Mutual diffusion coefficient calculations

For a binary mixture of two components A and B in the gas phase, the mutual diffusion coefficient such as defined in 4.3.2.3, does not depend on composition. It can be calculated by the Fuller (1966) method ... 

Keizer [455a, 498] has applied non-equilibrium statistical mechanics to the calculation of the reaction rate between two species which can both diffuse with mutual diffusion coefficient ) and encounter distance R. The partially reflecting boundary condition can be incorporated, but in the limit of fast reaction of encounter pairs for identical species... 

In Table 6-5 the mutual diffusion coefficients of a binary mixture of n-heptane and n-hexadecane at 25 °C are calculated for different molar fractions of the solutes and compared with experimental values (Landolt-Bornstein, 1969). 

As shown above for the homologous series for n-alkanes the tracer-diffusion coefficient, D ba, of a compound B in the solvent A at temperature Tis obtained within two steps first the ratio of mutual diffusion coefficients D AB DTBABis calculated using Eq. (6-30). Then D LrljA is calculated at the reference temperature Tm using Vl B instead of Vc.wA inEq. (6-30). Finally the value of results as ... 

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... 

In this experiment the mutual diffusion coefficients for the Ar—CO2 and He—CO2 systems are to be measured using a modified Loschmidt apparatus. These transport coefficients are then compared with theoretical values calculated with hard-sphere collision diameters. 

It is possible that the lower than required values of D2 reflect a problem with incorrect values of Q, which if too large would result in smaller values of D2. In an interferometric study of the diffusion of toluene in an uncrosslinked natural rubber sample, Mozisek (15) reported results for the mutual diffusion coefficient which were similar to the results of Hayes and Park. In the absence of thermodynamic data from Mozisek s work, correction factors calculated for the present work were applied to his data. The results are shown in Figure 7, which reproduces Mozisek s data along with the values for D2. The extrapolated value at 1, would exceed the self diffusion coefficient for toluene by about two orders of magnitude, similar to the discrepancy seen with Hayes and Park s data. This indicates that the fault with the results in the present case is not due to overly high values of the correction factors. Moreover, the method of calculating D from D12 has been confirmed experimentally by Duda and Vrentas (16) in a comparison of vapor sorption results for toluene diffusion in molten polystyrene with the values of D1 obtained directly using radio-labeled toluene. 

There is no simple relationship that permits calculation of the mutual diffusion coefficient from the tracer diffusion coefficient for a binary mixture the diffusion coefficients represent molecular mixing in different physical situations. However, the mutual diffusion coefficient should reduce to the tracer diffusion coefficient as the solute concentration approaches zero. For example, a frequently used formula for determining the mutual diffusion coefficient from the tracer diffusion coefficients is ... 

Finally, if we have measured or estimated both tracer diffusion coefficients as a function of composition and we know the thermodynamic factor for our mixture, we can now ask whether we are in a position to calculate the kinetics of interfacial broadening that occurs when two pure pol5miers are placed in contact and annealed. To do this it is sufficient to know the mutual diffusion coefficient as a function of composition and use this to solve the non-linear version of Pick s second law, equation (4.4.8). 

This equation does have some convincing experimental support for the case of polymers of high relative molecular mass. Figure 4.24 shows the results of an experiment in which both tracer diffusion coefficients were measured as a function of composition (Composto et al. 1988). Equation (4.4.11) was then used to calculate the mutual diffusion coefficient, given a value of x that had been measured independently. These calculated mutual diffusion coefficients were compared with the directly measured values and good agreement was found. 

Figure 4.25. (a) The mutual diffusion coefficient in the miscible polymer blend poly(vinyl chloride)-polycaprolactone (PVC-PCL) at 91 °C, as measured by x-ray microanalysis in the scanning electron microscope (Jones et al. 1986). The solid line is a fit assuming that the mutual diffusion coefficient is given by equation (4.4.11), with the composition dependence of the tracer diffusion coefficient of the PCL given by a combination of equations (4.4.9) and (4.4.10). The tracer diffusion coefficient of the PVC is assumed to be small in comparison, (b) The calculated profile of diffusion between pure PVC and pure PCL, on the basis of the concentration dependence of the mutual diffusion coefficient shown in (a). The reduced length u — where the... 

The primary reason for the discrepancy is the fact that as two particles approach one another, it is necessary for solvent molecules between the particles to be moved out of the way. This process is accounted for by the viscosity term in Equation (10.20) for large distances of separation, but at smaller distances, of the order of molecular dimensions, the simple viscosity relationships no longer strictly apply, so that the mutual diffusion coefficient D is no longer equal to Di + D2 (see Chapter 4). One could say that the microscopic viscosity of the solvent increases so that diffusion is slowed and the particles approach at a reduced velocity. The exact calculation of this hydrodynamic effect represents a difficult problem in fluid dynamics. However, a relatively simple formula for two spheres of equal diameter is... 

Schoen, M. and C. Hoheisel. 1984. The mutual diffusion coefficient-Dij in binary-liquid model mixtures—Molecular-dynamics calculations based on Lennard-Jones (12-6) potentials. 1. The method of determination. Molecular Physics. 52, 33. 

There are two routes to derive the chemical (or mutual) diffusion coefficient. One has been taken for HNC calculations and the other for an MSA treatment. It should be noted that while the HNC calculation is generally regarded as more accurate, it does not lead to explicit analytical expressions, in contrast with the MSA which is simpler, but perhaps not as accurate, and leads to explicit formulas that are reasonably accurate when the energy route for the thermodynamic quantities is used. 

The calculation of diffusion coefficients from equations based on some models describing the movement of matter in electrolyte solutions, in the end, a process contributing to the knowledge of their stmcture, provided we have accurate experimental data to test these equations. Thus, to understand the behavior of transport process of these aqueous systems, experimental mutual diffusion coefficients have been compared with those estimated using several equations, resulting from different models. 

It what concerns binary systems involving non-electrolytes, we have been measuring mutual diffusion coefficients of some cyclodextrins (a-CD, P-CD, HP-a-CD, and HP-p-CD) [12, 13] and some dmgs (e.g., caffeine and isoniazid [14]) in aqueous solutions. Also, from comparison of these experimental diffusion coefficients with the related calculated values, it is possible to give some stmctural information, such as diffusion coefficients at infinitesimal concentration at different temperatures, estimation of activity coefficients by using equations of Hartley and Gordon, estimation of hydrodynamic radius, and estimation of activation energies, Ea, of the diffusion process at several temperatures. 

The best way to measure the mutual-diffusion coefficient of a dilute polymer solution is light scattering. Instead of averaging the scattered intensity over time, the time-correlation function is calculated ... 

The concentration correlation function can be calculated from Pick s Law. When Equation 5.64 is inserted into Equation 5.63 and the mutual-diffusion coefficient is treated as independent of concentration over the small range of the fluctuations. Pick s Second Law is obtained ... 

High-molecular-weight chains in concentrated polymer soluhons diffuse at a very slow rate. A theory developed by deGennes to calculate the self-diffusion coefficient of highly entangled chains is presented in Sechon 7.6. The mutual-diffusion coefficient for concentrated polymer soluhons is also derived and discussed. 

The mutual-diffusion coefficient can also be calculated using Equation 6.21. The numerator is obtained from Equation 7.21, and the denominator is given by ... 

The 1/(1 - CO a) term is commonly referred to as the frame of reference term. For many cases of importance in polymeric systems such as in gas permeation, coa is relatively small, and the 1/(1 - >a) factor can safely be neglected so that the flux relative to fixed coordinates is equal to the flux relative to moving coordinates. Even for intermediate concentrations (0.1 < coa < 0.5), this factor may often be of second-order importance compared to difficulties in accurately determining the mutual diffusion coefficient due to strong concentration dependencies. However, not accounting for the factor 1/(1 — coa) can lead to very significant errors in flux calculations in highly swollen systems (eg, 90-95% solvent), even if the mutual diffusion coefficient is accurately determined (6). 

Figure A.9 Refractive index profiles. Broken line is the refractive index profile calculated from Equation A21 using the measured dependence of the mutual diffusion coefficient for a PMMA-DPS system. Solid line is...

Experimental diffusion coefficients, as obtained from time-lag measiu ements, report a transport diffusion coefficient which carmot be obtained from equilibrium MD simulation. Comparisons made in the simulation literatme are typically between time-lag diffusion coefficients (even calculated for glassy polymers without correction for dual-mode contributions and self-diffusion coefficients. As discussed above, mutual diffusion coefficients can be obtained directly from equilibrium MD simulation but simulation of transport diffusion coefficients require the use of NEMD methods, that are less commonly available and more computationally expensive [117]. 

Ellipsometry is a powerful tool [16] for measuring the interfacial thickness between two polymers, whether in the case of immiscible or miscible polymer blends. In the case of miscible blends, investigations of changes in interfadal thickness with time at a fixed temperature allow the calculation of mutual diffusion coefficients [22]. In contrast, for immiscible blends the Flory-Huggins interadion parameter x can be deduced by measuring the interfacial thickness in an equilibrium state, and using the theory of Helfand [41] and its extended version [42]. 

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