Effect of Long-Chain Branching on Viscosity

Several methods of estimating LCB levels using linear viscoelastic data are described in Section 5.12. A technique based on the shape of the viscosity curve has been proposed for single-site polyethylenes with low levels of LCB. Lai etal [80] found that for strictly linear polyethylenes prepared using single-site catalysts, the Cross equation (Eq. 10.55) gives a reasonably good fit to viscosity data. They further showed that the characteristic time A of the cross equation is proportional to the zero shear viscosity for these materials  

They noted that the Cross model could also be fitted to viscosity data for similar polymers into which a small level of long chain branching has been introduced but that the presence of the branches caused a departure from Eq. 10.55. They proposed the use of this departure as an indicator of the level of long-chain branching in such materials. To this end, they defined the Dow Rheology Index, DRI, as follows  

Complex viscosity as function of frequency for one linear and four branched ethylene/ 1-butene copolymers at 190 °C. For all samples = 155 kg/mol and M /M 2. Sample A is linear, and the level of LCB increases in the orcier B-C-D-E. As branching increases,the onset of shear thinning shifts to lower frequencies, and the low-shear-rate viscosity increases.At high frequencies,the data forall the samples come together. From Robertson efa/.[79]. 

We note that for any material obeying Eq. 10.58, the value of the DRI will be zero, and Lai etal. [80] found that the introduction of long chain branching resulted in positive values of the DRI and that the value increased with the level of LCB. 

Highly-branched, heterogeneous polymers such as LDPE have very broad relaxation time spectra, and this results in a very broad range of shear rates over which viscosity data make the transition from the Newtonian limiting value to a power-law region. In fact, it is often impossible to reach the Newtonian value using commercial rheometers, because the shear rate required is extremely low. The LDPE viscosity data shown in Fig. 10.10 were obtained using a specially modified rotational rheometer, the operation of which required exceptional skill. 

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