Polyethylene relaxation calculations

In Fig. 15.27, the transient extensional viscosity of a low-density polyethylene, measured at 150 °C for various extensional rates of strain, is plotted against time (Munstedt and Laun, 1979). Qualitatively this figure resembles the results of the Lodge model for a Maxwell model in Fig. 15.26. For small extensional rates of strain (qe < 0.001 s ) 77+(f) is almost three times rj+ t). For qe > 0. 01 s 1 r/+ (f) increases fast, but not to infinite values, as is the case in the Lodge model. The drawn line was estimated by substitution of a spectrum of relaxation times of the polymer (calculated from the dynamic shear moduli, G and G") in Lodge s constitutive equation. The resulting viscosities are shown in Fig. 15.28 after a constant value at small extensional rates of strain the viscosity increases to a maximum value, followed by a decrease to values below the zero extension viscosity. 

These pareuneters were calculated for the two polyethylenes, at 160 °C, using a procedure given by Carrot et al. [41], which enables the recovery of a minimum number of relaxation times (Table 1). 

Results of calculations for the relaxation rates v are shown by the empty circles in Fig. 8 as a function of 1/n [10]. Unlike the predictions of the Rouse model, a linear dependence between the relaxation rates and the size of the kinetic unit is observed. The filled circles are from the BD simulations of Fixman [99,100] for a polyethylene-like chain. Dependence of Vkk on the mode number for a 16 bond polyethylene chain are shown by the empty circles in Fig. 9. The filled circles are from the BD simulations of Fixman [99, 100] for a similar chain. The curve is the best fitting curve through the empty circles. It is interesting to note that both the DRIS approach and Fixman s simulations yield a plateau value for the rate of most of the relaxational modes, while a few slowest modes exhibit distinct lower values. 

The polymer is a branched polyethylene melt with = 1.55 x 10 and Mw/M = 11.9, flowing at 170 °C in a 3.3 1 planar contraction. The hnear viscoelastic properties and the nonlinear parameters for a four-mode PTT equation are shown in Table 10.1. Different values of e were used for each mode, but with a constant value of f = 0.08. These parameters provide a reasonable fit to the transient and steady-state shear and extensional data, although the nonlinear parameters for the two longest relaxation times cause small oscillations in startup of simple shear that are not observed experimentally using parameters that ehmi-nate the shear oscillations causes the calculated extensional stresses to be too low, and the contraction flow results are sensitive to the extensional stresses. The mean relaxation time was 1.74 s, the average velocity in the downstream channel was 7.47 mm/s, and the downstream channel half-width (the characteristic length) was 0.775 mm. The Weissenberg number based on downstream channel properties was therefore 16.8. 

Bruce Bersted [1, 2] in 1975 proposed a method for calculating the viscosity of a linear polyethylene from its MWD. The basic idea was that each molecule makes a contribution to the bulk viscosity equal to its zero-shear viscosity but that as the shear rate increases from zero, the maximum length of molecule that makes such a contribution decreases. In other words, as the shear rate increases, the effects of progressively shorter molecules on the viscosity are eliminated. He said that this is equivalent to cutting off the relaxation spectrum at progressively shorter times as the shear rate increases. 

Figure 2. Crystal packing in polyethylene (a, h projection), one coordination shell about a central chain is shown. Unit cell boundaries also are shown. In defect calculations, center chain (1) is replaced by one containing a conformational defect (see Figure 1). In the hybrid Newton-Raphson relaxation method, one chain at a time has its energy minimized with the other chains fixed. In the free chain, all internal degrees of freedom participate (torsional angles, etc.) as well as the position and orientation of the chain. After a cycle through all of the chains, the total energy of the assembly is computed. The cycles are repeated until the energy is stable.

The crux of the present method is how well the relaxation portion converges. Some experience has now been gained with it (11,12,13). Figure 4 shows results of calculations made in our laboratory (11,13) on the energies of three conformational defects in polyethylene crystals. These are a kin)c (15, ), a Reneker twist (Ig) and a smooth twist (12) (see Figure 1). The first two have been proposed as stable point defects cind the last one as a transition state for the motion that accomplishes... 

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