Maxwell–Stefan diffusion

The Maxwell–Stefan diffusion (or Stefan–Maxwell diffusion) is a model for describing diffusion in multicomponent systems. The equations that describe these transport processes have been developed independently and in parallel by James Clerk Maxwell for dilute gases and Josef Stefan for liquids. The Maxwell–Stefan equation is a i ∇ μ i R T = ∇ a i = ∑ j = 1 j ≠ i n χ j D i j ( v → j − v → i ) = ∑ j = 1 j ≠ i n c j c D i j ( J → j c j − J → i c i ) {\displaystyle a_{i}{\frac {\nabla \mu _{i}}{R\,T}}=\nabla a_{i}=\sum _{j=1 \atop j\neq i}^{n}{{\frac {\chi _{j}}{{\mathfrak {D}}_{ij}}}({\vec {v}}_{j}-{\vec {v}}_{i})}=\sum _{j=1 \atop j\neq i}^{n}{{\frac {c_{j}}{c{\mathfrak {D}}_{ij}}}\left({\frac {{\vec {J}}_{j}}{c_{j}}}-{\frac {{\vec {J}}_{i}}{c_{i}}}\right)}} ∇: vector differential operator χ: Mole fraction μ: Chemical potential a: Activity i, j: Indexes for component i and j n: Number of components D i j {\displaystyle {\mathfrak {D}}_{ij}} : Maxwell–Stefan-diffusion coefficient v → i {\displaystyle {\vec {v}}_{i}} : Diffusion velocity of component i c i {\displaystyle c_{i}} : Molar concentration of component i c: Total molar concentration J → i {\displaystyle {\vec {J}}_{i}} : Flux of component i The equation assumes steady state, 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. The molecular friction between two components is proportional to their difference in speed and their mole fractions. In the simplest case, the gradient 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...