Introduction to Polymer Rheology and Processing

Nicholas P. Cheremisinoff

• Publisher: CRC Press
• Year: 2018
• Print ISBN: 9781315894720
• eBook ISBN: 9781351090728
• Topic: Physical Sciences
• Subtopic: Industrial & Technical Chemistry

An Introduction to Polymer Rheology and Processing is a practical desk reference providing an overview of operating principles, data interpretation, and qualitative explanation of the importance and relationship of rheology to polymer processing operations. It covers full-scale processing operations, relating industrial processing operations and design methodology to laboratory-scale testing. Hundreds of design formulas applicable to scaling up the processing behavior of polymeric melts are presented. The book also provides a "working knowledge" description of major rheological test methods useful in product development and includes a useful glossary of polymer and test method/instrumentation definitions. Lavishly illustrated and featuring numerous sample calculations and modeling approaches, An Introduction to Polymer Rheology and Processing is a "must have" book for polymer engineers and rheologists.

Chapter 1
Introduction to Rheology

I. General Remarks

The subject of rheology is concerned with the study of flow and deformation. From a broad perspective, rheology includes almost every aspect of the study of the deformation of matter under the influence of imposed stress. In other words, it is the study of the internal response of materials to forces. Between the extremes of the conceptual views of the Newtonian fluid and the Hookean solid lie materials of great interest. Commercial interest in synthetic polymeric materials has given the greatest impetus to the science of rheology.

When a small stress is suddenly exerted on a solid, a deformation begins. The material will continue to deform until molecular (internal) stresses are established, which balance the external stresses. The term “deformation” refers to the equilibrium deformation that is established when the internal and external stresses are in balance. Most solids exhibit some degree of elastic response, in which there is complete recovery of deformation upon removal of the deforming stresses. The simplest such body is the Hookean elastic solid, for which the deformation is directly proportional to the applied stress. Elastic response may also be exhibited by non-Hookean materials, for which the deformation is not linearly related to the applied stress.

Not all materials reach an equilibrium deformation. In a fluid, if an external stress is exerted, deformation occurs, and continues to occur indefinitely until the stress is removed. A fluid response is one in which no resistance to deformation occurs. Internal frictional forces retard the rate of deformation, however, and an equilibrium can be established in which the rate of deformation is constant and related to the properties of the fluid. The simplest such fluid is the Newtonian, in which the rate of deformation is directly proportional to the applied stress. Many fluids exist which exhibit a nonlinear response to stress and are referred to as non-Newtonian fluids. Most synthetic polymer solutions and melts exhibit some degree of non-Newtonian behavior.

Between the extremes of elastic and fluid response lies a spectrum of combinations of these basic types of material behavior. For example, there is plastic response, wherein a material deforms like an elastic solid as long as the applied stress is below some limit, called the yield stress. If the applied stress exceeds the yield stress the material behaves as a fluid. A common example is paint. Brushing imposes stresses sufficiently large that paint behaves like a fluid. When paint lies on a vertical surface in a thin film, however, the stresses that arise from the weight of the fluid are below the yield stress, and the paint remains on the surface to dry as a uniform film.

Another important class of materials is the viscoelastic fluid. Such a material resists deformation, but at the same time resists a time rate of change of deformation. These materials exhibit a combination of both elastic and fluid response.

A materials response to a stress not only depends on the material, but also on the time scale of the experiment. For example, water behaves like a Newtonian fluid in ordinary experiments, but, if subjected to ultrahigh frequency vibrations, it will propagate waves as if it were a solid. The reason for this apparent change in the type of behavior lies in the fact that response is ultimately molecular in nature, and involves the stretching of intermolecular bonds and the motion of molecules past one another. In general, bonds can be stretched very quickly by an imposed stress, since little motion is involved. On the other hand, considerably more time is involved in causing molecules to “flow”. Thus, in a stress field with a very short time scale (high frequency vibration is an example) the stress may reverse itself before molecules have time to move appreciably, and only mechanisms giving rise to elasticity may have time to be excited. Because behavior can depend upon the type of stress field imposed, it is important that both the rheological properties of a material as well as the range of conditions over which these properties were measured be recorded. Only in this way can rheological properties be used with any assurance that they have pertinence to the application at hand.

II. Constitutive Equations

The response of any element of a body to the forces acting upon that element must satisfy the principle of conservation of momentum. In a continuum this principle is embodied in the “dynamic equations”, written in Cartesian coordinates as

$\text{ρ}(\frac{\partial {\text{ν}}_{\text{i}}}{\partial \text{t}}+{\text{υ}}_{\text{j}}\frac{\partial {\text{ν}}_{\text{i}}}{\partial {\text{x}}_{\text{j}}})=\text{ρ}{\text{f}}_{\text{i}}+\frac{\partial {\text{τ}}_{\text{ij}}}{\partial {\text{x}}_{\text{j}}}(1)$

for an incompressible fluid of density ρ subject to an external force field f. The stress tensor τ is usually interpreted as being comprised of the mean normal stress ρ = −⅓τ_ii, and the excess over the mean due to dynamic stresses $\overline{\text{τ}}$. For a fluid at rest, $\overline{\text{τ}}=0$, and the mean normal stress is just the hydrodynamic pressure. Thus, with $\text{τ}=-\text{p}\text{δ}+\overline{\text{τ}}$, the dynamic equations become

$\text{ρ}(\frac{\partial {\text{ν}}_{\text{i}}}{\partial \text{t}}+{\text{υ}}_{\text{j}}\frac{\partial {\text{ν}}_{\text{i}}}{\partial {\text{x}}_{\text{j}}})=\text{ρ}{\text{f}}_{\text{i}}-\frac{\partial \text{P}}{\partial {\text{x}}_{\text{i}}}+\frac{\partial {\text{τ}}_{\text{ij}}}{\partial {\text{x}}_{\text{j}}}(2)$

Of course, Equation 2 represents three equations, one for each direction of the coordinate system. Note that repeated subscripts imply summation over that subscript, δ is the Kronecker delta, or unit tensor. The negative sign in front of p corresponds to the convention that a pressure is a negative stress and a tension is a positive stress. By these definitions of p and $\overline{\text{τ}}$, τ_ii = 0.

In addition to the dynamic equations and equation that expresses the principle of conservation of mass (known as the continuity equation), we may write for an incompressible material (in Cartesian coordinates)

$\frac{\partial {\text{ν}}_{\text{i}}}{\partial {\text{x}}_{\text{i}}}=0(3)$

Often one wishes to use cylindrical or spherical coordinates in the solution of a problem. In that case, Equations 2 and 3 must be transformed. Standard textbooks on fluid mechanics provide Equations 2 and 3 in Cartesian, cylindrical, and spherical coordinate systems. The number of unknowns in these equations is much greater than the number of equations at hand. Normally f is given in a dynamic problem and p is known. Hence, the unknowns are the three components of v, the pressure p, and the nine components of $\overline{\text{τ}}$. For most real fluids, $\overline{\text{τ}}$ is a symmetric tensor and has only six independent components. Hence, ten unknowns are to be found from only four equations. The additional six equations required are the so-called constitutive equations for the material, which relate the components of τ to the velocity and its derivatives.

For example, a simple constitutive equation is that of the Newtonian fluid:

${\text{τ}}_{\text{ij}}={\text{η}}_0(\frac{\partial {\text{ν}}_{\text{i}}}{\partial {\text{x}}_{\text{j}}}+\frac{\partial {\text{ν}}_{\text{j}}}{\partial {\text{x}}_{\text{i}}})(4)$

where η₀ is a constant and is known as the coefficient of viscosity.

Once the constitutive equation is established, the dynamic problem is determinate. Unfortunately, it may not be amenable to solution for complex materials (defined by the constitutive equation) or for complex boundary conditions.

Fluids respond to stress by deforming or flowing. Flow is basically a process in which the material deforms at a finite rate. The basic kinematic measure of the response of a fluid is the rate of deformation tensor $\overline{\text{Δ}}$,

${\text{Δ}}_{\text{ij}}=(\frac{\partial {\text{ν}}_{\text{i}}}{\partial {\text{x}}_{\text{j}}}+\frac{\partial {\text{ν}}_{\text{j}}}{\partial {\text{x}}_{\text{i}}})={\text{Δ}}_{\text{ji}}(5)$

Motion may exist in a fluid even if $\overline{\text{Δ}}$ is identically zero. Each element of fluid may be translating at the same linear velocity, and each element may, in addition, have the same angular velocity about some axis, due to a rig...

Table of contents

• Cover
• Title Page
• Copyright Page
• Table of Contents
• Chapter 1 Introduction to Rheology
• Chapter 2 Conventional Viscometers and General Concepts
• Chapter 3 Viscometric Techniques and Analysis of Steady-State Flows
• Chapter 4 Normal Stress Measurements
• Chapter 5 Rheometric Techniques for Polymer Melts
• Chapter 6 Torque Rheometers and Processability Testing
• Chapter 7 Rheology and Introduction to Polymer Processing
• Chapter 8 Polymer Processing Operations
• Appendix A: Abbreviations of Polymers
• Appendix B: Glossary of Polymers and Testing
• Appendix C: Description of Professional and Testing Organizations
• Index