diffuse
出典: meddic
 vt.
 (気体・液体など)を放散する
 vi.
 adj.
 広汎性の、びまん性の
WordNet ［license wordnet］
「spread out; not concentrated in one place; "a large diffuse organization"」PrepTutorEJDIC ［license prepejdic］
「〈光・熱・液体など〉‘を'散らす,放散する,拡散させる / 〈学問・知識など〉‘を'広める,普及させる / 散る,放散する,拡散する / 広まる,普及する / 広く散った,広がった / 〈文体などが〉締まりのない,散漫な」WordNet ［license wordnet］
「lacking conciseness; "a diffuse historical novel"」
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出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/03/30 04:57:38」(JST)
wiki en
[Wiki en表示]Diffusion is the net movement of molecules or atoms from a region of high concentration (or high chemical potential) to a region of low concentration (or low chemical potential). This is also referred to as the movement of a substance down a concentration gradient. A gradient is the change in the value of a quantity (e.g., concentration, pressure, temperature) with the change in another variable (usually distance). For example, a change in concentration over a distance is called a concentration gradient, a change in pressure over a distance is called a pressure gradient, and a change in temperature over a distance is a called a temperature gradient.
The word diffusion derives from the Latin word, diffundere, which means "to spread out" (a substance that “spreads out” is moving from an area of high concentration to an area of low concentration). A distinguishing feature of diffusion is that it results in mixing or mass transport, without requiring bulk motion (bulk flow).^{[1]} Thus, diffusion should not be confused with convection, or advection, which are other transport phenomena that utilize bulk motion to move particles from one place to another.
Contents
 1 Diffusion vs. bulk flow
 2 Diffusion in the context of different disciplines
 3 History of diffusion in physics
 4 Basic models of diffusion
 4.1 Diffusion flux
 4.2 Fick's law and equations
 4.3 Onsager's equations for multicomponent diffusion and thermodiffusion
 4.4 Nondiagonal diffusion must be nonlinear
 4.5 Einstein's mobility and Teorell formula
 4.5.1 Teorell formula for multicomponent diffusion
 4.6 Jumps on the surface and in solids
 4.7 Diffusion in porous media
 5 Diffusion in physics
 5.1 Elementary theory of diffusion coefficient in gases
 5.2 The theory of diffusion in gases based on Boltzmann's equation
 5.3 Diffusion of electrons in solids
 6 Random walk (random motion)
 6.1 Separation of diffusion from convection in gases
 6.2 Other types of diffusion
 7 See also
 8 References
 9 External links
Diffusion vs. bulk flow
An example of a situation in which bulk flow and diffusion can be differentiated is the mechanism by which oxygen enters the body during external respiration (breathing). The lungs 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 in the lungs, which causes a decrease in pressure in the alveoli. This creates a pressure gradient between the air outside the body (relatively high pressure) and the alveoli (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 (i.e., the movement of air by bulk flow stops once there is no longer a pressure gradient).
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 in the alveoli decreases (air has a very low concentration of carbon dioxide compared to the blood in the body). This creates a concentration gradient for carbon dioxide to diffuse from the blood into the alveoli.
The pumping action of the heart 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). As the thoracic cavity contracts during expiration, the volume of the alveoli decreases and creates a pressure gradient between the alveoli and the air outside the body, and air moves by bulk flow down the pressure gradient.
Diffusion in the context of different disciplines
The concept of diffusion is widely used in: physics (particle diffusion), chemistry, biology, sociology, economics, and finance (diffusion of people, ideas and of price values). However, in each case, the object (e.g., atom, idea, etc.) that is undergoing diffusion is “spreading out” from a point or location at which there is a higher concentration of that object.
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.^{[2]}
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 of concentrations. It goes from regions of higher concentration to regions of lower concentration. Some time later, various generalizations of Fick's laws were developed in the frame of thermodynamics and nonequilibrium thermodynamics.^{[3]}
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 selfpropelled 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.^{[4]} The concept of diffusion is typically applied to any subject matter involving random walks in ensembles of individuals.
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 and if there is a higher concentration of oxygen outside the cell than inside, oxygen molecules diffuse into 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 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 (Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colours of stained glass or earthenware and Chinese ceramics.
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:^{[5]}
"...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 deriving, in 1867, the coefficient of diffusion for CO_{2} in air. The error rate is less than 5%.
In 1855, Adolf Fick, the 26yearold 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 that is similar to Fourier's law for heat conduction (1822) and Ohm's law for electric current (1827).
Robert Boyle demonstrated diffusion in solids in the 17th century^{[6]} 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 RobertsAusten, the wellknown British metallurgist, and former assistant of Thomas Graham, studied systematically solid state diffusion on the example of gold in lead in 1896. :^{[7]}
"... 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 JeanBaptiste 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.^{[8]}
In 1920–1921 George de Hevesy measured selfdiffusion using radioisotopes. He studied selfdiffusion of radioactive isotopes of lead in liquid and solid lead.
Yakov Frenkel (sometimes, Jakov/Jacov 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.
Some time later, Carl Wagner and Walter H. Schottky developed Frenkel's ideas about mechanisms of diffusion further. Presently, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals.^{[7]}
Henry Eyring, with coauthors, applied his theory of absolute reaction rates to Frenkel's quasichemical model of diffusion.^{[9]} The analogy between reaction kinetics and diffusion leads to various nonlinear versions of Fick's law.^{[10]}
Basic models of diffusion
Diffusion flux
Each model of diffusion expresses the diffusion flux through concentrations, densities and their derivatives. Flux is a vector $\mathbf {J}$. The transfer of a physical quantity $N$ through a small area $\Delta S$ with normal $\nu$ per time $\Delta t$ is
 $$
Δ N = ( J , ν ) Δ S Δ t + o ( Δ S Δ t ) , {\displaystyle \Delta N=(\mathbf {J} ,\nu )\Delta S\Delta t+o(\Delta S\Delta t)\,,}
where $(\mathbf {J} ,\nu )$ is the inner product and $o(...)$ is the littleo notation. If we use the notation of vector area $\Delta \mathbf {S} =\nu \Delta S$ then
 $$
Δ N = ( J , Δ S ) Δ t + o ( Δ S Δ t ) . {\displaystyle \Delta N=(\mathbf {J} ,\Delta \mathbf {S} )\Delta t+o(\Delta \mathbf {S} \Delta t)\,.}
The dimension of the diffusion flux is [flux]=[quantity]/([time]·[area]). The diffusing physical quantity $N$ may be the number of particles, mass, energy, electric charge, or any other scalar extensive quantity. For its density, $n$, the diffusion equation has the form
 $$
∂ n ∂ t = − ∇ ⋅ J + W , {\displaystyle {\frac {\partial n}{\partial t}}=\nabla \cdot \mathbf {J} +W\,,}
where $W$ is intensity of any local source of this quantity (the rate of a chemical reaction, for example). For the diffusion equation, the noflux boundary conditions can be formulated as $(\mathbf {J} (x),\nu (x))=0$ on the boundary, where $\nu$ is the normal to the boundary at point $x$.
Fick's law and equations
Fick's first law: the diffusion flux is proportional to the negative of the concentration gradient:
 $$
J = − D ∇ n , J i = − D ∂ n ∂ x i . {\displaystyle \mathbf {J} =D\nabla n\ ,\;\;J_{i}=D{\frac {\partial n}{\partial x_{i}}}\ .}
The corresponding diffusion equation (Fick's second law) is
 $$
∂ n ( x , t ) ∂ t = ∇ ⋅ ( D ∇ n ( x , t ) ) = D Δ n ( x , t ) , {\displaystyle {\frac {\partial n(x,t)}{\partial t}}=\nabla \cdot (D\nabla n(x,t))=D\Delta n(x,t)\ ,}
where $\Delta$ is the Laplace operator,
 $$
Δ n ( x , t ) = ∑ i ∂ 2 n ( x , t ) ∂ x i 2 . {\displaystyle \Delta n(x,t)=\sum _{i}{\frac {\partial ^{2}n(x,t)}{\partial x_{i}^{2}}}\ .}
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^{[11]} included the multicomponent transport processes in the general context of linear nonequilibrium thermodynamics. For multicomponent transport,
 $$
J i = ∑ j L i j X j , {\displaystyle \mathbf {J} _{i}=\sum _{j}L_{ij}X_{j}\,,}
where $\mathbf {J} _{i}$ is the flux of the ith physical quantity (component) and $X_{j}$ is the jth thermodynamic force.
The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the entropy density s (he used the term "force" in quotation marks or "driving force"):
 $$
X i = g r a d ∂ s ( n ) ∂ n i , {\displaystyle X_{i}={\rm {grad}}{\frac {\partial s(n)}{\partial n_{i}}}\ ,}
where $n_{i}$ are the "thermodynamic coordinates". For the heat and mass transfer one can take $n_{0}=u$ (the density of internal energy) and $n_{i}$ is the concentration of the ith component. The corresponding driving forces are the space vectors
 $$
X 0 = g r a d 1 T , X i = − g r a d μ i T ( i > 0 ) , {\displaystyle X_{0}={\rm {grad}}{\frac {1}{T}}\ ,\;\;\;X_{i}={\rm {grad}}{\frac {\mu _{i}}{T}}\;(i>0),} because $$
d s = 1 T d u − ∑ i ≥ 1 μ i T d n i {\displaystyle {\rm {d}}s={\frac {1}{T}}{\rm {d}}u\sum _{i\geq 1}{\frac {\mu _{i}}{T}}{\rm {d}}n_{i}}
where T is the absolute temperature and $\mu _{i}$ is the chemical potential of the ith component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore, the terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients.
For the linear Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium:
 $$
X i = ∑ k ≥ 0 ∂ 2 s ( n ) ∂ n i ∂ n k  n = n ∗ g r a d n k , {\displaystyle X_{i}=\sum _{k\geq 0}\left.{\frac {\partial ^{2}s(n)}{\partial n_{i}\partial n_{k}}}\right_{n=n^{*}}{\rm {grad}}n_{k}\ ,}
where the derivatives of s are calculated at equilibrium n^{*}. The matrix of the kinetic coefficients $L_{ij}$ should be symmetric (Onsager reciprocal relations) and positive definite (for the entropy growth).
The transport equations are
 $$
∂ n i ∂ t = − d i v J i = − ∑ j ≥ 0 L i j d i v X j = ∑ k ≥ 0 [ − ∑ j ≥ 0 L i j ∂ 2 s ( n ) ∂ n j ∂ n k  n = n ∗ ] Δ n k . {\displaystyle {\frac {\partial n_{i}}{\partial t}}={\rm {div}}\mathbf {J} _{i}=\sum _{j\geq 0}L_{ij}{\rm {div}}X_{j}=\sum _{k\geq 0}\left[\sum _{j\geq 0}L_{ij}\left.{\frac {\partial ^{2}s(n)}{\partial n_{j}\partial n_{k}}}\right_{n=n^{*}}\right]\Delta n_{k}\ .}
Here, all the indexes i, j, k=0,1,2,... are related to the internal energy (0) and various components. The expression in the square brackets is the matrix $D_{ik}$of the diffusion (i,k>0), thermodiffusion (i>0, k=0 or k>0, i=0) and thermal conductivity (i=k=0) coefficients.
Under isothermal conditions T=const. The relevant thermodynamic potential is the free energy (or the free entropy). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, $(1/T)\nabla \mu _{j}$, and the matrix of diffusion coefficients is
 $$
D i k = 1 T ∑ j ≥ 1 L i j ∂ μ j ( n , T ) ∂ n k  n = n ∗ {\displaystyle D_{ik}={\frac {1}{T}}\sum _{j\geq 1}L_{ij}\left.{\frac {\partial \mu _{j}(n,T)}{\partial n_{k}}}\right_{n=n^{*}}}
(i,k>0).
There is intrinsic arbitrariness in the definition of the thermodynamic forces and kinetic coefficients because they are not measurable separately and only their combinations $\sum _{j}L_{ij}X_{j}$ can be measured. For example, in the original work of Onsager^{[11]} the thermodynamic forces include additional multiplier T, whereas in the Course of Theoretical Physics^{[12]} this multiplier is omitted but the sign of the thermodynamic forces is opposite. All these changes are supplemented by the corresponding changes in the coefficients and do not affect the measurable quantities.
Nondiagonal diffusion must be nonlinear
The formalism of linear irreversible thermodynamics (Onsager) generates the systems of linear diffusion equations in the form
 $$
∂ c i ∂ t = ∑ j D i j Δ c j . {\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}\Delta c_{j}.}
If the matrix of diffusion coefficients is diagonal, then this system of equations is just a collection of decoupled Fick's equations for various components. Assume that diffusion is nondiagonal, for example, $D_{12}\neq 0$, and consider the state with $c_{2}=\ldots =c_{n}=0$. At this state, $\partial c_{2}/\partial t=D_{12}\Delta c_{1}$. If $D_{12}\Delta c_{1}(x)<0$ at some points, then $c_{2}(x)$ becomes negative at these points in a short time. Therefore, linear nondiagonal diffusion does not preserve positivity of concentrations. Nondiagonal equations of multicomponent diffusion must be nonlinear.^{[10]}
Einstein's mobility and Teorell formula
The Einstein relation (kinetic theory) connects the diffusion coefficient and the mobility (the ratio of the particle's terminal drift velocity to an applied force)^{[13]}
 $$
D = μ k B T , {\displaystyle D=\mu \,k_{\text{B}}T,}
where D is the diffusion constant, μ is the "mobility", k_{B} is Boltzmann's constant, T is the absolute temperature.
Below, to combine in the same formula the chemical potential μ and the mobility, we use for mobility the notation ${\mathfrak {m}}$.
The mobility—based approach was further applied by T. Teorell.^{[14]} In 1935, he studied the diffusion of ions through a membrane. He formulated the essence of his approach in the formula:
 the flux is equal to mobility × concentration × force per gramion.
This is the socalled Teorell formula. The term "gramion" ("gramparticle") is used for a quantity of a substance that contains Avogadro's number of ions (particles). The common modern term is mole.
The force under isothermal conditions consists of two parts:
 Diffusion force caused by concentration gradient: $RT{\frac {1}{n}}\nabla n=RT\nabla (\ln(n/n^{\text{eq}}))$.
 Electrostatic force caused by electric potential gradient: $q\nabla \varphi$.
Here R is the gas constant, T is the absolute temperature, n is the concentration, the equilibrium concentration is marked by a superscript "eq", q is the charge and φ is the electric potential.
The simple but crucial difference between the Teorell formula and the Onsager laws is the concentration factor in the Teorell expression for the flux. In the Einstein–Teorell approach, If for the finite force the concentration tends to zero then the flux also tends to zero, whereas the Onsager equations violate this simple and physically obvious rule.
The general formulation of the Teorell formula for nonperfect systems under isothermal conditions is^{[10]}
 $$
J = m exp ( μ − μ 0 R T ) ( − ∇ μ + ( external force per mole ) ) , {\displaystyle \mathbf {J} ={\mathfrak {m}}\exp \left({\frac {\mu \mu _{0}}{RT}}\right)(\nabla \mu +({\text{external force per mole}})),}
where μ is the chemical potential, μ_{0} is the standard value of the chemical potential. The expression $a=\exp \left({\frac {\mu \mu _{0}}{RT}}\right)$ is the socalled activity. It measures the "effective concentration" of a species in a nonideal mixture. In this notation, the Teorell formula for the flux has a very simple form^{[10]}
 $$
J = m a ( − ∇ μ + ( external force per mole ) ) . {\displaystyle \mathbf {J} ={\mathfrak {m}}a(\nabla \mu +({\text{external force per mole}})).}
The standard derivation of the activity includes a normalization factor and for small concentrations $a=n/n^{\ominus }+o(n/n^{\ominus })$, where $n^{\ominus }$ is the standard concentration. Therefore, this formula for the flux describes the flux of the normalized dimensionless quantity $n/n^{\ominus }$:
 $$
∂ ( n / n ⊖ ) ∂ t = ∇ ⋅ [ m a ( ∇ μ − ( external force per mole ) ) ] . {\displaystyle {\frac {\partial (n/n^{\ominus })}{\partial t}}=\nabla \cdot [{\mathfrak {m}}a(\nabla \mu ({\text{external force per mole}}))].}
Teorell formula for multicomponent diffusion
The Teorell formula with combination of Onsager's definition of the diffusion force gives
 $$
J i = m i a i ∑ j L i j X j , {\displaystyle \mathbf {J} _{i}={\mathfrak {m_{i}}}a_{i}\sum _{j}L_{ij}X_{j},}
where ${\mathfrak {m_{i}}}$ is the mobility of the ith component, $a_{i}$ is its activity, $L_{ij}$ is the matrix of the coefficients, $X_{j}$ is the thermodynamic diffusion force, $X_{j}=\nabla {\frac {\mu _{j}}{T}}$. For the isothermal perfect systems, $X_{j}=R{\frac {\nabla n_{j}}{n_{j}}}$. Therefore, the Einstein–Teorell approach gives the following multicomponent generalization of the Fick's law for multicomponent diffusion:
 $$
∂ n i ∂ t = ∑ j ∇ ⋅ ( D i j n i n j ∇ n j ) , {\displaystyle {\frac {\partial n_{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {n_{i}}{n_{j}}}\nabla n_{j}\right),}
where $D_{ij}$ 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 paces 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):
 $$
J i = − D i [ z ∇ c i − c i ∇ z ] . {\displaystyle \mathbf {J} _{i}=D_{i}[z\nabla c_{i}c_{i}\nabla z]\,.}
The corresponding diffusion equation is:^{[10]}
 $$
∂ c i ∂ t = − d i v J i = D i [ z Δ c i − c i Δ z ] . {\displaystyle {\frac {\partial c_{i}}{\partial t}}=\mathrm {div} \mathbf {J} _{i}=D_{i}[z\Delta c_{i}c_{i}\Delta z]\,.}
Due to the conservation law, $z=b\sum _{i=1}^{n}c_{i}\,,$ and we have the system of m diffusion equations. For one component we get Fick's law and linear equations because $(bc)\nabla cc\nabla (bc)=b\nabla c$. 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
 $$
J i = − ∑ j D i j [ c j ∇ c i − c i ∇ c j ] {\displaystyle \mathbf {J} _{i}=\sum _{j}D_{ij}[c_{j}\nabla c_{i}c_{i}\nabla c_{j}]}  $$
∂ c i ∂ t = ∑ j D i j [ c j Δ c i − c i Δ c j ] {\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}[c_{j}\Delta c_{i}c_{i}\Delta c_{j}]}
where $D_{ij}=D_{ji}\geq 0$ 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:^{[15]}
 $$
J = − D ∇ n m {\displaystyle \mathbf {J} =D\nabla n^{m}}  $$
∂ n ∂ t = D Δ n m , {\displaystyle {\frac {\partial n}{\partial t}}=D\Delta n^{m}\,,}
where D is the diffusion coefficient, n is the concentration, m>0 (usually m>1, the case m=1 corresponds to Fick's law).
For diffusion of gases in porous media this equation is the formalisation of Darcy's law: the velocity of a gas in the porous media is
 $$
v = − k μ ∇ p {\displaystyle v={\frac {k}{\mu }}\nabla p}
where k is the permeability of the medium, μ is the viscosity and p is the pressure. The flux J=nv and for $p\sim n^{\gamma }$ Darcy's law gives the equation of diffusion in porous media with m=γ+1.
For underground water infiltration the Boussinesq approximation gives the same equation with m=2.
For plasma with the high level of radiation the ZeldovichRaizer equation gives m>4 for the heat transfer.
Diffusion in physics
Elementary theory of diffusion coefficient in gases
The diffusion coefficient $D$ is the coefficient in the Fick's first law $J=D{\partial n}/{\partial x}$, where J is the diffusion flux (amount of substance) per unit area per unit time, n (for ideal mixtures) is the concentration, x is the position [length].
Let us consider two gases with molecules of the same diameter d and mass m (selfdiffusion). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient
 $$
D = 1 3 ℓ v T = 2 3 k B 3 π 3 m T 3 / 2 P d 2 , {\displaystyle D={\frac {1}{3}}\ell v_{T}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}
where k_{B} is the Boltzmann constant, T is the temperature, P is the pressure, $\ell$ is the mean free path, and v_{T} is the mean thermal speed:
 $$
ℓ = k B T 2 π d 2 P , v T = 8 k B T π m . {\displaystyle \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}
We can see that the diffusion coefficient in the mean free path approximation grows with T as T^{3/2} and decreases with P as 1/P. If we use for P the ideal gas law P=RnT with the total concentration n, then we can see that for given concentration n the diffusion coefficient grows with T as T^{1/2} and for given temperature it decreases with the total concentration as 1/n.
For two different gases, A and B, with molecular masses m_{A}, m_{B} and molecular diameters d_{A}, d_{B}, the mean free path estimate of the diffusion coefficient of A in B and B in A is:
 $$
D A B = 2 3 k B 3 π 3 1 2 m A + 1 2 m B 4 T 3 / 2 P ( d A + d B ) 2 , {\displaystyle D_{\rm {AB}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}}}}{\sqrt {{\frac {1}{2m_{\rm {A}}}}+{\frac {1}{2m_{\rm {B}}}}}}{\frac {4T^{3/2}}{P(d_{\rm {A}}+d_{\rm {B}})^{2}}}\,,}
The theory of diffusion in gases based on Boltzmann's equation
In 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 ith component of the mixture. Each component has its mean velocity $C_{i}(x,t)={\frac {1}{n_{i}}}\int _{c}cf(x,c,t)\,dc$. If the velocities $C_{i}(x,t)$ do not coincide then there exists diffusion.
In the ChapmanEnskog approximation, all the distribution functions are expressed through the densities of the conserved quantities:^{[8]}
 individual concentrations of particles, $n_{i}(x,t)=\int _{c}f_{i}(x,c,t)\,dc$ (particles per volume),
 density of moment $\sum _{i}m_{i}n_{i}C_{i}(x,t)$ (m_{i} is the ith particle mass),
 density of kinetic energy $\sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right)$.
The kinetic temperature T and pressure P are defined in 3D space as
 $$
3 2 k B T = 1 n ∫ c m i ( c i − C i ( x , t ) ) 2 2 f i ( x , c , t ) d c {\displaystyle {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc} ; $$
P = k B n T {\displaystyle P=k_{\rm {B}}nT} ,
where $n=\sum _{i}n_{i}$ is the total density.
For two gases, the difference between velocities, $C_{1}C_{2}$ is given by the expression:^{[8]}
 $$
C 1 − C 2 = − n 2 n 1 n 2 D 12 { ∇ ( n 1 n ) + n 1 n 2 ( m 2 − m 1 ) n ( m 1 n 1 + m 2 n 2 ) ∇ P − m 1 n 1 m 2 n 2 P ( m 1 n 1 + m 2 n 2 ) ( F 1 − F 2 ) + k T 1 T ∇ T } {\displaystyle C_{1}C_{2}={\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\}} ,
where $F_{i}$ is the force applied to the molecules of the ith component and $k_{T}$ is the thermodiffusion ratio.
The coefficient 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 {n_{1}}{n}}\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=1n_{1}/n);
 ${\frac {n_{1}n_{2}(m_{2}m_{1})}{n(m_{1}n_{1}+m_{2}n_{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;
 ${\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{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.
 $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,^{[8]}
 $D_{12}={\frac {3}{2n(d_{1}+d_{2})^{2}}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}$ for rigid spheres;
 $D_{12}={\frac {3}{8nA_{1}({\nu })\Gamma (3{\frac {2}{\nu 1}})}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}\left({\frac {2kT}{\kappa _{12}}}\right)^{\frac {2}{\nu 1}}$ for repulsing force $\kappa _{12}r^{\nu }$.
The number $A_{1}({\nu })$ is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book^{[8]})
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 = ∑ i ρ i C i ρ . {\displaystyle V={\frac {\sum _{i}\rho _{i}C_{i}}{\rho }}\,.}
where $\rho _{i}=m_{i}n_{i}$ is the mass concentration of the ith species, $\rho =\sum _{i}\rho _{i}$ is the mass density.
By definition, the diffusion velocity of the ith component is $v_{i}=C_{i}V$, $\sum _{i}\rho _{i}v_{i}=0$. The mass transfer of the ith component is described by the continuity equation
 $$
∂ ρ i ∂ t + ∇ ( ρ i V ) + ∇ ( ρ i v i ) = W i , {\displaystyle {\frac {\partial \rho _{i}}{\partial t}}+\nabla (\rho _{i}V)+\nabla (\rho _{i}v_{i})=W_{i}\,,}
where $W_{i}$ is the net mass production rate in chemical reactions, $\sum _{i}W_{i}=0$.
In these equations, the term $\nabla (\rho _{i}V)$ describes advection of the ith component and the term $\nabla (\rho _{i}v_{i})$ represents diffusion of this component.
In 1948, Wendell H. Furry proposed to use the form of the diffusion rates found in kinetic theory as a framework for the new phenomenological approach to diffusion in gases. This approach was developed further by F.A. Williams and S.H. Lam.^{[16]} For the diffusion velocities in multicomponent gases (N components) they used
 $$
v i = − ( ∑ j = 1 N D i j d j + D i ( T ) ∇ ( ln T ) ) ; {\displaystyle v_{i}=\left(\sum _{j=1}^{N}D_{ij}\mathbf {d} _{j}+D_{i}^{(T)}\nabla (\ln T)\right)\,;}  $$
d j = ∇ X j + ( X j − Y j ) ∇ ( ln P ) + g j ; {\displaystyle \mathbf {d} _{j}=\nabla X_{j}+(X_{j}Y_{j})\nabla (\ln P)+\mathbf {g} _{j}\,;}  $$
g j = ρ P ( Y j ∑ k = 1 N Y k ( f k − f j ) ) . {\displaystyle \mathbf {g} _{j}={\frac {\rho }{P}}\left(Y_{j}\sum _{k=1}^{N}Y_{k}(f_{k}f_{j})\right)\,.}
Here, $D_{ij}$ is the diffusion coefficient matrix, $D_{i}^{(T)}$ is the thermal diffusion coefficient, $f_{i}$ is the body force per unite mass acting on the ith species, $X_{i}=P_{i}/P$ is the partial pressure fraction of the ith species (and $P_{i}$ is the partial pressure), $Y_{i}=\rho _{i}/\rho$ is the mass fraction of the ith species, and $\sum _{i}X_{i}=\sum _{i}Y_{i}=1$.
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), electron 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 ∂ n / ∂ x , {\displaystyle 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 [length].
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 on in the left panel has a “random” motion, but this motion is not random as it 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.
Separation of diffusion from convection in gases
While Brownian motion of multimolecular 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 nontrivial 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 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 the temperature [inhomogeneity]. If ink is dropped in water, one usually observes an inhomogeneous evolution of the spatial distribution, which clearly indicates convection (caused, in particular, by this dropping).^{[citation needed]}
In contrast, heat conduction through solid media is an everyday occurrence (e.g. a metal spoon partly immersed in a hot liquid). This explains why the diffusion of heat was explained mathematically before the diffusion of mass.
Other types of diffusion
 Anisotropic diffusion, also known as the PeronaMalik equation, enhances high gradients
 Anomalous diffusion,^{[17]} in porous medium
 Atomic diffusion, in solids
 Eddy diffusion, in coarsegrained 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
 Levy flights and walks
 Momentum diffusion ex. the diffusion of the hydrodynamic velocity field
 Photon diffusion
 Plasma diffusion
 Random walk,^{[18]} model for diffusion
 Reverse diffusion, against the concentration gradient, in phase separation
 Rotational diffusion, random reorientations of molecules
 Surface diffusion, diffusion of adparticles on a surface
 Turbulent diffusion, transport of mass, heat, or momentum within a turbulent fluid
See also
 Advection
 Anomalous diffusion
 Diffusionlimited aggregation
 Fick's laws of diffusion
 False diffusion
 Isobaric counterdiffusion
 Sorption
 Osmosis
 Random walk
References
 ^ J.G. Kirkwood, R.L. Baldwin, P.J. Dunlop, L.J. Gosting, G. Kegeles (1960)Flow equations and frames of reference for isothermal diffusion in liquids. The Journal of Chemical Physics 33(5):15051513.
 ^ J. Philibert (2005). One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2, 1.1–1.10.
 ^ S.R. De Groot, P. Mazur (1962). Nonequilibrium Thermodynamics. NorthHolland, Amsterdam.
 ^ A. Einstein (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen" (PDF). Ann. Phys. 17 (8): 549–560. Bibcode:1905AnP...322..549E. doi:10.1002/andp.19053220806.
 ^ Diffusion Processes, Thomas Graham Symposium, ed. J.N. Sherwood, A.V. Chadwick, W.M.Muir, F.L. Swinton, Gordon and Breach, London, 1971.
 ^ L.W. Barr (1997), In: Diffusion in Materials, DIMAT 96, ed. H.Mehrer, Chr. Herzig, N.A. Stolwijk, H. Bracht, Scitec Publications, Vol.1, pp. 1–9.
 ^ ^{a} ^{b} H. Mehrer; N.A. Stolwijk (2009). "Heroes and Highlights in the History of Diffusion" (PDF). Diffusion Fundamentals. 11 (1): 1–32.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} S. Chapman, T. G. Cowling (1970) The Mathematical Theory of Nonuniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press (3rd edition), ISBN 052140844X.
 ^ J.F. Kincaid; H. Eyring; A.E. Stearn (1941). "The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid State". Chem. Rev. 28 (2): 301–365. doi:10.1021/cr60090a005.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} A.N. Gorban, H.P. Sargsyan and H.A. Wahab (2011). "Quasichemical Models of Multicomponent Nonlinear Diffusion". Mathematical Modelling of Natural Phenomena. 6 (5): 184–262. arXiv:1012.2908. doi:10.1051/mmnp/20116509.
 ^ ^{a} ^{b} Onsager, L. (1931). "Reciprocal Relations in Irreversible Processes. I". Physical Review. 37 (4): 405–426. Bibcode:1931PhRv...37..405O. doi:10.1103/PhysRev.37.405.
 ^ L.D. Landau, E.M. Lifshitz (1980). Statistical Physics. Vol. 5 (3rd ed.). ButterworthHeinemann. ISBN 9780750633727.
 ^ S. Bromberg, K.A. Dill (2002), Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology, Garland Science, ISBN 0815320515.
 ^ T. Teorell (1935). "Studies on the "Diffusion Effect" upon Ionic Distribution. Some Theoretical Considerations". Proceedings of the National Academy of Sciences of the United States of America. 21 (3): 152–61. Bibcode:1935PNAS...21..152T. doi:10.1073/pnas.21.3.152. PMC 1076553. PMID 16587950.
 ^ J. L. Vázquez (2006), The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press, ISBN 0198569033.
 ^ S. H. Lam (2006). "Multicomponent diffusion revisited" (PDF). Physics of Fluids. 18 (7): 073101. Bibcode:2006PhFl...18g3101L. doi:10.1063/1.2221312.
 ^ D. BenAvraham and S. Havlin (2000). Diffusion and Reactions in Fractals and Disordered Systems (PDF). Cambridge University Press. ISBN 0521622786.
 ^ Weiss, G. (1994). Aspects and Applications of the Random Walk. NorthHolland. ISBN 0444816062.
External links
 Diffusion in a Bipolar Junction Transistor Demo
 Diffusion Furnace for doping of semiconductor wafers. POCl3 doping of Silicon.
 A Java applet implementing Diffusion
 [1]
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Wikipedia preview
出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2017/07/20 23:44:45」(JST)
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[Wiki en表示]Diffusion is the net movement of molecules or atoms from a region of high concentration (or high chemical potential) to a region of low concentration (or low chemical potential). This is also referred to as the movement of a substance down a concentration gradient.
A gradient is the change in the value of a quantity (e.g., concentration, pressure, temperature) with the change in another variable (usually distance). For example, a change in concentration over a distance is called a concentration gradient, a change in pressure over a distance is called a pressure gradient, and a change in temperature over a distance is a called a temperature gradient.
The word diffusion derives from the Latin word, diffundere, which means "to spread out" (a substance that “spreads out” is moving from an area of high concentration to an area of low concentration).
A distinguishing feature of diffusion is that it is dependent on particle random walk and results in mixing or mass transport, without requiring directed bulk motion. Bulk motion (bulk flow) is the characteristic of advection.^{[1]} The term convection is used to describe the combination of both transport phenomena.
Contents
 1 Diffusion vs. bulk flow
 2 Diffusion in the context of different disciplines
 3 History of diffusion in physics
 4 Basic models of diffusion
 4.1 Diffusion flux
 4.2 Fick's law and equations
 4.3 Onsager's equations for multicomponent diffusion and thermodiffusion
 4.4 Nondiagonal diffusion must be nonlinear
 4.5 Einstein's mobility and Teorell formula
 4.5.1 Teorell formula for multicomponent diffusion
 4.6 Jumps on the surface and in solids
 4.7 Diffusion in porous media
 5 Diffusion in physics
 5.1 Elementary theory of diffusion coefficient in gases
 5.2 The theory of diffusion in gases based on Boltzmann's equation
 5.3 Diffusion of electrons in solids
 5.4 Diffusion in geophysics
 6 Random walk (random motion)
 6.1 Separation of diffusion from convection in gases
 6.2 Other types of diffusion
 7 See also
 8 References
 9 External links
Diffusion vs. bulk flow
An example of a situation in which bulk motion and diffusion can be differentiated is the mechanism by which oxygen enters the body during external respiration (breathing). The lungs 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 in the lungs, which causes a decrease in pressure in the alveoli. This creates a pressure gradient between the air outside the body (relatively high pressure) and the alveoli (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 (i.e., the movement of air by bulk flow stops once there is no longer a pressure gradient).
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 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 in the body.
The pumping action of the heart 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). As the thoracic cavity contracts during expiration, the volume of the alveoli decreases and creates a pressure gradient between the alveoli and the air outside the body, and air moves by bulk flow down the pressure gradient.
Diffusion in the context of different disciplines
The concept of diffusion is widely used in: physics (particle diffusion), chemistry, biology, sociology, economics, and finance (diffusion of people, ideas and of price values). However, in each case, the object (e.g., atom, idea, etc.) that is undergoing diffusion is “spreading out” from a point or location at which there is a higher concentration of that object.
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.^{[2]}
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 of concentrations. It goes from regions of higher concentration to regions of lower concentration. Some time later, various generalizations of Fick's laws were developed in the frame of thermodynamics and nonequilibrium thermodynamics.^{[3]}
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 selfpropelled 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.^{[4]} The concept of diffusion is typically applied to any subject matter involving random walks in ensembles of individuals.
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 and if there is a higher concentration of oxygen outside the cell than inside, oxygen molecules diffuse into 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 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 (Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colours of stained glass or earthenware and Chinese ceramics.
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:^{[5]}
"...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 deriving, in 1867, the coefficient of diffusion for CO_{2} in air. The error rate is less than 5%.
In 1855, Adolf Fick, the 26yearold 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 that is similar to Fourier's law for heat conduction (1822) and Ohm's law for electric current (1827).
Robert Boyle demonstrated diffusion in solids in the 17th century^{[6]} 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 RobertsAusten, the wellknown British metallurgist, and former assistant of Thomas Graham, studied systematically solid state diffusion on the example of gold in lead in 1896. :^{[7]}
"... 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 JeanBaptiste 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.^{[8]}
In 1920–1921 George de Hevesy measured selfdiffusion using radioisotopes. He studied selfdiffusion of radioactive isotopes of lead in liquid and solid lead.
Yakov Frenkel (sometimes, Jakov/Jacov 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.
Some time later, Carl Wagner and Walter H. Schottky developed Frenkel's ideas about mechanisms of diffusion further. Presently, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals.^{[7]}
Henry Eyring, with coauthors, applied his theory of absolute reaction rates to Frenkel's quasichemical model of diffusion.^{[9]} The analogy between reaction kinetics and diffusion leads to various nonlinear versions of Fick's law.^{[10]}
Basic models of diffusion
Diffusion flux
Each model of diffusion expresses the diffusion flux through concentrations, densities and their derivatives. Flux is a vector $\mathbf {J}$. The transfer of a physical quantity $N$ through a small area $\Delta S$ with normal $\nu$ per time $\Delta t$ is
 $$
Δ N = ( J , ν ) Δ S Δ t + o ( Δ S Δ t ) , {\displaystyle \Delta N=(\mathbf {J} ,\nu )\Delta S\Delta t+o(\Delta S\Delta t)\,,}
where $(\mathbf {J} ,\nu )$ is the inner product and $o(...)$ is the littleo notation. If we use the notation of vector area $\Delta \mathbf {S} =\nu \Delta S$ then
 $$
Δ N = ( J , Δ S ) Δ t + o ( Δ S Δ t ) . {\displaystyle \Delta N=(\mathbf {J} ,\Delta \mathbf {S} )\Delta t+o(\Delta \mathbf {S} \Delta t)\,.}
The dimension of the diffusion flux is [flux]=[quantity]/([time]·[area]). The diffusing physical quantity $N$ may be the number of particles, mass, energy, electric charge, or any other scalar extensive quantity. For its density, $n$, the diffusion equation has the form
 $$
∂ n ∂ t = − ∇ ⋅ J + W , {\displaystyle {\frac {\partial n}{\partial t}}=\nabla \cdot \mathbf {J} +W\,,}
where $W$ is intensity of any local source of this quantity (the rate of a chemical reaction, for example). For the diffusion equation, the noflux boundary conditions can be formulated as $(\mathbf {J} (x),\nu (x))=0$ on the boundary, where $\nu$ 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:
 $$
J = − D ∇ n , J i = − D ∂ n ∂ x i . {\displaystyle \mathbf {J} =D\nabla n\ ,\;\;J_{i}=D{\frac {\partial n}{\partial x_{i}}}\ .}
The corresponding diffusion equation (Fick's second law) is
 $$
∂ n ( x , t ) ∂ t = ∇ ⋅ ( D ∇ n ( x , t ) ) = D Δ n ( x , t ) , {\displaystyle {\frac {\partial n(x,t)}{\partial t}}=\nabla \cdot (D\nabla n(x,t))=D\Delta n(x,t)\ ,}
where $\Delta$ is the Laplace operator,
 $$
Δ n ( x , t ) = ∑ i ∂ 2 n ( x , t ) ∂ x i 2 . {\displaystyle \Delta n(x,t)=\sum _{i}{\frac {\partial ^{2}n(x,t)}{\partial x_{i}^{2}}}\ .}
Onsager's equations for multicomponent diffusion and thermodiffusion
Fick's law describes diffusion of an admixture in a medium. The concentration of this admixture should be small and the gradient of this concentration should be also small. The driving force of diffusion in Fick's law is the antigradient of concentration, $\nabla n$.
In 1931, Lars Onsager^{[11]} included the multicomponent transport processes in the general context of linear nonequilibrium thermodynamics. For multicomponent transport,
 $$
J i = ∑ j L i j X j , {\displaystyle \mathbf {J} _{i}=\sum _{j}L_{ij}X_{j}\,,}
where $\mathbf {J} _{i}$ is the flux of the ith physical quantity (component) and $X_{j}$ is the jth thermodynamic force.
The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the entropy density s (he used the term "force" in quotation marks or "driving force"):
 $$
X i = g r a d ∂ s ( n ) ∂ n i , {\displaystyle X_{i}={\rm {grad}}{\frac {\partial s(n)}{\partial n_{i}}}\ ,}
where $n_{i}$ are the "thermodynamic coordinates". For the heat and mass transfer one can take $n_{0}=u$ (the density of internal energy) and $n_{i}$ is the concentration of the ith component. The corresponding driving forces are the space vectors
 $$
X 0 = g r a d 1 T , X i = − g r a d μ i T ( i > 0 ) , {\displaystyle X_{0}={\rm {grad}}{\frac {1}{T}}\ ,\;\;\;X_{i}={\rm {grad}}{\frac {\mu _{i}}{T}}\;(i>0),} because $$
d s = 1 T d u − ∑ i ≥ 1 μ i T d n i {\displaystyle {\rm {d}}s={\frac {1}{T}}{\rm {d}}u\sum _{i\geq 1}{\frac {\mu _{i}}{T}}{\rm {d}}n_{i}}
where T is the absolute temperature and $\mu _{i}$ is the chemical potential of the ith component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore, the terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients.
For the linear Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium:
 $$
X i = ∑ k ≥ 0 ∂ 2 s ( n ) ∂ n i ∂ n k  n = n ∗ g r a d n k , {\displaystyle X_{i}=\sum _{k\geq 0}\left.{\frac {\partial ^{2}s(n)}{\partial n_{i}\partial n_{k}}}\right_{n=n^{*}}{\rm {grad}}n_{k}\ ,}
where the derivatives of s are calculated at equilibrium n^{*}. The matrix of the kinetic coefficients $L_{ij}$ should be symmetric (Onsager reciprocal relations) and positive definite (for the entropy growth).
The transport equations are
 $$
∂ n i ∂ t = − d i v J i = − ∑ j ≥ 0 L i j d i v X j = ∑ k ≥ 0 [ − ∑ j ≥ 0 L i j ∂ 2 s ( n ) ∂ n j ∂ n k  n = n ∗ ] Δ n k . {\displaystyle {\frac {\partial n_{i}}{\partial t}}={\rm {div}}\mathbf {J} _{i}=\sum _{j\geq 0}L_{ij}{\rm {div}}X_{j}=\sum _{k\geq 0}\left[\sum _{j\geq 0}L_{ij}\left.{\frac {\partial ^{2}s(n)}{\partial n_{j}\partial n_{k}}}\right_{n=n^{*}}\right]\Delta n_{k}\ .}
Here, all the indexes i, j, k=0,1,2,... are related to the internal energy (0) and various components. The expression in the square brackets is the matrix $D_{ik}$of the diffusion (i,k>0), thermodiffusion (i>0, k=0 or k>0, i=0) and thermal conductivity (i=k=0) coefficients.
Under isothermal conditions T=const. The relevant thermodynamic potential is the free energy (or the free entropy). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, $(1/T)\nabla \mu _{j}$, and the matrix of diffusion coefficients is
 $$
D i k = 1 T ∑ j ≥ 1 L i j ∂ μ j ( n , T ) ∂ n k  n = n ∗ {\displaystyle D_{ik}={\frac {1}{T}}\sum _{j\geq 1}L_{ij}\left.{\frac {\partial \mu _{j}(n,T)}{\partial n_{k}}}\right_{n=n^{*}}}
(i,k>0).
There is intrinsic arbitrariness in the definition of the thermodynamic forces and kinetic coefficients because they are not measurable separately and only their combinations $\sum _{j}L_{ij}X_{j}$ can be measured. For example, in the original work of Onsager^{[11]} the thermodynamic forces include additional multiplier T, whereas in the Course of Theoretical Physics^{[12]} 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
 $$
∂ c i ∂ t = ∑ j D i j Δ c j . {\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}\Delta c_{j}.}
If the matrix of diffusion coefficients is diagonal, then this system of equations is just a collection of decoupled Fick's equations for various components. Assume that diffusion is nondiagonal, for example, $D_{12}\neq 0$, and consider the state with $c_{2}=\ldots =c_{n}=0$. At this state, $\partial c_{2}/\partial t=D_{12}\Delta c_{1}$. If $D_{12}\Delta c_{1}(x)<0$ at some points, then $c_{2}(x)$ becomes negative at these points in a short time. Therefore, linear nondiagonal diffusion does not preserve positivity of concentrations. Nondiagonal equations of multicomponent diffusion must be nonlinear.^{[10]}
Einstein's mobility and Teorell formula
The Einstein relation (kinetic theory) connects the diffusion coefficient and the mobility (the ratio of the particle's terminal drift velocity to an applied force)^{[13]}
 $$
D = μ k B T , {\displaystyle D=\mu \,k_{\text{B}}T,}
where D is the diffusion constant, μ is the "mobility", k_{B} is Boltzmann's constant, T is the absolute temperature.
Below, to combine in the same formula the chemical potential μ and the mobility, we use for mobility the notation ${\mathfrak {m}}$.
The mobility—based approach was further applied by T. Teorell.^{[14]} 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 gramion.
This is the socalled Teorell formula. The term "gramion" ("gramparticle") is used for a quantity of a substance that contains Avogadro's number of ions (particles). The common modern term is mole.
The force under isothermal conditions consists of two parts:
 Diffusion force caused by concentration gradient: $RT{\frac {1}{n}}\nabla n=RT\nabla (\ln(n/n^{\text{eq}}))$.
 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 nonperfect systems under isothermal conditions is^{[10]}
 $$
J = m exp ( μ − μ 0 R T ) ( − ∇ μ + ( external force per mole ) ) , {\displaystyle \mathbf {J} ={\mathfrak {m}}\exp \left({\frac {\mu \mu _{0}}{RT}}\right)(\nabla \mu +({\text{external force per mole}})),}
where μ is the chemical potential, μ_{0} is the standard value of the chemical potential. The expression $a=\exp \left({\frac {\mu \mu _{0}}{RT}}\right)$ is the socalled activity. It measures the "effective concentration" of a species in a nonideal mixture. In this notation, the Teorell formula for the flux has a very simple form^{[10]}
 $$
J = m a ( − ∇ μ + ( external force per mole ) ) . {\displaystyle \mathbf {J} ={\mathfrak {m}}a(\nabla \mu +({\text{external force per mole}})).}
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 $n/n^{\ominus }$:
 $$
∂ ( n / n ⊖ ) ∂ t = ∇ ⋅ [ m a ( ∇ μ − ( external force per mole ) ) ] . {\displaystyle {\frac {\partial (n/n^{\ominus })}{\partial t}}=\nabla \cdot [{\mathfrak {m}}a(\nabla \mu ({\text{external force per mole}}))].}
Teorell formula for multicomponent diffusion
The Teorell formula with combination of Onsager's definition of the diffusion force gives
 $$
J i = m i a i ∑ j L i j X j , {\displaystyle \mathbf {J} _{i}={\mathfrak {m_{i}}}a_{i}\sum _{j}L_{ij}X_{j},}
where ${\mathfrak {m_{i}}}$ is the mobility of the ith component, $a_{i}$ is its activity, $L_{ij}$ is the matrix of the coefficients, $X_{j}$ is the thermodynamic diffusion force, $X_{j}=\nabla {\frac {\mu _{j}}{T}}$. For the isothermal perfect systems, $X_{j}=R{\frac {\nabla n_{j}}{n_{j}}}$. Therefore, the Einstein–Teorell approach gives the following multicomponent generalization of the Fick's law for multicomponent diffusion:
 $$
∂ n i ∂ t = ∑ j ∇ ⋅ ( D i j n i n j ∇ n j ) , {\displaystyle {\frac {\partial n_{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {n_{i}}{n_{j}}}\nabla n_{j}\right),}
where $D_{ij}$ is the matrix of coefficients. The Chapman–Enskog formulas for diffusion in gases include exactly the same terms. Earlier, such terms were introduced in the Maxwell–Stefan diffusion equation.
Jumps on the surface and in solids
Diffusion of reagents on the surface of a catalyst may play an important role in heterogeneous catalysis. The model of diffusion in the ideal monolayer is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure.
The system includes several reagents $A_{1},A_{2},\ldots A_{m}$ on the surface. Their surface concentrations are $c_{1},c_{2},\ldots c_{m}$. The surface is a lattice of the adsorption places. Each reagent molecule fills a place on the surface. Some of the places are free. The concentration of the free places is $z=c_{0}$. The sum of all $c_{i}$ (including free places) is constant, the density of adsorption places b.
The jump model gives for the diffusion flux of $A_{i}$ (i=1,...,n):
 $$
J i = − D i [ z ∇ c i − c i ∇ z ] . {\displaystyle \mathbf {J} _{i}=D_{i}[z\nabla c_{i}c_{i}\nabla z]\,.}
The corresponding diffusion equation is:^{[10]}
 $$
∂ c i ∂ t = − d i v J i = D i [ z Δ c i − c i Δ z ] . {\displaystyle {\frac {\partial c_{i}}{\partial t}}=\mathrm {div} \mathbf {J} _{i}=D_{i}[z\Delta c_{i}c_{i}\Delta z]\,.}
Due to the conservation law, $z=b\sum _{i=1}^{n}c_{i}\,,$ and we have the system of m diffusion equations. For one component we get Fick's law and linear equations because $(bc)\nabla cc\nabla (bc)=b\nabla c$. For two and more components the equations are nonlinear.
If all particles can exchange their positions with their closest neighbours then a simple generalization gives
 $$
J i = − ∑ j D i j [ c j ∇ c i − c i ∇ c j ] {\displaystyle \mathbf {J} _{i}=\sum _{j}D_{ij}[c_{j}\nabla c_{i}c_{i}\nabla c_{j}]}  $$
∂ c i ∂ t = ∑ j D i j [ c j Δ c i − c i Δ c j ] {\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}[c_{j}\Delta c_{i}c_{i}\Delta c_{j}]}
where $D_{ij}=D_{ji}\geq 0$ is a symmetric matrix of coefficients that characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration $c_{0}$.
Various versions of these jump models are also suitable for simple diffusion mechanisms in solids.
Diffusion in porous media
For diffusion in porous media the basic equations are:^{[15]}
 $$
J = − D ∇ n m {\displaystyle \mathbf {J} =D\nabla n^{m}}  $$
∂ n ∂ t = D Δ n m , {\displaystyle {\frac {\partial n}{\partial t}}=D\Delta n^{m}\,,}
where D is the diffusion coefficient, n is the concentration, m>0 (usually m>1, the case m=1 corresponds to Fick's law).
For diffusion of gases in porous media this equation is the formalisation of Darcy's law: the velocity of a gas in the porous media is
 $$
v = − k μ ∇ p {\displaystyle v={\frac {k}{\mu }}\nabla p}
where k is the permeability of the medium, μ is the viscosity and p is the pressure. The flux J=nv and for $p\sim n^{\gamma }$ Darcy's law gives the equation of diffusion in porous media with m=γ+1.
For underground water infiltration the Boussinesq approximation gives the same equation with m=2.
For plasma with the high level of radiation the ZeldovichRaizer equation gives m>4 for the heat transfer.
Diffusion in physics
Elementary theory of diffusion coefficient in 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 [length].
Let us consider two gases with molecules of the same diameter d and mass m (selfdiffusion). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient
 $$
D = 1 3 ℓ v T = 2 3 k B 3 π 3 m T 3 / 2 P d 2 , {\displaystyle D={\frac {1}{3}}\ell v_{T}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}
where k_{B} is the Boltzmann constant, T is the temperature, P is the pressure, $\ell$ is the mean free path, and v_{T} is the mean thermal speed:
 $$
ℓ = k B T 2 π d 2 P , v T = 8 k B T π m . {\displaystyle \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}
We can see that the diffusion coefficient in the mean free path approximation grows with T as T^{3/2} and decreases with P as 1/P. If we use for P the ideal gas law P=RnT with the total concentration n, then we can see that for given concentration n the diffusion coefficient grows with T as T^{1/2} and for given temperature it decreases with the total concentration as 1/n.
For two different gases, A and B, with molecular masses m_{A}, m_{B} and molecular diameters d_{A}, d_{B}, the mean free path estimate of the diffusion coefficient of A in B and B in A is:
 $$
D A B = 2 3 k B 3 π 3 1 2 m A + 1 2 m B 4 T 3 / 2 P ( d A + d B ) 2 , {\displaystyle D_{\rm {AB}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}}}}{\sqrt {{\frac {1}{2m_{\rm {A}}}}+{\frac {1}{2m_{\rm {B}}}}}}{\frac {4T^{3/2}}{P(d_{\rm {A}}+d_{\rm {B}})^{2}}}\,,}
The theory of diffusion in gases based on Boltzmann's equation
In Boltzmann's kinetics of the mixture of gases, each gas has its own distribution function, $f_{i}(x,c,t)$, where t is the time moment, x is position and c is velocity of molecule of the ith component of the mixture. Each component has its mean velocity $C_{i}(x,t)={\frac {1}{n_{i}}}\int _{c}cf(x,c,t)\,dc$. If the velocities $C_{i}(x,t)$ do not coincide then there exists diffusion.
In the ChapmanEnskog approximation, all the distribution functions are expressed through the densities of the conserved quantities:^{[8]}
 individual concentrations of particles, $n_{i}(x,t)=\int _{c}f_{i}(x,c,t)\,dc$ (particles per volume),
 density of moment $\sum _{i}m_{i}n_{i}C_{i}(x,t)$ (m_{i} is the ith particle mass),
 density of kinetic energy $\sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right)$.
The kinetic temperature T and pressure P are defined in 3D space as
 $$
3 2 k B T = 1 n ∫ c m i ( c i − C i ( x , t ) ) 2 2 f i ( x , c , t ) d c {\displaystyle {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc} ; $$
P = k B n T {\displaystyle P=k_{\rm {B}}nT} ,
where $n=\sum _{i}n_{i}$ is the total density.
For two gases, the difference between velocities, $C_{1}C_{2}$ is given by the expression:^{[8]}
 $$
C 1 − C 2 = − n 2 n 1 n 2 D 12 { ∇ ( n 1 n ) + n 1 n 2 ( m 2 − m 1 ) n ( m 1 n 1 + m 2 n 2 ) ∇ P − m 1 n 1 m 2 n 2 P ( m 1 n 1 + m 2 n 2 ) ( F 1 − F 2 ) + k T 1 T ∇ T } {\displaystyle C_{1}C_{2}={\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\}} ,
where $F_{i}$ is the force applied to the molecules of the ith component and $k_{T}$ is the thermodiffusion ratio.
The coefficient 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:
 $\nabla \left({\frac {n_{1}}{n}}\right)$ 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=1n_{1}/n);
 ${\frac {n_{1}n_{2}(m_{2}m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P$ 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;
 ${\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}F_{2})$ 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.
 $k_{T}{\frac {1}{T}}\nabla T$ 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,^{[8]}
 $D_{12}={\frac {3}{2n(d_{1}+d_{2})^{2}}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}$ for rigid spheres;
 $D_{12}={\frac {3}{8nA_{1}({\nu })\Gamma (3{\frac {2}{\nu 1}})}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}\left({\frac {2kT}{\kappa _{12}}}\right)^{\frac {2}{\nu 1}}$ for repulsing force $\kappa _{12}r^{\nu }$.
The number $A_{1}({\nu })$ is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book^{[8]})
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:
 $$
V = ∑ i ρ i C i ρ . {\displaystyle V={\frac {\sum _{i}\rho _{i}C_{i}}{\rho }}\,.}
where $\rho _{i}=m_{i}n_{i}$ is the mass concentration of the ith species, $\rho =\sum _{i}\rho _{i}$ is the mass density.
By definition, the diffusion velocity of the ith component is $v_{i}=C_{i}V$, $\sum _{i}\rho _{i}v_{i}=0$. The mass transfer of the ith component is described by the continuity equation
 $$
∂ ρ i ∂ t + ∇ ( ρ i V ) + ∇ ( ρ i v i ) = W i , {\displaystyle {\frac {\partial \rho _{i}}{\partial t}}+\nabla (\rho _{i}V)+\nabla (\rho _{i}v_{i})=W_{i}\,,}
where $W_{i}$ is the net mass production rate in chemical reactions, $\sum _{i}W_{i}=0$.
In these equations, the term $\nabla (\rho _{i}V)$ describes advection of the ith component and the term $\nabla (\rho _{i}v_{i})$ represents diffusion of this component.
In 1948, Wendell H. Furry proposed to use the form of the diffusion rates found in kinetic theory as a framework for the new phenomenological approach to diffusion in gases. This approach was developed further by F.A. Williams and S.H. Lam.^{[16]} For the diffusion velocities in multicomponent gases (N components) they used
 $$
v i = − ( ∑ j = 1 N D i j d j + D i ( T ) ∇ ( ln T ) ) ; {\displaystyle v_{i}=\left(\sum _{j=1}^{N}D_{ij}\mathbf {d} _{j}+D_{i}^{(T)}\nabla (\ln T)\right)\,;}  $$
d j = ∇ X j + ( X j − Y j ) ∇ ( ln P ) + g j ; {\displaystyle \mathbf {d} _{j}=\nabla X_{j}+(X_{j}Y_{j})\nabla (\ln P)+\mathbf {g} _{j}\,;}  $$
g j = ρ P ( Y j ∑ k = 1 N Y k ( f k − f j ) ) . {\displaystyle \mathbf {g} _{j}={\frac {\rho }{P}}\left(Y_{j}\sum _{k=1}^{N}Y_{k}(f_{k}f_{j})\right)\,.}
Here, $D_{ij}$ is the diffusion coefficient matrix, $D_{i}^{(T)}$ is the thermal diffusion coefficient, $f_{i}$ is the body force per unite mass acting on the ith species, $X_{i}=P_{i}/P$ is the partial pressure fraction of the ith species (and $P_{i}$ is the partial pressure), $Y_{i}=\rho _{i}/\rho$ is the mass fraction of the ith species, and $\sum _{i}X_{i}=\sum _{i}Y_{i}=1$.
Diffusion of electrons in solids
When the density of electrons in solids is not in equilibrium, diffusion of electrons occurs. For example, when a bias is applied to two ends of a chunk of semiconductor, or a light shines on one end (see right figure), electron diffuse from high density regions (center) to low density regions (two ends), forming a gradient of electron density. This process generates current, referred to as diffusion current.
Diffusion current can also be described by Fick's first law
 $$
J = − D ∂ n / ∂ x , {\displaystyle 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 [length].
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, wavecut terrace/shoreline retreat, alluvial channel incision, coastal shelf retreat, and delta progradation^{[17]}. 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 the 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^{[18]}.
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 on in the left panel has a “random” motion, but this motion is not random as it 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.
Separation of diffusion from convection in gases
While Brownian motion of multimolecular 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 nontrivial 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 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 the temperature [inhomogeneity]. If ink is dropped in water, one usually observes an inhomogeneous evolution of the spatial distribution, which clearly indicates convection (caused, in particular, by this dropping).^{[citation needed]}
In contrast, heat conduction through solid media is an everyday occurrence (e.g. a metal spoon partly immersed in a hot liquid). This explains why the diffusion of heat was explained mathematically before the diffusion of mass.
Other types of diffusion
 Anisotropic diffusion, also known as the PeronaMalik equation, enhances high gradients
 Anomalous diffusion,^{[19]} in porous medium
 Atomic diffusion, in solids
 Eddy diffusion, in coarsegrained 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
 Levy flights and walks
 Momentum diffusion ex. the diffusion of the hydrodynamic velocity field
 Photon diffusion
 Plasma diffusion
 Random walk,^{[20]} 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
 Diffusionlimited aggregation
 False diffusion
 Isobaric counterdiffusion
 Sorption
 Osmosis
References
 ^ J.G. Kirkwood, R.L. Baldwin, P.J. Dunlop, L.J. Gosting, G. Kegeles (1960)Flow equations and frames of reference for isothermal diffusion in liquids. The Journal of Chemical Physics 33(5):1505–13.
 ^ J. Philibert (2005). One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2, 1.1–1.10.
 ^ S.R. De Groot, P. Mazur (1962). Nonequilibrium Thermodynamics. NorthHolland, Amsterdam.
 ^ A. Einstein (1905). "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen" (PDF). Ann. Phys. 17 (8): 549–60. Bibcode:1905AnP...322..549E. doi:10.1002/andp.19053220806.
 ^ Diffusion Processes, Thomas Graham Symposium, ed. J.N. Sherwood, A.V. Chadwick, W.M.Muir, F.L. Swinton, Gordon and Breach, London, 1971.
 ^ L.W. Barr (1997), In: Diffusion in Materials, DIMAT 96, ed. H.Mehrer, Chr. Herzig, N.A. Stolwijk, H. Bracht, Scitec Publications, Vol.1, pp. 1–9.
 ^ ^{a} ^{b} H. Mehrer; N.A. Stolwijk (2009). "Heroes and Highlights in the History of Diffusion" (PDF). Diffusion Fundamentals. 11 (1): 1–32.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} S. Chapman, T. G. Cowling (1970) The Mathematical Theory of Nonuniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases, Cambridge University Press (3rd edition), ISBN 052140844X.
 ^ J.F. Kincaid; H. Eyring; A.E. Stearn (1941). "The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid State". Chem. Rev. 28 (2): 301–65. doi:10.1021/cr60090a005.
 ^ ^{a} ^{b} ^{c} ^{d} ^{e} A.N. Gorban, H.P. Sargsyan and H.A. Wahab (2011). "Quasichemical Models of Multicomponent Nonlinear Diffusion". Mathematical Modelling of Natural Phenomena. 6 (5): 184–262. arXiv:1012.2908 . doi:10.1051/mmnp/20116509.
 ^ ^{a} ^{b} Onsager, L. (1931). "Reciprocal Relations in Irreversible Processes. I". Physical Review. 37 (4): 405–26. Bibcode:1931PhRv...37..405O. doi:10.1103/PhysRev.37.405.
 ^ L.D. Landau, E.M. Lifshitz (1980). Statistical Physics. Vol. 5 (3rd ed.). ButterworthHeinemann. ISBN 9780750633727.
 ^ S. Bromberg, K.A. Dill (2002), Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology, Garland Science, ISBN 0815320515.
 ^ T. Teorell (1935). "Studies on the "Diffusion Effect" upon Ionic Distribution. Some Theoretical Considerations". Proceedings of the National Academy of Sciences of the United States of America. 21 (3): 152–61. Bibcode:1935PNAS...21..152T. PMC 1076553 . PMID 16587950. doi:10.1073/pnas.21.3.152.
 ^ J. L. Vázquez (2006), The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press, ISBN 0198569033.
 ^ S. H. Lam (2006). "Multicomponent diffusion revisited" (PDF). Physics of Fluids. 18 (7): 073101. Bibcode:2006PhFl...18g3101L. doi:10.1063/1.2221312.
 ^ Pasternack, Gregory B.; Brush, Grace S.; Hilgartner, William B. (20010401). "Impact of historic landuse change on sediment delivery to a Chesapeake Bay subestuarine delta". Earth Surface Processes and Landforms. 26 (4): 409–27. ISSN 10969837. doi:10.1002/esp.189.
 ^ Gregory B. Pasternack. "Watershed Hydrology, Geomorphology, and Ecohydraulics :: TFD Modeling". pasternack.ucdavis.edu. Retrieved 20170612.
 ^ D. BenAvraham and S. Havlin (2000). Diffusion and Reactions in Fractals and Disordered Systems (PDF). Cambridge University Press. ISBN 0521622786.
 ^ Weiss, G. (1994). Aspects and Applications of the Random Walk. NorthHolland. ISBN 0444816062.
External links
 Diffusion in a Bipolar Junction Transistor Demo
 Diffusion Furnace for doping of semiconductor wafers. POCl3 doping of Silicon.
 A Java applet implementing Diffusion
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 Yavasoglu Irfan,Çakiroglu Umut
 Internal Medicine 53(1), 7576, 2014
 NAID 130003383775
 Corpus Callosum Atrophy in Patients with Hereditary Diffuse Leukoencephalopathy with Neuroaxonal Spheroids: An MRIbased Study
 Kinoshita Michiaki,Kondo Yasufumi,Yoshida Kunihiro,Fukushima Kazuhiro,Hoshi Kenichi,Ishizawa Keisuke,Araki Nobuo,Yazawa Ikuru,Washimi Yukihiko,Saitoh Banyu,Kira Junichi,Ikeda Shuichi
 Internal Medicine 53(1), 2127, 2014
 … Objective Hereditary diffuse leukoencephalopathy with neuroaxonal spheroids (HDLS) is an adultonset white matter disease that presents clinically with cognitive, mental and motor dysfunction. …
 NAID 130003383770
関連画像
■★リンクテーブル★
リンク元  「100Cases 96」「spread」「widespread」「pervasive」「広汎性」 
拡張検索  「diffusely」「diffuse double layer」「diffuse alveolar hemorrhage syndrome」「diffuse uterine adenomyosis」 
関連記事  「diffused」「diffusing」 
「100Cases 96」
 96
 35歳 女性
 【主訴】息切れ
 【現病歴】6ヶ月から次第に増悪する息切れを訴え来院。息切れは進行しており、今では同年代の人に比べ階段を登ったり平地を歩くのがおそい。3ヶ月間空咳が出現してきている。
 【既往歴】喘息(子供の頃、中等度)
 【家族歴】父：40歳時に胸部疾患(chest problem)で死亡(と、この患者は思いこんでいる)。
 【服用薬】パラセタモール。やせ薬は過去に使ったことがある
 【嗜好歴】タバコ：吸わない。飲酒：週に10 units以下。
 【職業歴】印刷会社(学校卒業後ずっと)
 【生活歴】8歳と10歳の子供がいる。ペットとして自宅にネコとウサギを飼育している。
 【身体所見】
 バチ指、貧血、チアノーゼは認めない。心血管系正常。呼吸器系で、両側性に肺拡張能低下。胸部打診音、および触覚振盪音は正常。聴診上、両肺の肺底部に呼気終期年発音を聴取する。
 【検査所見】
 呼吸機能試験
 測定値 予測値
 FEV1(L) 3.0 3.64.2
 FVC(L) 3.6 4.55.3
 FER(FEV1/FVC)(%) 83 7580
 PEF(L/min) 470 450550
 胸部単純X線写真
 胸部単純CT(肺野条件)
 Q1 診断は？Q2 追加の検査と治療法
 (解説)
 ・6ヶ月から次第に増悪する息切れ
 ・本当に6ヶ月から息切れが始まったかは分からない！！実際にはもっと前から存在する可能性を考えよう。
 ・喘息の既往歴 → また喘息か・・・
 ・聴診上笛音 weezingなく、また呼吸機能検査で閉塞性肺障害は認められず否定的
 ・職業性喘息 ← 肺の疾患では職業歴が重要なんだよ、うん。
 ・職業に関連した特定の物質に曝露され引き起こされる気管支喘息。
 ・気管支喘息だと閉塞性換気障害でしょっ。
 ・本症例の病態
 ・拘束性病変(restrictive problem)：拡張制限 ＋ ラ音(呼気時に閉鎖していた気道の再開通によるラ音。このラ音は(1)肺が硬い＋(2)肺容量低下による起こるんじゃ)
 ・呼吸機能検査の結果
 ・中等度の拘束性換気障害(FEV1とFVCの低下、slightly high ratio)
 → 硬い肺と胸郭を示唆している → 肺コンプライアンス低下じゃな。
 → 拡散能低下が予想される。
 ・胸部単純X線写真
 ・小さい肺野、中肺野～下肺野にかけて結節性網状陰影(nodular and reticular shadowing)を認める ← 教科書的には「線状網状影」
 ・HRCT
 ・胸膜下嚢胞形成(subspleural cyst formationの直訳。教科書的には「胸膜直下の蜂巣肺所見」)を伴う線維化
 これらの所見→diffusing pulmonary fibrosis(fibrosing alveolitis)
 ・肺の線維化病変を見たら限局性かびまん性かを見なさい！マジで？
 ・びまん性：diffuse fine pulmonary fibrosis
 ・原因：膠原病、薬剤性、中毒性、特発性
 ・限局性：例えば肺炎感染後の瘢痕
 ・IPF
 ・まれに家族性の病型有 → 本症例で父がchest problemで無くなっている事と関連があるかもしらん。
 ・IPFの良くある病型：UIP ＋ CT上胸膜下に認められる蜂巣肺
 ・膠原病に合併する場合、NSIPの様に広い範囲に斑状の病変が出現
 ・CT上、ground glass shadowingに見える所はactive cellular alveolitisであり、反応に反応する確率が高いことと関連している。
 ・追加の検査の目的：原因と合併症の検索、肺生検施行の有無を決定
 ・肺生検：経気管支鏡生検は試料が少ないために負荷。VATは良く使用されており、若年の肺組織を得るのに適切な方法。
 ・治療
 ・低用量～中等量の副腎皮質ステロイド ± 免疫抑制薬(アザチオプリン)：数ヶ月経過観察し、反応性を観察。
 ・UIP症例でこのレジメンに対する反応性は乏しく、治療による利益より重大な副作用を起こさないことが重要
 ・アセチルシステイン：antioxidant
 ・予後を改善するというエビデンス有り
 ・ステロイド＋アザチオプリン＋アセチルシステインというレジメンで用いられることがある
 ・肺移植
 ・本症例の様にナウでヤングな患者には適応を考慮しても良い
 ・予後
 ・疾患の進展速度は症例により様々
 ・6ヶ月で死亡する急性増悪も起こることがある。
 Progression rates are variable and an acute aggressive form with death in 6 months can occur.
 ・UIP症例では多くの場合2～3年かけて確実に進行していく。
 ■glossary
 clubbing n. バチ指
 restrictive ventilatory defect 拘束性換気障害
 transfer factor = 拡散能/拡散能力 diffusion capacity
 subpleural bleb 胸膜下嚢胞
 diffuse fine pulmonary fibrosis びまん性微細肺線維症
 warrant vt. (正式)(SVO/doing)S(事)からするとO(事)は「～することは」当然のことである(justify)
 relevant adj. 直接的に関連する、関連性のある(to)
 ground glass すりガラス
 'ground glass' shadowingすりガラス陰影
「spread」
 n.
 v.
 関
 broaden、diffuse、diffusion、diffusional、diffusive、expand、extend、prevalence、prevalent、propagate、propagation、spreading、stretch
WordNet ［license wordnet］
「prepared or arranged for a meal; especially having food set out; "a table spread with food"」WordNet ［license wordnet］
「a tasty mixture to be spread on bread or crackers or used in preparing other dishes」WordNet ［license wordnet］
「two facing pages of a book or other publication」 同
 spread head, spreadhead, facing pages
WordNet ［license wordnet］
「process or result of distributing or extending over a wide expanse of space」PrepTutorEJDIC ［license prepejdic］
「〈たたんだ物など〉‘を'『広げる』,伸ばす《+『out』+『名,』+『名』+『out』》 / (…に)‥‘を'『薄く塗る;』(…に)‥‘を'かける,かぶせる《+『名』+『on(over)』+『名』》 / (…を)…‘に'『薄く塗る;』(で)…‘を'おおう《+『名』+『with』+『名』》 / …‘を'引き離す,押し広げる / 《しばしば副詞[句]を伴って》(…に)…‘を'『まき散らす』;〈知識・ニュースなど〉‘を'広める;〈病気など〉‘を'まん延させる《+『名』+『over(among)』+『名』》 / 〈仕事・支払いなど〉‘を'引き延ばす《+『名』+『out,』+『out』+『名』》;(ある期間に)〈支払いなど〉‘を'わたらせる《+『名』+『over(for)』+『名』》 / …‘を'広げて見せる,展示する / (食事ができるように)〈食卓〉‘を'用意する,〈食卓〉‘に'料理を並べる / 〈物が〉『広がる,』伸びる / 《しばしば副詞[句]を伴って》(…に)〈うわさなどが〉『広まる』〈病気などが〉まん延する《+『over』+『名』》 / (時間的に)延びる;(…の期間に)わたる《+『over(for)』+『名』》 / 〈U〉《the~》(…が)『広がること』,(…の)普及《+『of』+『名』》 / 〈U〉《しばしばa~》(…の)広がり,広がった距離(程度)《+『of』+『名』》 / 〈C〉(食卓掛けなどの)掛け布;(特に)ベッドカバー / 〈C〉(食卓に出された)たくさんの食へ物,ごちそう / 〈U〉〈C〉スプレッド(パンに塗るバター,ジャム,ゼリー類) / 〈C〉(新聞,雑誌などの数段ぬきまたは2ページにわたる)大広告,大見出し記事」WordNet ［license wordnet］
「act of extending over a wider scope or expanse of space or time」WordNet ［license wordnet］
「the expansion of a person''s girth (especially at middle age); "she exercised to avoid that middleaged spread"」WordNet ［license wordnet］
「distribute or disperse widely; "The invaders spread their language all over the country"」WordNet ［license wordnet］
「spread across or over; "A big oil spot spread across the water"」WordNet ［license wordnet］
「become distributed or widespread; "the infection spread"; "Optimism spread among the population"」WordNet ［license wordnet］
「strew or distribute over an area; "He spread fertilizer over the lawn"; "scatter cards across the table"」 同
 scatter, spread out
WordNet ［license wordnet］
「cover by spreading something over; "spread the bread with cheese"」WordNet ［license wordnet］
「distribute over a surface in a layer; "spread cheese on a piece of bread"」
「widespread」
 adj.
 広範な、広汎な、広汎性の
WordNet ［license wordnet］
「widely circulated or diffused; "a widespread doctrine"; "widespread fear of nuclear war"」PrepTutorEJDIC ［license prepejdic］
「(翼など)広げた / 広範囲にわたる,行き渡った」「pervasive」
 pervas (普及する) + ive(～の傾向がある)
 adj.
 (やたらに)広がる。浸透性の。普及力のある
 (医)広汎性の
 関
 diffuse, impregnate, impregnation, osmosis, penetrate, penetration, permeance, permeate, permeation, prevail, prevalence, widespread
PrepTutorEJDIC ［license prepejdic］
「広がる,みなぎる;普及する」
「広汎性」
「diffusely」
 散在性に、拡散性に
 関
 diffuse、diffusibility、diffusible、diffusive、disseminated
WordNet ［license wordnet］
「in a diffuse manner; "the arteries were diffusely narrowed"」
「diffuse double layer」
「diffuse alveolar hemorrhage syndrome」
「diffuse uterine adenomyosis」
「diffused」
WordNet ［license wordnet］
「(of light rays) subjected to scattering by reflection from a rough surface or transmission through a translucent material; "diffused light"」
「diffusing」