Introduction to Polymer Viscoelasticity E-Book

Montgomery T. Shaw

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 536

Veröffentlichungsjahr: 2018

Beschreibung

Completely revised and updated, the fourth edition of this classic text continues to offer the reader a thorough understanding of viscoelastic behavior, essential for the proper utilization of polymers.

• Explains principles, corresponding equations, and experimental methods with supporting real-life applications
• Adds coverage of measurement techniques (nano-indentation, atomic force microscopy (AFM), and diffusing wave spectroscopy (DWS)), biopolymer viscoelasticity, and the relationship between mechanical polymer properties and viscoelastic functions
• Has two new ections to address modern areas of viscoelastic measurement: large amplitude oscillatory shear (LAOS) and microrheology
• Includes problems in the text and an Instructor’s Manual (including solutions) available for adopting professors
• Prior edition reviews: "The book is clear written and…[is] appropriate for students in introductory undergraduate courses and for others wanting introduction to the fundamentals of the subject." (CHOICE, December 2005); "This book is invariably well written, logically organized and easy to follow...I highly recommend this book to anyone studying polymer viscoelasticity." (Polymer News, December 2005)

Leseprobe

Table of Contents

Cover

Dedication

Preface to the Fourth Edition

Preface to the Third Edition

Preface to the Second Edition

Preface to the First Edition

About the Companion Website

Chapter 1: Introduction

PROBLEMS

GENERAL REFERENCE TEXTS

REFERENCES

Chapter 2: Phenomenological Treatment of Viscoelasticity

A. ELASTIC MODULUS

1. LINEAR‐MOTION GEOMETRIES

REFERENCES

Chapter 3: Viscoelastic Models

A. MECHANICAL ELEMENTS

B. DISTRIBUTION

S OF RELAXATION AND RETARDATION TIMES

C. MOLECULAR THEORIES—THE ROUSE MODEL

D. APPLICATIONS OF FLEXIBLE‐CHAIN

MODELS TO SOLUTIONS

E. THE ZIMM MODIFICATION

F. EXTENSION TO BULK POLYMER

G. REPTATION

APPENDIX 3‐1: MANIPULATION OF THE ROUSE MATRIX

PROBLEMS

REFERENCES

Chapter 4: Time–Temperature Correspondence

A. FOUR REGIONS OF VISCOELASTIC BEHAVIOR

B. TIME–TEMPERATURE SUPERPOSITION

C. MASTER CURVES

D. THE WLF

EQUATION

E. MOLECULAR INTERPRETATION OF VISCOELASTIC RESPONSE

PROBLEMS

REFERENCES

Chapter 5: Transitions and Relaxation in Amorphous Polymers

A. PHENOMENOLOGY OF THE GLASS TRANSITION

B. THEORIES OF THE GLASS TRANSITION

C. STRUCTURAL PARAMETERS AFFECTING THE GLASS TRANSITION

D. RELAXATIONS IN THE GLASSY STATE

E. RELAXATION PROCESSES IN NETWORKS

F. BIOPOLYMER VISCOELASTICITY

PROBLEMS

REFERENCES

Chapter 6: Elasticity of Rubbery Networks

1. DERIVATION

2. ENERGY CONTRIBUTION

1. EFFECT OF DEGREE OF CROSSLINK

ing

2. EFFECT OF SWELLING

3. EFFECT OF FILLER

s

4. EFFECT OF STRAIN‐INDUCED CRYSTALLIZATION

PROBLEMS

REFERENCES

Chapter 7: Dielectric and NMR Methods

A. DIELECTRIC METHODS

B. NUCLEAR MAGNETIC RESONANCE

METHODS

PROBLEMS

REFERENCES

Answers to Selected Problems

CHAPTER 2

CHAPTER 3

CHAPTER 4

CHAPTER 5

CHAPTER 6

CHAPTER 7

List of Major Symbols

List of Files on the Website

INTRODUCTION

LIST OF FILES

Author Index

Subject Index

End User License Agreement

List of Tables

Chapter 2

Table 2-1 Test Geometries for Instruments that Generate Axial (Linear) Motion

Table 2-2 Geometries for Instruments that Generate Rotational Motion

Chapter 3

Table 3-1 Behavior in Extension of Simple Viscoelastic Models in Various Experiments.

a

Table 3-2 Behavior of Generalized Maxwell and Voigt–Kelvin Models in Various Experiments

Table 3-3 Transition Region Slope for Several Polymers

16

Chapter 4

Table 4-1 WLF

Parameters

a

Chapter 5

Table 5–1 Pressure Dependence of

T

g

Table 5–2 Glass Transition Temperatures of Selected Polymers

Chapter 6

Table 6-1 Thermoelastic Data of Selected Elastomer

s at 30 °C

Chapter 7

Table 7-1 Methods for Dielectric Measurements

List of Illustrations

Chapter 1

Figure 1-1 Deformation of samples made from plastic vs. rubber. As a reference, the undeformed shape for both samples is shown on the left side.

A

0

refers to the cross‐sectional area of the undeformed sample.

Figure 1-2 Schematics of the modulus vs. temperature behavior for a rubber and a plastic over a broad temperature range.

Figure 1-3 Schematic modulus–time curve for a polymer at constant temperature.

Figure 1-4 Specific volume data for poly(vinyl acetate)

used to determine its

T

g

. [Kovacs (1958).

1

Copyright 1958, John Wiley & Sons.]

Chapter 2

Figure 2-1 Tension (a) and shear (b) of a three‐dimensional sample subjected to very small deformations.

Figure 2-2 Forcing a sample to shear by fastening it to a solid surface and pulling on a plate adhered to its top surface.

Figure 2-3 Forces applied to the sample to cause shear will also lead to rotation unless the sample is constrained.

Figure 2-4 Rotation of the sample can be stopped by applying another pair of forces.

Figure 2-5 To keep the volume of the sample constant during stretching, the sample must become thinner.

Figure 2-6 Free‐body force diagram for the clamp used to stretch a sample. The clamp (dotted vertical line) is fastened to the end of the sample, which extends to the left.

Figure 2-7 Free‐body diagram of a virtual clamp pasted to the side of the sample.

Figure 2-8 This gasket material exhibits a negative Poisson ratio. On pulling, the rectangle marked on the sample increased in width from 7.1 to 7.6 mm, or about 7%. The thickness direction (not shown) expanded even more. The length of the rectangle increased ˜10%. Thus, Poisson's ratio is about −0.07/0.1 = −0.7.

Figure 2-9 A simple apparatus to measure shear creep.

Figure 2-10 A simple apparatus to measure tensile stress

relaxation

. The spring must be strong enough to stretch the sample very quickly.

Figure 2-11 Schematic of a dynamic experiment in shear. The electromechanical driver provides a sinusoidal

motion of fixed frequency and amplitude.

Figure 2-12 Dynamic experiment in shear wherein the disk‐shaped sample is twisted sinusoidally. The dotted line in the shaded sample depicts a line of material points, showing the shearing of the sample from right to left. In some designs, one plate is connected to a servomotor that produces the sinusoidal motion while the other plate is connected to the load cell. Other manufacturers use one plate for both functions, with the other solidly fastened to the frame of the machine. This design, while simple, introduces error at high frequencies due to inertia.

Figure 2-13 The response of a sample to a sinusoidal shear strain

γ

(

t

) is a sinusoidal shear stress

σ

(

t

) (dashed line) that leads the strain by a phase angle

δ

. Arrows show the physical meaning of the stresses

σ

′ and

σ

″ corresponding to the elastic or in‐phase component

G

′ of the dynamic

shear modulus

and the viscous

, out‐of‐phase

or loss

component

G

″, that is,

G

′ =

σ

′/

γ

0

and

G

″ =

σ

″/

γ

0

.

Figure 2-14 Strain sweep using a silicone polymer melt (left) and a polyurethane foam (right). For the latter, the response is very linear at low strains (see derivative curve), but

G

′ starts to drop at a strain of 0.01 (barely visible) and rapidly around 5% (−1.3 on log scale). The PDMS melt response seems quite linear throughout the strain range, but analysis shows a slight but significant trend for both

G

′ (down) and

G

″ (up). At higher frequencies, some of the changes might well be due to viscous heating. (PDMS data courtesy of Dr. Si‐Wan Li.)

Figure 2-15 Simulated LAOS result. The curve labeled LAOS has a small portion of a third harmonic added to the linear stress signal. This harmonic will have a phase and amplitude.

Figure 2-16 Lissajous–Bowditch patterns generated by assuming the presence of a third harmonic, contrasted with what would happen if an even harmonic were present. Generally, even harmonics are not present.

Figure 2-17 Example of a Pipkin diagram. These data were generated using the frequency and phase characteristics of a Maxwell model for the fundamental and a proportional response for the third harmonic. Note the progression with strain to more extreme nonlinearity. Log

De

varies from 0 to 2 along the abscissa, whereas log

Wi

varies from −1 to 2 along the diagonal.

Figure 2-18 Schematic of setup for finding diffusivity

of scattering objects (particles). The random‐walk path of the light under the condition of diffuse scattering involves many scattering events, so only a very small fraction of the light gets to the detector, which must be very sensitive and fast. Often a photomultiplier tube (PMT) is used for detection.

Figure 2-19 Schematic of the linear addition of strains resulting from sequentially applied stresses.

Figure 2-20 Stress history used in calculations.

Figure 2-21 An example of the relationship between creep compliance and a stress relaxation modulus. On the log–log plot shown, the two would be mirror images if reciprocally related.

Chapter 3

Figure 3-1 Spring, dashpot, and Maxwell model. In subsequent depictions of the dashpot, the “fluid” hatching will be removed for simplicity.

Figure 3-2 Creep response of a Maxwell body displayed using linear (left) and log–log (right) scales.

Figure 3-3 Maxwell body behavior using stress relaxation

conditions: (a) linear plot, (b) linear–log plot, and (c) log–log plot.

Figure 3-4 Log–log plots of

E

′(

ω

)/

E

and

E

″(

ω

)/

E

versus

ωτ

for a Maxwell model.

Figure 3-5 Voigt model.

Figure 3-6 Frequency dependence of the complex dynamic compliance for the Voigt model.

Figure 3-7 Generalized Maxwell model.

Figure 3-8 Behavior of a two‐component generalized Maxwell model in stress relaxation.

Figure 3-9 Voigt−Kelvin model.

Figure 3-10 Discrete distribution of relaxation times and associated partial modulus values. The continuous line represents the stress relaxation modulus

based on this distribution.

Figure 3-11 Continuous distribution of relaxation times expressed as

H

(

τ

) and the corresponding tensile stress relaxation modulus

E

(

t

).

Figure 3-12 Stress relaxation

master curve for NBS polyisobutylene at 25 °C (dotted line) and corresponding box‐and‐wedge distribution

(solid lines). [Tobolsky (1960).

6

Copyright 1960, John Wiley & Sons.]

Figure 3-13 Bead‐and‐spring representation of a real polymer molecule in dilute solution.

Figure 3-14 Motions of submolecules with varying orientations in a medium subjected to a uniform extensional deformation. A polymer chain will contain many submolecules.

Figure 3-15 Experimental observation (PS in toluene at 30.3 °C) and prediction of the Rouse theory. The molecular weights (kDa) and concentrations (kg /m

3

) from left to right are: 6200, 1.44; 520, 8.9; and 253, 14.6. [Rouse and Sittel (1953.)

11

Copyright 1943, AIP Publishing.]

Figure 3-16 Predictions of the Rouse and Zimm theories. The subscript

R

indicates that the modulus is divided by

NkT

. [De Mallie et al., (1962).

12

Copyright 1962, American Chemical Society.]

Figure 3-17 Experimental results for 1% polystyrene

in Arochlor

®

compared with predictions according to the Zimm theory. [De Mallie et al., (1962).

12

Copyright 1962, American Chemical Society.]

Figure 3-18 Rouse theory in the rubbery plateau

and flow region

s. Varying numbers of segments are considered. Note increasing region of power‐law behavior (slope = −½) as the number of segments is increased.

Figure 3-19 Calculations of the viscosity from the stress relaxation

master curve for NBS PIB at 25°C (see Figure 3‐12) via equation (3‐90), showing the important contribution to the integral from the long‐time data and the negligible contribution from the short‐time modes in the glassy region.

Figure 3-20 Two transitions predicted using the Ferry, Landel, and Williams

8

modification of the Rouse theory. The parameters required to get this result are quite extreme; see text.

Figure 3-21 Comparison between master curve of polystyrene

and predictions of the modified Rouse theory in the primary transition region.

Figure 3-22 Relaxation in the primary transition region considering varying numbers of relaxing elements. The number of elements in the transition region in each case is

z – p

c

.

Figure 3-23 Comparison of the relaxation behavior of the empirical KWW expression of equation (3‐106) with data for PIB in the rubbery flow region

. Data: NBS PIB; see Ferry

3

for a table of these data. Copies of the data are also on the website (PIB‐Rel‐1.TXT and PIB‐Rel‐2.TXT).

Figure 3-24 Reptation model view of a polymer chain: (a) with obstacles (dots); (b) in a tube.

Figure 3-25 Tube model with definitions of important length scales. [Larson et al. (2003).

28

Copyright 2003, The Society of Rheology.]

Figure 3-26 The molecular‐weight dependence of the viscosities of a series of polyethylene melts. It can be seen that the slope of 3.0 predicted by equation (3‐115) falls short, while a slope of 3.4 is virtually perfect. (Data from various sources.)

Chapter 4

Figure 4-1 Schematic modulus–temperature curve showing various regions of viscoelastic behavior.

Figure 4-2 Shear modulus

†

vs. temperature for two polycarbonate samples of different molecular weights, along with the response of a partially crystallized sample. (See Chapter 5 for additional discussion.) [Mercier, et al. (1965).

12

Figure 4-3 Comparison of 10‐s modulus vs. temperature curves for three common thermoplastics: polyvinyl chloride (PVC), polystyrene

(PS), and polyethylene (PE).

Figure 4-4 Panel (a) shows the influence of plasticizer

content on the modulus–temperature response of PVC through the glass transition

region. Note that increasing plasticizer content tends to broaden the transition markedly at first and then the effect decreases at even higher concentrations. [Schmieder and Wolf (1952).

2

Copyright 1952, Springer.] The data in panel (b) suggest that the broadening effect is highly dependent on the chemical nature of the plasticizer as well..

Figure 4-5 Construction of a master curve using tensile stress

relaxation data gathered at five temperatures. In this case, the reference temperature

T

0

is

T

4

.

Figure 4-6 The WLF equation using the constants for polystyrene

listed in Table

4‐1

.

Figure 4-7 (a) Plot of the relaxation master curve for each of the temperatures indicated. The line through each of the data sets indicates the relaxation behavior over the entire time range expected for that temperature. The curves are moved only in the horizontal direction to fit the data. The empirical description for the curves follows an equation suggested by Smith,

11

i.e.,

E

=(

E

g

−

E

r

) /[1+(

τ

/

τ

1

)

a

]+

E

r

/[1+(

τ

/

τ

2

)

b

], where

τ

is shifted time. The constants were obtained by a fit as shown in (b), to the original superposed data (open circles) listed in Ferry

5

while the line is from the equation of Smith.

Figure 4-8 Simulations of master curves

and modulus vs. temperature curves for a glassy polymer. (a) The master curves, shown at increments of 5 °C, tend to be spaced more widely as the temperature is lowered because of the nature of the WLF relationship used for the temperature dependence (see Figure 4‐6). (b) Demonstration of the influence of measurement time on the shape of the modulus–temperature curve. As the measurement time increases (by 1‐decade increments), the apparent

T

g

decreases but the sharpness of the transition increases. (Simulation uses Smith empiricism2 for glass transition

and the KWW function for the rubbery flow region.)

Chapter 5

Figure 5–1 Specific volume versus temperature for a poly(vinyl acetate)

sample quick‐quenched from well above

T

g

to the experimental temperatures. Volumes measured 0.02 and 100 h after quenching, as indicated. Filled points represent equilibrium behavior. [Kovacs (1958). [2] Copyright 1958, John Wiley and Sons.]

Figure 5–2 Thermal scans of polystyrene

at 5.4 K /min (0.09 K/s) after cooling from well above

T

g

at the rates indicated. Curves have been shifted vertically for clarity. [Wunderlich et al. (1964). [3] Copyright 1964, AIP Publishing.]

Figure 5–3 Temperature dependencies of

G

′,

G

″ and tan

δ

for styrene butadiene copolymer. [Adapted from Nielsen (1962). [4] ]

Figure 5–4 Temperature dependencies of

E

′ and tan

δ

for polyvinyl chloride at various frequencies. [Becker (1955). [5] Copyright 1955, Springer.]

Figure 5–5 Temperature dependence of viscosity for a cyclic polystyrene

sample of molecular weight 106,000 g /mol. The line is a fit with the WLF relationship, equation (4–6). [McKenna et al. (1987). [7] Copyright 1987, American Chemical Society.]

Figure 5–6 (a) Dependence of

T

g

on

ϕ

p

for mixtures of poly(methyl methacrylate)

with diethyl phthalate. Comparison of experimental results with equation (5–8). Parameters found were:

α

d

/

α

p

= 2.32,

T

g,d

= −57 °C,

†

T

g,p

= 104 °C. [Kelley and Bueche (1961). [8] Copyright 1961, John Wiley & Sons.]; (b) Variation of

T

g

for a miscible polymer blend of polycaprolactone

(PCL) and poly(styrene‐

co

‐acrylonitrile)

(SAN), with a description of the data using the Gordon–Taylor relationship, equation (5–27). The two points at low SAN content have a higher‐than‐expected

T

g

because of crystallization of the PCL. [Chiu and Smith (1984). [9] Copyright 1984, John Wiley & Sons.]

Figure 5–7 Schematic representation of conformational entropy vs. temperature for a glass‐forming substance. The temperature at which the entropy attains a value of zero is the

T

2

of Gibbs and DiMarzio. The dotted line is the extrapolation of Kauzmann designed to illustrate the infeasibility of a glassy equilibrium state, and the proposal of Gibbs and DiMarzio. [Gibbs (1960). [13] Copyright 1960, Elsevier]

Figure 5–8 The influence of molecular weight on the glass transition temperatures

of cyclic and linear

oligomers of poly(dimethyl siloxane)

, and the comparison of these data with the prediction of the Gibbs–DiMarzio theory. [DiMarzio and Guttman (1987). [16] Copyright 1987, American Chemical Society.]

Figure 5–9 Contraction isotherms for glucose, a low‐molecular‐weight glass‐forming substance. Samples were quenched from equilibrium at 40 °C to the temperatures indicated. Solid line through the points at 24.9 °C was calculated using equations (5–23) and (5–25) with

T

g

= 35°C,

f

g

= 0.025,

α

f

= 3.6 × 10

−4

K

−1

, and

τ

(

T

g

,

δ

= 0) = 0.015 h. Fits to the other temperatures are possible using the same equations, but require small changes in the parameters. [Kovacs (1964). [17] Copyright 1964, Springer.]

Figure 5–10 Composition dependence of

T

g

for a series of random styrene–butadiene copolymer

s compared to the predicted curve calculated on the basis of equation (5–27) with

k

= 0.34. [Gordon and Taylor (1952).

21

Copyright 1952, John Wiley & Sons.]

Figure 5–11

T

g

versus reciprocal molecular weight for polystyrene

fractions. The solid line is a plot of equation (5–28) with

c

= 1.7 × 10

5

K Da. [Fox and Flory (1950). [23] Copyright 1950, AIP Publishing]

Figure 5–12 Glass transition change in the system poly(styrene‐

co

‐divinylbenzene) as a function of the crosslink density.

N

0

is the number density (e.g., mmol /cm

3

) of crosslinks, whereas

ρ

is the mass density (e.g., g /cm

3

). The dotted line is the fit of equation (5–29) with the result that

kρ

is 130 ± 10 g /mol. [Fox and Loshaek (1955). [24] Copyright 1955, John Wiley & Sons.]

Figure 5–13 Temperature dependence of damping for polymethyl methacrylate showing the

α

and

β

relaxations. [Adapted from Nielsen, (1962). [28] ]

§

Figure 5–14 The crankshaft

mechanism according to Schatzki.

29

Figure 5–15 Temperature dependence of tan

δ

for copolymers of methyl methacrylate and cyclohexyl methacrylate. Open circles are pure polymethyl methacrylate. [Heijboer, (1960).

30

Copyright 1960, Springer.]

Figure 5–16 Physical stress relaxation

at 25 °C in a crosslinked network made by a zinc oxide cure of a halobutyl rubber. The elastomer has been soaked in oil to speed the relaxation. The line is the fit to the data using the Chasset–Thirion equation

31

with

E

∞

= 0.018 MPa,

τ

= 1.04 s, and

m

= 0.31. The nature of the polymer and the cure makes for a chemically stable structure; otherwise, the plot would begin to curve downward at long time.

Figure 5–17 Chemical stress relaxation of natural rubber cured with dicumyl peroxide. [Adapted from Takahashi and Tobolsky (1970). [33] ]

Figure 5–18 Dynamic mechanical

behavior during the chemical cure of a phenolic resole resin at 125

°

C. The noise at the beginning of the experiment results from the rapid evolution of moisture, which causes some bubbles to form. [From Rose (2001). [36] ]

Figure 5–19 Schematic of device for controlling humidity in a small test chamber. The gas sources and meters are not shown. Also water must be supplied from a constant‐level source.

§§

Supplying vaporization heat to the water might be as simple as having a tall, narrow container, at least for low gas flow. In operation, the water‐saturated gas is mixed with a controlled amount of dry gas to give the desired humidity. Roughly, if the two gas streams have the same flow rate, the humidity will be 50%.

Figure 5–20 Diffusivity measured during the drying of thin slices of wood analyzed assuming molecular diffusion obeying Fick's second law. The higher diffusivity

at higher water concentrations is very typical of water‐sensitive polymers.

Figure 5–21 Creep of European beech wood, along with time‐dependent Poisson's ratio. Left figure is the test sample. (a) With the grain as shown, tension or compression will be applied in the radial direction. The narrow straight areas in the center are sprayed with speckle patterns to serve as the strain‐measuring method. The positions of all speckles are recorded by cameras as functions of load and time. (b) Creep of specimen loaded in compression in the radial (

R

) direction, with strain measurements in radial and tangential directions. The Poisson's ratio

ν

TR

was calculated from these. [Ozyhar et al. (2013). [37] Copyright 2013, The Society of Rheology.]

Figure 5–22 Example of creep for wood with a constant tensile stress of 3.4 MPa. The presentation is quite unusual (although common for Civil Engineering applications) in that the values shown depict a normalized extra creep beyond the instantaneous and presumably recoverable elastic deformation. The curve is an attempt to fit the Voigt model to the data; a 3‐element fluid model would give a slightly better fit. [Data from Green, et al. (1999). [38] ]

Figure 5–23 Viscoelastic response of sodium alginate solution on exchanging calcium ions for sodium. At high calcium concentrations, the exchange is extremely rapid. [Data courtesy of Dr. C. Subramanian, 2010.]

Figure 5–24 (a)

G

′(

ω

) at several temperatures that bring the sample from a nearly perfect solid to a typical polymer solution. Note that the curve at 37.5 °C is nearly power‐law, i.e., neither liquid nor solid. The dotted rectangle outlines the area where data were taken for Figure 5‐24b. (b) Response of tan

δ

with frequency plotted as a vertical set of points at a fixed temperature. The pinch point of the lines of fixed frequency defines the critical gel temperature very precisely; at this temperature, tan

δ

is invariant with frequency. [From Liu et al. (2016).

40

Copyright 2016, The Society of Rheology.]

Chapter 6

Figure 6-1 Stress–temperature curves of natural rubber. Extension ratios

λ

are indicated on the right of the figure. [Shen et al. (1967).† Copyright 1967, AIP Publishing.]

Figure 6-2 Schematic diagram of an ensemble of linear polymer chains being crosslink

ed into an infinite network

. Note that the entanglement can become an extra crosslink.

Figure 6-3 A unit cube of elastomer: (a) in the unstrained state; (b) in the homogeneous strained state; (c) under uniaxial extension, assuming a homogeneous, incompressible material.

Figure 6-4 Stress–strain

curve for natural rubber. The theoretical curve was calculated from equation (6‐60) with

G

= 0.4 MPa. [Treloar (1944).6 Copyright 1944, Royal Society of Chemistry.]

Figure 6-5 Determination of shear moduli for natural rubber at 10 °C and 60 °C by plotting nominal tensile stress

σ

E

0

against

λ

− 1/

λ

2

as suggested by equation (6‐66). [Shen and Blatz (1968).7 Copyright 1968, AIP Publishing.]

Figure 6-6 Simulated Mooney−Rivlin plot, equation (6‐82), for

C

2

/

C

1

=2. Solid line is the undistorted response of the material; “data” are the results with a 1% (standard deviation) random error incorporated into both the force and length “measurements.”

Figure 6-7 Variation of shear moduli of natural rubber with reciprocal initial molecular weight for various degrees of crosslinking. See equation (6‐81) for explanation of

C

1

. [Mullins (1959).14 Copyright 1959, John Wiley & Sons.]

Figure 6‐8 Effect of swelling on the Mooney−Rivlin plot of natural rubber where Vr is the volume fraction of elastomer in the swollen sample, (swelling liquid, n‐decane). [Mullins (1959).14 Copyright 1959, John Wiley & Sons.]

Figure 6-9 Effect of filler concentration on the moduli of natural rubber samples (filler: MT carbon black). The curve was calculated from equation (6‐94). [Mullins and Tobin (1965).19 Copyright 1965, John Wiley & Sons.]

Figure 6-10 Mooney−Rivlin plots of natural rubber filled with MT carbon black: Top set: actual data without using the strain amplification factor. Bottom curves: after reduction using the strain amplification factor, equation (6‐97). [Mullins and Tobin (1965).19 Copyright 1965, John Wiley & Sons.]

Figure 6-11 Stress softening

of natural rubber

filled with MPC carbon black (Mullins effect). Numerals indicate the stress−strain cycles. [Bueche (1960).21 Copyright 1960, John Wiley & Sons.]

Figure 6-12 Schematic diagram of polymer chains attached to filler particles in an elastomer. The short chain will likely be the first to break on stretching the elastomer. [Bueche (1960).21 Copyright 1960, John Wiley & Sons.]

Figure 6-13 Stress−strain curves of natural rubber at 0 °C and 60 °C. The increase in stress at 0 °C at

λ

> 3 is attributed to strain‐induced crystallization. [Smith et al. (1964).22 Copyright 1964, Springer.]

Figure 6-14 Conformation of a polymer chain with one end fixed at the origin of a Cartesian coordinate system.

Figure 6-15 Radial distribution function of a chain of 10

4

freely orienting segments each of length 0.25 nm.

Figure 6-16 Polymethylene backbone in the fully extended conformation.

Chapter 7

Figure 7-1 Schematic representation of an

RC

circuit comprising a resistor

R

in series with a capacitor

C

.

Figure 7-2 Debye plots for dielectric relaxation.

Figure 7-3 Alignment of dipoles in a field.

Figure 7-4 Polarization at the interface between a polymer and a polar inclusion. The interfacial charges are illustrated as effective dipoles, showing that distinguishing interfacial from dipolar polarization will depend critically on the characteristic time of the former.

Figure 7-5 Cole−Cole plot for a single relaxation time,

ε

R

= 8,

ε

u

= 3.

Figure 7-6 Cole−Cole plot for the

α

relaxation in poly(2‐chlorostyrene). [From Alexandrovich et al., (1980).

15

Copyright 1980, Elsevier.]

Figure 7-7 Schematic of transformer ratio arm bridge.

Figure 7-8 (a) Impedance spectroscopy result for an ionically conductive epoxy composite [Courtesy of S. B. Boob (2003)

18

]; (b) the equivalent circuit (inset) for the circular part of the response; and (c) same as (b), but plotted on a frequency scale.

Figure 7-9 TSC results for polycarbonate compared with NMR line width and dynamic mechanical

(DMTA) measurements. [TSC data from Aoki and Brittain (1977).

20

Copyright 1977, John Wiley & Sons; NMR data from Matsuoka and Ishida (1966).

21

Copyright 1966, John Wiley & Sons; DMTA data from Illers and Breuer (1961).

22

Copyright 1966, Springer.]

Figure 7-10 Comparison between tan

δ

ε

and tan

δ

for PMMA. [Dielectric and mechanical data from McCrum et al. (1967)

5

by permission of copyright owner, N. G. McCrum. Original dielectric data were from Mikhailov and Borisova (1961).

23

Copyright 1961, Elsevier. Original mechanical data from Heijboer (1965).

24

Copyright 1965, North Holland.]

Figure 7-11 Correlation map for PMMA. Solid symbols, mechanical; open symbols, dielectric; and crosses, NMR

. [McCrum et al. (1967).

5

Copyright 1967, John Wiley & Sons.]

Figure 7-12 Broadline NMR investigation of plasticized PVC. [Boo and Shaw (1982).

29

Copyright 1982, John Wiley & Sons.]

Figure 7-13 Crossplot of broadline

1

H and CP‐MAS (narrow line)

13

C spectra for a 43% mixture of erucamide with

i

‐PP. Erucamide is a fatty acid amide used with PP film to reduce adhesion of adjacent layers of film. The contours indicated by arrows are due to polypropylene –CH

3

, >CH– and –CH

2

– groups, respectively. This shows that the >CH– group at ˜ 26 ppm on the

13

C resonance axis has the lowest mobility, as its proton resonance is the broadest. [Quijada‐Garrido et al. (1998).

30

Copyright 1998, John Wiley & Sons.]

Figure 7-14 Changes in NMR

spin–spin and spin–lattice relaxation times with temperature for a low‐

T

g

polymer PVME and a high‐

T

g

polymer PS. [Kwei et al. (1974). [32] Copyright 1974, American Chemical Society.]

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Introduction to Polymer Viscoelasticity

Fourth Edition

Montgomery T. ShawWilliam J. MacKnight

Copyright

This fourth edition first published 2018

© 2018 JohnWiley & Sons, Inc

Edition History

Third Edition: 2005, Copyright John Wiley & Sons, inc.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

The right of Montgomery T. Shaw and William J. MacKnight to be identified as the authors of this work has been asserted in accordance with law.

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In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

Library of Congress Cataloging‐in‐Publication Data

Names: Shaw, Montgomery T., author.

Title: Introduction to polymer viscoelasticity / Montgomery T. Shaw, William

J. MacKnight.

Description: Fourth edition. | Hoboken, NJ, USA : Wiley, 2018. | Original

1972 edition by John J. Aklonis, William J. MacKnight, and Mitchel Shen |

Includes bibliographical references and indexes. |

Identifiers: LCCN 2017058166 (print) | LCCN 2017059024 (ebook) | ISBN

9781119181811 (pdf) | ISBN 9781119181828 (epub) | ISBN 9781119181804

(cloth)

Subjects: LCSH: Polymers. | Viscoelasticity.

Classification: LCC TA455.P58 (ebook) | LCC TA455.P58 A35 2018 (print) | DDC

620.1/9204232‐‐dc23

LC record available at https://lccn.loc.gov/2017058166

Cover Design: Wiley

Cover Image: © stellalevi/Gettyimages

Dedication

To A. V. Tobolsky, who first introduced us to the mysteries of polymer viscoelasticity and to the art of scientific research

Preface to the Fourth Edition

For the last dozen years, the 3rd Edition of Introduction to Polymer Viscoelasticity has served well. During this time, the authors have received many constructive suggestions from students and instructors, including conflicting ones such as “simplify; it's a bit too complicated” paired with “expand the theoretical parts.” In response to these and other comments, rather modest goals were set for the 4th Edition, namely

Correct all known errors reported by many kind readers.

Errors fall into two categories: (1) the usual grammatical slips and (2) serious mistakes such as equations with missing parts and poorly posed problems. Indeed, one problem was found that we could not solve, and obviously didn't belong in an introductory text. It has been removed.

Provide more unanswered problems.

Since the birth of this text in the early 1970s, answers have been provided for many of the chapter‐end problems. The reason for this is these problems typically review fundamental concepts, serving as additional solved examples. Some introduce new concepts. However, most instructors want unsolved problems that can be assigned as homework, adapted for active learning exercises and short projects, or used for tests or quizzes.

Introduce the fields of microrheology and large‐amplitude oscillatory strain (LAOS).

The former attempts to examine dynamics around very tiny particles, including molecules, to gain information about the local environment, while the latter pushes the material hard enough to cause structural changes. LAOS is particularly important for complex materials where the concepts of structural strength and healing are of significant practical importance. Professors Eric Furst and Randy Ewoldt provided valuable advice on these sections, respectively.

Open the very large topic of biopolymer viscoelasticity.

There is no end to the examples that could be chosen, so we settled on some very different polymers: wood at the high‐modulus extreme and hydrogels for the low‐modulus extreme. Guidance for this section from Professor Kelly Burke is gratefully acknowledged.

Provide an Instructor's Guide with detailed answers to problems presented in the text.

The answers are much more than a key for grading homework, in that they provide guidance for what to say to the students who are having trouble approaching a problem solution. Nearly 80 new problems were added, including a selection of problems that do not appear in the text.

Items we have resisted including in this introductory text are tensorial representation of stress and strain, viscoelastic properties and analysis of composite materials (with wood being an exception), self‐structured materials such as liquid crystals and block copolymers with long‐range order, and the more baffling characterization methods and analyses such as transform, multiwave, and sonic methods. As the general level of student training and understanding increases, such topics may become more appropriate for an introductory text.

This book is about polymers. We assume some knowledge of polymer structure and behavior, but hope to add to that knowledge by viewing familiar behavior from a different angle. No attempt is made to explore admittedly important areas of suspension and colloid rheology, two‐phase flow, particulate flow, foams, etc. All are good topics for advanced texts.

Finally, a word about the authors. Because of advancing age, Prof. MacKnight has become a nonparticipating author, but is included in the author list in recognition of his significant contributions to the original versions and to the entire field of polymer properties and structure. Thanks to advancing technology, I have only myself to blame for errors and shortcomings. Please mail notice of such, along with general comments and advice, to [email protected]. (This will, I am told, be my e‐mail address in perpetuity, a somewhat daunting thought.) Granddaughter Katherine Shaw diligently checked Web links, for which the authors are grateful. Finally, I wish to thank my wife, Maripaz N. Shaw, for once again coping with the issues created by any large writing project.

MONTGOMERY T. SHAW

Storrs, Connecticut

November 2017

Preface to the Third Edition

More than 20 years have passed since the publication of the 2nd Edition of Introduction to Polymer Viscoelasticity. Although many of the fundamental aspects of the field remain unchanged, there have been a number of significant developments. Many have to do with instrumentation and the revolution in data collection and analysis, which in no small part has been due to the advent of the personal computer and the associated progression of instrumentation of all types.

In recognition of these changes, we have included descriptions of newer techniques for studying molecular motion along with updated descriptions of the classical experimental methods. An example of the latter is a new appendix that describes in more detail the advantages and disadvantages of the various geometries commonly used for measuring mechanical response. Included are tables with the working equations for these geometries. Those familiar with the earlier editions will also notice an increased emphasis on shear properties, which is an understandable response to the wide availability of instruments that can measure viscoelastic properties in simple shear. We have also, where possible, changed nomenclature to follow the recommendations of the Society of Rheology, and updated all the figures to increase readability and consistency.

Inexpensive computation hardware along with accessible software has impacted not only the acquisition of viscoelastic data, but also its interpretation. In the spirit of these changes, the 3rd Edition features many examples and problems that involve numerical modeling and analysis. To relieve the student of the drudgery of data entry, a CD with data files for most numerical problems has been included.

The authors' experience has shown that by far the most effective way to master the material in the text is to work as many problems as possible, hence the increased emphasis on this aspect in the 3rd Edition. The problems range from relatively straightforward use of an equation included in the book, to far more challenging problems requiring detailed analysis and/or numerical methods. Some of these would even be suitable for term projects. Problems requiring the use of the computer are clearly marked, as are open‐ended problems that have no one “correct” answer. This type of problem, which is required in many undergraduate curricula, provides the student with an opportunity to search, assume, approximate and innovate. As in past editions, answers to many of the problems are provided in an appendix. These answers form an important part of the book, and contain in some cases more details concerning the subject phenomena.

New topics have been introduced such as interfacial polarization, thermally stimulated currents (TSC), impedance spectroscopy for highly conducting polymers, nuclear magnetic resonance (NMR) relaxation techniques, and the physical relaxation of elastomers. Because impedance spectroscopy has not been used extensively as a tool for examining polymer motion, this short section is included more to explain the similarities and differences between this spectroscopy and the related dielectric spectroscopy. On the other hand, NMR techniques have undergone rapid development in the last few decades, not only in fields such as imaging and high‐resolution studies of the structures of biological macromolecules, but also as a tool for studying the relaxation behavior of polymers, particularly in identifying the molecular motions responsible for a given relaxation process. While the new section describing NMR techniques is necessarily introductory, we have attempted to compare its capabilities with dielectric and mechanical spectroscopy in a direct fashion.

As for other changes, we have with considerable trepidation moved the description of deformation in materials from one to three dimensions. Perhaps the main impetus for doing this was to simplify the rather complex explanation in previous editions of the relationship between tensile and shear properties. As an admitted expense, we now have double‐subscripted variables in several sections. However, we have refrained from including nonlinear strain theory, which should properly be left for more advanced courses.

It is always a difficult task to select material appropriate for inclusion and exclusion in an introductory text of modest size and cost. Because of the discussion of the topics mentioned above and a somewhat expanded treatment of the phenomenology of viscoelasticity, it was felt appropriate to eliminate the chapter on chemical stress relaxation. In its place, a discussion of this topic has been included in the chapter entitled Transitions and Relaxations in Polymers; and, of course, in several problems at the end of this chapter.

Professor Aklonis did not participate in the preparation of the 3rd Edition, but, as was the case with Professor Shen in the 2nd Edition, his influence is clearly present and it is a pleasure for us to acknowledge it.

Many individuals and organizations have been involved with the assembly of the 3rd edition. We wish to thank Ms. Jennifer Chudy and Mr. Alvin A. Altamirano for help with data entry and equation editing; Dr. Mark Poliks, Dr. Lou Madsen and Prof. Marcel Utz for critically reviewing the NMR section, Mr. Antonio Senador for checking several problem solutions and Mr. Gerald Ling who provided assistance with the challenging task of finding authors of classical publications for courtesy permission to reproduce figures. We especially want to acknowledge the patience displayed by our spouses, Maripaz N. Shaw and Carol B. MacKnight, as they endured through this lengthy project.

Errors in the text are, of course, the sole responsibility of the authors. It is to be hoped that we have recognized and corrected at least some of the errors in the 2nd Edition, (many of which were pointed out to us by friends and colleagues) and have refrained from introducing a significant number of new ones in the 3rd Edition.

M. T. SHAW

W. J. MACKNIGHT

Storrs, Connecticut

Amherst, Massachusetts

December 2004

Preface to the Second Edition

In the decade since the 1st of Introduction to Polymer Viscoelasticity appeared, we have noted a number of significant scientific developments. We also suffered a personal tragedy with the passing of Professor M. C. Shen.

Among the major developments are a new approach to long‐range relaxational motions known as the theory of reptation and the further elucidation of the kinetic theory of rubber elasticity. In this 2nd edition, we have attempted to take account of some of these developments on a level consistent with the introductory nature of the text. We have also added an entirely new chapter on dielectric relaxation, a technique now widely used to investigate molecular motions in polar polymers. Finally, we have tried to strengthen and clarify several other sections as well as eliminate errors or inconsistencies in the 1st edition that have been pointed out to us by colleagues and students.

Both of us felt very deeply the untimely death of Professor Shen, as did many others who valued his friendship and respected his scientific prowess. He made important and lasting contributions to such diverse areas as rubber elasticity theory, the understanding of mechanical properties of block copolymers, and plasma polymerization, to name but a few. His collaboration in the preparation of the 2nd Edition was sorely missed, but we feel that his influence remains clear and we are proud to acknowledge it.

We wish to thank Dr. Richard M. Neumann, who read the manuscript critically, and Ms. Teresa M. Wilder, who drew many of the figures. We are also indebted to Dr. Neumann and Professor L. L. Chapoy for furnishing some of the new problems contained in this edition.

Once again we accept sole responsibility for any errors in the text, be they old ones remaining from the 1st Edition or new ones that may appear in the second.

J. J. AKLONIS

W. J. MACKNIGHT

Los Angeles, California

Amherst, Massachusetts

December 1982

Preface to the First Edition

The viscoelastic response of polymeric materials is a subject that has undergone extensive development over the past 20 years and still accounts for a major portion of the research effort expended. It is not difficult to understand the reason for this emphasis in view of the vast quantities of polymeric substances that find applications as engineering plastics and the still greater volume that are utilized as elastomers. The central importance of the time and temperature dependence of the mechanical properties of polymers lies in the large magnitudes of these dependencies when compared to other structural materials such as metals. Thus, an understanding of viscoelastic behavior is fundamental for the proper utilization of polymers.

Viscoelasticity is a subject of great complexity fraught with conceptual difficulties. It is possible to distinguish two basic approaches to the subject which we shall designate as the continuum mechanical approach and the molecular approach. The former attempts to describe the viscoelastic behavior of a body by means of a mathematical schema, which is not concerned with the molecular structure of the body, while the latter attempts to deduce bulk viscoelastic properties from molecular architecture. The continuum mechanical approach has proven to be very successful in treating a large number of problems and is of very great importance. However, it is not our intention to treat this approach rigorously in this text. Rather, we shall be concerned with the molecular approach and attempt to present a basic foundation upon which the reader can build. The fundamental difficulty encountered with the molecular approach lies in the fact that polymeric materials are large molecules of very complex structures. These structures are too complex, even if they were known in sufficient detail, which, in general, they are not, to lend themselves to mathematical analysis. It is therefore necessary to resort to simplified structural models, and the results deduced from these are limited by the validity of the models adopted.

Several excellent treatments of molecular viscoelasticity are available. (See the references of Chapter 1.) The book by Professor Ferry, in particular, is an exhaustive and complete exposition. The question may then be asked, why the necessity for still another text and one restricted to bulk amorphous polymers, at that? Such a question must send each of the authors scurrying in quest of an “apologia pro vita sua.” The answer to the question lies in the use of the word “introduction” in the title. What we have attempted to do is to provide a detailed grounding in the fundamental concepts. This means, for example, that all derivations have been presented in great detail, concepts and models have been presented with particular attention to assumptions, simplifications, and limitations, and problems have been provided at the end of each chapter to illustrate points in the text. The level of mathematical difficulty is such that the average baccalaureate chemist should be able to readily grasp it. Where more advanced mathematical techniques are required, such as transform techniques, the necessary methods are developed in the text.

Having attempted to delineate what this book is, it may be well to remind the reader what it is not. First of all, it is not a complete treatment, lacking among other topics discussions of crystalline polymers, solution behavior, melt rheology, and ultimate properties. It is also not written from the continuum mechanics approach and thus is not mathematically sophisticated. Finally, it is not a primer of polymer science. Familiarity with the basic concepts of the field is presumed.

The authors' first acquaintance with the literature of viscoelastic behavior in polymers evoked a response much like that experienced by neophytes in the literary arts on a first reading of Finnegan's Wake, by James Joyce. It is immediately apparent that one is in the presence of a great work, but somehow it will be necessary to master the language before appreciation, let alone understanding, may be achieved. Recognizing the nature of the problem, Joycean scholars came to the rescue with works of analysis to provide a skeleton key to Finnegan's Wake. Proper utilization of this skeleton key will open the door to an understanding of that forbidding masterwork. It was thus our intent to provide a similar skeleton key to the literature of molecular viscoelasticity. How well we have succeeded must be left to the judgment of our readers.

We are grateful to the students at our respective institutions who have suffered our attempts to present the material in this text in coherent form at various stages of development. Their criticisms and suggestions have led to significant improvements. We are also grateful to Mrs. William Jackson, who translated many rough sketches into finished drawings. It is hardly to be expected that a work of this nature could be free from errors. We have attempted to eliminate as many as possible but, of course, bear full responsibility for those remaining.

JOHN J. AKLONIS

WILLIAM J. MACKNIGHT

MITCHEL SHEN

Los Angeles, California

About the Companion Website

This book is accompanied by a companion website:

www.wiley.com/go/Shaw/Intro_to_polyviscoelasticity

Available on the Wiley website is the Instructor's Guide for the 4th edition of the textbook Introduction to Polymer Viscoelasticity. Intended for legitimate instructors using the text in courses they are teaching, the Guide offers advice on using the text along with scores of unpublished problems of all sorts, many of which are suitable for homework, exams, or small projects. The answers are not just answers but an explanation of how the problem can be approached and solved. Thus, the solutions can be used almost directly for lecture examples.

Chapter 1Introduction

The subject matter of this book is the response that polymers exhibit when they are subjected to external forces of various kinds. Almost without exception, polymers belong to a class of substances known as “viscoelastic bodies.” As the name implies, these materials respond to external forces in a manner intermediate between the behavior of an elastic solid and a viscous liquid. To set the stage for what follows, it is necessary to describe in very general terms the types of forces to which the viscoelastic bodies might be subjected for characterization purposes.

Consider first the motion of a rigid body in space. This motion can be thought of as consisting of translational and rotational components. If no forces act on the body, it will maintain its original state of motion indefinitely in accordance with Newton’s first law of motion. However, if a single force or a set of forces whose vector sum is nonzero act on the body, it will experience acceleration or a change in its state of motion. Consider, however, the case where the vector sum of forces acting on the body is zero and the body experiences no change in either its translational or rotational component of motion. In such a condition, the body is said to be stressed. If the requirement of rigidity is removed, the body will in general undergo a deformation as a result of the application of these balanced forces. If this occurs, the body is said to be strained. It is the relationship between stress and strain that is our main concern. Depending on the types of stress and strain applied to a body, we can use these quantities to define new quantities—material properties—that ultimately relate to the chemical and physical structure of the body. These material properties are referred to using the terms “modulus” and “compliance.” To understand in rough terms the physical meaning of the modulus of a solid, consider the following simple experiment.

Suppose we have a piece of rubber (e.g., natural rubber), ½ cm × ½ cm × 4 cm, and a piece of plastic (e.g., polystyrene) of the same dimensions. The experiment to be performed consists of suspending a weight (applying a force) of, say 1 kg, from each sample as shown in Figure 1‐1.

Figure 1-1 Deformation of samples made from plastic vs. rubber. As a reference, the undeformed shape for both samples is shown on the left side. A0 refers to the cross‐sectional area of the undeformed sample.

As is obvious, the deformation of the rubber will be much greater than that of the plastic. Using this experiment, we might define a spring constant k as the applied force f divided by the change in length ΔL

(1-1)

and use this number to compare the samples. However, to obtain a measure that is independent of the sample size, that is, a material property, as opposed to a sample property, we must divide the applied force by the initial cross‐sectional area A0 and divide the ΔL by the initial sample length L0. Then, the modulus M is

(1-2)

Because ΔL is much larger for the rubber than for the plastic, from equation (1‐2) it is clear that the modulus of the rubber is much lower than the modulus of the plastic. Thus, the particular modulus defined in equation (1‐2) specifies the resistance of a material to elongation at small deformations and is called the Young’s modulus. It is normally given the symbol E. (See www.rheology.org for suggestions on standard nomenclature for viscoelastic quantities.)

Further experimentation, however, reveals that the situation is more complicated than is initially apparent. If, for example, one were to carry out the test on the rubber at liquid nitrogen temperature, one would find that this “rubber” undergoes a much smaller elongation than with the same force at room temperature. In fact, the extension would be so small as to be comparable to the extension exhibited by the plastic at room temperature. A more dramatic demonstration of this effect is obtained by immersing a rubber ball in liquid nitrogen for several minutes. The cold ball, when bounced, no longer has the characteristic properties of a rubbery object but, instead, is indistinguishable from a hard sphere made of plastic.

On the other hand, if the piece of plastic is heated in an oven to 130 °C and then subjected to the modulus measurement, it is found that a much larger elongation, comparable to the elongation of the rubber at room temperature, results.

These simple experiments indicate that the modulus of a polymeric material is not invariant, but is a function of temperature T, that is, M=M(T).

An investigation of the temperature dependence of the modulus of our two samples is now possible. At temperature T1, we measure the modulus as before, and then increase the temperature to T2, and so on. Schematic data from such an experiment are plotted in Figure 1‐2. The temperature dependence of the modulus is so great that it must be plotted using a logarithmic scale. (This large variation in modulus presents experimental problems that will be treated subsequently.) The region between the vertical dashed lines represents normal‐use temperatures and, consistent with the opening experiment, we find that in this range the plastic has a high modulus while the rubber has a relatively low modulus. Upon cooling, the modulus of the rubber rises markedly, by as much as four orders of magnitude, indicating that the rubber at lower temperatures behaves like a plastic. The nearly constant modulus for the rubber is evidenced at higher temperatures. This behavior is discussed in detail in Chapter 6. At low temperatures, the modulus–temperature behavior for the plastic is seen to be quite similar in shape to that of the rubber except that the large drop, called the glass transition, occurs at higher temperatures, resulting in the high modulus observed at room temperature. At ~100 °C it is clear that the modulus of the plastic is close to that of a rubber, agreeing with the results of one of the earlier “experiments” in this discussion. At yet higher temperatures, the plastic’s modulus drops once again; this is the region where the material can be easily molded.

Figure 1-2 Schematics of the modulus vs. temperature behavior for a rubber and a plastic over a broad temperature range.