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with s 0 approximately independent of both T and a, and B =[C/TSc(0)] only dependent on temperature, on the assumption that the variation of C with a is negligible – i.e., the relaxation time data vs a should lie on a Vogel–Fulcher-like curve that extrapolates to infinity at a = a0. Contrary to what is done by Johari [1], the consequences of Eq. (1) should be analyzed within the physical conditions that allow this relation to hold valid. In particular, a tendency of S c to zero (i.e., of s to diverge) at a finite a 0, as extrapolated from data at a < a0, does not imply a true vanishing of Sc (i.e., a divergence of s) ata0. The reaction, in fact, is not prevented from departing from Eq. (1) and from proceeding to an extent greater than a0, if the conditions for Eq. (1) are progressively removed. As a matter of fact any polymerization can be brought to its full extent by progressively increasing the reaction temperature, and this is normally used to estimate the total heat of reaction via calorimetry. It is also possible when the reaction temperature is high that the system remains fluid at even full reaction, as recognized in [1]. While data at a close to or higher than a0 are of no use for testing a correlation between Sc and s through a test of Eq. (2), a large amount of relaxation time data in the expected range of validity of Eq. (1) have been analyzed, from which an excellent agreement with Eq. (2) has turned out with the parameter a0 predicted on a molecular basis [2,6,7]. The observation that this relationship applies to a wide range of systems investigated, in different conditions, using different experimental methods, by numerous authors is hard to be seen as coincidental. Instead, it strongly suggests that the essence of the relaxational slowdown on polymerization can be traced back to a bond-controlled reduction in the number of configurations available to the system. Further experiments by our group – in which the characteristics of the systems investigated and the experimental conditions have been chosen to meet the requirements for Eq. (2), mentioned above – have strengthened the basis for Eq. (1) in combination with the Adam–Gibbs relation, particularly persuasive being the results for a number of epoxy–amine mixtures composed of two reagents (DGE- BA–DETA) in different molar ratios [8]. Because a0 may be controlled by varying the molar ratio of reagents in the initial mixture (solid line in Fig. 1), according to Eq. (2) the extent of reaction at which the relaxation time behavior appears to diverge would also be varied in the same way. We have found, for each reaction carried out isothermally, that Eq. (2) represents the relaxation time data well, in a wide relaxation range (at least five decades for s >10 4 s), where xn(a) is few units [8]. The derived values for the fitting parameter a0 are reported in Fig. 1. Indeed, Fig. 1 reveals that a0 not only changes with the mixture composition but (within the experimental error) actually follows the expected variation with the molar ratio (N a/N e) of the reacting molecules, in a wide N a/N e range. Moreover, the same a 0 is obtained by reacting a given mixture at different temperatures [9], despite the time-depen- S. Corezzi et al. / Chemical Physics 323 (2006) 622–624 623 α 0 1.0 0.9 0.8 0.7 0.6 10:3 5:2 5:2.8 10:9 0.2 0.3 0.4 0.5 0.6 Na /Ne 0.7 0.8 0.9 1.0 Fig. 1. Variation of the parameter a 0 with the molar ratio N a/N e between the amino and the epoxy monomer constituents of the system DGEBA– DETA Ne:Na. The experimental points for different compositions are labelled by their Ne:Na values. Open symbols are used to distinguish a different reaction temperature. The solid line is a 0 =[1+(N a/N e)]/2, calculated from the functionality and the molar ratio of the reacting molecules. dent evolution of the reaction is greatly affected. This is shown in Fig. 1 by closed and open symbols. In the same figure, isoabscissa closed-points refer to reactions repeated at the same temperature to check reproducibility. (2) A number of other observations reported by Johari deserve our comment. It is asserted in [1] that a connection between the relaxation time and a by using the Adam– Gibbs equation can be made only by allowing Dl, and therefore, the quantity B / Dl to change with a, being Dl a free-energy barrier per cooperative unit. We notice that this is the opposite of what the same author maintained in previous studies of polymerization [10–12]. Apart from this contradiction, we observe that as long as multimers behave effectively like cooperative units a variation of Dl with a seems not to be required by the theory [3]. On the other hand, if one would be forcing Eq. (2) with wrong but constant values of B, to fit the experimental data for polymerizations with Dl strongly varying with a, the values a0 which describe the data could scarcely correspond to the values of a physical quantity, i.e., the molecular parameter a0 calculated from the functionality of the reagents. This suggests that a sensible variation of Dl with a is not required either by the experimental data. A minor variation can, of course, account for details. Moreover, it is asserted in [1] that s 0 in Eq. (2) should necessarily correspond to the time scale of phonons, i.e., 10 13 –10 14 s. There is no question that in simple glassforming liquids analyses using the Adam–Gibbs approach usually do give pre-exponents not too far from the phonon time scale, though the values obtained notoriously span several decades (e.g., from 10 10.5 to 10 17.3 sin[14]). We remark that for it to be a general expectation the pre-exponent s 0 should represent the true value of s in the limit T !1, and different spectroscopic techniques (such as dielectric and photon-correlation spectroscopy) should measure the same s. Neither of these is the case. It is 4:3
624 S. Corezzi et al. / Chemical Physics 323 (2006) 622–624 well-known that relaxation times measured by different techniques all sensitive to structural dynamics may even differ by a few decades, which obviously reflects in different values of s0. It is also well-known that at some intermediate temperature on heating above Tg, the relaxation mechanism becomes different from cooperative transitions governed by the configurational entropy, and a marked change in the dynamics is observed [13]. Because of this change the relaxation time of a liquid at high temperature, which actually corresponds to the phonon time scale, may differ from s0 in Eq. (2), which is the value extrapolated from the cooperative regime. The difference between the two could be less pronounced if the process under analysis is a simple physical process. However, if it is a chemical process like polymerization then chemical step activation entropies could become involved and then all bets should be off about the value of the pre-exponent s 0. (3) In [1], the author reports as a feature of Eq. (1) that ‘‘... CP, c of a polymerizing melt at a fixed T would not change as a increases from zero to one’’, and similarly, ‘‘... the product bcVc, [where bc and Vc are the configurational contribution to the thermal expansion coefficient and the volume of the melt] would not change’’. On the contrary, we note that Eq. (1) yields a factor in C P, c and b c V c explicitly dependent on a [e.g., C P, c(a) =T(oS c(0)/oT) P (1 a/a0) =CP, c(0)(1 a/a0)], which factor has been erroneously omitted in [1]. Therefore, our Sc(a) relation does not conflict with the observation that changes in CP, b, and V during a meltÕs polymerization are also to be ascribed to changes in the configurational contribution to these quantities. (4) In our opinion, the calculations of S c performed by Wang and Johari [15] in a lattice–hole model are interesting but cannot prevail over the indications coming from experiments, as we have described. In addition, we observe that Sc, calculated with plausible values of the adjustable parameters, in the Wang and JohariÕs model would be expected to initially increase and then decrease with a in such a way that the value at the full extent of polymerization is nearly the same as (or even higher than) in the monomeric liquid (see Fig. 2 in [15]). This result can hardly be reconciled with an entropy-based picture of the process of polymerization; which picture, instead, Johari upholds. On the other hand, different lattice models exist and provide different results for the statistics of polymer chains. Interestingly, in a Flory–Huggins model of equilibrium polymerization [16] the average length of polymer chains scales inversely with the excess entropy of the polymer solution, similarly to our premise Sc / 1/xn. In conclusion, we believe there are sufficient experimental indications that Eq. (2) captures the principal physics behind the dynamical slowdown that occurs in a wide class of polymerizations, and therefore, goes beyond a mere phenomenological description of the process. We stress once more that it is appropriate in a limited range of a and T but this range usually covers the most part of relaxation data from experiments. Acknowledgement S.C. acknowledges financial support from MIUR- FIRB. References [1] G.P. Johari, Chem. Phys. 305 (2004) 231. [2] S. Corezzi, D. Fioretto, P. Rolla, Nature 420 (2002) 653. [3] G. Adam, J.H. Gibbs, J. Chem. Phys. 43 (1965) 139. [4] xn represents the average number of monomers per molecule in the system. a, also called the chemical conversion, provides a measure of the extent of reaction. [5] High temperature also tends: (i) to favor cyclization, which changes the variation of xn with a (e.g., xn would grow less rapidly and tend to diverge at a > a 0), and (ii) to favor departure from the quasiequilibrium condition. [6] S. Corezzi, L. Comez, D. Fioretto, J. Non-Cryst. Solids 345–346 (2004) 537. [7] S. Corezzi, AIP Conf. Proc. 708 (2004) 604. [8] S. Corezzi, D. Fioretto, J.M. Kenny, Phys. Rev. Lett. 94 (2005) 065702. [9] For all the polymerizations studied, the temperature was near the onset of the exothermic release in a calorimetric scan at the heating rate of 10 °C/min. [10] G.P. Johari, C. Ferrari, G. Salvetti, E. Tombari, Phys. Chem. Chem. Phys. 1 (1999) 2997. [11] E. Tombari, C. Ferrari, G. Salvetti, G.P. Johari, J. Phys. Condens. Matter 9 (1997) 7017. [12] C. Ferrari, E. Tombari, G. Salvetti, G.P. Johari, J. Chem. Phys. 110 (1999) 10599. [13] F. Stickel, E.W. Fischer, R. Richert, J. Chem. Phys. 104 (1996) 2043. [14] R. Richert, C.A. Angell, J. Chem. Phys. 108 (1998) 9016. [15] J. Wang, G.P. Johari, J. Chem. Phys. 116 (2002) 2310. [16] J. Dudowicz, K.F. Freed, J.F. Douglas, J. Chem. Phys. 111 (1999) 7116.
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