ORGANIC AND PHYSICAL CHEMISTRY OF POLYMERS phần 8 ppsx

Thus, a sample subjected to shearing undergoes a stretching λx in direction x and a compressionλy= 1/λx in the other direction y , with its volume λx λy λz= 1 Gelas= NυkT / 2λ − λ−12= Nυ

pressure to compensate Such an experiment is thus not practical Because Flory

showed that (∂S /∂L)T,V corresponds to (∂F /∂T )p,λ where λ denotes the relative

elongation L/L0, F can accordingly be expressed as follows:

the measurement of F as a function of the temperature (L0 should be measured

at each temperature) at constant pressure and elongation The experimenter thenobserves that the contribution of internal energy is negligible compared to that ofentropy

12.2.2 Statistical Theory of Rubber Elasticity

As indicated previously, an elastomer can be identified with an assembly of Nυchains connected through Nµ randomly distributed cross-links that are separatedfrom one another by a quadratic average end-to-end distance ( r20) satisfying a

Gaussian distribution function P (n, r ) (see Chapter 5) In the following treatment,

the network is considered ideal, without dangling chains and entanglements

P (n, r) = [3/(2πr20)]3/2exp[−3r2/2r20] (12.16)

On the one hand, this function can be used to calculate the probability of finding a

given chain end of the network in the spherical envelope of radius r and thickness

dr , with the other end corresponding to the origin; on the other hand, the function

can be used to calculate the entropy of the same chain:

S = k ln P (n, r)

where k is Boltzmann’s constant.

Thus the free energy Gi(r ) of this chain is written as

G i (r) = H − kT ln[3/(2πr20)]3/2 + kT [3r2/2r20] (12.17)

After regrouping the first two terms in the constant C (T ) depending on temperature, one obtains for an assembly of Nυ elastic chains

G(r) = C(T ) + Nυ kT[3r2/2r20]

The stretching of an elastomeric network changes its Gibbs free energy (
Gelas)

in two ways: on the one hand, by inducing conformational changes within each

elastic chain of the network [G(r ) − G(r0)] and, on the other hand, by modifying

the spatial distribution of the cross-links A term corresponding to the dispersion

(
Gdisp) and similar to that found in the case of gases where there is also volume

expansion from V0 to V has to be introduced:

Gdisp= Nµ kT ln(V /V0)

In this relation Nµ is the number of v-valent cross-links and can be written as

for an ideal network

Under these conditions,
Gelastis expressed as

Gelast= G(r) − G(r)0 +
Gdisp (12.19)Initially, when the network is at restr2 = r20, which gives for
Gelast

Gelast= Nυ kT[3r2/2r20]− Nµ kT ( 3/2) − Nµ kT ln(V /V0)

an expression that can be simplified into

Gelast= (3Nυ / 2)kT [( r2/r20) − 1] − (2Nυ /v)kT ln(V /V0) (12.20)Unless resorting to coherent neutron scattering, the experimenter does not havedirect access to information contained in this expression, that is, to the variation ofthe dimension of elastic chains as a result of a macroscopic deformation Modelsthat express the impact of a stress applied at a macroscopic scale —and thus of thedeformation undergone —on the molecular level have been proposed

12.2.3 ‘‘Affine’’ and ‘‘Phantom’’ Models

The two models postulate an affine displacement of the positions occupied bythe cross-links of the network resulting from a deformation, but differ about themovements undergone by these cross-links For the Flory– Rehner affine model,cross-links move proportionally to the macroscopic deformation and remain in agiven position of space at constant deformation In the James –Guth “phantom”model, cross-links are assumed to freely move or fluctuate around an averageposition corresponding to the affine deformation The amplitude of such fluctuations

is independent of the deformation but depends on the valence of the cross-linksand the length of elastic chains:


r2 = (2/v)r20 (12.21)Actually these two models correspond to two extreme situations The affine model iswell appropriate to describe the case of networks made of short elastic chains —inthis case, the fluctuation of cross-links is hindered by the presence of adjacentchains —whereas the “phantom” model is better suited to networks comprising

long elastic chains In the affine model, the macroscopic elongation λx in the

direction x is translated in a corresponding deformation at the molecular level:

λ2

x0 , y0, z0 being the coordinates of the vector r0 connecting the two ends of a given

elastic chain at rest, and x , y , z those of the same stretched chain Since the sample

considered is isotropic in the initial state, one can write

where Nυ denotes the total number of chains present in the sample

Due to the isotropic character, expression (12.22) can be rewritten in the form

Taking into account these two last expressions forr2/r20 and V /V0, the

expres-sion for
Gelas (12.20) can be reexpressed as

Gelas= Nυ (kT / 2){[λ2

x+ λ2

y+ λ2

z − 3] − (4/v) ln(λ xλyλz )} (12.24)The variation of the Gibbs free energy due to the deformation of the networkthus depends on the number of elastic chains, the valence of its cross-links, theelongationλ, and the temperature, but not on the chemical nature of the network

12.2.4 Uniaxial Stretching of Elastomers

In an uniaxial stretching experiment in the direction x , elongation is expressed as

Assuming that such stretching occurs without change in volume, one can deduce

λy = λz = (1/λ x ) 1/2 (12.26)and in the opposite case,

λy = λz = [(V /V0 )( 1/λx )]1/2 (12.27)The expression for the variation of the free energy resulting from a deformationthen simplifies to give

molar concentrations with k = R/N a, one obtains the expression below for the

applied stress at constant volume,

The swelling of a network by a solvent present in large excess —and hence thenetwork deformation—can be treated in the same manner as indicated previously.Such a swelling occurs in an isotropic manner, with the volume of the network

changing from V0to V The volume fraction of polymer is then equal to 2= V0 /V

If λx, λy, λz denote the variations in dimensions induced simultaneously by theswelling and the deformations resulting from application of an uniaxial stretching,then one deduces thatλxλyλz = 1/2 In other words, λxcan be expressed as the

product of L 0,s /L 0,d andλ, the deformation due uniquely to the stress applied to

the sample (L 0,s and L 0,d being lengths of the unstretched sample in swollen and

dry states, respectively) Since L 0,s /L 0,d is equal to (V /V0)1/3, one obtains

λx = λ(V /V0 ) 1/3 = λ/ 1/3

(12.32)

From the following equalityλxλyλz = 1/2, it can be deduced that

λy = λz = (2λ x ) −1/2 = 1/λ 1/2  1/32 (12.33)After introducing the terms corresponding toλx,λy, andλz in the relation (12.28)

giving
Gelas, and expressing the derivative of
Gelaswith respect to the length L (∂(
Gelas/∂L)T,V), the expression of the force F and hence of the stressσ11 for aswollen network now becomes

σ11= RT υ 1/3

12.2.5 Real Behavior of an Elastomer

In practice, none of the two models —“affine” and “phantom” —accounts factorily for the behavior of elastomers in the entire spectrum of the strains Theaffine model is generally appropriate for small deformations, under conditions oflimited motion of cross-links due to the presence of neighboring cross-links and

satis-of the entanglements For larger deformations, when chains are disentangled, theexperimental behavior is better described by the “phantom” model Thus, the ten-sile modulus tends to decrease with the applied deformation and gradually comesclose to that predicted by the “phantom” model (Figure 12.2) Mooney and Rivlintook into account this nearly general behavior in elastomers and thus proposed asemiempirical model At rest, the network is considered isotropic and incompress-ible and is assumed to behave like a Hookean material upon shearing The authorsproposed the following expression for the stress:

The coefficient C2takes higher values when the network contains entanglements

and the applied deformation is small On the other hand, C2 is close to zero in the

case of highly swollen networks ( ≤ 0.2) As for the coefficient C1, it can be

identified with the RTυ term of statistical models

12.2.6 Shearing of the Networks

Contrary to an uniaxial stress, shearing implies deformations in two opposite

direc-tions (x , y ) of space, with the third (z ) being preserved from any change Thus,

a sample subjected to shearing undergoes a stretching λx in direction x and a

compressionλy= 1/λx in the other direction (y ), with its volume (λx λy λz= 1)

Gelas= Nυ(kT / 2)(λ − λ−1)2= Nυ(kT / 2)γ2 (12.37)

The calculation of the derivative of
Gelas with respect toγ gives

Q = (∂
Gelas /∂ γ) T ,V = Nυ kT (λ − λ−1) = Nυ kTγ (12.38)The shear stress (σ12, denoted by T in order to avoid any confusion with tensile

stress σ11) applies not to a surface as in the case of the tensile stress, but to the

entire volume (V0) so that

Substituting the molar concentration of elastic chains (υ) for their number (Nυ)

(k = R/N a andυ = Nυ/NaV0) results in

The shearing modulus (G) corresponds to RTυ This term is also found in the

expression of the tensile stress but with a different strain component [σ11=

υRT (λ − λ−2)] To express the latter relationship in the form of Hooke’s law

(σ11= Eε), one can observe that ε = λ − 1 and hence λ − λ−2= 1 + ε − (1 + ε)−2

≈ 3ε Thus the Young modulus (E) corresponds to

E = 3υRT

The relation between the two moduli, G and E , is expressed as

where ˙γ denotes the rate of shearing and η the viscosity of this liquid

A body exhibiting simultaneously elastic properties (which are independent of

time) and a viscous behavior (which depends by definition on time) is called

vis-coelastic (Figure 12.3) Depending on the time scale of the experiment, either the

elastic or the viscous component dominates

Viscoelasticity is typical of polymers; it can be characterized through three types

of experiments, creep, stress relaxation, and dynamic mechanical analyses

In a creep experiment, a body is subjected to a constant stress (σ0) under mal conditions and the variation of its dimensions is followed as a function of time.After a rapid application of the stress to the sample —whatever the nature of this

isother-t

strain

purely elastic

purely viscous visco-elastic

Figure 12.3 Typical behavior of viscoelastic materials.

(a) (b)

s0 s

e

Figure 12.4 Principle of a creep experiment: (a) Application of constant stress (σ0) between t0

and tf, (b) Measurement of the deformation as a function of time.

stress, uniaxial or caused by shearing—, the strain [ε(t)] is recorded as a function

of time (Figure 12.4) One can then define the elongational compliance [D (t )] and the shearing compliance [J (t )] of this sample in the following manner:

D(t ) = ε(t)/σ11 and J (t ) = γ(t)/T (12.42)

In a stress relaxation test, the sample is subjected to a sudden deformation

(ε0) that is maintained constant and the variation of the stress —σ11(t ) or T (t )— is

followed as a function of time (Figure 12.5)

Then the tensile [E (t )] or shearing relaxation modulus [G(t)] can be defined as

e0 e

t

s0 s

t

Figure 12.5 Principle of a stress relaxation experiment: (a) Application of a constant

deforma-tion ( ε0 ), (b) Variation of the stress as a function of time.

12.3.2 Dynamic Mechanical Analysis

A complete description of the viscoelastic behavior of a polymer through creepand relaxation tests would require monitoring over long periods of time This

limitation can be overcome through dynamic experiments The latter involve stressand strain that vary periodically in a sinusoidal manner The sinusoidal oscillation

of frequency (υ) corresponds to cycles/s or ω (= 2πυ) corresponds to rad/s

Prac-tically, a sinusoidal experiment at frequencyυ is equivalent to creep or relaxation

experiments at a time t= 1/ω

When a dynamic stress is applied, the latter is directly proportional to the strainonly in the limit of small deformations; stress and strain then vary sinusoidallyand, in certain cases, completely in phase When submitted at sufficiently highfrequencies, a polymer network also behaves in an exclusively elastic manner withinthe limit of small deformations

In contrast, stress and strain can be 90◦ out of phase when sufficiently low quencies are used, a situation that is characteristic of liquid bodies At intermediatefrequencies, the phase difference between stress and strain is less pronounced.Sinusoidal variations of the stress can be represented as a rotating vector (0A)(Figure 12.6) whose projection (0B) on the vertical axis corresponds to the stressapplied at a given time in that direction

fre-In such a representation, the vector 0A rotates at frequency ω, which is that

of the sinusoidal stress, with the direction 0A thus corresponding to that of themaximum stress The cycle of deformation undergone by the sample is symbolized

by the vector 0C, which rotates at the same frequency (ω); its projection 0D on

the vertical axis denotes the deformation of the sample The strain thus lags behindthe stress by the phase angleδ, also called the loss angle.

In such a diagram, the strain vector can be divided into two components: thefirst, 0E, in-phase with the stress and the second, 0F, out-of-phase The projection

of 0E on the vertical axis (0H) reflects the strain in-phase with the stress, andthe projection of 0F on the same axis (0I) corresponds to the strain out-of-phase(90◦) with the stress in that direction If a sinusoidal stress is applied by uniaxialshearing, this stress will be defined as a function of time by the following relation:

A B

C d D

E H

T0 denotes the maximum amplitude of this stress andω denotes its frequency For

purely elastic “Hookean” bodies with no energy dissipated, the strain is written as

γt = γ0sinωt

whereγ0 is the maximum amplitude of strain

For real viscoelastic materials, shear deformations trail the applied stress, withsome energy being dissipated in viscous resistance (Figure 12.7) which gives therelation

As shown earlier, the stress includes two components: one in-phase with the strainand the other one in-advance with respect to the latter; the in-phase component isexpressed as

The ratio of the out-of-phase (in-advance of 90◦) component of the stress to the

strain corresponds to the loss modulus G(ω):

G( ω) = T( ω)/γ0 = (T0 /γ0)sinδ

Likewise, one can also define the instantaneous compliance or storage compliance

J(ω); this corresponds to the ratio of the in-phase strain component to the stress;

the loss compliance [J(ω)] then corresponds to the ratio of the out-of-phase

com-ponent of the strain lagging behind the stress to the latter G(ω) and J(ω) reflect

the propensity of a sample to retain a supplied mechanical energy and to restore

it in the form of an elastic strain; G(ω) and J(ω), on the other hand, reflect the

loss of this same energy due to viscous dissipation (flow)

Under the conditions of a brief shearing of low amplitude, the storage

mod-ulus [G(ω)] is equivalent to the previously defined shearing modulus (G) At a

given temperature and frequency, a sample can thus be identified by

characteris-tics such as G(ω), G(ω), J(ω), J(ω), and tan δ For the sake of didactics, two

different representations —rotating vector (Figure 12.6) and sinusoidal shear stress(Figure 12.7)—have been used above to describe the behavior of a system sub-mitted to a dynamic experiment; in practice, the experimenter prefers to resort tocomplex variables to express characteristics such as the dynamic mechanical mod-

ulus Thus, for a sinusoidal shearing, one can define a complex modulus [G*(ω)]

whose real component is the storage modulus (G) and the imaginary one is the

loss modulus (G):

G∗( ω) = G( ω) + iG( ω) (12.49)

In a similar manner, the complex compliance J *(ω) is the sum of two components:

one of these is real, corresponding to the storage component [J(ω)], while the other

is imaginary, corresponding to the loss compliance:

J∗( ω) = J( ω) − iJ( ω) = 1/G∗= 1/[G( ω) + iG( ω)] (12.50)

Using complex notations, the complex shear modulus (G*) becomes

|G∗| = [(G)2+ (G)2]1/2 (12.51)which gives

tanδ = G( ω)/G( ω) = J( ω)/J( ω) (12.52)tan δ is thus a measure of the ratio of the energy dissipated by the sample in the

form of heat to the energy retained and subsequently restored during one cycle ofsinusoidal deformation The heat produced results from the chainsflow and fromtheir friction; it mirrors the damping capacity of the sample The energy dissipated

by cycle is written asπγ2G

Dissipative phenomena can also be accounted for through the viscosity, whichcorresponds to the ratio of a stress to the rate of deformation—in fact the

shearing—and can also be expressed in a complex manner The complex viscosity(η*) is written as follows:

η∗( ω) = G∗( ω)/iω = [(G( ω)/ω) − i(G( ω)/ω)] = η( ω) − iη( ω) (12.53)From this it follows thatη(ω) is equal to G(ω)/ω and η(ω) is equal to G(ω)/ω

Like G,η(ω) (also called dynamic viscosity) reflects the dissipation of energy and

is equivalent toη0(viscosity in continuous flow) within the range of low-frequencyshearing

12.3.3 Linear Viscoelasticity

Linear viscoelasticity is characterized at a given temperature and frequency by alinear dependence of stress relaxation on strain even if they are not necessarilyin-phase It can be accounted for by combining both the Hooke and Newton lawswhich stipulate a linear variation of the stress with the strain for the first and ofthe stress with the rate of deformation for the second

This proportionality between the cause and the effect can be expressed in a more

general manner by the Boltzmann superposition principle The effect A produced by

the sum of several causes or loads is equivalent to the sum of the effects produced

by each of these causes taken separately:

In addition to the Boltzmann superposition principle, the second consequence

of linear viscoelasticity is the time– temperature equivalence, which will be

described in greater detail later on This equivalence implies that functions such

as σ = f (ε), but also moduli, behave at constant temperature and various

exten-sional rates similarly to analogues that are measured at constant extenexten-sional ratesand various temperatures Time- and temperature-dependent variables such as the

tensile and shear moduli (E, G) and the tensile and shear compliance (D, J ) can be transformed from E = f (t) into E = f (T ) and vice versa, in the limit of small defor-

mations and homogeneous, isotropic, and amorphous samples These principles areindeed not valid when the sample is anisotropic or is largely strained

12.3.4 Boltzmann Superposition Principle

The Boltzmann superposition principle states that:

1 The stress or strain status of a sample at a given instant mirrors all the stresses

or strains it has undergone

2 Each new stress contributes independently to the final deformation, whichthus corresponds to the algebric sum of the various loads applied In otherwords, a deformation or a recovery caused by an additional load or its removal

is independent of previous loadings or unloadings

In a tensile experiment, if the initial constraint σ11,0 is followed by a secondoneσ11,1at time t1, the deformation resulting only fromσ11,1is written as

ε(t) = ε11,1 D(t − t1 ) (12.55)with the total strain being expressed as

In the particular case where the initial constraint and thus the stress are stopped at

time t1 (creep experiment Figure 12.8), ε (t) becomes

ε(t) = σ11,0 [D(t) − D(t − t1 )] (12.57)The stressσ11,0applied at time t= 0 results in a deformation equal to ε1 at time t1

Discontinuing this stress at time t1 causes the initial deformation (ε1) to decrease

to a value equal to ε at t2 The difference in deformation
ε1 is expressed as

e s

Figure 12.8 Illustration of the Boltzmann superposition principle for a creep experiment.

ε2 can also be written as

ε2 = ε2− ε1+
ε1 =
ε1+ ε2−1

a relation in which ε2−1 denotes the additional deformation— with respect toε1—that this sample would undergo at time t2, if a putative constraint σ11,1 isapplied in addition toσ11,0 at time t1

In other words, the deformation caused by an additional load or stress is pendent of the preceding ones By adopting a similar reasoning for the case of arelaxation experiment, one obtains the following relation for the stress versus time:

12.3.5 Empirical Analogical Models

As indicated previously, the behavior of a polymer satisfying the criterion of linearviscoelasticity and hence the Boltzmann principle can be described as a linearcombination of Newtonian viscous and Hookean elastic behaviors

To illustrate this linear combination, various analogical models with no tionship with the molecular nature of the phenomena were proposed, identifying apolymer with a combination of springs and dashpots The spring can be strainedwithout inertia and thus reflects a purely elastic mechanical behavior whereas dash-pots, which are pistons moving in cylinders filled with a viscous liquid, cannotrespond instantaneously to a stress These two elements were thus associated undervarious combinations to simulate the response of a viscoelastic body to a mechan-ical stress

rela-12.3.5.1 The Maxwell Model In this model the spring and the dashpot are

connected linearly in series:

σ , ε

σr, εr , E σa, εa, η

As was previously mentioned, stress (σ) and strain (ε) undergone by a spring can

be related by the Hooke law:

In a dashpot, stress and strain obey the Newton law:

σa = ηdεa dt

In this model, the total stress (σ) felt by the spring and by the dashpot is identical:

After noting that the ratio η/E = trelax has the dimensions of time and σ11,0/ε0

corresponds to E , one obtains

and trelaxis called the relaxation time.

For very short times, the Maxwell model predicts that the material behavesessentially like a spring for a relaxation experiment Over long times, it actually

t/trelax0

1

Figure 12.9 Behavior of a material agreeing with Maxwell model for a relaxation test.

behaves only like a dashpot, where the stress decreases exponentially with time in

a sample subjected to a constant strainε0 For a time comparable to trelax, which

is the time required to reduce the stress by a factor (1/e) (i.e., by 36.8%) with

respect to initial one, the model predicts a response including simultaneously thecontribution of both the spring and the dashpot (see Figure 12.9)

The treatment of a creep experiment by the same Maxwell model leads tothe following equations In such an experiment, the stress is constant: σ = σ11,0,

d σ/dt = 0 and thus the relation (12.61) can be written as

Upon removal of the load (or stress) at time t , a sample corresponding to the

Maxwell model will retract by a value equal to its elastic contribution (ε0= σ11,0/E ),

but will be permanently strained by a valueε(t) = (σ11,0/η)t In a creep experiment,

such a sample behaves at the onset like an elastic solid and then like a viscousliquid; thus it exhibits the characteristics of a viscoelastic liquid However, thisMaxwell model also predicts a linear deformation as a function of time whensubjected to a constant stress which is not realistic, because no such example could

be found in the field of polymers (see Figure 12.10)

12.3.5.2 The Voigt Model It consists of a spring and a dashpot connected in

the spring and the dashpot (σ = σr+ σa).

In the context of Hookean solids and Newtonian liquids, the Voigt model thuspredicts

where tret= retardation time, which is different from the relaxation time The creep

compliance is described by the following function:

s11,0 /E

Maxwell

Voigt e

Figure 12.10 Behavior of polymer meeting the requirements of either Maxwell model or Voigt

model in a creep experiment.

and then more slowly in a second step The retardation time (tret) denotes the timerequired for the sample to attain (1− 1/e)th (≡ 63%) of the deformation (ε f) at the final time tfif the load is maintained constant At infinite time, the deformation ofthe sample is equal to ε0= σ11,0/E (see Figure 12.10).

A sample complying with the Voigt model behaves like a viscoelastic solid,

and thus like a liquid at short times and like a solid material between 0 < t < tf.This behavior makes the Voigt model more relevant than the Maxwell model fordescribing a creep experiment If the load applied to the Voigt solid is suppressed

at the time tf, its deformation then reduces and this sample recovers its initialdimension (ε = 0) If the Maxwell model is more appropriate to account for a stress

relaxation experiment and the Voigt model to describe a creep experiment, neither ofthese two analogical models is accurate enough to describe the viscoelastic response

of a polymer in its complexity These models comprise one unique retardation orrelaxation time, implying that all chains rearrange themselves in a same lapse oftime, which is not realistic

To overcome these limitations, more elaborate models combining a greater ber of springs and dashpots have been proposed with the idea of generalizing thetwo previously described types of models

num-For a relaxation experiment, the generalization of the Maxwell model—that is,

a given number of Maxwell elements connected in parallel—implies a relaxationmodulus that is the sum of the moduli of all the elements:

Upon associating a Maxwell element and a spring connected in parallel, one

obtains the Zener model, which describes very satisfactorily the behavior of highly

cross-linked polymers Such behavior is characterized by an instantaneous ity followed by a phase of retarded elasticity, with the deformation exhibiting acompletely reversible character

elastic-The generalization of the Voigt model corresponds to the connection of a givennumber of Voigt models As the regular version, the generalized Voigt model isinappropriate to describe the relaxation of a polymeric material In the context of

a creep experiment, the creep compliance is written as the sum of the compliance

of the various Voigt elements:

Upon associating a Voigt element and a Maxwell element in series, one obtains the

Burgers model, which is well-suited to describe the creep behavior of a

thermo-plastic polymer: the behavior of such a material is characterized by an instantaneouselasticity followed by a phase of retarded elasticity, but the strain retains in thiscase an irreversible character

12.3.6 Principle of Time– Temperature Superposition

Leaderman first suggested the existence of time –temperature equivalence in a coelastic material This principle can be illustrated through the case of a samplesubjected to a tensile experiment during which the rate of stretching is maintainedconstant If this sample is identified with a Maxwell element, the variation of thestress (which is zero at the beginning of the experiment) that it undergoes withtime can be written as

vis-dσ

dt = E dε

dt −

σ

The profile of the variation ofσ versus ε shows a large initial increase which then

softens independently of η (dε/dt), a term that normally remains constant

In the extreme situation of η(dε/dt) tending toward infinity, the Hooke law

(σ = Eε) applies; in contrast, if η(dε/dt) tends toward 0, σ is equal to zero At high

drawing rates, the sample thus behaves like a rigid material (E is high) and like a rubber (E is low) at low rates With the help of relation (12.78), the variation ofσ

versusε can be calculated for various values of η(dε/dt), with the two families of

curves shown below indicating through their similarity that establishing E = f (t)

or E = f (T ) are equivalent The function E = f (t) can thus be transformed into a

function E = f (T ) and vice versa (see Figure 12.11).

The simplest method to carry out this transformation would be to measure the

relaxation modulus [E (t )] for constant deformation at various temperatures and to

plot its variation as a function of time For the lowest temperatures, very long times

Figure 12.11 Stress– strain diagram: (a) Variation calculated for a Maxwell body for various

values ofηdε/dσ (b) Experimental curves obtained for a PMMA sample at various temperatures.

are required to establish a complete curve; on the other hand, at higher tures measurements at short times are sufficient This means that it is practicallyimpossible to plot a complete curve at a given temperature due to the limitationsrelated to measurement times

tempera-However, for measurements arbitrarily carried out at T= 25◦C, one observesthat the value of the relaxation modulus at short times coincides with that measured,for example, at 0◦C at longer times: this equality of the modulus for different pairs

of time and temperature can thus be used to build a whole curve by successive shifts

and superpositions and then to establish a so-called master curve (see Figures 12.12

and 12.13) The latter is drawn starting from a reference temperature that should bechosen in a judicious way (near the polymer service temperature) To each shift of a

curve segment along the time axis is associated a shift factor atwhich corresponds

to the time gap with respect to the reference temperature (T) (see Section 12.2.2)

Figure 12.13 (a) Variation of the Young modulus (E) of polyisobutene (PIB) versus time at

various temperatures (b) Master curve for a reference temperature T= 25 ◦C.

The time –temperature superposition principle can be written in the followingmathematical form:

(T r being the temperature of reference)

In other words, a change of temperature is equivalent to multiplying the time scale

by an appropriate factor To be completely rigorous, it is necessary to take intoaccount the variations of volume with temperature and, accordingly, to rewrite theprevious relation in the form

and for variables derived from dynamic measurements

In addition, Williams, Landel, and Ferry proposed a relation describing the

variation of aT as a function of (T − T r); insofar as Tr is suitably selected thefollowing relation can be relevant:

ln aT = − C1(T − T r )

where C1 and C2 are two constants depending on the polymer and on the selectedreference temperature From measurements carried out on 17 polymers at different

reference temperatures, they proposed a “universal” curve of the variation of aT

as a function of (T − T r) (see Figure 12.14) The same authors also showed that

if the reference temperature (Tr) selected is the glass transition temperature (Tg), then C1 and C2 can be taken equal to 17.4 and 51.6 These constants actually varylittle with the nature of the polymer under consideration, and thus the expression(12.81) becomes

ln aT = − 17.4(T − T g )

51.6 + T − T g

(12.82)

which is called the WLF relation.

Another situation to consider is the response of polymers to a dynamic stressapplied near thermal transitions

As previously shown, the storage modulus (E) characterizes the rigidity of aviscoelastic material It corresponds to the energy retained and restored by the poly-mer; under the conditions of a recoverable deformation, it is equivalent to Young’s

modulus (E ) In a glassy state —that is, below the glass transition temperature —the

modulus takes values in the range of 103–104MPa As previously discussed, this

(T-Tr) ° C

0 4 8 12

Figure 12.14 ‘‘Universal’’ curve giving the variation of aTversus (T − Tr) (in◦C), according toWilliams, Landel and Ferry.

elasticity is of enthalpic origin and thus implies the reversible modification of both

interatomic distances and valence angles The loss modulus (E) which corresponds

to the energy (in the form of heat) dissipated is almost equal to zero as molecularmotions are frozen, thus resulting in an absence of internal frictions Indeed, thechains cannot change their shape in response to the stress imposed

The expression of the energy dissipated is written as

The value ofη is very high, but the rate of deformation of the chains (˙ε) is zero

The energy dissipated and hence the loss modulus are then close to 0

As for the tangent of the loss angle (tan δ), also called the loss factor, it

cor-responds to the ratio of the loss modulus to the storage modulus and is thus ameasure of the energy dissipated versus the energy stored Its value reflects thecapacity of a material to absorb a stress in an elastic manner or to dissipate it byinternal friction A zero or low value of tan δ mirrors a purely elastic behavior,

whereas a high value corresponds to a pronounced nonelastic response

When a polymer sample undergoes a transition from the glassy state to therubbery one upon heating, the polymer segments become more mobile Their move-ments, caused by the softening of the material, now follow the stress imposed: theconversion of the internal friction and of the nonelastic deformation into energy

is now maximal Even if the viscosity of the medium tends to decrease, the rate

of deformation of the chains becomes significant Under these conditions, the loss

modulus (E or G), which measures the energy dissipated, reaches its highestvalue whereas the storage modulus decreases considerably by a factor of 1000 Atstill higher temperature,η decreases to such extent that G tends toward 0 even if

˙ε continues to grow

However, maxima of the loss modulus (E or G) and of the loss factor(tan δ) do not coincide perfectly: as both E or G and E or G moduli change

dramatically in this area, the maximum of tan δ is slightly shifted toward higher

temperatures compared to that of E (or G) at low frequency of shearing (1 Hz).The glass transition temperature can then be identified as the temperature at

which the loss modulus (E, G) or the loss factor reach their maximum value

Dynamic measurements thus provide an efficient means to measure Tg Othertransitions —for instance, second-order transitions —which are related to the move-ments of the chain substituents or due to certain segments can also be detected bydynamical stimuli

12.4 MECHANICAL PROPERTIES AT LARGE DEFORMATIONS

12.4.1 Polymers in Practical Situations

Within the limit of small deformations (the case that has been treated until now),

a polymer can respond in three different ways to an external mechanical stimulus:

• Instantaneous elastic deformation implying a spontaneous reversibility

• Time-dependent viscoelastic deformation, implying simultaneously both ation and reversibility

relax-• Viscous and irreversible deformation above Tg, varying with time

When departing from the limit of small deformations, solid-state polymers (T < Tg)lose their capacity to sustain a reversible deformation beyond the proportionality

limit The permanent deformation undergone is said to be plastic and the limit

of unrecoverable deformations is called the yield point (see Figure 12.15d) The

plastic deformation of a polymer sample results in a reorientation of its chains

on a large scale and can possibly lead to its degradation It concerns both phous and semicrystalline polymers In the latter, sequences of the amorphousparts and the crystalline lamellae tend to align under the effect of an uniaxialelongation which can reorganize the crystalline zones and even break them uponincreasing

amor-Such a chain orientation in the direction of deformation is accompanied by theirplastic deformation; depending upon the proportion of entanglements, it occurs by adisentanglement process The tensile test is the most convenient way to characterizethe mechanical strength of a polymer Before fracture which can be either fragile orductile, four different scenarios of the stress –strain behavior can be contemplated,depending on the scale considered

ductile fracture

Figure 12.15 Stress– strain curves for four typical situations: (a) soft and strong polymer; (b)

soft and ductile polymer; (c) rigid and fragile polymer; (d) rigid and ductile polymer.

A brittle fracture occurs in the elastic range and is associated with small fracture

elongation and high stresses The sample —typically fibers or thermoplastics —breaks perpendicularly to the stress direction without flowing

A ductile fracture corresponds to an irreversible plastic deformation; fracture

elongations may reach in the latter case several hundred percent and are found inlightly cross-linked elastomers

Typically, four different situations can be described:

(a) Polymers of low molar mass and thus free of entanglements are characterized

by a low elastic modulus (E ) and a low tensile strength Their fracture is

generally observed soon after the limit of proportionality of the stress –straincurve (Figure 12.15a);

(b) Elastomers also exhibit a relatively low elastic modulus, but they canundergo high deformations before break Their deformation is almost elastic

until the break point ; the considerable increase in modulus, which precedes

breaking, mirrors an orientation of the sample chains along the deformationaxis (Figure 12.15b).

(c) Thermoplastic polymers, such as polystyrene or poly(methyl methacrylate),are prone to a brittle fracture when elongated below their glass transitiontemperature Their Young modulus is high and their fracture occurs in thelinear part of the stress –strain curve so that their fracture deformation isnecessarily small (Figure 12.15c)

(d) Other thermoplastics, such as polyethylene and certain polyesters or amides, are subject to a ductile fracture: beyond the limit of proportionality

poly-of the stress –strain curve, chains flow, causing the sample to deform

irre-versibly and undergo the so-called necking phenomenon Under the pressure

exerted by the clamps of the tensile testing machine, which holds the testspecimen, a stress concentration builds up near the clamps, causing a lowflow and thus a decrease of the specimen cross section As chain elementsflow, their internal frictions provoke an increase in temperature which, inturn, reduces further the viscosity and favors the flow Beyond the yieldpoint, the necking phenomenon propagates and the sample deforms under

an equal or even lower stress At the end of the test and before the break,

an increase in the stress –strain curve is observed in certain cases, due tothe existence of entanglements The latter behave like physical cross-linksand limit the chain motion (Figure 12.15d)

The previously described typical behaviors are also determined by the rate ofdeformation: a given material can be ductile at low rate of deformation and at hightemperature or undergo a brittle fracture at low temperature and at a high rate ofdeformation

12.4.2 Physical Damaging

Damages undergone by a polymer during plastic deformation appear ically under two forms —that is, through the formation of shear bands or that ofcrazes (see Figure 12.16)

macroscop-The existence of shear bands reveals a plastic deformation resulting from theshearing of polymer chains When the deformation is highly localized, these shearbands can even be seen with a naked eye; in general they form a 45◦angle with thedirection of stress Experimentally, compression tests (not elongation ones) favorthe formation of shear bands in thermoplastic polymers

Figure 12.16 Mechanism of damaging during a plastic deformation: (a) formation of shear

bands; (b) formation of crazes comprising fibrillae, or (c) constituted by microcavities.

The second damaging mechanism generates crazes inside the material, moreparticularly in amorphous and even semicrystalline thermoplastic materials during

an uniaxial tensile stress Contrary to cracks (which are, by definition, open fissureswithout matter between their lips that form and grow from a flow or a defect), thetwo edges of a craze contain a proportion of matter close to 50% The interior

of a craze indeed comprises either fibrillae (diameter 20 nm) of highly stretchedchains and separated by voids or spherical microcavities of 10-nm size, embedded

in the polymer matrix Upon applying an uniaxial stress, crazes will grow dicularly to the direction of stretching, with the fibrillae being oriented parallel tothis same axis The formation of crazes comprises two stages, namely, nucleationand growth (terms used by analogy with those of crystallization) The existence ofstress concentrations inside the matter— in the form of fissures, surface defects, orinclusions —causes the nucleation of crazes The latter grow by extension of thevoids lying between the polymer fibrillae The kinetics of deformation affects therole played by the crazes during an uniaxial stress; at a slower rate of deformation,the chains have time to reorient inside the crazes, thus creating microcavities inthe whole sample The propagation of crazes is then slowed and a large plasticdeformation is obtained before the final ductile fracture

perpen-12.4.3 Brittle Fracture

In such a case, the sample elongation is small (a few percent) at the break point.The ultimate stress (σr) of chains perfectly aligned in the direction of the stress can

be easily calculated from the force required to break a chain and its cross section

From data relative to the bond energies (348 kJ/mol and L= 0.154 nm for a C–C

bond), a theoretical value ofσrequal to 40.8 GPa is found for polyethylene chains

In reality, the ultimate strength measured is much lower than the one calculatedfor an assembly of perfectly oriented chains; this is due to the presence of defectssuch as cracks in the analyzed sample The fracture of a material thus originatesfrom the existence, the growth, and the propagation of cracks which are voids; onlyone of such cracks is enough to cause a macroscopic failure of a sample subjected

to a tensile test One can then speak of a brittle fracture, because of the failurethat occurs in the elastic range of the stress –strain curve for samples that were notdamaged in a prior plastic macroscopic deformation

Griffith established the connection between brittle fractures and the presence ofcracks When working on glassy materials, he noticed that small samples show anultimate stress higher than that of bigger specimens He concluded that the smallerfracture strength of the latter results from the presence of structural defects in largernumber and bigger size At the origin of the Griffith theory, there is an observationmade by Inglis, who found that the presence of a hole inside a material enhancesthe stresses around its circumference This observation played a significant role

in the understanding of the mechanism of propagation of cracks upon applying astressσa Inglis treated mathematically the case of a plate-type specimen including

an ellipse-shaped hole and subjected to an uniaxial stress: the calculation of thenormal stresses (σxx,σyy) in the vicinity of the hole shows that the normal stress

A X sa

sa

sa s

sxx syy

Y

X B A

2a 2b

Figure 12.17 (a) Plates containing an elliptic hole whose dimensions are small compared to

those of the sample (b) Variation of normal stresses in the vicinity of the hole, starting from the

point A of the sample for an applied stressσa.

(σyy) reaches a maximal value (σm ) at the focus point (A), and then it decreases to

take an average value in the rest of the sample Theσxxstress in the perpendiculardirection increases from zero to take a value higher than the applied stress (σa)

before decreasing again (Figure 12.17)

The stress is thus maximal at the edge of the sample hole and can be determinedusing the following ratio:

A stress concentration factor K t = 1 + 2√a/ρ can even be defined where ρ, the

radius of the ellipse curvature, is equal to b2/a.

For a circular hole, the stress amplification would be only equal to 3, but, for a

highly stretched ellipse (a/b= 500), it can reach a factor of 1000 In the presence of

a sharp crack, the local stress can reach values capable of breaking carbon–carbonbonds

12.4.4 Crack Propagation: Griffith’s Theory

Cracks can only propagate if the energy of the system decreases and minimizes,and if the local fracture stress is equal to, or even higher than, the theoretical value;this is the basis of Griffith’s theory

The energy balance is expressed as follows: on the one hand, the introduction

of cracks and their propagation contribute to lower the elastic energy (U1) of the

system; on the other hand, the formation of new surfaces requires a work (W ) that tends to oppose the propagation of cracks The difference (U1− W ) is thus

available for the creation of new crack surfaces; and, on the whole, the energy (U )

of the system studied is written as

where U0 is the elastic energy of a plate-shaped specimen without cracks

As the material considered is elastic, the elastic energy of deformation per ume unit can be identified withσa /2E (E being Young’s modulus).

vol-Since the plate-shaped specimen has a volume V , U0is equal to U0 = σ2/ 2EV

To calculate U1, Griffith specified that the introduction of a crack into a dimensional plate-type specimen entails in its immediate vicinity i.e., in a vol-umeπa2l around this crack, a density of elastic energy close to zero; l represents

two-the sample thickness assumed equal to l Under two-these conditions, U1, which responds to the elastic energy lowered by the introduction of a crack, is in firstapproximation

where q is the Poisson ratio.

Besides, the work (W ) required for the formation of the crack surface can be

expressed as equal to

where  corresponds to the surface free energy of the material per surface unit, and 4a corresponds to the surface of the two lips of an elliptical crack On the whole, U , the potential energy of the system, is established as follows:

U= σ2a 2E V −πa2σ2a

According to Griffith, the equilibrium is reached when the energy released by the

crack propagation (da) is equal to the energy required for the creation of two new

surfaces resulting from this growth:

0 and there is no possibility of the crack to propagate; on the contrary, when

dU /da < 0, the crack is unstable and the sample can experience a catastrophic

fracture (see Figure 12.18)

U1

W Energy

Length of the crack

U

Figure 12.18 Energy criteria determining the propagation of the cracks or their halting.

According to this analysis, the threshold corresponding to equilibrium is ated to a critical stress beyond which the crack is likely to propagate irreversibly:

For a thick plate

σc= π(1 − q 2E2

In the same way and using Griffith’s line of reasoning, a crack critical lengthcan be defined for a given stress, beyond which this crack becomes unstable andgrows

Griffith’s theory describes satisfactorily the behavior of brittle polymers such as

PS or PMMA, and σc as well as a −1/2 vary in accordance with relation (12.93)

However, when these relations are used to calculate , very high values are

found (from 200 to 1500 J·m2), which are well beyond the real surface energies( ∼= 1 J/m2) Such a difference is due to the fact that the Griffith theory does nottake into account the plastic deformation In fact, a plastic deformation occurs inthese polymers at the bottom of their cracks, which act like energy absorbers and

induce a considerable increase of the value of  In this case, H is used instead

of , being more universal than ; H is defined as the elastic energy released per

crack growth unit and is written as

H =1

2F

2dD

where D is the compliance of the sample and F is the load applied.

The temperature at which the mechanical stress is applied is the first factor that

determines the type of fracture of the material For T T g the sample undergoes a

brittle fracture, and for T > Tg it undergoes a ductile one The value of the ultimatestress in a brittle fracture depends not only on the temperature but also on the chain

molar mass and orientation; indeed the ultimate stress varies even for T slightly lower than Tg and is independent of the size of the chains only beyond a criticalmolar mass The transition between a purely brittle behavior at low temperatureand a primarily ductile one at higher temperature occurs in an intermediate zone

(T < Tg) where the two types of mechanisms of fracture can occur, according to

• the thickness of the sample (greater brittleness in case of thick samples), or

• the presence of a notch in the sample (the latter entailing an increase in thefragile–ductile transition)

All these factors indicate that there is no strict correlation between the fragile –

ductile transition temperature and Tg Beyond Td and Tg, polymers are prone to a

pure ductile fracture, which means that they undergo an irreversible plastic mation before breaking