Diffusion

Each model of diffusion expresses the diffusion flux with the use of concentrations, densities and their derivatives. Flux is a vector ${\displaystyle \mathbf {J} }$ representing the quantity and direction of transfer. Given a small area ${\displaystyle \Delta S}$ with normal ${\displaystyle {\boldsymbol {\nu }}}$, the transfer of a physical quantity ${\displaystyle N}$ through the area ${\displaystyle \Delta S}$ per time ${\displaystyle \Delta t}$ is

${\displaystyle \Delta N=(\mathbf {J} ,{\boldsymbol {\nu }})\,\Delta S\,\Delta t+o(\Delta S\,\Delta t)\,,}$

where ${\displaystyle (\mathbf {J} ,{\boldsymbol {\nu }})}$ is the inner product and ${\displaystyle o(\cdots )}$ is the little-o notation. If we use the notation of vector area ${\displaystyle \Delta \mathbf {S} ={\boldsymbol {\nu }}\,\Delta S}$ then

${\displaystyle \Delta N=(\mathbf {J} ,\Delta \mathbf {S} )\,\Delta t+o(\Delta \mathbf {S} \,\Delta t)\,.}$

The dimension of the diffusion flux is [flux] = [quantity]/([time]·[area]). The diffusing physical quantity ${\displaystyle N}$ may be the number of particles, mass, energy, electric charge, or any other scalar extensive quantity. For its density, ${\displaystyle n}$, the diffusion equation has the form

${\displaystyle {\frac {\partial n}{\partial t}}=-\nabla \cdot \mathbf {J} +W\,,}$

where ${\displaystyle W}$ is intensity of any local source of this quantity (for example, the rate of a chemical reaction). For the diffusion equation, the no-flux boundary conditions can be formulated as ${\displaystyle (\mathbf {J} (x),{\boldsymbol {\nu }}(x))=0}$ on the boundary, where ${\displaystyle {\boldsymbol {\nu }}}$ is the normal to the boundary at point ${\displaystyle x}$.

Fick's first law: the diffusion flux is proportional to the negative of the concentration gradient:

${\displaystyle \mathbf {J} =-D\,\nabla n\ ,\;\;J_{i}=-D{\frac {\partial n}{\partial x_{i}}}\ .}$

The corresponding diffusion equation (Fick's second law) is

${\displaystyle {\frac {\partial n(x,t)}{\partial t}}=\nabla \cdot (D\,\nabla n(x,t))=D\,\Delta n(x,t)\ ,}$

where ${\displaystyle \Delta }$ is the Laplace operator,

${\displaystyle \Delta n(x,t)=\sum _{i}{\frac {\partial ^{2}n(x,t)}{\partial x_{i}^{2}}}\ .}$

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, ${\displaystyle -\nabla n}$.

In 1931, Lars Onsager[13] included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For multi-component transport,

${\displaystyle \mathbf {J} _{i}=\sum _{j}L_{ij}X_{j}\,,}$

where ${\displaystyle \mathbf {J} _{i}}$ is the flux of the ith physical quantity (component) and ${\displaystyle X_{j}}$ 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 ${\displaystyle s}$ (he used the term "force" in quotation marks or "driving force"):

${\displaystyle X_{i}=\nabla {\frac {\partial s(n)}{\partial n_{i}}}\,,}$

where ${\displaystyle n_{i}}$ are the "thermodynamic coordinates". For the heat and mass transfer one can take ${\displaystyle n_{0}=u}$ (the density of internal energy) and ${\displaystyle n_{i}}$ is the concentration of the ${\displaystyle i}$th component. The corresponding driving forces are the space vectors

${\displaystyle X_{0}=\nabla {\frac {1}{T}}\ ,\;\;\;X_{i}=-\nabla {\frac {\mu _{i}}{T}}\;(i>0),}$ because ${\displaystyle \mathrm {d} s={\frac {1}{T}}\,\mathrm {d} u-\sum _{i\geq 1}{\frac {\mu _{i}}{T}}\,{\rm {d}}n_{i}}$

where T is the absolute temperature and ${\displaystyle \mu _{i}}$ is the chemical potential of the ${\displaystyle i}$th component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore, the terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients.

For the linear Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium:

${\displaystyle X_{i}=\sum _{k\geq 0}\left.{\frac {\partial ^{2}s(n)}{\partial n_{i}\,\partial n_{k}}}\right|_{n=n^{*}}\nabla n_{k}\ ,}$

where the derivatives of ${\displaystyle s}$ are calculated at equilibrium ${\displaystyle n^{*}}$. The matrix of the kinetic coefficients ${\displaystyle L_{ij}}$ should be symmetric (Onsager reciprocal relations) and positive definite (for the entropy growth).

The transport equations are

${\displaystyle {\frac {\partial n_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=-\sum _{j\geq 0}L_{ij}\operatorname {div} X_{j}=\sum _{k\geq 0}\left[-\sum _{j\geq 0}L_{ij}\left.{\frac {\partial ^{2}s(n)}{\partial n_{j}\,\partial n_{k}}}\right|_{n=n^{*}}\right]\,\Delta n_{k}\ .}$

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 ${\displaystyle D_{ik}}$ 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 = constant. The relevant thermodynamic potential is the free energy (or the free entropy). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, ${\displaystyle -(1/T)\,\nabla \mu _{j}}$, and the matrix of diffusion coefficients is

${\displaystyle D_{ik}={\frac {1}{T}}\sum _{j\geq 1}L_{ij}\left.{\frac {\partial \mu _{j}(n,T)}{\partial n_{k}}}\right|_{n=n^{*}}}$

(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 ${\textstyle \sum _{j}L_{ij}X_{j}}$ can be measured. For example, in the original work of Onsager[13] the thermodynamic forces include additional multiplier T, whereas in the Course of Theoretical Physics[14] 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.

The formalism of linear irreversible thermodynamics (Onsager) generates the systems of linear diffusion equations in the form

${\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}\,\Delta c_{j}.}$

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 non-diagonal, for example, ${\displaystyle D_{12}\neq 0}$, and consider the state with ${\displaystyle c_{2}=\cdots =c_{n}=0}$. At this state, ${\displaystyle \partial c_{2}/\partial t=D_{12}\,\Delta c_{1}}$. If ${\displaystyle D_{12}\,\Delta c_{1}(x)<0}$ at some points, then ${\displaystyle c_{2}(x)}$ 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.[12]

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)[15]

${\displaystyle D={\frac {\mu \,k_{\text{B}}T}{q}},}$

where D is the diffusion constant, μ is the "mobility", k_B is Boltzmann's constant, T is the absolute temperature, and q is the elementary charge, that is, the charge of one electron.

Below, to combine in the same formula the chemical potential μ and the mobility, we use for mobility the notation ${\displaystyle {\mathfrak {m}}}$.

The mobility-based approach was further applied by T. Teorell.[16] 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: ${\displaystyle -RT{\frac {1}{n}}\,\nabla n=-RT\,\nabla (\ln(n/n^{\text{eq}}))}$.
2. Electrostatic force caused by electric potential gradient: ${\displaystyle q\,\nabla \varphi }$.

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[12]

${\displaystyle \mathbf {J} ={\mathfrak {m}}\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)(-\nabla \mu +({\text{external force per mole}})),}$

where μ is the chemical potential, μ₀ is the standard value of the chemical potential. The expression ${\displaystyle a=\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)}$ 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[12]

${\displaystyle \mathbf {J} ={\mathfrak {m}}a(-\nabla \mu +({\text{external force per mole}})).}$

The standard derivation of the activity includes a normalization factor and for small concentrations ${\displaystyle a=n/n^{\ominus }+o(n/n^{\ominus })}$, where ${\displaystyle n^{\ominus }}$ is the standard concentration. Therefore, this formula for the flux describes the flux of the normalized dimensionless quantity ${\displaystyle n/n^{\ominus }}$:

${\displaystyle {\frac {\partial (n/n^{\ominus })}{\partial t}}=\nabla \cdot [{\mathfrak {m}}a(\nabla \mu -({\text{external force per mole}}))].}$

Fluctuation-dissipation theorem based on the Langevin equation is developed to extend the Einstein model to the ballistic time scale.[17] According to Langevin, the equation is based on Newton's second law of motion as

${\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-{\frac {1}{\mu }}{\frac {dx}{dt}}+F(t)}$

where

• x is the position.
• μ is the mobility of the particle in the fluid or gas, which can be calculated using the Einstein relation (kinetic theory).
• m is the mass of the particle.
• F is the random force applied to the particle.
• t is time.

Solving this equation, one obtained the time-dependent diffusion constant in the long-time limit and when the particle is significantly denser than the surrounding fluid,[17]

${\displaystyle D(t)=\mu \,k_{\rm {B}}T(1-e^{-t/(m\mu )})}$

where

• k_B is Boltzmann's constant;
• T is the absolute temperature.
• μ is the mobility of the particle in the fluid or gas, which can be calculated using the Einstein relation (kinetic theory).
• m is the mass of the particle.
• t is time.

The Teorell formula with combination of Onsager's definition of the diffusion force gives

${\displaystyle \mathbf {J} _{i}={\mathfrak {m_{i}}}a_{i}\sum _{j}L_{ij}X_{j},}$

where ${\displaystyle {\mathfrak {m_{i}}}}$ is the mobility of the ith component, ${\displaystyle a_{i}}$ is its activity, ${\displaystyle L_{ij}}$ is the matrix of the coefficients, ${\displaystyle X_{j}}$ is the thermodynamic diffusion force, ${\displaystyle X_{j}=-\nabla {\frac {\mu _{j}}{T}}}$. For the isothermal perfect systems, ${\displaystyle X_{j}=-R{\frac {\nabla n_{j}}{n_{j}}}}$. Therefore, the Einstein–Teorell approach gives the following multicomponent generalization of the Fick's law for multicomponent diffusion:

${\displaystyle {\frac {\partial n_{i}}{\partial t}}=\sum _{j}\nabla \cdot \left(D_{ij}{\frac {n_{i}}{n_{j}}}\nabla n_{j}\right),}$

where ${\displaystyle D_{ij}}$ 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.

Diffusion in the monolayer: oscillations near temporary equilibrium positions and jumps to the nearest free places.

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 ${\displaystyle A_{1},A_{2},\ldots ,A_{m}}$ on the surface. Their surface concentrations are ${\displaystyle c_{1},c_{2},\ldots ,c_{m}.}$ 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 ${\displaystyle z=c_{0}}$. The sum of all ${\displaystyle c_{i}}$ (including free places) is constant, the density of adsorption places b.

The jump model gives for the diffusion flux of ${\displaystyle A_{i}}$ (i = 1, ..., n):

${\displaystyle \mathbf {J} _{i}=-D_{i}[z\,\nabla c_{i}-c_{i}\nabla z]\,.}$

The corresponding diffusion equation is:[12]

${\displaystyle {\frac {\partial c_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=D_{i}[z\,\Delta c_{i}-c_{i}\,\Delta z]\,.}$

Due to the conservation law, ${\displaystyle z=b-\sum _{i=1}^{n}c_{i}\,,}$ and we have the system of m diffusion equations. For one component we get Fick's law and linear equations because ${\displaystyle (b-c)\,\nabla c-c\,\nabla (b-c)=b\,\nabla c}$. 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

${\displaystyle \mathbf {J} _{i}=-\sum _{j}D_{ij}[c_{j}\,\nabla c_{i}-c_{i}\,\nabla c_{j}]}$

${\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}[c_{j}\,\Delta c_{i}-c_{i}\,\Delta c_{j}]}$

where ${\displaystyle D_{ij}=D_{ji}\geq 0}$ is a symmetric matrix of coefficients that characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration ${\displaystyle c_{0}}$.

Various versions of these jump models are also suitable for simple diffusion mechanisms in solids.

For diffusion in porous media the basic equations are (if Φ is constant):[18]

${\displaystyle \mathbf {J} =-\phi D\,\nabla n^{m}}$

${\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\,,}$

where D is the diffusion coefficient, Φ is porosity, n is the concentration, m > 0 (usually m > 1, the case m = 1 corresponds to Fick's law).

Care must be taken to properly account for the porosity (Φ) of the porous medium in both the flux terms and the accumulation terms.[19] For example, as the porosity goes to zero, the molar flux in the porous medium goes to zero for a given concentration gradient. Upon applying the divergence of the flux, the porosity terms cancel out and the second equation above is formed.

For diffusion of gases in porous media this equation is the formalization of Darcy's law: the volumetric flux of a gas in the porous media is

${\displaystyle q=-{\frac {k}{\mu }}\,\nabla p}$

where k is the permeability of the medium, μ is the viscosity and p is the pressure.

The advective molar flux is given as

J = nq

and for ${\displaystyle p\sim n^{\gamma }}$ Darcy's law gives the equation of diffusion in porous media with m = γ + 1.

In porous media, the average linear velocity (ν), is related to the volumetric flux as:

${\displaystyle \upsilon =q/\phi }$

Combining the advective molar flux with the diffusive flux gives the advection dispersion equation

${\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\ -\nu \cdot \nabla n^{m},}$

For underground water infiltration, the Boussinesq approximation gives the same equation with m = 2.

For plasma with the high level of radiation, the Zeldovich–Raizer equation gives m > 4 for the heat transfer.

Random collisions of particles in a gas.

The diffusion coefficient ${\displaystyle D}$ is the coefficient in the Fick's first law ${\displaystyle J=-D\,\partial n/\partial x}$, 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].

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

${\displaystyle D={\frac {1}{3}}\ell v_{T}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}$

where k_B is the Boltzmann constant, T is the temperature, P is the pressure, ${\displaystyle \ell }$ is the mean free path, and v_T is the mean thermal speed:

${\displaystyle \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}$

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:

${\displaystyle D_{\rm {AB}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}}}}{\sqrt {{\frac {1}{2m_{\rm {A}}}}+{\frac {1}{2m_{\rm {B}}}}}}{\frac {4T^{3/2}}{P(d_{\rm {A}}+d_{\rm {B}})^{2}}}\,,}$

In Boltzmann's kinetics of the mixture of gases, each gas has its own distribution function, ${\displaystyle f_{i}(x,c,t)}$, 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 ${\textstyle C_{i}(x,t)={\frac {1}{n_{i}}}\int _{c}cf(x,c,t)\,dc}$. If the velocities ${\displaystyle C_{i}(x,t)}$ 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:[10]

• individual concentrations of particles, ${\textstyle n_{i}(x,t)=\int _{c}f_{i}(x,c,t)\,dc}$ (particles per volume),
• density of momentum ${\textstyle \sum _{i}m_{i}n_{i}C_{i}(x,t)}$ (m_i is the ith particle mass),
• density of kinetic energy
  $${\displaystyle \sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right).}$$

  

The kinetic temperature T and pressure P are defined in 3D space as

${\displaystyle {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc;\quad P=k_{\rm {B}}nT,}$

where ${\textstyle n=\sum _{i}n_{i}}$ is the total density.

For two gases, the difference between velocities, ${\displaystyle C_{1}-C_{2}}$ is given by the expression:[10]

${\displaystyle C_{1}-C_{2}=-{\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}-m_{1})}{Pn(m_{1}n_{1}+m_{2}n_{2})}}\nabla P-{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\},}$

where ${\displaystyle F_{i}}$ is the force applied to the molecules of the ith component and ${\displaystyle k_{T}}$ 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. ${\displaystyle \nabla \,\left({\frac {n_{1}}{n}}\right)}$ 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. ${\displaystyle {\frac {n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P}$ 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;
3. ${\displaystyle {\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})}$ 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.
4. ${\displaystyle k_{T}{\frac {1}{T}}\nabla T}$ 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,[10]

• $${\displaystyle D_{12}={\frac {3}{2n(d_{1}+d_{2})^{2}}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}}$$

  
  for rigid spheres;
• $${\displaystyle D_{12}={\frac {3}{8nA_{1}({\nu })\Gamma (3-{\frac {2}{\nu -1}})}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}\left({\frac {2kT}{\kappa _{12}}}\right)^{\frac {2}{\nu -1}}}$$

  
  for repulsing force ${\displaystyle \kappa _{12}r^{-\nu }.}$

The number ${\displaystyle A_{1}({\nu })}$ is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book[10])

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:

${\displaystyle V={\frac {\sum _{i}\rho _{i}C_{i}}{\rho }}\,.}$

where ${\displaystyle \rho _{i}=m_{i}n_{i}}$ is the mass concentration of the ith species, ${\textstyle \rho =\sum _{i}\rho _{i}}$ is the mass density.

By definition, the diffusion velocity of the ith component is ${\displaystyle v_{i}=C_{i}-V}$, ${\textstyle \sum _{i}\rho _{i}v_{i}=0}$. The mass transfer of the ith component is described by the continuity equation

${\displaystyle {\frac {\partial \rho _{i}}{\partial t}}+\nabla (\rho _{i}V)+\nabla (\rho _{i}v_{i})=W_{i}\,,}$

where ${\displaystyle W_{i}}$ is the net mass production rate in chemical reactions, ${\textstyle \sum _{i}W_{i}=0}$.

In these equations, the term ${\displaystyle \nabla (\rho _{i}V)}$ describes advection of the ith component and the term ${\displaystyle \nabla (\rho _{i}v_{i})}$ 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.[20] For the diffusion velocities in multicomponent gases (N components) they used

${\displaystyle v_{i}=-\left(\sum _{j=1}^{N}D_{ij}\mathbf {d} _{j}+D_{i}^{(T)}\,\nabla (\ln T)\right)\,;}$

${\displaystyle \mathbf {d} _{j}=\nabla X_{j}+(X_{j}-Y_{j})\,\nabla (\ln P)+\mathbf {g} _{j}\,;}$

${\displaystyle \mathbf {g} _{j}={\frac {\rho }{P}}\left(Y_{j}\sum _{k=1}^{N}Y_{k}(f_{k}-f_{j})\right)\,.}$

Here, ${\displaystyle D_{ij}}$ is the diffusion coefficient matrix, ${\displaystyle D_{i}^{(T)}}$ is the thermal diffusion coefficient, ${\displaystyle f_{i}}$ is the body force per unit mass acting on the ith species, ${\displaystyle X_{i}=P_{i}/P}$ is the partial pressure fraction of the ith species (and ${\displaystyle P_{i}}$ is the partial pressure), ${\displaystyle Y_{i}=\rho _{i}/\rho }$ is the mass fraction of the ith species, and ${\textstyle \sum _{i}X_{i}=\sum _{i}Y_{i}=1.}$

As carriers are generated (green:electrons and purple:holes) due to light shining at the center of an intrinsic semiconductor, they diffuse towards two ends. Electrons have higher diffusion constant than holes leading to fewer excess electrons at the center as compared to holes.

When the density of electrons in solids is not in equilibrium, diffusion of electrons occurs. For example, when a bias is applied to two ends of a chunk of semiconductor, or a light shines on one end (see right figure), electrons diffuse from high density regions (center) to low density regions (two ends), forming a gradient of electron density. This process generates current, referred to as diffusion current.

Diffusion current can also be described by Fick's first law

${\displaystyle J=-D\,\partial n/\partial x\,,}$

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].

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.[21] 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.[22]

Schematic of semipermeable membrane during hemodialysis, where blood is red, dialysing fluid is blue, and the membrane is yellow.

Dialysis works on the principles of the diffusion of solutes and ultrafiltration of fluid across a semi-permeable membrane. Diffusion is a property of substances in water; substances in water tend to move from an area of high concentration to an area of low concentration.[23] Blood flows by one side of a semi-permeable membrane, and a dialysate, or special dialysis fluid, flows by the opposite side. A semipermeable membrane is a thin layer of material that contains holes of various sizes, or pores. Smaller solutes and fluid pass through the membrane, but the membrane blocks the passage of larger substances (for example, red blood cells and large proteins). This replicates the filtering process that takes place in the kidneys when the blood enters the kidneys and the larger substances are separated from the smaller ones in the glomerulus.[23]

While Brownian motion of multi-molecular mesoscopic particles (like pollen grains studied by Brown) is observable under an optical microscope, molecular diffusion can only be probed in carefully controlled experimental conditions. Since Graham experiments, it is well known that avoiding of convection is necessary and this may be a non-trivial task.

Under normal conditions, molecular diffusion dominates only at lengths in the nanometre-to-millimetre range. On larger length scales, transport in liquids and gases is normally due to another transport phenomenon, convection. To separate diffusion in these cases, special efforts are needed.

Therefore, some often cited examples of diffusion are wrong: If cologne is sprayed in one place, it can soon be smelled in the entire room, but a simple calculation shows that this can't be due to diffusion. Convective motion persists in the room because of the temperature [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 (for example, a metal spoon partly immersed in a hot liquid). This explains why the diffusion of heat was explained mathematically before the diffusion of mass.

• Anisotropic diffusion, also known as the Perona–Malik equation, enhances high gradients
• Atomic diffusion, in solids
• Bohm diffusion, spread of plasma across magnetic fields
• Eddy diffusion, in coarse-grained description of turbulent flow
• Effusion of a gas through small holes
• Electronic diffusion, resulting in an electric current called the diffusion current
• Facilitated diffusion, present in some organisms
• Gaseous diffusion, used for isotope separation
• Heat equation, diffusion of thermal energy
• Itō diffusion, mathematisation of Brownian motion, continuous stochastic process.
• Knudsen diffusion of gas in long pores with frequent wall collisions
• Lévy flight
• Molecular diffusion, diffusion of molecules from more dense to less dense areas
• Momentum diffusion ex. the diffusion of the hydrodynamic velocity field
• Photon diffusion
• Plasma diffusion
• Random walk,[24] model for diffusion
• Reverse diffusion, against the concentration gradient, in phase separation
• Rotational diffusion, random reorientation of molecules
• Surface diffusion, diffusion of adparticles on a surface
• Taxis is an animal's directional movement activity in response to a stimulus
  • Kinesis is an animal's non-directional movement activity in response to a stimulus
• Trans-cultural diffusion, diffusion of cultural traits across geographical area
• Turbulent diffusion, transport of mass, heat, or momentum within a turbulent fluid