6 Rheological and Dielectric Monitoring of Network Formation 6.1 INTRODUCTION Two main transformations can take place during the formation of a poly- mer network: gelation and vitrification. Gelation corresponds to the inci- pient formation of an infinite network, while vitrification involves the transformation from a liquid or rubbery state to a glassy state (Chapter 4). Gelation is characterized by the divergence of the mass-average molar mass, M w , and the radius of gyration, and by the formation of an insoluble gel. Vitrification occurs when the increasing glass transition temperature, T g , becomes equal to the reaction temperature. Below T g,gel , the temperature at which the time to gel is the same as the time to vitrify, the reactive system vitrifies in an ungelled (liquid) state. Although conversion may increase at a very slow rate during storage, the polymer can be processed as long as it remains in the ungelled state. The thermosetting polymer will not vitrify during an isothermal cure at a reaction temperature higher than T g 1 , the glass transition temperature of the fully cured material (Chapter 4). The experimental determination of gelation and vitrification is very important for the design of cure cycles (Chapters 4 and 9) and to control the morphology of inhomogeneous polymer networks (Chapter 7) and modi- fied-thermosetting polymers that undergo a phase-separation process during cure (Chapter 8). The rheological and dielectric monitoring of these trans- formations is analyzed in this chapter. 6.2 EQUILIBRIUM MECHANICAL MEASUREMENTS A typical evolution of equilibrium mechanical properties during reaction is shown in Fig. 6.1. The initial reactive system has a steady shear viscosity that grows with reaction time as the mass-average molar mass, M w , increases and it reaches to infinity at the gel point. Elastic properties, char- acterized by nonzero values of the equilibrium modulus, appear beyond the gel point. These quantities describe only either the liquid (pregel) or the solid (postgel) state of the material. Determination of the gel point requires extra- polation of viscosity to infinity or of the equilibrium modulus to zero. Accurate measurement of the equilibrium modulus is extremely diffi- cult because its value remains below the detection limit for a considerable time and, theoretically, an infinite time is required to perform the measure- ment. Steady shear viscosity measurements are very simple and are often used in practice. Very often a viscosity of 10 3 Pa s is arbitrarily identified with the gel point. But the determined gelation time, t gel , depends on the shear rate, and extrapolation to zero shear rate meets the following difficul- ties: 1. t gel may depend on the shear rate due to shear thinning at high rates. 2. The network structure near the gel point is very fragile and can be broken by the shear flow experiment. Rheological and Dielectric Monitoring of Network Formation 187 FIGURE 6.1 Schematic evolution of steady-state mechanical properties of a thermoset as a function of reaction time or conversion. Representative prop- erties are the steady shear viscosity for the liquid state and the equilibrium modulus for the solid state. 3. Infinite viscosity is not an unambiguous indicator of gelation: it can equally be caused by vitrification. This means that com- plementary methods, such as sol fraction measurements, are necessary to distinguish between both phenomena. For all these reasons, steady-state mechanical measurements, even if they are very simple and very often used in practice, lead to an apparent gel point. 6.3 DYNAMIC MECHANICAL MEASUREMENTS 6.3.1 Search for an Experimental Criterion for Gelation Dynamic mechanical measurements describe both the liquid and solid states. A freely oscillating torsion pendulum can be used to provide shear moduli data of solid specimens versus temperature or time. A composite specimen made by impregnating a glass-fiber braid with liquid thermoset precursors is used to study liquid systems that change to solids (Babayevsky and Gillham, 1973). Two maxima of mechanical damping are observed during reaction and assigned to gelation and vitrification of the material (Fig. 6.2, Enns and Gillham, 1983). As the measurement is performed at the resonant frequency of the pendulum, this technique is very sensitive. But one big disadvantage is that experiments are not performed at controlled frequencies. The evolution of the dynamic viscosity Z* ðo; xÞ or of the dynamic shear complex modulus G* (o,x) as a function of conversion, x, can be followed by dynamic mechanical measurements using oscillatory shear deformation between two parallel plates at constant angular frequency, o ¼ 2f(f¼ frequency in Hz). In addition, the frequency sweep at certain time intervals during a slow reaction (x $ constant) allows determination of the frequency dependence of elastic quantities at the particular conversion. During such experiments, storage G 0 ðo), and loss G 00 ðoÞ shear moduli and their ratio, the loss factor tandðoÞ, are obtained: G*ðoÞ¼G 0 ðoÞþiG 00 ðoÞð6:1Þ tan d ¼ G 00 =G 0 ð6:2Þ Typical rheological curves obtained during a diepoxy–diamine reac- tion are shown in Fig. 6.3. The cure temperature (T i ¼ 908C) is well above the glass transition temperature of the fully cured network (T g 1 $ 358C), which means that only gelation occurs. Three typical regions are observed during cure (Matejka, 1991). 188 Chapter 6 (a) The pregel region is characterized by an increase in the loss mod- ulus, G 00 , corresponding to the increase of the real part of dynamic viscosity Z 0 ð¼ G 00 oÞ, due to the increasing molar mass of the thermosetting polymer. The storage modulus, G 0 , is very low and tends to zero at low frequencies. In this region the loss modulus (G 00 ) is higher than the elastic modulus (G 0 ), and the loss factor, tan d > 1. (b) The ‘‘critical region’’ begins with a sudden increase in the storage modulus, G 0 , by several orders of magnitude. At the intersection of the G 0 (t) and G 00 (t) curves, tan d ¼ 1. After the intersection point, G 0 becomes higher and tan d becomes less than 1. The viscous properties are dominant in the liquid state, i.e., G 00 >G 0 and tan d > 1, while the elastic properties predominate in the solid state, where G 0 >G 00 and tan d < 1. For this reason, the G 0 –G 00 crossover (tan d ¼ 1Þ was firstly identified as the gel point (Tung and Dynes, 1982). The Rheological and Dielectric Monitoring of Network Formation 189 FIGURE 6.2 Torsional braid analysis (TBA) during isothermal reaction of a diepoxy, DGEBA, with a diamine, bis(p-aminocyclohexyl) methane, from À158C to 2208C: (a) relative rigidity; (b) logarithmic decrement (T g0 ¼À198C, T g 1 ¼ 1708C, and T g,gel ¼ 498C). (Enns and Gillham, 1983 – Copyright 2001 – Reprinted by permission of John Wiley & Sons, Inc.) problem is that when a reaction like the one represented in Fig. 6.3 is followed at different angular frequencies, it is found that the reaction time to reach tan d ¼ 1 increases with angular frequency. As the gel point is a material constant and should not depend on experimental conditions, the crossover point between G 0 (t) and G 00 (t) cannot correspond to gelation. The value of tan d decreases in the (b) region and the rate of this decrease depends on the angular frequency, o (Fig. 6.4). As a rough approx- imation: 1. As G 0 $ o 2 and G 00 $ o, tan d $ 1=o in the Newtonian liquid state. 2. In the solid state G 0 $ constant and G 00 $ o, thus, tan d $ o (Ferry, 1980). Therefore, the drop of tan d during the reaction is steeper at low angular frequencies. Figure 6.4 reveals that for a particular tan d value higher than 1 (tan d $ 2 in the case of Fig. 6.4), the time to reach the value is independent of frequencies in the 0.1–50 Hz range. This crossover of the tan d curves at various frequencies can be used as a criterium for the identification of the gel point. 190 Chapter 6 FIGURE 6.3 Evolution of the storage modulus, G 0 (*), loss modulus, G 00 (*), and loss factor, tan dðÁ), during the reaction of pure diglycydyl ether of bisphenol A (DGEBA) with poly(oxypropylene) diamine (PPO). M n ¼ 370 g/mol at T i ¼ 908C, [NH] / [epoxy] ¼ 2, and f ¼ 10 Hz. a) Pregel region, b) ‘‘critical region’’, c) postgel region. (Matejka, 1991 – Copyright 2001 – Reprinted by permission of Springer-Verlag) (c) Finally, in the postgel region, a slow increase in G 0 , that levels off in the final stages of the reaction is observed; tan d < 1 for the fully cured rubbery network (T i >T g 1 ). Additional experiments show that the stoichio- metric mixture has the highest final modulus and the lowest final loss factor (tan d) because it forms the most perfect network, with the highest crosslink density. 6.3.2 Experimental Evidence of Singular Power Laws at the Gel Point The frequency dependence of dynamic mechanical results is of primary importance for the interpretation of data. The dependence of G 0 and G 00 versus o can be evaluated from experiments. Chambon and Winter (1985) first revealed that the G 0 (o) and G 00 (o) curves, in logarithmic scales, were parallel over a wide range of angular frequencies at the gel point. The validity of a power law G 0 ðoÞ$G 00 ðoÞ$o Á ð6:3Þ over the entire frequency range was assumed to be an inherent property of the gel state. An example is given in Fig. 6.5. The critical exponent in this case is Á ¼ 0:72 Æ 0:02. Before and after the gel time, the storage modulus G 0 (o) decreases rapidly to zero or shows a rubbery plateau at a low angular frequency, respectively. Rheological and Dielectric Monitoring of Network Formation 191 FIGURE 6.4 Decrease in the loss factor (tan d) during cure, for the same epoxy–diamine system as that represented in Fig. 6.3, at different frequencies of the dynamic measurements. T i ¼ 708C. (Matejka, 1991 – Copyright 2001 – Reprinted by permission of Springer-Verlag) Similar evolution of the frequency dependence of tan d and the complex viscosity, Z*, during the reaction near the gel point can also be obtained. 6.3.3 Scaling Laws and Gel Point Determination: a Search for Exponents Many authors have studied the rheological behavior of chemical gels theo- retically and experimentally. From the original mean-field theories of Flory and Stockmayer (Flory, 1953) describing the gelation phenomena (Chapter 3), current emphasis has shifted to the utilization of the fractal geometry concepts and the connectivity transition model of percolation (De Gennes, 1979; Stauffer et al., 1982). Under the percolation model, predictions for numerous physical properties are made by assimilating the thermosetting polymer as a polydisperse self-similar distribution of clusters (fractals), which grow from the beginning of the reaction through the gel point. An 192 Chapter 6 FIGURE 6.5 Storage G 0 and loss G 00 moduli as a function of frequency, at different cure times, for the same epoxy–diamine system as that represented in Figs 6.3 and 6.4. T i =908C. The parameter is the reaction distance from the gel point. && t ¼ t gel þ 5 min; ** t ¼ t gel ; ~ ~ t ¼ t gel À5 min; ——G 0 ; G 00 . (Matejka, 1991 – Copyright 2001 – Reprinted by permission of Springer-Verlag) infinite cluster, with a size and a mass that diverges, appears at the gel point. Above gelation, clusters connect to the infinite cluster and, as a conse- quence, a strengthening of the network is produced. As the reaction pro- ceeds, finite clusters with a decreasing average size are joined to the infinite cluster, and the process continues to the end of the reaction. Both the Flory–Stockmayer mean-field theory and the percolation model provide scaling relations for the divergence of static properties of the polymer species at the gelation threshold. The correlation length in the thermosetting polymer is defined by the z-average cluster radius, which scales with conversion x as R $ jx À x gel j x gel ! Àn for x < x gel ð6:4Þ The mass-average molar mass of clusters in solution diverges as M w $ jx À x gel j x gel ! Àg for x < x gel ð6:5Þ For the Flory–Stockmayer theory the predicted exponents are g ¼ 1 and n ¼ 1 2 , while for percolation they are predicted as 1.76 and 0.88, respec- tively (Stauffer et al., 1982). Percolation theory has been quite successful in predicting the static properties. For example, R z and M w can be determined by static light scattering, and reported values for a polyurethane are g ¼ 1:65 Æ 0:1 and n ¼ 0:86 Æ 0:1 (Adam et al., 1987). Scaling predictions for the steady-state viscoelastic properties have also been developed. The growth of the equilibrium modulus after the gel point can be described as a function of x by G 0 $ jx À x gel j x gel ! u for x > x gel ð6:6Þ and, similarly, the zero shear-rate viscosity as Z 0 $ jx À x gel j x gel ! Às for x < x gel ð6:7Þ But it is difficult to apply zero shear predictions to measurements that have been performed at low, but nonzero, shear rates. Neither s nor u can be decisively determined at a fixed frequency near the gel point: the only way to truly obtain s and u exponents from experiments is through the use of a theory capable of predicting the entire frequency dependence of the visco- elastic response. Consideration of the dynamics near the gel point led to predictions for the frequency dependence of the G 0 ðo) and G 00 ðoÞ moduli. At the gel point, Rheological and Dielectric Monitoring of Network Formation 193 the behavior given by Eq. 6.3 – G 0 $ G 00 $ o Á – is predicted (Hess et al., 1988; Martin et al., 1989; Rubinstein et al., 1989; Hodgson and Amis, 1990). Calculations based on a Rouse-like dynamics (Ferry, 1980), applied to the percolation clusters, give Á ¼ u s þ u ð6:8Þ This model leads to Á ¼ 0:67 at the gel point, using the zero-frequency values for s and u. Use of the values for s and u calculated by treating the gelation phenomena as a three-dimensional percolation model of a supra- conductor/resistor network (electrical analogy), gives Á ¼ 0:72 Æ 0:02. The use of the Rouse model is questionable (Martin et al., 1989). A variety of predictions have been made for the critical exponents, but such a discussion is beyond the scope of this book. It can be concluded that when thermosetting polymers are cured at a temperature T i above T g 1 þ 508C (as the results presented in Figs 6.4 and 6.5), the polymeric material can be described by the percolation theory, with chains obeying the Rouse model (Eloundou et al., 1996a). An explanation for the lower experimental critical exponent values is that the experimental gel times, t gel , may be affected by the viscoelastic behavior of the polymer. The largest relaxation time of the polymer and the width of the distribution of relaxation times increase with increasing conversion and diverge at the gel point (Chambon and Winter, 1987; Winter, 1987). According to the width of the relaxation time spectrum, the relaxation times should be of the same order of magnitude as the obser- vation times. In conclusion, the relative magnitude of the polymer relaxa- tion time t and the characteristic time of the mechanical test, t exp $ 1=o, are the crucial factors governing the experimental dynamic results. It seems that this molecular mobility effect is more important in the case of polyurethane chemistry than in the case of epoxy-diamine systems (Izuka et al., 1994; Prochazka et al., 1996; Nicolai et al., 1997). An example of results obtained during polyurethane synthesis is given in Fig. 6.6. 6.3.4 Determination of the Second Phenomenon: Vitrification Thermosetting polymers represented in Figs 6.4, 6.5, and also 6.6 were cured at T i >T g 1 . In many cases and particularly for high-T g networks, a precure step is performed at T i <T g 1 . In this case the two transformations, gelation and vitrification, occur and the rheology in the vicinity of the critical gel point could be affected by vitrification. 194 Chapter 6 Figure 6.7 gives the experimental curves of the loss factor (tan d)at different angular frequencies, as a function of reaction time at a cure tem- perature, T i ¼ 1508C, for a reactive epoxy system exhibiting a T g 1 ¼ 1778C (Eloundou et al., 1996b). In this case, T i $ T g 1 À308C. The vitrification phenomenon is revealed by a peak at t $ 350 min, which is frequency dependent. In these conditions, gelation occurs at about 200 min, as revealed by the crossover of tan d versus time curves, recorded at various frequencies. A value of Á ¼ 0:69 was found. As T i decreases, the viscoelastic characterization of the gelation pro- cess becomes more and more disturbed by the vitrification phenomenon, and the scaling laws are no longer well verified. At T i ¼ 808C, which is close to T g 1 À 1008C or $ T g,gel þ 308C, vitrification and gelation are in the same range of reaction times and the highest frequency curve does not participate in the crossover (Fig. 6.8). Rheological and Dielectric Monitoring of Network Formation 195 FIGURE 6.6 Evolution of tan d during polyurethane synthesis at 1108C, at different angular frequencies, o (s À1 Þ¼1(!), 3.162 (X), 10(~), 31.62 (&), and 100 (þ). A polycaprolactone diol, M n ¼ 700 g mol À1 was stoichiometri- cally reacted in bulk with a triisocyanate (the trimer of isophorone diisocya- nate). The time t c at which tan d is independent of frequency determines the gel point. The critical gel exhibits values of tan d ¼ 1:4 and Á ¼ 0:61. (Reprinted with permission from Izuka et al., 1994. Copyright 2001. American Chemical Society) [...]... 2 067 –2088 (1973) Chambon F, Winter HH, Polym Bull., 13, 499–504 (1985) Chambon F, Winter HH, J Rheol., 31, 68 3 69 7 (1987) De Gennes PG, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, New York, 1979 ´ Eloundou JP, Feve M, Gerard JF, Harran D, Pascault JP, Macromolecules, 29, 69 07 69 16 (1996a) ´ Eloundou JP, Gerard JF, Harran D, Pascault JP, Macromolecules, 29, 69 17 69 27 (1996b)... 25 36 2542 (1988) Hodgson DF, Amis EJ, Macromolecules, 23, 2512–2519 (1990) Izuka A, Winter HH, Hashimoto T, Macromolecules, 27, 68 83 68 88 (1994) Johari GP, Wasylyshyn DA, J Polym Sci., Polym Phys., 38, 122–1 26 (2000) Kranbuehl DE, Delos S, Yi E, Jarvie T, Winfree W, Han T, Polym Eng Sci., 26, 338–345 (19 86) Martin JE, Adolf D, Wilcoxon JP, Phys Rev A., 39, 1325–1332 (1989) Matejka L, Polym Bull., 26, ... 109–1 16 (1991) Rheological and Dielectric Monitoring of Network Formation 205 Nicolai T, Randrianantoandro H, Prochazka F, Durand D, Macromolecules, 30, 5897–5904 (1997) Prochazka F, Nicolai T, Durand D, Macromolecules, 29, 2 260 –2 264 (19 96) Rubinstein M, Colby RH, Gillmor JR, In Chemical Physics, Springer, Berlin, 1989, 51, 66 –74 Senturia SD, Sheppard NF Jr, Lee HI, Day DR, J Adhesion, 15, 69 –90 (1982)...1 96 Chapter 6 FIGURE 6. 7 Evolution of tan d during an epoxy network synthesis at 1508C, for various angular frequencies, o (from 1 to 100 sÀ1 ) The diepoxy, DGEBA, was stoichiometrically reacted with 4,40 -methylene bis [3-chloro-2 ,6- diethylaniline], MCDEA The characteristics of this system are Tg,gel ¼ 508C and Tg1 ¼ 1778C (Reprinted with permission from Eloundou et al., 1996b Copyright... Chemical Society) FIGURE 6. 8 Evolution of tan d during an epoxy network synthesis at 808C, for various angular frequencies o (from 1 to 100 sÀ1 ) The epoxy system is the same as that in Fig 6. 7) (Reprinted with permission from Eloundou et al., 1996b Copyright 2001 American Chemical Society) Rheological and Dielectric Monitoring of Network Formation 197 6. 4 DIELECTRIC MONITORING 6. 4.1 Dielectric Parameters... Network Formation 203 FIGURE 6. 11 Curves of conductivity s as a function of conversion, x ; (a) and (b) are the same epoxy systems as those in Figs 6. 8 and 6. 9, for (a) Tg1 ¼ À128C; (b) Tg1 ¼ 1778C; (c) DGEBA reacting with diaminodiphenyl sulfone, DDS, Tg1 ¼ 2108C; (d) DGEBA reacting with 4,40 -methylene bis (2 ,6- diethylaniline), MDEA, Tg1 ¼ 1708C For the four systems xgel $ 0 .6 (Eloundou et al., 1998a... this precaution is respected, the conclusions may be quite different (Eloundou, 1998b): 200 Chapter 6 FIGURE 6. 10 Experimental curves of conductivity, s, as function of reaction time, t, at different cure temperatures, Ti: (a) same epoxy system as that in Fig 6. 9a and (b) same epoxy system as that in Fig 6. 9b (Eloundou et al., 1998a Copyright 2001 Reprinted with permission of Wiley-VCH) 1 2 The time and... Eloundou JP, Gerard JF, Pascault JP, Boiteux G, Seytre G, D Angew Makromol Chem., 263 , 57–70 (1998a) ´ Eloundou JP, Ayina O, Ntede Nga H, Gerard JF, Pascault JP, Boiteux G, Seytre G, J Polym Sci., Phys., 36, 2911–2921 (1998b) Enns JB, Gillham JK, J Appl Polym Sci., 28, 2 567 –2591 (1983) Ferry JD, Viscoelastic Properties of Polymers, 3rd ed, John Wiley & Sons, New York, 1980 Flory PJ, Principles of Polymer... 0 .6) Furthermore, when conductivity is expressed in terms of conversion using the corresponding kinetic information, there is no inflection point on log s(x) curves, as can be seen in Fig 6. 11 for the four epoxy–diamine systems (Eloundou et al., 1998a, Rheological and Dielectric Monitoring of Network Formation 201 1998b) In particular, a smooth behavior is observed around the gel point, xgel $ 0 .6 3... including polymers, is between 10 6 and 1011 Hz Values of e0 and e00 are calculated using equations in which the contributions of dipolar effects, ionic displacements, and electrode polarization effects are additive If conduction effects predominate, i.e., when neither interfacial effects nor dipolar effects are significant, the loss factor is given by (Kranbuehl et al., 19 86) : e00 ¼ s e0 o 6: 10Þ where . JF, Harran D, Pascault JP, Macromolecules, 29, 69 07 69 16 (1996a). Eloundou JP, Ge ´ rard JF, Harran D, Pascault JP, Macromolecules, 29, 69 17 69 27 (1996b). Eloundou JP, Ge ´ rard JF, Pascault JP,. al., 19 96; Nicolai et al., 1997). An example of results obtained during polyurethane synthesis is given in Fig. 6. 6. 6. 3.4 Determination of the Second Phenomenon: Vitrification Thermosetting polymers. Phenomenon: Vitrification Thermosetting polymers represented in Figs 6. 4, 6. 5, and also 6. 6 were cured at T i >T g 1 . In many cases and particularly for high-T g networks, a precure step is performed