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1298 FANG, KRÖGER, AND ÖTTINGER TABLE I. Values for the maximum possible stretching ratio max via Eq. 6, the entanglement molecular weight M e , the average end-to-end distance of the entanglement segment at equilibrium d t , and the length of a Kuhn step l K for three important melts at 140 °C. Melts M e gmol 1 d t (Å) l K (Å) max PS 13000 76 16.2 4.7 HDPE 860 33 12.9 2.6 PP atactic 5400 61 10.7 5.7 In Part I, the spring coefficient was deduced from an entropy expression containing a term squared in the relative stretch (1), together with the constraint of positive extension, thus motivating two additional parameters, called c 1 ,c 2 . Equation 6 is compatible with Part I, e.g., for a rather particular choice for c 1 together with c 2 0. The two approaches for the suggested setting c 1 1, c 2 0 possess measurable differences in their prediction of rheological quantities at startup of shear flow slightly increased stress amplitude at overshoot and slightly decreased strain at overshoot—for the model based on Eq. 6, steady shear flow weak decrease of steady stresses, and extensional flow critical strain rate postponed by a factor of 1.3–1.5, where the current setting has overall advantages when compared to experimental data for more details see Sec. IV. The parameter max clearly is the maximum possible stretching ratio of the chain contour length, which is equal to the square root of the number of Kuhn steps per entanglement N KE , max N KE d t l K , 7 where d t is the average end-to-end distance of the entanglement segment at equilibrium and l K is the length of the Kuhn step which is twice the persistence length. If we denote the molecular weight between entanglements for polymer melts as M e , then M e for polymer solutions can be estimated by using the relation M e M e / 1.2 , where is the volume fraction of polymer Ferry 1980. The values of d t and l K for some polymer melts can be calculated from the relevant experimental data tabulated in the literature Fetters et al. 1996, where the values of M e are also given. The value of l K is calculated from the characteristic ratio C by using the relationship C l K l 1, 8 where l is the bond length Flory 1988, p.111. Based on these available data, the values of max for several important melts are given in Table I. The diffusion equation for the configurational distribution function takes the form Df Dt u • 1 uu u 2 •–uf u •D 1 uu u 2 • u f , ˙ s ṡ tot f dissip f 1 2 d 2 f 2 s 9
THERMODYNAMICALLY ADMISSIBLE REPTATION MODEL 1299 where ṡ tot 1 s 1 2˙ dissip . 10 This drift velocity for s means there is only a rescaling of the position label for the tube segment when the chain relaxes in the tube. The third term creation/destruction term on the right side of Eq. 9 compensates for configurations lost or gained at the boundaries. The terms involving second-order derivatives in Eq. 9 are of irreversible nature and express the erratic reptational motion along the chain contour second-order derivative with respect to s with the reptation time d and constraint release second-order derivative with respect to u with the orientational diffusion coefficient D, respectively. The form of D is D 1 6 1 1 ˙ dissip 2 d H ˙ dissip , 11 where H(x) is the Heaviside step function. The 1 term is interpreted as representing ‘‘double reptation,’’ and the 2 term represents the CCR mechanism. The quantities 1/ d and ˙ dissip / determine the constraint release rate due to the loss of entanglements caused by reptation motion and chain retraction of side chains, respectively. The parameters 1 and 2 determine the transfer rate from the constraint release rate to the relaxation rate of chain orientation, here we take 1 2 1/. This choice is motivated by the work of Mead et al. 1998 in connection with the appearance of their switch function. The argument is that the constraint release causes not only chain segments reorientation, but also contour length shortening this effect is not explicitly taken into account here. The role of the parameters 1 and 2 is to apportion the effects of constraint release between these two effects. When the chain is unstretched, the constraint release causes only chain segments reorientation; when the chain is highly stretched, the constraint release causes mainly chain contour length shortening. Hence, the parameters must be chosen in such a way that they approach unity when is near unity, and approach zero when is large. Conservation of the total probability implies the boundary conditions f˜ f˜ 0, ss 0 ss 1 12 where f˜(s,r) f (u,s,r)d 3 u 1. At the chain ends, we assume random orientation by specifying the distribution fu,s,r 1 u1, s 0,1, 13 4 which is common practice, but has been opened to discussion by <strong>Kröger</strong> and Hess 1993. The extra stress tensor consists of two contributions, 1 2 , namely, the original Doi–Edwards contribution 1 r 3Zn p k B T 0 1 uuf u,s,rd 3 uds, 14
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