Diffusion Deborah number

Diffusion-controlled drug delivery, 9 77 Diffusion Deborah number, 23 101 Diffusion flames... 

The relative rates of diffusion and polymer relaxation can conveniently be examined using the diffusion Deborah number (9) defined as ... 

The timescale of the diffusion process is determined by the use of the diffusion Deborah, number De, given by the following equation ... 

If the diffusion Deborah number is small (small moleeular relaxation time or large diffusion time) moleeular relaxation is mueh faster than diffusive transport (in fact, it is almost instantaneous). In this case the diffusion process is similar to simple liquids. For example, diluted solutions and polymer solutions above glass transition temperature fall in fliis category. 

Vrentas, JS Duda, JL, Diffusion in Polymer-Solvent Systems, in. Construction of Deborah Number Diagrams, Journal of Polymer Science Polymer Physics Edition 15,441,1977. Wakao, N Smith, JM, Diffusion in Catalyst Pellets, Chemical Engineering Science 17, 825, 1962. 

Figure 4 A Deborah number diagram for the polystyrene-ethylbenzene system showing the diffusion behavior as a function of weight fraction and temperature. (From Ref. 33.)...

JS Vrentas, JL Duda. Diffusion in polymer-solvent systems. III. Construction of Deborah number diagrams. J Polym Sci, Polym Phys Ed 15 441-453, 1977. 

This relative importance of relaxation and diffusion has been quantified with the Deborah number, De [119,130-132], De is defined as the ratio of a characteristic relaxation time A. to a characteristic diffusion time 0 (0 = L2/D, where D is the diffusion coefficient over the characteristic length L) De = X/Q. Thus rubbers will have values of De less than 1 and glasses will have values of De greater than 1. If the value of De is either much greater or much less than 1, swelling kinetics can usually be correlated by Fick s law with the appropriate initial and boundary conditions. Such transport is variously referred to as diffusion-controlled, Fickian, or case I sorption. In the case of rubbery polymers well above Tg (De < c 1), substantial swelling may occur and... 

JS Vrentas, CM Jarzebski, JL Duda. Deborah number for diffusion in polymer-solvent systems. AIChE J 21 894-902, 1975. 

JC Wu, NA Peppas. Modeling of penetrant diffusion in glassy polymers with an integral sorption Deborah number. J Polym Sci Polym Phys Ed 31 1503-1518,... 

Although a mechanism for stress relaxation was described in Section 1.3.2, the Deborah number is purely based on experimental measurements, i.e. an observation of a bulk material behaviour. The Peclet number, however, is determined by the diffusivity of the microstructural elements, and is the dimensionless group given by the timescale for diffusive motion relative to that for convective or flow. The diffusion coefficient, D, is given by the Stokes-Einstein equation ... 

In order to observe linear viscoelasticity, structural relaxation by diffusion must occur on a timescale comparable to our measurement time. The ratio of these times is the Deborah number. When this is of the order of unity our experiment will follow the relaxation processes in the material and the material will appear to be viscoelastic ... 

Figure 2. Effect of Deborah number (DEB)d on the Characteristic time-dependent diffusion coefficient.

Figure 8. Time dependence of relative penetration of the diffusion front as a function of release Deborah number, Doo/ldl.2, for a swellable polymer sheet containing dispersed drug.

Set of ion concentrations in gel and outer solution Concentration of buff in acid form in outer solution Total buffer concentration in outer solution Concentration of i-th ionic spec insMe gel Concentration of i-th ionic species in outer solution Diffusion coefficient Deborah number Equilibrium water fraction Ionic strength... 

Other dimensionless groups similar to the Deborah number are sometimes used for special cases. For example, in a steady shearing flow of a polymeric fluid at a shear rate y, the Weissenberg number is defined as Wi = yr. This group takes its name from the discoverer of some unusual effects produced by normal stress differences that exist in polymeric fluids when Wi 1, as discussed in Section 1.4.3. Use of the term Weissenberg number is usually restricted to steady flows, especially shear flows. For suspensions, the Peclet number is defined as the shear rate times a characteristic diffusion time to [see Eq. (6-12) and Section 6.2.2]. 

Figure 11.23—Comparison of theo-retical and experimental first and second normal stress differences N and N2. The theoretical results (a) were calculated from the Smoluchowski equation (11-3) using the Onsager potential with U = 10.67, the minimum value for a fully nematic state, y/ >r is the dimensionless shear rate (or Deborah number), where Dr is the rotary diffusivity of a hypothetical isotropic fluid at the same concentration. Only the molecular-elastic contribution to the stress tensor was considered. The experimental results (b) are for 12.5% (by weight) PBLG (molecular weight = 238,000) in w-cresol. (Reprinted with permission from Magda et al., Macromolecules 24 4460. Copyright 1991, American Chemical Society.)...

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