Thermosetting Polymers Part 11 pot

11 Effect of Crosslink Density on Elastic and Viscoelastic Properties 11.1 INTRODUCTION This chapter is devoted to a short description of low-strain mechanical properties of polymers in the solid state and in the glass transition region, with an emphasis on the effect of crosslinking on these properties. There are three degrees of complexity in the description of this behavior, depending on the number of variables taken into account in the constitutive equations under consideration. Lowest Level. At the lowest level, these equations could involve only two variables, the stress s and the strain e: fðs; eÞ¼0 This means that the mechanical behavior is regarded in relatively sharp intervals of time and temperature. Then, engineering moduli are gen- erally sufficient to describe the material’s behavior at low strains. They are interrelated by the following relationships: E ¼ 3Kð1  2nÞ;G¼ 3 2 ð1  2nÞ ð1 þ nÞ K ¼ E 2ð1 þ nÞ ð11:1Þ where E, G, and K are the tensile (Young), shear (Coulomb), and bulk modulus, respectively and n is the Poisson’s ratio. For an ideal elastic body, n ¼ 0:33, leading to E ¼ KandG¼ 3E=8. For an ideal rubber, n ! 0:50 and E ! 3GandK!1. E can be determined from a uniaxial tensile (E ¼ s=e), or a uniaxial compressive test, or a flexural test (Fig. 11.1a and Chapter 12); G can be determined from a shear test, G ¼ s=g, where s is the shear stress and g is the shear strain (Fig. 11.1c); K can be determined from a compressibility test, K ¼ 1 V dV dp  1 where V is the volume and p is the hydrostatic pressure (Fig. 11.1b); and n can be determined from two independently determined values of modulus, or from a tensile test using a bidimensional extensometer. In practice, the values of these elastic quantities are needed for rough evaluations in mechanical design of parts working at ambient temperature, as illustrated for instance by Ashby (1992). Second Level. At the second level, the constitutive equations must involve two (or more) additional variables. For instance: fðs; e; _ ee; TÞ¼0 where _ ee is the strain rate and T is the temperature. These new variables are necessary to take into account viscoelastic effects linked to molecular motions. These effects are non-negligible in the glassy domain between boundaries a and b in the map of Fig. 11.2, and they are very important in the glass transition region (around boundary a). Here, we need relationships that express the effects of _ ee, _ ss (the stress rate may be used instead of the strain rate), and T on the previously defined elastic properties. Also numerical boundary values of elastic properties are required, characterizing unrelaxed and relaxed states (see Chapter 10). 324 Chapter 11 FIGURE 11.1 Mechanical tests to determine (a) E; (b) K, and (c) G. There are three main experimental approaches for mechanical char- acterization in this domain. They correspond to particular solutions of the material’s state equation: . static tests: e ¼ e 0 ¼ constant (relaxation) or s ¼ s 0 ¼ constant (creep) . monotonous tests with loading rate _ ee or _ ss ¼ constant (for instance tensile tests): _ ee ¼ 1 l dl dt . dynamic tests: e ¼ e 0 sin otors ¼ s 0 sin ot: Polymers are generally assumed to obey the Boltzmann superposition principle in the domain of small strains. When there are changes of loading conditions, the effects of these changes are additive when the corresponding responses are considered at equivalent times. For instance, if different stres- ses s 0 ; s 1 ; s 2 ; are applied at different times 0; t 1 ; t 2 ; ; respectively, the resulting strain is eðtÞ¼JðtÞs 0 þ Jðt t 1 Þs 1 þþJðt  t i Þs i ð11:2Þ where J(t) is the time-dependent creep compliance. In the same way, if different strains e 0 ; e 1 ; e 2 , are applied at times 0, t 1 ,t 2 , , the resulting stress is sðtÞ¼EðtÞe 0 þ Eðt t 1 Þe 1 þþEðt  t i Þe i ð11:3Þ where E(t) is the time-dependent relaxation modulus. Thus, what is needed is the knowledge of J(t) or E(t). It is generally convenient to use dynamic tests to determine J(o)orE(o), and then apply adequate mathematical transformations to obtain J(t) or E(t). Effect of Crosslink Density on Elastic and Viscoelastic Properties 325 FIGURE 11.2 Shape of relaxation maps (coordinates of transitions a; b; and g in a graph ln (frequency) – reciprocal temperature). Left: polymers having their a and b transitions well separated (example: polycarbonate, amine-crosslinked epoxy). Right: polymers with close a and b transitions (example : polystyrene, unsaturated polyester). In general, polymers obey a time–temperature superposition principle: P r ðt; TÞ¼P r t a T ; T R  ð11:4Þ where P r is a property, T R is a reference temperature, and a T is a shift factor that depends only on temperature. An interesting characteristic of polymers is that a T ¼ fðTÞ takes distinct mathematical forms below and above T g . Third Level. At the third level of complexity, the unsteady character of the polymer linked to the fact that it is out of equilibrium in the glassy state must be taken into account. The behavior of the material depends not only on the mechanical stimuli and environmental conditions but also on its thermomechanical history since its processing. In other words, time must be added as a variable to the constitutive equations: fðs; e; _ ee; T; tÞ¼0 From a practical point of view, the main consequence of physical ageing by structural relaxation is embrittlement (decrease in fracture resis- tance; Chapter 12). For the other aspects of mechanical behavior, ageing has either no effect or a favourable effect (increase of relaxation times, leading to a decrease of creep or relaxation rates). This is the reason why, in most thermoset applications, the knowledge of short-term properties is consid- ered to be sufficient for engineering design, as far as fracture and durability are not concerned. Thus, the present chapter contains essentially two sections, devoted to the first and second degree of complexity, respectively. 11.2 ENGINEERING ELASTIC PROPERTIES IN GLASSY STATE 11.2.1 Bulk Modulus a. Short Theoretical Survey The bulk modulus K can be derived directly from an expression for the intermolecular energy potential u(d), where d is the intermolecular distance (Fig. 11.3). The most usual expression for u(d) is the Lennard–Jones relationship: uðdÞ¼ðCEDÞ d 0 d  12 2 d 0 d  6 "# ð11:4Þ where CED is the cohesive energy density (Chapter 10). 326 Chapter 11 The bulk modulus can be derived from this equation, since K ¼ 1 V @V @p ! 1 and p ¼ @u @V with d 0 d ¼ V 0 V  1=3 ð11:5Þ These relationships lead to the following results: (i) The bulk modulus is proportional to the cohesive energy density: K ¼ a c ðCEDÞ where the theoretical value of a c is 8 (Tobolsky, 1960). (ii) However, since the intermolecular distances increase with tempera- ture, the intermolecular energy potential and the CED are temperature- dependent. It is then convenient to consider rather a temperature-indepen- dent quantity (CED) W , defined by ðCEDÞ W ¼ CED r* ¼ E coh V W ð11:6Þ where V W is the van der Waals volume and r* the packing density (Chapter 10). It has been shown that K (at T ¼ 0K)¼ 11(CED) W and K(T g )= 5.7(CED W ), just below T g (Porter, 1995). K at a given temperature T < T g can be interpolated by the following relationship: Effect of Crosslink Density on Elastic and Viscoelastic Properties 327 FIGURE 11.3 Variations of the intermolecular potential energy u(d) with inter- molecular distance d: d 0 is the equilibrium distance at which uðdÞ¼CED (CED ¼ cohesive energy density); d W is the van der Waals distance at which V ¼ V W ; and d g is the abscissa of the inflection point and corresponds to the glass transition. KðTÞ¼Kð0KÞ 1  a K T T g  ð11:7Þ where a K  0:5: (iii) There is apparently no discontinuity of the temperature depen- dence of K at secondary (sub-glass) transitions. In contrast, K varies by a factor of about 2 at T g :K¼ 3(CED) W , just above T g . In thermosets this gap is expected to be reduced as the crosslink density increases. b. Experimental Data Ultrasonic measurements (5 MHz) made at ambient temperature on amine- crosslinked epoxies (Morel et al., 1989), or styrene-crosslinked vinyl esters (Bellenger et al., 1994), show that K is effectively proportional to the cohesive energy density: K ¼ð11  1ÞðCEDÞð11:8Þ CED is determined by calculation, using the Van Krevelen increment values (Van Krevelen, 1990). There is no apparent effect of the crosslink density on K, which seems to depend (as CED), only on the molecular scale structure (polarity, hydrogen bonding). The following results were obtained with quasi-static tensile measurements (10 3 –10 4 s 1 strain rate) in a tempera- ture range 200 K–T g , using a bidimensional extensometer to determine E and n (from which K and G could be determined) (Verdu and Tcharkhtchi, 1996): 1. K values are of the same order as ultrasonic ones (typically 5–7 GPa). 2. K values are almost constant in the temperature interval between 200 K and T g 30, whereas the above theory would predict sig- nificant variations (typically 15–25% between 200 and 400 K). 3. The bulk modulus (K) is unaffected by the b transition; it is thus not surprising to find that it is independent of the frequency/ strain rate. To summarize, the bulk modulus of thermosets is proportional to the cohesive energy density and does not depend practically on temperature in the 200 K – (T g  30 K) temperature range. There is no significant effect of crosslink density on K, which can be predicted (in the temperature interval under consideration) using K ¼11 (CED), with an incertitude of about 10%. 328 Chapter 11 11.2.2 Shear Modulus The shear modulus can also be derived from an intermolecular energy potential, and its value at 0 K is also proportional to the cohesive energy density. Typically, at very low temperatures, n  1 3 , so that G  3 8 K, and thus Gð0KÞ4 ðCEDÞ W ð11:9Þ But the big difference with K is that G is directly affected by molecular motions and decreases almost discontinuously at sub-glass (secondary) tran- sitions (Fig. 11.4). Thus, G is more difficult to predict than K, since the prediction must take into account the characteristics of secondary transi- tions, which are difficult to establish theoretically (Chapter 10). From the engineering point of view, there are at least two relatively simple approaches to the problem. For Porter (1995), the important variable is the cumulative loss tan- gent, tan Á b , defined by tan Á b ¼ tan d b þ X tan d b ð11:10Þ where tan d b is the dissipation factor in the unrelaxed state and P tan d b is the sum of dissipation factors for all the secondary transitions occurring at temperatures lower than the temperature under consideration and in the selected timescale; tan d b is generally lower than 0.01, so that tan Á b is essentially representative of the mechanical activity of the secondary transi- Effect of Crosslink Density on Elastic and Viscoelastic Properties 329 FIGURE 11.4 Typical shape of the temperature variation of the bulk (K) and shear (G) modulus in the glassy state, around a relatively intense b transition. tions in the range of (T, t) conditions under consideration. Let us remind ourselves (see Chapter 10) that certain secondary transitions, such as T b in amine-crosslinked epoxies, are very active and lie at a relatively low tem- perature, for instance 200–240 K at 1 Hz. Other secondary transitions, such as Tg in polyesters (180 K at 1 Hz), have a low activity, whereas others, such as T b in polyesters ( 350 K at 1 Hz) or in polyimides (390–420 K at 1 Hz), are obviously inactive at ambient temperature, except eventually in long- term creep or relaxation experiments. In the frame of this analysis, the Poisson’s ratio depends only on the cumulative loss tangent: n ¼ 0:5½1  0:33ð1 tan Á 1=2 b Þ 2 ð11:11Þ Then, the knowledge of K and n allows us to calculate all the other elastic constants. Typically, the Poisson’s ratio of glassy thermosets at tem- peratures close to 208C ranges between 0.37 (styrene-crosslinked polyesters) and 0.42 (certain amine-crosslinked epoxies), which corresponds to tan Á b values ranging between about 0.01 and 0.1. The shear modulus is then given by G ¼ K 3 1  tan Á 1=2 b  2  0:33 ð11:12Þ The second method (in fact the method is applied to Young’s modulus but it can be transposed to G) starts from the following equation (Gilbert et al., 1986): G ¼ G 0 1  a G T T g   X ÁG b ð11:13Þ where a G has values in the order of 0.3–0.5 and P ÁG b is the sum of modulus gaps at the sub-glass transitions located at temperatures below the test temperature; ÁG b , ÁGg, etc., have to be experimentally determined. If the hierarchy of low-temperature/high-frequency (unrelaxed) G values corresponds more or less to the hierarchy of cohesive energy densi- ties, it is completely modified at ambient temperature/low frequency – i.e., in the usual mechanical testing conditions – owing to the importance of the modulus gap at T b (Fig. 11.5). 11.2.3 Tensile Modulus If one considers that K (cohesive energy density + packing density (expan- sion)) and G (cohesive energy density + packing density + molecular 330 Chapter 11 mobility) are the basic elastic quantities, then, the other elastic quantities, E and n, are fully determined by K and G values. By eliminating n, E may be expressed as a function of G and K: E ¼ 3Kð1  2nÞ¼ 3G 1 þ G 3K ð11:14Þ G  3K/8, so that G/3K < 1/8 and, roughly, E  3G. It is thus expected that E varies with loading conditions as G, display- ing noticeable gaps ÁE g ÁE b , etc. at the g; b, etc., transitions. As in the case of G, the unrelaxed tensile modulus will be found more or less proportional to the cohesive energy density, whereas the relaxed modulus will depend sharply on the activity of local motions. Polystyrene (PS) and bisphenol A polycarbonate (PC), are good exam- ples of linear polymers with respectively low (PS) and high (PC) local mobi- lity. Their characteristics are summarized in Table 11.1 (Porter, 1995). It is observed that E and G are lower for PC than for PS, despite the fact that PC is more cohesive than PS. This is because the b-dissipation peak is considerably more intense for PC than for PS. Unsaturated polyesters (UP) or vinyl esters (VE) on one side, and amine-crosslinked epoxies on the other side, can be considered as the ther- moset counterparts of, respectively, polystyrene and polycarbonate. Tensile moduli determined at relatively low strain rates (10 3 –10 4 s 1 ) are typically 3.0 < E < 4.5 GPa for UP and VE and 2.4 < E < 3.0 GPa for epoxies, Effect of Crosslink Density on Elastic and Viscoelastic Properties 331 FIGURE 11.5 Temperature variation of G for a network with low local mobility (UP) and a network with high local mobility (EPO); for the latter, (ÁG=G)  1/2. despite the fact that amine-crosslinked epoxies are more cohesive than UP or VE. 11.2.4 Poisson’s Ratio As for tensile modulus, n can be expressed as a function of G and K: n ¼ 3K  2G 6K þ 2G ¼ 1 2 1  G=K 1 þ 2 3 G=K 2 6 4 3 7 5 ð11:15Þ Derivation with respect to G gives dn dG ¼ 1 2K 1 1 þ 2 3 G K  2  1 2K ð11:16Þ As n varies in the opposite way as G, it increases with T (almost discontinuously at secondary transitions), and reaches a value of the order of 0:45  0:01 at T g  20 K. Then, it undergoes a rapid increase and attains a value very close to 0.50 in the rubbery region. The shape of temperature variations of n is represented in Fig. 11.6. The role of local mobility on the temperature or frequency variations of n appears clearly when one compares amine-crosslinked epoxies to unsaturated polyesters, at ambient temperature under usual tensile testing conditions ( _ ee  10 4 s 1 Þ: n (UP) 0.38–0:39, whereas n (epoxies)  0:41 0:02, the area of the sub-glass transition A b ¼ ð 270 150 tan d dT is of the order of 1–2 K for epoxies against 0.1 K for UP. 332 Chapter 11 TABLE 11.1 Comparison of low-frequency elastic properties of linear polymers with low (PS) and high (PC) local mobility, respectively (After Porter, 1995.) Polymer K (GPa) tan Á b G (GPa) E (GPa) n PS 3.5 0.008 1.2 3.0 0.36 PC 3.8 0.050 0.9 2.3 0.40 [...]... superposition principle (Sec 11. 1) In the case of programmed strain: ðt Eðt À t0 Þ_ ðtÞdt0 e 11: 17Þ Gðt À t0 Þ_ ðt0 Þdt0 g 11: 18Þ eðtÞ ! sðtÞ ¼ À1 or ðt gðtÞ ! sðtÞ ¼ À1 In the case of programmed stress: ðt sðtÞ ! eðtÞ ¼ À1 _ Dðt À t0 Þsðt0 Þdt0 11: 19Þ Effect of Crosslink Density on Elastic and Viscoelastic Properties 337 or ðt sðtÞ ! sðtÞ ¼ Jðt À t0 Þ_ ðt0 Þdt0 s 11: 20Þ À1 where E and G are moduli... epoxies) Actually, polymer relaxations are complex processes characterized by a relaxation spectrum (coexistence of many relaxation time) Their Cole– Cole plots are generally nonsymmetric (Fig 11. 11) FIGURE 11. 11 Cole–Cole plots for networks resulting from the condensation of diglycidyl ether of bisphenol A (DGEBA) and diethyltoluenediamine (ETHA), with various amine/epoxide molar ratios (numbers on... packing density, e.g., the ‘‘holes.’’ The model leads to the following equation for the complex modulus: E* ¼ E1 À E1 À E0 1 þ ðiotÞÀx þ QÞiotÞÀx 0 11: 26Þ FIGURE 11. 12 Shape of the Cole–Cole plots in the case of overlapping of a and b transitions 344 Chapter 11 11.3.4 Practical Use of Cole–Cole Plots Characteristics of Thermosets When Cole–Cole plots are established from the usual DMTA experiments, care... transition region in which tan d increases almost continuously with temperature 334 Chapter 11 Since n is a viscoelastic quantity, it is expected to increase with the loading time in the case of static loading (Theocaris and Hadjijoseph, 1965) All the above observations seem to justify Porter’s approach (Eq 11. 11)), according to which the Poisson’s ratio should depend only on the cumulative loss tangent... a-transition curve It can be recalled that there is no systematic difference between the relaxation spectra of linear and network polymers (they always have the shape of Fig 11. 8) From the practical point of view, the following characteristics are important: 340 Chapter 11 1 2 3 4 5 The Ta value measured at a low frequency (Ta % Tg, see the WLF equation below) It is a key characteristic of the polymer... and E00 ¼ 1 þ o2 t2 1 þ o2 t2 11: 23Þ The curve E00 ¼ f(E)0 is then a half circle intercepting the abscissa at E ¼ 0 and E0 ¼ E, with a maximum at E00 ¼ E/2 for E0 ¼ E/2 (ot ¼ 1) In the general case where the relaxed modulus takes a nonzero value, the following relationships hold: 0 342 Chapter 11 E0 ¼ E0 þ ðE1 À E0 Þo2 t2 ðE À E0 Þot and E00 ¼ 1 1 þ o 2 t2 1 þ o 2 t2 11: 24Þ where E0 and E1 are the... (0.59 < CED . many relaxation time). Their Cole– Cole plots are generally nonsymmetric (Fig. 11. 11). 342 Chapter 11 FIGURE 11. 11 Cole–Cole plots for networks resulting from the condensation of diglycidyl ether. consideration) using K 11 (CED), with an incertitude of about 10%. 328 Chapter 11 11.2.2 Shear Modulus The shear modulus can also be derived from an intermolecular energy potential, and its value. Right: polymers with close a and b transitions (example : polystyrene, unsaturated polyester). In general, polymers obey a time–temperature superposition principle: P r ðt; TÞ¼P r t a T ; T R  11: 4Þ where