Einstein Thermal Conductivity

The derivation presented in this section and the next is based on that given in [100, 52], with some of the mathematical steps given in more detail. The formulation of the Einstein thermal conductivity, kE, presented here is that given in [232], which is extended to arrive at the C-P high scatter limit, kCP, in Section C.2.

In the Einstein approach, the vibrational states do not correspond to phonons, but to the atoms themselves, which are assumed to be on a simple cubic lattice as shown in Figure C.1. As will be discussed, the choice of the crystal structure does not affect the final result. Each atom is treated as a set of three harmonic oscillators in mutually perpendicular directions. Although the atomic motions are taken to be independent, an atom is assumed to exchange energy with its first, second, and third nearest neighbors. The coupling is realized by modeling the atomic interactions as being a result of linear springs (with spring constant Γ) connecting the atoms. A given atom has 6 nearest neighbors at a distance of a, 12 second-nearest neighbors at a distance of 2½a, and 8 third-nearest neighbors at a distance of 3½a.

Heat transfer physics describes the thermodynamics and kinetics (mechanisms and rates) of energy storage, transport, and transformation by means of principal energy carriers. Heat is energy that is stored in the temperature-dependent motion and within the various particles that make up all matter in all of its phases, including electrons, atomic nuclei, individual atoms, and molecules. Heat can be transferred to and from matter by combinations of one or more of the principal energy carriers: electrons† (either as classical or quantum entities), fluid particles (classical particles with quantum features), phonons (lattice-vibration waves), and photons‡ (quasi-particles). The state of the energy stored within matter, or transported by the carriers, can be described by a combination of classical and quantum statistical mechanics. The energy is also transformed (converted) between the various carriers. All processes that act on this energy are ultimately governed by the rates at which various physical phenomena occur, such as the rate of particle collisions in classical mechanics. It is the combination of these various processes (and their governing rates) within a particular system that determines the overall system behavior, such as the net rate of energy storage or transport. Controlling every process, from the atomic level (studied here) to the macroscale (covered in an introductory heat transfer course), are the laws of thermodynamics, including conservation of energy.

Introduction

The extruder, shown schematically in Figure 1.1, is central to most melt processing operations. We can achieve considerable insight into the operation and design of single-screw extruders by remarkably simple models, despite the mechanical complexity. We begin this chapter by obtaining velocity, stress, and temperature distributions for flow in straight channels with parallel walls of “infinite” length. The infinite channel results are important in and of themselves, but we shall see here that they lead immediately to a model for the single-screw extruder as well. The results also provide an important framework for the modeling of flows in situations in which the walls are not parallel, which we address in Chapter 5.

Plane Channel

Stress Distribution

Let us suppose we have steady isothermal flow (i.e., the temperature is constant throughout the flow field and all ∂/∂ t = 0) between two infinite parallel planes, as shown in Figure 3.1. The flow is in the x direction. We assume for generality that there is a finite pressure gradient (∂ p/∂ x ≠ 0) and that the surface at y = 0 moves relative to the surface at y = H with a constant velocity V. We shall see subsequently that the results obtained here will form the foundation for the modeling of single-screw extrusion and the extrusion coating of flat sheets.

Introduction

The examples we have studied thus far have had rather simple kinematics: flow parallel or nearly parallel to a wall and ideal or nearly ideal extension. Thus, we have been able to obtain exact solutions for the flow or to obtain approximate solutions based on the small difference between the actual flow and an ideal case for which an exact solution is available. Even for the case of fiber spinning, where an analytical solution to the thin filament equations cannot be obtained under conditions relevant to industrial practice, we simply need to obtain a numerical solution to a pair of ordinary differential equations, which is a task that can be accomplished using elementary and readily available commercial software.

The flow in many real processing geometries is too complex for us to apply the analytical methods utilized in the preceding chapters. Indeed, even when the flow field is a simple one, the coupled heat transfer problem may not be amenable to a simple treatment; the elementary extruder in Chapter 3 is an example of a case in which we are unable to obtain an exact or even approximate solution for the spatial development of the two-dimensional temperature field.

Complex coupled flow and heat transfer problems can be solved using numerical techniques in which the partial differential equations are converted to a large set of coupled algebraic equations, and the algebraic equations are then solved using conventional methods developed specifically to be efficient on digital computers.

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.