Feature Article:May 1, 2000

Progress and Challenges in the Process Modeling
of Multiphase Polymer Systems

Chris E. Scott
Associate Professor of Polymer Engineering
Department of Materials Science and Engineering
Massachusetts Institute of Technology



Overview

Multiphase polymer systems such as blends and composites provide unique advantages for enhancing material value. As with any multiphase material, performance is critically dependent on the structure or morphology in the final part. Significant progress has been made in modeling the influence of processing on the morphology of multiphase systems. However, substantial challenges remain.

Introduction

There are numerous examples of multicomponent polymer systems where key performance characteristics depend critically on the material morphology. For instance, the phase domain morphology in a multicomponent polymer blend has a critical influence on the mechanical properties of the material. Wu [1985] shows the effect of dispersed phase size on the impact strength of nylon/rubber blends. This morphology is controlled by the properties of the constituent polymers and the processing conditions utilized in the manufacturing operation. In general, interest is not limited to the size scale of the morphology because control of domain shape can impart useful properties, such as reinforcing effects for cylindrical domains.

The vast majority of models for processing operations such as injection molding, single screw extrusion, twin screw extrusion, and die flows consider the material to be a homogeneous continuum. Substantial advances have been made in modeling these processes and this approach is quite successful for the determination of velocity, stress, and temperature fields along with useful macroscopic characteristics like pressure drop and flow rate. Commercial software packages are routinely used for equipment and process design. Unfortunately, with rare exceptions, these models do not even consider the morphology of the material and its interaction with the process.

The next major advance in polymer process modeling will be the inclusion of morphological characteristics, including both changes during the process itself and morphology of the final part. Large amounts of both experimental and numerical work are required before this goal can be achieved and substantial efforts are currently being made in a number of research laboratories. From a more encompassing perspective, this major advance is just one of the steps required to complete the process-structure-property connection for multiphase polymer systems. Due to the great breadth of this field, the following sections are intended to provide examples of progress and challenges rather than a comprehensive review.

Progress

Probably the most successful implementation of morphological simulation in polymer process modeling has been the prediction of fiber orientation in injection molding and compression molding. The basis for such predictions can be found in Advani and Tucker [1987, 1990] and Bay and Tucker [1992]. Commercial injection molding simulation software has provided this capability for some time. Although fiber orientation is only one aspect of composite morphology, it is critical for mechanical properties and thus its prediction is quite useful for part and process design. Other interesting morphological changes in chopped fiber composites such as attrition [Wolf, 1994] and dispersion have yet to be studied or modeled thoroughly.

Dispersion of carbon black in rubbers during batch intensive mixing is another area which has been successfully modeled [Manas-Zloczower et al., 1982, 1985, 1994]. The combination of a macroscopic description of the flow field along with a microscopic description of carbon black agglomerate strength and rupture constitutes the core of this approach, as with other fillers and reinforcements [Potente and Flecke, 1997].

Challenges

The commercial importance of polymer blends has led to substantial efforts to model multiphase fluid flow in polymer processing. Construction and implementation of these models is particularly difficult due to the desire to capture domain deformation, breakup, and coalescence in complex flow fields. Unfortunately, appropriate tools for modeling of multiphase fluid flows under conditions relevant to polymer processing operations do not currently exist. Most modeling capabilities are limited to predictions of droplet deformation and breakup in infinitely diluted, monodisperse Newtonian systems [Utracki and Shi, 1992]. This is a severe deficiency, considering the fact that most commercially relevant systems involve significant coalescence, high dispersed phase concentrations, and non-Newtonian fluids. Lack of fundamental understanding and applicable models impedes technological progress in this field. For example, new mixing devices in the polymer industry are currently designed on a trial-and-error basis. Successful development of a model capable of accurately simulating multiphase flows in polymer processing would dramatically advance our understanding of these processes as well as our ability to design and optimize them.

Computer simulations have recently been able to address many fundamental flow problems which include complicating features like inertial effects, multiple components that may be only partially miscible, and viscoelasticity. Molecular dynamics is probably the most fundamental numerical approach to fluid problems and it has been applied to problems like the determination of appropriate boundary conditions for the interaction of a fluid with a wall [Koplik and Banavar, 1998]. For simulations of flows on a scale much larger than the particle scale, however, molecular dynamics is both impractical and wasteful.

Most numerical fluid simulations rely on discretizations of the Navier-Stokes equations or some approximation of these equations in finite element or boundary integral methods and do so with great success. These simulations can be extended to include two-component systems, for example with the help of boundary-tracking methods [Falcovitz et al., 1997] or the boundary integral method even when viscoelastic effects are included [Noh et al., 1993]. Innovations such as implementation of diffuse interface theory [Verschueren et al., 1998, 1999] or emphasis of averaged global parameters such as the anisotropy tensor, interfacial area per unit volume, or area tensor [Doi and Ohta, 1991; Grmela and Ait-Kadi, 1994; Lee and Park, 1994; Wetzel and Tucker, 1999] provide promising new approaches to this problem.

During the last ten years a different type of simulation has developed starting from lattice gas automata [Frisch et al., 1986]. In these systems particles stream along lattice vectors in a discrete, spatial lattice and collide on lattice nodes. One can think of these systems as simplified microscopic models that are designed to exhibit the appropriate macroscopic behavior without devoting substantial resources to calculating the details a molecular dynamics calculation would provide when these are not essential to the phenomenon one is interested in. The averaged continuum behavior of these systems is governed by a mass and a momentum conservation equation which, apart from possible lattice artifacts, resembles the continuity and momentum conservation equations used in fluid dynamics. Lattice gas automata have the advantage of being unconditionally stable, but it is difficult to remove all lattice artifacts. This is because the collisions are severely constrained by the conservation laws due to the discreteness of the particles. Therefore particles were replaced by continuous particle densities in the lattice Boltzmann method, which greatly enhances the flexibility of the collision term.

Other new methods that rely on a microscopic representation of the fluid rather than a discretization of partial differential equations are dissipative particle dynamics [Hogerbrugge and Koelman, 1992], the smoothed particle method [Okuzono, 1997], and the Malevanets method [Malevanets and Kapral, 1999]. All these models have in common that they, like molecular dynamics, do not rely on a lattice and therefore do not suffer from lattice artifacts.

Because all of these new methods rely on some kind of microscopic dynamic it is not always easy to predict their macroscopic behavior. Expansion techniques equivalent to the Chapman Enskog method can be used to calculate the continuum equations that approximate their macroscopic behavior in some limit. But it is often much easier to introduce a certain physical phenomenon by introducing sensible microscopic rules. This is why these methods have been very successful for complicated applications like multiphase flow [Hogerbrugge and Koelman, 1992; Orlandini et al., 1995] and in the case of dissipative particle dynamics and the Malevanets method also viscoelastic fluids.

Lattice Boltzmann methods have been shown to be very effective for the calculation of two component systems, especially when handling complex geometries or complex morphologies [Wagner and Yeomans, 1997; 1998; 1999; Chen and Doolen, 1998]. For lattice Boltzmann methods, however, it is not a priori clear how one might represent polymers. It was all the more surprising when Giraud et al. [1997] first showed that lattice Boltzmann methods can have viscoelastic properties.

One goal of work in our laboratory has been to develop the capability of accurately modeling multiphase fluid flows in polymer processing using the lattice-Boltzmann method. We have enhanced the lattice-Boltzmann method itself in order to address issues of particular interest in polymer blends, such as viscoelasticity. This has required combining the multiphase approach of Orlandini et al. [1995] with the viscoelastic implementation of Giraud et al. [1998]. Application of this method has been demonstrated on two viscoelastic flow problems.

A bubble rising in a viscoelastic fluid exhibits several characteristics which are quite distinct from its behavior in a Newtonian fluid [Bird et al., 1987; Joseph et al., 1991]. Our simulations have succeeded in reproducing the cusp at the end of the bubble, as shown in Figure 1. Qualitative changes in bubble shape with driving force have been reproduced. In addition, our simulations suggest that the experimentally observed "velocity jump" is not due to improved streamlining upon formation of the cusp.

In addition, our lattice-Boltzmann simulations have demonstrated substantial effects of viscoelasticity in spinodal decomposition of binary mixtures. The late-time scaling state in spinodal decomposition is shown to be non-unique. Viscoelasticity even in just the early stages of the process influences not only the scaling exponent but also the connectedness of the domains, Figure 2. Very different morphologies are generated by mechanical mixing and spinodal decomposition. The results suggest that fine-tuning of the morphology is possible through a combination of mechanical mixing and spinodal decomposition.

Of course, even a complete computer model for viscoelastic multiphase fluid flow would not be sufficient for polymer blending. Manufacturing operations require solids conveying and melting as well and, unfortunately, our understanding and modeling capability for these regimes is even less developed than for melt flow. Experimental work is necessary to identify the key mechanisms of morphological change. This then provides the foundation for computer models of the process. Complete models for polymer blending including all three of the classical processing regimes: solids conveying, melting, and melt flow are in their infancy [Scott, 1996; Potente and Bastian, 1997; Potente et al., 1999; Ratnagiri, 2000]

Conclusion

The next major advance in polymer process modeling will be the inclusion of morphological characteristics, including both changes during the process itself and morphology of the final part. While significant progress has been achieved, substantial challenges remain. This major advance is just one of the steps required to complete the process-structure-property connection for multiphase polymer systems.

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