8.8 EFFECT OF COPOLYMERIZATION ON Tg 399 Figure 8.27 The thermosetting process transformation reaction diagram (112). as illustrating by the time–temperature– to provide an intellectual framework for understanding and comparing the cure and glass transition properties of thermosetting systems. Figure 8.27 illustrates the TTT diagram. Besides Tg•, the diagram also displays gelTg, the temperature at which gelation and vitrification occur simultaneously, and Tg0, the glass transition temperature of the reactants. The particular S-shaped curve between Tg0 and Tg• results because the reaction rate is increased with increasing temperature. At a temperature intermediate between gelTg and Tg•, the reacting mass first gels, forming a network. Then it vitrifies, and the reaction stops, incomplete. To the novice, the reaction products may appear complete. This last may result in material failure if the temperature is suddenly raised. The TTT diagram explains why epoxy and similar reactions are carried out in steps, each at a higher temperature. The last step, the postcure, must be done above Tg•. Other points shown in Figure 8.27 include the devitrification region, caused by degradation, and the char region, at still higher temperatures. 8.8 EFFECT OF COPOLYMERIZATION ON Tg The discussion above relates to simple homopolymers. Addition of a second component may take the form of copolymerization or polymer blending. R R R R (8.66) may be simplified: X1 {Ú Tg Tg 1 G R (C p 1 . when their values are denoted as S 0 1 and S 2. This condition and the use of appropriate superscripts G and R lead to the equation G 0 .DS m = 0 } (8. because the measure of the heat absorbed provides a direct measure of the increase in molecular motion. equation (8. and their respective mole fractions (moles of mers for the polymers) as X1 and X2.400 GLASS–RUBBER TRANSITION BEHAVIOR Addition of low-molecular-weight compounds results in plasticization.1 One-Phase Systems Based on the thermodynamic theory of the glass transition. The mixed system molar entropy may be written S = X1 S1 + X 2 S 2 + DS m (8. For later convenience.65) The mixed-system glass transition temperature.R (i = 1.64) where DSm represents the excess entropy of mixing. 8.G + Ú C p + Ú Cp 2 d ln T + DS m 1 d ln T + X 2 S 2 Tg 1 Tg 2 R 0 .G = Si0. Experimentally.67) . The treatment that follows is easily generalized to the case for statistical copolymers (113). S1 and S2 are referred to their respective pure-component glass transition tem0 peratures of Tg1 and Tg2.R + ÚTg 1 C p + Ú Cp 1 d ln T + X 2 S 2 2 d ln T + DS m Tg 2 { Tg } { Tg } { Tg } { Tg } Since Si0. Tg. The use of classical thermodynamics leads to an easy introduction of the pure-component heat capacities at constant pressure. Couchman derived relations to predict the Tg composition dependence of binary mixtures of miscible high polymers (113) and other systems (114–116).66) = X1R S10 .8. at Tg. Heat capacities are of fundamental importance in glass transition theories.C p1 )d ln T + X 2 } {Ú Tg Tg 2 G R G R (C p 2 .G G G G X1G S10 . or one mer and one plasticizer) having pure-component molar entropies denoted as S1 and S2.C p 2 )d ln T + DS m . is defined by the requirement that S for the glassy state be identical to that for the rubbery state. Consider two polymers (or two kinds of mers. two general cases may be distinguished: where one phase is retained and where two or more phases result. Cp1 and Cp2: 0 S = X1 S10 + Ú C p1 d ln T + X 2 S 2 + Ú C p 2 d ln T + DS m Tg 1 Tg 2 { T } { T } (8. 2) and X iG = XiR = Xi. for small x.71) is shown to fit Tg data of thermodynamically miscible blends (see Figure 8. and equation (8. where X denotes X1 and X2. DSm = DSm . Four particular nontrivial cases of the general mixing relation may be derived. If DCpi Tgi = constant (76–78.X ) ln (1 .28). For random copolymers these quantities are also equal. originally derived for random copolymers.69) becomes ln Tg = or equivalently Ê Tg ˆ M 2 DC p 2 ln(Tg 2 Tg1 ) ln = Ë Tg1 ¯ M1 DC p1 + M 2 DC p 2 (8. and noting that Tg1/Tg2 usually is not greatly different from unity yield Tg Ӎ M1 DC p1 Tg1 + M 2 DC p 2 Tg 2 M1 DC p1 + M 2 DC p 2 (8.70) Equation (8.X ).71) M1 DC p1 ln Tg1 + M 2 DC p 2 ln Tg 2 M1 DC p1 + M 2 DC p 2 (8.73) . Ê Tg ˆ Ê Tg ˆ =0 + X 2 DC p 2 ln X1 DC p1 ln Ë Tg 2 ¯ Ë Tg1 ¯ (8.72) which has the same form as the Wood equation (117). DSm is proportional to X ln X + (1 . After integration the general relationship emerges.8 EFFECT OF COPOLYMERIZATION ON Tg 401 In regular small-molecule mixtures. the familiar Fox equation (119) appears after suitable crossmultiplying: M1 M 2 1 = + Tg Tg1 Tg 2 (8. the increase in the heat capacity at Tg reflects the increase in the molecular motion and the increased temperature rate of these motions. Mi (recall that the DCpi are then per unit mass). Then X1 Ú DC p1 d ln T + X 2 Ú Tg 1 Tg Tg Tg 2 DC p 2 d ln T = 0 (8.8. Again. Similar relations hold for polymer– solvent (plasticizer) and polymer–polymer combinations.69) For later convenience the Xi are exchanged for mass (weight) fractions. Making use of the expansions of the form ln(1 + x) = x. Combined with the G R continuity relation.68) where D denotes transition increments.118). 28).75) which predicts a linear relation for the Tg of the blend.0671 cal K-1 · g-1.4-phenylene oxide)– blend–polystyrene (PPO/PS) blends versus mass fraction of PPO. Tg2 = 489 K.70): ln Tg = M1 ln Tg1 + M 2 ln Tg 2 (8.0528 cal K-1 · g-1. Tg. The Fox equation (119) was also originally derived for statistical copolymers (120). PPO was designated as component 2 (114. If DCp1 Ӎ DCp2. random copolymer. Previously they were used on a semiempirical basis. In this case the plasticizer behaves as a com- . Tg = M1Tg1 + M 2Tg 2 (8.120).6-dimethyl-1. the equation of Pochan et al. MPPO. This equation usually predicts Tg too high. DCp1 = 0. Tg1 = 378 K. (121) follows from equation (8.71) as circles. This equation predicts the typically convex relationship obtained when T is plotted against M2 (see Figure 8. Couchman’s work (113–116) shows the relationship between them. DCp2 = 0. or plasticized system. These equations also apply to plasticizers.74) Finally.28 Glass-transition temperatures.73) and (8. a low-molecular-weight compound dissolved in the polymer.75) are widely used in the literature. of poly(2. if both pure-component heat capacity increments have the same value and the log functions are expanded.402 GLASS–RUBBER TRANSITION BEHAVIOR Figure 8. The full curve was calculated from equation (8. Equations (8. 124).29 Dynamic mechanical behavior of polystyrene–block–polybutadiene–block– polystyrene. the material gradually softens. As the elastomer component increases (small spheres. Figure 8. softening it through much of the temperature range of interest. then alternating lamellae). The effect is to lower the glass transition temperature. especially in the loss spectra (E≤). is indicative of the mass fraction of that phase. When the rubbery phase becomes the only continuous-phase. In this case each phase will exhibit its own Tg. . then cylinders. are phase-separated (122) (see Section 4.” 8.124) illustrates two glass transitions appearing in a series of triblock copolymers of different overall compositions. a function of the styrene–butadiene mole ratio (123. A secondary effect is to lower the modulus. The storage modulus in the plateau between the two transitions depends both on the overall composition and on which phase is continuous. The intensity of the transition.3).2 Two-Phase Systems Most polymer blends.8. the storage modulus will decrease to about 1 ¥ 108 dynes/cm2.29 (123. as well as their related graft and block copolymers and interpenetrating polymer networks. An example is the plasticization of poly(vinyl chloride) by dioctyl phthalate to make compositions known as “vinyl.8. Figure 8. Electron microscopy shows that the polystyrene phase is continuous in the present case.8 EFFECT OF COPOLYMERIZATION ON Tg 403 pound with a low Tg. 91 0.09 0.97 0. frequently referred to as polymethylene.1 (125).3 3. Sometimes Tg appears to be masked. increases mixing. The overall composition was 80/20 epoxy/acrylic. with atactic polystyrene being the most frequently studied polymer. Using equation (8. Tg(U ). 8.9.404 GLASS–RUBBER TRANSITION BEHAVIOR Table 8.73).18 — Matrix Phase Weight Fraction PnBA 0.82 — Epoxy 0.30 Epoxy 0. and glycidyl methacrylate is shown to enhance molecular mixing between the chains. that refers to the completely amorphous state and that should be used in all correlations with chemical structure (this transition correlates with the molecular phenomena discussed in previous sections).70 Grafting mer. Because of the high degree of crystallinity. especially for highly crystalline polymers.0 a b Dispersed Phase Weight Fraction PnBAb 0.1 The Glass Transition of Polyethylene Linear polyethylene. and (b) an upper value.12 0. the extent of mixing within each phase in a simultaneous interpenetrating network of an epoxy resin and poly(n-butyl acrylate) was calculated (see Table 8.10).88 0. though only in the amorphous portions of these polymers.8. Boyer (9) points out that many semicrystalline polymers appear to possess two glass temperatures: (a) a lower one. that occurs in the semicrystalline material and varies with extent of crystallinity and morphology. The Tg is often increased in temperature by the molecular-motion restricting crystallites. the inward shift in the Tg of the two phases can each be expressed by the equations of Section 8. usually in excess of 80%.03 0. 8.9 EFFECT OF CRYSTALLINITY ON Tg The previous discussion centered on amorphous polymers. If appreciable mixing between the component polymers occurs. molecular motions associated with Tg are partly masked. offers a complete contrast with polystyrene in that it has no side groups and has a high degree of crystallinity. Poly(n-butyl acrylate).10 Phase composition of epoxy/acrylic simultaneous interpenetrating networks (125) Glycidyl Methacrylatea (%) 0 0. Tg(L). leading . Semicrystalline polymers such as polyethylene or polypropylene or of the polyamide and polyester types also exhibit glass transitions. Chapter 13 provides additional material on the glass transition behavior of multicomponent materials.