434 MECHANICAL PROPERTIES OF POLYMERS 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/L 0 , F can accordingly be expressed as follows: F =  ∂E ∂L  T,V +T  ∂F ∂T  p,λ (12.15) The latter equation is useful because it allows one to determine the variation of entropy and internal energy due to the chain orientation at constant volume from the measurement of F as a function of the temperature (L 0 should be measured at each temperature) at constant pressure and elongation. The experimenter then observes that the contribution of internal energy is negligible compared to that of entropy. 12.2.2. Statistical Theory of Rubber Elasticity As indicated previously, an elastomer can be identified with an assembly of N υ chains connected through N µ randomly distributed cross-links that are separated from one another by a quadratic average end-to-end distance ( r 2  0 ) 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/(2πr 2  0 )] 3/2 exp[−3r 2 /2r 2  0 ] (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 G i (r) of this chain is written as G i (r) = H − kT ln[3/(2πr 2  0 )] 3/2 +kT [3r 2 /2r 2  0 ] (12.17) After regrouping the first two terms in the constant C (T) depending on temperature, one obtains for an assembly of N υ elastic chains G(r) = C(T) +N υ kT [3r 2 /2r 2  0 ] The stretching of an elastomeric network changes its Gibbs free energy (G elas ) 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 ELASTIC BEHAVIOR OF ELASTOMERS 435 (G disp ) and similar to that found in the case of gases where there is also volume expansion from V 0 to V has to be introduced: G disp = N µ kT ln(V /V 0 ) In this relation N µ is the number of v-valent cross-links and can be written as N µ = 2N υ /v (12.18) for an ideal network. Under these conditions, G elast is expressed as G elast = G(r) −G(r) 0 +G disp (12.19) Initially, when the network is at rest r 2 =r 2  0 , which gives for G elast G elast = N υ kT [3r 2 /2r 2  0 ] −N µ kT (3/2) −N µ kT ln(V/V 0 ) an expression that can be simplified into G elast = (3N υ /2)kT [(r 2 /r 2  0 ) −1] −(2N υ /v)kT ln(V /V 0 ) (12.20) Unless resorting to coherent neutron scattering, the experimenter does not have direct access to information contained in this expression, that is, to the variation of the dimension of elastic chains as a result of a macroscopic deformation. Models that express the impact of a stress applied at a macroscopic scale—and thus of the deformation 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 by the cross-links of the network resulting from a deformation, but differ about the movements undergone by these cross-links. For the Flory–Rehner affine model, cross-links move proportionally to the macroscopic deformation and remain in a given position of space at constant deformation. In the James–Guth “phantom” model, cross-links are assumed to freely move or fluctuate around an average position corresponding to the affine deformation. The amplitude of such fluctuations is independent of the deformation but depends on the valence of the cross-links and the length of elastic chains: r 2 =(2/v)r 2  0 (12.21) Actually these two models correspond to two extreme situations. The affine model is well appropriate to describe the case of networks made of short elastic chains—in this case, the fluctuation of cross-links is hindered by the presence of adjacent chains—whereas the “phantom” model is better suited to networks comprising 436 MECHANICAL PROPERTIES OF POLYMERS long elastic chains. In the affine model, the macroscopic elongation λ x in the direction x is translated in a corresponding deformation at the molecular level: λ 2 x =x 2 /x 2  0 (12.22) x 0 , y 0 , z 0 being the coordinates of the vector r 0 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 N υ  i x 2 0 = N υ  i y 2 0 = N υ  i z 2 0 = 1 3 N υ  i r 2 0 = N υ  r 2  0 3 (12.23) where N υ denotes the total number of chains present in the sample. Due to the isotropic character, expression (12.22) can be rewritten in the form λ x 2 =x 2 /(r 2  0 /3) In the three directions of space, this gives r 2 =x 2 +y 2 +z 2 =(λ x 2 +λ y 2 +λ z 2 )(r 2  0 /3) Because the volume of the sample before deformation is equal to V 0 = L 3 0 with L 0 = L x,0 = L y,0 = L z,0 , one obtains after deformation V = L x L y L z = λ x λ y λ z L 3 0 = λ x λ y λ z V 0 Taking into account these two last expressions for r 2 /r 2  0 and V /V 0 , the expres- sion for G elas (12.20) can be reexpressed as G elas = N υ (kT /2){[λ 2 x +λ 2 y +λ 2 z −3] −(4/v) ln(λ x λ y λ z )} (12.24) The variation of the Gibbs free energy due to the deformation of the network thus depends on the number of elastic chains, the valence of its cross-links, the elongation λ, 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 λ x = L/L 0 (12.25) ELASTIC BEHAVIOR OF ELASTOMERS 437 Assuming that such stretching occurs without change in volume, one can deduce λ y = λ z = (1/λ x ) 1/2 (12.26) and in the opposite case, λ y = λ z = [(V /V 0 )(1/λ x )] 1/2 (12.27) The expression for the variation of the free energy resulting from a deformation then simplifies to give G elas = N υ (kT /2){[λ 2 +2(V /V 0 )λ −1 −3] −(4/v) ln(V /V 0 )} (12.28) By definition, the force F responsible for the elongation of a sample is written as F = ∂(G elas /∂L) T,V = ∂(G elas /∂λ) T,V /L 0 that is, F = N υ (kT /L 0 )[λ −(V /V 0 )λ −2 ] (12.29) Substituting the stress (σ 11 =F/A 0 with V 0 =A 0 L 0 ) for the force and introducing molar concentrations with k =R/N a, one obtains the expression below for the applied stress at constant volume, σ 11 = υRT (λ −λ −2 ) (12.30) where υ is the molar concentration of elastic chains per unit volume, which is equal to N υ /N a V 0 . Using a slightly different calculation for the variation of the free energy (G elast ), the phantom model leads to a very simple yet slightly modified expression: σ 11 = (υ −µ)RT (λ −λ −2 ) (12.31) where µ denotes the molar concentration of cross-links per unit volume. The swelling of a network by a solvent present in large excess—and hence the network 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 V 0 to V . The volume fraction of polymer is then equal to  2 =V 0 /V . If λ x , λ y , λ z denote the variations in dimensions induced simultaneously by the swelling and the deformations resulting from application of an uniaxial stretching, then one deduces that λ x λ y λ z =1/ 2 .Inotherwords,λ x can be expressed as the product of L 0,s /L 0,d and λ, 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 /V 0 ) 1/3 , one obtains λ x = λ(V /V 0 ) 1/3 = λ/ 1/3 2 (12.32) 438 MECHANICAL PROPERTIES OF POLYMERS From the following equality λ x λ y λ z =1/ 2 , it can be deduced that λ y = λ z = ( 2 λ x ) −1/2 = 1/λ 1/2  1/3 2 (12.33) After introducing the terms corresponding to λ x , λ y ,andλ z in the relation (12.28) giving G elas , and expressing the derivative of G elas with respect to the length L (∂(G elas /∂L) T,V ), the expression of the force F and hence of the stress σ 11 for a swollen network now becomes σ 11 = RT υ 1/3 2 (λ −λ −2 ) (12.34) 12.2.5. Real Behavior of an Elastomer In practice, none of the two models—“affine” and “phantom”—accounts satis- factorily for the behavior of elastomers in the entire spectrum of the strains. The affine model is generally appropriate for small deformations, under conditions of limited motion of cross-links due to the presence of neighboring cross-links and of the entanglements. For larger deformations, when chains are disentangled, the experimental behavior is better described by the “phantom” model. Thus, the ten- sile modulus tends to decrease with the applied deformation and gradually comes close to that predicted by the “phantom” model (Figure 12.2). Mooney and Rivlin took into account this nearly general behavior in elastomers and thus proposed a semiempirical model. At rest, the network is considered isotropic and incompress- ible and is assumed to behave like a Hookean material upon shearing. The authors proposed the following expression for the stress: σ 11 = [2C 1 +(2C 2 /λ)](λ −λ −2 ) where C 1 and C 2 are coefficients that vary with the material under consideration and its molecular characteristics. Contrary to molecular models, the stress is supposed to vary with the deformation applied. 1/l s/(l 2 −l −1 ) affine Mooney-Rivlin phantom Figure 12.2. Variation of the reduced stress versus the reverse of elongation for various models of elastomeric networks. ELASTIC BEHAVIOR OF ELASTOMERS 439 The coefficient C 2 takes higher values when the network contains entanglements and the applied deformation is small. On the other hand, C 2 is close to zero in the case of highly swollen networks ( ≤ 0.2). As for the coefficient C 1 , it can be identified with the RT υ 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 λ x in direction x and a compression λ y =1/λ x in the other direction (y), with its volume (λ x λ y λ z =1) remaining constant. A strain due to shearing is defined as γ = λ x −λ y = λ −λ −1 (12.35) In such a context the second term of expression (12.24) is equal to 0 and the first term (λ x 2 +λ y 2 +λ z 2 −3) becomes (λ 2 +λ −2 −2) = (λ −λ −1 ) 2 (12.36) Thus the variation of the free energy resulting from shearing is written as G elas = N υ (kT /2)(λ − λ −1 ) 2 = N υ (kT /2)γ 2 (12.37) The calculation of the derivative of G elas with respect to γ gives Q = (∂G elas /∂γ) T,V = N υ kT (λ −λ −1 ) = N υ kT γ (12.38) The shear stress (σ 12 , denoted by T in order to avoid any confusion with tensile stress σ 11 ) applies not to a surface as in the case of the tensile stress, but to the entire volume (V 0 )sothat T = Q/V 0 (12.39) Substituting the molar concentration of elastic chains (υ) for their number (N υ ) (k =R/N a and υ =N υ /N a V 0 ) results in T = RT υγ (12.40) 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 [σ 11 = υRT(λ −λ −2 )]. To express the latter relationship in the form of Hooke’s law (σ 11 =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 440 MECHANICAL PROPERTIES OF POLYMERS The relation between the two moduli, G and E, is expressed as E = 3G in agreement with expression (12.4), which was established on the basis of a deformation with constant volume, with a Poisson ratio equal to 1/2. 12.3. VISCOELASTICITY OF POLYMERS 12.3.1. A Specific Property Beyond short periods of time, a polymer cannot be regarded as a purely elastic material for the ratio of the stress applied to the strain undergone does not remain constant with time. Polymers indeed behave simultaneously like Hookean objects and like Newtonian (or non-Newtonian) viscous liquids. The matter in the latter case is not elastic and does not strain but flows under mechanical forces: T = η ˙γ (12.41) 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 (σ 0 ) under isother- 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 t strain purely elastic purely viscous visco-elastic Figure 12.3. Typical behavior of viscoelastic materials. VISCOELASTICITY OF POLYMERS 441 (a) (b) s 0 s t f t f e Figure 12.4. Principle of a creep experiment: (a) Application of constant stress (σ 0 ) between t 0 and t f , (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)/σ 11 and J(t) = γ(t)/T (12.42) In a stress relaxation test, the sample is subjected to a sudden deformation (ε 0 ) that is maintained constant and the variation of the stress—σ 11 (t)orT (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 E(t) = σ 11 (t)/ε 0 (12.43) G(t) = T (t)/γ 0 (12.44) ( a )( b ) e 0 e t s 0 s t Figure 12.5. Principle of a stress relaxation experiment: (a) Application of a constant deforma- tion (ε 0 ), (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 creep and relaxation tests would require monitoring over long periods of time. This 442 MECHANICAL PROPERTIES OF POLYMERS limitation can be overcome through dynamic experiments. The latter involve stress and 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 frequency υ 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 strain only in the limit of small deformations; stress and strain then vary sinusoidally and, in certain cases, completely in phase. When submitted at sufficiently high frequencies, a polymer network also behaves in an exclusively elastic manner within the limit of small deformations. In contrast, stress and strain can be 90 ◦ out of phase when sufficiently low fre- quencies are used, a situation that is characteristic of liquid bodies. At intermediate frequencies, 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 stress applied at a given time in that direction. 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 the maximum 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 behind the stress by the phase angle δ, also called the loss angle. In such a diagram, the strain vector can be divided into two components: the first, 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, and the 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 uniaxial shearing, this stress will be defined as a function of time by the following relation: T (t) = T 0 sin ωt (12.45) A B C d D E H F 0 I Figure 12.6. Stress vector and strain vector split into two components in a dynamic experiment. VISCOELASTICITY OF POLYMERS 443 T 0 denotes the maximum amplitude of this stress and ω denotes its frequency. For purely elastic “Hookean” bodies with no energy dissipated, the strain is written as γ t = γ 0 sin ωt where γ 0 is the maximum amplitude of strain. For real viscoelastic materials, shear deformations trail the applied stress, with some energy being dissipated in viscous resistance (Figure 12.7) which gives the relation γ t = γ 0 sin(ωt −δ) (12.46) As shown earlier, the stress includes two components: one in-phase with the strain and the other one in-advance with respect to the latter; the in-phase component is expressed as T  (ω) = T 0 cos δ (12.47) and the one in-advance is written as T  (ω) = T 0 sin δ (12.48) Similarly, the modulus can be expressed through two components: one in-phase with the direction of deformation (0C) and the other one out-of-phase in advance of 90 ◦ . From these elements, the storage modulus can be defined as the ratio of the in-phase component of the stress to the maximum amplitude of the strain. This storage modulus G  (ω) can be written as G  (ω) = T  (ω)/γ 0 = (T 0 /γ 0 ) cos δ 2p/w d/w t t t g t 0 g 0 −t 0 −g 0 Figure 12.7. Sinusoidal stress (τ) and the corresponding out-of-phase strain (γ). [...]... of a suitable analytical technique Organic and Physical Chemistry of Polymers, by Yves Gnanou and Michel Fontanille Copyright  20 08 John Wiley & Sons, Inc 467 4 68 RHEOLOGY, FORMULATION, AND POLYMER PROCESSING TECHNIQUES Different regimes can be distinguished in the dynamics of an isolated polymer chain that can be monitored using probes sensitive to different time and length scales: • The long-time... the ratio of the out -of- phase component 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... equal to 10/4 of Me (see Table 13.1) 13.1.4 Rheological Behavior of Polymer Melts To determine the movements of the whole chain and those of subchains, various techniques are accessible and available to the experimenter, including dielectric spectroscopy and mechanical spectrometry Dielectric techniques are suitable for the study of polymers in a wide range of frequencies (between 10−2 Hz and 1010 Hz),... expression of the energy dissipated is written as ˙ ε h = σ21 ε = η˙ 2 (12 .83 ) The value of η is very high, but the rate of deformation of the chains (˙ ) is zero ε 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 corresponds to the ratio of the loss modulus to the storage modulus and is thus a measure of the... of shear bands or that of crazes (see Figure 12.16) The existence of shear bands reveals a plastic deformation resulting from the shearing of polymer chains When the deformation is highly localized, these shear bands can even be seen with a naked eye; in general they form a 45◦ angle with the direction of stress Experimentally, compression tests (not elongation ones) favor the formation of shear bands... temperature and various extensional rates similarly to analogues that are measured at constant extensional rates and 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 deformations and homogeneous, isotropic, and amorphous... Tr D T, ρ(T ) T aT (12 .80 ) 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 − Tr ); insofar as Tr is suitably selected the following relation can be relevant: ln aT = − C1 (T − Tr ) C2 + (T − Tr ) (12 .81 ) where C1 and C2 are two constants depending on the polymer and on the selected reference... character 452 MECHANICAL PROPERTIES OF POLYMERS The generalization of the Voigt model corresponds to the connection of a given number of Voigt models As the regular version, the generalized Voigt model is inappropriate 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: n D(t)... the presence of defects such as cracks in the analyzed sample The fracture of a material thus originates from the existence, the growth, and the propagation of cracks which are voids; only one 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 failure that occurs in the elastic range of the stress–strain... capable of breaking carbon–carbon bonds 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 462 MECHANICAL PROPERTIES OF POLYMERS The energy balance is expressed as follows: on the one hand, the introduction of . λ y ,and z in the relation (12. 28) giving G elas , and expressing the derivative of G elas with respect to the length L (∂(G elas /∂L) T,V ), the expression of the force F and hence of the. stress (τ) and the corresponding out -of- phase strain (γ). 444 MECHANICAL PROPERTIES OF POLYMERS The ratio of the out -of- phase (in-advance of 90 ◦ ) component of the stress to the strain corresponds. materials. VISCOELASTICITY OF POLYMERS 441 (a) (b) s 0 s t f t f e Figure 12.4. Principle of a creep experiment: (a) Application of constant stress (σ 0 ) between t 0 and t f , (b) Measurement of the deformation