Glass transition cooperativity from broad band heat capacity spectroscopy

Molecular dynamics is often studied by broad band dielectric spectroscopy (BDS) because of the wide dynamic range available and the large number of processes resulting in electrical dipole fluctuations and with that in a dielectrically detectable relaxation process. Calorimetry on the other hand is an effective analytical tool to characterize phase and glass transitions by its signatures in heat capacity. In the linear response scheme, heat capacity is considered as entropy compliance. Consequently, only processes significantly contributing to entropy fluctuations appear in calorimetric curves. The glass relaxation is a prominent example for such a process. Here, we present complex heat capacity at the dynamic glass transition (segmental relaxation) of polystyrene (PS) and poly(methyl methacrylate) (PMMA) in a dynamic range of 11 orders of magnitude, which is comparable to BDS. As one of the results, we determined the characteristic length scale of the corresponding fluctuations. The dynamic glass transition measured by calorimetry is finally compared to the cooling rate dependence of fictive temperature and BDS data. For PS, dielectric and calorimetric data are similar but for PMMA with its very strong secondary relaxation process some peculiarities are observed.     

Heat capacities and entropies of liquid,high-melting-point polymers containing phenylene groups (PEEK,PC, and PET)

The heat capacity at constant pressure of liquid PEEK, poly(oxy-1,4-phenylene-oxy-1,4-phenylene-carbonyl-1,4-phenylene), has been measured by scanning calorimetry from 420 to 680 K, and that of PC, poly(4,4′-isopropylidenediphenylene carbonate), from 325 to 610 K. These new data were combined with data-bank data for PC and PET, poly(ethylene terephthalate), over wide temperature ranges. An addition scheme for liquid heat capacities of similar macromolecules has been obtained. In addition, values of absolute entropy, residual entropy for the glassy state, enthalpy, and Gibbs function are estimated for these three polymers. Both melting and glass transition temperatures have been confirmed. The heat capacity increases at the glass transition temperature have been determined by making use of previously calculated solid-state heat capacities.     

Gaussian excitations model for glass-former dynamics and thermodynamics

We describe a model for the thermodynamics and dynamics of glass-forming liquids in terms of excitations from an ideal glass state to a Gaussian manifold of configurationally excited states. The quantitative fit of this three parameter model to the experimental data on excess entropy and heat capacity shows that "fragile" behavior, indicated by a sharply rising excess heat capacity as the glass transition is approached from above, occurs in anticipation of a first-order transition--usually hidden below the glass transition--to a "strong" liquid state of low excess entropy. The distinction between fragile and strong behavior of glass formers is traced back to an order of magnitude difference in the Gaussian width of their excitation energies. Simple relations connect the excess heat capacity to the Gaussian width parameter, and the liquid-liquid transition temperature, and strong, testable, predictions concerning the distinct properties of energy landscape for fragile liquids are made. The dynamic model relates relaxation to a hierarchical sequence of excitation events each involving the probability of accumulating sufficient kinetic energy on a separate excitable unit. Super-Arrhenius behavior of the relaxation rates, and the known correlation of kinetic with thermodynamic fragility, both follow from the way the rugged landscape induces fluctuations in the partitioning of energy between vibrational and configurational manifolds. A relation is derived in which the configurational heat capacity, rather than the configurational entropy of the Adam-Gibbs equation, controls the temperature dependence of the relaxation times, and this gives a comparable account of the experimental observations without postulating a divergent length scale. The familiar coincidence of zero mobility and Kauzmann temperatures is obtained as an approximate extrapolation of the theoretical equations. The comparison of the fits to excess thermodynamic properties of laboratory glass formers, and to configurational thermodynamics from simulations, reveals that the major portion of the excitation entropy responsible for fragile behavior resides in the low-frequency vibrational density of states. The thermodynamic transition predicted for fragile liquids emerges from beneath the glass transition in case of laboratory water and the unusual heat capacity behavior observed for this much studied liquid can be closely reproduced by the model.     

Freezing-in and production of entropy in vitrification

Following the classical concepts developed by Simon [Z. Anorg. Allg. Chem. 203, 219 (1931)], vitrification in the cooling of glass-forming melts is commonly interpreted as the transformation of a thermodynamically (meta)stable equilibrium system into a frozen-in, thermodynamically nonequilibrium system, the glass. Hereby it is assumed that the transformation takes place at some well-defined sharp temperature, the glass transition temperature Tg. However, a more detailed experimental and theoretical analysis shows that the transition to a glass proceeds in a broader temperature range, where the characteristic times of change of temperature, tauT=-(TT), and relaxation times, tau, of the system to the respective equilibrium states are of similar order of magnitude. In this transition interval, the interplay of relaxation and change of external control parameters determines the value of the structural order parameters. In addition, irreversible processes take place in the transition interval, resulting both in an entropy freezing-in as well as in an irreversible increase of entropy and, as a result, in significant changes of all other thermodynamic parameters of the vitrifying systems. The effect of entropy production on glass transition and on the properties of glasses is analyzed here for the first time. In this analysis, the structural order-parameter concept as developed by de Donder and van Rysselberghe [Thermodynamic Theory of Affinity (Stanford University Press, Stanford, 1936)] and Prigogine and Defay [Chemical Thermodynamics (Longmans, London, 1954)] is employed. In the framework of this approach we obtain general expressions for the thermodynamic properties of vitrifying systems such as heat capacity, enthalpy, entropy, and Gibbs' free energy, and for the entropy production. As one of the general conclusions we show that entropy production has a single maximum upon cooling and two maxima upon heating in the glass transition interval. The theoretical concepts developed allow us to explain in addition to the thermodynamic parameters also specific features of the kinetic parameters of glass-forming melts such as the viscosity. Experimental results are presented which confirm the theoretical conclusions. Further experiments are suggested, allowing one to test several additional predictions of the theory.     

Heat capacity functions of polystyrene in glassy and in liquid amorphous state and glass transition

The heat capacity or the specific heat is for any crystalline, partially amorphous or completely amorphous substance or material a significant thermodynamic property. The glass transition may be regarded as the melting point of amorphous substances and materials, a transition property of an outstanding technical importance. A crucial point is the fact that the presence of a glass transition is an unequivocal proof of an amorphous content of a material. Furthermore, the change of the specific heat at the glass transition temperature enables the quantitative determination of the amorphicity on a relative or absolute level of any substance or material. The absolute determination of the amorphicity affords a calibration with a reference corresponding to the material under investigation. The crystallinity for this reference substance must be known from the preparation and or by any independent analytical method. The literature data for the specific heat and the glass transition of polystyrene were collected and evaluated. Data were found for the specific heat in literature from 10 to 470 K. The data were unified for each of the reported temperature in a mean value and the corresponding standard deviation was determined. An excellent conformity was found in the glassy state of polystyrene with standard deviations lower than 0.7%. The standard deviations above the glass transition were considerably higher.     

Thermodynamic properties of methane hydrate in quartz powder

Using the experimental method of precision adiabatic calorimetry, the thermodynamic (equilibrium) properties of methane hydrate in quartz sand with a grain size of 90-100 microm have been studied in the temperature range of 260-290 K and at pressures up to 10 MPa. The equilibrium curves for the water-methane hydrate-gas and ice-methane hydrate-gas transitions, hydration number, latent heat of hydrate decomposition along the equilibrium three-phase curves, and the specific heat capacity of the hydrate have been obtained. It has been experimentally shown that the equilibrium three-phase curves of the methane hydrate in porous media are shifted to the lower temperature and high pressure with respect to the equilibrium curves of the bulk hydrate. In these experiments, we have found that the specific heat capacity of the hydrate, within the accuracy of our measurements, coincides with the heat capacity of ice. The latent heat of the hydrate dissociation for the ice-hydrate-gas transition is equal to 143 +/- 10 J/g, whereas, for the transition from hydrate to water and gas, the latent heat is 415 +/- 15 J/g. The hydration number has been evaluated in the different hydrate conditions and has been found to be equal to n = 6.16 +/- 0.06. In addition, the influence of the water saturation of the porous media and its distribution over the porous space on the measured parameters has been experimentally studied.     

Configurational entropy of binary hard-disk glasses: nonexistence of an ideal glass transition

We study the thermodynamics of a binary hard-disk mixture in which the ratio of disk diameters is kappa=1.4. We use a recently developed molecular dynamics algorithm to calculate the free-volume entropy of glassy configurations and obtain the configurational entropy (degeneracy) of the supercompressed liquid as a function of density. We find that the configurational entropy of the glasses near the kinetic glass transition is very close to the mixing entropy, suggesting that the degeneracy is zero only for the phase-separated crystal. We explicitly construct an exponential number of jammed packings with densities spanning the spectrum from the accepted "amorphous" glassy state to the phase-separated crystal, thus showing that there is no ideal glass transition in binary hard-disk mixtures. This construction also demonstrates that the ideal glass, defined as having zero configurational entropy, is not amorphous, but instead is nothing more than a phase-separated crystal. This critique of the presumed existence of an ideal glass parallels our previous critique of the idea that there is a most-dense random (close) packing for hard spheres [Torquato et al., Phys. Rev. Lett. 84, 2064 (2000)]. We also perform free-energy calculations to determine the equilibrium phase behavior of the system. The calculations predict a first-order freezing transition at a density below the kinetic glass transition. However, this transition appears to be strongly kinetically suppressed and is not observed directly. New simulation techniques are needed in order to gain a more complete understanding of the thermodynamic and kinetic behavior of the binary disk mixture and, in particular, of the demixing process during crystallization.     

Heat capacity and chemical equilibria of liquid selenium

Heat capacities of liquid selenium have been measured by computer interfaced differential scanning calorimetry in the metastable region with an accuracy of ± 1% from 330 to 520°K. To avoid crystallization, the measurements were done on cooling. A semiquantitative fitting of the heat capacity to vibrational energy contributions, free volume (hole) effects, and heats of reaction from the changes in the ring-chain and depolymerization equilibria was possible to within ±5% of the newly measured and literature data between the glass transition temperature (ca. 303°K) and 1000°K. It could be established that the shift in the ring-chain equilibrium is not the major reason for the overall decrease in heat capacity above the glass transition temperature. The floor temperature, which was earlier placed at about 356°K, is possibly below the glass transition temperature. The increase in heat capacity beyond 800°K has been linked with the depolymerization reaction.     

The thermodynamic properties of polytetrafluoroethylene

Polytetrafluoroethylenes of different crystallinity were analyzed between 220 and 700 K by differential scanning calorimetry. A new computer coupling of the standard DSC is described. The measured heat capacity data were combined with all literature data into a recommended set of thermodynamic properties for the crystalline polymer and a preliminary set for the amorphous polymer (heat capacity, enthalpy, entropy, and Gibbs energy; range 0–700 K). The crystal heat capacities have been linked to the vibrational spectrum with a θ₃ of 54 K, and θ₁ of 250 K, and a full set of group vibrations. C_v to C_p conversion was possible with a Nernst–Lindemann constant of A = 1.6 × 10^?3 mol K/J. The glass transition was identified as a broad transition between 160 and 240 K with a ΔC_p of 9.4 J/K mol. The room-temperature transitions at 292 and 303 K have a combined heat of transition of 850 J/mol and an entropy of transition of 2.90 J/K mol. The equilibrium melting temperature is 605 K with transition enthalpy and entropy of 4.10 kj/mol and 6.78 J/K mol, respectively. The high-temperature crystal from is shown to be a condis crystal (conformationally disordered), and for the samples discussed, the crystallinity model holds.     

Analysis of a symmetric neopolyol ester

Quantitative thermal analysis was carried out for tetra[methyleneoxycarbonyl(2,4,4-trimethyl)pentyl]methane. The ester has a glass transition temperature of 219 K and a melting temperature of 304 K. The heat of fusion is 51.3 kJ mol^?1, and the increase in heat capacity at the glass transition is 250 J K^?1 mol^?1. The measured and calculated heat capacities of the solid and liquid states from 130 to 420 K are reported and a discussion of the glass and melting transitions is presented. The computation of the heat capacity made use of the Advanced Thermal Analysis System, ATHAS, using an approximate group-vibration spectrum and a Tarasov treatment of the skeletal vibrations. The experimental and calculated heat capacities of the solid ester were compared over the whole temperature range to detect changes in order and the presence of large-amplitude motion. An addition scheme for heat capacities of this and related esters was developed and used for the extrapolation of the heat capacity of the liquid state for this ester. The liquid heat capacity for the title ester is well represented by 691.1+1.668T [J K^?1 mol^?1]. A deficit in the entropy and enthalpy of fusion was observed relative to values estimated from empirical addition schemes, but no gradual disordering was noted outside the transition region. The final interpretation of this deficit of conformational entropy needs structure and mobility analysis by solid state¹³C NMR and X-ray diffraction. These analyses are reported in part II of this investigation.     

A theoretical interpretation of free volume at glass transition

We device a relaxed lattice model (RLM) to study the mechanism of glass transition,which unifies the cageeffects from particle-particle interaction and entropy.By analyzing entropy in RLM with considering the influence of interactions on equilibrium,we demonstrate that glass transition is a second-order phase transition.For a perfect onedimensional linked particle system like linear polymer under normal pressure,the free volume at glass transition is rigorously deduced out to be 2.6％,which provides a theoretical basis for the iso-free volume of 2.5％ given by Willian,Landel and Ferry (WLF) equation.Extending to system with dead particles linked with higher dimensions like branched or cross-linked chains under positive or negative pressure,free volume at glass transition is varied,based on which we construct a phase diagram of glass transition in the space of free volume-dead particle-pressure.This demonstrates that free volume is not the single parameter determining glass transition,while either dead particles like cross-linked points or external force fields like pressure can vary free volume at the glass transition.     

A new formula for the entropy of mixing in polymer systems and free volume

The famous equations of Flory-Huggins for the entropy of mixing with one highmolecular component are of great importance for polymer physics. But Gujrati stated in 1980 [12] that these equations cannot be exact. This is why we derived a new formula for the dependence of the entropy from the fraction of vacant sites in a quasi-lattice. It differs significantly from that of Huggins and still more from that of Flory in the case of low free volume. The equations of Flory-Huggins are correct with reference to low polymer content only.If our formula for entropy is used instead of that of Huggins an important result of the theory of Gibbs-DiMarzio is called in question. The increase of thermal expansion at the glass transition cannot be explained by an increase of vacant sites. A growth of the number of unoccupied sites according to the thermodynamic equilibrium condition would bring about a far too great thermal expansion coefficient. From estimations of the energy of interaction between polymer molecules, which can be found in literature, it follows that the increase of entropy is far too small to enable the formation of vacant sites above the glass transition. It is unambiguously shown that the free volume, commonly regarded to be the decisive quantity with respect to glass transition, cannot consist of holes as considered in the quasi-lattice model and in many theoretical treatments.     

The thermal properties of poly(oxy-1,4-benzoyl), poly(oxy-2,6-naphthoyl), and its copolymers

The thermal properties, i.e., heat capacity, enthalpy, entropy, and Gibbs function, and the transition behavior of the copolymer system of 4-hydroxybenzoic acid and 2,6-hydroxynaphthoic acid have been studied based on differential scanning calorimetry. The heat capacities of the glass, crystal, and anisotropic melt are shown to be largely additive on a molar basis. Additivity is lost in the two transition regions, glass transition and disordering transition. Isothermal crystallization experiments on the copolymers revealed the existence of two types of crystals which melt at high temperature (fast-grown crystals) and low temperature (slowly grown crystals). The ATHAS computation method is used to bring heat capacities of the solid state into agreement with approximate frequency spectra. The changes in heat capacity at the glass transitions occur at 434°K for the poly(oxy-1,4-benzoyl) [33.2 J/(K mol)] and at 420°K for poly(oxy-2,6-naphthoyl) [46.5 J/(K mol)]. The copolymers have a transition range of above 100°K. The anisotropic melt is linked to the well-known condis state of poly(oxy-1,4-benzoyl) by a continuous changes in disorder and mobility without an additional first-order transition.     

Temperature modulated calorimetry and dielectric spectroscopy in the glass transition region of polymers

The results from temperature modulated DSC in the glass transition region of amorphous and semicrystalline polymers are described with the linear response approach. The real and the imaginary part of the complex heat capacity are discussed. The findings are compared with those of dielectric spectroscopy. The frequency dependent glass transition temperature can be fitted with a VFT-equation. The transition frequencies are decreased by 0.5 to 1 orders of magnitude compared to dielectric measurements. Cooling rates from standard DSC are transformed into frequencies. The glass transition temperatures are also approximated by the VFT-fit from the temperature modulated measurements. The differences in the shape of the curves from amorphous and semicrystalline samples are discussed.Dedicated to Professor Bernhard Wunderlich on the occasion of his 65th birthday     

Polymer solidification as a dissipative process

Because their conformational entropy is a nonlinear function of size, polymer molecules can only solidify under conditions far from equilibrium with the formation of dissipative structures. A dissipative structure is nonhomogeneous, it evolves with time from an initially space-dependent fluctuation, and has a characteristic size that depends upon the initial perturbation and the conditions that apply during its entire history. The glass transition is envisioned as the formation of the ultimate dissipative structure where all of the latent heat of crystallization is dissipated within the structure. The mechanism of glass formation can be described as the spinodal decomposition of a homogeneous liquid into two interpenetrating networks of high- and low-energy solids with a characteristic distance and a gradient of density and/or composition extending through both high- and low-energy regions.     

Analysis of the residual entropy of amorphous polyethylene at 0 K*

The residual entropy of amorphous polyethylene (PE) at 0 K is discussed within the framework of the heat capacity (C_p). The measured C_p of the liquid was extended from the glass transition to low temperature by separately finding its three parts—the vibrational, conformational, and external contributions—and extrapolating each to low temperature. The vibrational C_p was calculated from the frequency distributions of the group vibrations on the basis of force constants obtained from experimental infrared and Raman spectra as well as the skeletal vibrations in the amorphous solid (glass) obtained from fitting of the appropriate experimental C_p to Debye functions in the form suggested by Tarasov. The conformational part of C_p was evaluated from a fit of the heat capacity of the liquid, decreased by the contributions of the vibrational and external parts, to a one‐dimensional Ising model that can be extrapolated to 0 K and requires two discrete states described by stiffness, cooperativity, and a degeneracy parameter. The external part was computed from the experimental data for expansivity and compressibility, fitted to an empirical equation of state, and modified at low temperatures in accordance with the Nernst–Lindemann approximation. The computed C_p of the liquid PE agreed with the experiment from 600 K to the beginning of the glass transition at about 260 K. Extending the heat capacity to 0 K, bypassing the freezing of the large‐amplitude conformational motion in the glass transition, led to a positive residual entropy and enthalpy and avoided the so‐called Kauzmann paradox. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1245–1253, 2002     

Low-temperature heat capacity of room-temperature ionic liquid, 1-hexyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide

Heat capacities of liquid, stable crystal, and liquid-quenched glass of a room-temperature ionic liquid (RTIL), 1-hexyl-3-methylimidazolium bis(trifluromethylsulfonyl)imide were measured between 5 and 310 K by adiabatic calorimetry. Heat capacity of the liquid at 298.15 K was determined for an IUPAC project as (631.6 +/- 0.5) J K(-1) mol(-1). Fusion was observed at T(fus) = 272.10 K for the stable crystalline phase, with enthalpy and entropy of fusion of 28.34 kJ mol(-1) and 104.2 J K(-1) mol(-1), respectively. The purity of the sample was estimated as 99.83 mol % by the fractional melting method. The liquid could be supercooled easily and the glass transition was observed around T(g) approximately 183 K, which was in agreement with the empirical relation, T(g) approximately ((2)/(3)) T(fus). The heat capacity of the liquid-quenched glass was larger than that of the crystal as a whole. In the lowest temperature region, however, the difference between the two showed a maximum around 6 K and a minimum around 15 K, at which the heat capacity of the glass was a little smaller than that of crystal.