Prediction of mutual diffusion coefficient

Since the prediction of mutual diffusion coefficients from self-diffusion coefficients is not accurate enough to be used for modeling of chemical processes, complete data sets of mutual and self-diffusion coefficients are necessary and valuable. 

In addition to temperature and concentration, diffusion in polymers can be influenced by the penetrant size, polymer molecular weight, and polymer morphology factors such as crystallinity and cross-linking density. These factors render the prediction of the penetrant diffusion coefficient a rather complex task. However, in simpler systems such as non-cross-linked amorphous polymers, theories have been developed to predict the mutual diffusion coefficient with various degrees of success [12-19], Among these, the most notable are the free volume theories [12,17], In the following subsection, these free volume based theories are introduced to illustrate the principles involved. 

Dullien, F. A. L. and Asfour, A-F. A., Concentration Dependence of Mutual Diffusion Coefficients in Regular Binary Solutions A New Predictive Equation, Ind. Eng. Chem. Fundam., 24, 1-7 (1985). 

Vrentas and Duda s theory formulates a method of predicting the mutual diffusion coefficient D of a penetrant/polymer system. The revised version ( 8) of this theory describes the temperature and concentration dependence of D but requires values for a number of parameters for a binary system. The data needed for evaluation of these parameters include the Tg of both the polymer and the penetrant, the density and viscosity as a function of temperature for the pure polymer and penetrant, at least three values of the diffusivity for the penetrant/polymer system at two or more temperatures, and the solubility of the penetrant in the polymer or other thermodynamic data from which the Flory interaction parameter % (assumed to be independent of concentration and temperature) can be determined. An extension of this model has been made to describe the effect of the glass transition on the free volume and on the diffusion process (23.) ... 

The need to predict mutual diffusion coefficients from self-diffusion coefficients often arises, and many efforts have been made to understand and predict mutual diffusion data, through approaches such as, for example, the following extension of the Darken equation [5j ... 

Fig. 8.8 Free volume theory prediction of mutual binary diffusion coefficient for the toluene-PS system based on parameters (19). [Reproduced by permission from J. L. Duda, J. S. Vrentas, S. T. Ju and H. T. Liu, Prediction of Diffusion Coefficients, A.I.Ch.E J., 28, 279 (1982).]...

In nonideal mixtures, the thermodynamic nonideality of the mixture has to be considered. We still need to predict the concentration dependence of the mutual diffusion coefficient Dt] of a binary pair of nonelectrolytes. The concentration dependency of l)u in liquid mixtures may be calculated by using the Vignes equation or the Leffler and Cullinan equation. Besides these, we may also use a correlation suggested by Dullien and Asfour (1985), given by... 

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

Graf D. L., Anderson D. E., and Woodhouse J. E. (1983) Ionic diffusion in naturally-occurring aqueous solutions transition-state models that use either empirical expressions or statistically-derived relationships to predict mutual diffusion coefficients in the concentrated-solution regions of 8 binary systems. Geochim. Cosmochim. Acta 47, 1985—1998. 

Rg is the polymer radius of gyration, Xs is the value of the x parameter (see Section 2.3.1) at the spinodal point, and D is the mutual diffusion coefficient of the two polymer components. Bates and Wiltzius (1989) have confirmed the predictions of Eqs. (9-4) and (9-5) for early-time SD of binary blends of perdeuterated and protonated 1,4-polybutadiene. Neutron-scattering studies of SD on a similar system by Jiimai et al. (1993a, 1993b) also confirm the Cahn theory at early times, but the spinodal growth rates deviate somewhat from Eq. (9-5). 

Figure 4.21. Mutual diffusion coefficients as functions of the concentration for blends of deuterated polystyrene N = 9.8 X 10 ) and normal polystyrene N = 8.7 X 10 ). The diffusion temperatures were 166 °C (o), 174 °C (A), 190 °C (O) and 205 °C ( ). The solid lines are the predictions of equation (4.4.11). The decrease in diffusion coefficient for volume fractions aroimd a half is a direct result of the unfavourable thermodynamies of mixing - thermodynamic slowing down . After Green and Doyle (1987).

Figure 4.24. Diffusion coefficients as functions of the composition in the miscible blend polystyrene-poly(xylenyl ether) (PS-PXE) at a temperature 66 °C above the (concentration-dependent) glass transition temperature of the blend, measured by forward recoil spectrometry. Squares represent tracer diffusion coefficients of PXE (VpxE = 292), circles the tracer diffusion coefficients of PS and diamonds the mutual diffusion coefficient. The upper solid line is the prediction of equation (4.4.11) using the smoothed curves through the experimental points for the tracer diffusion coefficients and an experimentally measured value of the Flory-Huggins interaction parameter. The dashed line is the prediction of equation (4.4.11), neglecting the effect of non-ideality of mixing, illustrating the substantial thermodynamic enhancement of the mutual diffusion coefficient in this miscible system. After Composto et al. (1988).

Tables 7.2 and 7.3 display the heats of transports and thermal diffusion ratio (Kj) of chloroform in binary mixtures with selected alkanes and of toluene (1), chlorobenzene (2), and bromobenzene at 30 °C and 1 atm. Concentration-dependent thermal conductivity, mutual diffusion coefficients, and heats of transport of alkanes in chloroform and in carbon tetrachloride are given by Rowley et al. (1988). The polynomial fits to these coefficients for the alkanes in chloroform and in carbon tetrachloride are used to estimate the degree of coupling and the thermal diffusion ratio Kn from Eqns (7.46) and (7.47), and shown in Figures 7.1 and 7.2 (Demirel and Sandler, 2002). The thermal conductivity and the thermodynamic factors for the hexane-carbon tetrachloride mixture have been predicted by the local composition model of NRTL.

Except for supercritical extraction conditions, hydrostatic pressure effects are typically of negligible importance for simple solvent vapors diffusing in polymers, since the saturation vapor pressure is low, <101.3 kPa (1 atm), in most applications. The predictive power of the approach is indicated by the results for the mutual diffusion coefficient of toluene in a toluene-polystyrene system (Fig. 14) (35). 

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