Diffusion coefficient volume-fixed

By selecting the reference properly, the diffusion coefficients DA and DB can be made equal to each other. This value is termed the mutual diffusion (or interdiffusion) coefficient Dab- The reference frame is one across which no change in volume occurs (fixed volume) ... 

When applied to a volume-fixed frame of reference (i.e., laboratory coordinates) with ordinary concentration units (e.g., g/cm3), these equations are applicable only to nonswelling systems. The diffusion coefficient obtained for the swelling system is the polymer-solvent mutual diffusion coefficient in a volume-fixed reference frame, Dv. Also, the single diffusion coefficient extracted from this analysis will be some average of concentration-dependent values if the diffusion coefficient is not constant. 

D)v mutual diffusion coefficient in a volume-fixed frame of reference. 

This result is generally applicable to any situation in which a fixed-width diffusion medium is partitioned into (Jmax - 1) volume elements. In this case, the selection of Jmax fixes the magnitude of the model diffusion coefficient DM. The constraints of Equation 20.12 still apply, however. Therefore, one must resist the temptation to increase the size of Jmax (without limit) to reduce the size of Ax. The maximum value of Jmax selected must be such that DM is still less than /2. 

Here (DJp is the polymer-fixed diffusion coefficient of the penetrant component (see section 1.1) and at is the activity of the penetrant in the given polymer-diluent mixture. Combining Eqs. (2) and (31) and noticing that the product c, , is equal to the volume fraction vt of the penetrant component in the mixture, one obtains... 

Dependencies of the three main parameters on the electrolyte volume portion are depicted in Fig. 15. Here, a percolation threshold Xc = 0.1 is taken, implying a highly interconnected composite structure. The carbon/catalyst volume portion is kept fixed at Xm = 0.3, a value that lies well above the percolation threshold. Consistent with the small percolation threshold a large value M = 8-10 was presumed. The residual diffusivity coefficient was taken... 

The quantity directly associated with the translation of the polymer component relative to the solvent is not Dm, which is the volume-fixed diffusion coefficient, but the solvent-fixed diffusion coefficient D, which is defined by [19]... 

In the course of a diffiisional experiment, the concentrations of the diffusing components vary in a restricted pliysical space (a diffusional cell), which leads to changes in the density and volume in parts of the diffusional cell with respect to a immovable coordinate system. That is why choosing a location of the reference plane of the section, through which passes a unit of the substance amount, is not a simple matter, and the coordinate system for flux density is fixed in different ways (Erdey-Gruz, 1974 Read et al., 1977 Malkin and Chalykh, 1979), which leads to different diffusion coefficients, according to Equation 31. 

For molecular interpretation, D (also called the mutual diffusion coefficient, or diffusion coeflBcient of the penetrant and polymer relative to the local enter of volume) has to be modified to take into account the bulk flow of the penetrant. Accordingly, a new coefficient, referred to as intrinsic diffusion coefficient, has been defined in terms of the rate of mass transfer relative to the center of mass. Crank (1968) shows that for negligible polymer mobility, the intrinsic diffusivity of the penetrant (designated as Z j, which is also the diffusivity of the penetrant based on the polymer frame of reference) is related to the volume-fixed mutual diffusion coefficient, D, by... 

In principle, each of the coefficients in equation 22 can be evaluated independently by observing the effects on Da over a sufficiently wide range of temperatures, external hydrostatic pressures, and sorbed penetrant concentrations. Hydrostatic effects on p can be decoupled from penetrant sorptive effects on y by using a very low sorbing penetrant, such as helium in the presence of a fixed partial pressure of the penetrant of interest. On the one hand, hydrostatic pressure is expected to have a rather small effect on Da since solid pol5uners are only slightly compressible. On the other, increases in temperature and sorbed penetrant concentration cause large increases in the free-volume fraction Vf and in the self-diffusion coefficient Da. ... 

The process of intersite electron hopping has been discussed in terms of a quasi-diffusional process. We now take a more detailed view of the intersite electron transfer reaction in a fixed-site redox polymer. The approach adopted here is due to Fritsch-Faules and Faulkner. These researchers developed a microscopic model to describe the electronhopping diffusion coefficient Z>e in a rigid three-dimensional polymer network as a function of the redox site concentration c. The model takes excluded volume effects into consideration, and it is based on a consideration of probability distributions and random-walk concepts. The microscopic approach was adopted by these researchers to obtain parameters that could be readily understood in the context of the polymer s molecular architecture. A previously published related approach was given by Feldberg. ... 

Therefore one must be very careful in the case in which the xes are measured in respect to a frame of reference fixed with the cell in fact in this case one cannot apply the Fick s law to derive the diffusion coefficient unless the partial specific volumes are nearly independent from the composition or the concentration gradients are sufficiently small. It is worthwhile to notice that also in the case in which we can neglect the integral in eq. (1.21)the Pick s law will not give the diffusion coefficient by a linear combination among them with a weight given by the concentrations and the partial molar volumes. We now go back to eq. (1.12), i.e. we suppose to measure the fluxes in the mass fixed frame of reference, and we want to discuss the case of a binary mixture (i.e. r=2). In such a situation we have from eq. 

Fig.3. Diffiision coefficients D as derived from a fit of the experimental data in Fig.2 with the solution of the diffusion equation for cylindrical sample (open circles) in addition, values corrected for the amount of gas to be transported (D d>pseudo, fuU circles) are depicted. For comparison the theoretical diffusion coefficients for gas phase diffusion in cylindrical pores are also included (dashed lines) hereby the value of the macroporosity (50%) and a tortuosity factor of 3 are taken into account. The macroporosity was calculate fix)m the bulk density of the sample and the micropore volume (macroporDsity=total porosity—microporosity= 86 % - 36%).

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