the capillaries that surround the alveoli. Oxygen then moves by diffusion, down the concentration gradient, into the blood. The other consequence of the air arriving in alveoli is that the concentration of
carbon dioxide Carbon dioxide ( chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is t ...
in the alveoli decreases. This creates a concentration gradient for carbon dioxide to diffuse from the blood into the alveoli, as fresh air has a very low concentration of carbon dioxide compared to the
blood Blood is a body fluid in the circulatory system of humans and other vertebrates that delivers necessary substances such as nutrients and oxygen to the cells, and transports metabolic waste products away from those same cells. Blood in the cir ...
in the body. Third, there is another "bulk flow" process. The pumping action of the
heart The heart is a muscular Organ (biology), organ in most animals. This organ pumps blood through the blood vessels of the circulatory system. The pumped blood carries oxygen and nutrients to the body, while carrying metabolic waste such as ca ...
then transports the blood around the body. As the left ventricle of the heart contracts, the volume decreases, which increases the pressure in the ventricle. This creates a pressure gradient between the heart and the capillaries, and blood moves through blood vessels by bulk flow down the pressure gradient.


Diffusion in the context of different disciplines

The concept of diffusion is widely used in:
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
( particle diffusion),
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
, biology, sociology,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
, and finance (diffusion of people, ideas and of price values). However, in each case the substance or collection undergoing diffusion is "spreading out" from a point or location at which there is a higher concentration of that substance or collection. There are two ways to introduce the notion of ''diffusion'': either a phenomenological approach starting with Fick's laws of diffusion and their mathematical consequences, or a physical and atomistic one, by considering the '' random walk of the diffusing particles''. In the phenomenological approach, ''diffusion is the movement of a substance from a region of high concentration to a region of low concentration without bulk motion''. According to Fick's laws, the diffusion flux is proportional to the negative
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of concentrations. It goes from regions of higher concentration to regions of lower concentration. Sometime later, various generalizations of Fick's laws were developed in the frame of
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
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, who found that minute particle suspended in a liquid medium and just large enough to be visible under an optical microscope exhibit a rapid and continually irregular motion of particles known as Brownian movement. The theory of the Brownian motion and the atomistic backgrounds of diffusion were developed by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
. The concept of diffusion is typically applied to any subject matter involving random walks in ensembles of individuals. In
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
and materials science, diffusion refers to the movement of fluid molecules in porous solids. Molecular diffusion occurs when the collision with another molecule is more likely than the collision with the pore walls. Under such conditions, the diffusivity is similar to that in a non-confined space and is proportional to the mean free path. Knudsen diffusion, which occurs when the pore diameter is comparable to or smaller than the mean free path of the molecule diffusing through the pore. Under this condition, the collision with the pore walls becomes gradually more likely and the diffusivity is lower. Finally there is configurational diffusion, which happens if the molecules have comparable size to that of the pore. Under this condition, the diffusivity is much lower compared to molecular diffusion and small differences in the kinetic diameter of the molecule cause large differences in
diffusivity Diffusivity is a rate of diffusion, a measure of the rate at which particles or heat or fluids can spread. It is measured differently for different mediums. Diffusivity may refer to: * Thermal diffusivity, diffusivity of heat *Diffusivity of mas ...
. Biologists often use the terms "net movement" or "net diffusion" to describe the movement of ions or molecules by diffusion. For example, oxygen can diffuse through cell membranes so long as there is a higher concentration of oxygen outside the cell. However, because the movement of molecules is random, occasionally oxygen molecules move out of the cell (against the concentration gradient). Because there are more oxygen molecules outside the cell, the
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
that oxygen molecules will enter the cell is higher than the probability that oxygen molecules will leave the cell. Therefore, the "net" movement of oxygen molecules (the difference between the number of molecules either entering or leaving the cell) is into the cell. In other words, there is a ''net movement'' of oxygen molecules down the concentration gradient.


History of diffusion in physics

In the scope of time, diffusion in solids was used long before the theory of diffusion was created. For example, Pliny the Elder had previously described the cementation process, which produces steel from the element
iron Iron () is a chemical element with symbol Fe (from la, ferrum) and atomic number 26. It is a metal that belongs to the first transition series and group 8 of the periodic table. It is, by mass, the most common element on Earth, right in ...
(Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colors of stained glass or earthenware and
Chinese ceramics Chinese ceramics show a continuous development since Chinese Neolithic, pre-dynastic times and are one of the most significant forms of Chinese art and ceramics globally. The first pottery was made during the List of Palaeolithic sites in China, ...
. 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 measurements of Graham contributed to
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
deriving, in 1867, the coefficient of diffusion for CO2 in the air. The error rate is less than 5%. In 1855,
Adolf Fick Adolf Eugen Fick (3 September 1829 – 21 August 1901) was a German-born physician and physiologist. Early life and education Fick began his work in the formal study of mathematics and physics before realising an aptitude for medicine. He ...
, the 26-year-old anatomy demonstrator from Zürich, proposed his law of diffusion. He used Graham's research, stating his goal as "the development of a fundamental law, for the operation of diffusion in a single element of space". He asserted a deep analogy between diffusion and conduction of heat or electricity, creating a formalism similar to Fourier's law for heat conduction (1822) and Ohm's law for electric current (1827).
Robert Boyle Robert Boyle (; 25 January 1627 – 31 December 1691) was an Anglo-Irish natural philosopher, chemist, physicist, alchemist and inventor. Boyle is largely regarded today as the first modern chemist, and therefore one of the founders ...
demonstrated diffusion in solids in the 17th century by penetration of zinc into a copper coin. Nevertheless, diffusion in solids was not systematically studied until the second part of the 19th century.
William Chandler Roberts-Austen Sir William Chandler Roberts-Austen (3 March 1843, Kennington – 22 November 1902, London) was an English metallurgist noted for his research on the physical properties of metals and their alloys. The austenite class of iron alloys is named aft ...
, the well-known British metallurgist and former assistant of Thomas Graham studied systematically solid state diffusion on the example of gold in lead in 1896. :
"... 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 Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's princip ...
introduced the concept of the mean free path. In the same year,
James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
developed the first atomistic theory of transport processes in gases. The modern atomistic theory of diffusion and Brownian motion was developed by
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
, Marian Smoluchowski and Jean-Baptiste Perrin. Ludwig Boltzmann, in the development of the atomistic backgrounds of the macroscopic transport processes, introduced the Boltzmann equation, which has served mathematics and physics with a source of transport process ideas and concerns for more than 140 years.S. Chapman, T. G. Cowling (1970) ''The Mathematical Theory of Non-uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases'', Cambridge University Press (3rd edition), . In 1920–1921, George de Hevesy measured self-diffusion using radioisotopes. He studied self-diffusion of radioactive isotopes of lead in the liquid and solid lead. Yakov Frenkel (sometimes, Jakov/Jacob Frenkel) proposed, and elaborated in 1926, the idea of diffusion in crystals through local defects (vacancies and interstitial atoms). He concluded, the diffusion process in condensed matter is an ensemble of elementary jumps and quasichemical interactions of particles and defects. He introduced several mechanisms of diffusion and found rate constants from experimental data. Sometime later,
Carl Wagner Carl Wilhelm Wagner (May 25, 1901 – December 10, 1977) was a German Physical chemist. He is best known for his pioneering work on Solid-state chemistry, where his work on oxidation rate theory, counter diffusion of ions and defect chemistry ...
and
Walter H. Schottky Walter Hans Schottky (23 July 1886 – 4 March 1976) was a German physicist who played a major early role in developing the theory of electron and ion emission phenomena, invented the screen-grid vacuum tube in 1915 while working at Siemen ...
developed Frenkel's ideas about mechanisms of diffusion further. Presently, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals. Henry Eyring, with co-authors, applied his theory of absolute reaction rates to Frenkel's quasichemical model 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 with the use of concentrations, densities and their derivatives. Flux is a vector \mathbf representing the quantity and direction of transfer. Given a small area \Delta S with normal \boldsymbol, the transfer of a physical quantity N through the area \Delta S per time \Delta t is :\Delta N = (\mathbf,\boldsymbol) \,\Delta S \,\Delta t +o(\Delta S \,\Delta t)\, , where (\mathbf,\boldsymbol) is the inner product and o(\cdots) is the
little-o notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
. If we use the notation of vector area \Delta \mathbf=\boldsymbol \, \Delta S then :\Delta N = (\mathbf, \Delta \mathbf) \, \Delta t +o(\Delta \mathbf \,\Delta t)\, . The
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of the diffusion flux is
lux The lux (symbol: lx) is the unit of illuminance, or luminous flux per unit area, in the International System of Units (SI). It is equal to one lumen per square metre. In photometry, this is used as a measure of the intensity, as perceived by ...
nbsp;=  uantity(
ime Ime is a village in Lindesnes municipality in Agder county, Norway. The village is located on the east side of the river Mandalselva, along the European route E39 highway. Ime is an eastern suburb of the town of Mandal. Ime might be considered to ...
rea. 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= - \nabla \cdot \mathbf +W \, , where W is intensity of any local source of this quantity (for example, the rate of a chemical reaction). For the diffusion equation, the no-flux boundary conditions can be formulated as (\mathbf(x),\boldsymbol(x))=0 on the boundary, where \boldsymbol is the normal to the boundary at point x.


Fick's law and equations

Fick's first law: the diffusion flux is proportional to the negative of the concentration gradient: :\mathbf=-D \,\nabla n \ , \;\; J_i=-D \frac \ . The corresponding diffusion equation (Fick's second law) is :\frac=\nabla\cdot( D \,\nabla n(x,t))=D \, \Delta n(x,t)\ , where \Delta is the Laplace operator, :\Delta n(x,t) = \sum_i \frac \ .


Onsager's equations for multicomponent diffusion and thermodiffusion

Fick's law describes diffusion of an admixture in a medium. 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, :\mathbf_i=\sum_j L_ X_j \, , where \mathbf_i is the flux of the ''i''th physical quantity (component) and X_j is the ''j''th 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= \nabla \frac \, , 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= \nabla \frac\ , \;\;\; X_i= - \nabla \frac \; (i >0) , because \mathrms = \frac \,\mathrmu-\sum_ \frac \, 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_ \left.\frac\_ \nabla n_k \ , where the derivatives of s are calculated at equilibrium n^*. The matrix of the ''kinetic coefficients'' L_ should be symmetric ( Onsager reciprocal relations) and
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite fu ...
( for the entropy growth). The transport equations are :\frac= - \operatorname \mathbf_i =- \sum_ L_\operatorname X_j = \sum_ \left _\right\, \Delta n_k\ . Here, all the indexes are related to the internal energy (0) and various components. The expression in the square brackets is the matrix D_ of the diffusion (''i'',''k'' > 0), thermodiffusion (''i'' > 0, ''k'' = 0 or ''k'' > 0, ''i'' = 0) and
thermal conductivity The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa. Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
() coefficients. Under isothermal conditions ''T'' = constant. 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_=\frac\sum_ L_ \left.\frac \_ (''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_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 affect 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 = \sum_j D_ \, \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_ \neq 0, and consider the state with c_2 = \cdots = c_n = 0. At this state, \partial c_2 / \partial t = D_ \, \Delta c_1. If D_ \, \Delta c_1(x) < 0 at some points, then c_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.


Einstein's mobility and Teorell formula

The
Einstein relation (kinetic theory) In physics (specifically, the kinetic theory of gases), the Einstein relation is a previously unexpected connection revealed independently by William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchowski in 1906 in their works o ...
connects the diffusion coefficient and the mobility (the ratio of the particle's terminal drift velocity to an applied force) : D = \frac, where ''D'' is the diffusion constant, ''μ'' is the "mobility", ''k''B is Boltzmann's constant, ''T'' is the absolute temperature, and ''q'' is the elementary charge, that is, the charge of one electron. Below, to combine in the same formula the chemical potential ''μ'' and the mobility, we use for mobility the notation \mathfrak. The mobility-based approach was further applied by T. Teorell. In 1935, he studied the diffusion of ions through a membrane. He formulated the essence of his approach in the formula: :the flux is equal to mobility × concentration × force per gram-ion. This is the so-called ''Teorell formula''. The term "gram-ion" ("gram-particle") is used for a quantity of a substance that contains
Avogadro's number The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining con ...
of ions (particles). The common modern term is mole. The force under isothermal conditions consists of two parts: # Diffusion force caused by concentration gradient: -RT \frac \, \nabla n = -RT \, \nabla (\ln(n/n^\text)). # Electrostatic force caused by electric potential gradient: q \, \nabla \varphi. Here ''R'' is the gas constant, ''T'' is the absolute temperature, ''n'' is the concentration, the equilibrium concentration is marked by a superscript "eq", ''q'' is the charge and ''φ'' is the electric potential. The simple but crucial difference between the Teorell formula and the Onsager laws is the concentration factor in the Teorell expression for the flux. In the Einstein–Teorell approach, if for the finite force the concentration tends to zero then the flux also tends to zero, whereas the Onsager equations violate this simple and physically obvious rule. The general formulation of the Teorell formula for non-perfect systems under isothermal conditions is :\mathbf = \mathfrak \exp\left(\frac\right)(-\nabla \mu + (\text)), where ''μ'' is the
chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...
, ''μ''0 is the standard value of the chemical potential. The expression a = \exp\left(\frac\right) is the so-called activity. It measures the "effective concentration" of a species in a non-ideal mixture. In this notation, the Teorell formula for the flux has a very simple form :\mathbf = \mathfrak a (-\nabla \mu + (\text)). The standard derivation of the activity includes a normalization factor and for small concentrations a = n/n^\ominus + o(n/n^\ominus), where n^\ominus is the standard concentration. Therefore, this formula for the flux describes the flux of the normalized dimensionless quantity n/n^\ominus: :\frac = \nabla \cdot mathfrak a (\nabla \mu - (\text))


Fluctuation-dissipation theorem

Fluctuation-dissipation theorem based on the Langevin equation is developed to extend the Einstein model to the ballistic time scale. According to Langevin, the equation is based on Newton's second law of motion as :m \frac = -\frac\frac + F(t) where * ''x'' is the position. * ''μ'' is the mobility of the particle in the fluid or gas, which can be calculated using the
Einstein relation (kinetic theory) In physics (specifically, the kinetic theory of gases), the Einstein relation is a previously unexpected connection revealed independently by William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchowski in 1906 in their works o ...
. * ''m'' is the mass of the particle. * ''F'' is the random force applied to the particle. * ''t'' is time. Solving this equation, one obtained the time-dependent diffusion constant in the long-time limit and when the particle is significantly denser than the surrounding fluid, : D(t) = \mu \, k_ T(1-e^) 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) In physics (specifically, the kinetic theory of gases), the Einstein relation is a previously unexpected connection revealed independently by William Sutherland in 1904, Albert Einstein in 1905, and by Marian Smoluchowski in 1906 in their works o ...
. * ''m'' is the mass of the particle. * ''t'' is time.


Teorell formula for multicomponent diffusion

The Teorell formula with combination of Onsager's definition of the diffusion force gives :\mathbf_i = \mathfrak a_i \sum_j L_ X_j, where \mathfrak is the mobility of the ''i''th component, a_i is its activity, L_ is the matrix of the coefficients, X_j is the thermodynamic diffusion force, X_j= -\nabla \frac. For the isothermal perfect systems, X_j = - R \frac. Therefore, the Einstein–Teorell approach gives the following multicomponent generalization of the Fick's law for multicomponent diffusion: :\frac = \sum_j \nabla \cdot \left(D_\frac \nabla n_j\right), where D_ is the matrix of coefficients. The Chapman–Enskog formulas for diffusion in gases include exactly the same terms. Earlier, such terms were introduced in the Maxwell–Stefan diffusion equation.


Jumps on the surface and in solids

Diffusion of reagents on the surface of a catalyst may play an important role in heterogeneous catalysis. The model of diffusion in the ideal monolayer is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure. The system includes several reagents A_1,A_2,\ldots, A_m on the surface. Their surface concentrations are c_1,c_2,\ldots, c_m. The surface is a lattice of the adsorption places. Each reagent molecule fills a place on the surface. Some of the places are free. The concentration of the free places is z=c_0. The sum of all c_i (including free places) is constant, the density of adsorption places ''b''. The jump model gives for the diffusion flux of A_i (''i'' = 1, ..., ''n''): :\mathbf_i=-D_i \, \nabla c_i - c_i \nabla z, . The corresponding diffusion equation is: :\frac=- \operatorname\mathbf_i=D_i \, \Delta c_i - c_i \, \Delta z\, . Due to the conservation law, z=b-\sum_^n c_i \, , and we have the system of ''m'' diffusion equations. For one component we get Fick's law and linear equations because (b-c) \,\nabla c- c\,\nabla(b-c) = b\,\nabla c. For two and more components the equations are nonlinear. If all particles can exchange their positions with their closest neighbours then a simple generalization gives :\mathbf_i=-\sum_j D_ _j \,\nabla c_i - c_i \,\nabla c_j/math> :\frac=\sum_j D_ _j \, \Delta c_i - c_i \,\Delta c_j/math> where D_ = D_ \geq 0 is a symmetric matrix of coefficients that characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration c_0. Various versions of these jump models are also suitable for simple diffusion mechanisms in solids.


Diffusion in porous media

For diffusion in porous media the basic equations are (if Φ is constant): :\mathbf=- \phi D \,\nabla n^m :\frac = D \, \Delta n^m \, , where ''D'' is the diffusion coefficient, Φ is porosity, ''n'' is the concentration, ''m'' > 0 (usually ''m'' > 1, the case ''m'' = 1 corresponds to Fick's law). Care must be taken to properly account for the porosity (Φ) of the porous medium in both the flux terms and the accumulation terms. For example, as the porosity goes to zero, the molar flux in the porous medium goes to zero for a given concentration gradient. Upon applying the divergence of the flux, the porosity terms cancel out and the second equation above is formed. For diffusion of gases in porous media this equation is the formalization of Darcy's law: the volumetric flux of a gas in the porous media is :q=-\frac\,\nabla p where ''k'' is the permeability of the medium, ''μ'' is the
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the int ...
and ''p'' is the pressure. The advective molar flux is given as ''J'' = ''nq'' and for p \sim n^\gamma Darcy's law gives the equation of diffusion in porous media with ''m'' = ''γ'' + 1. In porous media, the average linear velocity (ν), is related to the volumetric flux as: \upsilon= q/\phi Combining the advective molar flux with the diffusive flux gives the advection dispersion equation \frac = D \, \Delta n^m \ - \nu\cdot \nabla n^m, For underground water infiltration, the Boussinesq approximation gives the same equation with ''m'' = 2. For plasma with the high level of radiation, the Zeldovich–Raizer equation gives ''m'' > 4 for the heat transfer.


Diffusion in physics


Diffusion coefficient in kinetic theory of gases

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 ength 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 \ell v_T = \frac\sqrt \frac\, , where ''k''B is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, ''T'' is the temperature, ''P'' is the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
, \ell is the mean free path, and ''vT'' is the mean thermal speed: :\ell = \frac\, , \;\;\; v_T=\sqrt\, . 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_=\frac\sqrt\sqrt\frac\, ,


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

In Boltzmann'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 ''i''th component of the mixture. Each component has its mean velocity C_i(x,t) = \frac \int_c c f(x,c,t) \, dc. If the velocities C_i(x,t) do not coincide 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 momentum \sum_i m_i n_i C_i(x,t) (''mi'' is the ''i''th particle mass), * density of kinetic energy \sum_i \left( n_i\frac + \int_c \frac f_i(x,c,t)\, dc \right). The kinetic temperature ''T'' and pressure ''P'' are defined in 3D space as :\frack_ T=\frac \int_c \frac f_i(x,c,t)\, dc; \quad P=k_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=-\fracD_\left\, where F_i is the force applied to the molecules of the ''i''th component and k_T is the thermodiffusion ratio. The coefficient ''D''12 is positive. This is the diffusion coefficient. Four terms in the formula for ''C''1−''C''2 describe four main effects in the diffusion of gases: # \nabla \,\left(\frac\right) describes the flux of the first component from the areas with the high ratio ''n''1/''n'' to the areas with lower values of this ratio (and, analogously the flux of the second component from high ''n''2/''n'' to low ''n''2/''n'' because ''n''2/''n'' = 1 – ''n''1/''n''); # \frac\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; # \frac(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. # k_T \frac\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_=\frac\left frac \right for rigid spheres; * D_=\frac \left frac\right \left(\frac \right)^ for repulsing force \kappa_r^. The number A_1() 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 \rho \, . where \rho_i =m_i n_i is the mass concentration of the ''i''th species, \rho=\sum_i \rho_i is the mass density. By definition, the diffusion velocity of the ''i''th component is v_i=C_i-V, \sum_i \rho_i v_i=0. The mass transfer of the ''i''th component is described by the continuity equation :\frac+\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 ''i''th component and the term \nabla (\rho_i v_i) represents diffusion of this component. In 1948,
Wendell H. Furry Wendell Hinkle Furry (February 18, 1907 – December 17, 1984) was a professor of physics at Harvard University who made contributions to theoretical and particle physics. The Furry theorem is named after him. Early life Furry was born in Pra ...
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_^N D_ \mathbf_j + D_i^ \, \nabla (\ln T) \right)\, ; :\mathbf_j=\nabla X_j + (X_j-Y_j)\,\nabla (\ln P) + \mathbf_j\, ; :\mathbf_j=\frac \left( Y_j \sum_^N Y_k (f_k-f_j) \right)\, . Here, D_ is the diffusion coefficient matrix, D_i^ is the thermal diffusion coefficient, f_i is the body force per unit mass acting on the ''i''th species, X_i=P_i/P is the partial pressure fraction of the ''i''th species (and P_i is the partial pressure), Y_i=\rho_i/\rho is the mass fraction of the ''i''th species, and \sum_i X_i=\sum_i Y_i=1.


Diffusion of electrons in solids

When the density of electrons in solids is not in equilibrium, diffusion of electrons occurs. For example, when a bias is applied to two ends of a chunk of semiconductor, or a light shines on one end (see right figure), electrons diffuse from high density regions (center) to low density regions (two ends), forming a gradient of electron density. This process generates current, referred to as diffusion current. Diffusion current can also be described by Fick's first law :J=- D \, \partial n/\partial x\, , where ''J'' is the diffusion current density ( amount of substance) per unit area per unit time, ''n'' (for ideal mixtures) is the electron density, ''x'' is the position ength


Diffusion in geophysics

Analytical and numerical models that solve the diffusion equation for different initial and boundary conditions have been popular for studying a wide variety of changes to the Earth's surface. Diffusion has been used extensively in erosion studies of hillslope retreat, bluff erosion, fault scarp degradation, wave-cut terrace/shoreline retreat, alluvial channel incision, coastal shelf retreat, and delta progradation. Although the Earth's surface is not literally diffusing in many of these cases, the process of diffusion effectively mimics the holistic changes that occur over decades to millennia. Diffusion models may also be used to solve inverse boundary value problems in which some information about the depositional environment is known from paleoenvironmental reconstruction and the diffusion equation is used to figure out the sediment influx and time series of landform changes.


Dialysis

Dialysis works on the principles of the diffusion of solutes and ultrafiltration of fluid across a semi-permeable membrane. Diffusion is a property of substances in water; substances in water tend to move from an area of high concentration to an area of low concentration.'' Mosby’s Dictionary of Medicine, Nursing, & Health Professions''. 7th ed. St. Louis, MO; Mosby: 2006 Blood flows by one side of a semi-permeable membrane, and a dialysate, or special dialysis fluid, flows by the opposite side. A semipermeable membrane is a thin layer of material that contains holes of various sizes, or pores. Smaller solutes and fluid pass through the membrane, but the membrane blocks the passage of larger substances (for example, red blood cells and large proteins). This replicates the filtering process that takes place in the kidneys when the blood enters the kidneys and the larger substances are separated from the smaller ones in the glomerulus.


Random walk (random motion)

One common misconception is that individual atoms, ions or molecules move randomly, which they do not. In the animation on the right, the ion in the left panel appears to have "random" motion in the absence of other ions. As the right panel shows, however, this motion is not random but is the result of "collisions" with other ions. As such, the movement of a single atom, ion, or molecule within a mixture just appears random when viewed in isolation. The movement of a substance within a mixture by "random walk" is governed by the kinetic energy within the system that can be affected by changes in concentration, pressure or temperature. (This is a classical description. At smaller scales, quantum effects will be non-negligible, in general. Thus, the study of the movement of a single atom becomes more subtle since particles at such small scales are described by probability amplitudes rather than deterministic measures of position and velocity.)


Separation of 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 at lengths in the nanometre-to-millimetre range. On larger length scales, transport in liquids and gases is normally due to another transport phenomenon, convection. To separate diffusion in these cases, special efforts are needed. Therefore, some often cited examples of diffusion are ''wrong'': If cologne is sprayed in one place, it can 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 of the temperature nhomogeneity 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). In contrast, heat conduction through solid media is an everyday occurrence (for example, 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 In image processing and computer vision, anisotropic diffusion, also called Perona–Malik diffusion, is a technique aiming at reducing image noise without removing significant parts of the image content, typically edges, lines or other details t ...
, also known as the Perona–Malik equation, enhances high gradients *
Atomic diffusion Atomic may refer to: * Of or relating to the atom, the smallest particle of a chemical element that retains its chemical properties * Atomic physics, the study of the atom * Atomic Age, also known as the "Atomic Era" * Atomic scale, distances com ...
, in solids * Bohm diffusion, spread of plasma across magnetic fields * Eddy diffusion, in coarse-grained description of turbulent flow *
Effusion In physics and chemistry, effusion is the process in which a gas escapes from a container through a hole of diameter considerably smaller than the mean free path of the molecules. Such a hole is often described as a ''pinhole'' and the escap ...
of a gas through small holes * Electronic diffusion, resulting in an electric current called the diffusion current *
Facilitated diffusion Facilitated diffusion (also known as facilitated transport or passive-mediated transport) is the process of spontaneous passive transport (as opposed to active transport) of molecules or ions across a biological membrane via specific transmembra ...
, present in some organisms * Gaseous diffusion, used for
isotope separation Isotope separation is the process of concentrating specific isotopes of a chemical element by removing other isotopes. The use of the nuclides produced is varied. The largest variety is used in research (e.g. in chemistry where atoms of "marker" ...
* 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 *
Lévy flight A Lévy flight is a random walk in which the step-lengths have a Lévy distribution, a probability distribution that is heavy-tailed. When defined as a walk in a space of dimension greater than one, the steps made are in isotropic random directi ...
* Molecular diffusion, diffusion of molecules from more dense to less dense areas *
Momentum diffusion Momentum diffusion most commonly refers to the diffusion, or spread of momentum between particles ( atoms or molecules) of matter, often in the fluid state. This transport of momentum can occur in any direction of the fluid flow. Momentum diffusion ...
ex. the diffusion of the
hydrodynamic In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) ...
velocity field *
Photon diffusion Photon diffusion is a situation where photons travel through a material without being absorbed, but rather undergoing repeated scattering events which change the direction of their path. The path of any given photon is then effectively a random wal ...
* Plasma diffusion * Random walk, model for diffusion *
Reverse diffusion Reverse diffusion refers to a situation where the transport of particles ( atoms or molecules) in a medium occurs towards regions of higher concentration gradients, opposite to that observed during diffusion. This phenomenon occurs during phase ...
, against the concentration gradient, in phase separation * Rotational diffusion, random reorientation of molecules * Surface diffusion, diffusion of adparticles on a surface * Taxis is an animal's directional movement activity in response to a stimulus ** Kinesis is an animal's non-directional movement activity in response to a stimulus * Trans-cultural diffusion, diffusion of cultural traits across geographical area *
Turbulent diffusion Turbulent diffusion is the transport of mass, heat, or momentum within a system due to random and chaotic time dependent motions. It occurs when turbulent fluid systems reach critical conditions in response to shear flow, which results from a combin ...
, transport of mass, heat, or momentum within a turbulent fluid


See also

* * * * * * * *


References

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