Concentration dependence of diffusion

Many practical adsorption processes involve multicomponent systems, so the problem of micropore diffusion in a mixed adsorbed phase is both practically and theoretically important. Major progress in understanding the interaction effects has been achieved by Krishna and his coworkers through the application of the Stefan-Maxwell approach. The diverse patterns of concentration dependence of diffusivity that have been observed for many systems can, in most cases, be understood on this basis. The reader is referred, for details, to the review articles cited in the bibliography. 

The models most frequently used to describe the concentration dependence of diffusion and permeability coefficients of gases and vapors, including hydrocarbons, are transport model of dual-mode sorption (which is usually used to describe diffusion and permeation in polymer glasses) as well as its various modifications molecular models analyzing the relation of diffusion coefficients to the movement of penetrant molecules and the effect of intermolecular forces on these processes and free volume models describing the relation of diffusion coefficients and fractional free volume of the system. Molecular models and free volume models are commonly used to describe diffusion in rubbery polymers. However, some versions of these models that fall into both classification groups have been used for both mbbery and glassy polymers. These are the models by Pace-Datyner and Duda-Vrentas [7,29,30]. 

Suzuki F, Onozato K, and Takahashi N. Pervaporation of thermal mixture of benzene/toluene by polyfethylene terephthalate) membrane and synergetic effect on concentration dependence of diffusion rate. J Appl Polym Sci 1982 27 2179-2188. 

Concentration dependence of diffusion coefficient in good solvent. Filled ... 

The reptation prediction of the concentration dependence of diffusion coefficient in athermal solvent is slightly weaker than in 0-solvent, since... 

Y. P. Tang Diffusion of Carbon Dioxide in Liquids—Concentration Dependence of Diffusion Coefficient and Diffusion in Mixed Solvents, Ph.D. Dissertation, University of Texas, 1963. 

The concentration dependence of diffusivity is accounted for with a number of different models. The diffusivity behavior has been variously described as... 

Another common way to express the concentration dependence of diffusivity is the Fujita-Doolittle equation, which takes the form... 

C. E. Rogers and D. Machin, "The Concentration Dependence of Diffusion Coefficients in Polymer-Penetrant Systems", CRC Critical Rev, in Macromol. Sci., April 1972, pp. 245-313. 

The derivation of Dt from coherent QENS is similar to a computation of Dt from the fluctuations in an equilibrium density distribution. This was accomplished by Tepper and co-workers for Ar in AIPO4-5 [6]. Using the Green-Kubo formahsm, they were able to extract this non-equilibrium quantity from just one equihbrium simulation. Moreover, the calculations being performed in reciprocal space, the variation of the diffusivity upon the wave vector was used to check when the system was in the linear regime [6]. The first application of non-equihbrium molecular dynamics (NEMD) to zeolites was performed by Maginn et al. on methane in sihcalite [7]. Standard equi-libriiun MD techniques were later used by Sholl and co-workers to determine the concentration dependence of diffusivities [8]. 

The discrepancies between the values of the activation energies provided by different authors can be attributed to the different alkane partial pressures. Several theoretical and experimental studies indicate a significant concentration dependence of diffusion in zeohtes [4,19,77,78,80]. Coppens et al. [19] have shown for MFI zeolite with Monte-Carlo simulations that the diffusivity can drop by a factor of ten when the occupancy is close to saturation. In this work we performed our experiments under a hydrocarbon partial pressure of 6.6 kPa, which is higher than the pressures in TEOM, gravimetric and volumetric measurements. 

Figure 6.1.3. Kinetic curves of plane layer swelling at different values of concentration dependence of diffusion coefficient (e=0.1,d = 2/9) a - power law mode (l-s = 0 2-s=l 3-s = 1.5 4-s = 2.5) b - exponential (5 - s = 5) and blow-up mode (6 - s = 5.5). [Adapted, by permission, from E. Ya. Denisyuk, V. V. Tereshatov, Vysokomol. soed., X42,74 (2000)].

Finally, it is found that the coefficient p appearing in the concentration dependence of diffusivity. Equation (7.12), is related to the FFV of the polymer phase of the MMM at infinite dilution (Figure 7.7c) through an exponential law ... 

Figure 15a. Concentration dependence of diffusion coefficients for SDS micelles in media of different ionic strength at 25 °C. b. Relaxation dme distributions using DLT for SDS micelles at Cerent concentrations in 10 mM NaQ at 25 °C.

At the high pore fillings conventionally used, product molecules can be assumed to be equilibrated in the micropores. This implies, as we have seen, that their relative concentration is determined by the molecular chemical potential in the micropores. Since the diffusion constants can also be a strong function of concentration, one has to apply these expressions with care, or even better one should use expressions that include the concentration dependence of diffusion constants such as the Maxwell-Stefan expression. 

Based on the assumption of complete dissociation, the concentration dependence of diffusion coefficient D is described e.g., by (2)... 

Scaling laws predict that the exponent of p for M should vary from 2.25 in good solvents to 3 in theta solvent conditions for the concentration dependence of diffusion coefficients the same exponents are 0.75 and 1 respectively. At 308 K an exponent of 1.17 was observed, which within the experimental error agreed with predictions. However, at 333 K, the exponent was 0.46, much lower than theory predicts. This observation and the high exponent observed for M were attributed to the presence of dangling chains in the network, since the correlation functions were observed to become more non-exponential as the temperature... 

Zhang et al. [71] employed a packed-column IGC technique to determine the activity coefficients and diffiisivities of small molecules in phenol resins. For this, they applied the Romdhane-Danner approach for data analysis, and the Braun-Guillet approach to determine the thickness of the polymer layer on the support Subsequently, the authors reported that the IGC-derived difiusivity data were consistent with the theoretical data obtained from the Romdhane-Danner approach, and might be further used to predict the temperature- and concentration-dependence of diffusivity by the free volume theory. 

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