Polymers Deborah number

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. 

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,... 

With the average elongational strain rate of the flow field between the eddies and the relaxation time of the polymer molecules, one can define a dimensionless characteristic number, the Deborah number, which represents the ratio of a characteristic time of flow and a characteristic time of the polymer molecule, and thus one can transfer considerations in porous media flow to the turbulent flow region. 

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

Figure 4. Fractional release vs. square root of dimensionless time as a function of release Deborah number, D ,/k 2, for a swellable polymer sheet containing dispersed drug + solvent fronts meet.

Vrentas and Duda did outstanding work in the 1970s and 1980s. For this we have to refer to their contribution in the Encyclopaedia of Polymer Science and Engineering (1986) and their papers since 1976. They made use of the Deborah-number, introduced by Reiner, but defined in their special way ... 

Whether a polymer exhibits elastic as well as viscous behavior depends in part on the time scale of the imposition of a load or deformation compared to the characteristic response time of the matei ial. This concept is expressed in the dimensionless Deborah number ... 

In extensional flows at high Deborah numbers, szi > 1, long polymer molecules can fracture (Keller and Odell 1985 Rabin 1987 Kausch 1985). If the strain is large, the breakage usually is near the middle of the molecule, indicating that fracture occurs after the molecule has undergone a coil-stretch transition (Keller and Odell 1985). For polystyrene of molecular weight 20 million in a low-viscosity solvent (dekalin), fracture occurs at extension rates above about 1000 sec In the crossed-slot device, the critical extension rate for... 

The term Peclet number is common in the suspension literature, while the corresponding quantity is usually called the Deborah number or Weissenberg number in tbe polymer literature. From Eqs. (6-30) through (6-33) we find, in general, for a solvent of viscosity 1 cP, that Drd /b, where b — d or L) is the particle s longest dimension in units of pm, and is in sec. Since typical shear rates are in the range 10 > 10 sec , ... 

Another criterion for predicting if the transport in polymeric gels is controlled by diffusion (Fickian) or by relaxation, is to determine the diffusional Deborah number De), which is a ratio between the characteristic polymer relaxation time of the polymer (2) when it is subject to a swelling stress and a characteristic diffusion time (6), defined as the coefficient between the square of the sample thickness (h) and the coefficient of water diffusion in the polymeric gel... 

When describing dilatant behavior, the maximum stretch rate, e, in the converging flow at the contraction is a better parameter, but more difficult to be calculated. Instead of the term stretch rate, other authors also used deformation rate (e.g., Chauveteau, 1981) or elongational rate (e.g.. Sorbic, 1991). The shear-thickening viscosity is also called elongational viscosity (often referred to as the Trouton viscosity Sorbie, 1991) or extensional viscosity in the literature. James and McLaren (1975) reported that for a solution of polyethylene oxide (a flexible coil, water-soluble polymer physically similar to HPAM), the onset of elastic behavior at maximum stretch rates was of the order of 100 s and shear rates of the order of 1000 s. In this instance, the stretch rate is about 10 times lower than the shear rate. However, some authors use shear rate instead of stretch rate in defining the Deborah number—for example, Delshad et al. (2008). 

In polymer flooding, some authors—for example, Masuda et al. (1992) and Delshad et al. (2008)—defined the Deborah number as the ratio of a polymer molecule s relaxation time to its average residence time (fie) between pore body and pore throat, and the residence time is defined as the inverse of the equivalent shear rate (Yeq) ... 

The flow velocity is in the order of 10 m/s. The radius of an oil thread is about 10 m. The relaxation time of polymer solution used in the oil displacement process is about 10 to 10 s. Under these conditions, the range of Deborah number, Noe, is between 0.1 and 10. Figure 6.26 shows the normal stress of the viscoelastic fluid with different Deborah numbers. The stress acts on the undulated oil/water interface. When the representation in Figure 6.26 was constructed, the fluid velocity of 3.47 x 10 m/s and the relaxation time of 0.247 s were used. In the figure, negative stress represents that the stress direction is opposite to the external normal line of the acting surface. We can... 

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