Negative diffusion

Since it leads to a diffusion equation with a negative diffusion coefficient (Pope 2000), the use of this expression directly in the composition PDF transport equation is of little... 

For solutions which present a mixing gap, the locus of points where the second derivative of G vanishes is called the spinodal. Within the spinodal, this second derivative is negative which results in negative diffusion coefficients or uphill diffusion (Figure 8.10). 

Thus, the Maxwell-Stefan diffusion coefficients satisfy simple symmetry relations. Onsager s reciprocal relations reduce the number of coefficients to be determined in a phenomenological approach. Satisfying all the inequalities in Eq. (6.12) leads to the dissipation function to be positive definite. For binary mixtures, the Maxwell-Stefan dififusivity has to be positive, but for multicomponent system, negative diffusivities are possible (for example, in electrolyte solutions). From Eq. (6.12), the Maxwell-Stefan diffusivities in an -component system satisfy the following inequality... 

Estimations for nonideal mixtures need six diffusivities at infinite dilution and three diffusivities of the type D kl x. Negative diffusion coefficients can exist in ternary systems and are consistent with the nonequilibrium thermodynamics approach. 

The last term is a thermodynamic correction factor for nonideal solutions. Equation M 5-23di predicts that Dab is not constant even for ideal systems [term in brackets in Eq. tl5-23di has a value of 1.0]. Equation ri5-23dl can also predict a negative diffusion coefficient, which is an indication of the formation of two liquid phases and is very important for liquid-liquid extraction. 

Fickian model. The Fickian model is a widely accepted model for diffusion, which means no one will laugh if you use it. It works well for ideal and close to ideal binary systems and can be used for nonideal binary systems if data are available. Most diffusivity data and correlations for mass-transfer coefficients are based on the Fickian model. This model is very difficult to use for nonideal ternary systems and can require negative diffusion coefficients to predict data. This model works well for dilute binary systems. 

During the growth of the deformation the flow of Uquid is from the thinner zones of the film to the thicker parts, which is an example of negative diffusion. 

The Maxwell-Stefan theory is more comprehensive than the classical Pick s diffusion theory, as the former does not exclude the possibility of negative diffusion coefficients. 

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