Introduction

Viscoelasticity will clearly have a large effect in some processing operations and little or none in others, and we require a way to discriminate between these cases. One clue follows from the linear viscoelastic experiments shown in Figures 9.2 and 9.3 and the accompanying spectral description in Equations 9.11a–b. The entangled network is able to relax at low frequencies, so the elastic contribution to the stress is negligible and the deformation is mostly dissipative (G′ ‎ 0). The stress at high frequencies cannot relax, so dissipation is negligible and the deformation is recoverable (η′ ‎ 0). The transition between these two extremes is sharp for a liquid with a single Maxwell mode and occurs in the neighborhood of λω ~ 1.ω-1 is the characteristic time for the oscillatory deformation, so we may think of the two limiting cases as representing processes that are slow and fast, respectively, relative to the characteristic time of the fluid. The transition is murkier for most polymer melts, where there are many dynamical modes, but there will be some relaxation time – a mean value like that given by Equation 9.20 or the longest relaxation time in the spectrum – such that the same criterion can be usefully applied. The ratio of the characteristic time of the fluid to the characteristic time of the process is known as the Deborah number and is usually denoted De. The time scale for the process is usually the residence time.

Laminar Mixing

Mixing and blending in polymer processing applications almost always takes place in the laminar regime. The basic idea in laminar mixing is straightforward: Adjacent laminae of dissimilar materials are stretched – let us say doubled in length – so that the thicknesses of the laminae are reduced by a factor of two. The stretched sections are then folded back to create a block of the same thickness as the original, but it now contains four lamina instead of the original two. This process is repeated, and the number of lamina grows as 2N, where N is the number of stretching/folding steps, while the thicknesses decrease as 2-N. This process is known as the baker's transformation, for it is precisely the sequence of steps that is carried out in kneading a loaf of bread. (Push down to stretch, then fold back, turn 90°, and repeat.) Laminar mixing is related to the theory of chaotic dynamical systems, and it has been widely studied in the context of dynamical systems since the late 1980s.

The implementation of this methodology in polymer processing long predates the development of the theoretical tools currently in use for analysis. Static mixers are commonly employed to effect the baker's transformation. A cutaway view of a Kenics static mixer is shown in Figure 14.1. This device consists of a series of helically twisted blades that divide the circular channel into two twisted semicircular ducts.

Introduction

The preceding chapters addressed flows with a single velocity component that is parallel to the conduit walls. Most confined polymer processing operations are characterized by flows in thin gaps, but in many cases the walls are not parallel, so there must be more than one component of velocity. It is often the case, however, that the gap between the confining surfaces changes slowly in the direction of mean flow, a situation we call nearly parallel. Such flows can be treated analytically, and we can gain considerable insight into process performance and design. We will illustrate the approach in this chapter with an application of polymer coating of a sheet, but the methodology applies equally well to calendaring, extrusion, and compression and injection molding.

The analysis of nearly parallel flows originated in the study of problems of lubrication, and the approach is often called the lubrication approximation. The terminology is unfortunate from our perspective, given that this approach is at the heart of all analytical treatments of polymer processing operations – we would prefer that it be called the polymer processing approximation – but the historical name is well established. The major figure in the analysis of lubrication flows was Osborne Reynolds, and one widely used form of the resulting equations is often called the Reynolds lubrication equation.

Basic Equations, Newtonian Liquid

We restrict ourselves to two-dimensional flows, where all changes occur in the xy plane and there is no flow in the “neutral” z direction.

Most of the shaping in the manufacture of polymeric objects is carried out in the melt state, as is a substantial part of the physical property development. Melt processing involves an interplay between fluid mechanics and heat transfer in rheologically complex liquids, and taken as a whole it is a nice example of the importance of coupled transport processes. This is a book about the underlying foundations of polymer melt processing, which can be derived from relatively straightforward ideas in fluid mechanics and heat transfer; the level is that of an advanced undergraduate or beginning graduate course, and the material can serve as the text for a course in polymer processing or for a second course in transport processes. The book is based on a course that has evolved over thirty years, which I first taught at the University of Delaware and subsequently at the University of California, Berkeley; the Hebrew University of Jerusalem; and the City College of New York. The target audience is twofold: engineers and physical scientists interested in polymer processing who seek a firm command of basic principles without getting into details of the process geometry or the fluid rheology, and students who wish to apply the basic material from courses in transport processes to practical processing situations. The only background necessary is some prior study of the fundamentals of fluid flow and heat transfer and a command of mathematics at a level typically expected of an advanced undergraduate student in engineering or the physical sciences; the text is otherwise self-contained.

This is the conclusion of the book, and an appropriate point to look back and reflect. Our goal throughout has been to establish the foundations of polymer melt processing in fluid mechanics and heat transfer without introducing unnecessary complexity. In doing so we have avoided geometrical detail of the equipment; such detail is important for specific applications, but its inclusion adds little to our overall understanding of the essential interplay between fluid mechanics and heat transfer in basic process performance, which was our primary objective. Similarly, we initially developed the subject in terms of the flow of inelastic liquids; many polymer processes are characterized by a low Deborah number, either as a consequence of the nature of the flow or the properties of the polymer being processed, and the essential behavior in this case does not depend on the fact that the melt is viscoelastic. We subsequently introduced viscoelasticity for those applications where it is needed, using viscoelastic constitutive equations that have been found to be effective in describing melt flow in complex geometries but fall short of the state of the art in polymer rheology. Viscoelasticity can be quite significant in some processing situations, notably in steady flows with substantial elongation and in all flows when dynamical response is of interest, and the rôle of viscoelasticity – when it is important and when it is not – must be understood for a complete and accurate picture.

Introduction

Many polymeric liquids have a microstructure even at rest. This might be a consequence of the presence of dispersed particulates or, in the case of liquid crystalline polymers, because of the rigidity of the polymer molecules. Continuum equations describing the stress and microstructure evolution are available for some limiting cases, permitting calculations of flow in complex geometries. The levels of description of the stress states are not comparable to that for entangled flexible polymer melts, so the resulting calculations are less likely to be in quantitative agreement, but they are still very useful for gaining insight into the development of morphology. We address three cases of structured fluids in this chapter: fiber suspensions, such as those that might be used for thermoplastic composites; liquid crystalline polymers; and fluids that exhibit a yield stress, which might include nanoparticle-filled melts.

Fiber Suspensions

The continuum approach to the rheology of fiber suspensions is based on a 1922 solution by Jeffery for the creeping-flow mechanics of a single ellipsoid in a shear flow. The ellipsoid rotates in a nonsinusoidal fashion, spending most of the period near a fixed angle to the flow direction. The ellipsoid aligns with the flow direction at all times in the limit of an infinite aspect ratio. The key assumptions in deriving a constitutive equation for a fiber suspension from Jeffery's result for the ellipsoid are that the suspending fluid is Newtonian and the suspension is dilute.

Introduction

Our analysis of polymer melt processing operations has thus far assumed that the polymer melt can be described as an inelastic liquid, and in fact we have generally assumed for simplicity that the melt is Newtonian. An inelastic liquid has no memory; that is, the stress in the fluid at a given time and place depends only on the deformation rate at that time and place. Entangled polymers should have memory, since the response to a deformation must depend on the reorganization of the entangled macromolecules, which cannot be instantaneous. We saw a manifestation of such memory in Figure 1.8, where a silicone polymer being squeezed between two plates under constant force “bounced,” causing transient increases in the gap spacing. Another way to think about memory is to imagine the polymer melt at rest, with the chains forming an entangled network. The chains cannot respond instantaneously if we attempt to deform the melt rapidly because they are entangled, so the initial short-time response must be that of a rubberlike network, not a viscous fluid, including shape recovery if the stress causing the deformation is quickly removed. In general, we expect to see a superposition of two responses: the short-time rubberlike response caused by deformation of the entangled network and the long-time viscous response caused by the dissipative process of relative chain motion in the flowing melt. Hence, polymer melts are viscoelastic liquids.

Introduction

Our discussion of continuous processes like extrusion and spinning has focused thus far on steady operation. The dynamical response of these processes is also an important processing consideration. The field of process dynamics has paid little attention to polymer processing, other than to apply classical control system methodology to implement temperature control loops. In particular, models of continuous processes have not been used extensively, and there is considerable scope for dynamical analyses to improve operation and control.

There are two fundamental issues in considering the dynamics of a process. One is operational stability: If we design a process to operate under given conditions, and the process moves away from the design conditions for any reason, will it ultimately return or will it move further away? The other is operational sensitivity: If the process is operating under the design conditions, and disturbances enter the system, will the disturbances attenuate or will they grow as they propagate through the process? These are different questions, although they are often treated as the same because, up to a point, they share a common mathematical framework.

The dynamics of melt spinning has received more attention than any other process, and it is the primary focus of this chapter, although other processes are also briefly addressed. Instabilities in rectilinear flow through an extrusion die are confounded by issues regarding boundary conditions at high stress levels, and they are addressed in the next chapter.

Introduction

The processes we have considered thus far – extrusion, wire coating, and injection and compression molding – are dominated by shear between confined surfaces. By contrast, in fiber and film formation the melt is stretched without confining surfaces. It is still possible to gain considerable insight from very elementary flow and heat transfer models, but we must first parallel Section 2.2 and develop some basic concepts of extensional flow. The remainder of the chapter is then devoted to an analysis of fiber formation by melt spinning.

Our analysis of fiber spinning in this chapter will be based on an inelastic rheological model of the stresses. This rheological description appears to be adequate for polyesters and nylons, which comprise the bulk of commercial spinning applications, and our spinning model is essentially the one used in industrial computer codes. This is a process in which melt viscoelasticity can sometimes play an important role, however, and we will revisit the process in Chapter 10.

Uniaxial Extensional Flow

Consider a cylindrical rod of a very viscous polymer melt, as shown in Figure 7.1, with radius R and length L. We impose a stress σzz in the axial direction in order to stretch the rod; hence, R and L are both functions of time, but R2L is a constant for an incompressible melt. We assume that the rod draws down uniformly as it is stretched, so R is independent of z.

Introduction

Polymeric materials – often called plastics in popular usage – are ubiquitous in modern life. Applications range from film to textile fibers to complex electronic interconnects to structural units in automobiles and airplanes to orthopedic implants. Polymers are giant molecules, consisting of hundreds or thousands of connected monomers, or basic chemical units; a polyethylene molecule, for example, is simply a chain of covalently bonded carbon atoms, each carbon containing two hydrogen atoms to complete the four valence sites. The polyethylene used to manufacture plastic film typically has an average molecular weight (called the number-average molecular weight, denoted Mn) of about 29,000, or about 2,000 –CH2– units, each with a molecular weight of 14. The symbol “–” on each side of the CH2 denotes a single covalent bond with the adjacent carbon atom. (The monomer is actually ethylene, CH2=CH2, where “=” denotes a double bond between the carbons that opens during the polymerization process, and a single “mer” is –CH2–CH2–; hence, the molecular weight of the monomer is 28 and the degree of polymerization is about 1,000.) The ultra-high molecular weight polyethylene used in artificial hips and other prosthetic devices has about 36,000 –CH2– units. Polystyrene is also a chain of covalently bonded carbon atoms, but one hydrogen on every second carbon is replaced with a phenyl (benzene) ring. Two or more monomers might be polymerized together to form a copolymer, appearing on the chain in either a regular or random sequence.

Introduction

The creeping flow approximation to the momentum equation, which we obtain in a formal way by setting ρ = 0 on the left side of the equations in Table 2.2 or 2.4, has the interesting property that time never appears explicitly. Thus, creeping flow solutions to time-dependent problems are quasi-steady in the sense that they correspond to the steady-state solution for the given geometry at each time. This property can be exploited to obtain analytical solutions to simple transient problems in mold filling, and the same concepts are utilized for numerical solutions to more complex problems. We illustrate the use of the quasi-steady character of the creeping flow equations with two model mold filling problems, one in injection molding and one in compression molding.

Center-Gated Disk Mold

Isothermal Newtonian Liquid

A mold to form a thin circular disk is shown in Figure 6.1. Molten polymer is fed through a small circular hole at the center of the mold (the gate) and then flows out radially to fill the mold cavity. We assume that the mold is vented, allowing air to escape as the polymer fills the cavity, so the pressure at the polymer/air interface is always close to atmospheric. The disk has a thickness H and a radius RD. The radius of the circular gate is RG. The pressure at the gate is Po, and the polymer enters with a volumetric flow rate Q; Po and Q may vary with time.

Introduction

Polymer flow in any melt processing geometry is governed by three fundamental principles of physics: conservation of mass, conservation of linear momentum, and conservation of energy. Linear momentum is a vector quantity that must be conserved in each of three independent coordinate directions, so we must expect five conservation statements in the most general case. The natural language of these conservation statements is differential and integral calculus (recall that Newton invented the calculus to enable him to describe problems of motion); in particular, because we have four independent variables – time and three spatial variables – our language will employ partial derivatives, and the conservation equations will be stated as partial differential equations. The problems we will address in this text do not generally require familiarity with methods of solution of partial differential equations because the equations will usually simplify to forms that can be analyzed using elementary concepts of the calculus of one independent variable. (An apt linguistic analogy might be the contrast between understanding basic prose – our task here – and writing poetry.) Hence, the subject is open to any student who has completed a basic sequence in calculus.

The conservation equations are derived in many textbooks on fluid mechanics and transport phenomena, and we shall simply state them here, with an explanation of the meanings of terms where appropriate.

INTRODUCTION

The two combustion modes of deflagration (flame) and detonation can be generally distinguished from each other in a number of ways: by their propagation speed, the expansion (deflagration) versus compression (detonation) nature of the wave, subsonic (deflagration) versus supersonic (detonation) speed relative to the mixture ahead of the wave, and the difference in the propagation mechanism. A deflagration wave propagates via the diffusion of heat and mass from the flame zone to effect ignition in the reactants ahead. The propagation speed is governed by heat and mass diffusivity, and the diffusion flux is also dependent on the reaction rate that maintains the steep gradient across the flame. On the other hand, a detonation wave is a supersonic compression shock wave that ignites the mixture by adiabatic heating across the leading shock front. The shock is in turn maintained by the backward expansion of the reacting gases and products relative to the front, thus providing the forward thrust needed to drive the shock. A propagating flame generally has a precursor shock ahead of it, and the flame can therefore propagate at supersonic speeds with respect to a stationary coordinate system. In theory, there is still a pressure drop across the flame itself, but there may still be a net pressure increase in the products across the shock–flame complex with respect to the initial pressure of the reactants.