Darken relation, diffusion

An adsorption kinetic model was developed to evaluate the adsorption rates of five pure gases (Nj, O2, Ar, CO, and CH4) on a Takeda-3A CMS over a wide range of pressures up to ISatm. The kinetic characteristics of adsorption on the CMS were studied by using the adsorption equilibrium of five pure gases measured at three different temperatures and their physical properties. Since the diffiisional time constants of all the components showed much stronger dependence of pressure than those expected by the traditional Darken relation, a structural diffusion model was applied to predict the strong pressure dependence. The proposed model successfully predicted the dif ional time constant up to high pressure on the CMS. 

The exponential increases of the effective difiusional time constant might be explained by the following Darken-relation and this relation was derived under the assumption that the chemical potential gradient is the driving force of the diffusion (Ruthven, 1992). 

The transport diffusivity and self-diffusivity are related by the Darken relation,... 

This is now known as the Darken relation, and it depends on the equilibrium isotherm between the two phases. It basically states that the surface diffusivity at any loading is equal to the value at zero loading multiplied by a thermodynamic correction factor... 

Let us now turn to the case where both the Langmuir isotherm and surface diffusion are included. We assume that the surface diffusivity takes the following Darken relation form ... 

Caldwell-Babb Darken observed that sohd-state diffusion in metallurgical applications followed a simple relation. His equation related the tracer diffusivities and mole fractious to the mutual diffusivity ... 

This coefficient describes a diffusion process under the influence of a gradient in the chemical composition. When two diffusing species mix together, their rate of mixing depends on the diffusion rates of both species. Consequently, the interdiffusion coefficient is defined to measure this rate of mixing in relation to a laboratory frame of reference [13]. In this sense, the relation defining the interdiffusion coefficient, deduced by Darkens [29], is... 

Neither the tracer diffusivity nor the self-diffusivity has much practical value except as a means to understand ordinary diffusion and as order-of-magnitude estimates of mutual diffusivities. Darken s equation [Eq. (5-230)] was derived for tracer diffusivities but is often used to relate mutual diffusivities at moderate concentrations as opposed to infinite dilution. 

However, if we can assiune that the non-diffusive relaxation of the less mobile polymers is fast compared with the diffusion of the more mobile chains, we can derive a simple equation relating the mutual diffusion coefficient to the tracer diffusion coefficient and a thermodynamic factor. This equation has been applied to liquid systems, in which context it is known as the Hartley-Crank equation (Tyrell and Harris 1984), and to metals, in which context it is known as the Darken equation (Haasen 1984). The polymer version was first stated by Kramer (Kramer et al. 1984) and is often known as the fast theory . It is most compactly written in the form... 

The interdifTusion coefficient relates to partial diffusion coefficients according to Darken (remembering ffiat partial volumes have different values) ... 

Concentration in the simulation cell is extremely low and its diffusion coefficient is an order of magnitude larger than that of the polymeric segments. Under these circumstances, the self-diffusion and mutual diffusion coefficients of the penetrant are approximately equal, as related by the Darken... 

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