Comment on 'Direct determination of the fragility ...

Response to “Comment on ‘Direct determination of the fragility indices of glassforming liquids by differential scanning calorimetry: Kinetic versus thermodynamic fragilities’” [J. Chem. Phys. 118, 10351 (2003)] Li-Min Wang and C. Austen Angell Citation: J. Chem. Phys. 118, 10353 (2003); doi: 10.1063/1.1571815 View online: http://dx.doi.org/10.1063/1.1571815 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v118/i22 Published by the AIP Publishing LLC.

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JOURNAL OF CHEMICAL PHYSICS

VOLUME 118, NUMBER 22

8 JUNE 2003

Response to ‘‘Comment on ‘Direct determination of the fragility indices of glassforming liquids by differential scanning calorimetry: Kinetic versus thermodynamic fragilities’ ’’ †J. Chem. Phys. 118, 10351 „2003…‡ Li-Min Wang and C. Austen Angella) Department of Chemistry and Biochemistry, Arizona State University, Tempe, Arizona 85287-1604

共Received 13 February 2002; accepted 12 March 2003兲 We give a brief review of how activation energies from scan-rate-dependent T g studies have been correlated with those for viscosity and other relaxation processes, in order to place the present study in context with previous work, including that referred to in the comment. Then we examine the ensuing question of the extent to which kinetic fragility can be predicted from thermodynamic data. We present a new correlation involving only the entropy of fusion and the jump in heat capacity at T g , that is followed by most nonchain liquids. Polymer liquids, as the comment authors have shown, have different behavior, consistent with earlier failures to identify Kauzmann and Vogel temperatures in these materials. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1571815兴

The Comment by Roland et al. has two elements to which we would like to respond. The first concerns the relation between the calorimetric method for determining fragility that we reported1 and the work done by others, including the authors of the Comment.2– 4 The second concerns the observation that we made concerning the lack of correlation of the fragility with the heat capacity jump at T g , and, more importantly, the question of what is dominating the kinetic behavior of glassforming liquids. In both cases, the matter of precedence seems to be involved. Concerning the former, we regret that we did not cite the interesting work of Robertson et al.2 and also of Hempel et al.3 and Chebli et al.4 Certainly, the idea of using activation energies, obtained from DSC conducted at different scan rates, to obtain the so-called ‘‘m-fragilities’’ of liquids and polymers, is not new. For instance, it was used in 1994 in our own group’s work on poly-l-asparagine,5 which we also did not cite. We note that that work also correlated the activation energy, and fragility, with the reduced width of the glass transition ⌬T g /T g , though the first use of this correlation was made by Ma et al. in 1992,6 and the association was only tentative. Although the designation ‘‘m-fragility’’ was not used 共as it had not yet been invented兲, Moynihan’s DSCbased activation energy7 was used even earlier in the discussion of fragilities by Tatsumisago et al.,7 who made direct comparisons with the viscosity data on the same systems, showing how each measurement gave extrema at a mean coordination number of 2.4. That work was also not cited in our paper. Instead we gave precedence to the original work of Moynihan and co-workers,7–9 who were the first to show that the activation energy obtained from Q-dependent DSC was the same as the activation energy for viscosity—but who were not cited in Ref. 2. Moynihan’s group was also the first to use the fictive temperature, T f , as a more precisely determined quantity than the glass temperature, for this purpose.9 Moynihan’s a兲

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group preferred to determine T f from scans employing equal cooling and heating rates and, for reasons given in their papers, discouraged the use of a single ‘‘standard’’ upscan with variable downscan rates. The first to ignore, apparently successfully, this advice were indeed Robertson et al.2 as far as we know, though neither they nor those 共including ourselves兲 who followed immediately3,4,10,11 after, justified the approach except by its success in reproducing the activation energies obtained by other means. A particular virtue, claimed for our use of the fictive temperature for obtaining fragilities, was the method of plotting data so as to obtain the m-fragility directly, both from the slope and the intercept of the plot. In our paper we provided a mathematical account of the origin of this previously10,11 puzzling coincidence. A further use of DSC to obtain information on the activation energy for relaxation near T g is to be noted. The characteristic time, determined by DSC, for crystallization in the diffusion-dominated regime, approaches but does not reach the activation energy for viscosity at the lowest temperatures of measurement.12 This supports the position of Ngai et al.13 in their argument that diffusivity rather than viscosity controls crystallization rates, and their suggestion that this observation was the first to demonstrate decoupling of diffusion from viscosity. Turning to the second matter, is it the thermodynamic behavior of liquids near their glass temperatures that dominates the kinetics of relaxation 共fragility兲, as the concentration on ‘‘free volume’’ among polymer physicists would suggest? Or is it something quite distinct encapsulated in the term ‘‘intermolecular cooperativity’’—as Roland et al. suggest? An attempt was made by Martinez and Angell14 共not cited in the Comment兲 to answer this question—for liquids. They showed that the order of viscosity departures from the Arrhenius law 共which defines the fragility兲 was the same as the order of temperature dependence of the excess entropy, S ex , scaled by its value at the glass transition. This suggested thermodynamic dominance of the kinetics. The scaling by

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FIG. 1. The measured m fragility plotted vs the value calculated from purely thermodynamic data, determined at T g and at T m , respectively, using the formula given in the legend. ⌬C p is the value determined at T g . The exceptions in Fig. 1 are less pronounced if the fragility F 1/2 is plotted instead of the m value determined at T g . The substances for which data are plotted are identified by number. The numbered substances and the sources of data for m meas and m calc are available as supplementary material from the journal. Substance 42 is cisdecalin. We note that for this substance, the m-fragility proves to be anomalously larger than the F 1/2 fragility 共Ref. 1兲, F 1/2 being determined near the melting point where ⌬H m is determined.

the property S ex at T g 共suggested by the Adam–Gibbs equation15兲 removed the discrepancies in the correlation of fragility and ⌬C p which had been noted in the first paper on fragility16 and frequently thereafter, e.g., Ref. 17, notwithstanding the implications of Roland et al. to the contrary. One outstanding exception to the Martinez–Angell correlation, that of SiO2 , has since been removed thanks to simulation studies18 that avoided any dependence on a crystalline reference state. Because this surprising19 correspondence might involve too much data processing to be convincing to many readers, we give here a further correlation that involves more transparent measured quantities. Figure 1 employs a combination of the enthalpy of fusion ⌬H m and the jump in heat capacity at T g , ⌬C p 共data that are widely available兲 on the vertical axis. The form of m calculated from these data m calc.⫽cT g ⌬C p /⌬H m

共1兲

was suggested to us by the combination of Eqs. 共8兲 and 共11兲 of Ref. 1. The adjustable constant c is given the value 56 to simplify the presentation of data. The figure shows that, with not many exceptions, the measured m-fragility is predictable from purely thermodynamic quantities—for liquids. In fact the deviations are not much greater than the variations in m itself reported by different authors using different measurement techniques— though they may be more significant. On the other hand for polymer liquids, on which Roland et al. concentrate their attention, it is clear that such thermodynamic predictions are quite unreliable—and we have never made any contrary suggestion. The failure of the Vogel and Kauzmann temperatures to correspond in the case of many polymers has long been known,20 so T g /T 0 共one measure of fragility兲 is not the same as T g /T K . Indeed, the Adam– Gibbs equation,15 which has enjoyed much success with liquids, both in simulation and experiment, predicts that the purely kinetic parameter ⌬␮ 共the activation energy per rear-

rangeable unit兲 will play a quite independent role in determining the fragility, as we have always pointed out. The surprise has been only that the role of ⌬␮ should prove so unimportant for liquids. For polymers it seems, ⌬␮ comes into its own.21 Presumably a plot like that of Fig. 1 for polymers would be a scatter pattern. In energy landscape terms, the heights of the saddle points on the potential energy landscape for polymers are no longer constrained to scale with the depths of connected minima. It now becomes a challenge to the theorist to explain why, and at what polymer chain length, this distinction between polymers and liquid becomes manifested. 1

L.-M. Wang, V. Velikov, and C. A. Angell, J. Chem. Phys. 117, 10184 共2002兲. 2 C. G. Robertson, P. G. Santangelo, and C. M. Roland, J. Non-Cryst. Solids 257, 135 共2000兲. 3 E. Hempel, G. Hempel, A. Hensel, C. Schick, and E. Donth, J. Phys. Chem. B 104, 2460 共2000兲. 4 K. Chebli, J. M. Saiter, J. Grenet, A. Hamou, and G. Saffarini, Physica B 304, 228 共2001兲. 5 J. L. Green, J. Fan, and C. A. Angell, J. Phys. Chem. 98, 13780 共1994兲. 6 H.-L. Ma, J. Lucas, X. H. Zhang, H. Senapati, R. Bo¨hmer, and C. A. Angell, J. Solid State Chem. 96, 181 共1992兲. 7 C. T. Moynihan, A. J. Easteal, J. Wilder, and J. C. Tucker, J. Phys. Chem. 78, 2674 共1974兲. 8 M. Tatsumisago, B. L. Halfpap, J. L. Green, S. M. Lindsay, and C. A. Angell, Phys. Rev. Lett. 64, 1549 共1990兲. It was Tatsumisago who first wrote an expression for fragility incorporating the Arrhenius activation energy 共‘‘Transformation Range Viscosity for Various Kinds of Glassy Liquids,’’ M. Tatsumisago and C. A. Angell, Proceedings of 30th Glass Meeting, Japan, 28 September 1989兲. 9 M. A. DeBolt, A. J. Easteal, P. B. Macedo, and C. T. Moynihan, J. Am. Ceram. Soc. 59, 16 共1976兲. 10 V. Velikov, S. Borick, and C. A. Angell, Science 294, 2335 共2001兲. 11 V. Velikov, S. Borick, and C. A. Angell, J. Phys. Chem. 106, 1069 共2002兲. 12 H. Senapati, K. K. Kadiyala, and C. A. Angell, J. Phys. Chem. 95, 7050 共1991兲. 13 K. L. Ngai, J. H. Magill, and D. J. Plazek, J. Chem. Phys. 112, 1887 共2000兲. 14 L.-M. Martinez and C. A. Angell, Nature 共London兲 410, 663 共2001兲. 15 G. Adam and J. H. Gibbs, J. Chem. Phys. 43, 139 共1965兲.


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C. A. Angell, in Relaxations in Complex Systems, edited by K. Ngai and G. B. Wright, National Technical Information Service, U.S. Department of Commerce, Springfield, VA 22161. 17 S. W. Martin and C. A. Angell, J. Phys. Chem. 90, 6736 共1986兲. We note now that, although the very strong liquid, P2 O5 , has a relatively large jump in heat capacity at T g 共and we predict also a compensating large enthalpy of fusion兲, it does have the broadest glass transition width, as

Response to ‘‘Comment on ‘Direct determination’ ’’

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predicted by Fig. 1 共inset兲 of K. Ito, C. T. Moynihan, and C. A. Angell, Nature 共London兲 398, 492 共1999兲. 18 I. Saika-Voivod, P. H. Poole, and F. Sciortino, Nature 共London兲 412, 514 共2001兲. 19 F. H. Stillinger and P. G. Debenedetti, J. Chem. Phys. 116, 3353 共2002兲. 20 A. A. Miller, Macromolecules 11, 859 共1978兲. 21 C. M. Roland, P. G. Santangelo, and K. L. Ngai, J. Chem. Phys. 111, 5593 共1999兲.
