To be able to understand a polymer's properties, it is essential to have a good understanding of polymer behavior, which greatly differs from that of other materials. Polymer behavior is described with the aid of constitutive equations. The constitutive equations used to describe polymers and polymer flow are relatively complex. These equations are almost always nonlinear. They needs to describe nonlinear phenomena such as normal stress differences, shear thinning, and extensional thickening.
The general nonlinear behavior of a material is characterized by finding a fitting constitutive equation for that material. The appropriateness of such an equation depends on many factors such as the equation's accuracy in predicting data, its simplicity, the soundness of its mathematical basis, and the range of phenomena one wishes to address. The weighting of these factors is very subjective, resulting in a wide variety of constitutive equations over the years, many of which are still in use.
In the development of new products, computer simulations are becoming increasingly more important. With the aid of computer simulations it is possible to process a polymer and test whether its properties meet the requirements needed for a particular application. All in a virtual environment. It is no longer needed to produce and test the polymer in the laboratory. For these computer simulations to be applicable, we need accurate underlying mathematical models that describe how polymers behave under certain conditions.
From the wide range of constitutive equations available in literature, only a few are selected for this project. They are the PTT model presented by Phan-Thien and Tanner, the pom-pom model presented by McLeish and Larson, and the newly developed double-convected pom-pom (DCPP) model presented by Clemeur et al. Before we can use these three models, we need a solver that is both robust and efficient. A solver exists for the pom-pom model and has been implemented in an existing software package. A solver for the PTT and the DCPP models still need to be developed and implemented.
It is chosen to estimate the model parameters for all the models investigated in this project with a particle swarm optimizer, from a data set containing rheological information. The parameters for the pom-pom model are currently estimated manually, which is a time-consuming and subjective process. The particle swarm optimizer automates the process of parameter estimation. The swarm optimizer calculates the model parameters by a search routine. To calculate the parameters, we need to link the pom-pom solver implementation to the swarm optimizer. Once we have developed and implemented the PTT and DCPP solver we estimate the parameters of both models by linking the solver implementations to the swarm optimizer.
After obtaining the parameters for the pom-pom, the DCPP, and the PTT models, we test the models by means of extrudate swell simulations in a finite element package, namely Polyflow. Unfortunately, only the DCPP and PTT modes are implemented in Polyflow. Hence it is not possible to test the pom-pom model with Polyflow. The DCPP and PTT models need to be tested to confirm statements about the their application made in literature, and to investigate the influence of numerical procedures, mesh generation, and parameter estimations on complex flow geometry. In the complex flow simulation, i.e. the extrudate swell simulation, we consider only the die and the extrudate. The barrel and the plunger preceding the die will not be considered. We test the performance of the DCPP and the PTT models by comparing the extrudate geometry from the simulation results with experimental results.
We estimate the parameters in the pom-pom, DCPP and PTT models by means of the particle swarm optimizer. From this implementation we are able to fit the pom-pom, DCPP and PTT model parameters. The results found for the parameters compare well with results found in literature. The results are obtained within an acceptable number of iterations. From our results we conclude that the swarm optimizer is an efficient tool for parameter estimation.