Fundamentals of Polymer Engineering Part 13 doc

Given these differences, the twovariables that influence the mechanical properties of amorphous polymers themost are molecular weight and temperature.. However, the elastic moduli andothe

of the material to crack In this chapter, we will consider mechanical properties ofpolymers at small strains as well as large strains In general, the mode ofdeformation could be tension, compression, shear, flexure, torsion, or a combina-tion of these To keep the discussion manageable, we will restrict ourselves totension and shear Note, however, that we can use viscoelasticity theory [1],

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especially at small strains, to predict the behavior in one mode of deformationfrom measurements made in another mode of deformation As with metals, weexpect that the measured properties depend on the chemical nature of the polymerand the temperature of measurement However, what makes data analysis andinterpretation both fascinating and challenging are the facts that results alsodepend on time of loading or the rate of deformation, polymer molecular weight,molecular-weight distribution, chain branching, degree of cross-linking, chainorientation, extent of crystallization, crystal structure, size and shape of crystals,and whether the polymer was solution cast or melt processed These variables arenot all independent; molecular weight, for example, can determine chainorientation and crystallinity in a particular processing situation To explain theseparate influence of some of these variables, we present data on polystyrene, apolymer that can be synthesized in narrow molecular-weight fractions usinganionic polymerization Methods of improving polymer mechanical propertiesare again illustrated using polystyrene This chapter therefore focuses on the(glassy) behavior of polymers below their glass transition temperature.

When discussing the theory of rubber elasticity inChapter 10, we were concernedwith fairly large extensions or strains These arose because polymer moleculescould uncoil at temperatures above Tg For materials used as structural elements(such as glassy polymers), we usually cannot tolerate strains of more than afraction of 1% Therefore, it is customary to employ measures of infinitesimalstrain In a tensile test, we usually take a specimen with tabs at the ends andstretch it, as shown in Figure 12.1 One end of the sample is typically fixed,whereas the other is moved outward at a constant velocity The force F necessary

to carry out the stretching deformation is monitored as a function of time alongwith the instantaneous sample length, L From the measured load versusextension behavior, we can calculate the stress and strain as follows:

Stress ðsÞ ¼ Force ðFÞ

If the cross-sectional area is the undeformed, original cross-sectional area, thestress is called engineering stress, and if the actual, instantaneous area is used, thetrue stress is measured

Strain ðEÞ ¼L
L0

where L0is the initial sample length, and the strain, so defined, is known as theengineering strain Note that this strain is related to the Hencky strain, also calledthe true strain, as follows:

and the two strain measures are identical for small strains

As the sample is stretched in the z direction, its cross-sectional areadecreases, and this implies that the material suffers a negative strain in the xdirection, which is perpendicular to the stretching direction This is quantifiedusing the Poisson ratio n, defined as

For incompressible materials such as rubber, it is easy to show that Poisson’s ratioequals 0.5 For glassy polymers the sample volume increases somewhat onstretching, and Poisson’s ratio ranges from 0.3 to 0.4

Typical stress–strain data for glassy polystyrene are shown inFigure 12.2inboth tension and compression [2] The slope of the stress–strain curve evaluated

at the origin is termed the elastic modulus, E, and is taken to be a measure of thestiffness of the material It is seen in this particular case that the modulus intension differs from that in compression The two curves end when the samplefractures The stress at fracture is called the strength of the material Becausematerials fracture due to the propagation of cracks, the strength in tension isusually less than that in compression because a compressive deformation tends toheal any cracks that form (provided the sample does not buckle) The strain at

FIGURE12.1 Typical specimen for a tensile test

fracture is known as the elongation-to-break; the larger the value of this quantity,the more ductile is the material being tested Glassy polystyrene is not ductile intension; indeed, it is quite brittle Finally, the area under the stress–strain curve iscalled the toughness and has units of energy per unit volume For designpurposes, the materials generally sought are stiff, strong, ductile, and tough.For materials that are liquidlike, such as polymers above their softeningpoint, it is easier to conduct shear testing than tensile testing This conceptuallyinvolves deforming a block of material, as shown in Figure 12.3 The force F is

FIGURE12.2 Stress–strain behavior of a normally brittle polymer such as polystyreneunder tension and compression

(Reprinted from Nielsen, L E., and R F Landel: Mechanical Properties of Polymers andComposites, 2nd ed., Marcel Dekker, Inc., New York, 1994, p 250, by courtesy of MarcelDekker, Inc.)

FIGURE12.3 Shear deformation

again monitored, but now as a function of the displacementDux Stress and strainare now defined as follows:

Shear stress ðtÞ ¼ Force ðFÞ

in shear and tension are related by the following expression:

so that E equals 3 G for incompressible, elastic polymers Note that when materialproperties are time dependent (i.e., viscoelastic), the modulus and strengthincrease with increasing rate of deformation [3], whereas the elongation-to-break generally reduces Viscoelastic data are often represented with the help ofmechanical analogs

Example 12.1: A polymer sample is subjected to a constant tensile stress s0.How does the strain change with time? Assume that the mechanical behavior ofthe polymer can be represented by a spring and dashpot in series, as shown inFigure 12.4

Solution: The stress-versus-strain behavior of a Hookean spring is given by

s ¼ EE

FIGURE12.4 A Maxwell element

For a Newtonian dashpot, the relation is

s ¼ Z dE

dtThe terms E and Z are the spring modulus and dashpot viscosity, respectively.For the spring and dashpot combination, often called a Maxwell element,the total elongation or strain is the sum of the individual strains The stress for thespring and for the dashpot is the same,

Total strain ¼s0

E þs0

Z tand it is seen that the strain increases linearly with time This behavior is known ascreep Although a simple mechanical analog such as a Maxwell element cannot beexpected to portray true polymer behavior, it does illustrate the usually undesir-able phenomenon of creep A better model for the quantitative representation ofcreep is a four-element model which is a linear combination of a Maxwell elementand a Voigt element; the latter is composed of a spring and a dashpot in parallel

A polymer sample creeps because polymer molecules are held in place bysecondary bonds only, and they can rearrange themselves under the influence of

an applied load This is especially easy above the polymer glass transitiontemperature, but it also happens below Tgand strain gauges have to be employedfor accurate measurements To illustrate the latter point, we show long-term creepdata, in the form of circles in Figure 12.5, on samples of polyvinyl chloride(PVC) at constant values of tensile stress, temperature, and relative humidity [4].Note that data for the first 1000 h are shown separately, followed by all of the datausing a compressed time scale It is seen that the total creep can be severalpercent, and a steady state is not reached even after 26 years! These and similardata can be represented by the following simple equation shown by solid lines inFigure 12.5:

EðtÞ ¼ E0þ Eþ

in which E0, Eþ, and n are constants Although n is often independent oftemperature and imposed stress, the other two constants are stress and tempera-ture dependent If creep is not arrested, it can lead to failure, which may occureither by the process of crazing or by the formation of shear bands; these failuremechanisms are discussed later in the chapter Equation (12.2.8) is an empiricalequation that is known as the Findley model It may sometimes contain a secondtime-dependent term if failure can occur by two different mechanisms Creep cangenerally be reduced by lowering the test temperature, raising the polymer Tg,cross-linking the sample, or adding either particulates or short fibers Conversely,anything that lowers the Tg, such as exposure to atmospheric moisture, promotescreep Physical aging (described later) also affects the extent of creep

FIGURE12.5 Creep curves for polyvinyl chloride at 75F, 50% relative humidity (From Ref 4.)

12.2.1 In£uence of Variables such as Molecular

Weight and Temperature

The strength and stiffness of one glassy polymer can be expected to differ fromthat of another glassy polymer due to differences in intermolecular forces as aresult of differences in chemical structure and the presence or absence ofsecondary bonds (e.g., hydrogen-bonding) Given these differences, the twovariables that influence the mechanical properties of amorphous polymers themost are molecular weight and temperature However, the elastic moduli andother small-strain properties of strain-free glassy polymers such as polystyrene(PS) are found not to depend on the molecular weight or molecular-weightdistribution, except at very low molecular weights [5–7] The tensile strength, sf,

of polymers having a narrow molecular-weight distribution, however, is gible at low molecular weight, increases with increasing molecular weight, and,ultimately reaches an asymptotic value [8] This behavior can often be repre-sented by the following equation [6, 9]:

where A and B are constants Data for polystyrene, shown in Figure 12.6, supportthese conclusions [10] From an examination of this figure, it is obvious that theaddition of a low-molecular-weight fraction is bound to affect the tensile strength

of any polymer However, for polydisperse samples, data do not follow Eq.(12.2.9) exactly; results vary with the polydispersity index, even when thenumber-average molecular weight is held fixed

The data just discussed are related to amorphous polymer samples forwhich the polymer chains were randomly oriented One method of increasingboth strength and stiffness is to use samples wherein polymer chains are oriented

FIGURE12.6 Tensile strength of monodisperse polystyrene as a function of molecularweight From Hahnfeld, J L., and B D Dalke: General purpose polystyrene, inEncyclopedia of Polymer Science and Engineering, 2nd ed., vol 16, H F Mark, N M.Bikales, C G Overberger, and G Menges (eds.) Copyright# 1989 by John Wiley &Sons, Inc This material is used by permission of John Wiley & Sons, Inc

along the stretching direction By using this technique, we can very significantlyincrease the modulus of polystyrene and hope to get strength that approaches thestrength of primary chemical bonds [11] Indeed, as discussed in Chapter 11,mechanical property enhancement using chain alignment is the reason for thepopularity of polymers that possess liquid-crystalline order Properties in adirection perpendicular to the chain axis, however, are likely to be inferior tothose along the chain axis.

When the Young’s modulus of any polymer is plotted as a function oftemperature, we find that this quantity is of the order of 105–106psi at lowtemperatures and decreases slowly with increasing temperature This region isknown as the glassy region At the glass transition temperature Tg(see also Chap

2), which varies for different polymers, the modulus drops suddenly by at leastthree orders of magnitude and can reach extremely low values for low-molecular-weight polymers Figure 12.7 shows the Young’s modulus of polystyrene in atemperature range of
200C to 25C [12].Figure 12.8shows shear stress versusshear strain data for an entangled polystyrene in a temperature range of 160C–

210C [13] If we disregard the numerical difference between the Young’smodulus and the shear modulus and note that 1 MPa equals 145 psi, we findthat the modulus calculated from data in Figure 12.8 is several orders ofmagnitude smaller than the number expected on the basis of extrapolating thecurve in Figure 12.7 This happens because the Tg of polystyrene is 100C Thebehavior of the Young’s modulus, in qualitative terms, is sketched inFigure 12.9

over a temperature range that includes Tg If the polymer molecular weight isabove that needed for entanglement formation (for polystyrene, this is approxi-mately 35,000), the presence of these entanglements temporarily arrests the fall inmodulus on crossing Tg This region of almost constant modulus is called therubbery plateau, and the result is a rubbery polymer Because crystals act in amanner similar to entanglements, the modulus of a semicrystalline polymer does

FIGURE12.7 Effect of temperature on Young’s modulus of polystyrene (From Ref.12.) Reprinted with permission from J Appl Phys., vol 28, Rudd, J F., and E F Gurnee:Photoelastic properties of polystyrene in the glassy state: II Effect of temperature, 1096–

1100, 1957 Copyright 1957 American Institute of Physics

FIGURE12.8 Effect of temperature on the stress–strain curves of polystyrene melts.(From Ref 13.)

FIGURE12.9 Qualitative effect of temperature on the elastic modulus of polymers

not fall as precipitously as that of amorphous polymers for temperatures betweenthe Tgand the melting point of the crystals Of course, if chemical cross-links arepresent, the polymer cannot flow and the temperature variation of the modulusabove Tg is given by the theory of rubber elasticity Understanding and relatingmechanical properties of a semicrystalline polymer to the different variables thatcharacterize its structure has been discussed inChapter 11and is treated in detail

by Samuels [14]

As discussed in Chapter 2, the glass transition temperature separates regions ofdramatically different polymer properties In particular, a polymer behaves like ahard, brittle, elastic solid below Tg In this glassy region, the motion of polymerchains is frozen and strain occurs by the stretching of bonds The elastic modulusdecreases with increasing temperature On heating above Tg, an entangled,amorphous polymer displays a rubbery region in which it is soft and pliabledue to the ability of polymer chain segments and entire polymer chains to movepast each other in a reversible manner In this region, the elastic modulus canincrease with an increase in temperature; this property has been explainedtheoretically in Chapter 10 Structural applications clearly require a polymer Tgabove room temperature, whereas applications where material flexibility isimportant, such as in films used for packaging, require that the Tg be belowroom temperature

Although we can use observations of the change in mechanical properties

as a means of measuring Tg, we also find that thermodynamic properties changeslope on going through the glass transition Thus, if we plot the volume of asample or its enthalpy as a function of temperature, behavior depicted qualita-tively inFigure 12.10is observed: The slope in the liquid phase is larger than theslope in the solid phase By contrast, for a crystalline solid, there would be adiscontinuity or jump in the value of these thermodynamic variables at thecrystalline melting point Note that all polymers exhibit a Tg, but only crystal-lizable ones show a Tm(melting temperature); the latter phenomenon is called afirst-order transition, whereas the former is called a second-order transition.Clearly, the specific heat of the rubbery phase exceeds that of the glassy phase.The exact temperature where the change in slope occurs, though, depends on thecooling rate, and we obtain a range, albeit a narrow one, for the transitiontemperature This happens because the rearrangement of polymer molecules into

a glassy structure is a kinetic process The greater the time available for thetransition is, the more orderly the packing and the lower the observed Tg Thiseffect, however, is reversed on rapid heating, and the slowly cooled materialovershoots the original Tg This change in Tg can be related to the free volume

mentioned inChapter 2 To recapitulate, the polymer free volume is the difference

in the sample volume and the actual volume occupied by the atoms andmolecules The free volume is zero at absolute zero temperature and it increases

as the temperature increases Slow cooling allows for a closer approach toequilibrium and a lower free volume relative to material subjected to rapidcooling Thus, the slowly cooled sample has to be heated to a higher temperature

in order that there be enough free volume for the molecules to move around, andthis implies a higher Tg In addition to changes in Tg with cooling rate, we alsoobserve volume relaxation when a polymer sample that was rapidly cooled issubsequently heated to a temperature close to Tg and held there for some time.Material shrinkage also occurs, accompanied by changes in the mechanicalproperties of the solid polymer The phenomenon is known as physical aging[15] and is the subject of considerable research because of its influence onproperties such as creep [16]

The glass transition temperature of a polymer depends on a number offactors, including the polymer molecular weight The molecular-weight depen-dence can be seen in Figure 12.11, where the Tg of polystyrene is plotted as afunction of the number-average molecular weight [3,17] These data can berepresented mathematically by the following equation [18]:

Tg¼ Tg1
K

M

Mn

ð12:3:1ÞThis variation of Tgwith molecular weight can again be related to the free volume[19] As the molecular weight decreases, the number density of chain endsincreases Because each chain end is assumed to contribute a fixed amount of free

FIGURE12.10 Variation of volume or enthalpy of polymers with temperature

volume, the total free volume increases on lowering the molecular weight, whichexplains the data of Figure 12.11 On increasing the chain length beyond a certainvalue, the contribution of chain ends becomes negligible and Tg becomesconstant.

If it is assumed that the ratio of the volume of the polymer chain segmentthat moves to the free volume associated with that segment is the same for allpolymers at the glass transition temperature, the variation of Tg with chemicalstructure becomes easy to understand [19] Any structural change that increasesthe segmental volume requires a larger free volume per segment and results in alarger Tg because, as previously explained, the free volume increases onincreasing temperature Thus, Tg increases as a result of increasing chainstiffness, adding stiff or bulky side groups, and introducing steric hindrances.Similarly, hydrogen-bonding raises the Tg because such a polymer expands lessthan a non-hydrogen-bonded polymer on increasing temperature Consequently, ahigher temperature is necessary to get the same free-volume level Finally, thepresence of plasticizers or low-molecular-weight additives increases the freevolume and lowers the Tg; plasticizers such as dioctyl phthalate are routinelyadded to PVC to convert it from a rigid to a more flexible material The glasstransition temperatures of common polymers are listed in the Polymer Handbook[20], and selected values are given inTable 12.1

One of the most convenient methods of measuring Tg is through the use of

a differential scanning calorimeter (DSC) [21] The principle of operation of thisinstrument is shown schematically inFigure 12.12 A DSC contains two sampleholders, each provided with its own heater The actual sample is placed in one ofthe sample holders in an aluminum pan and the other sample holder contains an

FIGURE12.11 Glass transition temperature of polystyrene as a function of Mn asdetermined by various methods: (s) and (d) dilatometry, (j) Differential thermal analysis(DTA), (u) differential scanning calorimetry, () electron spin resonance (From Ref 3.)

empty pan The temperature of both the sample holders is increased at a constantrate, such as 10C=min, and we measure the difference in the energy H supplied

to the two pans to keep them at the same temperature at all times From an energybalance, it is obvious that the rate of differential heat flow must be as follows:dH

of the kind shown in Figure 12.13 for a sample of amorphous nylon If the

TABLE12.1 Glass Transition Temperature ofCommon Polymers

polymer is semicrystalline, it must be quenched from the melt state rapidly to give

a wholly amorphous structure; otherwise, the presence of crystals can impede themotion of polymer chains and result in a Tg value that is higher than the truevalue For some very crystallizable polymers such as nylon 66, amorphoussamples cannot be obtained and a DSC fails to even pick up a glass transition Insuch a case, we turn to dynamic mechanical analysis, wherein a polymer sample,whether glassy or rubbery, is deformed in an oscillatory manner (in tension orshear, as appropriate) such that the maximum strain amplitude is infinitesimal inmagnitude

If a polymer is subjected to a sinusoidal strain g of infinitesimal amplitude g0andfixed frequency o,

t ¼ ðt0cos dÞ sin ot þ ðt0sin dÞ cos ot ð12:4:3Þ

FIGURE12.13 DSC thermogram of an amorphous nylon (Tg¼ 153C)

On dividing the stress by the strain amplitude, one obtains the modulus G as

G ¼ G0ðoÞ sin ot þ G00ðoÞ cos ot ð12:4:4Þwhere G0¼ t0cos d=g0 and G00¼ t0sin d=g0 The term G0, called the storagemodulus, is the in-phase component of the modulus and represents storage ofenergy, whereas G00, the loss modulus, is the out-of-phase component and is ameasure of energy loss The ratio of the loss to storage modulus, G00=G0, is tan dand is an alternate measure of energy dissipation One may conduct dynamicexperiments in an isochronal manner by varying the temperature at a fixedfrequency, or in an isothermal manner by varying the frequency at a fixedtemperature The former kinds of experiment are discussed in this section,whereas the latter are considered in the next section

For a perfectly elastic material, stress and strain are always in phase and G0equals the elastic modulus and G00is zero For viscoelastic polymers, on the otherhand, the work of deformation is partly stored as potential energy, and theremainder is converted to heat and shows up as mechanical damping This isindependent of the mode of deformation, which could be extension, shear,bending, or torsion If a polymer is glassy, it will act essentially as an elasticsolid and dynamic experiments will allow us to measure the modulus or stiffness.This value is typically of the order of 109Pa Similarly, in the rubbery region, thepolymer is again elastic but with a much smaller modulus of the order of 106Pa

FIGURE12.14 Polystyrene data: dynamic modulus versus temperature for fractions.Numbers on curves are fraction numbers (Reprinted with permission from Merz, E H., L

E Nielsen, and R Buchdahl: ‘‘Influence of Molecular Weight on the Properties ofPolystyrene,’’ Ind Eng Chem., vol 43, pp 1396–1401, 1951 Copyright 1951 AmericanChemical Society.)

Thus, a plot of storage modulus with temperature will mirror the plot of Young’smodulus versus temperature and allow us to determine the glass transitiontemperature.Figures 12.14and 12.15 show typical data for the storage modulusand tan d values of various polystyrene fractions as a function of temperature Thefrequency range here is 20–30 Hz As expected, the glass transition temperature isapproximately 100C Note that both G00 and tan d go through a maximum at the

Tgbecause the ability of a spring to store energy depends on its modulus [22] Onpassing through the Tg, the polymer goes from a stiff spring to a soft one thatcannot store as much energy The difference in energy is dissipated in thetransition from the glassy to the rubbery states Note that Tg measured usingdynamic mechanical analysis is usually slightly larger than that measured using aDSC This discrepancy increases with increasing frequency of oscillation.Figures 12.14 and 12.15 show data obtained in tension using cast filmsoscillated with the help of an electromagnetic reed vibrator operating atresonance Commercial instruments available today use forced vibrations withoutresonance These are desirable because they allow the user to vary temperatureand frequency over wide intervals For example, in the dynamic mechanicalthermal analyzer (DMTA), an instrument made by the Rheometrics Company, abar sample is clamped rigidly at both ends and its central point is vibratedsinusoidally by the drive clamp The stress experienced by the sample isproportional to the current supplied to the vibrator The strain in the sample isproportional to the sample displacement and is monitored by a nonloading eddycurrent transducer and a metal target on the drive shaft In this instrument, the

FIGURE12.15 Polystyrene data: mechanical dissipation factor versus temperature forfractions Fractions 1, 4, 9, 29, and 34 were tested (Reprinted with permission from Merz,

E H., L E Nielsen, and R Buchdahl: ‘‘Influence of Molecular Weight on the Properties ofPolystyrene,’’ Ind Eng Chem., vol 43, pp 1396–1401, 1951 Copyright 1951 AmericanChemical Society.)

frequency can be varied from 0.033 to 90 Hz and the temperature changed from

150C to 300C Descriptions of other instruments can be found in the book byNielsen and Landel [2] Note that liquidlike materials are often supported on glassbraids [23]

Example 12.2: Determine the storage and loss moduli of a polymer whosemechanical behavior can be represented by the Maxwell element shown earlier in

Figure 12.4

Solution: Because the total strain g is the sum of the individual strains, we have

_gg ¼ _ss

EþsZSubstituting for the strain using Eq (12.4.1) and rearranging gives

_ss þE

Zs ¼ Eg0o cos otwhose solution for t ! 1 is

G0ðoÞ ¼ Eo2y

2

1 þ o2y2; G00ðoÞ ¼ Eoy

1 þ o2y2Dynamic mechanical analysis is an extremely powerful and widely used analy-tical tool, especially in research laboratories In addition to measuring thetemperature of the glass transition, it can be used to study the curing behavior

of thermosetting polymers and to measure secondary transitions and dampingpeaks These peaks can be related to phenomena such as the motion of sidegroups, effects related to crystal size, and different facets of multiphase systemssuch as miscibility of polymer blends and adhesion between components of acomposite material [24] Details of data interpretation are available in standardtexts [1,2,25] In the next section, we consider time–temperature superposition,which is another very useful application of dynamic mechanical data

molecular-weight distribution at temperatures from 130C to 220C The remarkable feature

of these data and similar data on other polymer molecular-weight fractions orother polymer melts is that all of the different curves can be made to collapse into

a single curve by means of a horizontal shift Thus, if we move the curve for

180C to the left until it bumps into the 160C curve, we find that it overlaps with

it nicely and the composite curve extends to lower frequencies The range of data

at 160C, taken to be the reference temperature, can be extended further towardlower frequencies by shifting the 200C and 220C curves to the left as well Tomake the 130C, 140C, and 150C curves line up with the 160C data, though,these curves have to be moved to the right The final result is a single mastercurve, as shown inFigure 12.17 Note that sometimes the different curves have to

be moved slightly in the vertical direction as well to obtain perfect alignment.Figure 12.17 shows master curves for data on other molecular-weight fractionsalso; the molecular weights range from 8900 (curve L9) to 581,000 (curve L18).The reference temperature in each case is 160C Because changes in temperatureappear to be equivalent to changes in frequency or time, the process of generating

a master curve is called time–temperature superposition

FIGURE12.16 Frequency dependence of G0 for narrow-distribution polystyrene L27(molecular weight 167,000) at various temperatures (Reprinted with permission fromOnogi, S., T Masuda, and K Kitagawa: ‘‘Rheological Properties of Anionic Polystyrenes:

I Dynamic Viscoelasticity of Narrow-Distribution Polystyrenes,’’ Macromolecules, vol 3,

pp 109–116, 1970 Copyright 1970 American Chemical Society.)