(164f) Comparison of Rheological Models for Low-Density Polyethylene (LDPE) Produced in High-Pressure Tubular Reactors.
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To understand the effect of reactor operation conditions on molecular structure and consequently on rheological and processing properties of the molten polymer, it is necessary to describe how the molecules react to the imposition of flow. Several mathematical models have been proposed in the literature to describe the rheological behavior of both linear and branched polymer melts in terms of molecular characteristics, such as the weight-average molecular weight, MWD and LCBD. However, this is still today a topic of continuous research . Linear viscoelastic parameters, such as the relaxation modulus, dynamic moduli and the complex viscosity, are commonly used to characterize polymer melts, because they are experimentally measured under low-frequency rotational tests that ensure that the rheological response obtained is completely due to the molecular architecture of the polymer. Another important rheological property is the shear viscosity as a function of the shear rate. This property, also called flow curve of the polymer, is usually employed as an input for the estimation of useful end-use parameters such as the melt index . The above mentioned rheological properties are measured using different types of rheometers, which are usually expensive and need trained operators. Therefore, it would be very useful for polymer producers to count with a computational tool capable of predicting the rheological properties of the polymer as a function of the reactor operating conditions.
In the present work, two different rheological models for the prediction of the rheological properties: relaxation modulus, dynamic moduli, complex viscosity, and the shear viscosity curve, are compared. The first one is an empirical model developed by Bersted  and subsequently adapted by Pedersen and Ram  to predict the shear viscosity curve of LDPE. This model uses as input the MWD as well as long chain branching information expressed as the branching index. Bersted  developed an empirical model relating the shear dependence of the steady shear viscosity of linear high-density polyethylene (HDPE) melts to the MWD. Pedersen and Ram  extended this model to branched LDPE melts using the mean-square radius of gyration instead of the weight-average molecular weight to describe the melt viscosity behavior. The model by Bersted  requires a shear-rate dependent parameter, Mc, which separates molecular weights into two classes: molecules with molecular weight below Mc contribute to the viscosity as they do at zero shear rate, and molecules with molecular weight above Mc contribute to the viscosity as if they had the same relaxation times as molecules of molecular weight . Each molecular weight species contributes according to its weight fraction. In the extended model by Pedersen and Ram , Mc is replaced by gMc, introducing the effect of the branches in LDPE trought the branching index g.
The second rheological model studied in this work is based on molecular theories that describe how sufficiently long molecules participate in entanglements with other molecules restricting the motion of the polymer melt. The first theoretical molecular model for predicting rheological properties in terms of the MWD was originally proposed by de Gennes  and by Doi and Edwards  for linear polymers. The underlying theory of these models represents the molecular environment of an entangled chain employing the concept of a tube, in which the chain relaxes after a strain imposition through the mechanism of reptation. Subsequently, this theory was extended by Tsenoglou  and Des Cloizeaux , who introduced the concept of double repatriation (DR). This last author proposed a model called time-dependent diffusion reptation (TDD). Several authors have used the TDD model in conjunction with a double reptation model obtaining excellent predictions of the relaxation modulus, dynamic moduli, the complex viscosity, and the shear viscosity curve, for linear and slightly branched polyethylene . In the present work, the TDD model is updated to incorporate the influence of long-chain branches of LDPE on molecular entanglements and in the relaxation mechanism, using information about MWD and LCBD.
Each rheological model was added to a deterministic model of the high-pressure ethylene polymerization in tubular reactors previously developed by the authors . The reactor model predicts the MWD and the branching index, as well as several other molecular properties of LDPE, such as the bivariate molecular weight-long chain branching distribution (MWD-LCBD), the bivariate molecular weight-short chain branching distribution (MWD-SCBD),the weight-average molecular weight, the number-average molecular weight, and LCB/1000C. The calculated MWD and branching index are used as input for the rheological models. Model parameters corresponding to each rheological approach were adjusted using experimental data of shear viscosity of several LDPE samples. The results of both models were compared to evaluate the effectiveness of the predictions concerning the complexity and the computational time of each model.
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