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
, where
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
.
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 CO
2 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
representing the quantity and direction of transfer. Given a small
area with normal
, the transfer of a
physical quantity through the area
per time
is
:
where
is the
inner product and
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 then
:
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
may be the number of particles, mass, energy, electric charge, or any other scalar
extensive quantity. For its density,
, the diffusion equation has the form
:
where
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
on the boundary, where
is the normal to the boundary at point
.
Fick's law and equations
Fick's first law: the diffusion flux is proportional to the negative of the concentration gradient:
:
The corresponding diffusion equation (Fick's second law) is
:
where
is the
Laplace operator,
:
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,
.
In 1931,
Lars Onsager included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For
multi-component transport,
:
where
is the flux of the ''i''th physical quantity (component) and
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
(he used the term "force" in quotation marks or "driving force"):
:
where
are the "thermodynamic coordinates".
For the heat and mass transfer one can take
(the density of internal energy) and
is the concentration of the
th component. The corresponding driving forces are the space vectors
:
because
where ''T'' is the absolute temperature and
is the chemical potential of the
th 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:
:
where the derivatives of
are calculated at equilibrium
.
The matrix of the ''kinetic coefficients''
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
:
Here, all the indexes are related to the internal energy (0) and various components. The expression in the square brackets is the matrix
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,
, and the matrix of diffusion coefficients is
:
(''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
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
:
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,
, and consider the state with
. At this state,
. If
at some points, then
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)
:
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 .
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: .
# Electrostatic force caused by electric potential gradient: .
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[
:
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 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[
:
The standard derivation of the activity includes a normalization factor and for small concentrations , where is the standard concentration. Therefore, this formula for the flux describes the flux of the normalized dimensionless quantity :
:
]
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
:
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,
:
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
:
where is the mobility of the ''i''th component, is its activity, is the matrix of the coefficients, is the thermodynamic diffusion force, . For the isothermal perfect systems, . Therefore, the Einstein–Teorell approach gives the following multicomponent generalization of the Fick's law for multicomponent diffusion:
:
where 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 on the surface. Their surface concentrations are 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 . The sum of all (including free places) is constant, the density of adsorption places ''b''.
The jump model gives for the diffusion flux of (''i'' = 1, ..., ''n''):
:
The corresponding diffusion equation is:[
:
Due to the conservation law, and we
have the system of ''m'' diffusion equations. For one component we get Fick's law and linear equations because . 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
:]