Diffusivity, theoretical prediction

Wool [32] has considered the fractal nature of polymer-metal and of polymer-polymer surfaces. He argues that diffusion processes often lead to fractal interfaces. Although the concentration profile varies smoothly with the dimension of depth, the interface, considered in two or three dimensions is extremely rough [72]. Theoretical predictions, supported by practical measurements, suggest that the two-dimensional profile through such a surface is a self-similar fractal, that is one which appears similar at all scales of magnification. Interfaces of this kind can occur in polymer-polymer and in polymer-metal systems. 

Many theoretical embellishments have been made to the basic model of pore diffusion as presented here. Effectiveness factors have been derived for reaction orders other than first and for Hougen and Watson kinetics. These require a numerical solution of Equation (10.3). Shape and tortuosity factors have been introduced to treat pores that have geometries other than the idealized cylinders considered here. The Knudsen diffusivity or a combination of Knudsen and bulk diffusivities has been used for very small pores. While these studies have theoretical importance and may help explain some observations, they are not yet developed well enough for predictive use. Our knowledge of the internal structure of a porous catalyst is still rather rudimentary and imposes a basic limitation on theoretical predictions. We will give a brief account of Knudsen diffusion. 

Figure 13 Theoretical predictions of Eq. (31) for diffusion in a hydrogel (jlVfMv -1.8 nnT3) for solutes of different hydrodynamic radii. (From Ref. 174.)... 

Fig. 44. Double logaritmic plot of Q(Q)/Q2 vs. Q for various dilute solutions under good solvent conditions visualize the crossover from segmental to monomer diffusion [119]. The solid lines result from fitting the theoretical predictions of Akcasu et al. [94] to the experimental data using B = 0.38 and T s and a = / as adjustable parameters. The dotted lines are the corresponding predictions for 0 conditions. (Reprinted with permission from [119]. Copyright 1981 American Chemical Society, Washington)...

Fig. 46. Segmental diffusion in a dilute PDMS/d-bromobenzene solution at the crossover from to good solvent conditions. Reduced characteristic frequencies Qred (Q, x) vs. Q at different x-values. Comparison between experimental results ( ) and theoretical predictions (-). according to [98].

With respect to the segmental diffusion, the characteristic frequencies of the cyclic systems vary with Q3 as in the case of the linear chains in dilute solution (see Sect. 5.1.2). The absolute values are independent of the topology of the polymers and their molecular masses, and thus exhibit the same deviations from the theoretical predictions that have just been pointed out for dilute solutions of linear homopolymers. 

While fitting five adjustable parameters to four sets of experimental data may not seem surprising, the strength of the diffusion model lies in predicting a much wider body of experimental results. Of these, the most important are the variations of molecular yields with LET and solute concentration. Since these calculated variations agree quite well with experiment, no further comment is necessary except to note that calculations often require normalization, so that only relative yields can be compared with experiment. One main reason is that the absolute yields often differ from laboratory to another for the same experiment. Thus, Schwarz s theoretical predictions have reasonable normalization constants, which, however, are not considered as new parameters. In the next subsection, we will consider some experimental features that could possibly be in disagreement with the diffusion model. 

Experimental measurements of surface diffusion are usually calculated by subtracting from the measured total diffusion that predicted theoretically for Knudsen and molecular diffusion. 

Fig. 3.5. Dependence of A0m on the activity of I", at various activities of Cr, when the ions form insoluble Ag salts and there is no diffusion potential across the membrane. The solid line represents the theoretical prediction and the dotted line the commonly observed experimental dependence.

The observed constancy of y/t / is in accord with theoretical predictions for one dimensional diffusion with reaction of the diffusant (15). In that case y(t) is given by... 

Platelet attachment to subendotheliumf is determined predominantly by physical factors controlling the rate of platelet transport to the subendothelium at low shear rates (800 s ). In addition, platelet deposition was found to be highly dependent on the concentration of red cells, an effect attributed in part to the fact that red cells, by increasing the radial movement of platelets, enhance their diffusivity by several orders of magnitude compared to that theoretically predicted and experimentally measured in platelet-rich plasma. 

In accordance with theoretical predictions of the dynamic properties of networks, the critical concentration of dextran appears to be independent of the molecular weight of the flexible polymeric diffusant although some differences are noted when the behaviour of the flexible polymers used is compared e.g. the critical dextran concentrations are lower for PEG than for PVP and PVA. For ternary polymer systems, as studied here, the requirement of a critical concentration that corresponds to the molecular dimensions of the dextran matrix is an experimental feature which appears to be critical for the onset of rapid polymer transport. It is noteworthy that an unambiguous experimental identification of a critical concentration associated with the transport of these types of polymers in solution in relation to the onset of polymer network formation has not been reported so far. Indeed, our studies on the diffusion of dextran in binary (polymer/solvent) systems demonstrated that both its mutual and intradiffusion coefficients vary continuously with increasing concentration 2. ... 

Thus, we find a remarkable lack of sensitivity for the calculated burning rates of an adiabatic droplet-burning process in which the reactions go to completion. This observed lack of sensitivity to reaction rates may well be related to the known successes (11,12, 22) of simplified diffusion-flame theories in theoretical predictions of droplet burning rates. 

Recent work has extended the use of a diffusion based instrument to one in which the diffusion tube has been rotated 90° with respect to that of Figure 13. In this configuration, there is a possibility of flow directly into the diffusion tube. This tube has dimensions L = 1.6 cm and d = O.87 cm. Substituting these values into Equation (2) gives a predicted diffusion cell response of 1.5 jua/ppm CO. Observed values of 1.1 -1.2 jua/ppm CO were again in good agreement with theoretical predictions. 

It can be seen that a theoretical prediction of values is not possible by any of the three above-described models, because none of the three parameters - the laminar film thickness in the film model, the contact time in the penetration model, and the fractional surface renewal rate in the surface renewal model - is predictable in general. It is for this reason that the empirical correlations must normally be used for the predictions of individual coefficients of mass transfer. Experimentally obtained values of the exponent on diffusivity are usually between 0.5 and 1.0. 

Theoretical prediction of the diffusivity thus depends only on estimating the ratio of partition functions///// and the potential energy of a molecule at the center of the window (Ur). 

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