Entropy, Landscapes, and Fragility in Liquids and Polymers, and the

Chapter 3

Entropy, Landscapes, and Fragility in Liquids and Polymers, and the ΔCp Problem

Downloaded by UNIV OF MINNESOTA on October 14, 2014 | http://pubs.acs.org Publication Date: January 28, 1999 | doi: 10.1021/bk-1998-0710.ch003

C. A. Angell Department of Chemistry, Arizona State University, Tempe, AZ 85287

We examine the relation of the mode coupling theory of glassforming liquid dynamics to the Gibbs-Goldstein landscape picture of relaxation, and identify, using both scaling relationships and thermodynamic calculations, where the crossover between the two domains occurs. This "landscape" approach is then applied to chain polymers, using a neglected relation between the W L F C parameter and the high frequency limit for relaxation, to establish the characteristic temperature of the landscape ground state and the appropriate fragility parameters. A problem with the relation between thermodynamic to relaxational assessments of the landscape "height" is used to focus attention on a problem with the heat capacity changes at T in the case of chain polymer melts and their glasses. 1

g

The problem of viscous liquids and the glass transition is currently enjoying one of its characteristic thirty-year cycles of high activity, stimulated jointly by the review of Anderson in 1979 (1), and the advent of Götze and colleagues' mode coupling theory of the glass transition in 1984 (2) The latter gave, as its central features, the existence of a two-step response to a perturbation of the equilibrium state about which detailed predictions were made, and the prediction of a dynamical "jamming," i.e. vitrification at finite temperature due to power law divergences in the response times with decreasing temperature. The first of these has been extensively verified for liquids and polymers whereas the second has not.

©1998 American Chemical Society In Structure and Properties of Glassy Polymers; Tant, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1999.

37


38 The failure to jam in the predicted manner has been attributed to the crossover in relaxation mechanism at some temperature between normal liquid (high fluidity, short relaxation times) temperatures and the glass transition temperature, an occurrence which was anticipated by Goldstein in a 1969 paper which is now a classic. Here we will (1) give a brief review of what might be called the Gibbs-Goldstein picture of the phenomenology, thermodynamic and dynamic, of glassforming liquids, (2) add to this a classification of liquid types ranging from the highly non-Arrhenius glassformers (to which the mode coupling description was expected to apply) to the almost Arrhenius liquids (to which it was not expected to apply but, according to recent studies, does), and then (3) to examine the extent to which the phenomenology of polymer liquids and glasses can be interpreted in terms of (1) and (2). This will lead us to recognize a problem in the polymer case for which we will then supply a partial resolution and a proposal for further work. In the process, we will encounter not only a variety of insights into the utility of the "landscape paradigm" for discussions of glassformers but also both thermodynamic and dynamic markers for the crossover temperature between "landscape-dominated" and "free diffusion" domains. The

Gibbs-Goldstein

Picture

The fact that glasses are brittle solids at temperatures below their glass transition temperatures implies that the arrangement of particles taken up as a liquid cools below T can be described by a point in configuration space near the bottom of a potential energy minimum in this space (3,4). If this were not so, the system would move in the direction dictated by the collective unbalanced force acting on it, and some sort of flow would occur. On the other hand, the existence of the annealing phenomenon, in which the density and energy of a glass formed during steady cooling can change with time on holding at a temperature below but close to the "glass transition temperature" means that there is more than one such mechanically stable minimum available to the system. Indeed, there would appear to be a huge number, of order e , where Ν is the number of particles in the system (5, 6, 7). The minima, or "basins of probability" (5) obviously are distributed over a wide range of energies, usually scaling with density. However, there are also many ways of organizing the same collection of particles into g

N

In Structure and Properties of Glassy Polymers; Tant, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1999.


39 minima which differ negligibly in energy from one another. Each minimum is a configurational microstate, or "configuron," of the system. Each laboratory "glass" is thus a palpable configuron, though there is evidence that the distinguishing properties of a glass, v i z . its structure, heat capacity, and diffusivity are determined within very small distances, so that the properties of a glassformer can be evaluated by consideration of only a tiny subset of its particles. The fact that annealing ("aging" for polymers) proceeds more slowly the lower the temperature at which the annealing is carried out suggests that the process of finding deeper minima becomes more difficult statistically as the temperature is decreased. One arrives at the notion of an interconnected series of minima on a landscape of inconceivable complexity, in which increasing depth is associated with decreasing configuron population. The important implication of Kauzmann's paradox (8) is that for each system, at least for each "fragile" system, there must exist a statistically small number of minima at energies still well above that representing the crystal and that these must set an absolute limit on the energy lowering achievable by annealing the amorphous system. It is into one of these last few minima that the system is tending to settle at the temperature where the excess entropy is tending to vanish. The energy of these lowest minima ksTjc defines the ground state for the amorphous system, to be reached at the Kauzmann temperature Τ κ on indefinitely slow cooling. Hence at this temperature, the excess entropy would vanish. The Adam-Gibbs equation for viscous liquid relaxation asserts that the time scale for re-equilibration after some perturbation is related to the excess entropy of liquid over crystal, according to τ = τ exp 0 Δ μ / Τ 8 0

(D

ς

14

where τ is a quasi-lattice vibration time, 1 0 " s, Δμ is a free energy barrier per particle to cooperative rearrangements, and S is the excess (configurational) entropy. Evaluating S as the entropy generated above Τ κ according to 0

c

c

*-dT-

(2)

leads to the well-known Vogel-Fulcher-Tamann (VFT) equation as an identity or as a good approximation depending on how the


40 excess heat capacity vs. Τ relation is approximated (9). Although the V F T equation is itself only a rough description of the behavior of supercooling liquids over the 16 orders of magnitude of relaxation times which can now be measured (10), a fitting of data to that equation under the constraint that τ have a physical value 10" s, yields an agreement between the V F T relaxation time divergence temperature T and the thermodynamically determined Τ κ , within a variance of ~2% for 40 liquids for which the T data are available with T ranging from 50 to 1000 Κ (11). The departure from Arrhenius behavior of the relaxation time arises, i n the Adam-Gibbs theory, from the temperature dependence of the S term in equation (1), which itself is a consequence of the excess heat capacity A C of equation (2). For constant Δ μ in equation (2), the degree of non-Arrhenius character, now called the fragility, is determined by the magnitude of Δ 0 . A general but incomplete accord seems to exist between the fragility and Δ Ο and exceptions, like the alcohols, can be rationalized by the presence of unusual Δμ terms. 0

1 4

0

K


g

c

P

Ρ

ρ

Thus, in broad brush, the Adam-Gibbs approach within the landscape paradigm, which is summarized pictorially in Figure 1, provides a good basic understanding of key features of the behavior of glassforming liquids and polymers in their ergodic states above T . Furthermore, by setting S in equation (1) at a value fixed by cooling rate or annealing time, the Adam-Gibbs equation goes far towards a description of behavior in the non-ergodic regime below g

c

The viscosity of polymers above T can be incorporated in this picture by inclusion of a molecular weight-dependent preexponent in equation 1, so as to obtain a local viscosity relaxation g

(3) time, where χ = 1 for pre-entanglement conditions, χ = 3.2 for the entanglement range of M , and τ again has values of - 1 0 - s. The exponent in equation (3) becomes identical with that of the VogelFulcher-Tammann equation (exp(B/[T-T ]) with Β = D T , when (9) S is developed using the hyperbolic temperature dependence of the excess heat capacity A C found so frequently for molecular liquids. D is an inverse measurement of the "fragility" of the liquid or polymer, to which more reference will be made below. 1 4

0

0

0

c

p



41

> α.-

TEMPERATURE

Figure 1. Summary of phenomenology of glassformers showing diverging relaxation times (in part (a)) being related, by points 1, 2, 3. ... on the plot, to vanishing excess entropy (in part (b)) and finally, in part (c), to the level of energy minima on the potential energy hypersurface (or "landscape"). The temperature Τκ corresponds with the energy of the lowest minimum on the amorphous phase hypersurface. Many vertical spikes, corresponding to configurations in which particle core coordinates overlap, are excludedfromthis diagram for clarity. The number of minima on the surface is of order exp(N), where Ν is the number of particles. The height of the landscape relative to kfiTx is a measure of the "strength" of the liquid and seems to be about 1.4-1.6Τκ forfragileliquids. Near the top of the landscape is a crossover region to a domain in which only disconnected remnants of the low temperature "structure" remain to provide the "caging" effects responsible for the slow relaxation described by mode coupling theory. At higher temperatures, >2Τκ, (above the melting points of glassforming liquids, but still below the melting points of non-glassformers) the cages are dismantled, relaxation becomes exponential, and diffusion becomes "free." Boiling typically occurs near 3.5-4Τκ·


42 Scaling Schemes, and "Height*' of the Landscape In a recent scaling proposal, that of Rôssler and Sokolov (12,13) , the α-glass transition temperature used by the author in developing the fragility concept from scaled Arrhenius plotting (14,15), is retained as a scaling parameter and a second characteristic temperature, T , is introduced in order to collapse all liquid viscosities on to the same V T F curve. This second characteristic temperature is higher relative to the first ( T ) for the stronger liquids, and indeed lies close to the crossover temperature of mode coupling theory (12). In the following argument, we show that, if this second characteristic temperature is a measure of T , then both are measures of T , the crossover temperature where landscape domination of the relaxation dynamics takes over. This is because, as we suggested some time ago (14) but now have effectively proved (16,17), T is near the point where the system encounters the "top" of the landscape. We "prove" it by establishing the nature of A C ρ in terms of landscape exploration and then integrating A C over a temperature interval sufficient to exhaust a l l the configurational states (configurons) of the system. This upper limit of the integration, T , is found (16,17) not only to relate closely to Τχ (* T ) , but also with the α - β bifurcation temperature, and the Stickel temperature, T . The "height" of the landscape k e T can be estimated (17) by accepting, after Speedy and Debenedetti (7) and Stillinger (5) that there are, to good approximation, e configurons per mole of heavy atoms, and then calculating the temperature T in the expression, c


g

c

x

x

p

u

c

B

u

N

u

S = k l n W = _ k l n (eN)= R/mole = Hu κ χ c

B

B

d T

.

(4)

τ

Here A C is the heat capacity increment associated with exploration of the landscape (i.e. the jump in C as Τ > T ) . Of course the relative height T / T K is then greater for liquids with small A C (i.e. strong liquids), in accord with the higher T / T found by Rôssler et al (12,13) for the stronger liquids. The value of T turns out to depend only weakly on the functional form assumed for A C . The two simple choices, p

p

g

U

p

c

g

u

p

A C = constant, p

and

A C = constant/T p


43 yield, respectively from equation (3), T =T U

exp(RMCp)

K

(5 a)

and T = T /[1 - R/AC (T )]


u

K

P

(5b)

K

For simple laboratory glassformers like S2CI2 (16), and also for Lennard-Jones argon (18) A C at T is found to be about 17J/K per mole of heavy atoms, and T is then 1.59Τκ (which is about 1.27T ) for each of Eqs. (5a) and (5b). Interestingly enough, this is almost the same as the value of T / T found by Rôssler and co-workers (we prefer T / T ) for the ratio of upper scaling temperature T to T , which means it is the same as the ratio T / T where T is the mode coupling theory T . The identification of T ( T ) with the temperature characterizing the top of the landscape is consistent with the long-standing idea that M C T fails at low temperature because of the crossover to landscape-dominated dynamics (which must be the real meaning of the term "hopping" used in many M C T papers (2,19,20). Our estimate of T is probably a minimum value because a part of the A C at T (of unknown magnitude, but thought to be small in the general case) is vibrational in nature, and there are also other possible non-configurational contributions to the heat capacity difference between liquid and crystal (21). A value of T somewhat above T would seem appropriate because T presumably represents a crossover in dominant relaxation mechanism rather than a real end-point in structural character. In some model systems for which data are available, e.g o-terphenyl, T is also found to correspond with the bifurcation temperature into distinguishable a - and β-relaxations (22) and with the lower limit temperature for accurate data-fitting by the "high temperature Vogel-Fulcher law" in the Stickel-plot analysis (10. The reason that a V T F law (with unphysical parameters) should fit in the free diffusion domain is not at all clear. This is the domain in which a power law fit of the same data yields the T found by the other criteria, so the V T F fit may be a trivial consequence of its relation to the power law through the Bardeen singularity (23). If A C drops to a value like that of Z n C h , 7.5 J/m.K of heavy atoms (24), then p

g

u

c

x

g

g

g

c

g

c

c

g

c

c

x

u

p

g

u

c

c

c

c

p


44 T / T K = 2.94 (equation 5a) (equation 5b becomes inappropriate because for low A C substances, the value of A C is either constant above T or increases with increasing temperature). Using Τ κ = 250 Κ, we obtain T = 736 Κ from equation (5a). This compares poorly with the value of 580K obtained by Rôssler scaling, but this would be improved i f the initial increase in A C observed in the laboratory studies (24) were taken into account. Referring to the relation between entropy, relaxation time, and the position of k T on the energy surface summarized in Figure 1, we note the presence of a regime of free diffusion above the highest features of the landscape but below the boiling point. T falls somewhat below the highest energy features of the landscape because, as noted above, this is merely the temperature about which the landscape dominance of relaxation, not the landscape itself, terminates. This raises a problem which has so far not been addressed in our discussions (11,17) of the landscape limits assessed by the above argument. The problem is that no thermodynamic signature of T , where the landscape contribution to the excess C would be expected to drop out, is seen in the heat capacity vs temperature relation. Since T falls at a value where there is still a six decade difference between vibrational and structural relaxation time scales, the separability of degrees of freedom should still be clear. Either the landscape concept must be at fault, the non-landscape contributions to the A C are seriously underestimated, or the landscape limit is much more tenuously demarcated than a simple endpoint, at T , would suggest. Some information on this point may be forthcoming from the discussion of the " A C problem" for polymers given in the next section. There we will suggest that the presence of the polymer chain has an effect comparable to that of examining A C in liquids at a very short relaxation time hence in the vicinity of T . U

p

p

g

u

p


B

c

u

p

u

P

u

P

P

u

The Polymer Case:

Fragility and the A C

p

Problem

To incorporate polymers into the above scheme, we need to recognise that the special effects of polymer chain length on viscosity enter the problem in the pre-exponents of Vogel-Fulcher or Adam-Gibbs-like expressions (see equation (2)) and, accordingly, to scale them out. This is achieved by use of the W L F representation. To get the appropriate landscape ground state


45 temperature Τ κ in the face of the uncrystallizability of many polymer systems, one must turn to the relation between T of the Vogel-Fulcher relation and the Kauzmann temperature. Τκ can be obtained from the W L F parameter C2 via T = Τκ = T - C2, p r o v i d e d C i is set at the number 16 or 17 (depending on the relaxation time at the reference temperature, t ) . This is because of the little recognised (25) identity 0

0

g

g

C i = log(t /t ) Downloaded by UNIV OF MINNESOTA on October 14, 2014 | http://pubs.acs.org Publication Date: January 28, 1999 | doi: 10.1021/bk-1998-0710.ch003

r

(6)

0

where τ is the Vogel Fulcher pre-exponent which must be on the phonon time scale for the other fit parameters to be physically acceptable. With Τκ identified, a convenient representation of the fragility which varies between extremes of 1 and 0, can be obtained from the ratio Τ κ / Τ = 1 - C 2 / T according to the commonly recognized relation between W L F and V T F equation parameters, C and T (C2 = T - T ) . To see i f polymer behavior is consistent with the small molecule scenario, a second measure of fragility can be obtained from the upper limit of the landscape using either assessments of T from mode coupling theory (e.g., breaks in the quadratic behavior of the Debye Waller factor) , fits to the high temperature relaxation time data, or the α - β bifurcation temperature. According to the recent report of Frick and Richter (26) on polybutadiene, a fairly fragile polymer (the D value (D = B / T ) extracted from the Vogel Fulcher parameters in ref 26 is 10.9 ) there is indeed a close accord between T (215K) and the α - β bifurcation temperature (217K). The data are reproduced in Figure 2. Furthermore, T from Vogel -Fulcher fits of the monomeric friction coefficient for viscosity yield (26) a T of 126K which then gives T / T = 1.69, a little higher than the value for the height of the landscape for fragile liquids given above. In footnote 26 we give evidence that T should actually be higher, hence the ratio Τς/ T lower. When we look for a thermodynamic confirmation of the landscape height, however, we encounter a problem. Instead of finding that the jump in heat capacity at T is large in proportion to how fragile the polymer is, we find that it is much the same for all polymers (despite wide variation in fragility (27,28), and usually rather small compared with the polymer glass heat capacity at T . If this value is adopted for calculations using equation (3), all 0

δ

g

2

0

g

0

c

0

c

0

0

c

0

0

0

g

g



46

2

ίο !

I

ι

ι

3.0

3.5

4.0

1

«

1

4.5 5.0 1000/7(Κ)

5.5

6.0

1

Figure 2. The α-β relaxation bifurcation in the case of the polymer polybutadiene as observed in inelastic neutron scattering^) and dielectric relaxation (open symbols Δ, ±). Note that the bifurcation temperature accords with the MCT T , as in the case of molecular liquids. T /T (VFT) is 1.69 in this case[26], or 1.59T if T is taken to be T -50. (reproduced form ref. 26, by permission). c

C

0

0

g


0

47 polymers would have comparable values of T / T K far above their T / T values. Clearly there is something to be understood about the relation of A C in polymers to the corresponding quantity in molecular liquids. Some insight into, if not resolution of, this problem can be obtained by examining the relation between the A C and the chain length. As the latter decreases and the T correspondingly drops quickly, one finds (29,52) that the value of A C rapidly increases. As with the increase of T , most of the effect is obtained with the first few repeat unit additions. The relation is shown in Figure 3. However the chain length of polymers has generally been found not to affect their fragilities (27,28) (though there is some recent conflict from different estimates in the case of polystyrene (31,32)). Clearly, though, the correlation between A C and fragility seen in the inorganic ionic (33), and covalent (34) network polymers does not hold for carbon chain polymers (in which the in-chain bonds are inviolable). U

c

0

p

p

g

p


g

P

To look into this further, we show in Fig 4, in part (a), the behavior of the heat capacity of polypropylene, in units of J/K.(mol of -CH2-CH2(CH3)- repeat units) (35,36) in comparison with that of the molecular liquid 3-methyl pentane (37) (divided by 2 to have the same mass basis as the polymer repeat unit) (38). It is seen that the liquid heat capacity of the hexane isomer (x 0.5) falls not much above the natural extrapolation to lower temperatures of the heat capacity per repeat unit of the polymer. This implies that the main effect of polymerization, as far as the change in heat capacity at Tg is concerned, is to postpone the glass transition until a much higher vibrational heat capacity has been excited. This not only reduces the value of A C but has a disproportionate effect on the ratio C , l / C , at T . This happens despite a lower glassy heat capacity in the polymer than in the molecular liquid at the same temperature. The latter effect is a direct consequence of the lower Debye temperature (and lower vibrational anharmonicity) at a given temperature for in-chain interactions in the polymer than for mtermolecular interactions in the same mass of molecules. A somewhat similar effect is seen in the effect of pressure on the heat capacity of 3-methyl pentane (37) where the origin of the effect is a little clearer. Takara et al (37), in a study which established the correctness of the Adam-Gibbs conclusion that the glass transition should occur at constant T S , found that, as pressure acting on the glass state increased, the heat capacity of the glass rose to higher values before the glass transition occurred. This can p

p

p

g

g

C



48

0.1 ι 1

1 10

1 10

2

1 10

3

» 10

4

η

Figure 3. The heat capacity of several polymer glasses and liquids as a function of increasing number or repeat units in the chain, showing the rapid decrease in ACp which accompanies MW increase (and consequent T increase). The cases illustrated are PDMS (triangles), polycarbonate (filled circles) and polymethyl siloxane (open circles), (reproduced from ref.29, by permission) g



02

0.4

temperature/ 1000K 200

3-Methylpentane

100

1-Propanol

50

100

150

Τ I Κ Figure 4. (a) Heat capacity of polypropylene crystal, glass and liquid in J/K per mole of -CH2-CH2(CH3>-repeat units, from refs. 35 and 36 compared with the same properties of 3-methyl pentane (1/2 mole) from ref. 37. The comparison shows how the postponement of the glass transition to the higher temperature, consequent on the extension of the carbon chain, completely changes the relation between liquid and glass heat capacities, causing polymers to appear "strong" by the C (l)/C (g) (at T ) criterion. The ACp/Cp, value observed for polymers is characteristic of that of a molecular liquid studied on the nanosecond time scale. p

p

g

g

(b) heat capacity of 3-methyl pentane glass and supercooled liquid at three different pressures 0.1, 108.4, and 198.6 MPa, from ref. 37, showing how the effect of increasing pressure is somewhat similar to the effect of increasing molecular weight in its influence on the magnitude of the vibrational heat capacity at the temperature of the glass transition, (reproduced from ref.37, by permission)



50 be understood as a result of increase of pressure, at constant temperature, rendering the vibrational motions more harmonic, since it raises the vibrational frequencies (positive Gruneisen constants dlnv/dV). Consequently i f also raises the glass transition temperature (since the glass transition is itself a consequence of anharmonicity (39-41)). While the effect of pressure in this respect is weaker than the effect of increase of molecular weight, both factors, through their effects on the mean vibrational anharmonicity, permit the glass (i.e. vibrational) heat capacity to build up to larger values before Tg intervenes, while less strongly affecting the liquid heat capacity. This progressively diminishes the observed jump in C and consequently, but misleadingly, makes the substance appear "stronger" by the change in heat capacity criterion. In fact the behavior observed has much in common with a molecular liquid which has been heated to far above its T (where its excess heat capacity has fallen to a fraction of its value at T ) . This itself is understandable since the between-chain interactions must be highly anharmonic in both crystal and liquid states at temperatures in excess of twice the glass transition temperature of the monomer liquid. A n energy hypersurface consequence should be the existence of many low energy modes of escape from any given minimum and a less well-defined landscape limit (hence dynamical crossover temperature) than in the case of fragile molecular liquids. This may be the reason that polymers tend to be more fragile than molecular liquids (43), and tend to obey the W L F or V F T equations over wider ranges of relaxation times and with more consistent pre-exponents (hence C i values) than do fragile molecular liquids. The extension of the crystalline polypropylene data to the melting point shows the difference in liquid and crystalline heat capacity almost disappearing, giving an appearance akin to that of the glass-forming liquid metals (44), the crystalline states of which melt to liquids of very low viscosity. A l l considered, it is not obvious how to obtain an appropriate ACp to use in a landscape height calculation or indeed if the concept of a landscape limit is a useful one for polymers (although the coincidence of T and α - β bifurcation temperatures seen in Fig. 2 would suggest that it should be). It is possible that one could use T / T g ( = Τχ/Tg) value to back-calculate an effective contributing A C p value. This is an area for further work, in which an investigation of the effect of diluent concentration on A C for the binary polymer diluent solutions, could play a useful role. p

g

g

c

c

p


51 Concluding

remarks.


The "polymer problem" brought out in the above discussion is provocative and deserves further exploration. Understanding could be helped considerably by a systematic investigation of the entropies and Kauzmann temperatures of a series of easily crystallized η-mers of increasing η value, and a concurrent determination of their fragilities by the F1/2 criterion (45). Acknowledgements. This work was supported by the NSF under Solid State Chemistry grant no. DMR9614531. The assistance of Vesselin Velikov in identifying the effect of chain length on the jump in heat capacity at T from the Russian literature is gratefully acknowledged. We also thank P.G. Santangelo and C. M . Roland for sharing information on their current studies of heat capacity and fragility in polymers in advance of publication. g

Literature

Cited

1.

Anderson, P. W. In Ill-Condensed Matter; Balian, R.; Maynard, R.; Toulouse, G., Ed.; Les Houches: North Holland, 1979; pp 171. 2. Götze, W. In Liquids, Freezing, and the Glass Transition, Hansen, J.-P.; Levesque, D., Ed.; NATO-ASI; Les Houches: North Holland (Amsterdam), 1989: Götze, W., and Sjögren, L., Rep. Progr. Phys.,1992,55,241, 3. Goldstein, M . J. Chem. Phys. 1969, 51, 3728. The configuration space is a space of 3N + 1 dimensions, Ν the number of particles. 4. Gibbs, J. H. In Modern Aspects of the Vitreous State; McKenzie, J. D., Ed.; Butterworths: London, 1960; ch. 7. 5. Stillinger, F. H.; Weber, T. A. Science 1984, 225, 983 (1984); 1995, 267, 1935. 6. Ball, K. D.; Berry, R. S.; Kuntz, R. E.; Li, F.-Y.; Proykova, Α.; Wales, D. J. Science 1996, 271, 963. 7. Speedy, R. J.; Debenedetti, P. G. Mol. Phys. 1996, 88, 1293. 8. Kauzmann, W. Chem. Rev. 1948, 43, 219. 9. Angell, C. Α.; Sichina, W. Ann. N.Y. Acad. Sci. 1976, 279, 53. 10. Stickel, F.; Fischer, E. W.; Richert, R. J. Chem. Phys. 1996, 104, 2043. 11. Angell, C. A. APS Symposium Proceedings, J. Res. NIST (in press). 12. Rôssler, Ε.; Sokolov, A. P. Chem. Geol. 1996, 128, 143. 13. Novikov, V. N.; Rössler, E.; Malinovsky, V. Κ.; Surovtev, Ν. V. Europhys. Lett. 1996, 35, 289. 14. Angell, C. A. J. Phys. Chem. Sol. 1988, 49, 863. 15. Angell, C. A. J. Non-Cryst. Sol., 1991, 131-133, 13. 16. Angell, C. Α.; Tucker, J. C. (to be published). 17. Angell, C. Α., in "Complex Behavior of Glassy Systems" (Proc. 14th Sitges Conference on Theoretical Physics, 1996) Ed. M . Rubi, Springer, 1997, p. 1. 18. Clarke, J. H. R. Trans. Faraday Soc. 2 1976, 76, 1667. 19. G. Li, W. Du, A. Sakai, and H. Z. Cummins, Phys. Rev A 1992, 46, 3343


52 20. W. Petry, E. Bartsch, F. Fujara, M . Kiebel, H. Sillescu, and B. Ferrago, Z. Phys. B, 1991, 83, 175 21. M . Goldstein, J. Chem. Phys., 1976, 64, 4767. 22. E. Rössler, Phys. Rev. Lett. 1990, 65, 1595 23. P. W. Anderson. Ann. N . Y. Acad. Sci, 1986, 484, 241 24. C. A. Angell, E. Williams, K. J. Rao and J. C. Tucker, J. Phys. Chem., 1977, 81, 238 25. Angell, C. A. Polymer, 1997, 38, 6261. 26. Frick, B., and Richter, D, Science, 1995, 267, 1939, (The D value (D = B/T ) extractedfromthe Vogel-Fulcher parameters provided in this ref. is 10.9, comparable to polypropylene oxide, however the T value is somewhat further below T than the common finding T = T - 50). Other assessments (see ref. 28) assign PED a very high fragility. 27. Angell, C. Α., Monnerie, L., and Torell, L. M., Symp. Mat. Res. Soc. Ed. J. M . O'Reilly, 1991, 215, 327. 28. Plazek, D. J. and Ngai, K.-L., Macromolecules, 1991, 24, 1222 29. Bershtein, V. Α.; Egorov, V. M. Differential Scanning Calorimetry of Polymers: Physics, Chemistry, Analysis, Technology; Ellis Horwood: New York. 30. Jacobsen, P., Borjesson, L., and Torell, L. M., J. Non-Cryst. Sol. 31. Sahoune Α., and Piche, L., in "Structure and Dynamics of Glasses and Glassformers, Eds., Angell, C. Α., Ngai, K.-L., Kieffer, J., Egami, T., and Nienhaus, G. U., MRS. Symp. Proc. 1997, 455, 183 32. Santangelo, P. G., and Roland, C. M., (preprint, 1997) 33. Angell, C. Α., J. Non-Cryst. Sol. 1985, 73, 1: Lee and Tatsumisago, M., J. Non-Cryst. Sol., 1996, 34. Tatsumisago, M., B. L Halfpap, J. L. Green, S. M . Lindsay, and C. A. Angell, Phys. Rev. Lett., 1990, 64, 1549 35. Gaur and Wunderlich, B., J. Phys. Chem. Ref. Data, 10, 1001, (1981) 36. Wunderlich, B., Polymer, 29, 1485, (1988) 37. S.Tanaka, O. Yamamuro, and H. Suga, J. Non-Cryst. Sol. 171,259, (1994) 38. Comparison with 43% of the heat capacity of 2,4, dimethyl heptane would arguably be a better choice but the C data are not available.) 39. Angell, C. A.J. Am. Ceram. Soc, 1968, 51, 117: C. A. Angell, P. H. Poole, and J. Shao, Nuovo Cimento, 1994, 16D, 993 : J. Shao and C.A. Angell, Proc. XVIIth Internat. Congress on Glass, (Beijing) 1995, Vol. 1, P. 311 40. Ngai, K. L., and Roland, C. M., in "Structure and Dynamics of Glasses and Glassformers, Eds., Angell, C. Α., Ngai, K.-L., Kieffer, J., Egami, T., and Nienhaus, G. U., MRS. Symp. Proc. 1997, 455, 81 41. Sokolov, A. P., "Structure and Dynamics of Glasses and Glassformers, Eds., Angell, C. Α., Ngai, K.-L., Kieffer, J., Egami, T., and Nienhaus, G. U., MRS. Symp. Proc. 1997, 455, 69 42. Buchenau, U., Phil. Mag. , 1992 65, 303-315. 43. Bohmer, R., Ngai, K-.L., Angell, C. Α., Plazek, D., J.J. Chem. Phys., 1993, 99, 4201 44. H. S. Chen and D. Turnbull, Appl. Phys. Lett. 1967, 10, 284, 45. R. Richert and C. A. Angell, J. Chem. Phys. (in press). 0

0


g

0

g

p


Entropy, Landscapes, and Fragility in Liquids and Polymers, and the

Recommend Documents