Simple Models in Polymer Processing

In addition, the continuity equation also tells us that the two inertia terms in the x-momentum equation are of similar magnitude, i.e., 

There are only a few exact or analytical solutions of the momentum balance equations, and most of those are for situations in which the flow is unidirectional that is, the flow has only one nonzero velocity component. Some of these are illustrated below. We end the section with a presentation of the, which today is widely accepted to model the flows that occur during mold filling processes. 

---

The field of transport phenomena is the basis of modeling in polymer processing. This chapter presents the derivation of the balance equations and combines them with constitutive models to allow modeling of polymer processes. The chapter also presents ways to simplify the complex equations in order to model basic systems such as flow in a tube or Hagen-Poiseulle flow, pressure flow between parallel plates, flow between two rotating concentric cylinders or Couette flow, and many more. These simple systems, or combinations of them, can be used to model actual systems in order to gain insight into the processes, and predict pressures, flow rates, rates of deformation, etc. 

In this example of model reactive polymer processing of two immiscible blend components, as with Example 11.1, we have three characteristic process times tD,, and the time to increase the interfacial area, all affecting the RME results. This example of stacked miscible layers is appealing because of the simple and direct connection between the interfacial layer and the stress required to stretch the multilayer sample. In Example 11.1 the initially segregated samples do create with time at 270°C an interfacial layer around each PET particulate, but the torsional dynamic steady deformation torques can not be simply related to the thickness of the interfacial layer, <5/. However, the initially segregated morphology of the powder samples of Example 11.1 are more representative of real particulate blend reaction systems. 

In Section 5.2.1.1 we provide an overview of the classical treatment of polymerization kinetics. Some aspects of termination kinetics are not well understood and no wholly satisfactory unified description is in place, Nonetheless, it remains a fact that many features of the kinetics of radical polymerization can be predicted using a very simple model in which radical-radical termination is characterized by a single rale constant. The termination process determines the molecular weight and molecular weight distribution of the polymer. In section... 

The modern discipline of Materials Science and Engineering can be described as a search for experimental and theoretical relations between a material s processing, its resulting microstructure, and the properties arising from that microstructure. These relations are often complicated, and it is usually difficult to obtain closed-form solutions for them. For that reason, it is often attractive to supplement experimental work in this area with numerical simulations. During the past several years, we have developed a general finite element computer model which is able to capture the essential aspects of a variety of nonisothermal and reactive polymer processing operations. This "flow code" has been Implemented on a number of computer systems of various sizes, and a PC-compatible version is available on request. This paper is intended to outline the fundamentals which underlie this code, and to present some simple but illustrative examples of its use. 

Radioprotection. The processes of crosslinking and degradation observed in polymers irradiated in the pure state can also be observed in polymers irradiated in solution. The presence of a solvent can intervene in the reaction in several ways thus it allows increased polymer mobility, and some of the radiolytic products of the solvent may react with the polymer or with the polymer radicals, etc. The polymer-water system is of particular interest in that it provides a simple model for some radiobiological systems and can be analyzed far more readily. 

The most simple model uses the gas-phase concentration of the monomer as driving force. Alternatively, if capillary condensation of monomers is to be taken into account, the monomer concentration can be calculated directly from the liquid density. A sorption balance, for example, should be used to measure the constants of Equations (5.4-7) and (5.4-8), because of their dependence on the changing polymer structure formed in the polymerization process. 

In this paper, an overview of the origin of second-order nonlinear optical processes in molecular and thin film materials is presented. The tutorial begins with a discussion of the basic physical description of second-order nonlinear optical processes. Simple models are used to describe molecular responses and propagation characteristics of polarization and field components. A brief discussion of quantum mechanical approaches is followed by a discussion of the 2-level model and some structure property relationships are illustrated. The relationships between microscopic and macroscopic nonlinearities in crystals, polymers, and molecular assemblies are discussed. Finally, several of the more common experimental methods for determining nonlinear optical coefficients are reviewed. 

During synthesis of a polymer, particularly of polyurethane, gaseous products can appear. Therefore, a complete model of the process must take into account (at least in some cases) the possibility of local evaporation and condensation of a solvent or other low-molecular-weight products. Such a complex model is discussed for chemical processing of polyurethane that results in formation of integral foams in a stationary mold.50 In essence, the model is an analysis of the effects of temperature in a closed cell containing a solvent and a monomer. An increase in temperature leads to an increase in pressure which influences the boiling temperature of the solvent and results in an increase in cell volume. The kinetics of polymerization is described by a simple second-order equation. The... 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

Although all polymer processes involve complex phenomena that are non-isothermal, non-Newtonian and often viscoelastic, most of them can be simplified sufficiently to allow the construction of analytical models. These analytical models involve one or more of the simple flows derived in the previous chapter. These back of the envelope models allow us to predict pressures, velocity fields, temperature fields, melting and solidification times, cycle times, etc. The models that are derived will aid the student or engineer to better understand the process under consideration, allowing for optimization of processing conditions, and even geometries and part performance. 

---