The Maxwell–Stefan diffusion (or Stefan–Maxwell diffusion) is a

model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...

for describing

diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical ...

in multicomponent systems. The equations that describe these transport processes have been developed independently and in parallel by

James Clerk Maxwell James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...

J. C. Maxwell: ''On the dynamical theory of gases'', The Scientific Papers of J. C. Maxwell, 1965, 2, 26–78. for dilute gases and Josef StefanJ. Stefan: ''Über das Gleichgewicht und Bewegung, insbesondere die Diffusion von Gemischen'', Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien, 2te Abteilung a, 1871, 63, 63-124. for liquids. The Maxwell–Stefan equation is

\frac
= \nabla \ln a_i
= \sum_^
= \sum_^

* ∇: vector differential operator * χ:

Mole fraction In chemistry, the mole fraction or molar fraction (''xi'' or ) is defined as unit of the amount of a constituent (expressed in moles), ''ni'', divided by the total amount of all constituents in a mixture (also expressed in moles), ''n''tot. This ex ...

* μ:

Chemical potential In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...

* a: Activity * i, j: Indexes for component i and j * n: Number of components *

\mathfrak_

: Maxwell–Stefan-diffusion coefficient *

\vec v_i

: Diffusion velocity of component i *

c_i

Molar concentration Molar concentration (also called molarity, amount concentration or substance concentration) is a measure of the concentration of a chemical species, in particular of a solute in a solution, in terms of amount of substance per unit volume of sol ...

of component i * c: Total molar concentration *

\vec J_i

Flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...

of component i The equation assumes

steady state In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ''p' ...

, i.e., the neglect of time derivatives in the velocity. The basic assumption of the theory is that a deviation from equilibrium between the molecular friction and thermodynamic interactions leads to the diffusion flux.S. Rehfeldt, J. Stichlmair: ''Measurement and calculation of multicomponent diffusion coefficients in liquids'', Fluid Phase Equilibria, 2007, 256, 99–104 The molecular friction between two components is proportional to their difference in speed and their mole fractions. In the simplest case, the

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

of chemical potential is the driving force of diffusion. For complex systems, such as electrolytic solutions, and other drivers, such as a pressure gradient, the equation must be expanded to include additional terms for interactions. A major disadvantage of the Maxwell–Stefan theory is that the

diffusion coefficient Diffusivity, mass diffusivity or diffusion coefficient is a proportionality constant between the molar flux due to molecular diffusion and the gradient in the concentration of the species (or the driving force for diffusion). Diffusivity is enc ...

s, with the exception of the diffusion of dilute gases, do not correspond to the Fick's diffusion coefficients and are therefore not tabulated. Only the diffusion coefficients for the binary and ternary case can be determined with reasonable effort. In a multicomponent system, a set of approximate formulas exist to predict the Maxwell–Stefan-diffusion coefficient. The Maxwell–Stefan theory is more comprehensive than the "classical" Fick's diffusion theory, as the former does not exclude the possibility of negative diffusion coefficients. It is possible to derive Fick's theory from the Maxwell–Stefan theory.

References

{{DEFAULTSORT:Maxwell-Stefan diffusion Diffusion James Clerk Maxwell

See also

References