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# Diffusion

in a glass of water. At first, the particles are all near one corner of the glass. If the particles randomly move around ("diffuse") in the water, they eventually become distributed randomly and uniformly from an area of high concentration to an area of low concentration, and organized (diffusion continues, but with no net flux).]] molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container. Top: A single molecule moves around randomly. Middle: With more molecules, there is a statistical trend that the solute fills the container more and more uniformly. Bottom: With an enormous number of solute molecules, all randomness is gone: The solute appears to move smoothly and deterministically from high-concentration areas to low-concentration areas. There is no microscopic force pushing molecules rightward, but there appears to be one in the bottom panel. This apparent force is called an entropic force.]] Diffusion is the net movement of molecules or atoms from a region of high concentration with high chemical potential to a region of low concentration with 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, or temperature with the change in another variable, usually distance. 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 dependends 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. 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. The term convection is used to describe the combination of both transport phenomena.

## 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 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.J. Philibert (2005). One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2, 1.1–1.10. 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 non-equilibrium thermodynamics.S.R. De Groot, P. Mazur (1962). Non-equilibrium Thermodynamics. North-Holland, Amsterdam. From the atomistic point of view, diffusion is considered as a result of the random walk of the diffusing particles. In molecular diffusion, the moving molecules are self-propelled by thermal energy. Random walk of small particles in suspension in a fluid was discovered in 1827 by Robert Brown. The theory of the Brownian motion and the atomistic backgrounds of diffusion were developed by Albert Einstein. 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 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 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.

## 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 \Delta N = (\mathbf{J},\nu) \,\Delta S \,\Delta t +o(\Delta S \,\Delta t)\, , where (\mathbf{J},\nu) is the inner product and o(\cdots) is the little-o notation. If we use the notation of vector area \Delta \mathbf{S}=\nu \, \Delta S then \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 \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 no-flux 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: \mathbf{J}=-D \,\nabla n \ , \;\; J_i=-D \frac{\partial n}{\partial x_i} \ . The corresponding diffusion equation (Fick's second law) is \frac{\partial n(x,t)}{\partial t}=\nabla\cdot( D \,\nabla n(x,t))=D \, \Delta n(x,t)\ , where \Delta is the Laplace operator, \Delta n(x,t) = \sum_i \frac{\partial^2 n(x,t)}{\partial x_i^2} \ .

### Nondiagonal diffusion must be nonlinear

The formalism of linear irreversible thermodynamics (Onsager) generates the systems of linear diffusion equations in the form \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 non-diagonal, for example, D_{12} \neq 0, and consider the state with c_2 = \cdots = 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 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) connects the diffusion coefficient and the mobility (the ratio of the particle's terminal drift velocity to an applied force)S. Bromberg, K.A. Dill (2002), Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology, Garland Science, . D = \mu \, k_\text{B} T, where D is the diffusion constant, μ is the "mobility", kB 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. 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 of ions (particles). The common modern term is mole. The force under isothermal conditions consists of two parts:
1. Diffusion force caused by concentration gradient: -RT \frac{1}{n} \, \nabla n = -RT \, \nabla (\ln(n/n^\text{eq})).
2. Electrostatic force caused by electric potential gradient: q \, \nabla \varphi.
Here R is the gas constant, T is the absolute temperature, n is the concentration, the equilibrium concentration is marked by a superscript "eq", q is the charge and φ is the electric potential. The simple but crucial difference between the Teorell formula and the Onsager laws is the concentration factor in the Teorell expression for the flux. In the Einstein–Teorell approach, If for the finite force the concentration tends to zero then the flux also tends to zero, whereas the Onsager equations violate this simple and physically obvious rule. The general formulation of the Teorell formula for non-perfect systems under isothermal conditions is \mathbf{J} = \mathfrak{m} \exp\left(\frac{\mu - \mu_0}{RT}\right)(-\nabla \mu + (\text{external force per mole})), where μ is the chemical potential, μ0 is the standard value of the chemical potential. The expression a = \exp\left(\frac{\mu - \mu_0}{RT}\right) is the so-called activity. It measures the "effective concentration" of a species in a non-ideal mixture. In this notation, the Teorell formula for the flux has a very simple form \mathbf{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: \frac{\partial (n/n^\ominus)}{\partial t} = \nabla \cdot 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 \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: \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): \mathbf{J}_i=-D_i \, \nabla c_i - c_i \nabla z\, . The corresponding diffusion equation is: \frac{\partial c_i}{\partial t}=- \operatorname{div}\mathbf{J}_i=D_i \, \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 (b-c) \,\nabla c- c\,\nabla(b-c) = b\,\nabla c. For two and more components the equations are nonlinear. If all particles can exchange their positions with their closest neighbours then a simple generalization gives \mathbf{J}_i=-\sum_j D_{ij} \,\nabla c_i - c_i \,\nabla c_j \frac{\partial c_i}{\partial t}=\sum_j D_{ij} \, \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:J. L. Vázquez (2006), The Porous Medium Equation. Mathematical Theory, Oxford Univ. Press, . \mathbf{J}=- D \,\nabla n^m \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=-\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 Zeldovich–Raizer equation gives m > 4 for the heat transfer. As we know that in porous media there are pores as well as solid material existing to gather all the way along the spacial dimensions of the porous media. So, there exist a combination of diffusion mechanisms in poroous media. To understand this, lets consider the example of ceramic membranes. when a gas enters into a ceramic membrane it undergoes the molecular diffusion when moving through the material of the porous media and knudsen diffusion ( if the diameter of pore is less than the length of the pore) and molecular difussion ( can be self diffusion or mutual diffusion depending on the number of species being diffused) in series when moving in the pores. while moving in the pore the poiseuille diffusion also happens in parrllel with knudsen and molecular diffusion. Depending on the nature of material and operating conditions one or more types of diffusion may not be aplicable be small and can be neglected but basic phenomenon is the same. for more information see

## 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 ( self-diffusion). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient D=\frac{1}{3} \ell v_T = \frac{2}{3}\sqrt{\frac{k_{\rm B}^3}{\pi^3 m}} \frac{T^{3/2}}{Pd^2}\, , where kB is the Boltzmann constant, T is the temperature, P is the pressure, \ell is the mean free path, and vT is the mean thermal speed: \ell = \frac{k_{\rm B}T}{\sqrt 2 \pi d^2 P}\, , \;\;\; v_T=\sqrt{\frac{8k_{\rm B}T}{\pi m}}\, . We can see that the diffusion coefficient in the mean free path approximation grows with T as T3/2 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 T1/2 and for given temperature it decreases with the total concentration as 1/n. For two different gases, A and B, with molecular masses mA, mB and molecular diameters dA, dB, the mean free path estimate of the diffusion coefficient of A in B and B in A is: 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 c f(x,c,t) \, dc. If the velocities C_i(x,t) do not coincide then there exists diffusion. In the Chapman–Enskog approximation, all the distribution functions are expressed through the densities of the conserved quantities:
• individual concentrations of particles, n_i(x,t)=\int_c f_i(x,c,t)\, dc (particles per volume),
• density of momentum \sum_i m_i n_i C_i(x,t) (mi is the ith particle mass),
• density of kinetic energy
: \sum_i \left( n_i\frac{m_i C^2_i(x,t)}{2} + \int_c \frac{m_i (c_i-C_i(x,t))^2}{2} f_i(x,c,t)\, dc \right). The kinetic temperature T and pressure P are defined in 3D space as \frac{3}{2}k_{\rm B} T=\frac{1}{n} \int_c \frac{m_i (c_i-C_i(x,t))^2}{2} f_i(x,c,t)\, dc; \quad P=k_{\rm B}nT, where n=\sum_i n_i is the total density. For two gases, the difference between velocities, C_1-C_2 is given by the expression: C_1-C_2=-\frac{n^2}{n_1n_2}D_{12}\left\{ \nabla \left(\frac{n_1}{n} \right)+ \frac{n_1n_2 (m_2-m_1)}{P n (m_1n_1+m_2n_2)}\nabla P- \frac{m_1n_1m_2n_2}{P(m_1n_1+m_2n_2)}(F_1-F_2)+k_T \frac{1}{T}\nabla T\right\}, where F_i is the force applied to the molecules of the ith component and k_T is the thermodiffusion ratio. The coefficient D12 is positive. This is the diffusion coefficient. Four terms in the formula for C1-C2 describe four main effects in the diffusion of gases:
1. \nabla \,\left(\frac{n_1}{n}\right) describes the flux of the first component from the areas with the high ratio n1/n to the areas with lower values of this ratio (and, analogously the flux of the second component from high n2/n to low n2/n because n2/n = 1 – n1/n);
2. \frac{n_1n_2 (m_2-m_1)}{n (m_1n_1+m_2n_2)}\nabla P describes the flux of the heavier molecules to the areas with higher pressure and the lighter molecules to the areas with lower pressure, this is barodiffusion;
3. \frac{m_1n_1m_2n_2}{P(m_1n_1+m_2n_2)}(F_1-F_2) describes diffusion caused by the difference of the forces applied to molecules of different types. For example, in the Earth's gravitational field, the heavier molecules should go down, or in electric field the charged molecules should move, until this effect is not equilibrated by the sum of other terms. This effect should not be confused with barodiffusion caused by the pressure gradient.
4. k_T \frac{1}{T}\nabla T describes thermodiffusion, the diffusion flux caused by the temperature gradient.
All these effects are called diffusion because they describe the differences between velocities of different components in the mixture. Therefore, these effects cannot be described as a bulk transport and differ from advection or convection. In the first approximation,
• D_{12}=\frac{3}{2n(d_1+d_2)^2}\left m_1m_2} \right^{1/2} for rigid spheres;
• D_{12}=\frac{3}{8nA_1({\nu})\Gamma(3-\frac{2}{\nu-1})} \left m_1m_2}\right^{1/2} \left(\frac{2kT}{\kappa_{12}} \right)^{\frac{2}{\nu-1}} for repulsing force \kappa_{12}r^{-\nu}.
The number A_1({\nu}) is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book) We can see that the dependence on T for the rigid spheres is the same as for the simple mean free path theory but for the power repulsion laws the exponent is different. Dependence on a total concentration n for a given temperature has always the same character, 1/n. In applications to gas dynamics, the diffusion flux and the bulk flow should be joined in one system of transport equations. The bulk flow describes the mass transfer. Its velocity V is the mass average velocity. It is defined through the momentum density and the mass concentrations: V=\frac{\sum_i \rho_i C_i} \rho \, . where \rho_i =m_i n_i is the mass concentration of the ith species, \rho=\sum_i \rho_i is the mass density. By definition, the diffusion velocity of the ith component is v_i=C_i-V, \sum_i \rho_i v_i=0. The mass transfer of the ith component is described by the continuity equation \frac{\partial \rho_i}{\partial t}+\nabla(\rho_i V) + \nabla (\rho_i v_i) = W_i \, , where W_i is the net mass production rate in chemical reactions, \sum_i W_i= 0. In these equations, the term \nabla(\rho_i V) describes advection of the ith component and the term \nabla (\rho_i v_i) represents diffusion of this component. In 1948, Wendell H. Furry proposed to use the form of the diffusion rates found in kinetic theory as a framework for the new phenomenological approach to diffusion in gases. This approach was developed further by F.A. Williams and S.H. Lam. For the diffusion velocities in multicomponent gases (N components) they used v_i=-\left(\sum_{j=1}^N D_{ij}\mathbf{d}_j + D_i^{(T)} \, \nabla (\ln T) \right)\, ; \mathbf{d}_j=\nabla X_j + (X_j-Y_j)\,\nabla (\ln P) + \mathbf{g}_j\, ; \mathbf{g}_j=\frac{\rho}{P} \left( Y_j \sum_{k=1}^N Y_k (f_k-f_j) \right)\, . Here, D_{ij} is the diffusion coefficient matrix, D_i^{(T)} is the thermal diffusion coefficient, f_i is the body force per unite mass acting on the ith species, X_i=P_i/P is the partial pressure fraction of the ith species (and P_i is the partial pressure), Y_i=\rho_i/\rho is the mass fraction of the ith species, and \sum_i X_i=\sum_i Y_i=1.

### 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 \, \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, wave-cut terrace/shoreline retreat, alluvial channel incision, coastal shelf retreat, and delta progradation. Although the Earth's surface is not literally diffusing in many of these cases, the process of diffusion effectively mimics the holistic changes that occur over decades to millennia. Diffusion models may also be used to solve inverse boundary value problems in which some information about the depositional environment is known from paleoenvironmental reconstruction and the diffusion equation is used to figure out the sediment influx and time series of landform changes.

## 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 multi-molecular mesoscopic particles (like pollen grains studied by Brown) is observable under an optical microscope, molecular diffusion can only be probed in carefully controlled experimental conditions. Since Graham experiments, it is well known that avoiding of convection is necessary and this may be a non-trivial task. Under normal conditions, molecular diffusion dominates only on length scales between nanometer and millimeter. On larger length scales, transport in liquids and gases is normally due to another transport phenomenon, convection, and to study diffusion on the larger scale, special efforts are needed. Therefore, some often cited examples of diffusion are wrong: If cologne is sprayed in one place, it 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). 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.