Polymer Characterization

MATERIALS SCIENCE

&

ENGINEERING 5473

POLYMER

CHARACTERIZATION

PROFESSOR JOHN A. NAIRNSPRING 2003

TABLE OF CONTENTS

1 Introduction 1

1.1 Definitions of Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Course Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Molecular Conformations 7

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Property Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Freely-Jointed Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.1 Freely-Jointed Chain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.2 Comment on Freely-Jointed Chain . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 Equivalent Freely Jointed Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.6 Vector Analysis of Polymer Conformations . . . . . . . . . . . . . . . . . . . . . . . . 24

2.7 Freely-Rotating Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.8 Hindered Rotating Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.9 More Realistic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.10 Theta (Θ) Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.11 Rotational Isomeric State Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Polymer Molecular Weight 43

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 Number Average Molecular Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Weight Average Molecular Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4 Other Average Molecular Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 A Distribution of Molecular Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6 Most Probable Molecular Weight Distribution . . . . . . . . . . . . . . . . . . . . . . 48

4 Polymer Solutions 57

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Solution Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Physical Origins of Entropy of Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 Flory-Huggins Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Colligative Properties 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2

5.2 Boiling Point Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Freezing Point Depression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4 Osmotic Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.5 Practical Aspects of Osmotic Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.6 Experimental Aspects of Membrane Osmometry . . . . . . . . . . . . . . . . . . . . 74

6 Viscosity 77

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.2 Viscosity and Viscosity Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.3 Viscosity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.4 Viscosity Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4.1 Polydisperse Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.4.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.5 Viscosity Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

7 Light Scattering 89

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.2 Rayleigh Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.3 Ideal Polymer Solutions with Small Particles . . . . . . . . . . . . . . . . . . . . . . 93

7.4 Non-Ideal Polymer Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.5 Large particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.6 Light Scattering Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

8 Miscellaneous Molecular Weight Methods 101

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8.2 Gel Phase Chromatography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8.3 Field Flow Fractionation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.4 End Group Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.5 Ultracentrifugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8.6 Insoluble Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

9 Thermal Analysis 109

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

9.2 Thermal Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

9.3 Thermal Analysis of Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

10 Dilatometry 115

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

10.2 Amorphous Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

0

10.2.1 Pressure Dependence of Tg . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

10.3 Semicrystalline Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

10.3.1 Pressure Dependence of Tm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

10.4 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

10.5 Dilatometry Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

11 Differential Scanning Calorimetry 127

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

11.2 Differential Scanning Calorimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

11.3 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

11.4 Results with Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

11.4.1 Heating of Semicrystalline Polymers . . . . . . . . . . . . . . . . . . . . . . . 130

11.4.2 Cooling of Semicrystalline Polymers . . . . . . . . . . . . . . . . . . . . . . . 137

11.4.3 Amorphous Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

11.5 DSC Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

11.6 Thermogravimetric Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

12 Mechanical Properties 141

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

12.2 Simple Elasticity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

12.3 Stress-Strain Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

12.3.1 Stress-Strain Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

12.3.2 Uses of Stress-Strain Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 149

12.3.3 Uniaxial Compression Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

12.3.4 Shear Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

12.3.5 Hydrostatic Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

12.4 Fracture and Impact Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

12.5 Time Dependence or Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

12.5.1 Creep Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

12.5.2 Free-Volume Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

12.5.3 Stress Relaxation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 158

12.5.4 Dynamic Mechanical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Chapter 1

Introduction

The subject of polymer characterization covers the techniques that are used to give us information

about polymer materials. The type of information we can use includes thermal properties, me-

chanical properties, microstructural information, time dependence of properties, and much more.

This information is obtained with a variety of experimental and theoretical approaches. In these

notes we will discuss many polymer characterization techniques. Perhaps more importantly we will

discuss the theory behind those techniques. Understanding the theory behind polymer characteri-

zation techniques is an essential ingredient to making correct use of those techniques. The subject

of polymer characterization is typically a second course in polymer science. As such it will be

assumed that all students have completed, as a prerequisite, an introduction to polymer materials

course (MS&E 3410 or equivalent).

Philosophically we choose to subdivide the field of polymer characterization into three types of

characterization methods — theoretical characterization, macroscopic property measurement, and

spectroscopic analysis. Theoretical characterization involves the detailed description of molecular

configurations and conformations. Polymers are random-coil molecules that can exist in a variety

of configurations and conformations. We can learn much about polymer conformations purely by

theoretical analysis. Many of the theoretical results can be verified by experiment, but most of our

insight is gained by the process of doing the theoretical analysis and not by learning about tech-

niques used to verify the analysis. Macroscopic property measurement is what might be referred

to as conventional polymer characterization. It involves taking a macroscopic polymer specimen

or polymer solution and doing experiments that give information about properties of the polymer.

Some of the more important properties include molecular weight, thermal properties, mechanical

properties, melt viscosity, friction and wear properties, and electrical properties. Spectroscopic

analysis is aimed at characterization on the molecular scale. Some common spectroscopic tech-

niques are UV-visible absorption spectroscopy, infrared spectroscopy (IR), Raman spectroscopy,

nuclear magnetic resonance (NMR), electron spin resonance (ESR), and mass spectrometry (MS).

1

2 CHAPTER 1. INTRODUCTION

These techniques are usually aimed at getting information about the chemical structure of polymer

materials.

1.1 Definitions of Terms

The most basic definition is that of a polymer. A polymer is molecule formed by covalent chemical

bonds between atoms (or groups of atoms) to give a large structure (linear chains, branched chains,

or cross-linked networks). The key word is “large.” The word polymer is usually reserved for high

molecular weight molecules. Historically, the fact that polymers are molecules with ordinary chem-

ical bonds (i.e., with chemical bonds identical to those found in low molecular weight molecules)

was not recognized and polymers were once thought to be a distinct state of matter. Because

this old thinking was wrong and instead polymers are large molecules (or macromolecules), we will

find that most of the principles of chemistry (e.g., chemical reactions) and physics (e.g., physical

properties) apply to polymers just as they do to conventional molecules. In the field of polymer

characterization, we can therefore draw on all the knowledge of the physical chemistry of small

molecules. Before applying any traditional physical chemistry analysis, however, we must first ask

about the effect that large molecular size has on the traditional analysis and then correct for those

effects. In one sense, physical chemistry of polymers and polymer characterization can be thought

of as a subset of physical chemistry. Fortunately the effect of large molecular size is of enough

significance that polymer science is not a trivial subset of physical chemistry — it is a challenging

and important subset.

Two types of polymers are natural polymers and synthetic polymers. Natural polymers are,

as expected, naturally occurring macromolecules. Natural polymers include DNA, RNA, proteins,

enzymes, collagen, silk, cotton, wool, and natural rubber. Despite the unquestioned importance

of natural polymers, most of the polymer and chemical industry is based on synthetic polymers

or polymers that can be synthesized by polymerization of low molecular weight monomers. Some

example synthetic polymers are

Polyethylene (PE): (CH2 CH2)

Polystyrene (PS): (CH2 C

��HHH

��HH

H)

Polyvinyl Chloride (PVC): (CH2 C

Cl

H)

1.1. DEFINITIONS OF TERMS 3

Polytetrafluoroethylene (PTFE or Teflon): (CF2 CF2)

Polypropylene (PP): (CH2 C

CH3

H)

Other examples of synthetic polymers include nylon, polycarbonate, polymethyl methacrylate (lu-

cite), epoxy, polyethylene terepthalate (polyester, mylar), and polyoxymethlyene.

The above structures show the repeat unit of the polymer. The repeat unit is usually the

smallest piece of the polymer that can be said to “repeat” periodically to give the polymer chain.

In polyvinyl chloride the repeat unit is (CH2 CHCl) . In PE, the repeat unit listed above is

(CH2 CH2) . From a topological point of view, the PE repeat unit could be (CH2) , but be-

cause PE is polymerized from ethylene or CH2 CH2, it is common practice to call (CH2 CH2)

the repeat unit although it is not the smallest periodically repeating unit.

The word polymer literally means many “mers” or many monomers. Monomers are the start-

ing materials used in synthesizing polymers. Polymers are made by combining many monomers.

The repeat unit and the monomer are usually closely related. Sometimes (e.g., in condensation

polymers) some atoms are lost (e.g., a molecule of water (H2O)) from the monomer during poly-

merization and the repeat unit will differ slightly from the monomer. The names of polymers

often indicate the starting monomer material. Thus polytetrafluoroetheylene is a polymer made

by polymerizing tetrafluoroethylene monomers.

If a polymer is made from only one type of monomer or if it has a single repeat unit, it is called

a homopolymer. If a polymer is made from more than one type of monomer or has more than a

single repeat unit, it is called a copolymer. Some polymers are made up of alternating monomers

or alternating repeat units. Such polymers are often made from two types of monomers and thus

are formally copolymers. However, it is possible to consider two consecutive monomer units in the

polymer chain as a single repeat unit. As there is then only one type of repeat unit, it is common

practice to refer to such alternately copolymers as homopolymers. A good example is Nylon

6/6. The polymerization of Nylon 6/6 is shown in Fig. 1.1. Nylon 6/6 is actually an alternating

copolymer polymerized from monomers of adipic acid and hexamethyl diamine. Although it is

a copolymer, the structure on the right side Fig. 1.1 can be viewed as the single repeat unit for

Nylon 6/6. The “6/6” in the polymer name denotes the number of carbon atoms in each of the

two monomers. The first gives the number of carbon atoms in the amine; the second gives the

number of carbon atoms in the acid. Besides Nylon 6/6, other commercial nylons include Nylon

6/12, Nylon 6/11, Nylon 6/9 and Nylon 4/6. Some nylons are followed by a single number as in

Nylon 6. These nylons are made from ω-amino acids (a methylene chain with an amine group on

one end and an acid group on the other) and the single number is the number of carbon atoms in

the ω-amino acid. Some examples of such commercial nylons are Nylon 6 and Nylon 11.


HO–C–(CH2)4–C–OH + H2N–(CH2)6–NH2

—(N–C–(CH2)4–C–N–(CH2)6)—

O O

O O

H H

Figure 1.1: Copolymerization of the two monomers, adipic acid and hexamethyl diamine, result in synthesisof the polymer Nylon 6/6.

– A – A – A – A – A – A – A – A –

Linear

Branched

Cross-Linked

– A – A – A – A – A – A – A – A –

A – A – A

AA

AA

– A – A – A – A – A – A – A – A –

– A – A – A – A – A – A – A – A –

AA

AA

Figure 1.2: A schematic view of monomer “A” connected to make a linear, a branched, or a cross-linkedpolymer.

The repeat unit can also be called the structural unit. Structural units can be connected to

make linear, branched, or cross-linked polymers. Cross-linked polymers are also called network

polymers. Figure 1.2 shows the various ways in which structural units “A” can be connected to

make a polymer molecule. The way in which structural units are connected has a profound effect

on polymer properties such as toughness, viscosity, glass transition temperature, etc..

Besides the way the units are connected, the total number of units also has a profound effect

on polymer properties. The more units that are connected, the higher the molecular weight. A

polymer must have a high molecular weight before it has useful properties, especially mechanical

properties. The number of monomers, or sometimes the number of repeat units, in a polymer is

called the degree of polymerization.

Polymers are often characterized as being thermoplastic or thermoset. A thermoplastic polymer

will soften as it is heated (i.e., thermal treatment leads to plastic flow). This behavior has important

1.2. COURSE GOALS 5

implications about processing such polymers. Thermoplastic polymers can usually be molded and

are typically used as injection molding resins. In brief, the polymer is heated until is softens and

then is injected into a mold where it cools and solidifies into a solid part. Thermoplastic behavior is

a physically change that is reversible; the polymer can be heated and soften and cooled and solidified

many times. In contrast, a thermoset polymer sets up when heated. Thermoset polymers undergo

irreversible chemical reactions on heating. Such polymers cannot be reheated and soften; instead

they normally degrade when reheated. They are never processed by thermal injection molding, but

instead required other processing methods such as liquid casting or reaction injection molding.

Thermoplastic polymers typically have linear polymers with few or no cross links. Thermoset

polymers are typically highly cross-linked or are network polymers. It is the cross linking reactions

that cause the “setting” at high temperatures. An important use of the thermoplastic and thermoset

terminology is to decide how to process a given polymer. Thermoplastic polymers are processed

by heating, molding, and solidification; thermoset polymers are processed by liquid or gel methods

followed by chemical reactions. Some polymers fall between these two categories. For example,

some polymers with no cross linking may have very high melting points and may undergo thermal

degradation before they soften enough to allow molding. With regards to polymer structure, such

polymers have more in common with thermoplastic polymers than with thermoset polymers. They

require, however, different processing methods than thermoplastic polymers. For example, Kevlar R©

aramid polymer is a linear polymer, but it cannot be processed by molding. It is processed into

high-modulus and high-strength fibers by a solution, fiber-spinning method.

1.2 Course Goals

These notes will emphasize polymer characterization of high molecular weight synthetic polymers.

The focus on high molecular weight is because high molecular weight is a prerequisite for an organic

material to have useful physical properties. It takes high molecular weight before a polymer has

sufficient stiffness or strength to be useful for making things ranging from clothing to airplanes.

One finds high molecular weight polymers in plastic parts, synthetic fibers (both textile fibers and

high performance fibers), elastomers (synthetic rubbers), glues, and composites. These uses are the

ones that are important to most of the polymer industry and thus these notes focus on polymers

of commercial interest. Likewise, the focus on synthetic polymers reflects the current emphasis of

the polymer and chemical industry. In fact, synthetic polymers not only dominate the polymer

industry, they even dominate the chemical industry as a whole. This dominance in illustrated by

the fact that more than half the chemists and chemical engineers employed in this country are

involved with polymers.

The ultimate goal of polymer characterization is to understand polymer properties, how those

properties relate to polymer structure, and how they relate to potential polymer applications.


Someone well versed in polymer characterization should be able to take any polymer and decide

whether or not that polymer is suitable for some contemplated application. A long-range goal of

polymer characterization is also to design new polymer materials. With knowledge of how various

polymer structures translate into polymer properties (as measured using polymer characterization),

the astute polymer engineer could recommend molecular and structural modifications that could

be used to create new and more useful polymers.

Problems

1–1. What are the repeat units for the following polymers: Nylon 6, Nylon 11, Polymethyl

methacrylate, and polypropylene?

1–2. What is the difference between natural polymers and synthetic polymers?

1–3. What is the difference between a branched polymer and a cross-linked polymer?

1–4. What is the difference between a thermoplastic polymer and a Thermoset polymer?

1–5. Search the Internet and find the top five polymers on the basis of volume sold.

1–6. Search the Internet and find product information on two commercial copolymers. Give the

chemical structures of the monomers used to synthesize the polymer.

Chapter 2

Molecular Conformations

2.1 Introduction

Polymers can exist in various conformations and various configurations. Two polymers which differ

only by rotations about single bonds are said to be two different conformations of that polymer.

A schematic view of two polymer conformations is show in Fig. 2.1. Two polymers which have the

same chemical composition but can only be made identical (e.g., superposable) by breaking and

reforming bonds are said to be two configurations of that polymer. Two examples in Fig. 2.2 are

polymers that contain asymmetric carbon atoms or that contain double bonds. Asymmetric carbon

atoms can exist in d or l states while double bonds can exist in cis or trans states. No manner of

rotations about single bonds can turn polymers in different configuration states into superposable

polymers.

The above definitions of conformation and configuration are standard, but they have not always

been rigorously followed in the literature. For example, Paul Flory, who won a Nobel prize for

studies of polymer conformations, used configuration in his writings when he meant conformation.

Fortunately a writer’s meaning is usually obvious from context. It is recommended that you strive

to use the correct terminology as defined above. These notes strive to follow that convention.

Figure 2.1: Two molecules with different conformations. These two molecules can be made identical witha rotation of 180◦ about the central single bond.

7

8 CHAPTER 2. MOLECULAR CONFORMATIONS

Cl H Cl H Cl H H Cl

trans cisFigure 2.2: Two distinct configurations of polymers can exist at asymmetric carbon atoms (top) and across

double bonds (bottom).

To understand polymers, we must understand the shape or the form that polymers have in

solution as well as in solid state. We will therefore spend some time studying the conformations

of isolated polymer chains. An isolated polymer chain is clearly a good model for a gas phase or

a dilute solution polymer. It turns out that the description of an isolated polymer chain is also a

good description for polymer chains in concentrated solutions, or even in the solid state.

The first thing to realize about polymers is that they are usually very long and are best de-

scribed as resembling a random coil . In a collection of polymers (gas phase, solution, or solid

state), the various individual polymer molecules will assume many possible conformations. Each

conformation will be a different variation on a random coil. The different conformations arise by

thermal energy and the relatively low barrier to rotations about single bonds. The shapes of the

possible conformations have a direct bearing on the physical properties of the polymer. Some

properties that can be influenced included solution viscosity, solution light scattering, and mechan-

ical properties. For example, rigid rod polymers generally assume more extended conformations

than polymers with more flexible backbones. These more extended shapes give rigid rod polymers

very different properties. To be specific, rigid rod polymers are generally stiffer and have a higher

glass-transition temperature.

Figure 2.3 shows an extended chain polymer and a random coil polymer. The extended chain

picture is the way we might choose to draw a polymer structure. The random coil picture, however,

is a more realistic view of the shape of real polymer molecules. In this chapter we will discuss various

theoretical methods for characterizing the shape of random coil polymers. The major results will

be predictions of the end-to-end distance for linear polymers. The ideas that enter into the analysis

give insight about the shape of polymers. As discussed in the Introduction chapter (chapter 1),

2.2. NOMENCLATURE 9

Figure 2.3: Extended chain polymer on the left. A more realistic picture of a polymer as a random coil onthe right. The colors indicate rotation angle about each bond. Blue is for trans bonds while read and greenare for gauche bonds.

this type of polymer characterization is theoretical characterization.

2.2 Nomenclature

We will restrict ourselves to linear polymers and we will consider all their possible conformations.

To describe any given conformation we must first define a nomenclature or coordinate system. We

begin with a polymer having n bonds. These n bonds connect n + 1 backbone atoms. We can thus

define any conformation by giving the 3(n + 1) Cartesian coordinates of the n + 1 atoms along the

polymer backbone. This nomenclature works but is normally more cumbersome than desired and

we thus make some simplifications.

We begin with the bond length (l). In many polymers the bonds in the polymer backbone are all

identical and therefore have a constant bond length. For example, in PE the bonds are all carbon-

carbon bonds and they are all typically about 1.53A long. For simplicity we will restrict ourselves

to polymers with constant bond lengths. A generalization to non-constant bond lengths can be

made later if necessary. With constant bond lengths, we can consider a polymer conformation as

a 3D random walk of n steps where each step has length l. Instead of listing absolute coordinates

of each atom in the backbone, we choose to describe a polymer by listing the relative directions of

each step in the random walk.

Directions in space are most conveniently described using polar angles. Figure 2.4 shows an

arbitrary direction in space emanating from the origin of a coordinate system. The angle with

respect to the z axis is called the polar angle and is usually denoted by θ. The angle that the


z

y

x

θ

φ

Figure 2.4: Definitions of the polar angle θ and the azimuthal angle φ for any vector in a right-handedcoordinate system.

projection of the direction onto the x–y plane makes with any consistently chosen reference point

in that plane is called the azimuthal angle and is usually denoted by φ. All possible directions in

space can be spanned by choosing θ from 0 to π and φ from 0 to 2π. In other words, any direction

from the origin can be defined by a unique pair of θ and φ.

We will represent a polymer as a 3D random walk of n steps where n is the number of bonds

(note that n is not necessarily the same as the degree of polymerization or the number of repeat

units; some repeat units have more than one bond and for n we count all of these bonds). In the

random walk, each step can be described by polar and azimuthal angles, θ and φ, where those

angles are given with respect to an axis system centered on the atom at the start of that bond. For

n bonds, each bond will have its own angles, θi and φi, and the complete chain will be described

with the 3 original coordinates for the first atom and the 2n angles for the steps of the random

walk. We thus require 2n + 3 variables to specify a conformation of a polymer with constant bond

length.

Normally we will not be concerned with the absolute location in space of the polymer chain.

If we do everything relative to the location of the first bond, then we do not need to know the 3

original coordinates nor the 2 polar angles of the first bond. Subtracting these five variables, we can

define an arbitrary polymer conformation with 2n− 2 or 2(n− 1) variables. The 2(n− 1) variables

are the polar and azimuthal angles for each bond except the first bond. If we ever generalize to

different l’s for each bond, we must add to these 2(n− 1) variables, n new variables which specify

the length of each bond.

2.3. PROPERTY CALCULATION 11

φ=0˚

θi+1

φ=180˚

bond i+1

bond i

bond i-1

xz

l

φi+1

Figure 2.5: Definition of polar and azimuthal angles for bond i. With the illustrated selection of x, y, andz axes, the polar angle is the bond angle for bond i + 1 and the azimuthal angle is the dihedral angle forbond i + 1.

It is convenient to choose a coordinate system that lends physical interpretations to the polar

and azimuthal angles of each bond in the polymer chain. As illustrated in Fig. 2.5, we consider

the central bond as bond i and take the z axis to point back along bond i. With this choice for

the z axis, the polar angle for bond i + 1 is just the bond angle between bond i and bond i + 1

(see Fig. 2.5). From now on, we will refer to the polar angle as the bond angle. The possible

orientations for bond i + 1 when the bond angle is θi+1 sweep out the cone illustrated in Fig. 2.5.

The azimuthal angle (φ) for bond i + 1 is the counter-clockwise angle around that cone from some

suitably selected reference point. We choose the x axis to define the reference point such that the

azimuthal angle for bond i + 1 is 180◦ when bond i is a trans bond. This choice is arbitrary, but

is consistent with the bulk of the modern literature (note: Flory choose φ = 0 to correspond to

trans bonds which makes his results shifted by 180◦ from these notes). Another term for such an

azimuthal angle is the dihedral angle for bond i + 1 — a term that we will adopt throughout these

notes. Finally, the y axis is chosen to be perpendicular to the x and z axes and directed to make

the x-y-z coordinates a right-handed coordinate system.

2.3 Property Calculation

The goal of theoretical characterization of polymers is to be able to predict certain properties of

those polymers. When a polymer exists in a single conformation, the task is simple — we merely

calculate the property for that conformation. Random coil polymers, however, can exist in many

different conformations. An observed macroscopic property of an ensemble of polymer chains will

be an average value of that property over the range of polymer conformations. We denote the

average value of any property over an ensemble of random coil polymer chains as 〈Property〉.


The way to find 〈Property〉 is to examine a large number of polymer chains by considering

a large number of random walks. For the simplest models (models of short chains) we will be

able to examine all possible random walks. When we can consider all possible random walks we

can assign to each random walk a probability which equals the probability that that conformation

will be selected when one polymer is selected from an ensemble of random coils. Assuming we

can calculate some polymer property (e.g., size, stiffness, etc.) for each specific conformation, we

can average them to get the average of that property for the bulk polymer sample. The average

property is defined by

〈Property〉 =∑

i

Property(conf i)× Probablity(conf i) (2.1)

where Property(conf i) is the value of the property calculated for conformation i and Probablity(conf i)

is the probability of that conformation occurring.

For small molecules you can often do the above averaging process exactly. In other words you

can enumerate all possible conformations, find the probability and property of each conformation,

and then find the average property by averaging the results. Some small molecules have only one

conformation and the task is relatively simple — the average property is equal to the property

of the single conformation. Other molecules have only a few conformations and the task is still

relatively simple. For a non chemistry example, consider the roll of a single dice and consider the

property of the number of pips showing on each role. A die has six faces which represent six possible

conformations of the die after each roll. When counting pips, the Property(conf i) = i. Assuming

the die is a fair die (i.e., not loaded) the probability of each conformation is the same and equal to

1/6 (thus Probablity(conf i) = 1/6). The property of the number of pips therefore has the exact

average value of

〈pips〉 =6∑i

i× 16

=16

+26

+36

+46

+56

+66

= 3.5 (2.2)

For polymer calculations there will usually be too many conformations to make the above

exact calculation procedure possible. Instead we will select conformations at random and use

a Monte Carlo procedure to get the average property. By the Monte Carlo procedure, if the

probability of selecting a particular conformation at random is proportional to the actual probability

of conformation i (selection probability ∝ Probablity(conf i)), than the average property for a

polymer sample can be approximated by

〈Property〉 ≈ 1N

∑Property(sample i) (2.3)

where N is the number of randomly generated polymer chains. The larger N , the more accurate

will be the calculated average property.

We can illustrate the Monte Carlo method with the dice problem. A Monte Carlo solution to

the dice problem would be to roll a die many times, total the pips, and divide by the number of

2.4. FREELY-JOINTED CHAIN 13

rolls. If the die was rolled sufficiently many times and if the die was fair (i.e., symmetric and not

loaded), the Monte Carlo solution would be very close to the exact answer of 3.5. After a few rolls,

the answer might differ from 3.5. After many rolls, however, the answer would be very unlikely to

show much deviation from 3.5.

The success of the Monte Carlo procedure is dependent on ones ability to select polymer confor-

mations with realistic probabilities that accurately reflect the true distribution of conformations.

This problem is easily solved in the dice problem by rolling a die. Unfortunately for polymer

problems we cannot physically select real polymers. Instead we have to generate conformations

mathematically or in a computer. The problem we must solve is the development of rules or al-

gorithms for realistically generating conformations. We will approach this problem in a series of

steps. We will begin with the simplest possible rules. At each subsequent step we will add more

realism to the procedure used to generate the random conformations. The final results can be used

to accurately predict many polymer properties.

2.4 Freely-Jointed Chain

In a freely-jointed chain all 2(n− 1) angular variables are allowed to assume any values with equal

probability. In others words the direction of any bond is equally likely to occur in any of the

possible directions of space — the joints at each bond thus move freely to allow all these possible

orientations.

Let’s begin with one particular property — the polymer size. Size can be characterized by

calculating the end-to-end distance, r, or the radius of gyration, s. As an average property, these

properties are usually calculated as a root mean squared end-to-end distance (or a root mean

squared radius of gyration). End-to-end distance is the distance from the beginning of the chain to

the end of the chain (see Fig. 2.6). The root mean squared end-to-end distance is the square root

of the average of the squared end-to-end distances:

rms r =√〈r2〉 =

√√√√ 1N

N∑i=1

r2i (2.4)

where N is the total number of possible conformations and ri is the end-to-end distance for confor-

mation i. The radius of gyration is the average of the distances of each of the atoms in the polymer

chain to the center of mass of the polymer. The root mean squared radius of gyration is the square

root of the average of the squared radius of gyrations:

rms s =√〈s2〉 =

√√√√ 1N

N∑i=1

s2i (2.5)

where si is the radius of gyration for conformation i.


r

←

Figure 2.6: The length of a vector (~r) from the first atom to the last atom on a linear polymer chain isthe end-to-end distance for that polymer conformation. This figure shows the end-to-end vector.

l

l cos θ

θ

Figure 2.7: The projection of a bond of length l onto the z axis is lz = l cos θ where θ is the angle betweenthe bond and the z axis.

2.4.1 Freely-Jointed Chain Analysis

We begin our analysis of the freely-jointed chain by analyzing a single direction in space — we

will analyze the z axis direction. For any of the bonds in the chain (or steps in the random walk),

the bond length along the z axis is lz = l cos θ (see Fig. 2.7) where θ is the angle between the

bond direction and the z-axis direction. θ is also the polar angle in the coordinate system defined

in Fig. 2.5 in which the z axis is along the previous bond. The average value of lz is found by

integrating over all possible bond projections or all possible bond directions:

〈lz〉 =∫ l

−llzp(lz)dlz =

∫ ∫l cos θ p(θ, φ)dS (2.6)

where p(θ, φ) is the probability that any given bond has directional angles of θ and φ. The inte-

gration is over the surface of a sphere centered on the origin and dS is the differential of surface

area.

For a freely-jointed chain all angles are all equally likely and therefore p(θ, φ) must be a constant;


z

x

y

dφ

dθ

r

r sin θ

dS = r2 sin θ dθ dφ

r sin θ dφr dθ

Figure 2.8: The differential of surface integration for polar coordinates is the area of the surface element onthe right side of the figure which is a top view of the left side. For small angles this area is dS = r2 sin θdθdφ.

we call it k. To find k, we integrate over the surface of sphere, which represents all possible directions

in space, and note that the integral over p(θ, φ), which is a probability distribution function, must

be 1. Because dS for surface integration over a sphere is r2 sin θdθdφ (see Fig. 2.8) we can integrate

over the unit sphere (r = 1) to get:

1 = k

∫ 2π

0dφ

∫ π

0dθsinθ = 4πk (2.7)

or

k = p(θ, φ) =14π

(2.8)

We can now insert p(θ, φ) into the expression for 〈lz〉 and integrate. The result is

〈lz〉 =∫ 2π

0dφ

∫ π

0dθ

l cos θ sin θ

4π=

l

2

∫ π

0cos θ sin θdθ = 0 (2.9)

This result could have been anticipated. When random walk steps in all directions are equally

likely, we are equally likely to jump in the positive direction as in the negative directions. These

equally likely jumps cancel out to give zero average jump size.

A simple average step size is not useful and we therefore instead consider the average squared

step length. It is for this reason that we constantly work with root-mean-squared end-to-end


distances in our discussion of polymer size. Squaring each step length makes all step size positive

and we are guaranteed to get a nonzero result. With the known p(θ, φ) function, we can easily

calculate the mean squared jump size:

〈l2z〉 =∫ 2π

0dφ

∫ π

0dθ

l2 cos2 θ sin θ

4π=

l2

2

∫ π

0cos2 θ sin θdθ (2.10)

=l2

2

[−1

3cos2 θ

∣∣∣∣π0

]=

l2

3(2.11)

The root mean squared distance per step is:√〈l2z〉 =

l√3

(2.12)

The above result gives as an average jump size per step, but we are concerned with the total

z axis root-mean-squared end-to-end distance. The solution to this problem is approached by

considering a set of typical jump directions. For a chain of n bonds, some of the bonds will point in

the positive z direction and some will point in the negative z direction. If n is large, the root mean

squared length of all positive jumps will be the same as that of all negative jumps and each will be

equal to the average of all jumps. We let n+ be the total number of jumps in the positive direction

and n− be the total number of jumps in the negative direction. Then the root-mean-squared

distance traveled in the z direction, denoted by z, is

z = (n+ − n−)√〈l2z〉 = (n+ − n−)

l√3

(2.13)

To solve for z we must determine (n+ − n−). The factor(n+ − n−) is like the result of a coin

toss experiment. Each step is considered a coin toss, if z increases on the step the coin toss result

is heads, if z decreases, the coin toss result is tails. In the coin toss results, the expected result is to

have equal numbers of heads and tails. If a large number of coin tosses are made the distribution

of (n+ − n−) will be a Gaussian function centered at zero (mean of zero). We can thus represent

the factor (n+ − n−), or more usefully the distance in the z direction, with the following Gaussian

distribution function:

W (z)dz =1√

2πσ2e−z2

2σ2 dz =β√π

e−β2z2dz (2.14)

where σ is the standard deviation in z-direction distance and the term β is defined in terms of

standard deviation by

β =1√2σ2

(2.15)

The freely-jointed chain problem is solved if we can find the standard deviation in z-direction

distance. For a single step the variance, or the standard deviation squared, follows simply from the

formula for variance:

σ21 = 〈l2z〉 − 〈lz〉2 = 〈l2z〉 (2.16)


Statistical analysis tells us that for n steps, the standard deviation in z is n times the standard

deviation for a single step or

σ2 = nσ21 = n〈l2z〉 (2.17)

Substituting the above result for 〈l2z〉 gives

β =1√

2n〈l2z〉=

√3

2nl2(2.18)

An alternate route to finding β is to find the variance by integration (i.e., find the average value

of z2 and subtract the square of the average value of z, which we know to be zero). The result is

〈z2〉n

= 〈l2z〉 =λ2

3=

1n

∫ ∞

−∞z2W (z)dz =

12nβ2

(2.19)

Solving for β again gives

β =

√3

2nl2(2.20)

The expression for β together with the Gaussian distribution function give the distribution

function for chain length in one direction. Now we need to solve the three-dimensional problem.

Because the chain is freely jointed, the three axes are independent of each other. From probability

theory, the probability that a given polymer chain jumps distances of x, y, and z in each of the three

Cartesian directions is the product of the probabilities for each of the axes considered separately.

The probability that a chain has an end-to-end distance characterized by a vector (x, y, z) is thus

W (x, y, z)dx dy dz = W (x)W (y)W (z)dx dy dz (2.21)

Because the analysis for W (z)dz given above applies equally well to the x and y directions, we have

W (x, y, z)dx dy dz =(

β√π

)3

e−β2r2dx dy dz (2.22)

where r2 = x2 + y2 + z2 is the square of the distance from the origin to the end of the chain at

(x, y, z).

As stated above, W (x, y, z)dx dy dz gives the probability that a chain’s end-to-end vector is

characterized by a vector (x, y, z). In other words, it is the probability that a chain that begins at

the origin ends in a box center at the point (x, y, z) or size dx dy dz (see Fig. 2.9). One dimension of

W (x, y, z)dx dy dz is plotted in Fig. 2.9. The function is a Gaussian distribution function centered

at the origin or centered about the mean value of zero.

The function W (x, y, z)dx dy dz solves the freely-jointed chain problem, but it is not in the most

useful form. We are normally not concerned with absolute end of the chain (i.e., location (x, y, z)),

but rather with the end-to-end distance r. To find this result we sum up all possible (x, y, z)

coordinates that give the same r value. In other words, we integrate over the volume element V


z

y

x

dV = dx dy dz

βr-3 -2 -1 0 1 2 3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 2.9: The left side shows a chain that starts at the origin and ends in a box centered a (x, y, z). Theright side is a one-dimensional plot of W (x, y, z)dx dy dz.

of width dr where√

x2 + y2 + z2 is between r and r + dr. The volume element of constant r is

a spherical shell as shown in Fig. 2.10. Integrating over this volume element yields a probability

distribution in terms of the end-to-end distance r:

W (r)dr =∫

VW (x, y, z)dx dy dz =

(β√π

)3

4πr2e−β2r2dr (2.23)

This type of distribution function is called a radial distribution function.

Figure 2.11 schematically plots the end-to-end distance distribution function, W (r)dr. We can

characterize the distribution function by finding some key points. The function W (r)dr always

increases to some maximum and then decrease towards zero. The peak value is found by finding

where the derivative of W (r)dr is zero. The maximum value, rmax, occurs at

rmax =1β

= l

√2n

3= 0.82l

√n (2.24)

The average value of r, 〈r〉, is found by integrating W (r)dr:

〈r〉 =∫ ∞

0rW (r)dr =

2β√

π= l

√8n

3π= 0.92l

√n (2.25)

Likewise, the mean-squared value of r, 〈r2〉, is

〈r2〉 =∫ ∞

0r2W (r)dr =

32β2

= l2n (2.26)

and the root mean squared end-to-end distance is√〈r2〉 = l

√n (2.27)


z

y

x

← ←

r + dr

←

rShell ofwidth dr

Figure 2.10: Cross section of a spherical shell between radii of r and r + dr.

length (units of l √n)

W(r

) dr

0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

rmax = 0.82 l √n<r> = 0.92 l √n

<r2> = l √n√

Figure 2.11: A typical plot of W (r)dr. The key values rmax, average r or 〈r〉, and root-mean-squared rare indicated on the figure.


The above key values are indicated in Fig. 2.11. For constant l and n, they always rank in the

order rmax < 〈r〉 < 〈r2〉.The variance in the end-to-end distance can be found from the mean and mean-squared end-

to-end distances:

σ2 = 〈r2〉 − 〈r〉2 =3

2β2− 4

β2π=

0.23β2

= 0.15nl2 (2.28)

From this results, the coefficient of variation (standard deviation divided by the mean) is

CV =σ

〈r〉= 42% (2.29)

This result can be characterized as a fairly large coefficient of variation.

2.4.2 Comment on Freely-Jointed Chain

We only used two facts in deriving W (r). First we assumed that the chain can be simulated by a

random walk. Second, we assumed there are enough steps to make the random walk a Gaussian

distribution. To find the Gaussian curve we therefore only had to find the mean (mean = 0) and the

standard deviation (σ =√

n〈l2z〉). The final result predicts that the root-mean-squared end-to-end

distance is √〈r2〉 = l

√n (2.30)

In other words the root mean squared end-to-end distance is proportional to the square root of the

number of bonds and linear in the bond length.

The linear dependence on bond length is a trivial result. It is merely a scaling parameter.

Thus if we double all bond lengths we double the end-to-end distance. It can also be thought of

as a consequence of units. If we solve the problem in inches and then in millimeters, we should

get answers that differ only by the units conversion factor for inches to millimeters of 25.4. This

expected result will only occur if the end-to-end distance is linear in bond length.

The dependence of root-mean-squared end-to-end distance on the square root of the number of

bonds is a profound, or at least a non-trivial, result. Let’s consider the origins of the square root

dependence on bond length. Our analysis is one of a completely random three dimensional random

walk. The square root of n dependence comes from the expression for the standard deviation of

the walk which contains√

n. If we repeated the analysis for one- or two-dimensional random walks

we would find the same result. The root-mean-squared end-to-end distance is√〈r2〉 = l

√n (2.31)

in any dimension. We thus conclude that the square root of n dependence is a property of the

random walk nature of polymers and unrelated to geometrical effects. Only polymer features that

alter the random walk nature of the chain can alter the square root dependence on n.


Walk to the right

Walk to the left

...

...

Figure 2.12: The only two possible one-dimensional, self-avoiding random walks.

To anticipate a future result that does alter the random walk nature of polymers, we consider

self-avoiding random walks. In self-avoiding random walks, the path cannot revisit any spot that

was previously visited. Because no two atoms in a polymer chain can occupy the same space,

a self-avoiding random walk is a better model for a polymer chain than the completely random

walk discussed previously. A self-avoiding random walk, however, is not a completely random walk

because some steps may be influenced by previous steps. In other words, some steps may be biased

away from moving in a given direction because doing so would revisit a previous part of the random

walk.

The exact analysis of two- and three-dimensional self-avoiding random walks is not possible.

One-dimensional, self-avoiding random walks, however, are trivial to analyze. As shown in Fig. 2.12

there are only two possible one-dimensional, self-avoiding random walks. A one-dimensional random

walk must begin with one step to the left or to the right. If the first step is to the left, the next step

must also be to the left because a step to the right would revisit the starting location. Continuing

on, the chain that starts to the left must make all steps to the left. The other possible chain is the

one that starts with its first step to the right. This chain can only continue with repeated steps

to the right. There are thus only two possible chains. One makes all n steps to the left and its

length is nl. The other makes all n steps to the right and its length is also nl. Averaging over

all possible chain conformations, the root-mean-squared end-to-end distance for a one-dimensional,

self-avoiding random walk is √〈r2〉 = nl (2.32)

In contrast to the completely random walk, this result is now linear in n. Because of scaling

requirements it remains linear in l.

In two- and three-dimensional random walks, the effect of imposing self avoiding characteristics

will be less dramatic. A one-dimensional, self-avoiding, random walk is hardly random. All steps

(except the first one) are determined by the requirement of being self avoiding and not by random

chance. Two- and three-dimensional, self-avoiding random walks will not restricted as much. Some

steps will be influenced by self-avoiding requirements, but most will have other options than can

be reached by random chance. Without proof, we state that the end-to-end distance for two- and

three-dimensional random walks will be proportional to n to some power between 0.5 and 1.0.

The two extremes are completely random walks (power equal to 0.5) and self-avoiding random


walks in which every step is determined by the self-avoiding requirement (power of 1.0). The

former extreme is the random walk result from above; the later extreme is the one-dimensional,

self-avoiding random walk result.

We now return to the random walk analysis and its assumption that the polymer chains are

long enough such that a Gaussian distribution function accurately represents the results. How big

do the chains have to be to be large enough? The Gaussian distribution was applied to the factor

(n+ − n−). For large chains with random n+ and n− jumps, we require |n+ − n−| � n. W (r)dr is

accurate as long as this inequality is true. W (r)dr, however, is supposed to work for all values of

r which includes r near rmax = nl. The maximum value of r can only occur when all jumps are in

the same direction which means either n+ = n and n− = 0 or n− = n and n+ = 0. In each of these

cases |n+ − n−| = n and the requirement that |n+ − n−| be much less than n is violated. In other

words, as r gets large, the chain becomes less Gaussian and W (r)dr becomes increasing inaccurate.

To assess the accuracy of W (r)dr, some researchers have constructed more detailed analyses

that account for deviations from Gaussian behavior at large r. One published result is

W (r)dr = (Const)4πr2 exp

{−β2r2

[1 +

310

(r

rmax

)2

+ . . .

]}dr (2.33)

The first term within the square brackets is 1 which is the previous result for a Gaussian distribution.

The rest of the terms within the square brackets are correction terms that are small for small r

and become more significant for large r. Only the first and largest correction term is given. Let’s

consider a chain of n bonds and say that the uncorrected W (r)dr is adequate as long as the

correction term is less than 0.01 (or less than 1%). W (r)dr will then be good for a chain that is

up to f times larger than the root-mean-squared end-to-end distance (i.e., good for r ≤ f√〈r2〉)

as long as

310

(f√〈r2〉

rmax

)2

=3f2

10n< 0.01 or f =

√n

30(2.34)

For example consider a chain of n = 100 bonds (a fairly short polymer chain). Solving this

equation for f results in f = 1.83. Therefore the first correction term in W (r)dr is less than 1% for

0 < r < 1.83l√

n = 18.3l. When 18.3l < r < nl = rmax = 100l the first correction term is greater

than 1%.

We can relate the range of small correction terms to the standard deviation in root mean

squared end-to-end distance. The upper limit in r for small corrections is fl√

n where l√

n is the

root mean squared end-to-end distance. From above, the standard deviation in end-to-end distance

is 0.15l√

n. Therefore the upper limit in r for small corrections is

fl√

n− l√

n

0.15l√

n=

√n30 − 10.15

(2.35)

standard deviations above the mean. For the above example with n = 100 bonds, this equation

shows the W (r)dr is accurate as long as r is less than 5.50 standard deviations above the mean. In

2.5. EQUIVALENT FREELY JOINTED CHAIN 23

Figure 2.13: The thin lines are a real polymer chain of n bonds. The thick lines are the superposedequivalent freely-jointed chain of n′ = n/m bonds.

a normal distribution very few chains will be more than 5.50 standard deviations above the mean.

Thus even for a relatively short 100 bond chain, the correction factor is insignificant. We conclude

that the correction factors are insignificant for most real polymer chains.

2.5 Equivalent Freely Jointed Chain

Real polymers are not freely jointed; the bond directions θ and φ are not free but are restricted

by local environment to be biased towards specific values. The major restrictions are caused by

nearby bonds. In other words, the conformation of any bond is correlated with the conformations

of nearby bonds. As the distance between any two bonds increases, the correlation decreases with

the conformations of distant bonds eventually becoming uncorrelated. Say that it takes m bonds

for the correlation to drop to zero and then construct from an n bond chain an n′ bond chain

with n′ = n/m. Let the average length of each new bond be l′ which will be proportional to l (or

l′ = kl). We call the new chain of n′ bonds of average length l′ the equivalent freely-jointed chain

for the original n bond chain. An example equivalent freely-jointed chain is shown in Fig. 2.13.

Because the bond conformations are uncorrelated over long distances, the random walk steps

in this new chain are freely-jointed. Using the freely-jointed chain result the root-mean-squared

end-to-end distance is √〈r2〉 = l′

√n′ = l

√Cn (2.36)

where

C =k√m

(2.37)

By using the equivalent freely-jointed chain, we have shown that the end-to-end distance of a

realistic chain is proportional to l√

n. The proportionality constant is√

C. C is normally greater


than one and therefore√

C is sometimes called the expansion factor and sometimes denoted as α.

It tells how much the real chain is expanded relative to a freely jointed chain. The square of the

expansion factor, or C, is known as the characteristic ratio. It is defined by

C =〈r2〉nl2

(2.38)

In the freely-jointed chain, C = 1. In real polymers the characteristic ratio is a function of n or

the number of bonds and is usually greater than 1. It is customary to write the bond-dependent

characteristic ratio as Cn. In relatively ideal polymers Cn approaches a limiting constant value for

large n. The value of Cn at large n for such polymers characterizes the random coil nature of that

polymer. Low values of Cn correspond to tightly coiled polymers and high values of Cn correspond

to polymers that are loosely coiled or extended.

In real polymers, self-avoiding requirements or excluded volume effects, to be described latter,

cause Cn to increase without bound as n increases without bound. A scaling law derived by

deGennes states that for large n, Cn increases in proportion to n0.2. Thus we have

Cn =〈r2〉nl2

∝ n0.2 (2.39)

or √〈r2〉 ∝ ln0.6 (2.40)

This result agrees with the previous discussion of self-avoiding random walks which claimed that

the power-law dependence on n for a real chain is between 0.5 (for a completely random chain) and

1.0 (for a chain with all steps controlled by self-avoiding requirements).

Our discussion on the equivalent freely-jointed chain shows that any real polymer can be reduced

to any expression for end-to-end distance that is proportional to l√

n (provided we ignore, for the

moment, excluded volume effects). The thing that distinguishes one polymer from another is

the characteristic ratio C. We thus embark on a series of models whose main purpose is to use

theoretical arguments for calculating the characteristic ratio.

2.6 Vector Analysis of Polymer Conformations

We will find it useful to develop an alternate approach to the analysis of end-to-end distance in

polymers. The new analysis will be based on vectors and analytical geometry. We associate with

each bond a vector. Let the vectors ~u1 to ~un be the vectors along the n bonds. Then a vector from

one end of the chain to the other is simple the sum of these vectors:

~r =n∑

i=1

~ui (2.41)

2.6. VECTOR ANALYSIS OF POLYMER CONFORMATIONS 25

The mean squared end-to-end distance is

〈r2〉 = 〈~r · ~r〉 = 〈n∑

i=1

~ui ·n∑

j=1

~uj〉 (2.42)

Expanding this expression for a polymer chain with a constant bond length, i, results in

〈r2〉 =n∑

i=1

~ui · ~ui + 2∑i<j

〈~ui · ~uj〉 (2.43)

where the sum over i < j means all combinations of i and j such that i is less than j. The factor of

2 in front of the sum includes, by symmetry, the terms when i is greater than j. The dot product

of a vector with itself is simply the square of the length of the vector. Thus we can write

〈r2〉 = nl2 + 2l2∑i<j

〈cos θ′ij〉 (2.44)

where θ′ij is the angle between the vectors along bond j and along bond i. If vector j is equal to

i + 1 (i.e., the bond next to bond i) then θ′ij is the supplement of the bond angle between bonds i

and i + 1.

For the freely-jointed chain, the bond directions are uncorrelated and 〈cos θ′ij〉 is zero which

yields√〈r2〉 = l

√n as before. This result was obtained with less work than the previous analysis.

The previous analysis, however gave a full distribution function (a Gaussian distribution) while this

one only gives the root-mean-squared end-to-end distance. For more complicated models, we will

need to evaluate the term 〈cos θ′ij〉.We can use a vector analysis to investigate the relation between end-to-end distance and radius

of gyration - s. In vector notation, we can write an expression for s2 in which ~ri is the location of

the ith atom along the polymer chain and ~z is the location of the center of mass:

s2 =1n

∑i

(~ri − ~z) · (~ri − ~z) (2.45)

Expanding this sum results in

ns2 =∑

i

r2i + nz2 − 2~z ·

∑i

~ri (2.46)

where ri is the distance from the origin to atom i and z is the distance from the origin to the center

of mass. The definition of the center of mass is

~z =1n

∑i

~ri or∑

i

~ri = n~z (2.47)

Substituting this expression in to the above results gives

ns2 =∑

i

r2i − nz2 =

∑i

r2i − n(~z · ~z) =

∑i

r2i −

1n

∑i

∑j

~ri · ~rj (2.48)


ririj

rj

rij2 = ri

2 + rj2 - 2 ri rj cos θ

Figure 2.14: A triangle framed by vectors ~ri and ~rj . The vector ~rij connects the two ends of vectors ~ri

and ~rj .

To evaluate the vector dot product ~ri · ~rj , consider the triangle in Fig. 2.14. By the cosine law

the length of the vector ~rij connecting the ends of vectors ~ri and ~rj is

r2ij = r2

i + r2j − 2rirj cos θ = r2

i + r2j − 2~ri · ~rj (2.49)

Solving for ~ri · ~rj gives

~ri · ~rj =r2i + r2

j − r2ij

2(2.50)

The expression for the radius of gyration then simplifies to:

s2 =1

2n2

∑j

∑i

r2ij =

1n2

∑j

∑i<j

r2ij (2.51)

In this step we made use of symmetry (rij = rji) and the fact that rii = 0.

Because the vectors ~ri are the coordinates of the atoms in the polymer chain, the distance rij

is the end-to-end distance for a sub-polymer chain of j − i bonds. By the discussion in the section

on the equivalent freely-jointed chain, we can write this distance as

r2ij = C(j − i)l2 (2.52)

We do not know the value of C but it is the characteristic ratio for the polymer under consideration

and may depend on j − i (especially for small j − i). Ignoring the j − i dependence of C, inserting

this result, and reordering the summation over i yields:

s2 =Cl2

n2

n∑j=1

j−1∑i=1

i =Cl2

n2

n∑j=1

j(j − 1)2

≈ Cl2

2n2

n∑j=1

j2 =Cl2

2n2

n(n + 1)(2n + 1)6

(2.53)

For large polymer chains (large n) this result simplifies to

√〈s2〉 =

√Cnl2

6=

√〈r2〉6

(2.54)

We conclude that whenever 〈r2〉 = Cnl2 for a polymer, that the radius of gyration is directly

proportional to (and smaller than) the end-to-end distance. This result holds for large molecules

2.7. FREELY-ROTATING CHAIN 27

when Cn is independent of n. An important corollary to this result is that everything we learn

while studying the end-to-end distance also applies to the radius of gyration. The reverse is also

true. Any analysis or experiments that give us information about the radius of gyration also give

us information about the end-to-end distance.

The approximations used to relate radius of gyration to end-to-end distance can be understood

using a matrix representation. Using the equivalent freely-jointed chain approximation, the radius

of gyration squared is the sum of all terms above the diagonal of the following matrix:

0 Cl2 2Cl2 3Cl2 4Cl2 . . .

− 0 Cl2 2Cl2 3Cl2 . . .

− − 0 Cl2 2Cl2 . . .

− − − 0 Cl2 . . .

− − − − 0 . . .

. . . . . . . . . . . . . . . . . .

(2.55)

The terms near the diagonal are random walk, long-chain approximations for short chains and may

be inaccurate. If the matrix is large, however, there will be many more terms far from the diagonal

then close to the diagonal. For large matrices, the inaccuracies in the terms close to the diagonal

will have only a negligible effect on the sum of all terms. Thus the relation between 〈s2〉 and 〈r2〉derived above will be accurate. In order for the derived relation, rewritten here as

〈r2〉〈s2〉

= 6 (2.56)

to be accurate, the chain must be slightly larger than is required for 〈r2〉 to be equal to Cnl2.

The relation will be inaccurate for short chains or for chains that are not random walk coils (e.g.,

extended chain polymers).

2.7 Freely-Rotating Chain

Commonly the bond angles in polymers are fixed or narrowly fixed to constant values. For example,

in polyethylene the bonds angles are all close to 112◦. It takes much more energy to distort bond

angles than it does to induce rotations about single bonds. We therefore consider a random walk

where all bond angles, θi, are constant and equal to one value θ. The more easily moved dihedral

or rotational angles are considered to all be equally likely. Thus there is free rotation about each

bond and hence we call this model the “freely-rotating chain” model. The conformation of a given

chain is reduced to specifying the dihedral angles for n− 1 bonds (all bonds except the first bond).

To analyze the freely-rotating chain we use the vector approach outlined in the previous section.

By taking successive projections of one bond onto another, we can show (See Flory book) that

〈cos θ′ij〉 = cosm θ′ where m = |j − i| where θ′ = π − θ is the supplement of the fixed bond angle.


The resulting vector analysis becomes

〈r2〉 = nl2 + 2l2i<j∑

cosj−i θ′ (2.57)

This expression can be imagined as the sum of all the terms in the matrix of size n × n in which

the element at row i and column j is cos|j−i| θ′:

l2 ×

1 cos θ′ cos2 θ′ cos3 θ′ . . . cosn−1 θ′

cos θ′ 1 cos θ′ cos2 θ′ . . . cosn−2 θ′

......

......

......

cosn−1 θ′ cosn−2 θ′ . . . cos2 θ′ cos θ′ 1

(2.58)

The n diagonal terms sum to n which give the first term (nl2) in the 〈r2〉 expression. Writing out

the required terms for the summation term by inspection of this matrix gives

〈r2〉 = nl2 + 2l2[(n− 1) cos θ′ + (n− 2) cos2 θ′ + . . . + (n− (n− 1)) cosn−1 θ′

](2.59)

which can be written as

〈r2〉 = nl2 + 2l2n−1∑i=1

(n− i) cosi θ′ (2.60)

We rewrite this summation in simpler terms by letting x = cosθ′:

〈r2〉 = nl2 + 2l2 (nS1 − xS2) (2.61)

where

S1 =n−1∑i=1

xi and S2 =n−1∑i=1

ixi−1 (2.62)

The terms S1 and S2 can be evaluated in closed form for all values of n. For polymer problems,

we are normally concerned with large n for which these sums converge to constant values and we

can treat the two sums as infinite series. The infinite geometric series S1 has the well known result

S1 ≈∞∑i=1

xi = x(1 + x + x2 + x3 + . . .

)=

x

1− x(2.63)

For S2, we use a trick by noticing that

S2 =d

dx

n−1∑i=1

xi =dS1

dx(2.64)

For large n we thus have

S2 ≈d(

x1−x

)dx

=1

(1− x)2(2.65)

2.8. HINDERED ROTATING CHAIN 29

Substituting S1 and S2 into the expression for 〈r2〉 gives

〈r2〉 = nl2 + 2l2[

nx

1− x− x

(1− x)2

](2.66)

For large n, the second term in the brackets quickly becomes negligible and we have

〈r2〉 = nl2 +2nl2x

1− x= nl2

1 + x

1− x(2.67)

Finally we substitute for x and note that cos θ = − cos θ′ (where θ is the bond angle and θ′ = π− θ

is the supplement of the bond angle). We get

〈r2〉 = nl21− cos θ

1 + cos θ(2.68)

The root-mean-squared end-to-end distance for large, freely-rotation chains is

√〈r2〉 = l

√n

√1− cos θ

1 + cos θ(2.69)

The characteristic ratio is

Cn =〈r2〉nl2

=1− cos θ

1 + cos θ(2.70)

Note that the characteristic ratio is independent of n. This independence is a consequence of our

assumption of large n. It is possible to evaluate S1 and S2 for any value of n and get an exact

result. The exact result gives a Cn that depends on n but rapidly converges to the above result for

large n.

For an example we consider polyethylene. All carbon bonds are in approximately tetrahedral

geometries. We can thus approximate all bond angles by the tetrahedral angle of θ = 109.47◦or

cos θ = −1/3 (note: it is easier to remember the tetrahedral angle as the cos−1(−1/3) than it is to

remember it as 109.47◦). For a freely-rotating polyethylene chain√〈r2〉 = l

√2n (2.71)

and the characteristic ratio is Cn = 2. Real polyethylene is not totally symmetric and thus the

C C C bonds deviate slightly from 109.47◦ to 112◦. The C C H and H C H bonds shrink

slightly to compensate. This slight increase in bond angle increases the characteristic ratio by 10%

to Cn = 2.20.

2.8 Hindered Rotating Chain

The dihedral angle is clearly not free to assume all possible values. Instead the angle is restricted

by steric interactions. Consider the potential energy for rotation about the central C C bond in

butane (CH3 CH2 CH2 CH3) shown in Fig. 2.15. The rotations about the central C C bond


Rotational Angle

Ene

rgy

0 60 120 180 240 300 360 0

1

2

3

4

C

C

C

H

HH

H

φ

Figure 2.15: Left side is an end view of the central C–C bond in butane. The right side plots to potentialenergy as a function of angle with φ = 180◦ defined as the trans position illustrated on the left side.

in butane are a reasonable model for the C C rotations in polyethylene. A preferred analogy

would be higher molecular weight alkanes. Those experiments and their analysis have been done,

but with more effort and less accuracy than available for butane.

The trans state illustrated on the left side of Fig. 2.15 is the lowest energy state. It is lowest

because the two CH3 groups, which are larger than the H atoms, are as far apart as possible. As

rotations occur the groups appended to each of the central C atoms come into closer proximity and

the steric energy increases. There are local maxima at φ = ±120◦. These maxima occur when the

CH3 groups are aligned with H atoms on the other carbon atom. There are local minima when

φ = ±60◦. These minima occur when all the groups are staggered (as in the left of Fig. 2.15) but

the CH3 groups are adjacent to each other rather than opposite each other as in the trans state.

These local minima are called the gauche+ and gauche- states. Finally there is a local maximum

at φ = 0◦, which occurs when the two CH3 groups on either end of the butane chain are aligned

and as close together as possible.

When rotations are hindered by potential energy functions such as those shown in Fig. 2.15,

it is clear that the rotations will not be free. All rotational angles are perhaps possible, but the

dihedral angles will have a preference for the low energy states. For butane, most bonds will be

at or near the trans state (φ = 180◦) or at or near the local gauche minima (φ = ±60◦). Thermal

energy will allow the angles to deviate from these states and to overcome the barriers to transitions

from one state to another. The populations of all rotational angles, however, will not be equal. We

thus replace the freely-rotating chain with a more realistic hindered-rotating chain model.

End-to-end distance for a hindered-rotation chain is more difficult to find than the freely-

2.9. MORE REALISTIC ANALYSIS 31

rotating chain model. We omit the details and give the results. The final equation is found using

the vector analysis and averaging the dot products ~ui ·~uj , while accounting for unequal populations

of the possible rotational angles. The result is

√〈r2〉 = l

√n

√1− cos θ

1 + cos θ

√1− 〈cos φ〉1 + 〈cos φ〉

(2.72)

where 〈cos φ〉 is the average value of the rotational angle. The characteristic ratio is

Cn =〈r2〉nl2

=1− cos θ

1 + cos θ

1− 〈cos φ〉1 + 〈cos φ〉

(2.73)

Note that if this hindered-rotation chain becomes freely-rotating chain then 〈cos φ〉 = 0 and the

results reduce to the freely-rotating chain results.

For a detailed example, let’s consider polyethylene with l = 1.53A, n = 5000 bonds, and

θ = 112◦. The simplest model, the freely-jointed chain model, gives√〈r2〉 (freely jointed) = 108A (2.74)

and a characteristic ratio of 1. The next most realistic model, the freely-rotating chain model, gives√〈r2〉 (freely rotating) = 108

√2.20A = 160A (2.75)

and a characteristic ration of 2.20. For the hindered-rotation chain model we need to know 〈cos φ〉.We thus need some new information. For example, we will say that experiments show that 60% of

the bonds have φ = 180◦ (trans) and 20% each have φ’s of +60◦ or −60◦ (gauche+ or gauche-).

Then the average dihedral angle gives

〈cos φ〉 = −0.6 + 0.2× (0.5) + 0.2× (0.5) = −0.4 (2.76)

The end-to-end distance for the hindered-rotation chain is√〈r2〉 (hindered rotation) = 108

√2.20

√2.33A = 245A (2.77)

and the characteristic ratio is Cn = 5.13. We can compare this result to the experimental result

for polyethylene which is 6 to 7 (depending on temperature). The hindered rotating chain gives

a result that is close to the correct results, but is still too low. In the next section we consider

improvements on the hindered rotating chain model.

2.9 More Realistic Analysis

We can list two general factors that influence the end-to-end distance, but are not included in the

hindered rotation chain model — short range interactions and long range interactions. The next

step in our analysis is to include some, or preferable all, of these effects.


We consider short-range interactions as interactions with neighboring bonds that affect the

relative probabilities of the dihedral angle. When we include short range interactions we cannot

take a simple average of cos φ as in the hindered rotation model. We need to account for the effect

of neighboring bonds. In its simplest implementation we might assume that the conformation of

one bond depends on relative energies that are influenced by the conformation of the previous

bond. Each step of the random walk then requires an adjustment of the jump-direction proba-

bilities according to the conformation of the previous bond. Some probability analysts call such

models Markov chains instead of random walks. After we develop a model that does a good job

of accounting for short-range interactions, we will have a good model for polymer conformations

that are unperturbed by the long-range interactions described below. We will call the polymers

generated by this modified random walk or Markov chain process “unperturbed polymer conforma-

tions.” The end-to-end distance of these chains will be called the unperturbed end-to-end distance

and will be denoted with a subscript zero — 〈r2〉0.Long-range interactions are all interactions that are not short-range interactions. To be more

specific, long-range interactions are interactions between atoms or molecular units that are not

covalently bonded to each other. Defined this way, long range interactions include interactions

between two parts of the polymer that are separated by many bonds or between the polymer and

solvent molecules. We discuss three types of long-range interactions.

The first important perturbation to random walks is that while random walks can cross their

own path, a polymer conformation cannot — random walks used to simulate polymer chains should

be self-avoiding random walks. This new feature is called the effect of excluded volume. The

elements of the chain exclude other elements from their occupied volume. If included in the analysis,

excluded volume will perturb the end-to-end distance and cause it to increase. Besides increasing

the end-to-end distance, excluded volume will cause the root-mean-squared end-to-end distance to

no longer be proportional to√

n. When excluded volume effects are significant, we can assume

that 〈r2〉 is proportional to some power of n such as p (where p > 0.5):

〈r2〉 ∝ npl2 (2.78)

According to deGennes, p = 1.2 and therefore the characteristic ratio increases as n0.2:

Cn =〈r2〉nl2

∝ n0.2 (2.79)

A second long-range interaction is interactions between the polymer and the solvent. A solvent

can cause the unperturbed end-to-end distance to increase or decrease. A good solvent will cause

it to increase. In effect the polymer will prefer interactions with the solvent over interactions with

itself and therefore will expand to maximize polymer-solvent interactions. A poor solvent will cause

the unperturbed end-to-end distance to contract. The polymer wants to avoid the solvent and thus

will contract on itself to avoid polymer-solvent interactions. In the extreme case, a poor solvent will

2.10. THETA (Θ) TEMPERATURE 33

cause the polymer to completely contract and precipitate out of solution. This extreme example of

a poor solvent is better characterized as a non-solvent.

The last types of long-range interactions are specific interactions. By specific interactions we

mean identifiable interactions between remote parts of the chains. Two examples are hydrogen

bonding and ionic interactions.

After considering both short-range and long-range interactions we are led to split the problem of

theoretical polymer characterization into two parts. The first part is the study of the unperturbed

end-to-end distance. We plan to modify the hindered rotation chain model to correctly account for

short-range interactions and thereby calculate the correct unperturbed end-to-end distance. After

solving this problem we will consider what influence the long-range interactions (including excluded

volume effects) have on the results. Let’s define α to be the factor by which the linear dimension

expands (α > 1) or contracts (α < 1) due to long-range interactions:√〈r2〉 = α

√〈r2〉0 (2.80)

The solution to the long-range interaction problem is to find α.

2.10 Theta (Θ) Temperature

We also need to consider temperature effects. Temperature can affect all types of interactions.

It can affect short-range interactions by influencing the relative probabilities of the various dihe-

dral angles. Rotations about single bonds are influenced by the presence of thermal energy and

thus higher temperature will make it more likely to find conformations in higher-energy rotational

angles. Temperature affects long-range interactions mainly through polymer-solvent interactions.

In general increasing the temperature improves solvent quality and lowering it decreases solvent

quality.

As discussed in the previous section, we will first attempt to solve the short-range interaction

problem. The first problem we encounter is how do we know when we have the right answer? Real

polymers have long-range interactions and short-range interactions. Thus real polymers have a

mean-squared end-to-end distance of 〈r2〉 and not 〈r2〉0. If we compare our theoretical analysis to

experimental results we will normally find disagreement, regardless of whether the analysis to find

〈r2〉0 was correct or not. Fortunately there is an experimental solution to this problem called the

theta (Θ) temperature. For any given polymer-solvent pair, there is a specific temperature, called

the Θ temperature, for which the true end-to-end distance is equal to the unperturbed end-to-end

distance. Thus experiments at the Θ temperature can be used to measure 〈r2〉0 which can be

compared to experimental predictions.

The concept of Θ temperature can be understood by considering the effect of temperature

on polymer-solvent interactions. In general the polymer-solvent interactions for a given polymer-


solvent pair is an increasing function of temperature. As the polymer-solvent interaction increases

the solvent becomes a better solvent. Thus increasing temperature leads to a better solvent and to

a higher α; decreasing temperature leads to a poorer solvent and a lower α. For a given polymer-

solvent pair, there will be some temperature for which α = 1. That temperature is called the theta

(Θ) temperature. At the Θ temperature 〈r2〉 = 〈r2〉0 and experiments at the Θ temperature can

be used to measure 〈r2〉0.Besides measuring 〈r2〉0, the Θ temperature has other useful properties. At the Θ temperature,

some physical properties (e.g., osmotic pressure) obey ideal laws up to high concentrations. Thus

simple theories for analysis of a physical property that do not account for long-range interactions

will still work well at the Θ temperature. There is an analogy between the Θ temperature and the

Boyle point of a gas. The Boyle point is the temperature where the repulsion between gas molecules

due to excluded volume is exactly compensated by their mutual attraction. At the Boyle point,

the ideal gas laws are obeyed to very high pressures. Note that conformity to ideal laws is one way

of determining the Θ temperature. The temperature can be varied until the measurements obey

ideal laws. That temperature is the Θ temperature.

Despite the advantages of the Θ temperature, we note that working in theta solvents is often

impractical. To get α down to one, you need a solvent that is sufficiently poor. The solvent may

become so poor the precipitation becomes imminent. It is often better to work in a good solvent

and correct the results to the unperturbed results. For measurements of physical properties, this

correction involves correcting ideal laws for large molecules or long-range interactions.

2.11 Rotational Isomeric State Model

The rotational isomeric state model will be our solution to the problem of accounting for short-range

interactions and some long range interactions (e.g., the effect of excluded volume). In general, any

polymer conformation is defined by a set of bond lengths, li, bond angles, θi, and dihedral angels,

φi. In the rotational isomeric state model, we assume that the polymer is restricted to certain

types of conformations or to certain isomeric states. Bond lengths are likely to vary only ±3% and

thus as before we consider them to be fixed at l. Likewise, bond angles are likely to vary only ±5

degrees and we consider them to be fixed at θ. The dihedral angles are not allowed to assume any

possible angle, but instead are assumed to be restricted to a small number of discrete values. The

possible discrete dihedral angles define the set of rotational isomeric states for each bond.

We will illustrate with polyethylene. These ideas will be extended to other polymers latter.

Due to symmetry of polyethylene there are three possible rotational isomeric states (see Fig. 2.16).

The trans state (φ = +180◦), the gauche+ state (φ = +60◦) and the gauche- state (φ = −60◦).

In the rotational isomeric state model, each bond of polyethylene is assumed to be in one of these

three states. No other values of φ are allowed. In reality, thermal fluctuations about the minima in

2.11. ROTATIONAL ISOMERIC STATE MODEL 35

C

C

C

H

H

H

H

C

C

C

H

HH

H

C

C

C

H

H

H

H

Trans Gauche -Gauche +Figure 2.16: The three rotational isomeric states in polyethylene. The dihedral angle, φ, is defined as the

angle between to white circles.

potential energy will lead to a distribution in φ’s around these three local minima. The rotational

isomeric state model asserts that we can realistically simulate a polymer chain by ignoring variations

in φ around the local minima.

In our random walk analysis of the rotational isomeric state, the probability of any bond existing

in one of the rotational state i characterized by angle φi, is given by a Boltzman factor:

P (φi) =e−E(φi)/RT

Z(2.81)

where Z is a normalizing factor or the sum of the Boltzman energy factors for all possible angles:

Z =3∑

i=1

e−E(φi)/RT (2.82)

where E(φi) is the energy associated with the state at dihedral angle φi and the sum is over the

three possible states in Fig. 2.16 (Note: this equation can easily be generalized to more than three

rotational isomeric states) Because of short-range interactions discussed above, this probability

factor is expected to be affected by neighboring bonds. The simplest model is to assume that the

energy, E(φi), is affected by the bond conformation of the previous bond. This important nearest-

neighbor interaction is ignored in the hindered rotation chain model. Its inclusion in the rotational

isomeric state model is the major advance in realism in this new model.

To account for nearest neighbor interactions, we must consider all bond pairs and all possible

conditional probabilities of the form P (φi|φi−1), which gives the conditional probability that bond

i has angle φi given that the previous bond or bond i − 1 has angle φi−1. By analogy with the

previous probability factor

P (φi|φi−1) =e−E(φi−1φi)/RT

Z(φi−1)(2.83)


where Z(φi−1) is a normalizing factor or the sum of the Boltzman energy factors for all angles φi

given that φi−1 is fixed:

Z =3∑

j=1

e−E(φi−1φj)/RT (2.84)

In these equations, E(φi−1φi) is the energy associated with two consecutive bonds with dihedral

angles φi−1 and φi. In this section we are restricting ourselves to three rotational states of trans

(t), gauche+ (g+), and gauche- (g−). The energy factors we care about are E(tt), E(tg+), E(tg−),

etc.. The conditional probabilities we are concerned with can conveniently be tabulated in a matrix

called the weighting factor matrix. For polyethylene (or for any three-state polymer) we write can

write a weighting factor matrix as

U =

P (t|t) P (g+|t) P (g−|t)

P (t|g+) P (g+|g+) P (g−|g+)

P (t|g−) P (g+|g−) P (g−|g−)

(2.85)

For example, the matrix element P (t|g+) gives the probability that the next bond is trans given

that the previous bond was gauche+.

Let’s begin with the first row and arbitrarily assign the tt state to the ground state. Thus we

assume E(tt) = 0. The tg+ and tg− bond pairs will each be higher in energy than the tt bond pair.

In polyethylene, the excess energy associated with the two possible gauche bonds will be the same.

We let that excess energy be Eg or the extra energy associated with a gauche bond. We further

define

σ = e−Eg/RT (2.86)

and we can quickly derive Z(t) = 1 + 2σ and

P (t|t) =1

1 + 2σP (g+|t) =

σ

1 + 2σP (g−|t) =

σ

1 + 2σ(2.87)

Z(t) appears in the denominator of each probability as a normalizing factor.

For the second row, we might just consider the energy associated with all the gauche bonds

and assume E(g+t) = Eg and E(g+g+) = E(g+g−) = 2Eg. The probabilities would become

P (t|g+) =σ

σ + 2σ2P (g+|g+) =

σ2

σ + 2σ2P (g−|g+) =

σ2

σ + 2σ2(2.88)

This simple analysis ignores an important short-range interaction that occurs when two consecutive

bonds are g+g− or g−g+. Figure 2.17 illustrates a g+g− conformation. Two parts of the polymer

chain separated by four bonds are in close proximity which causes a large steric interaction. This

interaction is known as the pentane effect and causes the energy of g+g− or g−g+ conformations to

be much higher than that g+g+ or g−g− conformations. If we let Ep be the energy of the pentane

effect interaction then E(g−g+) = 2Eg + Ep whereas E(g+g+) = 2Eg. Defining

ω = e−Ep/RT (2.89)


C C

C

CC

Gauche -

Gauche +

StrongStericInteractions

Figure 2.17: The steric interactions caused by the pentane effect when two consecutive bonds are g+|g−or g−|g+.

we derive the conditional probabilities for the second row of U to be

P (t|g+) =σ

σ + σ2 + σ2ω=

11 + σ + σω

(2.90)

P (g+|g+) =σ2

σ + σ2 + σ2ω=

σ

1 + σ + σω(2.91)

P (g−|g+) =σ2ω

σ + σ2 + σ2ω=

σω

1 + σ + σω(2.92)

Analyzing the third row similarly to the second row and collecting all probabilities in the matrix

we arrive at

U =

1

1+2σσ

1+2σσ

1+2σ1

1+σ+σωσ

1+σ+σωσω

1+σ+σω1

1+σ+σωσ

1+σ+σωσω

1+σ+σω

(2.93)

It usually suffices to work with relative probabilities rather than absolute probabilities. By conven-

tion we normalize each row to the first element in that row. For a simple polymer like polyethylene,

we write the weighting matrix of relative probabilities as

U =

1 σ σ

1 σ σω

1 σω σ

(2.94)

By theory and experiment, the energy terms for polyethylene are well known. The best results give

Eg = 2100 J and Ep = 8400 J. At 300◦C these lead to weighting factors of

σ = e−2100J/RT = 0.43 and ω = e−8400J/RT = 0.081 (2.95)

Thus gauche bonds are only about 46% as likely as trans bonds (2σ/(1 + 2σ)) and g+|g− confor-

mations are only about 2.4% as likely (σω/(1 + σ + σω)).


We next consider how to use the rotational isomeric state model to predict polymer properties.

Imagine a polymer chain of n bonds. A polymer conformation of this chain is determined by setting

the states of each bond; each bond can be t, g+, or g−. We can ignore the first two bonds and thus

we need to set the states of the terminal n−2 bonds. The first bond can be ignored because it only

serves to locate the start of the polymer in space. The second bond direction is determined by the

rotational state of bond 1, but this state only serves to locate the initial direction of the random

walk. The remaining n− 2 bonds need to be determined.

There are two possible solutions to the rotational isomeric state model. Because each bond

has a small number of discrete states, the total number of polymer conformations is finite. For

a polymer with n bonds of which each bond can be in one of three states, the total number of

polymer conformations is 3n−2. One possible approach is to enumerate all possible conformations

and calculate properties using the exact process discussed earlier in this chapter. This exact solution

can be done for small n, but soon becomes impractical for large n’s typical of real polymers.

When n is large the total number of conformations is too large to enumerate, even on the fastest

supercomputer.

For large polymers, a preferred approach is to use the Monte Carlo method. We generate a large

number of chains using a random walk or Markov chain process. For each bond the probabilities of

the next bond being in a given state are determined by the weighting matrix like the one given above

and by the state of the previous bond. For each chain we can calculate a property of interest (e.g.,

end-to-end distance, radius of gyration, etc.). The average value of that property after generating

sufficiently many chains will be our Monte Carlo solution to the problem.

The above Monte Carlo random walk process requires a computer. The computer program

LatticeTM discussed in class is an example of a program that can do Monte Carlo simulations using

the rotational isomeric state model. Programs like LatticeTM are, in effect, computer experiments.

You set polymer properties by selecting weighting factors, excluded volume, temperature, etc..

Then you run the program and calculate the end-to-end distance or the radius of gyration. These

computer experimental results can be compared to real experimental results. If the two results

agree, you have evidence that the rotational isomeric state is a valid model for polymers. Fur-

thermore the physical concepts of the rotational isomeric state model give you insight about the

polymer molecules and the factors that control conformations.

The following pseudo computer code is the program logic, or flow chart, of the main part of

any program that does Monte Carlo solutions using the rotational isomeric state model. This flow

chart assumes a three rotational isomeric states, but it could easily be generalized to handle any

number of states.

let v = (1,0,0) or the first bond is t

for bond=2 to number of bonds-1


let q=Uv be a vector giving the probabilities for t, g+, and g-

pick a random number between 0 and q1+q2+q3

if (random number is between 0 and q1) then

next bond is t and v = (1,0,0)

else if (random number is between q1 and q2) then

next bond is g+ and v = (0,1,0)

else if(random number between q2 and q3) then

next bond is g- and v = (0,0,1)

end if

if desired, check for excluded volume effects

if jump to occupied site, try another site, reject the entire

chain, or reject several previous bonds

if all sites occupied start new chain

end if

next bond

Now have set of n-2 bond conformations which completely define the chain

Calculate properties (e.g. end-to-end distance, radius of gyration, etc.)

Start over with another chain and continue until results converge

The key part of this loop is calculating the properties of each generated chain. You can calculate

any property than can be reliably calculated for a given polymer conformation. The simplest

and most reliable properties are size properties. The program LatticeTM calculates the following

properties:

1. Last r (end-to-end distance of current chain)

2. rms r

3. 〈r〉

4. rms s

5. r2

nl2— the characteristic ratio (equal to Cn)

6. r2

s2 — how close is it to 6?

7. % trans

8. Number of rejected steps

9. Relative entropy


In addition to the above calculations, LatticeTM can also plot many results. It can plot each

generated polymer in a three-dimensional view. These stick figures give a feel for typical polymer

conformations. It can plot the distribution of end-to-end distances and compare them to the radial

Gaussian function calculated for in the freely-jointed chain analysis.

There is one important aspect of LatticeTM and many rotational isomeric state solutions that

requires discussion. As implied by its name, LatticeTM generates chains on a tetrahedral lattice.

This requires all bonds angles to be the tetrahedral bond angle of 109.47◦. Because it is a regular

lattice, it also requires all bonds lengths to be the same. This type of simulation is reasonably

accurate for polyethylene and for many other carbon-backbone polymers. Lattice simulations,

however, cannot give you information about the effect of bond angle (e.g., 112◦ in real polyethylene

vs. the 109.47◦ in a tetrahedral lattice) and cannot simulate polymers with different bond lengths.

The advantage of lattice simulations is that they are fast. They can do most of the work with

integer arithmetic which is much faster in computers. The use of a lattice makes it possible to do

meaningful simulations in personal computers such as Macintoshes. An off-lattice calculation on a

Macintosh would be limited to much more rudimentary work and to much shorter chains.

We can simulate many polymers using the rotational isomeric state model. Some interesting

polymers are:

1. Polyethylene (see above)

2. Polytetrafluoroethylene (PTFE)

3. Copolymers

4. Polyoxymethylene (POM) (Delrin, Celcon, etc.)

5. Polypropylene (PP) — a vinyl polymer

Some results from these types of polymers are discussed in the LatticeTM manual and will be covered

in class.

Problems

2–1. The radial distribution function for the freely-jointed chain model was shown in class to be

W (r)dr =(

β√π

)3

4πr2e−β2r2dr (2.96)

where

β =

√3

2nl2(2.97)

Because the form of the radial distribution function even for non-freely-jointed chains is similar

to the freely-jointed chain result (except with a different value for β) it is worth studying this

function.


a. Show that the maximum in W (r)dr occurs at r = 1β

b. Show that the mean value of r is 〈r〉 = 2β√

π.

c. Show that the average value of r2 is 〈r2〉 = 32β2

d. What is the standard deviation in r?

e. Express the answers to a through d in terms of number of bonds (n) and bond length

(l).

2–2. Pretend that linear polyethylene (PE or (CH2 CH2)n ) can be modeled as a freely-jointed

chain in which each bond is 1.53A long. Using the results from problem 2-1, evaluate the

following quantities:

a. How many bonds are in a 500,000 g/mole PE polymer chain?

b. What is the maximum length of the chain?

c. What is the most likely end to end distance?

d. What is the mean end-to-end distance

e. What is the root mean squared end-to-end distance?

f. What is the standard deviation in r?

g. Plot the radial distribution function for this polymer chain and indicate on your plot

the answers to c, d, and e.

2–3. Polymer A contains x freely jointed segments each of length la and polymer B contains y

freely jointed segments each of length lb. One end of A is connected to one end of B to form

a two block, block copolymer.

a. What is the root-mean-squared end-to-end distance of this block copolymer?

b. If segments of A are randomly interspersed with segments of B and the segments are still

freely jointed, what would be the root-mean-squared end-to-end distance of the random

copolymer?

2–4. How do each of the following affect the radius of gyration?

a. Decreasing the molecular weight.

b. Decreasing the polymer-solvent interaction.

c. Adding the effects of excluded volume.

d. Reducing the hindrance to rotations about bonds.


2–5. Many polymers have more than one type of bond the different types of bonds may have

more the one value for bond length. One example is Nylon 66 whose repeat unit structure is

HO–C–(CH2)4–C–OH + H2N–(CH2)6–NH2

—(N–C–(CH2)4–C–N–(CH2)6)—

O O

O O

H H

This polymer has N–CH2 bonds, N–carbonyl bonds, carbonyl-CH2 bonds, and CH2–CH2

bonds. Explain, in general terms, how you would use the rotational isomeric state model to

calculate the characteristic ratio of Nylon 66.

Chapter 3

Polymer Molecular Weight

3.1 Introduction

Polymer molecular weight is important because it determines many physical properties. Some

examples include the temperatures for transitions from liquids to waxes to rubbers to solids and

mechanical properties such as stiffness, strength, viscoelasticity, toughness, and viscosity. If molec-

ular weight is too low, the transition temperatures and the mechanical properties will generally be

too low for the polymer material to have any useful commercial applications. For a polymer to be

useful it must have transition temperatures to waxes or liquids that are above room temperatures

and it must have mechanical properties sufficient to bear design loads.

For example, consider the property of tensile strength. Figure 3.1 shows a typical plot of

strength as a function of molecular weight. At low molecular weight, the strength is too low for

the polymer material to be useful. At high molecular weight, the strength increases eventually

saturating to the infinite molecular weight result of S∞. The strength-molecular weight relation

can be approximated by the inverse relation

S = S∞ − A

M(3.1)

where A is a constant and M is the molecular weight. Many properties have similar molecular

weight dependencies. They start at a low value and eventually saturate at a high value that is

characteristic for infinite or very large molecular weight.

Unlike small molecules, however, the molecular weight of a polymer is not one unique value.

Rather, a given polymer will have a distribution of molecular weights. The distribution will depend

on the way the polymer is produced. For polymers we should not speak of a molecular weight,

but rather of the distribution of molecular weight, P (M), or of the average molecular weight, 〈M〉.Polymer physical properties will be functions of the molecular weight distribution function as in

S = S∞ − A

F [P (M)](3.2)

43

44 CHAPTER 3. POLYMER MOLECULAR WEIGHT

Molecular Weight

Stre

ngth

S∞

S = S∞ – AM

Figure 3.1: A typical plot of tensile strength as a function of molecular weight.

where F [P (M)] is some function of the complete molecular weight distribution function. For some

properties, F [P (M)] my reduce to simply an average molecular weight. The property will thus be a

function of the average molecular weight, 〈M〉, and insensitive to other the details of the molecular

weight distribution function:

S = S∞ − A

〈M〉(3.3)

There are many ways, however, to calculate an average molecular weight. The question therefore

is how do you define the average molecular weight for a given distribution of molecular weights.

The answer is that the type of property being studied will determine the desired type of average

molecular weight. For example, strength properties may be influenced more by high molecular

weight molecules than by low molecular weight molecules and thus the average molecular weight

for strength properties should be weighted to emphasize the presence of high molecular weight

polymer. In this chapter we consider several ways of calculating molecular weights. We also

consider the meanings of those averages. Finally, we consider typical distributions of molecular

weights.

3.2 Number Average Molecular Weight

Consider a property which is only sensitive to the number of molecules present — a property that

is not influenced by the size of any particle in the mixture. The best example of such properties

are the colligative properties of solutions such as boiling point elevation, freezing point depression,

and osmotic pressure. For such properties, the most relevant average molecular weight is the total

3.2. NUMBER AVERAGE MOLECULAR WEIGHT 45

weight of polymer divided by the number of polymer molecules. This average molecular weight

follows the conventional definition for the mean value of any statistical quantity. In polymer science,

it is called the number average molecular weight — MN .

To get a formula for MN , we must first realize that the molecular weight distribution is not a

continuous function of M . Rather, only discrete values of M are allowed. The possible values of M

are the various multiples of the monomer molecular weight — M0. By monomer molecular weight

we mean the weight per monomer that appears in the polymer chain. For condensation reactions,

for example, where molecules of water are typically lost from the monomers during reaction, we will

take M0 as the monomer molecular weight less any weight loss due to the polymerization reaction.

The possible values of M make up a set of numbers with discrete values labeled Mi. Let Ni be the

number of polymers with molecular weight Mi. Then the total weight of all polymers is

Total Weight =∞∑i=1

NiMi (3.4)

and the total number of polymer molecules is

Total Number =∞∑i=1

Ni (3.5)

As discussed above, the number average molecular weight is

MN =∑∞

i=1 NiMi∑∞i=1 Ni

=Total Weight

Number of Polymers=

WeightPolymer

(3.6)

The term Ni/∑

Ni is physically the number fraction of polymers with molecular weight Mi. If we

denote number fraction as Xi (i.e., mole fraction) the number average molecular weight is

MN =∞∑i=1

XiMi (3.7)

In lab experiments it is more common to measure out certain weights of a polymer rather than

certain numbers of moles of a polymer. It is thus useful to derive an alternate form for MN in

terms or weight fraction of polymers with molecular weight Mi denoted as wi. First we note that

the concentration of polymer species i is (in weight per unit volume):

ci =NiMi

V(3.8)

Inserting ci for NiMi and expressing Ni in terms of ci results in

MN =∑∞

i=1 ci∑∞i=1

ciMi

(3.9)

Dividing numerator and denominator by∑

ci results in

MN =1∑∞

i=1wiMi

(3.10)


where wi is the weight fraction of polymer i or the weight of polymer i divided by the total polymer

weight:

wi =NiMi∑∞i=1 NiMi

=ci∑∞i=1 ci

(3.11)

3.3 Weight Average Molecular Weight

Consider of polymer property which depends not just on the number of polymer molecules but

on the size or weight of each polymer molecule. A classic example is light scattering. For such

a property we need a weight average molecular weight. To derive the weight average molecular

weight, replace the appearance of the number of polymers of molecular weight i or Ni in the

number average molecular weight formula with the weight of polymer having molecular weight i or

NiMi. The result is

MW =∑∞

i=1 NiM2i∑∞

i=1 NiMi(3.12)

By noting that NiMi/∑

NiMi is the weight fraction of polymer with molecular weight i, wi, an

alternative form for weight average molecular weight in terms of weight fractions

MW =∞∑i=1

wiMi (3.13)

Comparing this expression to the expression for number average molecular weight in terms of

number fraction (see Eq. (3.7)) we see that MN is the average Mi weighted according to number

fractions and that MW is the average Mi weighted according to weight fractions. The meanings of

their names are thus apparent.

3.4 Other Average Molecular Weights

To get MW from MN we replaced Ni by NiMi. We can generalize this process and replace Ni by

NiMki to get an average molecular weight denoted as Mk:

Mk =∑∞

i=1 NiMk+1i∑∞

i=1 NiMki

(3.14)

Thus M0 = MN , and M1 = MW . Several other Mk forms appear in experiments. Two examples

are M2 = Mz and M3 = Mz+1 which are used in analysis of ultracentrifugation experiments.

One average molecular weight which does not fit into the mold of Mk is the viscosity average

molecular weight or Mv. It is defined by

Mv =(∑∞

i=1 NiM1+ai∑∞

i=1 NiMi

) 1a

(3.15)

3.5. A DISTRIBUTION OF MOLECULAR WEIGHTS 47

Molecular Weight

Wei

ght F

ract

ion

MN MvMW

Mz

Mz+1

Figure 3.2: A schematic plot of a distribution of molecular weights along with the rankings of the variousaverage molecular weights.

where a is a constant that depends on the polymer/solvent pair used in the viscosity experiments.

Viscosity average molecular weight and viscosity experiments are discussed in Chapter 6.

For any molecular weight distribution, the various average molecular weights always rank in

the order

MN ≤ Mv ≤ MW ≤ Mz ≤ Mz+1 ≤ M4 ≤ . . . (3.16)

The equalities hold only when the polymer is monodisperse; i.e., only when all molecules have the

same molecular weight. For monodisperse polymers all molecular weight averages are the same and

equal to the one molecular weight. For polydisperse polymers, the average molecular weights will all

be different and will rank in the above order. Historically this fact was not always recognized thus

it was sometimes difficult to reconcile conflicting experimental results. Say two scientists measured

average molecular weight, but one used a colligative property which yields MN and the other used

light scattering which yields MW . Until it was recognizes that MN 6= MW , it was difficult to explain

differing experimental results on the same polymer solution.

3.5 A Distribution of Molecular Weights

Schematically, a typical molecular weight distribution might appear as in Fig. 3.2. It resembles

a probability distribution curve. The various average molecular weights are indicated in their

expected rank.

The spread of any distribution function can be characterized by its standard deviation, or

equivalently by its coefficient of variation. We can express the standard deviation of molecular


weight in terms of MN and MW . The definition of variance, σ2, is

σ2 = 〈M2〉 − 〈M〉2 (3.17)

where angle brackets (e.g., 〈M〉) denote conventional averaging. In terms of Ni and Mi the variance

is

σ2 =1N

∞∑i=1

NiM2i −

(1N

∞∑i=1

NiMi

)2

=∑∞

i=1 NiM2i

∑∞i=1 NiMi∑∞

i=1 Ni∑∞

i=1 NiMi−MN

2 (3.18)

which in terms of MN and MW is

σ2 = MW MN −MN2 = MN

2(

MW

MN

− 1)

(3.19)

or the standard deviation is

σ = MN

√MW

MN

− 1 (3.20)

The coefficient of variation is the mean divided by the standard deviation. Because MN is also the

conventional mean

C.V. =σ

MN

=

√MW

MN

− 1 (3.21)

A key term in the coefficient of variation is MW

MN. This term is known as the polydispersity

index. For the coefficient of variation to be real (as it must), the polydispersity index must be

greater than or equal to one. When it is equal to one, the coefficient of variation is zero which

means that the distribution is monodisperse. For all real polymers it is greater that one and the

amount that it is greater than one is a measure of the polydispersity of that polymer.

3.6 Most Probable Molecular Weight Distribution

Many condensation polymers are synthesized by the polymerization of bifunctional monomers. If

we denote two functional groups as A and B than a bifunctional monomer would have an A group

on one end and a B group on the other and be denoted A B. The polymerization reaction of

A B is

n(A B) → (A B)n (3.22)

For example if A is an acid group ( COOH) and B is an alcohol group ( OH), the A B monomer

can polymerize to a polyester. Or, if A is an acid group ( COOH) and B is an amine group

( NH2) the A B monomer can polymerize to a polyamide. Flory considered the polymerization

of A B type monomers and used simple statistical arguments to calculate the expected, or most

probable distribution of molecular weights. His results give us insight into typical molecular weight

distributions.

3.6. MOST PROBABLE MOLECULAR WEIGHT DISTRIBUTION 49

We define p as the fraction of functional groups of type A that have reacted at a given stage of

polymerization. Because A reacts by reacting with B, the fraction of functional groups of type B

that have reacted at the same stage of polymerization is also p. We define p in mathematical terms

but note that in practical terms it is often easily accessible by measurement. For example, to find

the fraction of reacted acid A groups of type COOH, one could use simple acid/base titration

experiments. Next, after some amount of polymerization (i.e., p 6= 0) we select a molecule at

random. We begin at one end of the molecule which will be an unreacted A group. The probability

that the adjacent B group is also unreacted is simply (1− p) — one minus the probability that a

B group has reacted. Thus the probability that the randomly selected molecule is a monomer is

P (i = 1) = (1− p) (3.23)

The probability that the randomly selected molecule is a dimer is equal the product of the inde-

pendent probabilities that the first group is reacted (p) and the second in unreacted (1− p):

P (i = 2) = p(1− p) (3.24)

Continuing on by induction, the probability that the randomly selected molecule has a degree of

polymerization i is

P (i) = pi−1(1− p) (3.25)

The pi−1 term is for the i− 1 reacted functional groups in the chain and the 1− p term is for

the terminal unreacted functional group.

If there are N molecules in the polymerizing mixture, then the number of polymer chains of

length i is N times the probability of having length i:

Ni = Npi−1(1− p) (3.26)

N is related to the initial number of monomers N0 by N = N0(1− p). This relation can easily be

derived be realizing that each reaction of a functional group reduces the total number of molecules

by one. For extent of reaction p, the total number of molecules is reduced by N0p. Now, in terms

of known quantities Ni is

Ni = N0pi−1(1− p)2 (3.27)

The above equation for Ni describes the complete polymer distribution. It is called the most

probable distribution or the Flory Distribution. Virtually all condensation polymers no matter how

they are formed will end up with a distribution resembling the most probable distribution. Plots of

Ni for various values of p are given in Fig. 3.3. At all values of p, all molecular weights are present

to some extent. The surprising results is that at all values of p, the most probable species is the

monomer. This monotonically decreasing function is not the type commonly drawn to illustrate

distribution functions.


Degree of Polymerization

Num

ber F

ract

ion

(Ni/N

0)

0 20 40 60 80 100 0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

p=0.94

p=0.98

p=0.96

Figure 3.3: The number fraction as a function of degree of polymerization for the most probable molecularweight distribution. The three curves are for three values of p.

A more familiar distribution function results if we consider the weight fraction of polymer with

length i. Weight fraction is defined by

wi =iM0Ni

N0M0=

iNi

N0= ipi−1(1− p)2 (3.28)

where M0 is the monomer molecular weight. When the repeat of the polymer has lower molecular

weight than the monomer, because of reaction products such as H2O loss due to condensation, M0

should be the molecular weight of the monomer that makes it into the polymer. In other words, M0

is the repeat unit molecular weight, iM0 is the molecular weight of a polymer of length i, and N0M0

is the total weight of monomer that ends up in a polymer. Some plots of weight fraction for various

values of p are given in Fig. 3.4. The most prevalent species is no longer the monomer. Although

there will be a lot of monomers, each monomer weighs very little. As time of reaction increases,

which increases p, the peak in the weight fraction shifts to higher values and the distribution curve

broadens. The peak molecular weight turns out to be very close to MN expecially as p is close to

1 (see problem 3 at end of this chapter)

Now that we have a complete distribution function (i.e., an equation for Ni) we can calculate

MN and MW . We can calculate MN for the most probable distribution using two methods. First

we evaluate the sums in the number average molecular weight formula:

MN =∑∞

i=1 iM0Ni∑∞i=1 Ni

= M0(1− p)∞∑i=1

ipi−1 (3.29)


Degree of Polymerization

Wei

ght F

ract

ion

(wi)

0 20 40 60 80 100 0.000

0.005

0.010

0.015

0.020

0.025

p=0.94

p=0.96

p=0.98

Figure 3.4: The weight fraction as a function of degree of polymerization for the most probable molecularweight distribution. The three curves are for three values of p.

The evaluation of the sum is nontrivial. The sum, however, can be expressed as the derivative of

another sum which is simpler to evaluate.

∞∑i=1

ipi−1 =d

dp

∞∑i=1

pi =d

dp

(p

1− p

)(3.30)

Evaluating the derivative gives∞∑i=1

ipi−1 =1

(1− p)2(3.31)

Multiplying by M0(1− p) gives

MN =M0

1− p(3.32)

An alternative and simpler method to MN is to realize that, by conservation of mass, the total

weight of material is always M0N0. From above, the total number of polymers is N0(1− p). Thus

MN =Total weight of polymer

Total number of polymers=

M0N0

N0(1− p)=

M0

1− p(3.33)

To get MW for the most probable distribution we use the weight average molecular weight

formula in terms of weight fractions:

MW =∞∑i=1

wiiM0 = M0(1− p)2∞∑i=1

i2pi−1 (3.34)


We evaluate the sum using the trick used to find MN and some additional work.

∞∑i=1

i2pi−1 =d

dp

∞∑i=1

ipi =d

dp

(p

∞∑i=1

pi−1

)=

d

dp

(p

(1− p)2

)(3.35)

The last step uses the result from the MN calculation. Evaluating the derivative gives

∞∑i=1

i2pi−1 =1 + p

(1− p)3(3.36)

Multiplying by M0(1− p)2 gives the final result:

MW = M01 + p

1− p(3.37)

Combining the results for MN and MW , the polydispersity index for the most probable distri-

bution isMW

MN

= 1 + p (3.38)

As the reaction nears completion, p approaches one and the polydispersity index approaches 2.

That is the coefficient of variation of the most probable distribution is 100%. That large of a

coefficient of variation means that the molecular weight distribution is relatively broad.

We also notice that as p approaches one, both MN and MW approach infinity. This limit

means that all the monomers will be in a single polymer molecule. It is usually not desirable to

have molecular weights that are too high. Such polymers would not be processible; they would

not flow when melted. To avoid unprocessible polymers, it is desirable to use methods to control

molecular weight. One way to control molecular weight would be to freeze the reaction at some p

less than one. This scheme, however, can produce a material that is unstable with time. Instability

occurs if over long times, there are more reactions (albeit at a slow rate) which cause p to increase.

When p increases, the polymer properties change with time and might eventually give a molecular

weight that is too high to be processible.

One solution to molecular weight control is to polymerize the two monomers A A and B B

instead of the single monomer A B. If the two monomers are mixed in equal proportions, the

analysis will be identical to the one above and there will be no molecular weight control (note:

although the analysis is the same, the meaning of M0 has to be changed to be half the repeat

unit molecular weight to account for the fact that the synthesis is from two monomers (A−A and

B−B) instead of from one monomer (A−B)). If the proportions are unequal and r = NA/NB < 1

then the results are different. A more complicated analysis gives the following MN :

MN =M0(1 + r)1 + r − 2rp

≈ M0(1 + r)1− r

(3.39)


where, as explained above, M0 is half the repeat unit molecular weight. The second part of this

equation assumes p is equal to one. Sample calculations for various values of r give

r = 1.00 MN = ∞r = 0.99 MN = 199M0

r = 0.95 MN = 39M0

r = 0.90 MN = 19M0

By selecting r, we see it is possible to control molecular weight to some finite value. Physically

what happens is that the monomer mixture runs out of A A and all polymers are end capped

with B B monomers. Because B can only react with A and no unreacted A remains, the reaction

stops at a finite molecular weight. The only problem is that small changes in r lead to large changes

in MN . For example a 5% deviation of r from 1.00 reduces the molecular weight from infinite to

39M0. But, 39M0 is not a very high molecular weight and may not be high enough to be useful.

To prevent polymerization from stopping at low molecular weights, you must have accurate control

over r. Also you must account for any side reactions and monomer volatility which might remove

monomer of one type and effectively change r.

Problems

3–1. Suppose you have n batches of polydisperse polymers. Let Ni,j be the number of polymers

of type j with degree of polymerization i and Mi,j be the molecular weight of that polymer.

The basic MN and MW equations for the total mixture of polymers now require double sums:

MN =

∑nj=1

∑i Ni,jMi,j∑n

j=1

∑i Ni,j

and MW =

∑nj=1

∑i Ni,jM

2i,j∑n

j=1

∑i Ni,jMi,j

(3.40)

Now, assume that the number average and weight average molecular weights of batch j are

MN j and MW j . and that you mix a weight wj of each batch to make a new polymer blend.

a. Starting from the above basic number average molecular weight definition, show that

the number average molecular weight of the blend is

MN =w1 + w2 + · · ·+ wn

w1

MN 1+ w2

MN 2+ · · ·+ wn

MN n

In other words, show that MN of the blend can be calculated from the individual MN j

of the components of the blend. Here MN j has the usual definition of

MN j =∑

i Ni,jMi,j∑i Ni,j

(3.41)

or a sum over just the polymers of component j.


b Starting from the above basic weight average molecular weight definition, show that the

weight average molecular weight of the blend is

MW =w1MW 1 + w2MW 2 + · · ·+ ωnMW n

w1 + w2 + · · ·+ wn

In other words, show that MW of the blend can be calculated from the individual MW j

of the components of the blend. Here MW j has the usual definition of

MW j =

∑i Ni,jM

2i,j∑

i Ni,jMi,j(3.42)

or a sum over just the polymers of component j.

3–2. Calcium stearate (Ca(OOC(CH2)16CH3)2, molecular weight = 607) is sometimes used as a

lubricant in the processing of poly(vinyl chloride). A sample of pure PVC polymer with a

polydispersity index of 2.8 is modifed by the addition of 3% by weight of calcium stearate.

TYhe mixture of PVC and salcium stearate is found to have MN = 15, 000 g/mol.

a. What is the MN of the PVC part of the compound? (Hint: use the blend MN result

from the previous problem.)

b. What is the MW of the blend?

c. What effect does the calcium stearate have on the light scattering and osmotic pressure

properties of the polymer? (Hint: light scattering measures MW while osmotic pressure

measures MN )

d. What is the highest possible MN for a polymer containing 3% by weight of calcium

stearate?

3–3. Consider the most probable molecular weight distribution:

a. Derive an expression for P (M) where P (M) is the probability that a randomly selected

polymer chain as molecular weight M . Express your result in terms of M (and not

degree of polymerization i).

b. What molecular weight has the maximum probability?

c. Derive an expression for w(M) where w(M) is the weight fraction of polymer that has

molecular weight M . Again, express your answer in terms of M (and not x).

d. What molecular weight has the largest weight fraction? Express your answer in terms

of the number average molecular weight.

3–4. Calculate the percentage conversion of functional groups required to obtain a polyester with

a number-average molecular weight of 24,000 g/mol from the monomer HO(CH2)14COOH.


3–5. A polyamide was prepared by bulk polymerization of hexamethyl diamine (9.22 g and molec-

ular weight 116) and adipic acid (13.2 g and molecular weight 166) at 280◦C. Analysis of the

whole reaction product showed that it contained 2.6× 10−3 moles of carboxylic acid groups.

Evaluate MN of the mixture. Assume it has a “most probable distibution” and also evaluate

MW .


Chapter 4

Polymer Solutions

4.1 Introduction

There are many reasons for studying polymer solutions. One obvious reason is to understand the

nature of those solutions. A practical reason is that polymer solutions are often used when measur-

ing molecular weight. Some common molecular weight determination methods are osmotic pressure

measurements, solution viscosity, solution light scattering, and gel-phase chromatography. Each of

these methods uses a polymer solution. Before discussing these or other polymer characterization

techniques, we therefore will spend some time discussing the thermodynamics of polymer solutions.

4.2 Solution Thermodynamics

Solutions are often characterized in terms of Gibbs free energy — G. G can be viewed as a

multivariable function that depends on temperature (T ), pressure (P ), and the number of moles

of each component (ni for i = 1 to number of components). By multivariable math analysis, the

differential in free energy following a change in any or all of the key variables is:

dG =(

∂G

∂P

)T,ni

dP +(

∂G

∂T

)P,ni

dT +∑

i

(∂G

∂ni

)T,P,nj 6=ni

dni (4.1)

The subscripts on the partial derivatives mean that those variables are held constant while taking

the derivative. We will be using lower case n to mean moles of some component and upper case

N to mean molecules. The number of molecules N is always equal to nL where L is Avagadro’s

number.

We know some of the terms in the dG expression from conventional thermodynamics. Consider

a solution with constant composition (i.e., dni = 0 for all i). The total free energy is

G = H − TS (4.2)

57

58 CHAPTER 4. POLYMER SOLUTIONS

The differential in free energy follows the well known relation

dG = V dP − SdT (4.3)

Comparing this expression to the solution results above gives(∂G

∂P

)T,ni

= V and(

∂G

∂T

)P,ni

= −S (4.4)

We do not have a similar result for ∂G/∂ni. Instead ∂G/∂ni is given a name; it is called the

chemical potential and denoted by µi or Gi. Specifically:(∂G

∂ni

)T,P,nj 6=ni

= µi = Gi (4.5)

Using these new results, the differential in G for a solution becomes

dG = V dP − SdT +∑

i

µidni (4.6)

Next we consider the change in free energy upon mixing two components (A and B) at constant

temperature and pressure (dP = dT = 0). Only the chemical potential terms cause a change in G.

The differential form reduces to

dG = µAdnA + µBdnB (4.7)

Integrating dG from 0 to nA and from 0 to nB which corresponds to the mixing process gives

G = µAnA + µBnB (4.8)

If instead of mixing A and B we consider pure components A and B, the expression for G will be

the same except that the chemical potentials will be different. If we denote the chemical potential

of pure components with a superscript “◦”, the free energy of the separated pure components is

G◦ = µ◦AnA + µ◦BnB (4.9)

Comparing the solution G and the pure component G◦, we find the change in free energy on mixing

or ∆Gmix to be:

∆Gmix = nA (µA − µ◦A) + nB (µB − µ◦B) = nA (GA −G◦A) + nB (GB −G◦

B) (4.10)

To understand solutions we need to evaluate terms like (µA − µ◦A). As expressed, chemical

potential is somewhat awkward for practical use. We thus redefine it in terms of activity. Ac-

tivity at first appears no better then chemical potential, but we will soon show that it has useful

interpretations under important special cases. We thus write chemical potential as

µA = µ•A + RT ln aA (4.11)

4.2. SOLUTION THERMODYNAMICS 59

where aA is the relative activity of component A and µ•A is the chemical potential in an arbitrarily

chosen reference state. In terms of chemical potential, the activity is

aA = exp(

µA − µ•ART

)(4.12)

We next apply the definition of activity to the analysis of a solution in equilibrium with its

vapor. At equilibrium the chemical potential of component A in the vapor is equal to its chemical

potential in the solution. We assume the vapor is an ideal gas and use the above result that

∂G/∂P = V to get

∂µvapA

∂P=

∂GvapA

∂P=

∂

∂P

∂Gvap

∂nA=

∂

∂nA

∂Gvap

∂P=

∂V

∂nA= VA (4.13)

where VA is the partial molar volume of component A. In an ideal vapor of A, V = nART/PAwhere

PA is the partial pressure due to component A. Differentiating with respect to nA to get VA gives

∂µvapA

∂PA=

RT

PAor dµvap

A = RTdPA

PA(4.14)

Integrating we get

µvapA = RT lnPA + const (4.15)

The integration constant is found by using the reference state where µA = µ•A. Finding the constant

results in

µvapA = µ•A + RT ln

PA

P •A

(4.16)

where P •A is the partial pressure of component A in the reference state.

We turn next to the chemical potential in the liquid. Because the liquid is in equilibrium

with the vapor, the chemical potential of component A in the liquid must be equal to its chemical

potential in the vapor. Thus

µlA = µvap

A = µ•A + RT lnPA

P •A

(4.17)

A good reference state for liquids is the chemical potential of the pure liquid. Defining µ◦A as the

chemical potential of the pure liquid we have

µ◦A = µ•A + RT lnP ◦

A

P •A

(4.18)

Solving for µ•A and substituting into the expression for µlA gives

µlA = µvap

A = µ◦A + RT lnPA

P ◦A

(4.19)

In general, PA < P ◦A which implies µl

A < µ◦A. Physically, this observation means that compounds

are less active in solution then they are in pure form. In the above analysis we have assumed

the vapor is an ideal vapor. In effect, we have proposed a physical model for the liquid/vapor


equilibrium. In this model, the activity is no longer an abstract property; it is defined in terms of

physical quantities. The activity of any component is the ratio of its partial pressure to the partial

pressure that exists over a pure liquid of that component. Thus, the activity of component A is

PA/P ◦A.

We can extend our model by introducing an assumption about the solution properties. Let’s

say that the solution (as well as the vapor) is ideal in that it obeys Raoult’s law which states

that PA = XAP ◦A where XA is the mole fraction of component A in the solution. With this new

assumption, the activity coefficient of component A is XA and its chemical potential is

µA = µ◦A + RT lnXA (4.20)

In examining the free energy of mixing, we noted that we need terms like (µA − µ◦A). For an

ideal solution and an ideal vapor

µA − µ◦A = RT lnXA (4.21)

Our previous result for free energy of mixing becomes

∆Gmix = RT (nA lnXA + nB lnXB) (4.22)

This free energy of mixing can be split up into an entropy part and an enthalpy part with

∆Gmix = ∆Hmix − T∆Smix (4.23)

We begin with entropy of mixing. A fundamental relation of thermodynamics is ∂G/∂T = −S.

Applying this relation to chemical potentials by taking the temperature derivative of µA gives

SA = −∂µA

∂T= S◦A −R lnXA (4.24)

Following the free energy of mixing analysis, the entropy of mixing is

∆Smix = nA(SA − S◦A) + nB(SB − S◦B) = −R (nA lnXA + nB lnXB) (4.25)

Finally, the enthalpy of mixing is

∆Hmix = ∆Gmix + T∆Smix = 0 (4.26)

∆Hmix for an ideal solution is zero. ∆Hmix is zero because the assumption of an ideal solution

implies no interactions between the components of the solution.

We next examine the sign of ∆Gmix. Both nA and nB, the numbers of moles, must be positive.

Both XA and XB, the mole fractions, must between 0 and 1, which means that both lnXA and

lnXB must be negative. These facts mean that ∆Gmix is always negative and ∆Smix is always

positive. In other words all two component ideal solutions are miscible in all proportions. The

driving force for this miscibility is the increase in entropy of mixed components vs. pure separated

4.3. PHYSICAL ORIGINS OF ENTROPY OF MIXING 61

A A B A B B A A

A B A A A A B A

B A A B B A A A

A B B A A B A B

A A A B B A B A

A B B A A A B A

A B A A A B A A

Figure 4.1: A simple lattice model for a solution of A and B.

components. The fact that not all materials are soluble in all other materials means that many

solutions deviate from ideality. Deviations from ideality often mean that enthalpy is important

and not zero. To make two components immiscible, ∆Gmix must be positive which implies that

∆Hmix 6= 0 and further that ∆Hmix is sufficiently positive that it overwhelms the entropy or that

∆Hmix > T∆Smix.

4.3 Physical Origins of Entropy of Mixing

The physical origins of the entropy of mixing can be modeled using a simple lattice model for the

structure of a solution. Say the simple lattice model in Fig. 4.1 represents a solution of A and B.

By statistical thermodynamics, entropy S is

S = k lnW (4.27)

where W is the number of possible configurations of A and B solution. In the simple lattice, it is

easy to count the number of ways of putting NA A units and NB B units into NA + NB lattice

sites. The result from combinatorial analysis is

W =

(NA + NB

NA

)=

(NA + NB)!NA!NB!

(4.28)

(Note we now use upper case N ’s because they refer to individual molecules of component A and

B).

For large NA and NB (i.e., for many molecules) we can apply Stirling’s approximation for

factorials (lnn! ≈ n lnn) to get

W = NA [ln (NA + NB)− lnNA] + NB [ln (NA + NB)− lnNB] (4.29)


A A B A A A A A

A A B A A A A A

A A B B B A B A

A A A A B A B A

A B B B B A B A

A B A A A A B A

A B B B B B B A

Figure 4.2: Polymer molecule B in a lattice model for a solution in solvent A.

We can even replace the ≈ by a straight equal sign because NA and NB will be so large that

Stirling’s approximation is effectively exact. By definition of mole fraction XA = NA/(NA + NB)

and XB = NB/(NA + NB). Introducing Avagadro’s number, L, and realizing that R = Lk,

NA = LnA, and NB = LnB, we quickly derive

∆Smix = −R(nA lnXA + nB lnXB) (4.30)

This result agrees exactly with the activity analysis for entropy of mixing. Thus the lattice model

recovers all details of the thermodynamics analysis. A lattice can be viewed as a good physical

picture of an ideal solution.

We find some interesting results when we apply the lattice model to a polymer solution. The

physics of mixing is unchanged, but now we must consider the case where one molecule is much

larger than the other molecule. Figure 4.2 shows a lattice model for one polymer (component B) in

solvent A. It is clear that the number of ways of introducing the polymer molecule into solvent A

will be much less than the number of ways of introducing the same amount of monomer molecules

B into the same solvent. To count the polymer lattice configurations we could go back to the lattice

model for a solution of A and B and examine all of the W configurations. For that large set of

configurations we must now reject all configurations that do not have connected B molecules. The

remaining configurations, which will be much less than W , will be the number of configurations of

polymer B in solvent A. The conclusion is that the entropy of polymer solutions is much less that

the corresponding entropy for monomer solutions of the same weight fractions A and B.

Compared to the monomer solution problem (Fig. 4.1), it is a much harder task to count the

number of configurations in which all B units are connected into chains of length x with excluded

volume on a lattice with solvent A. An approximate calculation gives the following result for the

4.4. FLORY-HUGGINS THEORY 63

entropy of mixing:

∆Smix = −R (nA ln vA + nB ln vB) (4.31)

where vA and vB are volume fractions of solvent (component A) and polymer (component B):

vA =NA

NA + xNBand vB =

xNB

NA + xNB(4.32)

Consider two solutions of equal weight fractions of component A and B — monomer B molecules

in solvent A and a polymer of connected B monomers in solvent A. The mole fractions of A and B

in the monomer solution are identical to the volume fractions of A and B in the polymer solution.

Likewise, nA will be the same for both solution. The ∆Smix expression differs only in the value for

nB. Because the polymer solution connects many B monomers into a single polymer, nB in the

polymer solution will be hundreds or thousands of times lower than nB in the monomer solution.

Furthermore, in dilute solutions, the nB lnXB and nB ln vB terms are the dominant entropy terms.

We thus conclude that the entropy of mixing for polymer solutions will be hundreds or thousands

of times lower than the entropy of mixing for monomer solutions.

One thing that monomer solutions and polymer solutions have in common is that they both obey

conventional thermodynamics. We do not need to derive new methods for understanding polymer

solutions. A major difference, however, is that polymer solutions are much more likely to deviate

from ideal behavior at comparable concentrations (in weight fraction solute). The deviations from

ideality are direct consequence of the size of polymer molecules. The large size reduces the entropy

of mixing which reduces the term T∆Smix. When the entropy of mixing term is small, it is much

easier for the ∆Hmix term to assume a greater role. It also easy for ∆Hmix to be greater than

T∆Smix and therefore for polymers to be insoluble in many solvents.

4.4 Flory-Huggins Theory

We introduce a simple model for dealing with polymer solutions called the Flory-Huggins theory.

In brief, the theory attempts to estimate ∆Hmix using the lattice theory for solutions. We consider

nearest neighbor interaction between A and B molecules. Then

∆Hmix = kTχNAvB = RTχnAvB (4.33)

where NA is the number of molecules of component A, nA is the number of moles of component

A, vB is the volume fraction of polymer, and χ is the Flory interaction parameter. The interaction

parameter describes the level of interaction between pairs of A and B. A high positive χ indicates

a repulsive interaction and causes a positive ∆Hmix which tends to inhibit the solution process.

A negative χ indicates an attractive interaction and causes a negative ∆Hmix which guarantees

dissolution.


The free energy of mixing in the Flory-Huggins theory is

∆Gmix = RT (nA ln vA + nB ln vB + χnAvB) (4.34)

Unfortunately, the Flory-Huggins theory is not very good. In practice solution property data cannot

be predicted by using a single value of the interaction parameter. The theory is particularly bad

for dilute solutions. Reasons for these problems will be discussed latter. Despite its short-comings,

the Flory-Huggins theory has some applications for physical understanding of solution properties.

We will make qualitative use of this theory when discussing osmotic pressure experiments.

Problems

4–1. Compare the entropy of mixing for two solution. The first is a solution 1% by weight of

monomer B in solvent A (assume the molecular weights of A and B are the same). The

second is a polymer of monomer B in a 1% by weight solution in solvent A where the polymer

has a degree of polymerization of 100,000.

4–2. Explain the physical origin of each term in then Flory-Huggins expression for free-energy of

mixing in polymer solutions.

4–3. a. Explain why two components that form an ideal solution are miscible in all proportions.

b. When two components form a nonideal solution, it is common to observe that the solute

will precipitate out at low temperature and only be soluble in the solvent at elevated

temperatures. Explain why increasing the temperature often makes it more likely for

one component to dissolve in another component even when the resulting solution is

nonideal.

Chapter 5

Colligative Properties

5.1 Introduction

Properties of solutions that depend on the number of molecules present and not on the kind of

molecules are called colligative properties. These properties include boiling point elevation, freezing

point depression, and osmotic pressure. Historically, colligative properties have been one means

for determining the molecular weight of unknown compounds. In this chapter we discuss using

colligative properties to measure the molecular weight of polymers. Because colligative properties

depend on the number of molecules, we expect, and will show, that colligative property experiments

give a number average molecular weight.

5.2 Boiling Point Elevation

Figure 5.1 shows the vapor pressure of a liquid for pure liquid and for a solution with that liquid as

the solvent. In an ideal solution, the vapor pressure of the solvent, PA, is reduced from the vapor

pressure of a pure liquid, P ◦A, to XAP ◦

A where XA is the mole fraction of liquid A. This reduction is

reflected in a shift to the right of the vapor-pressure curves in Fig. 5.1. By definition, boiling point

is the temperature at which the vapor pressure of the liquid reaches 1 atm. Thus, the right-shift

caused by the dissolution of component B in solvent A causes the boiling point to increase. This

increase, ∆Tb, is the boiling point elevation effect.

A well known result from introductory chemistry is that the boiling point elevation is propor-

tional to the molar concentration of solute particles

∆Tb = Kbm (5.1)

where m is the molality of solute molecules and Kb is the boiling point elevation coefficient that is

a function of only the solvent. Molality is the number of moles of component B per 1000 grams of

solvent. If we prepare a solution of an unknown compound of molecular weight B at a concentration

65

66 CHAPTER 5. COLLIGATIVE PROPERTIES

Vapor Pressure

Vapo

r Pre

ssur

e

Solution

Temperature

Pure Solvent1 atm

Tb

∆T

Figure 5.1: Boiling point elevation effect is a consequence of the effect of solute molecules on the vaporpressure of the solvent.

c in g/cm3, then

m =1000c

MBρ(5.2)

where ρ is the density of the solvent (in g/cm3). Substituting into the expression for ∆Tb gives

MB =1000Kbc

ρ∆Tb(5.3)

or∆Tb

c=

1000Kb

ρMB(5.4)

For a given solvent (e.g., water where Kb = 0.52 and ρ = 1.00) and concentration (c), all terms in

Eq. (5.4) are known except for MB. Thus, measuring ∆Tb can be used to determine the molecular

weight MB.

We can also express boiling point elevation in terms of mole fraction. Mole fraction is

XB =cVMB

ρVMA

+ cVMB

≈ cMA

ρMB(5.5)

where V is total volume and MA is molecular weight of the solvent. The boiling point elevation

becomes

∆Tb =1000Kb

MAXB (5.6)

To apply boiling point elevation to polymers, we begin by using solution thermodynamics to

derive an expression for ∆Tb. At equilibrium, the chemical potential of the vapor is equal to the

chemical potential of the liquid

µvapA = µliq

A = µ◦A + RT lnXA orµvap

A − µ◦ART

= ln XA (5.7)

5.2. BOILING POINT ELEVATION 67

where we have assumed an ideal solution. Differentiating both sides gives

∂(µA

T

)∂T

=∂(

GAT

)∂T

=1T

(∂GA

∂T

)P

− GA

T 2= −SA

T− HA

T 2+

SA

T= −HA

T 2(5.8)

which is used to get

−Hvap

A −H◦A

RT 2=

d

dTlnXA (5.9)

where HvapA −H◦

A is the heat of vaporization of the solvent or ∆Hvap.

Now consider the process of forming a solution. As the polymer is added, the mole fraction of

A will go from 1 at the start to XA which is the mole fraction of the final solution. During the

process, the boiling point will go from Tb to T where Tb is the boiling point of the pure liquid and

T is the boiling point of the solution. Integrating over this process gives

−∫ T

Tb

∆Hvap

RT 2dT =

∫ XA

1d lnXA (5.10)

The integrals are easily evaluated if we assume that ∆Hvap is independent of temperature over the

small temperature range from Tb to T . The result is

∆Hvap

R

(1T− 1

Tb

)= ln XA (5.11)

We can simplify this result using ∆Tb = T −Tb, TTb ≈ T 2b , and lnXA = ln(1−XB) ≈ −XB. These

simplifications apply when XB is small (which occurs when the solution is dilute) and when ∆Tb is

small. In general, ∆Tb will be small when the solution is dilute. The previous equation simplifies

to∆Hvap∆Tb

RT 2b

= XB (5.12)

or

∆Tb =RT 2

b

∆HvapXB (5.13)

Comparison of this result to Eq. (5.6) gives a theoretical expression of Kb:

Kb =MART 2

b

1000∆Hvap(5.14)

The result is often derived in physical chemistry books.

In applying boiling point elevation to polymer solutions, we should realize that polymer solu-

tions are really solutions of many components. The various components are the polymer species of

different molecular weights. Because boiling point elevation is a colligative property, we can write

the boiling point elevation of a polymer solution as a sum over the mole fractions of each molecular

weight component:∆Hvap∆Tb

RT 2b

=∑

i

Xi (5.15)


where Xi is the mole fraction of polymer with molecular weight Mi. We more conveniently rewrite

Xi in terms of concentration:

Xi =ciVMi

ρVMA

+∑

iciVMi

≈ ciMA

ρMi(5.16)

where ci is the concentration in weight/unit volume (e.g., g/cm3) of polymer with molecular weight

i. The approximation in this expression is valid for dilute solutions in which the number of moles

of solvent is much greater than the total number of moles of polymer. Summing the mole fractions,

Xi, results in ∑i

Xi =cMA

ρ

∑i

ciMi∑

i ci=

cMA

ρ

∑i

wi

Mi=

cMA

ρMN

(5.17)

where

c =∑

i

ci (5.18)

The final expression for the boiling point elevation becomes

∆Tb

c=

MART 2b

ρ∆HvapMN

(5.19)

It is common to express the boiling point elevation in terms of the latent heat of vaporization, lvap,

defined as energy or vaporization per unit weight or

lvap =∆Hvap

MA=

J/moleg/mole

= heat of vaporization in J/g (5.20)

The boiling point elevation becomes

∆Tb

c=

RT 2b

ρlvapMN

(5.21)

Except for incorporation of polydispersity, there is nothing new about the boiling point eleva-

tion expression for polymer solutions vs. the comparable expression for small molecule solutions.

In polymers, however, the solution is more likely to be non-ideal. For this equation to apply we

will probably need to use very low concentrations or techniques to extrapolate to very low concen-

trations.

For an example, let’s consider a solution of polystyrene in benzene. For benzene ρ = 0.8787 g/cm3,

Tb = 55◦C, and lvap = 104 cal/g. We assume a relatively concentrated solution of c = 1 g/cm3

of a polymer with molecular weight MN = 20, 000. The change in the boiling point elevation for

this solution is ∆Tb = 1.4 × 10−3 ◦C. This boiling point elevation is very small. It is probably

beyond the accuracy of most temperature measuring equipment. The small change arises despite

relatively ideal conditions of a fairly concentrated solution and a low molecular weight polymer.

More dilute solutions or higher molecular weight polymers would give an even smaller ∆Tb. The

problem with polymer solutions is that for a given weight of material, the polymer solution will

5.3. FREEZING POINT DEPRESSION 69

have many less molecules than the comparable small molecule solution. When there are a small

number of molecules, the change in boiling point (a colligative property) is small. The problem with

the boiling point elevation method applied to polymer solutions is that it is not sensitive enough.

It has found some use with polymers but it is limited to polymers with relatively low molecular

weights. (e.g., MN less than 20,000 g/mol).

5.3 Freezing Point Depression

A similar analysis (but with sign changes) can be applied to the freezing point depression of a

polymer solution. The final result is∆Tf

c=

RT 2f

ρlfMN

(5.22)

where Tf is the freezing point of the solvent and lf is the latent heat of fusion. We consider the

same example of polystyrene in benzene with Tf = 5.5◦C, lf = 30.45 cal/g for the freezing point

of benzene. For a c = 1 g/cm3 solution of polystyrene with molecular weight MN = 20, 000, the

change in the freezing point of the solution is ∆Tf = 2.9×10−3 ◦C. Like the boiling point elevation

effect, the freezing point depression effect is too small. The technique is insensitive and only useful

for low molecular weight polymer (e.g., MN less than 20,000 g/mol).

5.4 Osmotic Pressure

Another colligative property is osmotic pressure. Figure 5.2 illustrates the osmotic pressure ef-

fect. Imagine a pure solvent and a solution separated by a semipermeable membrane. An ideal

semipermeable membrane will allow the solvent molecules to pass but prevent the solute molecules

(polymer molecules) from passing. The different concentrations on the two sides of the membrane

will cause an initial difference in chemical potential. At equilibrium, this difference in potential

will be counteracted by an effective pressure across the membrane. As shown in Fig. 5.2, it can be

imagined that solvent molecules pass from the pure solvent side to the solution side. The excess

height in the column of liquid above the solution side is related to the osmotic pressure by π = ρgh.

Here π is the osmotic pressure, ρ is the density of the solution, g is the acceleration of gravity

(9.81 m/sec2) and h is the height of the column of liquid.

We begin with a thermodynamic analysis of osmotic pressure. At equilibrium the chemical

potential in the solution will be equal to the chemical potential in the pure solvent:

µsolventA = µsolution

A = µ◦A + RT ln aA (5.23)

where µsolventA is the chemical potential of the pure liquid or

µsolventA = µ◦A (5.24)


Solvent Solution

Membrane

h

π = ρgh

Figure 5.2: A schematic view of osmotic pressure across a semipermeable membrane.

The only way the chemical potentials will be equal will be if the activity of component A in the

solution is equal to 1. The activity can be raised to 1 by applying pressure. That is, activity is a

pressure dependent quantity. By applying the correct pressure, the osmotic pressure, the activity

in the solution can be changed to 1. The osmotic pressure that gets applied occurs naturally by

the tendency to approach equilibrium.

To get the pressure dependence of activity, we consider the pressure dependence of the chemical

potentialdµA

dP=

d

dP

dG

dnA=

d

dnA

dG

dP=

dV

dnA= VA = RT

d

dPln aA (5.25)

Rearranging and integrating the left hand side from the original activity aA to the final activity 1

and the right hand side from the initial pressure 0 to the final pressure — the osmotic pressure π

— results in: ∫ 1

aA

d ln aA =∫ π

0

VA

RTdP (5.26)

which integrates to

− ln aA =πVA

RT(5.27)

Now, in an ideal solution

− ln aA = − lnXA ≈ XB. (5.28)

This last approximation follows because ln(1 − x) ≈ −x for small x. Finally, as in the analysis of

boiling point elevation, we replace XB by∑

i Xi which was derived in Eq. (5.17) to be∑i

Xi =cMA

ρMN

(5.29)

givingcMA

ρMN

=πVA

RT(5.30)

5.5. PRACTICAL ASPECTS OF OSMOTIC PRESSURE 71

But, MA/ρ is the grams per mole of solvent divided by the grams per cm3 of solvent. The grams

cancel and we have cm3 per mole of the solvent or the partial molar volume of component A —

VA. Substituting into the osmotic pressure equation thus gives:

π

c=

RT

MN

(5.31)

Rewriting the osmotic pressure equation gives a result that is similar to the ideal gas law

cRT = π

∑i NiMi∑

i Nior

∑i NiMi

VRT = π

∑i NiMi∑

i Ni(5.32)

which simplifies to

πV =∑

i

NiRT (i.e. PV = nRT ) (5.33)

For an example, let’s consider the solution of polystyrene in benzene that was used for examples

of boiling point elevation and freezing point depression; i.e., a solution of polystyrene in benzene

with MN = 20, 000 and a concentration of c = 1 g/cm3. For the correct units we use R =

8.3143× 107 ergs/K/mol and calculate π = 1.24× 104 dynes/cm2. This pressure will be measured

by a difference in heights of liquids in columns. The height difference comes from π = ρgh or

h =1.24× 104 dynes/cm2

0.8787 g/cm3 981 cm/sec2 = 14.3 cm (5.34)

This height difference is large and is an easily measurable quantity. In fact we expect to be able

to measure distances at least 100 times smaller than this result. Thus osmotic pressure can, in

principle, be used to determine molecular weights in polymers with MN up to 2,000,000 g/mol.

5.5 Practical Aspects of Osmotic Pressure

Osmotic pressure measurements appear to be a suitable method for measuring number average

molecular weights in polymers. It is therefore worthwhile considering practical aspects of polymer

characterization by osmotic pressure. The first practical consideration is that we expect polymer

solutions to deviate from ideal behavior and thus the osmotic pressure expression will need to be

corrected. In the limit of zero concentration, the solution will eventually become ideal. We can

therefore take a series of measurements and extrapolate back to zero concentration to get the ideal

result. In other words

limc→0

π

c=

RT

MN

(5.35)

The question which remains is “how do we extrapolate?” A common approach in thermodynamics

is to use a virial expansion. We thus write πc as a sum of many terms:

π

c=

RT

MN

+ RTA2c + RTA3c2 + · · · (5.36)


orπ

c=

RT

MN

(1 + Γ2c + Γ3c

2 + · · ·)

(5.37)

Here A2, A3, ... and the related Γ2, Γ3, ... are called the virial coefficients. If we include enough

virial coefficients we will always be able fit experimental data. But, how many of these terms do

we need? Furthermore, how do we analyze experimental data when virial expansion terms are

required? We consider two approaches to this problem.

In the first approach, we assume that only the second virial coefficient — A2 or Γ2 — will be

needed. Then π/c is predicted to be linear in concentration:

π

c=

RT

MN

+ RTA2c (5.38)

A set of data for π/c vs. c can be plotted. If the results are linear, the assumption in the first

approach is valid. When the data is linear, the intercept of the data at zero concentration will be

RT/MN and thus can be used to determine MN .

Besides an intercept, we can measure the slope which is equal to RTA2. In other words the

slope of the π/c vs. c plot is proportional to the second virial coefficient — A2. We can make use

of the Flory-Huggins theory to get a physical interpretation of the second virial coefficient. The

Flory-Huggins theory includes non-ideal interactions through the Flory interaction parameter, χ.

Let’s use the Flory-Huggins theory to develop an osmotic pressure theory for nonideal solutions.

We begin with an early osmotic pressure formula:

π = −RT ln aA

VA= −

µA − µ◦AVA

(5.39)

The term µA − µ◦A is found by differentiating the free energy of mixing

d∆Gmix

dnA= µA − µ◦A (5.40)

To use the Flory-Huggins theory we differentiate the ∆Gmix from that theory. In performing the

integration we must realize that vA and vB also depend on nA. The work is left as an exercise to

the reader. The result is

µA − µ◦A = RT

[ln vA +

(1− 1

x

)vB + χv2

B

](5.41)

Substituting into the osmotic pressure formula and at the same time using the approximation

ln vA = ln(1− vB) ≈ −vB + v2B/2 (Note that in this approximation to ln(1− vB) we keep one more

term than we have used in the past. The reason for the extra term is that the µA − µ◦A expression

already includes terms with v2B), the osmotic pressure becomes:

π =RT

VA

[vB

x+(

12− χ

)v2B + · · ·

](5.42)

5.5. PRACTICAL ASPECTS OF OSMOTIC PRESSURE 73

The volume fraction of polymer, vB, is equal to the concentration of B in g/cm3 divided by the

density of polymer — ρB (vB = c/ρB). The osmotic pressure is then

π

c=

RT

VAxρB+

RT

VAρ2B

(12− χ

)c + · · · (5.43)

The first term is the ideal solution result (which can be deduced by noting that MN = VBρB and

VB = xVA in the lattice solution model).. The second term, which is proportional to concentration,

gives the second virial coefficient

We return now to the slope of the linear fit of π/c vs. c which gives the second virial coefficient.

From the Flory-Huggins analysis, the second virial coefficient is:

A2 =1

VAρ2B

(12− χ

)or A2 =

M0

ρB

(12− χ

)(5.44)

where M0 = MN/x is the monomer molecular weight. When χ is large and negative, the second

virial coefficient will be large and positive and the slope of π/c vs. c will be large and positive.

A negative interaction parameter also implies a favorable interaction (∆Gmix more negative) and

therefore a good solvent will give a large positive slope. In fact the slope of the osmotic pressure

data can be thought of as a direct measure of the solvent quality — the higher the slope the better

the solvent.

The second virial coefficient from the Flory-Huggins result is also proportional to 1/ρB or rather

is proportional to the specific volume of polymer. This result suggests an excluded volume effect.

When the excluded volume effect is absent, the solution will act as if the specific volume of the

polymer is zero (1/ρB = 0) and the second virial coefficient will therefore be zero. The excluded

volume effect disappears in a theta solvent and as a result the osmotic pressure data slope will

be zero in a theta solvent. In other words a theta solvent acts as an ideal solution to fairly high

concentrations. The fact that a zero slope is a low slope illustrates a result from earlier in the

course — theta solvents in general are not very good solvents. Although working in theta solvents

would simplify data analysis (i.e., give results that obey ideal solution laws) the fact that theta

solvents are poor solvents makes working with them difficult. It is usually more convenient to work

in good solvents and make use of extrapolation techniques. The observation of zero slope in osmotic

pressure data, however, is a useful method for determining theta solvent conditions.

Unfortunately, plots of π/c vs. c are often not linear. We thus need a second approach to

analysis of data from nonideal solutions. The obvious approach is to include both the second and

the third virial coefficients. In other words we assume that deviations from linearity are caused by

the third virial coefficient no longer being insignificant. Let’s take g as the ratio of the third virial

coefficient to the second virial coefficient squared (Γ3 = gΓ22) and let’s ignore terms beyond the

third virial coefficient. Then π/c becomes

π

c=

RT

MN

(1 + Γ2c + gΓ2

2c2)

(5.45)


We now have two parameters — Γ2 and g. Deriving two parameters for osmotic pressure data will

be more complicated than deriving the slope and intercept of simple linear fits. It requires more

advanced curve-fitting techniques. We can simplify the process by introducing some theoretical

calculations about g. For hard spheres, g can be calculated to be g = 5/8. For polymer molecules,

g has been estimated to be g = 0.25 to 0.28. The actual value of g depends on various properties such

as the expansion coefficient α, the characteristic ratio, etc.. Fortunately, however, g is restricted to

a relatively narrow range for most polymers.

Because g must be positive and a polymer cannot be more impenetrable than hard spheres, g

must be between 0 and 5/8. If we pick a value for g than we are left with only one parameter (Γ2)

and we calculate MN and Γ2 by simpler curve fitting analyses. Fortunately it has been found that

the results are not very sensitive to the exact value of g. Because polymers have g’s calculated to

be near 0.25, we will assume g = 0.25. The choice of g = 0.25 is desirable because it completes the

square and the data analysis can again be done by linear fits (Scientists, especially scientists that

worked before computers, like linear theories):

π

c=

RT

MN

(1 + Γ2c +

14Γ2

2c2

)or

√π

c=

√RT

MN

(1 +

Γ2c

2

)(5.46)

When g can be assumed to be 0.25, a plot of√

π/c vs. c should be linear. The slope will give Γ2

and the intercept will give√

RT/MN .

The advantage of setting g = 0.25 is that the data can be analyzed with a simple linear fit.

This advantage was important before computers were readily available. Now we can easily treat g

as a second parameter and do a two parameter fit to the data. You will try this type of analysis

in one of the class labs and be able to discuss whether the added complexity improves or weakens

the interpretation of the results.

5.6 Experimental Aspects of Membrane Osmometry

A simple type of osmometer is illustrated in Fig. 5.3. The solution is placed in a cell with membranes

on either side (one or two membranes, but two gives more area and faster equilibration). The entire

assembly is then immersed in pure solvent. The heights of the liquids in the capillaries are read and

the height difference gives the osmotic pressure. This apparatus is called a block type osmometer.

It is the type of osmometer used to get the data that will be given to you in a lab. This osmometer

uses a small cell and a large membrane. The membrane is supported by stainless steel plates with

holes. By supporting the membrane, the membrane can be made larger; with larger membrane

area equilibrium will be reached sooner.

Block osmometers are called static osmometers because they wait for the natural development

of equilibrium. The problem with static osmometers is that it can take hours (12-24 hrs) to reach

5.6. EXPERIMENTAL ASPECTS OF MEMBRANE OSMOMETRY 75

SIDE VIEW FRONT VIEW

Capillaries

Outerflange

Outerflange

Membrance

Exploded View

Innerflange Solution cell

Figure 5.3: A block type, static osmometer.

an accurate equilibrium. The time depends on many factors such as the membrane area and the

speed of transport through the membrane. To quicken osmotic pressure experiments, dynamic

osmometers are sometimes used. Recall that osmotic pressure develops for the purpose of raising

the activity of the solvent in the solution to 1. By applying a pressure it is possible to do the

same thing. You will know when you have applied the correct pressure by monitoring flow across

the membrane. When you apply enough pressure to stop the flow you have artificially reached

equilibrium. The pressure required can be used to get the equilibrium osmotic pressure. This quick

method, unfortunately, is less accurate.

Finally, we make a few comments about what makes a good semipermeable membrane. The

membrane must be permeable to solvent and impermeable to polymer. This requirement limits the

low-end applicability of osmometry to MN of 20,000 g/mol or more. Note that we really require

all polymers to be above 20,000 g/mol otherwise the low molecular weight tail will pass through

membrane and the measured MN will be too high (do you see why it would be too high?). Therefore,

polydisperse polymers probably require MN greater than about 50,000 g/mol; for monodisperse

polymers it might be possible to go down to 20,000 g/mol.

There are also some material concerns for the membrane. An obvious concern is that the mem-

brane not be soluble in the solvent. Perhaps the most common membrane material is gel cellulose.


Other membranes include cellulose hydrate, cellulose acetate, cellulose nitrate, polyurethanes, and

poly(chlorotrifluoroethylene).

Problems

5–1. In analyzing osmotic pressure data why is a plot of√

(π/c) versus c sometimes used rather

than a plot of (π/c) versus c?

5–2. The following are data from osmotic pressure measurements on a solution of polyester in

chloroform at 20◦C. The results are in terms of centimeters of solvent. The density of HCCl3

is 1.48 g/cm3. Plot π/c versus c and find MN under the assumption that you can neglect

terms beyond the second virial coefficient.

c (g/dl) h (cm of HCCl3)

0.57 2.829

0.28 1.008

0.17 0.521

0.10 0.275

5–3. a. Suppose that in a different universe that the boiling point elevation was given instead

by∆Hvap∆Tb

R∆T 2b

=wBMB

M0=

1M0

∑i

wiMi (5.47)

where wi is the weight fraction of polymer with molecular weight Mi and M0 is the

monomer molecular weight. By this law, what molecular weight average could by found

from boiling point elevation measurements and give a formula for calculating that molec-

ular weight average.

b. The measurement of boiling point elevation is not very useful for finding the molecular

weight of high molecular weight polymers. If boiling point elevation was given instead by

the formula in part a, would it be a more or less useful approach to finding the molecular

weight of high molecular weight materials?

Chapter 6

Viscosity

6.1 Introduction

Viscosity of a polymer solution depends on concentration and size (i.e., molecular weight) of the

dissolved polymer. By measuring the solution viscosity we should be able to get an idea about

molecular weight. Viscosity techniques are very popular because they are experimentally simple.

They are, however less accurate and the determined molecular weight, the viscosity average molec-

ular weight, is less precise. For example, Mv depends on a parameter which depends on the solvent

used to measure the viscosity. Therefore the measured molecular weight depends on the solvent

used. Despite these drawbacks, viscosity techniques are very valuable. This chapter discusses the

theory and methods for determining polymer molecular weight by measuring solution viscosity

6.2 Viscosity and Viscosity Nomenclature

Figure 6.1 shows a piece of a liquid moving at a strain rate γ under an applied shear stress of τ .

The viscosity of the liquid is the ratio of the applied shear stress to the resulting strain rate (or

equivalently, the ratio of the shear stress required to move the solution at a fixed strain rate to that

strain rate). The shear strain in Fig. 6.1 is

γ =du

dy(6.1)

where u is displacement in the x direction. The strain rate is therefore

γ =d

dt

du

dy=

d

dy

du

dt=

dvx

dy(6.2)

where vx is velocity in the x direction. The relations between viscosity (η), shear stress (τ), and

shear rate (γ) are

τ = ηγ or γ =τ

ηor η =

τ

γ(6.3)

77

78 CHAPTER 6. VISCOSITY

F

τ =F

Adγdt

γ = •

γ

Figure 6.1: A piece of a liquid moving at shear rate γ under an applied shear stress of τ .

A Newtonian fluid is one in which the viscosity is independent of the shear rate. In other words

a plot of shear stress versus shear strain rate is linear with slope η. In Newtonian fluids all the

energy goes into sliding molecules by each other. In non-Newtonian fluids, the shear stress/strain

rate relation is not linear. Typically the viscosity drops at high shear rates — a phenomenon known

as shear thinning. Although the following development will not discuss shear rate effects in detail,

the possibility of experimental results being affected by the shear rate of the measurement should

be kept in mind. Plots of shear force vs. shear rate for Newtonian and non-Newtonian fluids are

given in Fig. 6.2.

We let η0 be the viscosity of the pure solvent and η be the viscosity of a solution using that

solvent. Several methods exist for characterizing the solution viscosity, or more specifically, the

capacity of the solute to increase the viscosity of the solution. That capacity is quantified by using

one of several different measures of solution viscosity. The most common solution viscosity terms

are:

1. Relative viscosity

ηr =η

η0(6.4)

2. Specific viscosity

ηsp =η − η0

η0= ηr − 1 (6.5)

3. Inherent Viscosity

ηi =ln ηr

c(6.6)

4. Intrinsic Viscosity

[η] = limc→0

ηsp

c(6.7)

6.2. VISCOSITY AND VISCOSITY NOMENCLATURE 79

Shear Force

Non-Newtonian

Newtonian

Shear Rate

She

ar F

orce

Figure 6.2: Schematic plots of shear force vs. shear rate for Newtonian and non-Newtonian fluids.

In these equations, η is solution viscosity, η0 is viscosity of the pure solvent, and c is concentration.

Relative viscosity is self-explanatory. Specific viscosity expresses the incremental viscosity due

to the presence of the polymer in the solution. Normalizing ηsp to concentration gives ηsp/c which

expresses the capacity of a polymer to cause the solution viscosity to increase; i.e., the incremental

viscosity per unit concentration of polymer. As with other polymer solution properties, the solutions

used for viscosity measurements will be nonideal and therefore ηsp/c will depend on c. As with

osmotic pressure, it will probably be useful to extrapolate to zero concentration. The extrapolated

value of ηsp/c at zero concentration is known as the intrinsic viscosity — [η]. [η] will be shown to

be a unique function of molecular weight (for a given polymer-solvent pair) and measurements of

[η] can be used to measure molecular weight.

The remaining form for the viscosity is the inherent viscosity. Like ηsp, ln ηr is zero for pure

solvent and increases with increasing concentration, thus ln ηr also expresses the incremental vis-

cosity due to the presence of the polymer in the solution. Normalizing ln ηr to concentration or

ln ηr/c gives the inherent viscosity In the limit of zero concentration, ηi extrapolates the same as

ηsp/c and becomes equal to the intrinsic viscosity. This can be proved by:

limc→0

ln ηr

c= lim

c→0

ln (1 + ηsp)c

= limc→0

ηsp

c= [η] (6.8)

We can thus find [η] be extrapolating either ηsp/c or ηi to zero concentration. When c is not equal

to zero the specific viscosity and inherent viscosities will be different, even for an ideal solution. In

ideal solutions ηsp/c will be independent of concentration, but ni will depend on concentration.


6.3 Viscosity Theory

Einstein derived the specific viscosity for a dilute solution of hard spheres. His result is

ηsp = 2.5NBVη

V(6.9)

where NB is the number of particles (hard spheres), Vη is the hydrodynamic volume of each sphere,

and V is the total volume. For a first approximation to polymer solutions, we will use Einstein’s

theory and assume each polymer to be a hard sphere with Vη equal to some effective hydrodynamic

volume. Letting moles of polymer be nB = NB/L, we get

ηsp = 2.5nBVηL

V(6.10)

We begin with a monodisperse polymer solution with concentration c = nBM/V where M is

molecular weight. We can rewrite ηsp as

ηsp = 2.5nBM

V

VηL

M= 2.5c

VηL

M(6.11)

orηsp

c= 2.5L

Vη

M(6.12)

This result is for an ideal solution. We see that ηsp/c in an ideal solution is independent of

concentration. Extrapolating to zero concentration, which is trivial because ηsp/c is independent

of c, the intrinsic viscosity of an ideal solution is

[η] = limc→0

ηsp

c= 2.5L

Vη

M(6.13)

For a hard sphere we could take Vη to be the volume of that sphere or equal to 4πr3/3. Because a

polymer is not a hard sphere, we do not have so simple a result. We can, however, probably assume

that Vη is be proportional to root-mean-squared radius cubed —(√

〈r2〉)3

. Writing Q for the as

yet unknown proportionality constant we have Vη = Q(√

〈r2〉)3

. Lumping the proportionality

constant along with other constants in the above equation results in:

[η] = Φ〈r2〉

32

M= Φ

(〈r2〉M

) 32

M12 = Φ

(〈r2〉0M

) 32

M12 α3 (6.14)

where Φ = 2.5LQ. In the last step of the above equation we rewrote the root-mean-squared

end-to-end distance using the expansion factor α and the unperturbed end-to-end distance:

〈r2〉 = α2〈r2〉0 (6.15)

In a theta solvent the expansion factor is 1 (α = 1) and the above equation tells us that [η] is

proportional to the square root of molecule weight and the 3/2 power of 〈r2〉0/M

[η] = Φ(〈r2〉0M

) 32

M12 (6.16)

6.3. VISCOSITY THEORY 81

But, both 〈r2〉0 and M are proportional to the number of bonds or n. Thus 〈r2〉0/M is a constant

for a particular polymer that is independent of molecular weight. We get in the simplest form

[η] = KM12 (6.17)

where K is a constant.

The polymer size is determined by the ratio of the mean-squared unperturbed end-to-end

distance and its molecular weight. In terms of K and Φ, the result is

(〈r2〉0M

)=(

K

Φ

) 23

(6.18)

Recall that characteristic ratio is defined as the ratio of the mean-squared end-to-end distance,

〈r2〉0, to the freely-jointed chain mean-squared length, nl2. The characteristic ratio can thus be

written as

Cn =〈r2〉0nl2

or Cn =〈r2〉0M

Mb

l2(6.19)

where Mb is the molecular weight per bond or M/n. Inserting this result into the previous result

gives Cn in terms of K and Φ.

Cn =(

K

Φ

) 23 Mb

l2(6.20)

The viscosity constant Φ can be evaluated from theory. For polymers it is typically assumed to be

2.5-2.66×1021 (when r is in cm and [η] is in dl/g). Therefore a measurement of [η] for a polymer of

known molecular weight in a theta solvent will give K, which in turn can be used to give a direct

measurement of the characteristic ratio. These types of experiments were used to check early work

on the conformations of polymers and calculations of end-to-end distances.

For polymers in non-theta solvents, α will no longer be one and we lose the square-root depen-

dence of [η] on molecular weight. We instead have

[η] = KM12 α3 (6.21)

where α may have molecular weight dependence of its own. It is usually acceptable to treat non-

theta solvent results by lumping the molecular weight dependence of α into one molecular weight

term to get

[η] = KMa (6.22)

where a is some constant that depends on the polymer-solvent pair and on temperature. Values of

a are typically between 0.5 and 0.8. Close to 0.5 represents near theta solvent conditions. Farther

away from 0.5 represents large deviations from a theta solvent.


6.4 Viscosity Experiments

It is usually not convenient to work in theta solvents. We thus next discuss how to use viscosity

experiments in non-theta solvents to measure molecular weight. The key experimental property

is intrinsic viscosity — [η]. You measure [η] by measuring ηsp at several different concentrations

and extrapolating to zero concentration (we will discuss the extrapolation further below). Once

[η] is known, it is related to molecular weight through two unknown constants — K and a. These

constants must have been previously determined by a calibration experiment. For a particular

polymer-solvent pair, K and a can be measured by measuring [η] on a several batches of that poly-

mer with different, and known, molecular weights. The molecular weights might be known by some

other experimental technique such as osmotic pressure experiments. Plotting log [η] vs. log M will

easily give K and a for that polymer-solvent pair. Many K’s and a’s for common polymer-solvent

pairs can be found in handbooks of polymer science. Once K and a are known, a measurement

of [η] on a polymer of unknown molecular weight can be used to measure the molecular weight.

We note that this viscosity technique for measuring molecular weight is not an absolute technique

as was osmotic pressure. It requires calibration experiments using samples with known molecular.

To avoid errors resulting from the type of average molecular weights determined by osmotic pres-

sure or viscosity experiments, accurate calibration experiments should be done with monodisperse

polymers.

6.4.1 Polydisperse Polymers

We next consider experiments on polydisperse polymers. How do we use viscosity experiments to

measure an average molecular weight? What sort of average molecular weight do we measure?

We cannot be as rigorous as we were for osmotic pressure on polydisperse polymers. We can,

however, make some reasonable assumptions and get a reasonable result. We make the following

three assumptions

1. ηsp =∑

i

(ηsp)i (6.23)

2. (ηsp)i = [η]ici (6.24)

3. [η]i = KMai (6.25)

The first assumption says that the total specific viscosity is the sum of the specific viscosities for

each components (i.e., each molecular weight of the polymer). The second assumption says that

the specific weight of each component can be calculated from the intrinsic viscosity due to that

component. The third assumption says that the relation between intrinsic viscosity and molecular

weight is independent of molecular weight; i.e., K and a are independent of molecular weight.

6.4. VISCOSITY EXPERIMENTS 83

Using the above three assumptions we derive

ηsp

c=∑

i KciMai

c=∑

i KNiMi

V Mai∑

iNiMi

V

=∑

i KNiM1+ai∑

i NiMi(6.26)

Assumption two implies we are considering the limit of zero concentration. Thus we remove the

constant K from the summation terms and state the zero-concentration limit to get

limc→0

ηsp

c= KMv

a (6.27)

where

Mv =(∑

i NiM1+ai∑

i NiMi

) 1a

(6.28)

Mv is known as the viscosity average molecular weight (for obvious reasons). When a = 1, Mv

is equal to the weight average molecule weight, MW ; otherwise Mv is between the weight average

molecule weight and the number average molecule weight, MN ≤ Mv ≤ MW . For typical a’s (0.5

to 0.8), Mv is closer to MW . This result is to be expected because viscosity properties are expected

to be a function of size and not just the number of polymers.

6.4.2 Experimental Procedures

We next discuss the experimental procedures and the equipment used to measure solution viscosity.

The experiments are simple and can all be down with a relatively simple and relatively inexpensive

piece of glassware called a viscometer (see Fig. 6.3). The experiment is to fill the viscometer such

that the level on the left side is at point C and the level on the right side is at point A (i.e.,

a certain volume of liquid depending on the volume of the viscometer). Because of the height

difference between C and A (h), there is a hydrostatic head or driving pressure to cause the liquid

to flow through the capillary or narrow diameter section of the viscometer. The experiment is to

measure the time it takes the liquid to flow from point A to B. In other words the time it takes

the volume of liquid between A and B to flow through the capillary.

Flow of a Newtonian fluid through a capillary is relatively easy to analyze. The result can

be found in many fluids books. If we ignore kinetic energy terms, the time is proportional to the

solution viscosity through the following relation:

η =πr4

c∆P

8VfLt = Aρt (6.29)

where rc is the radius of the capillary, Vf if the volume of liquid that flows (i.e., the volume between

points A and B), ∆P is pressure difference, and L is the length of the capillary. When flowing

due to gravity only in a tube of length h, ∆P = ρgh where ρ is the solution density. The physical

constants of the viscometer can be then lumped into a single constant denoted above as A. If A is


h

A

B

Capillary

C

hFigure 6.3: Simple viscometer.

known for a particular viscometer, a measurement of flow time and solution density can be used to

determine solution viscosity.

A more accurate calculation of solution viscosity in terms of flow time would account for kinetic

energy of the moving liquid. The result with kinetic energy is

η = Aρ

(t− B

At

)(6.30)

where B is a constant. If the flow occurs rapidly (t is small), the liquid is moving fast and there

can be significant kinetic energy. In this case it is important to correct for kinetic energy. If the

flow is slow (t is large) the kinetic energy term, which is proportional to 1/t, gets small. Physically

the liquid is moving slowly and has little kinetic energy. In practice, it is simplest to use long flow

times and ignore the kinetic energy term.

For most accurate results, there are several important considerations. First, the viscometer

must be clean. Any contamination from previous solutions might add to the current solution and

change its viscosity. A clean tube also means a tube that is free from dust particles. Dust particles

can add to the solution as hard spheres with a large hydrodynamic volume. They can increase

the solution viscosity and affect the results, especially when working with very dilute solutions.

Third, it is best to choose a viscometer such that the flow time is greater than 100 seconds. When

t > 100 seconds, it is permissible to ignore the kinetic energy terms. It is then a simple matter to

calculate relative viscosity and specific viscosity by comparing the flow time of the solution, t, to

the flow time of the pure solvent, t0:

ηr =t

t0and ηsp =

t− t0t0

(6.31)

6.4. VISCOSITY EXPERIMENTS 85

To adjust flow times, viscometers are available with different radius capillaries. The smaller the

capillary, the longer the flow time. You thus select a viscometer to give good flow times in the

expected viscosity range for your solutions.

To measure intrinsic viscosity requires extrapolation of ηsp/c to zero concentration. As with

osmotic pressure, we are faced with the problem of how to extrapolate. The simplest approach is

to do a simple linear extrapolation:ηsp

c= [η] + kc (6.32)

Experience shows that a linear extrapolation works well as long as ηr is restricted to less than 2.

Another consideration for viscosity experiments, therefore, is to keep the relative viscosity less than

2.

Experimental results with polymer solutions has revealed that the slope of the ηsp/c vs. c curve,

k, depends on molecular weight of the polymer. Huggins found that a plot of k versus [η]2 was

linear and passed through the origin. In other words

d(ηsp

c

)dc

= k′[η]2 (6.33)

Integrating this empirical result gives the Huggins’ equation:

ηsp

c= [η] + k′[η]2c (6.34)

The new constant, k′, is a function of the polymer/solvent/temperature of the system, but is inde-

pendent of molecular weight. Therefore once k′ is known, a single relative viscosity measurement

at one concentration can be used to get the viscosity average molecular weight. The following steps

are used:

1. Measuring ηr leads to ηsp/c which using k′ gives [η]

2. With known K and a, the measured [η] gives Mv or the viscosity average molecular weight.

If k′ is not known, the typical viscosity experiment is to measure ηsp/c at several concentrations

and do a linear extrapolation to zero concentration to get [η]. With known K and a, the measured

[η] gives Mv.

An alternate technique is to extrapolate inherent viscosity or ln ηr/c to zero concentration to

get [η]. As we approach zero concentration, ηsp will get small. Using the approximation for small

x of ln(1 + x) ≈ x− x2/2 we can derive

ln ηr = ln (1 + ηsp) ≈ ηsp −η2

sp

2(6.35)

Thenln ηr

c≈ ηsp

c−

η2sp

2c2c (6.36)


[η]

Concentration

ηspc

ln ηrc

Figure 6.4: Typical plots of ηsp/c and ln ηr/c as a function of concentration. The curves extrapolate tothe same [η] at zero concentration, but approach zero concentration with different slopes.

Using Huggins’ equation, this result gives:

ln ηr

c≈ [η] + k′[η]2c− [η]2c

2− k′[η]2c2 − k′2[η]4c3

4(6.37)

Neglecting higher order terms of order c2 or higher gives

ln ηr

c≈ [η] +

(k′ − 1

2

)[η]2c = [η] + k′′[η]2c (6.38)

Thus a plot of ln ηr/c vs. c should be linear with slope k′′[η]2 and intercept [η]. The new constant,

k′′, is related to k′ by k′′ = k′−1/2. We see that plots of both ηsp/c vs. c and ln ηr/c vs. c will both

be linear and both extrapolate to [η] at zero concentration. The slopes of the two plots, however,

will be different.

The terms ηsp/c, ln ηr/c, and [η] all have units of 1/concentration (e.g., a common convenient

unit is deciliters per gram — dl/g). Because [η]2c also has units of 1/c, k′ and k′′ are dimensionless.

For many polymers k′ ≈ 1/3 which makes k′′ ≈ −1/6. This difference in signs makes plots of

both types of data useful. The two types of plots converge on the same intrinsic viscosity but from

different directions — one has a positive slope and the other a negative one. Plotting of both types

of data will often give a good feeling for the accuracy of the extrapolated intrinsic viscosity. Typical

plots will resemble the curves in Fig. 6.4.

One final complication arises if [η] is large. A high intrinsic viscosity probably means that the

viscosity depends on the shear rate of the test. Likewise, the extrapolation to zero concentration

will also depend on shear rate. To remove shear rate effect it is sometimes necessary to get [η] at

several shear rates and do a second extrapolation of [η] to zero shear rate. The zero shear-rate

6.5. VISCOSITY SUMMARY 87

intrinsic viscosity is the intrinsic viscosity that should be used in molecular weight calculations. If

[η] is not too high, the measured value of [η] at some non-zero shear rate can probably be assumed

to be a good estimate of the zero shear-rate viscosity.

6.5 Viscosity Summary

In summary, the work you need to do to get useful results from viscosity experiments depends on the

prior knowledge you might have about the constants relevant to your particular polymer/solvent

pair. We consider four situations:

1. Known K, a, k′ (or k′′): If all these constants are all known, then measurement of η0 and

η at one concentration will lead directly to Mv. Many of these constants are tabulated for

common polymers. Thus when you are working with common polymers it is relatively easy

to get a good estimate of molecular weight.

2. Known K and a, but unknown k′ (and k′′): Tables of K and a are common, but there is less

data about k′ (or k′′). When these constants are not known, you have to measure η0 and

measure η at several concentrations. These results can be used to linearly extrapolate ηsp/c

and ln ηr/c to zero concentration to get [η] which will lead to Mv. Although this situation

requires more work than using a known k′ (or k′′), it is less prone to errors; it is less prone to

uncertainties in the tabulated values of k′ (or k′′).

3. No constants known: If you are working with a new polymer or an unusual solvent, it is

likely that none of the constants (K, a, k′, or k′′) will be known. For such polymers you

can still measure [η] by extrapolating ηsp/c and ln ηr/c to zero concentration. Although you

cannot use [η] to determine molecular weight, you can compare values of [η] from the same

polymer/solvent pair to get relative molecular weights. The higher the value of [η], the higher

the molecular weight. Relative molecular weights are often enough. If they are not enough it

might be useful to do calibration experiments to find K and a. The calibration experiments

will require monodisperse polymers whose molecular weight has been absolutely determined

by some other method (e.g., osmotic pressure or light scattering). The use of monodisperse

polymers avoids problems of differences in MN , MW , and Mv.

4. Insoluble polymer: Solution viscosity techniques assume that you can dissolve the polymer

in some solvent. Obviously the techniques cannot be used with polymers that cannot be

dissolved in any solvent. For such polymers it is often possible to measure the zero shear-rate

viscosity of the polymer melt. The zero shear-rate melt viscosity increases as the molecular

weight increases. A crude, but standardized method of estimating melt viscosity is called the

melt index. The melt index test measures the mass of material that flows through a standard


orifice in a fixed amount of time. The more material that flows, and hence the higher the

melt index, the lower the melt viscosity and thus the lower the molecular weight.

Problems

6–1. Two “monodisperse” polystyrenes are mixed in equal quantities by weight. One polymer has

molecular weight 39,000 and the other molecular weight of 292,000. What is the intrinsic vis-

cosity of the blend in benzene at 25◦C. The Mark-Houwink constants for polystyrene/benzene

at 25◦C are K = 9.18× 10−5 dl/g and a = 0.74.

6–2. With enough predetermined constants for a particular polymer-solvent pair, it is possible to

determine molecular weight of a polymer by doing only two viscosity measurements. What

two viscosity measurements are required? What constants for the particular polymer-solvent

pair are required and how are they used?

6–3. Solution viscosities for a particular polymer and solvent are plotted in the form (η−η0)/(cη0)

versus c, where η is the viscosity of the polymer solution at concentration c in g/cm3 and η0

is the viscosity of the pure solvent. The plot is a straight line with intercept of 1.50 cm3/g

and a slope of 0.9 cm6/g2. Give the magnitude and the units for the Huggin’s constant for

this polymer-solvent pair.

Chapter 7

Light Scattering

7.1 Introduction

Figure 7.1 shows light scattering off a particle in solution or in vacuum. The incident light scatters

in all different directions. The intensity of the scattered light depends on the polarizability (to

be defined later) and the polarizability depends on the molecular weight. This property of light

scattering makes it a valuable tool for measuring molecular weight.

Because the intensity of scattered light depends on molecular weight of the particle, light

scattering will depend on weight average molecular weight. This result contrasts to colligative

properties, such as osmotic pressure, which only depended on number of particles and therefore

gave the number average molecular weight. Besides molecular weight dependence, light scattering

also has a direct dependence on particle size. For polymer solutions, this dependence on size can

be used to measure the radius of gyration of the polymer molecule. As with osmotic pressure, we

expect all light scattering experiments to be done in non-ideal solutions. Nonideality complicates

the data analysis, but, like osmotic pressure, allows you to determining a virial coefficient, A2.

In summary, light scattering experiments can be used to measure three things: weight average

molecular weight (MW ), mean-squared radius of gyration (〈s2〉), and the second virial coefficient

(A2 or Γ2).

To interpret light scattering experiments, we begin with a discussion of light scattering theories.

Classical light scattering theory was derived by Lord Rayleigh and is now called Rayleigh theory.

Rayleigh theory applies to small particles. By small particles, we mean particles whose size is much

less than λ or the wavelength of the light that is being scattered. By “much less” we mean√〈s2〉 < λ/20 (7.1)

Because visible light has λ between 4000A and 8000A, we need the root mean squared radius of

gyration√〈s2〉 < 200 to 400A. Many polymers will violate this criterion and the light scattering

results will have to be corrected for large particle size effects.

89

90 CHAPTER 7. LIGHT SCATTERING

Incident Light

Scattered Light

Figure 7.1: Scattering of incident light off a particle in solution or in vacuum.

A light scattering theory known as the Rayleigh-Gans theory was developed to extend Rayleigh

theory to particles that are not optically small. The correction method involves extrapolation

techniques that extrapolate light scattering intensity to zero scattering angle. This correction

technique is important for analyzing results on polymer solutions.

Analysis of osmotic pressure experiments requires extrapolation techniques to account for non-

ideal solutions. In light scattering there are two non-ideal effect — nonideal solutions and large

particle size effects. Thus, analysis or deconvolution of light scattering data requires two extrapo-

lations. One is an extrapolation to small particle size to remove the large particle size effect. The

other is an extrapolation to zero concentration to remove the effect of non-ideal solutions. The slope

of the first extrapolation gives the mean squared radius of gyration (〈s2〉). The slope of the second

extrapolation gives the second virial coefficient (A2). The intercept of the two extrapolations gives

the weight average molecular weight (MW ).

7.2 Rayleigh Theory

We begin by describing the theory for light scattering off a small particle in an ideal solution. Light

is an electromagnetic field. At the origin the field is time dependent and described by:

Ez = E0 cos(

2πct

λ

)(7.2)

where E0 is the amplitude of the electric field, c is the speed of light, and λ is the wavelength of

light. The subscript z on E means we are considering plane polarized light with the light polarized

along the z axis. An incident beam of light polarized in the z direction is shown in Fig. 7.2.

If the particle at the origin in Fig. 7.2 is polarizable, the incident electric field will induce a

dipole moment in that particle. The magnitude of the dipole moment is proportional to the field.

7.2. RAYLEIGH THEORY 91

Incident Polarized Light

z

Figure 7.2: Plane polarized light polarized in the z direction and incident on a small particle.

The proportionality constant is called the polarizability — αp. The higher a particle’s polarizability

the higher will be the magnitude of the dipole moment induced by a given electromagnetic field.

The dipole moment is

p = αpE0 cos(

2πct

λ

)(7.3)

The induced dipole moment will radiate light in all directions. We consider observing the radiated

or scattered light at a distance r from the origin along a line that makes an angle θz with the z

axis (see Fig. 7.3). The scattered light field will be proportional to (1/c2)(d2p/dt2). The second

derivative of p is the acceleration of the charge on the dipole moment. To include spatial effects,

the scattered light is also proportional to 1/r (electromagnetic fields die off as 1/r) and to sin θz

(the projection of the dipole moment on the observation direction). Combining all these effects,

the electric field for light scattered in the θz direction is

Es =1r

1c2

d2p

dt2= − 1

c2 αpE04π2c2

rλ2sin θz cos

(2πct

λ

)(7.4)

Equipment that measures scattered light is typically only sensitive to the intensity of light. The

intensity of light is equal to the amplitude of the electromagnetic field squared. Thus, squaring the

amplitude of Es gives the scattered light intensity at r and θz:

Is = α2pI0z

16π4

r2λ4sin2 θz (7.5)

where I0z is the intensity of the z polarized incident light.

I0z = E20 (7.6)


Incident Polarized Light

z

r

θz

Figure 7.3: Observation direction for light scattered off a particle at the origin in a direction that makesan angle θz with respect to the z axis. The observation distance is r.

The above results are for incident light polarized in the z direction. Experiments, however, are

usually done with unpolarized light. We can account for unpolarized incident light by summing

the intensity of equal parts of incident light polarized in both the z direction and the y direction.

The incident intensity becomes

I0 =12I0z +

12I0y (7.7)

and the intensity of scattered light becomes

Is =12Isz +

12Isy = I0

8π4α2p

r2λ4

(sin2 θz + sin2 θy

)(7.8)

where θy is the angle the observation direction makes with the y axis. Scattering of unpolarized

light is illustrated in Fig. 7.4.

By geometry the θz and θy terms can be related to the angle θx that the observation direction

makes with the x axis (see Fig. 7.4). This angle will simply be referred to as θ. Because the sum

of the direction cosines is 1:

cos2 θx + cos2 θy + cos2 θz = 1 (7.9)

the geometric result is easily derived to be

sin2 θz + sin2 θy = 1 + cos2 θ (7.10)

We now have the scattered light intensity for scattering off a single particle. For scattering off n

moles of particles or nL particles (L is Avagadro’s number) in a dilute solution of volume V , the

7.3. IDEAL POLYMER SOLUTIONS WITH SMALL PARTICLES 93

Incident Unpolarized Light

z

θ

Figure 7.4: Scattering of unpolarized light is analyzed by considering scattering of incident light polarizedin both the z and y directions.

scattered intensity at θ is:

i0θ =I0nL

V

8π4α2p

r2λ4

(1 + cos2 θ

)(7.11)

The superscript 0 on the i indicates that this is scattering due to small molecules.

The light scattering intensity depends on scattering angle. The shape of the diagram is deter-

mined by the (1 + cos2 θ) term. A plot of this term is given in Fig. 7.5. The maximum scattering

intensity is at θ = 0. The minimum scattering intensity is at θ = 90. The scattering intensity for

forward scattering is equal to the intensity for back scattering at the corresponding angle. In other

words, the scattering intensity at angle θ is equal to the scattering intensity at angle 180− θ.

As a function of λ, the scattered intensity is proportional 1/λ4. This strong wavelength depen-

dence makes short wavelength light scatter more than long wavelength light. This effect explains

why the sky is blue. Short wavelength or blue light scatters the most. Normally we do not look

at the sun and θ is not zero. When θ is not zero you see the scattered light or the blue light. At

sunset you normally do look in the direction of the sun and θ is zero or near zero. Because blue

light is scattered away, you are left with the red light and sunsets appear red.

7.3 Ideal Polymer Solutions with Small Particles

For practical results, we need to make a connection between the scattering intensity derived in the

previous section and molecular weight of the polymer particles in solution. The connection arises

because polarizability depends on molecular weight. First, the polarizability can be thought of as


-2 -1 1 2

1

-0.5

-1

θ

Figure 7.5: Shape of the scattering intensity as a function of scattering angle for scattering off a smallparticle.

a difference in the index of refraction between the polymer and the solvent. In other words light

scattering only occurs in mediums that have an inhomogeneous index of refraction. Specifically,

the polarizability of particles at concentration c is

αp =n0cV

2πnL

dn0

dc(7.12)

where n0 is the index of refraction of the solution and dn0/dc is the concentration dependence of

the index of refraction. Note that if the index of refraction of the solvent and of the polymer are

the same then dn0/dc will be zero and there would be no polarizability and therefore no scattered

light. Writing c as nM/V (in units of g/ml) yields

αp =n0M

2πL

dn0

dc(7.13)

and substituting into the scattered light intensity gives (where we also replace n/V by c/M):

i0θI0

=2π2

r2λ4

n20

L

(dn0

dc

)2

Mc(1 + cos2 θ

)(7.14)

In a given scattering experiment, I0 and r will be fixed and we will measure i0θ. These measured

quantities can be combined into one quantity called the Rayleigh ratio — R0θ:

R0θ =

r2i0θI0

(7.15)

The advantage of the Rayleigh ratio is that it is independent of the incident light intensity and

the distance to the scattered light detector (i.e., independent of I0 and r). From the scattering

equation, the Rayleigh ratio can be written as:

R0θ = KMc (7.16)

7.4. NON-IDEAL POLYMER SOLUTIONS 95

where

K =2π2n2

0

λ4L

(dn0

dc

)2 (1 + cos2 θ

)(7.17)

The constant K depends only on the solvent properties, on λ, and on θ. K is therefore a system

constant that is independent of the concentration of the solution and the molecular weight of the

polymer.

For a dilute, polydisperse polymer solution, the total Rayleigh ratio can be written as a sum of

the Rayleigh ratios for scattering of polymers of each possible molecular weight:

R0θ = K

∑i

ciMi (7.18)

orKc

R0θ

=∑

i ci∑i ciMi

=∑

i NiMi∑i NiM2

i

=1

MW

(7.19)

The Rayleigh ratio for an ideal polymer solution with small particles is thus directly related to the

weight average molecular weight (MW ).

7.4 Non-Ideal Polymer Solutions

As done with osmotic pressure, the possibility of non-ideal solutions is handled by adding virial

coefficients and concentration terms to the ideal result. Thus expanding Kc/R0θ gives

Kc

R0θ

=1

MW

+ 2A2c + 3A3c2 + · · · (7.20)

The virial coefficients A2 and A3 are the same as the virial coefficients we discussed in osmotic

pressure theory. The factors of 2, 3, etc., come from the thermodynamic theory of fluctuations

which can be used to show thatKc

R0θ

=1

RT

∂π

∂c(7.21)

where π is osmotic pressure. In a virial expansion

π =RT

MN

c + RTA2c2 + RTA3c

3 + · · · (7.22)

To convert to the light scattering experiment, the MN in the π expression must be changed to MW .

The only difference between the A2 in osmotic pressure and the A2 in light scattering is that the

light scattering A2 is formally a weight-average virial coefficient. Besides that difference, the light

scattering A2 gives similar information, notably information about the quality of the solvent.

Typically, we will ignore terms beyond the second virial coefficient. Then Kc/R0θ is predicted to

be linear in c. From experiments we can plot Kc/R0θ as a function of c. The slope will give the second

virial coefficient (slope = 2A2) and the intercept will give the molecular weight (intercept = 1/MW ).

This extrapolation, however, ignores any possible large particle size effects The extrapolated MW

will therefore be in error. The next section considers how to correct for large particle size.


Scattering from differentparts of the particle

Path length differencecauses destructive interference

Figure 7.6: Scattering of light of two different parts of a large polymer molecule.

7.5 Large particles

If a particle is not small compared to the wavelength of light, the light can scatter from different

parts of the particle. Fig. 7.6 shows a large polymer that is scattering light. Light scattering

from different parts of the particle will reach the detector by traveling different path lengths. The

difference in path lengths can lead to destructive interference that reduces the intensity of the

scattered light. The net effect is that the scattering diagram for large particles is reduced in

intensity from the scattering diagram for small particles (see Fig. 7.5).

The amount of intensity reduction or the amount of destructive interference depends on the

scattering angle. At θ equal to zero, the path lengths will always be identical. With identical path

lengths, there will be no destructive interference. In other words at θ = 0, the intensity of scattered

light will be identical to i0θ. At θ not equal to zero there will be destructive interference. As θ

increases, the interference will increase reaching a maximum and θ = 180◦. A comparison of the

scattering diagrams for large particles vs. small particles is given in Fig. 7.7. The large particle

scattering diagram shows the effect of large particles and now shows asymmetry in scattering; i.e.,

the back scattering intensity is much reduced from the forward scattering intensity.

To correct for large particles, we merely need to do the light scattering experiments at zero

scattering angle (θ = 0). Unfortunately, these experiments cannot be done. At θ = 0 most light

will be transmitted light that is not scattered. The transmitted light will swamp the scattered

light preventing its measurement. Because scattered light and transmitted light have the same

wavelength, there is no way to distinguish between them. Instead, we must do experiments at

θ > 0 and extrapolate to θ = 0. We thus do a second extrapolation, an extrapolation to zero

scattering angle.

To develop an extrapolation method, we define a new function, P (θ), that describes the large

7.5. LARGE PARTICLES 97

-2 -1 1 2

1

-0.5

-1

Large Particles

Small Particles

θ

Figure 7.7: Scattering diagrams for both small particles and large particles.

particle size effect. P (θ) is the ratio between the actual scattering (iθ) and the scattering that

would occur off small particles (i0θ)

P (θ) =iθi0θ

=Rθ

R0θ

(7.23)

From the above discussions we know that P (0) = 1 (there is no effect at zero scattering angle)

and P (θ) < 1 for all other θ (destructive interference can only cause a reduction in intensity). The

larger effect on back scattering than on forward scatter means that P (θ < 90) > P (180− θ).

First consider an ideal solution. The measured Rayleigh ratio, written as Rθ, includes the large

particle size effect. Using P (θ) we can write Rθ = P (θ)R0θ. The key measured quantity becomes

Kc

Rθ=

Kc

P (θ)R0θ

=1

MW P (θ)(7.24)

The second equality follows from the previously derived ideal solution result with small particles.

To use this equation, we need some information about P (θ). That information can sometimes be

derived by theoretical analysis of large-particle scattering. Fortunately, some theoretical results are

available for scattering off a large random coil. The results are accurate as long as the particle size

is not too large. Instead of requiring√〈s2〉 < λ/20 as done before for small particles, we can use

the theoretical result to handle particles with√〈s2〉 < λ/2. For scattering with visible light we

now can use√〈s2〉 < 2000A to 4000A. Most polymers fall within or below this range and thus we

can derive effective extrapolation methods for scattering off polymer molecules.

The theoretical result for P (θ) is

1P (θ)

= 1 +16π2

3λ2〈s2〉 sin2 θ

2+ · · · (7.25)

The “· · ·” means that there are higher order terms in sin(θ/2). Those terms are normally assumed

to be negligible. For a polydisperse polymer, the scattering intensity as a function of scattering


angle becomesKc

Rθ=

1MW

(1 +

16π2

3λ2〈s2〉w sin2 θ

2

)(7.26)

Note we have changed 〈s2〉 to 〈s2〉w, the weight average radius of gyration squared. In terms of the

various polymer weights, the relevant radius of gyration squared is

〈s2〉w =∑

i NiMi〈s2〉i∑i NiMi

=∑

i

wi〈s2〉i (7.27)

where 〈s2〉i is the average squared radius of gyration for polymers with molecular weight Mi

To find weight-average molecular weight (MW ) in ideal solutions, we truncate 1/P (θ) after the

sin2(θ/2) term and plot Kc/Rθ as a function of sin2(θ/2). That plot should be linear. The intercept

will give the molecular weight:

intercept =1

MW

(7.28)

The slope divided by the intercept will give the radius of gyration

slope/intercept =16π2〈s2〉w

3λ2(7.29)

7.6 Light Scattering Data Reduction

To handle both non-ideal solutions and large particle effects, we need to do two extrapolations.

First, we introduce non-ideal solution effects into the large particle analysis in the previous section.

Instead of using P (θ) to correct the ideal solution result, we use it to correct the non-ideal solution

result. Thus the actually measured Kc/Rθ is

Kc

Rθ=

Kc

P (θ)R0θ

=(

1MW

+ 2A2c

)1

P (θ)(7.30)

where we have truncated the non-ideal solution result to a single virial coefficient. Inserting the

theoretical result for P (θ) truncated after the sin2(θ/2) term gives

Kc

Rθ=(

1MW

+ 2A2c

)(1 +

16π2


2

)(7.31)

A set of light scattering experiments consists of measure Kc/Rθ for various concentrations and

at various scattering angles. To get MW , we do two extrapolations. First, plotting Kc/Rθ as a

function of sin2(θ/2) at constant c gives a straight line with the following slope and intercept:

slope =(

1MW

+ 2A2c

)16π2

3λ2〈s2〉w (7.32)

intercept =(

1MW

+ 2A2c

)(7.33)

7.6. LIGHT SCATTERING DATA REDUCTION 99

c=0 c1 c2 c3 c4 c5

θ=0

θ1θ2

θ3θ4

θ5θ6

sin2 θ + kc2

KcRθ

Figure 7.8: Typical Zimm plot. The experimental data points are at the grid intersection points exceptalong the θ = 0 and c = 0 lines.

Next we plot the intercepts of the first plots as a function of concentration. The resulting plot

should be a straight line with

slope = 2A2 (7.34)

intercept =1

MW

(7.35)

The slope and intercept of the second line gives us MW and A2. Substituting these results into the

slope of the first line allows us to find 〈s2〉w. We could achieve similar results by first plotting as a

function of concentration and then plotting the intercept of those plots as a function of sin2(θ/2).

The above analysis assumes that all c’s are low enough such that the concentration dependence

is linear in concentration and only requires the second virial coefficient. It also assumes all scattering

angles are low enough that terms higher than the sin2(θ/2) can be neglected. Both these conditions

are easy to satisfy for light scattering with polymer solutions.

The analysis method described above is easy to do in a personal computer. When light scatter-

ing techniques were first developed, however, computers were not available and the numerous linear

fits were tedious. To avoid the tedium the Zimm plot was developed. In the Zimm plot technique,

you plot Kc/Rθ versus sin2(θ/2)+ kc where k is a constant. k is chosen to spread out the plot and

give equal weights to each variable. For example sin2(θ/2) is always less than 1 or has a maximum

of 1. The range in kc should also go from 0 to 1 or kcmax = 1 which means a good k might be

1/cmax where cmax is the maximum concentration used. A typical Zimm plot is given in Fig. 7.8.

Plotting all Kc/Rθ points on a Zimm plot should result in a grid such as the one shown in


Fig. 7.8. There will be experimental points at all grid points except along the lower line (the θ = 0

line) and the left-most line (the c = 0) line. Connecting all the grid lines and extrapolating to the

lower-left corner, the intercept point gives the molecular weight (intercept = 1/MW ). Incorporating

the k constant, the Zimm plot is plotting

Kc

Rθ=(

1MW

+2A2

kkc

)(1 +

16π2


2

)(7.36)

The slopes of the two directions in the parallelogram have physical meaning. The lines labeled θ1,

θ2, etc., are lines at constant θ. Inspection of the Zimm equation shows that the slopes of these

lines are:

slope of the contant θ lines =2A2

k

(1 +

16π2


2

)(7.37)

Notice that these slopes are a function of θ. Thus the slope of the θ = 0 line and the θ5 (or any θi

line) are different. In other words the Zimm plot is not actually a parallelogram. The lines labeled

c1, c2, etc., are lines at constant concentration. Inspection of the Zimm equation shows that the

slopes of these lines are:

slope of the constant c lines =(

1MW

+ 2A2c

)16π2

3λ2〈s2〉w (7.38)

Notice that these slopes are a function of c. Thus the slope of the c = 0 line and the c5 (or any

ci line) are different. In other words the Zimm plot is not actually a parallelogram. The slopes

of constant θ and constant c lines both depend on A2 and on 〈s2〉w. The slopes of the constant θ

lines are mostly sensitive to A2. The slopes of the constant c lines are mostly sensitive to 〈s2〉w.

Because A2 and 〈s2〉w are independent physical quantities, it is possible to get Zimm plots that

are inverted from the plot in Fig. 7.8. If A2 increases and/or 〈s2〉w decreases, it is possible for the

steeper lines to be the constant θ lines and for the shallower lines to be the constant c lines.

Chapter 8

Miscellaneous Molecular Weight

Methods

8.1 Introduction

The previous three chapters described several methods for determining the molecular weight, or an

average molecular weight for a polymer dissolved in solution. This chapter briefly describes a few

more techniques. At the end there is a short discussion about polymers that cannot be dissolved

into any solvent.

8.2 Gel Phase Chromatography

Gel phase chromatography is also called size exclusion chromatography; the reason for this second

name will become apparent shortly. In gel phase chromatography, a standard chromatographic col-

umn is filled with beads of a rigid porous gel (e.g., porous glass or highly cross-linked polystyrene).

The beads are selected such that the size of its pores are in the range of the sizes of polymer

molecules. Figure 8.1 illustrates what happens when a dilute polymer solution is introduced into

the column. The larger molecules do not fit into the pores and thus they take the easier and faster

route around the filler particles and out the bottom of the column. The smaller molecules are able

to get into the pores and thus they follow a path that sometimes travels through the filler particles.

Because passage through a filler particle presents a more tortuous path, the smaller molecules spend

more time in the column before coming out the bottom.

A typical gel phase chromatography experiment consists of collecting elution fractions as they

elute from the bottom of the column. The first fractions will be more concentrated in high molecular

weight polymer. As a function of time (or as a function of total volume that has eluted — the

elution volume) the elution fraction will be more concentrated in increasingly lower molecular weight

101

102 CHAPTER 8. MISCELLANEOUS MOLECULAR WEIGHT METHODS

Figure 8.1: Schematic drawing of a gel phase chromatographic column. The line represents a path of apolymer. The solution that comes out the bottom can be collected in flasks.

polymer. By many means (e.g., absorption spectroscopy or index of refraction) it is possible to

determine the concentration of polymer in each fraction. Plotting these concentrations as a function

of time or elution volume will give the complete molecular weight distribution (see Fig. 8.2). Because

high molecular weight fractions come out first, the plot of raw data will be a mirror image of the

molecular weight distribution.

Like viscosity experiments, the results from gel phase chromatography experiments are not

absolute. You need a calibration experiment to convert the concentration data versus elution

volume to concentration data versus molecular weight or true molecular weight distribution. The

calibration procedure typically uses calibrated viscosity experiments. From the viscosity theory

(see Chapter 6), the intrinsic viscosity is

[η] = Φ(〈r2〉M

) 32

M12 = Φ

(〈r2〉

) 32 M−1 (8.1)

where Φ is a constant (relatively fixed for polymer solutions), 〈r2〉 is the mean squared end-to-end

distance of the polymer (hence (〈r2〉)3/2 is proportional to total volume of the polymer), and M is

molecular weight. This equation can be rearranged to give

[η]M = Φ(〈r2〉

) 32 (8.2)

Because Φ is a relatively polymer-insensitive constant and (〈r2〉)3/2 merely expresses volume, a

plot of [η]M as a function of any property that depends only on polymer volume should produce a

8.2. GEL PHASE CHROMATOGRAPHY 103

Concentration

High MW Time Low MWElution Volume

Con

cent

ratio

n

Figure 8.2: Molecular weight distribution curve from gel phase chromatography. The high molecularweight polymers come out at early time and the low molecular weight polymers come out a latter time.

universal or master curve. In other words the curve would be the same for all polymer/solvent/-

temperature systems.

Passage time through a gel phase chromatography column is one such property that depends

only on polymer volume. We can expect or assume that all polymers with the same value of

[η]M will elute at the same time in that column. The calibration procedure uses this assumption.

Calibration data consists of a plot of [η]M versus elution volume. [η]M is measured using viscosity

experiments on polymers of known molecular weight. For greatest accuracy and elimination of

ambiguity about the type of average molecule weight, the preferred experiments use monodisperse

polymers. The elution time is measured in a chromatographic column. A plot of [η]M should

be a universal calibration curve for that specific gel phase chromatographic column. Because the

calibration curve should be independent of polymer type, the calibration experiment can be done

with any convenient polymer. A typical calibration curve is shown in Fig. 8.3. As [η]M increases

the elution volume (or elution time) decreases. Because [η]M increases with molecular weight, the

curve expresses the tendency of high molecular weight polymer to elute faster than low molecular

weight polymer.

Once you have a calibration curve, you can use that column to measure the molecular weight

distribution of any soluble polymer. For the unknown polymer, data of concentration versus elution

volume is collected. For each elution volume, the calibration curve is used to calculate the [η]M

corresponding to that elution time. By viscosity theory, a given calibration value for [η]M (here

denoted as ([η]M)c is related to the molecular weight of the unknown polymer by

([η]M)c = KMa+1 (8.3)


Calibration Curve

Elution Volume

[η] M

Figure 8.3: A typical calibration curve for gel phase. chromatography

where K and a are the viscosity-molecular weight constants for the unknown polymer being studied.

Solving for molecular weight gives

M =(

([η]M)c

K

) 1a+1

(8.4)

Formally this M is a viscosity average molecular weight. Because each elution fraction, however, is

close to monodisperse polymer, we can treat it is an absolute molecular weight and not an average

molecular weight.

In summary, gel phase chromatography gives information about complete molecular weight

distribution. Its only drawbacks are that it requires calibration experiments, it requires the polymer

to be soluble, and requires a modest amount of specialized equipment.

8.3 Field Flow Fractionation

Field flow fractionation or FFF is a powerful and flexible technique that was developed at the

University of Utah. There is currently no information on FFF in these notes, but the technique

will be discussed in class.

8.4 End Group Analysis

Consider the polymer shown in Fig. 8.4. The two ends have acid groups ( COOH groups) and the

reminder of the polymer backbone has no acid groups. The situation in Fig. 8.4 is common to many

polymers. The common situation is not that many polymers have COOH end groups, but that

the end groups are often chemically distinguishable from the backbone of the polymer. When the

8.4. END GROUP ANALYSIS 105

COOH

HOOC

Figure 8.4: A typical polymer having chemically distinguishable end groups.

backbones are chemically unique it is sometimes possible to use analytical chemistry techniques to

measure the concentration of end groups. Some typical analytical methods are: chemical methods

(e.g., titration of acid groups as for Fig. 8.4 or of basic groups), spectroscopic measurements, or

radioactive measurements (e.g., for polymers with labeled end-groups).

Measuring the concentration of end groups allows you to measure the number of end groups.

Assuming the polymer is linear, each polymer will have two end groups. Thus dividing the number

of end groups by two will give the total number of polymer molecules. In some polymers the two

end groups are distinguishable from each other. In this case, measuring the number of one type of

end group will be equal the number of polymer molecules (without dividing by two). The average

molecular weight, which in this case will be the number average molecular weight, can be calculated

fromTotal Weight

Number of Polymers=∑

i NiMi∑i Ni

= MN (8.5)

When end-group analysis works, it can be a useful technique to measure an absolute molecular

weight. Unfortunately, it often has problems that make it impractical. First the ratio of end-

groups to total amount of polymer is small. Thus when using dilute concentrations of polymers,

an accurate end group analysis will require extremely sensitive techniques. The sensitivity problem

gets worse as the molecular weight gets higher, because the number of end groups decreases as the

molecular weight increases. As a result, end-group analysis is limited to relatively low molecular

weight polymers.

Another problem is that use of end-group analysis assumes you know the structure of the

polymer. You must know the chemical nature of the end groups and know that the polymer is linear.

Furthermore, the chemical nature of the end groups must be such that they can be distinguished

from the rest of the polymer. End-group analysis cannot be used when the end groups are uncertain

(a situation that arises in some polymer synthesis schemes) or when the polymer is branched or

cross-linked. In principle, if you knew the number of branches per polymer, you could divide the

total number of end groups by that number and still calculate the number of polymer molecules.

Unfortunately, the extent of branching is not often characterized so precisely. If the polymer is


linear you can be sure it has two ends; if it is branched you cannot be sure how many ends it has.

8.5 Ultracentrifugation

In the ultracentrifugation method, a polymer solution is placed in a centrifuge and the rate of

sedimentation or the equilibrium position of polymer after centrifugation is measured. These quan-

tities are typically measured by some spatially resolved optical means that is scanned along the

length of the centrifugation tube. The diffusion process under centrifugal force is related to the

size or molecular weight of the polymer. The polymer concentration along the tube as a function

of position can be reduced to give average molecular weights. At its best, the results can even give

information about distribution of molecular weights by finding MN , MW , Mz, and Mz+1, etc..

Ultracentrifugation works well and is used often when working with biological macromolecules

(e.g., proteins). These macromolecules usually have fixed structures and are close to being spherical.

As a result they do not entangle with each other and simple sedimentation theories are applicable.

In contrast polymers are often random coils. They will often entangle with each other and the

sedimentation will be strongly effected by the amount of entanglement. The technique is less useful

for high molecular weight random coil polymers.

8.6 Insoluble Polymers

If a polymer is insoluble, then most of the previously discussed techniques cannot be used to get

molecular weights. Colligative properties (e.g., osmotic pressure), solution viscosity, light scat-

tering, gel phase chromatography, field-flow fraction, end-group analysis, and ultracentrifugation

always use polymer solutions. If the polymer cannot be dissolved in any solvent, those techniques

will not be applicable. One technique available which is sensitive to molecular weight is melt

rheology (i.e., melt viscosity experiments).

On a theoretical basis one can take a known molecular weight distribution and make a reasonable

calculation of melt rheology data. Recent workers have suggested that the reverse calculation can

be made. Given melt rheology data, they claim to be able to calculate the molecular weight

distribution. These techniques make many assumptions and are probably only qualitative at best.

Their work indicates, however, that melt rheology data can be used to get at least qualitative and

comparative data about molecular weights in various insoluble polymers.

One crude standard technique is called melt index. It measures the amount of molten polymer

that flows through a standard opening in a fixed amount of time. The higher the melt index, the

lower the viscosity of the polymer melt and thus the lower the molecular weight. Although this

technique is crude, it is often used by industry and reported in company literature on various grades

of polymers.

8.6. INSOLUBLE POLYMERS 107

Problems

8–1. Nylon-11 is poly-(11-aminoundecanoic acid). This polymer has a crystal melting point around

190◦C and has lower water absorption than Nylon 6,6 or Nylon 6. It can be used to make

mechanical parts, packaging films, gristles, monofilaments, and sprayed and fluidized coatings.

It is formed via polymerization of the following monomer:

HO–C–(CH2)10–NH2

O

a. If 2.65 grams of nylon 11 was dissolved in 450 ml of Formic acid and it was subsequently

determined that the concentration of amine groups was [-NH2]=3.56× 10−4 moles/liter,

what would be the average molecular weight of the polymer?

b. In part a, what type of molecular weight average was determined?

c. Without any additional information, what would be the “most probable” polydispersity

index MW /MN . From your guessed polydispersity index, write down both MN and MW

for this polymer.


Chapter 9

Thermal Analysis

9.1 Introduction

As temperature changes polymers undergo transitions (e.g., melting, glass-transition, sub-glass

transitions, crystallization, etc.). In thermal analysis we do two things. First, we find the temper-

atures where the various transitions occur, and second, we evaluate the change in properties the

might occur as the temperature is varied from below the transition to above the transition.

9.2 Thermal Transitions

The well-known thermodynamicist Ehrenfest, developed a scheme for classifying thermal transi-

tions. He classified transitions as first order, second, order, third order, etc.. In practice, we only

encounter first and second order transitions. In this section we discuss Ehrenfest’s classification

scheme and consider specifically first order and second order transitions.

A first order transition is a transition that involves a discontinuous change in a first derivative

of the free energy. Free energy is a function of temperature (T ) and pressure (P ). From our earlier

solution thermodynamics, but now considering constant composition, the first differential of free

energy, or the change in free energy (dG) following a change in pressure (dP ) and/or a change in

temperature (dT ) is

dG = V dP − SdT (9.1)

From this differential expression, the first derivatives of free energy are(∂G

∂P

)T

= V (9.2)(∂G

∂T

)P

= −S (9.3)

These first derivatives have physical interpretations. The first derivative with respect to pressure

is volume (V ). The first derivative with respect to temperature is minus the entropy (−S). By

109

110 CHAPTER 9. THERMAL ANALYSIS

Volu

me

(V)

Ent

ropy

(S)

First Order Transitions

Temperature Temperature

Figure 9.1: Typical observation of a first order transition while measuring volume or entropy (first deriva-tives of free energy).

Ehrenfest’s classification scheme, a first order transition occurs when the volume or entropy of a

material undergoes a discontinuous change at some temperature. The temperature at which the

change occurs is called the transition temperature.

Volume can be measured by dilatometry and entropy can be measured by calorimetry (S =

q/T ). A typical first order transition observed by either dilatometry or calorimetry is given in

Fig. 9.1. Both volume and entropy are expected to be higher above the transition temperature than

below it. Thus there is a discontinuous change to a higher value at the transition temperature.

An example of a first order transition is the melting transition which occurs at the melting

point. The change in volume, ∆Vm, and the change in entropy, ∆Sm, describe the magnitude of

the discontinuous change. Other changes, such as the change in enthalpy can also be measured.

For melting, the change in enthalpy is the heat of fusion (∆Hm or ∆Hf ). Note that a change in

first derivative implies a change in slope in the function itself. Thus at a first order transition, G

will change in slope but not change in value. We then have the result that ∆Gm = 0. Because

∆Gm = 0 = ∆Hm − Tm∆Sm, we have

∆Hm = Tm∆Sm or Tm =∆Hm

∆Sm(9.4)

A second order transition is a transition that involves a discontinuous change in a second

derivative of the free energy. We can find the possible second derivatives by differentiating V or

S (the first derivatives) with respect to pressure or temperature. Of the four possible derivatives,

three are related to common physical properties:(∂2G

∂P 2

)T

=(

∂V

∂P

)T

= −V K (9.5)(∂2G

∂T 2

)P

= −(

∂S

∂T

)P

= −CP

T+

S

T(9.6)

9.2. THERMAL TRANSITIONS 111

Second Order Transitions

K Cp


Figure 9.2: Typical observation of a second order transition while measuring compressibility or constant-pressure heat capacity (second derivatives of free energy).

[∂

∂T

(∂G

∂P

)T

]P

=(

∂V

∂T

)P

= V β (9.7)

(9.8)

where K is compressibility, CP is the constant-pressure heat capacity, and β is the volumetric ther-

mal expansion coefficient. Second order transitions can be observed be measuring a discontinuous

change in K, in CP , or in β. K and β can be measured by dilatometry; CP can be measured by

calorimetry. Some typical dilatometry and calorimetry data at a second order transition are given

in Fig. 9.2. K, CP , and β are all expected to by higher above the transition temperature than

below it. Thus there is a discontinuous change to a higher value at the transition temperature.

Suppose that instead of measuring K, CP , or β, you were measuring V or S during a second

order transition. In other words you were measuring a first derivative of free energy instead of a

second derivative of G. From the mathematical definition of derivatives, when the second deriva-

tive of a function undergoes a discontinuous change, the first derivative will undergo a change in

slope. Thus when K, CP , and β undergo a discontinuous change at a second order transition, V

and S will change in slope (but not value, i.e., ∆V = ∆S = 0) at the transition temperature.

Figure 9.3 shows typical second order transitions when observed by measuring a first derivative of

free energy. Because K, CP , and β change to a higher value above the transition, the slopes of V

or S vs. temperature become higher above the transition.

Second order transitions are less common than first order transitions. Some typical second

order transitions are order-disorder transitions, rotational transitions, the disappearance of ferro-

magnetism at the Curie point, and a transition in Helium at 2.2K. In polymers, the glass transition

is often treated as a second order transition. We will see that it has much in common with sec-

ond order transitions, but on closer inspection it is not a true second order transition. The glass

transition is sometimes called a pseudo second order transition.


Volu

me

(V)

Ent

ropy

(S)

Second Order Transitions


Figure 9.3: Observation of a second order transition when observing volume or entropy (first derivativesof free energy).

First Order Transitions

K Cp


Figure 9.4: Observation of a first order transition when observing compressibility or constant pressureheat capacity (second derivatives of free energy).

Suppose you were measuring K, CP , or β, perhaps in search of a second order transition, and

your material goes through a first order transition. At the transition, the first derivative of free

energy would have just undergone a discontinuous change. When measuring a second derivative of

free energy, a discontinuous change in the first derivative of free energy will show up as a peak. If

the transition was very sharp, the peak would be very high and very narrow. For many practical

materials (especially polymers), the peak is less high and somewhat broad. Figure 9.4 shows typical

first order transitions that are observed while measuring a second derivative of free energy.

A generalization of Ehrenfest’s classification scheme is to call a transition an nth order transi-

tions when the material undergoes a discontinuous change in an nth derivative of free energy at the

transition temperature. Transitions higher than second order are rare. For polymers we will only

deal with first and second order transitions.

9.3. THERMAL ANALYSIS OF POLYMERS 113

9.3 Thermal Analysis of Polymers

This sections lists some of the common types of transitions observed in organic materials. Some of

them are common in polymers; others are rarely or never observed.

1. Vaporization: first order transition that occurs at the boiling point and marks the transition

from liquid to gas (vapor). Vaporization transitions are almost never studied in polymers.

Because polymers are high molecular weight materials, their boiling points are very high. In

fact, almost all polymers will thermally degrade before they vaporize.

2. Melting: first order transition that occurs at the melting point and marks the transition from

solid to liquid. Melting is a transition from solid to liquid, but is better characterized as a

disappearance of crystals. Because only semicrystalline polymers have crystals (amorphous

polymers have none), melting is only observed in semicrystalline polymers. There are some

important differences between disappearance of crystals on heating and their reappearance

on cooling. Melting is the disappearance of crystals and thus is observed while heating.

3. Crystallization: occurs when cooling from above the melting temperature to a low tempera-

ture. Like melting, crystallization is a first order transition. It corresponds to the formation

of crystals while cooling. As will be explained latter, the melting point and the crystallization

temperature are usually different. Because only semicrystalline polymers can form crystals

(amorphous polymers have none), crystallization is only observed in semicrystalline polymers.

4. Glass transition: pseudo second order transition that occurs in amorphous polymers or in

the amorphous phase of semicrystalline polymers. The glass transition occurs when long-

range motions become very slow with respect to the observation time scale. Formally, the

glass-transition is not a true thermodynamic second order transition but many aspects of

the thermal analysis of glass transitions are similar to true second order transitions. It is

thus often useful to treat it as a second order transitions. Because no polymers are 100%

crystalline, all polymers will have an amorphous fraction and will exhibit a glass transition.

The glass transition temperature is always lower than the melting or crystallization transition

temperature (can you explain why?).

5. Sub-glass transitions. a transition that occurs below the glass transition temperature. Sub-

glass transitions are typically second order transitions. The transitions can occur in either

the amorphous fraction or in the crystal fraction. The physical processes that correspond to

sub-glass transitions will vary from polymer to polymer.

The above transitions in polymers are observed using various thermal analysis techniques. Sev-

eral thermal analysis techniques will be discussed in the following three chapters. Those techniques

and what they measure are listed below:


1. Dilatometry (PVT or TMA): Measures volume as a function of temperature and/or pressure

(see Chapter 10).

2. Differential Scanning Calorimetry (DSC): Measures constant-pressure heat capacity as a func-

tion of temperature (see Chapter 11).

3. Thermogravimetric Analysis (TGA): Measures weight as a function of temperature and time.

This method is aimed at providing information about polymer degradation rather than ther-

modynamic transitions (see Chapter 11).

4. Dynamic Mechanical Analysis (DMA): Measures stiffness and energy losses as a function of

temperature. By measuring a mechanical property, DMA can determine transition tempera-

tures and simultaneously determine the effect of the transition on properties (i.e., on stiffness)

(see Chapter 12).

Chapter 10

Dilatometry

10.1 Introduction

The technique of dilatometry measures volume (V ) as a function of temperature (T ) and pressure

(P ). The method is sometimes referred to as PV T measurements. In equilibrium systems, the

V is a unique function of T and P , i.e., V does not depend on the path taken or on the sample

history. In equilibrium systems, dilatometry is used to study the relation between P , V , and T

and to develop or test equations of state. Polymer melts are close to equilibrium systems. PV T

experiments on polymer melts have been used to check theoretical equations of state for polymer

melts. Note that equilibrium systems also require that no chemistry is occurring. If cross linking or

polymerization occurs at the test temperature than volume will change because of these reactions

and not because of T and P . Such results will not be an equilibrium system and cannot be analyzed

with an equation of state.

Polymers in solid form will not be equilibrium (e.g., glassy and semicrystalline polymers). Their

volume as a function of T and P will depend on the formation history (e.g., cooling rate, formation

pressure, and time at temperature and pressure). The formation history can effect such things as

the degree of crystallinity, the crystal morphology, and the glass density. Much interesting PV T

work is done on non-equilibrium systems, and the results are useful despite the material being in

non-equilibrium conditions.

Many times it is possible to do quasi-equilibrium experiments on non-equilibrium systems. For

example, if you form a solid by lowering the temperature below the glass transition temperature

and then do all experiments on that solid, the results might be considered to be in a state of quasi-

equilibrium; i.e., a set of experiments on samples with the same formation history. The results

will be quasi-equilibrium if it can be demonstrated that the process of approaching equilibrium is

much slower than the time scale of observation for all observation temperatures. For polymers, this

condition holds as long as the sample remains far enough below the glass transition temperature

115

116 CHAPTER 10. DILATOMETRY

P=0Increasing P

Volu

me

Temperature

Figure 10.1: Typical PV T data for an amorphous polymer. The curves are data at constant pressure. Asthe pressure increases, the curves move down and to the right.

(perhaps 30◦C or more below). Such quasi-equilibrium data can give much useful information.

Because it is not a true equilibrium, the results might change with time. These changes, however,

are often very slow.

10.2 Amorphous Polymers

Some typical PV T data for an amorphous polymer are give in Fig. 10.1. There are several curves.

Each curve plots the volume as a function of pressure for some constant pressure. As the pressure

increases, the curves move down and to the right. The breaks in the curves are the glass transition

temperatures — Tg. At Tg, the slopes of the volume vs. temperature curves change. This change

in slope is an indication that the glass transition is a second order transition (see Chapter 9). At

higher pressures, the sample is more dense and the volume is lower. The Tg always shifts to a

higher temperature at higher pressures. The glass transition is associated with mobility. At higher

pressures, the transition to higher mobility is more difficult and thus requires higher thermal energy

or a higher transition temperature.

Besides a direct measure of volume as a function of temperature, dilatometry data can be

used to determine other physical properties of the material. We consider first determination of the

thermal expansion coefficient. Consider a transition from V1 to V2 and from T1 to T2 at constant

pressure P . The new volume can be expressed as

V2 =[1 + β(T2 − T1)

]V1 (10.1)

where β is the thermal expansion per degree — the volumetric thermal expansion coefficient.

10.2. AMORPHOUS POLYMERS 117

L0 L=(1+α∆T)L0

Figure 10.2: Linear thermal expansion. a is the linear thermal expansion coefficient.

Rearranging this expression gives

β =1V1

V2 − V1

T2 − T1(10.2)

For a small temperature change, this expression becomes a derivative and the thermal expansion

coefficient is defined as

β(P, T ) =1

V (P, T )

(∂V (P, T )

∂T

)P

=(

∂ lnV (P, T )∂T

)P

(10.3)

Any one of these expressions can be used with dilatometry data to measure β. The discrete forms

are easiest to use with discrete dilatometry data. Dilatometry data is sometimes reported as an

empirical fit to an equation of state (see below). In these cases the equation of state can be

differentiated to measure β using the definition in terms of derivatives.

Below Tg, β is usually nearly constant. Hence volume vs. temperature is close to a straight

line. At Tg, β undergoes a discontinuous change to a higher value. The higher value remains

roughly constant above Tg. Hence volume vs. temperature is close to a straight line above Tg. The

discontinuous change in β is another manifestation of the glass transition being a second order

transition — β is a second derivative of free energy and it undergoes a discontinuous change at Tg.

Consider thermal expansion of an isotropic material. Because of isotropy, the amount of ther-

mal expansion along each of the three axes will be the same. If we define α as the linear thermal

expansion coefficient, then the change in lineal dimension (L) is illustrated in Fig. 10.2. Follow-

ing the interpretation of volumetric thermal expansion coefficient, the linear thermal expansion

coefficient (or commonly just the thermal expansion coefficient) is defined by

α(P, T ) =1

L(P, T )

(∂L(P, T )

∂T

)P

=(

∂ lnL(P, T )∂T

)P

(10.4)

β and α both describe thermal expansion and thus can be related to each other. Consider

thermal expansion from V1 to V2 after a temperature change of 1◦C (T2 − T1 = 1) where

V1 = L1L2L3 and V2 = (1 + α)3L1L2L3 = (1 + β)V1 (10.5)

Comparing these expressions

1 + β = 1 + 3α + 3α2 + α3 (10.6)

Because α is usually small (≈ 10−5) we can ignore higher order terms (3α2 and α3) and write with

sufficient accuracy

β ≈ 3α (10.7)


This result assumes an isotropic material. It is easy to generalize it to an anisotropic material by

allowing the linear thermal expansion coefficients in the three directions to be different. Thus

β ≈ α1 + α2 + α3 (10.8)

where αi is the thermal expansion coefficient for thermal expansion in the ith direction.

The values of α and β do not vary dramatically from polymer to polymer. At most we might

expect a factor of two difference between any two polymers. Typical values for isotropic, amorphous

polymersBelow Tg : α = 50− 100 ppm/◦C (β = 150− 300 ppm/◦C)

Above Tg : α = 150− 300 ppm/◦C (β = 450− 900 ppm/◦C)

The designation “ppm/◦C” means parts per million per degree C. These units are convenient for

thermal expansion coefficients. Without the“ppm” designation the numbers should include a factor

of 10−6. For example the typical α below Tg is 5× 10−5 to 1× 10−4.

The pressure dependence of PV T data at constant temperature can be used to calculate the

isothermal compressibility. In analogy with volumetric thermal expansion coefficient, compressibil-

ity is defined by

K(P, T ) = − 1V (P, T )

(∂V (P, T )

∂P

)T

= −(

∂ lnV (P, T )∂P

)T

(10.9)

The minus sign is included to make K positive. When K is large, the material is compressible or

soft. When K is small, the material is incompressible or hard. K undergoes a discontinuous change

from a low to a higher value at Tg. The discontinuous change in K is another manifestation of the

glass transition being a second order transition — K is a second derivative of free energy and it

undergoes a discontinuous change at Tg.

Empirically, the PV T properties of many amorphous polymers can be fit using the Tait equa-

tion. The Tait equation was derived in the nineteenth century from experiments on sea water. It

empirically explains sea water results (volume as a function of pressure and temperature) and also

the PV T properties of many other amorphous materials such as amorphous polymers. The Tait

equation is:

V (P, T ) = V (0, T )[1− 0.0894 ln

(1 +

P

B(T )

)](10.10)

where V (0, T ) is the volume at zero applied pressure (or atmospheric pressure) and B(T ) is

B(T ) = B0e−B1T (10.11)

where B0 and B1 are constants. Over a restricted temperature range, V (0, T ) can often be expressed

as a polynomial. For example, we can write

V (0, T ) = A0 + A1T + A2T2 + A3T

3 + · · · (10.12)

10.2. AMORPHOUS POLYMERS 119

where A0, A1, . . . are constants. Alternatively, if we assume that the volumetric thermal expansion

coefficient is independent of temperature for some temperature range, we can integrate the definition

of β to get

V (0, T ) = A0eβ(0)T (10.13)

where A0 is a constant and β(0) is the volumetric thermal expansion coefficient at zero (or atmo-

spheric) pressure.

Fitting experimental PV T data to the Tait equation is often a useful exercise. The Tait

equation has several fitting constants. There are fitting constants when fitting V (0, T ) (A0, A1,

. . . or A0 and β(0)); there are two additional fitting constants in B(T ) (B1 and B0). Over some

restricted temperature range all the PV T data can be reduced to 4 to 6 Tait parameters. Once

these parameters are known, it is possible to calculate volume for any temperature and pressure

within that temperature range. By differentiated the Tait equation with respect to temperature

or pressure, it is also possible to calculate the volumetric thermal expansion coefficient and the

isothermal compressibility as functions of temperature and pressure. Because Tait equation fits

can typically reproduce all the experimental results to four significant figures, the Tait parameters

are a useful way to tabulate PVT data. Thus instead of storing a large amount of raw data,

one can record several sets of Tait parameters for temperature ranges of interest. Indeed, there

are polymer handbooks that list Tait parameters for common polymers. Another advantage of

using the Tait equation vs. the raw data is that it is more reliable to calculate volumetric thermal

expansion coefficient and the isothermal compressibility by differentiating the Tait equation than

it is to numerically differentiate the raw data.

10.2.1 Pressure Dependence of Tg

Although the glass transition is not a true second-order transtion, it is possible to calculate the

pressure dependence of Tg by using thermodynamics methods. If one is careful in analysis of

experiments, this result is correct for polymer glass transitions. Two key results, first derived by

Ehrenfest, are:dTg

dP=

∆K

∆βand

dTg

dP=

V T∆β

∆Cp(10.14)

where the “∆” terms are changes in thermodynamic properties at the glass transition. Each of

these properties (compressibility (K), volumetric thermal expansion coefficient (β), and constant-

pressure heat capacity (Cp)) has a discontinuous change at a second-order transition. Because both

results are exact thermodynamics results, they can be equated to define the Prigogine-Defray ratio

(R) or

R =∆K∆Cp

V T (∆β)2= 1 (10.15)

This ratio must hold at any second-order transition.


Tc

Volu

me

Temperature

TgTm

Cooling Heating

Figure 10.3: Typical PV T data for a semicrystalline polymer at a single pressure. The solid curve isobtained during heating, the dashed curved is obtained during cooling.

10.3 Semicrystalline Polymers

Some typical PV T data for a semicrystalline polymer are give in Fig. 10.3. The single curve

plots the volume as a function of pressure for some constant pressure. If we included curves for

several pressures (as in Fig. 10.1), the curves would move down and to the right as the pressure

increased. The break in the curve at low temperature is the glass transition temperatures — Tg.

Because all semicrystalline polymers are also partially amorphous, they exhibit a glass transition

that is characteristic of amorphous polymers. At Tg, the slope of the volume vs. temperature

curve changes. The magnitude of the change will reflect the amount of amorphous polymer that

is present. As the polymer gets more crystalline, the change in slope will get smaller. For some

highly crystalline polymers (e.g., polyethylene), the change in slope may be undetectable.

Figure 10.3 shows PV T results for both heating (solid curve) and cooling (dashed curve). In

addition to a glass transition, the heating curve shows a melting transition. The melting transition

temperature is always above the glass transition temperature. Melting of polymers is a first order

transition and we expect a discontinuous change in volume. The change is never exactly discontin-

uous but occurs over a relatively narrow temperature range of 20-40◦C. The melting temperature

is usually defined as the end of the melting range or the temperature at which all crystals have

completely melted. Most materials melt in a much narrower temperature ranges (< 1◦C). One

reason that polymers melt over a broad temperature range is that polymers have a distribution of

molecular weights. It can be demonstrated that melting point is a function of molecular weight.

Thus a polydisperse sample naturally shows a distribution in melting points or a temperature range

for melting.

10.4. EXPERIMENTAL TECHNIQUES 121

Above Tm, the material is called a polymer melt. The PV T properties of polymer melts are

nearly equilibrium properties. There are equations of state which can predict this data. Polymer

melts are also amorphous liquids and thus the data can be fit with the Tait equation if desired.

Below Tm, the results are largely empirical. The Tait equation can be used well below the melting

transition. Near the melting transition, the melting process is too complex to be described by the

Tait equation. The Tait equation is an empirical equation that can describe amorphous materials.

The Tait equation, however, is not a thermodynamic equation of state and thus says nothing about

the occurrence of transitions. When the Tait equation cannot be used, α and K can still be found

by numerical differentiation of the raw data.

Heating and cooling experiments on semicrystalline polymers are different. A thermodynamic

definition of Tm is the temperature at which the free energy of the crystals and the melt are equal.

Above Tm the free energy of the melt is lower. Below Tm, the free energy of the crystals is lower.

Thus, on cooling, it is theoretically possible (by thermodynamic data) for new crystals to form

when the temperature first reaches Tm. In polymers, however, the crystals never form at Tm.

Instead, there are energy barriers associated with nucleation and growth of the crystals that must

be overcome. These barriers are eventually overcome at some temperature below Tm called the

crystallization temperature Tc. The phenomenon whereby the temperature decreases below Tm to

Tc before crystallization occurs is called supercooling. The amount of supercooling depends on the

cooling rate, the properties of the sample, and the presence of heterogeneous nucleating agents.

The amount of supercooling can be large and in fact if cooled fast enough the crystallization can

be prevented entirely giving an amorphous polymer.

10.3.1 Pressure Dependence of Tm

Th pressure dependence of the melting point is easy to determine by thermodynamics analysis of

the first-order transition. The result is

dTm

dP=

Tm∆V

∆Hf(10.16)

where ∆V is the volume change at the melting point and ∆Hf is the heat of fusion of the pure

crystals (or the enthalpy change at the melting point).

10.4 Experimental Techniques

A simple, but not very accurate, dilatometer is called a thermal mechanical analyzer (TMA). A

TMA measures linear thermal expansion of a specimen. A schematic of a TMA apparatus is show

in Fig. 10.4. In brief, a probe rests on a sample with some weight (note: the probe shape may

be important). The probe is attached to a rod which passes into an linear variable displacement

transducer (LVDT). As the rod moves in the LVDT, the voltage output changes in proportion to


LVDT

Sample

Weights

Figure 10.4: Schematic drawing of a thermomechanical analyzer or TMA.

displacement. The LVDT thus converts a linear displacement to a voltage which can be recorded

by many types of scientific instruments. In a TMA experiment, the temperature is scanned and

the displacement is measured as a function of temperature.

The TMA method has several limitations. First, the probe must rest on the sample. The

sample must therefore remain solid over the entire temperature range of the tests. For polymers

TMA can be used as long as the temperature is kept well below Tg. At higher temperature, the

probe may start to penetrate the specimen and give erroneous expansion results. If the probe

penetrated deeply, the TMA may report an illogical negative thermal expansion coefficient. The

temperature at which penetration becomes significant depends on the shape of the probe and the

amount of weight used to hold the probe on the surface. A second disadvantage of TMA is that

it only measures linear thermal expansion. To calculate volumetric thermal expansion, you must

either assume the specimen is isotropic (same thermal expansion in all directions) or you must

measure thermal expansion in all directions. In summary, TMA is does not give quantitative

results about transition temperatures (because of probe penetration problems) or about thermal

expansion coefficient (because of uncertainty about specimen isotropy).

Unlike TMA, true dilatometry experiments measure total volume change. A simple dilatometer

can be constructed from simple glass ware — a beaker and a capillary (see Fig. 10.5). A sample is

immersed in a liquid. The experiment is to measure total volume of the liquid and polymer as a

function of temperature. The volume can be accurately measured by observing height of the liquid

in a narrow capillary on top of the beaker. To get the volume of the polymer, you must subtract the

volume of the pure liquid which can be measured by a calibration experiment. In the calibration

experiment you measure value of the pure liquid in the same dilatometer

To calibrate such a dilatometer you begin with an experiment on the pure liquid. Let ml0 be

the mass of the liquid in the calibration experiment. Then you can measure the liquid volume as

a function of temperature, V0(T ). It will be equal to

V0(T ) =ml0

ρl(T )(10.17)

10.4. EXPERIMENTAL TECHNIQUES 123

Capillary

Sample

Figure 10.5: A simple dilatometry for measure volume of a polymer at atmospheric pressure.

where ρl(T ) is the density of the liquid at temperature T . Now do a real experiment combining a

mass ms of a specimen with a mass ml of liquid. The specific volume of the specimen is

v =Vs(T )ms

=V (T )− Vl(T )

ms(10.18)

where Vs(T ) is the volume of the specimen, V (T ) is the total measured volume, and Vl(T ) is the

volume of the liquid, all at temperature T . The volume of the liquid is

Vl(T ) =ml

ρl(T )(10.19)

which gives

v =1

ms

(V (T )− ml

ρl(T )

)(10.20)

In terms of the calibration data, V0(T ), this equation can be rewritten as

v =1

ms

(V (T )− ml

ml0V0(T )

)(10.21)

Thus the specific volume of the specimen is given in terms of the masses (ml and ms), the mea-

sured total volume (V (T )), the mass used in the calibration experiment (ml0) and the measured

calibration volume (V0(T )).

There are two other considerations when doing dilatometry experiments. First, there must be

no chemical reactions, either in the polymer, or between the liquid and the polymer. A chemical

reaction would cause a volume change not associated with PV T physical properties. Second, the

liquid must not undergo any transitions which would mask the sample volume change. In other

words, the confining liquid must remain liquid over the entire temperature range of the experi-

ments. If the liquid started to boil, the large volume changes associated with vaporization would

mask the polymer volume changes. If the liquid froze, it would no longer communicate polymer

volume changes to height changes in the capillary. A good confining liquid for studying polymers


LVDT

Bellows

Sample

Heated Jacket

Hydraulic Fluid Inlet

Figure 10.6: Schematic drawing of a PVT apparatus.

is mercury. It is relatively inert and reacts with very few polymers (but can react with fluoropoly-

mers). It is also a liquid at room temperature and remains a liquid to very high temperature. Its

liquid temperature range makes its suitable for studying glass transitions, melting transitions and

crystallization transitions in all but the highest melting point polymers. The main problem with

mercury is that its vapors are toxic.

Neither TMA nor simple dilatometry in a beaker are suitable to applying pressure and therefore

to getting volume data as a function of temperature and pressure. Equipment that can vary both

temperature and pressure is called PV T equipment. A good PV T technique is to apply pressure

using hydraulic fluid. A schematic drawing of a high-pressure PV T apparatus is given in Fig. 10.6.

The specimen is confined in a liquid as described above for simple dilatometry. The same concerns

that apply to liquids for simple dilatometry also apply to high-pressure PV T experiments. The

specimen plus liquid volume is measured by converting the volume expansion into a linear expansion

through the use of a bellows. The linear change in the bellows is measured with an LVDT. The

pressure is applied by hydraulic fluid; the fluid surrounds the sample and cell and is inside the

bellows. As the hydraulic fluid is pressurized, movement of the bellows applies pressure to the

sample. Like simple dilatometry, the volume expansion of the confining liquid is subtracted out by

doing a calibration experiment on pure liquid.

10.5 Dilatometry Summary

Dilatometry can be used to measure thermal transitions, PV T properties, thermal expansion coef-

ficients and isothermal compressibility. Although not discussed above, dilatometry can be extended

to many other polymer processes that involve volume changes. Two examples are the kinetics of

crystallization or the kinetics of cure in thermoset polymers.

10.5. DILATOMETRY SUMMARY 125

Dilatometry results are potentially of great importance to the field of polymer processing. For

example, polymers are often heated, and than molded at high pressures. The amount of volume

change during the pressure and temperature conditions of the molding cycle are important. Size

can be of critical importance for plastic parts such as gears, LEGOs, etc.. For molding of such parts

the entire molding process must be controlled to tight tolerances and specifically to tolerances on

volume change. PV T data can help develop these controls. Unfortunately, many polymer processes

work by empiricism. They play with the molding conditions until they get acceptable part sizes.

There is probably room for a more scientific approach that would use PV T data to predict the

required molding conditions.

Another molding consideration is the pressure dependence of the thermal transitions (Tm and

Tg). PV T experiments can measure Tm and Tg as a function of pressure. The findings with poly-

mers is that the transition temperatures can change by 50-100◦C between atmospheric pressure

and molding pressure. If Tm is 50◦C higher at the molding pressure, than the molding temperature

better be more than 50◦C higher than Tm or the sample will never melt. Again, problems associ-

ated with molding temperature are often handled by empiricism. Some simple reflecting on PV T

experiments could save polymer molders a lot of trial and error experimentation.

Problems

10–1. Derive an expression for the compressibility of a material which obeys the Tait equation.

10–2. Derive an expression for the volumetric thermal expansion coefficient of a material which

obeys the Tait equation.


Chapter 11

Differential Scanning Calorimetry

11.1 Introduction

Calorimetry with polymers is normally done using differential scanning calorimetry or DSC. In

brief, DSC scans temperature and measures heat capacity of a specimen. The experiments are typ-

ically done at atmospheric pressure and thus DSC measures the constant-pressure heat capacity.

Another thermal analysis technique that typically involves temperature scanning is thermogravi-

metric analysis or TGA. In TGA experiments, the temperature is scanned while measuring the

weight of a specimen. Thermal degradation reactions typically cause the weight to decrease and

thus decreases in weight are a sign of thermal instabilities. This chapter discusses both DSC and

TGA.

11.2 Differential Scanning Calorimetry

DSC measures a specimen’s heat capacity at constant pressure — CP . Heat capacity units are

cal/◦K or J/◦K. It measures the amount of heat input (q) required to raise the temperature of the

specimen by one degree Celsius while at constant pressure. Heat capacity is usually normalized by

dividing the specimen heat capacity by the number of grams to get the heat required to raise one

gram of specimen by one degree Celsius. If desired, heat capacity can be normalized by the number

of moles. When observing thermal transitions, we note that heat capacity is related to the second

derivative of free energy:

(∂G

∂T

)P

= −S = − q

T(11.1)(

∂2G

∂T 2

)P

= −(

∂(q/T )∂T

)P

=S

T− 1

T

(∂q

∂T

)P

=S

T− CP

T(11.2)

127

128 CHAPTER 11. DIFFERENTIAL SCANNING CALORIMETRY

and constant-pressure heat capacity is defined by

CP =(

∂q

∂T

)P

= S − T

(∂2G

∂T 2

)P

(11.3)

At the second order transition, there will be a discontinuity in the second derivative of free energy.

Because S will be continuous at the second order transition (S is the first derivative of free en-

ergy), a second order transition will show a discontinuous change in heat capacity at the transition

temperature. A first order transition will show up as a peak in heat capacity.

DSC is a form of calorimetry and in fact gives the same information as conventional calorimetry.

In conventional calorimetry (e.g., bomb calorimeters you may have used in freshman chemistry),

you are usually concerned with measuring the heat of some reaction or some process. The method

is to carry out the reaction adiabatically (i.e., with no heat flow) and measure the change in

temperature within the calorimeter. If the temperature changes from T0 to T1, the enthalpy of the

reaction (∆H) is

∆H =∫ T1

T0

CP dT (11.4)

where CP the heat capacity of your system. CP of your system can be found by a calibration

experiment in which the temperature is changed by providing a known amount of heat in the form

of supplied electrical energy. Usually ∆T is small and CP is independent of temperature between

T0 and T1. The integral thus reduces to

∆H = CP (T1 − T0) = CP ∆T (11.5)

The experiment consists of measuring ∆T . The only problem is determining the final temperature

T1. Usually the calorimeter will not be perfectly adiabatic and there will be a slow decrease

in temperature as heat escapes to the surroundings. The standard approach is to record T0 at

the beginning and then to monitor temperature during and after the reaction. Eventually the

temperature will enter a steady decrease and return to T0. Extrapolating this portion of the curve

back to zero time will give the true T1 or the ideal adiabatic T1.

Conventional calorimetry is accurate and has been an important part of chemistry for several

centuries. Unfortunately, it is slow and not adaptable to scanned experiments or to studying thermal

transitions. When we want faster calorimetry and/or scanned results, DSC is the calorimetry

method of choice. DSC is fast and easy. It is not as accurate as conventional calorimetry but it

is still within 1-2%; an accuracy that is more than sufficient for most polymer applications. As

discussed above, DSC measures heat capacity which is a second derivative of free energy. Second

order transitions, such as the glass transition, will show up as a discrete change in CP . DSC will

give the ∆CP of transition and the temperature of the transition. First order transitions, such as

melting or crystallization, will appear as a spike or peak. The area under the peak will be the

heat evolved or absorbed during the transition. This information will give, for example, the heat of

11.3. INSTRUMENTATION 129

Sample Reference

Heaters Difference of twochannels torecorder

amp

amp

Error

Error

ProgrammedTemp

ProgrammedTemp

MeasuredTemp

MeasuredTemp

Figure 11.1: Schematic drawing of a DSC.

fusion, ∆Hf , or the heat of crystallization, ∆Hcrys. The area under the curve analysis is similar to

the conventional calorimetry method discussed above where T0 and T1 are the beginning and the

end of the thermal transition and CP is the measured heat capacity. During a transition, however,

CP does not remain constant and we must use the integral expression or the area under the CP

curve.

11.3 Instrumentation

A schematic drawing of a DSC instrument is given in Fig. 11.1. There are two sample chambers — a

“reference” chamber and the“sample” chamber. An empty pan is placed in the reference chamber

and a pan filled with a polymer specimen is placed in the sample chamber. The two chambers

are then heated or cooled until they each reach the selected starting temperature. A temperature

program is then begun; a typical temperature program would be to increase the temperature to

some higher temperature at some fixed rate or fixed number of degrees Celsius per minute. As the

program runs, the system monitors the temperature in each chamber. If the temperature differs

from the programmed temperature in either chamber, heat or coolant is supplied to that chamber

to make the temperature equal the program temperature. The difference in the energy supplied to

the two chambers per unit time (or dq/dt) is proportional to the heat capacity of the sample. That

energy difference is monitored electronically and output to the recorder.

The instrument shown in Fig. 11.1 is a true calorimetry. It measures heat flow which is converted

into heat capacity. An alternative approach to DSC is to measure temperature difference. In

particular, the reference and sample chamber can be supplied with the same amount of heat.


The amount of heat supplied can be adjusted to give various heating rates. Then instead of

maintaining equal temperatures and measuring differential heat flow, it is possible to maintain

constant heat flow and measure the difference in temperature between the reference and sample

chambers. The temperature difference will give a good measure of the heat capacity. When

the heat capacity of the sample chamber increases (because of a heat absorbing or endothermic

transition) the temperature in the sample chamber will lag behind the reference chamber and

there will be a negative temperature differential (Tsample − Tref < 0). Conversely, when the heat

capacity of the sample chamber decreases (because of a heat generating or exothermic transition)

the temperature in the sample chamber will be ahead of the reference chamber and there will be a

positive temperature differential (Tsample − Tref > 0).

DSCs that measure temperature differential are not true DSCs or true calorimeters. They are

more accurately labeled as differential thermal analyzers or DTAs. They were originally developed

by many manufacturers to avoid patent protection of the DSC mode of operation and because the

instrumentation is slightly simpler. Many such DTAs are inaccurately labels as DSCs by those

manufacturers. Fortunately both DSCs and DTAs can give suitable heat capacity results when

working with polymers. DTAs are perhaps slightly less quantitative than DSCs. When using a new

DSC, it is advisable to determine whether it is a DSC or a DTA.

11.4 Results with Polymers

11.4.1 Heating of Semicrystalline Polymers

A typical plot of dq/dT (or heat capacity) vs. T for the DSC of a semicrystalline polymer is given

in Fig. 11.2. If the same experiment was done on a DTA, it would instead be a plot of −∆T vs. T .

In such a DSC plot, higher heat capacity is up and lower heat capacity is down. Endothermic first

order transitions or transitions that absorb heat have a high heat capacity and thus show positive

peaks. Exothermic first order transitions or transitions that release heat have a low or negative

heat capacity and thus show negative peaks. Some DTAs plot ∆T instead of −∆T and thus the

entire plot in Fig. 11.2 would be inverted. Lower heat capacity is up, higher heat capacity is down,

endothermic first order transitions show negative peaks and exothermic first order transitions show

positive peaks. When interpreting DSC data, you must verify the sign convention used by your DSC

instrument. The sign used in Fig. 11.2 is the most logical one and the one that is used throughout

these notes.

The lowest temperature transition in Fig. 11.2 is the glass transition or Tg. Around Tg, CP

undergoes a quasi-discontinuous change from a lower value to a higher value. This behavior is

typical for a second order transition. Polymers always have a higher heat capacity above Tg than

below Tg because there are more degrees of freedom that can absorb energy above Tg. We called

11.4. RESULTS WITH POLYMERS 131

+

-Temperature

dqdT Tg Tc

Tm

Figure 11.2: Typical DSC data while heating a semicrystalline polymer. The low-temperature second ordertransition is the glass transition. The exothermic transition is crystallization. The endothermic transition ismelting.

the change at Tg a quasi-discontinuous change because it is not sharp, but happens over a range

of about 10◦C. Many polymer handbooks list a single Tg; the question remains: which part of

the curve defines the actual Tg? The answer depends on who is determining Tg. Some possible

answers are the temperature at the onset of the discontinuous change (the first line in Fig. 11.3),

the inflection point of the change (the second line in Fig. 11.3), or the end of the discontinuous

change (the last line in Fig. 11.3). The most common convention is to take the inflection point.

The results in the glass transition temperature depend on the rate of the test. One effect is a

material property; namely, the glass transition temperature is a rate dependent property. Its true

value will be different at different rates and thus DSC will be sensitive to rate. Tg increases with

increasing heating rate. Recall that Tg is the temperature at which long-range relaxations become

slow with respect to the observation time. If the observation time is shorter, the relaxations do not

need to be as fast to be at Tg; thus the glass transition occurs at a higher temperature. Conversely,

Tg decreases with decreasing heating rate. The effect of rate on Tg is not large. Tg changes about

3◦C for each order of magnitude change in heating rate.

Another heating rate effect depends on sample preparation. Before doing the heating experi-

ment, the sample must be cooled down to the testing temperature. Even if you do not cool the

sample down, the sample was cooled down by the material supplier before you bought it. The

glass transition region will depend on the heating rate (see above) and on the difference between

the heating rate and the cooling rate used in preparing the sample. The results might also depend

on aging time or the length of time that has elapsed since the polymer was cooled down. If you

bought the sample and have not yet heated it, the aging time could be several years. Speaking

qualitatively, if the heating rate is slower than or similar to the cooling rate and the sample has


Glass Transition

Temperature

Cp

Onset

End

Inflection

Figure 11.3: Blow up of the glass transition region. The vertical lines show two possible methods fordetermining Tg. The infection point method is the most common one.

not aged long, the glass transition will resemble the one in Fig. 11.3. In other words, there will be

a smooth transition from a low heat capacity below Tg to a higher heat capacity above Tg. If the

heating rate is much faster than the cooling rate or the sample has aged significantly before testing,

it is possible to get a glass transition that resembles the trace in Fig. 11.4. The heat capacity passes

through a peak on preceding from the low heat capacity below Tg to the higher heat capacity above

Tg. This peak resembles an endothermic, first order transition. Some researchers have, indeed,

misinterpreted such peaks as endothermic transitions. The peak, however, has nothing to do with

a first order transitions and can be completely explained by a detailed analysis of the kinetics of

the glass transition.

If a semicrystalline polymer did not crystallize to the limit of its ability when cooled, it is

possible to see some additional crystallization during the heating DSC. Crystallization shows up

as an exothermic peak (negative peak or peak that gives off heat); the crystallization temperature,

Tc, is always between Tg and the melting temperature Tm. To emphasize, crystallization will only

occur in samples that can crystallize (semicrystalline polymers) and in samples that are not already

crystallized as far as they can be crystallized (because of incomplete crystallization during cooling).

A typical crystallization peak is shown in Fig. 11.5. The crystallization temperature is sometimes

taken as the onset of crystallization, but is more commonly assigned to the peak temperature.

Both the onset and peak temperature are sensitive to rate. During heating, these temperatures

may increase with increasing rate. The total heat evolved, qc or ∆Hc, can be determined from the

area under the peak (the shaded portion in Fig. 11.5).

The highest temperature peak in Fig. 11.2 is the melting transition. Melting is a first order,

endothermic transition (i.e., it requires heat); it thus occurs as a positive peak on a DSC trace.

DSC can also be used to study low molecular weight samples. A typical small molecule would


Glass Transition

Temperature

Cp

Figure 11.4: Glass transition with an overshoot due to kinetic effects in the glass transition process. Theovershoot is accentuated by using faster heating times or by aging the sample before heating.

Crystallization

OnsetT1 T2

Peak

Cp

Temperature

Figure 11.5: A typical exothermic crystallization peak. The shaded area is the area under the curve.


Polymer Melting

Onset

Peak

Cp

Temperature

End

Small MW Melting

Cp

Temperature

Figure 11.6: The melting transition for typical polymer molecules and a typical monodisperse low molecularweight molecule.

show neither a glass transition (there are no amorphous parts of most small molecules) nor a

crystallization peak (because the sample could not be prevented from crystallizing during the initial

cool down). It would show a melting point, but the melting transition would be much sharper than

the corresponding transition in a polymer sample (see Fig. 11.6). For small molecules, the peak

is very sharp, and the melting temperature is taken as the peak temperature. For polymers, the

melting transition is broad. As explained in the dilatometry section, the breadth of the peak is due

to the polydispersity of polymer molecular weights. The melting temperature is sometimes taken

as the onset temperature, the peak temperature, or the end temperature (see Fig. 11.6). The end

temperature is perhaps the most logical because it defines the temperature at which all crystals

disappear. By convention, however, a melting temperature determined by DSC is usually taken as

the peak temperature. The discrepancy between the peak temperature and the end temperature is

minute for low molecular weight materials, but can be significant for polymer materials.

The area under a melting transition curve is the total amount of heat absorbed during the

melting process. The relevant area is shown as the shaded portion in Fig. 11.6. This area is used

below to calculate the fraction crystallinity that existed before the polymer was heated.

Some polymers can crystallize into more than one form of crystal. When there are two types

of crystals, it is possible for them to have different melting point. This complicating feature

in the polymer structure can lead to a complicated melting transition region in the DSC trace.

One possible effect is shown in the left half of Fig. 11.7 where the two types of crystals give two

independent melting transitions. There are thus two peaks in the DSC trace. It is even possible for

crystals that melt at the lower temperature to recrystallize into the higher melting point crystal and

then melt at the higher temperature. The melting point region would than be the superposition of

two melting peaks and one crystallization peak (see right half of Fig. 11.7)


Cp Cp


Two melting peaksTwo melting and onecrstallization peak

Figure 11.7: Melting transition region of a polymer with more than one type of crystal forms. The left sideshows two melting transitions. The right side shows two melting transitions and a crystallization transition.

The area under any first order transition peak in a DSC trace gives the heat evolved during

that transition (see Figs. 11.5 and 11.6). DSC data measures heat capacity or dq/dT . Integrating

dq/dT clearly gives heat or q: ∫ T2

T1

dq

dTdT = q (11.6)

Many DSCs print the trace onto chart paper (e.g., one of the DSCs in the Material Science De-

partment). If we let A be the area under the peak on the chart in square inches (in2), R be the

machine sensitivity or range in cal/sec/in, and S be the chart speed in in/sec, then the heat per

gram or enthalpy per gram (∆H) of the transition is

∆H =K A(in2) R(cal/sec/in)

m(g)S(in/sec)(11.7)

where ∆H is in cal/g and K is a calibration constant. The calibration constant is determined by

measuring the heat evolved in a transition with known ∆H. Now consider applying this formula

to the melting transition. The entire heat of evolution is due to melting of the crystal fraction. If

the sample is χ% crystalline by weight, you can get the heat of fusion of the crystals by replacing

total mass of the specimen (m) by the mass of the crystals which would be mχ/100. Substituting

into the above equation:

∆Hf =100KAR

mχS=

100∆H

χ(11.8)

or

χ =100∆H

∆Hf(11.9)

The equation for χ is perhaps the more intuitive of the two equations. It says that fraction

crystallinity on any sample is equal to the ratio of ∆H to ∆Hf . ∆Hf is the enthalpy of fusion


and is defined as the calories required to melt one gram of crystal or cal/g or crystal. ∆H is

experimental data and it is the calories required to melt one gram of sample or cal/g of sample.

Their ratio is

∆H cal/g of sample∆Hf cal/g of crystal

=g of crystalg of sample

(11.10)

or the mass fraction of crystal in the sample. Multiplying this fraction by 100 gives the percent

crystallinity.

The ease of measuring areas under melting transitions makes DSC a good tool for measuring

percent crystallinity. The method, however, requires that you know the heat of fusion for the

polymer, ∆Hf . If the polymer could be obtained in 100% crystalline form, ∆Hf could be measured

by a simple DSC experiment. Unfortunately, all crystalline polymers are semicrystalline and we

need some alternative method for determining ∆Hf or the enthalpy of fusion of the pure crystal.

There are two approaches. One is a noncalorimetric method for determining ∆Hf . The other is to

do DSC on a sample with known fraction crystallinity. We describe each method.

The Clausius-Clapeyron equation is a simple-to-derive thermodynamic relation that predicts

the pressure dependence of the melting point. Solving that equation for heat of fusion gives:

∆Hf =Tm(Va − Vc)

dTmdP

(11.11)

where Tm is the melting point, dTm/dP is the pressure dependence of the melting point, Va is

the specific volume of amorphous material at the melting point, and Vc is the specific volume of

the crystal at the melting point. Tm, dTm/dP , and Va can all be determine by PV T experiments

or dilatometry. Tm is easy; it is the temperature at which all crystals disappear. dTm/dP is

determined by determining Tm at several different pressures. Tm always increases with pressure

and thus dTm/dP is greater than zero. Va is determined by measuring the volume in the melt and

extrapolating back to the melting temperature. Although dilatometry measures volumes, it cannot

determine Vc. Vc is the specific volume of the pure crystal and polymers cannot be obtained in

a purely crystalline form. Vc can, however, be determined by X-ray crystallography. The specific

volume can be calculated from unit cell dimensions. The X-ray experiments should be done at

temperatures near Tm and extrapolated up to Tm to get the most accurate results.

Once Tm, Va, Vc, and dTm/dP are known, ∆Hf can be calculated using the Clausius-Clapeyron

equation. Note that neither dilatometry nor X-ray crystallography are calorimetric techniques.

This method thus provides an independent, noncalorimetric route to finding ∆Hf . Once ∆Hf is

known, DSC can be used on any sample of that polymer to find the percent crystallinity. The

Clausius-Clapeyron equation is an exact thermodynamic equation. The only potential problem

with applying it to polymers is difficulty in obtaining reliable results for Tm, Va, Vc, and dTm/dP .

Fortunately, dilatometry and X-ray of polymers gives data that is accurate enough for calculating

∆Hf .


Instead of measuring ∆Hf , an alternative approach is to do a DSC on a sample with a known

χ. If χ can be accurately determined by some noncalorimetric method, than the heat of fusion can

be calculated from DSC data by

∆Hf =100∆H

χ(11.12)

The problem thus reduces to finding a second method for determining the percent crystallinity. We

discuss two methods:

1. Sample density method: a sample of density ρ has a fraction crystallinity of

χ =ρcVc

ρV(11.13)

where Vc is the total volume of crystal, V is the total volume of sample, and ρc is the density

of a 100% crystalline polymer. The ratio of volumes is

Vc

V=

ρ− ρa

ρc − ρa(11.14)

where ρa is the density of a 100% amorphous polymer. Thus, in terms of densities

χ =ρc

ρ

ρ− ρa

ρc − ρa(11.15)

The amourphous density, ρa can often be determined by quenching or rapidly cooling a

sample to prevent crystallization and then measuring density. The crystalline density, ρc is

best determined by X-ray crystallography and unit cell dimensions. Note that the Vc/V or

the volume fraction of crystals was used above. It comes from a simple linear interpolation

between the volume of a pure crystal and the volume of a pure amorphous sample.

ρV = ρcVc + ρaVa = ρcVc + ρa(V − Vc) (11.16)

To be successful, the density method requires use of void-free samples.

2. Spectroscopic methods: many spectroscopic methods (e.g., IR, NMR, X-ray, etc.) sometimes

show peaks that result from the crystalline fraction and are not present in the amorphous

fraction. Conversely, they sometimes show peaks in the amorphous fraction that are not

present in the crystalline fraction. Careful analysis of these peaks with suitable normal-

ization can be used to determine the fraction crystallinity in a sample. To be successful,

spectroscopic methods require careful characterization of the origins of the observed signals.

This characterization might entail much extra work.

11.4.2 Cooling of Semicrystalline Polymers

Cooling of semicrystalline polymers will typically give a trace similar to the heating trace except

that there will never be a melting transition. There will probably also be significant differences


in the crystallization region. When approaching Tc from high temperature, the sample is 100%

noncrystalline and is thus ready to crystallize as much as possible. Typically the crystallization

peak will be more pronounced in cooling curves than in heating curves. Cooling rate will have a

significant effect. The rate will affect the crystallization temperature, with higher rates tending to

decrease Tc. More importantly, higher rates will affect the amount of crystals that form. Higher

rates tend to reduce the area under the peak and thus reduce the amount of crystallization. Many

polymers crystallize so sluggishly that they can be entirely prevented from crystallizing by cooling

at a high rate or by quenching to well below Tc. The DSC of a polymer that is prevented from

crystallization will not show any crystallization peak.

11.4.3 Amorphous Polymers

Amorphous polymers have no crystals and thus never show melting transitions or crystallization

transitions. Amorphous polymers show only the glass transition region. The discussion of the

glass transition region given above applies also to amorphous polymers. The only difference is that

because amorphous polymers are 100% amorphous, the glass transition will be more pronounced

or will show a larger change in heat capacity.

11.5 DSC Summary

DSC finds many uses in calorimetry and in polymer science. We list a few of the properties that

can be measured by DSC. Many of these were discussed above. Others can be found discussed in

the literature or in text books:

1. Temperature for thermal transitions: Tg, Tc, and Tm

2. Heat and entropy of transition: ∆H and ∆S (from q/T )

3. Heat capacity: CP

4. Percent crystallinity: χ (assuming ∆Hf is known)

5. Heat absorbed or evolved during cure reactions or decomposition reactions

6. Sub-glass or solid state transitions

7. Crystallization kinetics (heat evolved during isothermal crystallization)

11.6 Thermogravimetric Analysis

Thermogravimetric analysis or TGA involves heating a sample to some temperature and then

monitoring its weight as a function of time. It is commonly used to monitor polymer degradation

11.6. THERMOGRAVIMETRIC ANALYSIS 139

reactions. TGA only requires a sensitive method for monitoring weight and a method for setting

the temperature of the sample. Ideally the sample chamber should have a controllable environment

to allow you to monitor degradation under various conditions (e.g., in the presence of oxygen or

under dry nitrogen).

The most accurate TGA experiments are to set the temperature to some constant value and

follow weight as a function of time. The results will give quantitative information about the rate

of degradation at that temperature. Repeating this experiment for several temperatures will give

information about the temperature range under which this polymer can be used. Usually, people

(i.e., industry) are not patient enough to do all those required experiments. Therefore, most TGAs

are set up to scan temperature. Monitoring weight while scanning temperature will give useful

results, but it must be realized that if the degradation is slow, the polymer will appear to survive

to higher temperatures than it would actually survive under long term exposure. In other words,

the results will also be very sensitive to rate. The sensitivity is large because degradation rates

may be slow when compared to typical scanning rates such as 10-20◦C/min.

Coupling the by-products of a TGA experiment or the degradation products from the polymer

to other instruments such as mass spectrometer, infrared spectrometry, gas chromatography, etc.,

can be useful in studying the mechanisms of polymer degradation. The coupled instruments can

be used to identify the structure of the degradation products. Combining structural information

with kinetics information can give a complete picture of the degradation process.

Problems

11–1. Most polymers crystallize by heterogeneous crystallization at nucleating agents. Nucleating

agents help the crystallization process to get started. Suppose you wanted to add a nucleating

agent to promote rapid crystallization during processing and that you have several nucleating

agents to choose from. Describe some DSC experiments (including control experiments) that

would enable you to decide which nucleating agent would be the best. (Hints: nucleating

agents will change the crystallization temperature (specify which way) and will change the

level of crystallization for a given amount of crystallization time).

11–2. Because it is easy to measure sample density, the density method appears to provide and

ideal way to measure the weight fraction crystallinity. Unfortunately real-life experiments

are not always trouble free. Describe problems that might arise in the following parts of the

analysis:

a. Accurately knowing ρa:

b. Accurately knowing ρc:

c. Polymers that have ρa ≈ ρc:


Chapter 12

Mechanical Properties

12.1 Introduction

Polymers are often used in applications that involve stresses. Examples include housings, gears,

ropes, structures, composites, etc.. Before using polymers in load-bearing applications, we must

study the effect that stress has on them. In the field of polymer characterization, there are numerous

tests aimed as characterizing the mechanical properties of polymers. Some of tests and the type of

information measured are:

1. Static testing: tensile, compressive and shear properties; stress-strain curves; stiffness, strength,

and toughness.

2. Creep test: time dependencies of elongation; viscoelastic properties.

3. Relaxation test: time dependencies of stress; viscoelastic properties.

4. Dynamic mechanical test: viscoelastic properties; thermal transitions; molecular relaxations.

5. Fatigue test: lifetimes or durability

6. Impact tests: high-rate properties; impact strength.

All these mechanical characterization tests measure some mechanical properties. Because any

mechanical property may be affected by temperature and environment, the tests sometimes need to

be done at various temperatures and in various environments. The test results for many mechanical

properties (e.g., toughness and impact strength) depend on sample geometry. For such properties

it is important to have a detailed understanding of geometry effects and to avoid oversimplified

interpretations such as merely dividing by the cross-sectional area of the specimen.

141

142 CHAPTER 12. MECHANICAL PROPERTIES

12.2 Simple Elasticity Theory

Our basis for understanding the influence of stress on materials is the theory of elasticity. Although

polymers often deviate from small-strain elasticity theory, the theory is still useful. It can be directly

applied to many polymer problems and it helps in the understanding of other polymer problems.

The simplest elasticity problem is the one dimensional problem of a tensile load being applied

to a bar of constant cross section. According to Hooke’s law, such a bar made of a linear elastic

material acts like a spring. The force applied to the bar, F , is proportional to the extension of the

bar, x:

F = kx (12.1)

where x is the extension of the bar or (L− L0) where L is the extended length of the bar and L0

is the original length of the bar. Thus

F = k(L− L0) (12.2)

The spring constant, or the stiffness, k, will depend on the size of the sample (i.e., on the size of

the spring) and on the mechanical properties of the sample. To eliminate size effects, it is desirable

to normalize out the sample size effect using

k =EA0

L0(12.3)

where E is the Young’s modulus of the material and A0 is the initial cross-sectional area. Then

F

A0= E

L− L0

L0(12.4)

The term on the left is called the stress, σ, or the force per unit area. It has the same units

as pressure. The term (L − L0)/L0 is the strain, ε, or the fractional increase in length; it is

dimensionless. The final one-dimensional elasticity equation is

σ = Eε (12.5)

Analyzing one dimension is enough when you are working only with simple springs. For solid or

three dimensional materials, we must consider length changes in all directions. First assume that

a one-dimensional stress is applied along the z axis and that the z axis stress causes dimensional

changes only in the z direction. In this case the change in volume is

∆V = V − V0 = LxLy(1 + εz)Lz − LxLyLz = εzV0 (12.6)

Simple dilatometry experiments during loading can show that this large a volume change does not

occur. What must be happening is that the x and y dimensions contract slightly to reduce ∆V .

In essence, a solid material wants to resist a change in volume. Thus an application of a tensile

12.2. SIMPLE ELASTICITY THEORY 143

stress in one direction causes the other two direction to contract in order to minimize the increase

in volume. Poisson assumed that the contractions in the x and y directions are proportional to the

extension in the z direction or:

εx = −νεz and εy = −νεz (12.7)

where ν is called the Poisson’s ratio. The volume change on extension in the z direction is now

∆V = (1 + εx)(1 + εy)(1 + εz)V0 − V0 ≈ (εx + εy + εz)V0 = (1− 2ν)εzV0 (12.8)

where the indicated approximations assumes all strains are small and thus higher order terms in

strain can be neglected. When ν = 1/2, ∆V is equal to zero. This is approximately true for many

rubber-like materials. Most metals, and glassy or semicrystalline polymers have ν of about 1/3.

Considering strains in all directions, a stress of σz in the z direction induces the following

strains:

εx = − ν

Eσz (12.9)

εy = − ν

Eσz (12.10)

εz =1E

σz (12.11)

Of course we are not limited to applying stress in only one direction. We may apply stresses in the

x, y, and z directions simultaneously. Because linear elasticity is a linear theory, the response of

a solid to stress in all direction is just the superposition of the individual responses. We can thus

generalize Hooke’s law for tensile loading to get

εx =1E

[σx − ν(σy + σz)] (12.12)

εy =1E

[σy − ν(σx + σz)] (12.13)

εz =1E

[σz − ν(σx + σy)] (12.14)

Besides tensile loading, polymers may also be subjected to shear loading and shear stresses.

Unlike tensile loading, shear loading causes no change in volume and thus the responses due to

shear loading in the three directions are uncoupled. Hooke’s law for shear loading are

γxy = γyx =τxy

G(12.15)

γxz = γzx =τxz

G(12.16)

γyz = γzy =τyz

G(12.17)

where γij are shear strains and G is the shear modulus. For isotropic materials, the shear modulus

is related to E and ν by

G =E

2(1 + ν)(12.18)


The above discussion was all about isotropic materials. In polymer science, one often encoun-

ters anisotropic materials. Two examples are composite materials and oriented polymer specimens.

For anisotropic materials, the mechanical properties depend on the direction of the applied load.

Dealing with general anisotropy is very complex. Here we introduce a single example of an or-

thotropic material. An orthotropic material has different properties in the x, y, and z directions,

but still has no coupling between shear and tensile deformation. Orthotropic materials are usually

the most complex situation encountered in analysis of polymeric specimens. For an orthotropic

material, Hooke’s law becomes

εx =σx

Ex− νxyσy

Ex− νxzσz

Ex(12.19)

εy = −νxyσx

Ex+

σy

Ey− νyzσz

Ey(12.20)

εy = −νxzσx

Ex− νyzσy

Ey+

σz

Ez(12.21)

γxy =τxy

Gxy(12.22)

γxz =τxz

Gxz(12.23)

γyz =τyz

Gyz(12.24)

where the Ei’s are the tensile moduli for loading in the i direction, the Gij ’s are the shear moduli

for shear deformation in the i-j plane, and the νij ’s are the generalized Poisson ratios; νij is the

ratio of the strain in the j direction to the strain in the i direction when the specimen is under

uniaxial load in the i direction.

This discussion of elasticity theory has been extremely brief, It should suffice for the material

covered in this chapter. If you need more information, you may want to refer to any text of

mechanics of materials or elasticity theory.

12.3 Stress-Strain Curves

A stress-strain curve is measured by monotonically increasing stress or strain and measuring the

other quantity. Plotting the results gives a stress strain curve. Consider first a tensile stress-strain

curve measured by pulling in one direction. As the sample extends, the cross-sectional area changes

due to Poisson contraction effects. The “true” stress and “true” strain should be defined in terms

of the current sample dimensions. That refinement, however, is rarely used when characterizing

polymers. Instead, we measure engineering stress and stress. Engineering stress is the total load

divided by the original cross-sectional area. Engineering strain is the total elongation divided by the

original length. Some typical polymer engineering stress-strain curves are illustrated in Fig. 12.1.

Curve a in Fig. 12.1 is for a linear elastic polymer. The stress increases linearly with strain (or

very nearly linearly) until ultimate failure. Curve b is for a polymer that yields and then breaks.

12.3. STRESS-STRAIN CURVES 145

a

b c

d

Stre

ss

Strain

Figure 12.1: Some typical stress-strain curves. a: linear elastic; b. nonlinear with yield; c. yieldingfollowed by necking and high elongation; d. rubber-elastic curve.

Yielding shows up as deviations from linearity. In metals, yield stress is often defined as some fixed

amount of deviation from linearity. In polymers, this approach is often unworkable and yield stress

is, by convention, the peak stress. A peak stress will not always be observed, but it shows up barely

in curve b and clearly in curve c. Thus curve c is for a material that yields and then continues

to elongate. After yielding, a neck will form. Then the neck region elongates or grows, sometimes

to several hundred percent strains. Eventually the polymer chains can align in the necked region.

Upon alignment, the material will get stiffer and soon break. Figure 12.2 shows the process of neck

formation followed by neck elongation.

Curve d in Fig. 12.1 is typical for rubbers or for materials that exhibit rubber elasticity. Rubber

elastic deformation is characterized by high recoverable extension; there is no necking. At the

highest elongations, the polymer chains can align and crystallize. These effects will cause a stiffening

and the curve turns up before failure.

12.3.1 Stress-Strain Properties

The raw curves in Fig. 12.1 tell you much about the mechanical properties of polymers. It is useful,

however, to reduce the data in the raw curves to a set of mechanical properties. We next consider

typical properties measured from stress-strain curves, how they are measured, and what they mean.

The stiffness or modulus of a polymer is usually taken from the slope of the stress-strain curve at

very low strains — the initial slope. As the strain increases, the meaning of the slope of the stress-

strain curve decreases. The problem is that engineering stress and strain are no longer close to true

stress and strain. Thus, for example the negative slopes in curve c have no physical meaning. If we


YieldingNecking

Figure 12.2: The appearance of a specimen that necks and follows curve c in Fig. 12.1.

did plot true stress-true strain curves, the slopes would give the strain dependence of the stiffness.

We would also find that the true stress never decreases as in curve c. A high slope is a hard or

a stiff sample; a low slope is a soft or a compliant sample. The modulus of unoriented polymers

are almost universally in the range of 1 to 4 GPa. Polymers are all made up from similar organic

bonds and thus we never expect to find unoriented polymers with moduli significantly outside this

range. If a polymer is oriented, however, such that the backbone chains are all aligned with the

applied load, it is possible to have a very high modulus when the load is applied in the direction

of the orientation. Highly oriented polymers can have moduli as high as 300 GPa which is higher

than the modulus of steel. Unfortunately, you do not get something for nothing when orienting a

polymer. The increase in modulus in oriented polymers applies only to loading in the direction of

orientation. The oriented polymer is anisotropic and the modulus in the other directions decreases

as the orientation-direction modulus increases.

The strength of a polymer is usually defined as the maximum stress reached during the stress-

strain curve, or the stress when the sample breaks. Strength is often a poor way to characterize

polymer failure properties. The results are usually highly variable and are also a strong function of

the sample size and shape. The strengths of unoriented polymer are in the range of 10 to 100 MPa.

Highly oriented polymers can have strengths up to 4 GPa.

A formal definition of yielding is the point at which the stress-strain response becomes nonlinear

and irreversible. In polymers, it is difficult to define irreversibility, because if you wait long enough,

the viscoelastic nature of polymers can make extreme nonlinearities reversible. By convention,

yield strength in polymers is often assigned to the peak of the stress-strain curve that occurs prior

to necking. Measuring the yield strength of polymers that do not neck can be a problem. Many

such polymers, however, can be induced to neck when tested in sufficiently thin specimens and at

sufficiently slow rates. High yield strength is associated with brittle and linear elastic materials;


low yield strength is associated with ductile materials. Like strengths, yield strengths of unoriented

polymers are usually 10 to 100 MPa. Highly oriented polymers will not yield and thus do not have

a yield strength.

Elongation to break is the total elongation at the time the sample breaks. High elongation

means a ductile failure; low elongation means a brittle failure. Elongation is a nearly useless

material property. It is highly dependent on sample size and testing conditions. High elongation

is obtained by promoting necking. Necking is promoted by slow testing rates and by using thin

samples. Also many polymer applications will not involve large strains. If the designed part will

only experience 5% strain, who cares if the polymer can extend to 250% or to 1000%? In summary,

elongation to break is easy to measure and is commonly reported. Without additional experiments

and additional information about testing conditions, it is not a useful material property.

Toughness is defined as the amount of energy or work required to break the sample. Because

work is force × distance, stress × strain is work per unit volume. Thus the area under the stress-

strain curve is the total work per unit cross-sectional area to break the sample. This work is often

equated with toughness. High area is for tough samples or ductile samples; low area is for brittle

samples. Although toughness is an important material property, the area under the curve is not a

good way to measure the toughness. The main problem is that the total area is a sensitive function

of the elongation to break. As stated above, elongation to break is very sample size and test

condition dependent. Useful values of toughness are normally measured by techniques of fracture

mechanics and not by areas under the curve.

Unlike most metals, the stress-strain behavior of polymers is very sensitive to testing rate and

temperature. Large effects can be seen at ordinary temperatures and testing rates. The effect

of temperature is illustrated in Fig. 12.3. At low temperatures, the polymer becomes hard and

brittle. As the temperature increases, polymers typically become softer and more ductile. In short,

one polymer can exhibit all the possible stress-strain curve types by changing the temperature.

The effect is connected with the glass transition. Well below the glass transition temperature the

polymer is immobile and behaves as a brittle material. As the temperature is raised and the glass

transition is approached or passed, the material becomes more ductile and softer.

The testing rate effect is illustrated in Fig. 12.4. The rate effect is the opposite of the tempera-

ture effect. Increasing the rate is like decreasing the temperature; it causes the polymer to become

harder and more brittle. This effect is again associated with the glass transition. At slow rates the

polymer has time to rearrange and on the time scale of the test, it can be said to be effectively

above the glass transition temperature. At high rates, no motion is possible and it can be said to

be effectively below the glass transition temperature.

The similarity between temperature effects and rate effects suggests that time and temperature

are related. We will see later in the chapter that we can exploit the relation using a principle

known as time-temperature superposition. Suppose you want to do a very slow test at room


Low Temp

Room Temp

High Temp

Increasing Temp

Figure 12.3: The effect of temperature on the stress-strain curve of polymers.

High Rate

Moderate Rate

Low Rate

Increasing Rate

Figure 12.4: The effect of testing rate on the stress-strain curve of polymers.


temperature. One method would be to use the very slow rate that you are interested in. This

slow rate, however, might be very slow and in fact might involving testing over many years. A

better method is to use the time-temperature superposition principle which states that the effect of

slowing done the rate by some amount is equivalent to lowering the temperature by some amount.

You can therefore get reasonable very slow rate, room temperature data by doing higher rate tests

at a higher temperature. The only problem is knowing how much to increase the temperature to

simulate a given slow rate. This subject will be discussed latter.

When measuring stress-strain curves, there are many possible sources of error. One common

error is slippage of the specimen in the grips. If the specimen slips, the measured grip displacement

will be different that the actual specimen displacement resulting in large errors in the measured

strain. For this reason, in quantitative work it is important to measure displacements or strains

directly on the sample by using strain gages or extensometers instead of measuring strain from

grip displacement. Grip slippage will not affect measured load and therefore loads will always be

accurate assuming the load cell in accurate and well calibrated. Specimen geometry and quality

can have a large effect of the stress-strain curve. If necking is part of the specimen deformation,

the amount of deformation will be very dependent on sample thickness. In other words necking is a

specimen property and only partially a material property. Although some materials are more likely

to neck than other materials, the amount of necking will be determined by sample geometry. Thin

samples will neck more than thick samples. Sample geometry and quality can dramatically affect

ultimate failure loads. For example, samples with stress concentrations, such as scratches on the

surface, can prematurely fail at the sample imperfections. Thus, reproducibly measuring sample

strength requires great attention to specimen design and quality.

Polymer stress-strain curves are often nonlinear. The nonlinearity can be caused by plastic

deformation (yielding) or by rate/temperature effects (low rate or high temperature promotes

curvature). It can be difficult to separate plastic affects from viscoelastic effects. Additional

experiments, besides stress-strain experiments, may be needed to fully characterize the mechanical

response of a polymer.

12.3.2 Uses of Stress-Strain Properties

The reason for measuring stress-strain properties, such as modulus, strength, or toughness, is

because they are useful in polymer applications. We next give a few comments about using stress-

strain properties.

Stiffness or modulus is a key input into Hooke’s laws and they are used to calculate deformations

under load. The importance of knowing deformation for polymer design problems should be self

evident. An important aspect of stiffness is that it is a function of temperature. A common

and quick stiffness/temperature experiment is called the heat deflection temperature. In brief, a


cantilever beam is loaded with a fixed amount of load and the deformation of the beam is measured

as a function of temperature. A large deflection means that the polymer stiffness has become low

and will probably no longer be suitable for stiffness-critical applications. In general polymers should

only be used at temperatures well below the heat deflection temperature. In amorphous polymers,

the heat deflection temperature is always below, but near Tg; in semicrystalline polymers, the heat

deflection temperature can be above Tg, but it is always below Tm.

Failure properties describe the conditions under which a polymer will either break or become

so distorted that it can be considered failed. Of the previously discussed properties, strength, yield

strength, elongation to break, and toughness are failure properties. Although failure properties are

important, a stress-strain experiment is not the ideal way to study failure. Better information about

failure, especially about toughness, can be derived by using the techniques of fracture mechanics.

12.3.3 Uniaxial Compression Testing

If instead of loading the sample in tension, you load it in compression, you can record a compression

stress-strain curve. Many features of compression stress-strain curves are similar to tensile stress

strain curves. You can, for example measure stiffness, strength, and toughness. There are, how-

ever, some important differences. First yielding will be very different in compression. In particular,

there will be no necking and there will be no opportunity for very large elongations (the maximum

compression strain is -100% and this amount of strain means the sample has disappeared). Sec-

ond, when testing samples in compression, you are often limited by buckling rather than by the

true compressive strength of the material. Buckling is a geometric instability and not a material

property. If buckling can be prevented, most polymers will be able to sustain 2 to 3 times more

stress in compression than they can in tension.

12.3.4 Shear Testing

This topic may be discussed and involves loading the sample in shear and measuring shear strain.

12.3.5 Hydrostatic Compression

The pressure-dependent dilatometry experiments discussed in Chapter 10 involved immersing a

sample in a confining liquid and applying a pressure. This type of pressure application results in a

form of stress known as hydrostatic compression. In a state of hydrostatic compression, all three

axial stresses are equal and the shear stresses are all zero. Thus

σx = σy = σz = −p or p = −13(σx + σy + σz) (12.25)

where p is the pressure. Plugging this stress state into the generalized Hooke’s laws gives:

εx = εy = εz = −p(1− 2ν)E

(12.26)

12.4. FRACTURE AND IMPACT TESTING 151

and the change in volume is

∆V = (ex + ey + ex)V0 = −3p(1− 2ν)E

V0 (12.27)

In the discussion of dilatometry in Chapter 10, we defined the isothermal compressibility as

K = − 1V

(∂V

∂P

)T

= −∆V

V0p(12.28)

Substituting ∆V for hydrostatic compression gives K in terms of E and ν:

K =3(1− 2ν)

E(12.29)

Hydrostatic compression call also be viewed as plotting pressure vs. volumetric stain. By

analogy with tensile strain, we define volumetric strain as ∆V/V0. The volumetric strain/pressure

relation then becomes

−p =1K

∆V

V0(12.30)

Comparing this expression to the one-dimensional Hooke’s law, we see that 1/K plays the role of

modulus. The quantity 1/K is known as the bulk modulus of the material. In terms of E and ν:

1K

=E

3(1− 2ν)(12.31)

Note the factor of 1− 2ν in the denominator. Many rubbers which are very soft also have ν ≈ 1/2.

When ν is equal to 1/2, the bulk modulus is predicted to be infinite. In reality, ν is never quite 1/2

and it turns out that the increase in ν in soft materials is roughly compensated by the decrease in

E. In the end the variability of bulk moduli over a wide range of polymer materials is not large.

This is reasonable because most polymers are made up of units of similar chemical composition and

the compression of these units under hydrostatic compression in not overly sensitive to the nature

of how these units are connected together. Note this statement only applies to hydrostatic stress.

When the stresses in one or more directions are significantly different, then the nature in which the

chemical units are connected becomes more important.

Another unique aspect of hydrostatic stress is the absence or near absence of yielding. In

general, yielding requires shear stresses. When none are present, as in hydrostatic stress, the yield

strength is predicted to be infinite. In compression, a material under hydrostatic stress will not

yield. In hydrostatic tension, all polymers will be brittle and will not yield, or at least will not

yield until high stress.

12.4 Fracture and Impact Testing

This topic may be discussed and inolve loading a sample until in cracks or hitting with an object

in impact. The analysis of the results can be complicated.


12.5 Time Dependence or Viscoelasticity

In elastic materials (linear or nonlinear) the strain is a function of the stress only, or the stress is

a function of the strain only:

ε = f(σ) (12.32)

In polymers, the stress and strain are often time dependent; in other words, polymers are viscoelas-

tic:

ε = f(σ, t) (12.33)

The are several ways to observe and quantify time dependence of the mechanical properties of

polymers. One method is a creep test. In a creep test, the stress is held constant, and the strain

is measured as a function of time. Let’s assume that the stress-time dependencies of the strain can

be separated. In a creep test we then write the strain as the product of two functions, J(t) and

f(σ). One depends only on time and the other depends only on stress:

ε(t) = J(t)f(σ) (12.34)

We next assume the material is linear viscoelastic, that is, we assume that f(σ) includes only terms

that are linear in σ. Because the coefficient of the linear term can be included in J(t), we can write:

e(t) = J(t)σ (12.35)

J(t) is a new material property; it is called the creep compliance. It is time dependent and it can

be measured in a creep test. The stress is set to some constant value, σ0, and the strain is measured

as a function of time. The creep compliance is

J(t) =ε(t)σ0

(12.36)

If the material was linear elastic then ε = (1/E)σ. The creep compliance of a linear elastic material

is just 1/E or the compliance.

The above analysis describes the creep behavior of a linear viscoelastic material. But, what

does linear viscoelastic mean? By the equation above, J(t) was written to be independent of stress;

in other words, you would find the same J(t) no matter what value of σ0 was used in the creep

test. To the extent that this is true, the material under test is linear viscoelastic. When J(t) is

observed to depend on the level of constant stress, the material is said to be non-linear viscoelastic.

Most analyses of polymer viscoelasticity assume linear viscoelasticity. That assumption is good for

small strains. It could be called the analog of the small-strain assumption in linear elasticity.

The complementary experiment to a creep test is a stress relaxation test. In the stress relaxation

test, the strain is held constant, and the stress is measured as a function of time. In analogy with

the development above, the stress relaxation test is described by:

σ(t) = G(t)ε0 or G(t) =σ(t)ε0

(12.37)

12.5. TIME DEPENDENCE OR VISCOELASTICITY 153

where G(t) is the relaxation modulus and ε0 is the constant strain level used during the test.

For a linear elastic material, the relaxation modulus would be just the modulus E. For a linear

viscoelastic material, the relaxation modulus depends on time, but not on the level of applied strain,

ε0. For a nonlinear viscoelastic material, the relaxation modulus depends both on time and on the

level of the applied strain.

Creep and stress relaxation experiments can be done using a variety of loading conditions.

Some examples include tensile loading, shear loading, torsional loading, compressional loading, and

even hydrostatic compression. You can imagine that in hydrostatic compression where yielding

and therefore most flow is eliminated, that there is much less time dependence in the mechanical

properties. The other loading conditions, however often show significant time dependencies.

Time dependence of mechanical properties is an important piece of information about the

response of polymers to stress. Polymers are often used in applications that may experience long

term loads. It is important to know not only if the polymer can sustain a given load but if it

can sustain the load for a long period of time. For an example consider the recent development of

polyethylene ropes. By orienting PE, it is possible to create stiff and strong fibers. The fibers can

be woven into ropes. As with the fiber, the rope is very stiff and very strong in short time scale

static tests. Because it is made from PE, the rope is potentially cheap. There are many potential

problems, however, that have not yet been considered. First, PE is above Tg at room temperature

and therefore can be expected to be strongly viscoelastic. While the rope may be strong, it also

may rapidly creep. A large amount of creep would make the rope useless. The viscoelastic effect

will also be compounded by high levels of stress. A high uniaxial stress will increase the volume;

it will increase the volume by the amount equivalent to a hydrostatic tension equal to 1/3 of the

axial tensile stress. In other words, the uniaxial tensile stress can be considered to be a negative

pressure. While positive pressures always increase Tm and Tg, a negative pressure will cause Tm

and Tg to decrease. Therefore at a given temperature, the rope will be farther above Tg when

under stress than it will be when unloaded. If the stress is high enough the rope may even melt by

the application of load. This lowering of Tg and Tm will have the effect of magnifying any possible

viscoelastic effects.

12.5.1 Creep Experiments

Depending on temperature, it can take a long time to run a creep experiment that shows a significant

amount of creep. Therefore, it is typical to plot the creep compliance as a function of log time.

Some typical creep data at one specific temperature are given on the right side of Fig. 12.5. The

time scales in such plots can be so long that it is impractical to rely on isothermal creep tests. To

get the long time scales that might be of interest, we instead use the principle of time-temperature

superposition that was introduced earlier in this chapter. We call the temperature we care about the


J(t) J(t)

log t log tr

Figure 12.5: Left side is a set of creep experiments as a function of temperature. The lower curves are forlower temperatures; the higher curves are for higher temperatures. The right side is the master curve forcreep compliance. It is obtained from the creep data on the left by using time-temperature superposition.

reference temperature. By the principle of time-temperature superposition, the creep at very long

times at the reference temperature, can be expected to be equivalent to the creep at shorter times

at some higher temperature. Conversely, the creep at short times at the reference temperature,

can be expected to be equivalent to the creep at longer times at some lower temperature. A set of

creep experiments thus consists of measuring creep over a relatively short amount of time (105 to

106 seconds) at a series of different temperatures. The temperatures will probably be both above

and below the reference temperature. A typical set of data for creep as a function of time at many

temperatures is shown in the left side of Fig. 12.5.

The question is how do we relate the times at one temperature to the times at another tem-

perature? If the temperatures are not too far apart, and the measurement time is not too short,

than the creep data can be expected to overlap. That is, the creep compliance near the end of the

measurement time for one temperature will be identical or higher than the creep compliance near

the beginning of the measurement time at a slightly higher temperature. It has been experimentally

observed that the log time plots can usually be overlapped by a simple linear shift along the time

axis. A typical result is illustrated in the right side of Fig. 12.5. The master curve was obtained

from the raw data on the left by fixing the curve at the reference temperature and then shifting

all other curves horizontally until they achieve the maximum overlap with data at the reference

temperature.

The master curve on the right side of Fig. 12.5 is plot as a function of log tr where tr is called

the reduced time. Reduced time is actual time for experiments run at the reference temperature.


log aT

TREF

Temperature

log

a T

0

Figure 12.6: Typical shift factor plot for log aT as a function of temperature.

For experiments at other temperatures, reduced time is related to real time by:

log tr = logt

aT= log t− log aT (12.38)

To move a curve at a given temperature in the plot of the left half of Fig. 12.5 to the master curve

on the right half of Fig. 12.5, the curve is shifted horizontally by the amount − log aT ( the curve

is shifted to the right if log aT < 0 and to the left if log aT > 0). The term log aT is thus known

as the shift factor. Because for T > Tref , the curves are shifted to the right, log aT is less than

zero. At Tref , log aT is equal to zero. For T < Tref , log aT is greater than zero. A schematic plot

of log aT versus T is shown in Fig. 12.6. Plots such as Fig. 12.6 quantify the time-temperature

superposition principle. They tell you how to convert the time at one temperature to the time at

another temperature.

We next examine the physical meaning of the creep compliance plot of the right side of Fig. 12.5.

At early times J(t) is increasing. This region is known as viscoelastic response. If the experiments

included data to very short times (or low temperatures), J(t) would level off at a constant value

equal to 1/E for the material at that time (or temperature). This region of the curve would be the

elastic response region. The plateau region at middle times is known as the rubbery plateau. In this

region the polymer acts like a rubber or a polymer that is above the glass-transition temperature.

The transition of J(t) from a low value to the plateau is a measurement of the glass-transition on

a time-scale plot. The time to reach the rubbery plateau is termed the glass-transition time. At

late times, J(t) begins to rapidly increase. The polymer is now acting like a liquid and the region

is called the viscous flow region. If the material is cross-linked and thus prevented from flowing,

this terminal region would not occur.


12.5.2 Free-Volume Theory

With sufficient data, log aT can be found experimentally by shifting creep curves to obtain the max-

imum amount of overlap. This measured log aT can then be used to quantify the time-temperature

superposition principle. It would be nice, however, to be able to predict log aT . There is an empir-

ical equation, the WLF equation (for Williams-Landel-Ferry), and that equation can be justified

with free-volume theory. The theory was originally developed for amorphous polymers and the

glass-transition in those polymers. Because semicrystalline polymers also have amorphous regions,

the theory can be applied to semicrystalline polymers. The most quantitative results, however, are

obtained when working with amorphous polymers.

An amorphous polymer can be considered to be made up of occupied volume and free volume.

As the temperature is changed, the free volume and the occupied volume will both change in specific

volume or density. The occupied volume might be expected to expand with an expansion coefficient

appropriate to a solid polymer or close to the expansion coefficient that exists below Tg. Above

Tg, the solid part or occupied volume should continue expanding at the same rate and the excess

expansion observed above Tg can be attributed to expansion in the free volume. Expansion of the

total, free, and occupied volume is illustrated schematically in the dilatometry data in Fig. 12.7.

At any given temperature, the fraction of the total volume that is free volume is

f(T ) =Vf

Vf + V0≈

Vf

V0(12.39)

where the approximation is based on Vf << V0. Around Tg and above Tg, the fraction of free

volume can be expressed as

f(T ) = fg + αf (T − Tg) (12.40)

where fg is the fraction of free volume at Tg and αf is an expansion coefficient for the fraction

free volume (see Fig. 12.7). αf is approximately αm − αg, or the difference between the thermal

expansion coefficients of the polymer above (αm where m stands for melt) and below (αg where g

stands for glass) Tg. According to free volume theory, viscosity can be related to the fraction of

free volume by

ln η(T ) = A +B

f(T )(12.41)

This equation states that as the fraction of free volume increases, the viscosity decreases. This

equation is known as the Doolittle equation.

The shift factor can be expected to be related to the relative viscosities at two temperatures.

If we select Tg as the reference temperature, then the shift factor at any other T (≥ Tg) is given by

ln aT = lnη(T )η(Tg)

= ln η(T )− ln η(Tg) = B

(1

f(T )− 1

fg

)(12.42)


V

Vtotal

Vfree

Voccupied

αf

Temperature

Figure 12.7: Plots of total volume and occupied volume for an amorphous polymer both above and belowthe glass-transition temperature.


Substituting in f(T ) from above, this expression can be rearranged to get:

log aT =−(

B2.303fg

)(T − Tg)(

fg

αf

)+ (T − Tg)

(12.43)

or

log aT =−C1(T − Tg)C2 + (T − Tg)

(12.44)

where C1 and C2 are two constants for a given polymer. This shift-factor equation is the WLF

equation.

There is nothing special about the choice of Tg in the above derivation. In fact any reference

temperature, Tref , could be used as long as the fraction of free-volume can be expressed by the

equation give earlier. In general, this requirement limits T ’s to near Tg and equations like the WLF

equation can only be expected to be quantitative near and above Tg. There is an advantage to

choosing Tg as the reference temperature. If we restrict the reference temperature to Tg, it has

been observed that the constants in the expressions above (C1 and C2) appear to be universal for

all polymers. In other words, the constants C1 and C2 are relatively invariant to the nature of the

polymer. For a wide variety of polymers the shift factor is given by:

log aT =−17.44(T − Tg)51.6 + (T − Tg)

(12.45)

A plot of this equation would be similar to the plot in Fig. 12.6. The universal constants come from

fitting the WLF equation to a wide variety of polymers. We note that the variation from polymer

to polymer can be significant. In practice, it is recommended that the WLF universal constants

only be used as a last resort. It is better to determine shift factors experimentally or to find WLF

constants specific to the polymer of interest.

The WLF equation gives us information about the effect of rate on Tg. The predicted effect

is about a 3◦C change in Tg for each order of magnitude change in rate. The reasons behind this

result will be discussed in class.

In some cases, the shift factors cannot be observed experimentally. This can occur when the

creep compliance remains constant over a long period of time. The shift can only be measured if

there are features (i.e., changes) in the creep compliance data at all temperatures. If there is no

change at several temperatures, there is no way of knowing how far to shift.

12.5.3 Stress Relaxation Experiments

The complementary experiment to creep experiments is stress relaxation experiments. Like creep

tests, long-term stress relaxation tests are done using time-temperature superposition. The left side

of Fig. 12.8 shows several stress relaxation tests at various temperature. The higher curves are for

lower temperatures and the lower curves are for higher temperatures. The right side of Fig. 12.8


G(t) G(t)

log t log tr

Glassy

Viscoelastic

Rubbery

ViscousFlow

Figure 12.8: Left side is a set of stress relaxation experiments as a function of temperature. The highercurves are for lower temperatures; the lower curves are for higher temperatures. The right side is themaster curve for relaxation modulus. It is obtained from the raw data on the left by using time-temperaturesuperposition.

shows the master plot for relaxation modulus plot that was obtained by appropriately shifting the

raw data in the left side plot.

The master relaxation modulus curve resembles a flipped over version of the master creep

compliance curve. Furthermore the various regions of the relaxation modulus curve correspond to

the regions of the creep compliance curve. The earliest time data, where G(t) is flat, is the glassy

region. In this region the response is elastic and G(t) is approximately equal to the static tensile

test modulus, E. The next region is called the viscoelastic region. The middle region is called

the rubbery plateau. The transition from the glassy region to the rubbery plateau is the glass

transition visualized on a time scale. The final or terminal region represents viscous flow.

12.5.4 Dynamic Mechanical Analysis

While creep and stress relaxation tests can be used to characterize viscoelastic materials, they can

often be impractical because they require much time. They require several experiments at various

temperatures, each of which can take 105 to 106 seconds (1 to 10 days). A more common technique

for characterizing viscoelastic polymers is dynamic mechanical analysis (DMA). In DMA, instead

of keeping the strain constant, as in a stress relaxation test, the strain is oscillated at a constant

amplitude, ε0, and a constant frequency, ω. Thus the strain input is

ε(t) = ε0 sinωt (12.46)


As in most physical response problems, when the system is driven by a constant frequency, it will

respond at the same frequency but may have a phase difference and a different amplitude. In other

words, the stress will be given by:

σ(t) = σ0 sin(ωt + δ) = G(ω)ε0 sin(ωt + δ) (12.47)

where G(ω) is the dynamic modulus and δ is the phase angle.

In analogy with other definitions of moduli, the dynamic modulus G(ω) is the ratio of stress to

strain; in this case we take the ratio of the amplitudes of the stress and the strain:

G(ω) =σ0

ε0(12.48)

G(ω) or the dynamic modulus defines the modulus of a viscoelastic material. The phase angle,

δ, describes the viscous character of the material. If δ = 0, the stress is in phase with the strain and

the material is totally elastic. If δ = π/2, the stress is completely out of phase and is completely

viscous. In general, δ will be between 0 and π/2 and the material will be viscoelastic.

Some people prefer other ways to describe DMA data. To see alternative representations, we

can expand the stress function:

σ(t) = G(ω)ε0 cos δ sinωt + G(ω)ε0 sin δ cos ωt (12.49)

= G′(ω)ε0 sinωt + G′′(ω)ε0 cos ωt (12.50)

where

G′(ω) = G(ω) cos δ and G′′(ω) = G(ω) sin δ (12.51)

G′(ω) is known as the storage modulus. It is proportional to that part of the strain energy that

is stored and then recovered during one cycle. It also describes the elastic part of the stress

response, or the part of the stress in phase with the strain. G′′(ω) is known as the loss modulus.

It is proportional to that part of the strain energy that is lost during a cycle due to viscoelastic

dampening. It also describes the viscoelastic part of the stress response, or that part of the stress

that is out of phase with the strain.

The above expressions give G′(ω) and G′′(ω) in terms of G(ω) and δ. The inverse relations can

easily be derived:

G(ω) =√

G′(ω)2 + G′′(ω)2 (12.52)

tan δ =G′′(ω)G′(ω)

(12.53)

These two relations are reminiscent of complex numbers. It is common practice to define a complex

modulus:

G∗(ω) = G′(ω) + iG′′(ω) (12.54)


n

G'(ω)

tan δG'(ω

) or t

an δ

Tg

Temperature

Figure 12.9: Typical DMA data for an amorphous polymer.

The storage modulus is the real part of G∗(ω), the loss modulus is the imaginary part of G∗(ω), the

dynamic modulus is the amplitude of G∗(ω), and tan δ is the tangent of the angle between G∗(ω)

and the positive real axis when G∗(ω) is plotted in the complex plane.

We have defined four types of dynamic properties — G(ω), δ, G′(ω), and G′′(ω). Only two of

these are independent and it is only useful or rather necessary to measure two of them. Common,

but not universal practice, is to measure G′(ω) and tan δ. Some typical DMA data for an amorphous

polymer as a function of temperature are plotted in Fig. 12.9. Below Tg, G′(ω) is fairly constant,

and because tan δ and therefore also G′′(ω) are both small, G′(ω) is close to the modulus of the

material that would be measured in a static test. At Tg, G′(ω) drops dramatically, while tan δ goes

through a peak. The peak is often used to measure the glass transition temperature. Peaks in tan δ

can also be observed below Tg. Observing these smaller peaks is one of the best ways of observing

sub-glass relaxations.

Some typical DMA data for semicrystalline polymers as a function of temperature are given

in Fig. 12.10. Below Tg these data are similar to the amorphous polymer data including peaks in

tan δ at sub-glass relaxations and at Tg. At Tg, however, G′(ω) does not drop nearly as far. The

crystal fraction contributes stiffness and prevents overall viscous flow. At Tm, the crystals melt

and a second drop in G′(ω) results in a very low final value. The relative drops at Tg and Tm are

sensitive to the percent crystallinity.

The stress relaxation experiments measure a relaxation modulus known as G(t). DMA experi-

ments measure a new viscoelastic modulus called the complex dynamic modulus of G∗(ω). G(t) and

G∗(ω), however, are not independent quantities. It is easy to show that G∗(ω) can be calculated

from a Fourier transform of G(t). Thus G∗(ω) is the frequency domain version of the relaxation


n

G'(ω)

tan δ

G'(ω

) or t

an δ

Tg Tm

Temperature

Figure 12.10: Typical DMA data for an semicrystalline polymer.

modulus.

In analogy with creep experiments, we could imagine dynamic stress experiments in which the

sample is subjected to an oscillating stress and we measure the amplitude and the phase angle of

the strain response. These dynamic experiments will give a dynamic complex compliance, J∗(ω),

which can be shown to be the Fourier transform of J(t). Like dynamic modulus, J∗(ω) can be

broken up to component functions of J ′(ω) (the storage compliance), J ′′(ω) (the loss compliance),

J(ω) (the dynamic modulus), and δ (phase angle). The analysis of dynamic stress experiments

is identical to the above analysis of DMA experiments. Most commercial instruments are limited

to oscillating strains or displacements. Thus must dynamic tests on polymers measure dynamic

modulus instead of dynamic compliance.

Problems

12–1. Show that Eq. (12.27) is valid for any stress state and not just for hydrostatic compression.

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Polymer Characterization