Energy is stored within atoms (electrons and relativistic effects) and in the bonds among them when they form molecules, and also between these molecules when they form clusters and bulk materials. In this chapter we review the intraatomic and interatomic forces and energies (potentials). We examine electron orbitals of atoms and molecules and review ab initio computations of interatomic potentials and potential models. Then we discuss the dynamics (and scales) of molecular (or particle) interactions in classical (Newtonian and Hamiltonian) treatments of multi-body dynamics and the relationship between the microscopic and macroscopic states (particle ensembles). We also begin examining the quantum energy states of the simple harmonic oscillator, free-electron gas, and electrons in hydrogen atom orbitals, by seeking exact solutions to the Schrödinger equation.

Interatomic Forces and Potential Wells

Atoms and molecules, at sufficiently low temperatures, can aggregate into linked structures that form what we know as the condensed phases of matter. What causes this aggregation behavior are the attractive forces generated by the opposite electrical charges of each particle (be it individual atom, ion, or polar molecule) in the system. These forces are in turn caused by the innate charges of the constituent subatomic particles of each atom, ion, or molecule. All physical and chemical bonding is the result of such molecular forces that are electrical in origin (the Coulomb forces).

The solid electric thermal conductivity tensor Ke, in addition to the phonon thermal conductivity tensor (i.e., total conductivity K</b< = Ke = Kp), determines heat conduction in solids through the Fourier law qk = −K · ∇T. The average heat capacity of an electron cv,e is small, except at high temperatures. Electrons can also have a net motion under an applied electric field ee, thus creating opportunities for exchange of their gained kinetic energy, e.g., with the lattice through inelastic scattering in Joule heating. The coupling of electronic and thermal transport, known as thermoelectricity, leads to Peltier heating/cooling.

In Section 2.6.5, we examined the electronic energy states of an idealized electron gas by solving the Schrödinger equation for the case of a collection of free electrons. In Section 2.6.6 we also derived the electronic energy states of hydrogen-like atoms, along with the designation of the quantum numbers and atomic orbitals. As atoms gather in a cluster or a bulk phase, their orbiting electrons and their energy states are altered because of various nuclear and electronic interactions (Section 2.2), including representation as interatomic potentials. These interactions may increase or decrease the energy gaps between the electron orbital states of individual atoms. The electrons can gain sufficient energy to be free (conduction) electrons or lack this and be bounded (valence) electrons. If a significant electronic energy gap exists between these two electron states, then this cluster or bulk matter may become an electrical insulator.

The Fermi-Dirac (fermion) and Bose-Einstein (boson) statistics include quantum effects and apply to interacting, indistinguishable particles (Table 1.2). The M- B statistics apply to noninteracting indistinguishable particles (classical particles), whose wave functions do not overlap and quantum effects vanish (footnote of Section 2.6.5). When the particle concentration is much less than the quantum limit, quantum effects will vanish and all the particles can be treated as classical particles. The ratio of the particle concentration and the quantum limit is called the quantum concentration (footnote of Section) nq, which is defined by

where N is the number of particles, V is the volume of the system, m is the mass of the particle, and T is the temperature. Therefore both fermions and bosons become the M-B statistics at high temperatures or low concentrations.

Derivation of the particle (including quantum) statistics distribution functions are given in [119, 239].

Partition Functions

For different ensembles [Section 2.5.1(A)], the partition function may have different forms. For a canonical ensemble, the partition function (2.27) is defined as

where j designates the energy state of the system.

For a grand canonical ensemble (μ VT), the partition function is defined as

where N is the number of particles, and when it is a constant, then μ = 0, and Z(μ, V, T)becomes Z(N, V, T), relating (F.2) and (F.3).

Fluid particle refers to matter in a gas or liquid phase, with the particle being the smallest unit (made of atoms or molecules) in it, for which further breakdown would change the chemical identity of the particle. For a fluid in motion, the convection heat flux vector qu = ρ f cp, f u f T and the surface-convection heat flux vector qku (which describes the interfacial heat transfer between two phases in relative motion, in which at least one phase is a fluid) are influenced by the specific heat capacity cp, f of the fluid particle (whereas qku also depends on the isotropic fluid thermal conductivity kf, viscosity μf, and velocity). The fluid (gas or liquid) velocity u f can be subsonic or supersonic, and for contained gases at low pressures or in small spaces, it is possible for fluid particle–surface collisions to dominate over the interparticle collisions. In this chapter, we examine energy storage and transport in fluids, as well as fluid interactions with surfaces (and the associated fluid flow regimes).

Fluid particles can have five types (forms) of energy: potential, electronic, translational, vibrational, and rotational (Figure 1.1). The electronic energy is part of potential energy, however, here we use the potential energy for interparticle interactions only. Each form of energy can be considered separately, and the total energy is the summation of these five energies. The maximum vibrational and rotational energies a molecule can attain are limited by the dissociation energy ϕe.

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.