An equilibrium supercooled liquid’s entropy and enthalpy in the Kauzmann and the third law extrapolations, and a proposed experimental resolution

Application of advanced thermal analysis for characterization of crystalline and amorphous phases of carvedilol

The thermal behaviour of crystalline and amorphous carvedilol (CAR) phases was studied by advanced thermal analysis using Quantum Design Physical Property Measurement System and Differential Scanning Calorimetry. Theoretical functions describing crystalline carvedilol heat capacity at low temperatures and the Debye-Einstein function for high temperatures were obtained. Based on the experimental heat capacity values, solid and liquid baselines were established, and the state functions (H, S, G) for solid and liquid states were calculated. A comprehensive characterization of melting and glass transition processes was obtained. CAR is easily amorphizable by cooling the liquid. The residual entropy, which quantifies the extent of frozen-in disorder in the amorphous solid, for glassy CAR was estimated as 51 J·mol^-1·K^-1. The Kauzmann temperature (T_K) was estimated based on enthalpy and entropy. Molecular motions in the amorphous phase were also studied. The activation energy for structural relaxation (E_a = 539 kJ·mol^-1) and fragility parameter (m = 91) were obtained from the non-isothermal physical ageing. The isothermal physical ageing kinetics of amorphous CAR was studied by applying Kohlrausch-Williams-Watts (KWW) model. The mean molecular relaxation time constant (τ^KWW = 117 min) and relaxation constant (β^KWW = 0.33) were obtained. CAR was classified as a fragile glass-former. Furthermore, τ^KWW constant for samples aged at 303.15 K is very low, thus, the physical ageing will occur during the short- and long-term storage of amorphous CAR, potentially changing its physicochemical properties during the ageing process. However, the results of molecular mobility studies (high molecular motions) show that the relationship between molecular motions in a glassy solid and its tendency to crystallization does not seem to follow an expected pattern, i.e., no crystallization occurred by thermal treatment of glassy, supercooled liquid and liquid phases of CAR as one would expect. Modern calorimetry and quantitative thermal analysis provided the fundamental kinetic and thermodynamic information about the crystalline and amorphous states of CAR.

Instability and thermal conductivity of pressure-densified and elastically altered orientational glass of Buckminsterfullerene

We report on the temperature, pressure, and time (T, p, and t)-dependent features of thermal conductivity, κ, of partially ordered, non-equilibrium state of C₆₀-OG, the orientational glass of Buckminsterfullerene (at T below the orientational freezing temperature T_og) made more unstable (i) by partially depressurizing its high-p formed state to elastically expand it and (ii) by further pressurizing that state to elastically contract it. The sub-T_og effects observed on heating of C₆₀-OG differ from those of glasses because phonon propagation depends on the ratio of two well-defined orientational states of C₆₀ molecules and the density of the solid. A broad peak-like feature appears at T near T_og in the κ-T plots of C₆₀-OG formed at 0.7 GPa, depressurized to 0.2 GPa and heated at 0.2 GPa, which we attribute to partial overlap of the sub-T_og and T_og features. A sub-T_og local minimum appears in the κ-T plots at T well below T_og of C₆₀-OG formed at 0.1 GPa, pressurized to 0.5 GPa and heated at 0.5 GPa and it corresponds to the state of maximum disorder. Although Buckminsterfullerene is regarded as an orientationally disordered crystal, variation of its properties with T and p is qualitatively different from other such crystals. We discuss the findings in terms of the nature of its disorder, sensitivity of its rotational dynamics to temperature, and the absence of the Johari-Goldstein relaxation. All seem to affect the phenomenology of its glass-like transition.

Thermal behavior in Se-Te chalcogenide system: interplay of thermodynamics and kinetics

Heat capacity measurements were performed for Se, Se90Te10, Se80Te20, and Se70Te30 materials in the 230-630 K temperature range. Both glassy and crystalline Cp dependences were found to be identical within the experimental error. The compositional dependence of the N-type undercooled liquid Cp evolution was explained on the basis of free-volume theory; vibrational and chemical contributions to heat capacity were found to be roughly similar for all Se-Te compositions. The thermal behavior in the Se-Te chalcogenide system was thoroughly studied: glass transition, cold crystallization, and melting were investigated in dependence on composition and various experimental conditions (heating rate, particle size, and pre-nucleation period). The kinetics of the structural relaxation and crystallization processes are described in terms of the Tool-Narayanaswamy-Moynihan and Johnson-Mehl-Avrami models. The complexity of these processes is thoroughly discussed with regard to the compositionally determined changes of molecular structures. The discussion is conducted in terms of the mutual interplay between the thermodynamics and kinetics in this system.

Non-exponential relaxation, fictive temperatures, and dispersive kinetics in the liquid-glass-liquid transition range of acetaminophen, sulfathiazole, and their mixtures

To investigate the effects of added molecular heterogeneity on the hysteretic features of liquid-glass-liquid transition, we studied acetaminophen, sulfathiazole, and three of their mixtures by calorimetry, and determined the T(g) and the fictive temperature, T(f), from changes in the enthalpy and entropy on the cooling and heating paths, as well as the non-exponential parameter, β(cal). We find that, (i) T(f) for cooling is within 1-3 K of T(f) for heating and both are close to T(g), (ii) the closed loop entropy change in the liquid-glass-liquid range is negligibly small, (iii) T(g) and T(f) increase on increasing sulfathiazole in the mixture, (iv) β(cal) first slightly increases when the second component is added and then decreases, and (v) ageing causes deviations from a non-exponential, nonlinear behavior of the glass. In terms of fluctuations in a potential energy landscape, adding a solute heterogeneity would shift the state point to another part of the landscape with a different distribution of barrier heights and a different number of minima accessible to the state point. Part of the change in β(cal) is attributed to hydrogen-bond formation between the two components. Ageing changes the relaxation times distribution, more at short relaxation times than at long relaxation times, and multiplicity of relaxation modes implied by β(cal) < 1 indicates that each mode contributing to the enthalpy has its own T(g) or T(f). β(cal) differs from β(age) determined from isothermal ageing, and the distribution parameter of α-relaxation times would differ from both β(cal) and β(age).

Dynamics and thermodynamics of polymer glasses

The fate of matter when decreasing the temperature at constant pressure is that of passing from gas to liquid and, subsequently, from liquid to crystal. However, a class of materials can exist in an amorphous phase below the melting temperature. On cooling such materials, a glass is formed; that is, a material with the rigidity of a solid but exhibiting no long-range order. The study of the thermodynamics and dynamics of glass-forming systems is the subject of continuous research. Within the wide variety of glass formers, an important sub-class is represented by glass forming polymers. The presence of chain connectivity and, in some cases, conformational disorder are unfavourable factors from the point of view of crystallization. Furthermore, many of them, such as amorphous thermoplastics, thermosets and rubbers, are widely employed in many applications. In this review, the peculiarities of the thermodynamics and dynamics of glass-forming polymers are discussed, with particular emphasis on those topics currently the subject of debate. In particular, the following aspects will be reviewed in the present work: (i) the connection between the pronounced slowing down of glassy dynamics on cooling towards the glass transition temperature (Tg) and the thermodynamics; and, (ii) the fate of the dynamics and thermodynamics below Tg. Both aspects are reviewed in light of the possible presence of a singularity at a finite temperature with diverging relaxation time and zero configurational entropy. In this context, the specificity of glass-forming polymers is emphasized.

Molecular mobility, thermodynamics and stability of griseofulvin's ultraviscous and glassy states from dynamic heat capacity

PURPOSE

To determine the calorimetric relaxation time needed for modeling griseofulvin's stability against crystallization during storage.

METHODS

Both temperature-modulated and unmodulated scanning calorimetry have been used to determine the heat capacity of griseofulvin in the glassy and melt state.

RESULTS

The calorimetric relaxation time, tau cal, of its melt varies with the temperature T according to the relation, tau cal [s] = 10(-13.3) exp [2, 292 /(T[K] - 289.5)] , and the distribution of relaxation times parameter is 0.67. The unrelaxed heat capacity of the griseofulvin melt is equal to its vibrational heat capacity.

CONCLUSIONS

Griseofulvin neither crystallizes on heating to 373 K at 1 K/h rate, nor on cooling. Molecular mobility and vibrational heat capacity measured here are more reliable for modeling a pharmaceutical's stability against crystallization than the currently used kinetics-thermodynamics relations, and molecular mobility in the (fixed structure) glassy state is much greater than the usual extrapolation from the melt state yields. Molecular relaxation time of the glassy state of griseofulvin is about 2 months at 298 K, and longer at lower temperatures. It would spontaneously increase with time. If the long-range motions alone were needed for crystallization, griseofulvin would become more stable against crystallization during storage.

Determining vibrational heat capacity and thermal expansivity and their change at glass-liquid transition

The vibrational parts of a liquid's heat capacity Cp and thermal expansion coefficient alpha may be determined from dynamic measurements. The Cp data for Pd40Ni40P20 and 0.4 Ca(NO3)(2).0.6 KNO3 have been analyzed accordingly, and it is found that change in the vibrational part at liquid-glass transformation is negligible. Analysis for alpha of poly(styrene) leads to the same conclusion. There is no discontinuity in the vibrational parts of Cp and alpha on structural unfreezing in the Tg range, and hence the change in Cp and alpha at Tg is almost entirely due to change in the configurational part. Crystallization decreases the vibrational part not because the molecular mobility is lost but because the density increases.

Free volume of an oligomeric epoxy resin and its relation to structural relaxation: evidence from positron lifetime and pressure-volume-temperature experiments

From positron annihilation lifetime spectroscopy analyzed with the new routine LT9.0 and pressure-volume-temperature experiments analyzed employing the equation of state (EOS) Simha-Somcynsky lattice-hole theory (SS EOS) the microstructure of the free volume and its temperature dependence of an oligomeric epoxy resin (ER6, M(n) approximately 1750 g/mol , T(g)=332 K ) of diglycidyl ether of bisphenol-A (DGEBA) have been examined and characterized by the hole free-volume fraction h, the specific free and occupied volumes V(f)=hV and V(occ)=(1-h)V, and the size distribution (mean, <nu(h)>, and mean dispersion, sigma(h)) and the mean density N(h)'=V(f)/<nu(h)>, of subnanometer-size holes. The results are compared with those from a previous work [G. Dlubek, Phys. Rev. E 73, 031803 (2006)] on a monomeric liquid of the same resin (ER1, M(n) approximately 380 g/mol, T(g)=255 K ). In the glassy state ER6 shows the same hole sizes as ER1 but a higher V(f) and N(h)'. In the liquid V(f), <nu(h)>, dV(f)/dT, and dV(f)/dP are smaller for ER6. The reported dielectric alpha relaxation time tau shows certain deviations from the free-volume model which are larger for ER6 than for ER1. This behavior correlates with the SS EOS, which shows that the unit of the SS lattice is more heavy and bulky and therefore the chain is less flexible for ER6 than for ER1. The free-volume fraction h in the liquid can be described by the Schottky equation h proportional to exp(-H(h)/k(B)T) , where H(h)=7.8 - 6.4 kJ/mol is the vacancy formation enthalpy, which opens a different way for the extrapolation of the equilibrium part of the free volume. The extrapolated h decreases gradually below T(g) and becomes zero only when 0 K is reached. This behavior means that no singularity would appear in the relaxation time at temperatures above 0 K. To quantify the degree to which volume and thermal energy govern the structural dynamics, the ratio of the activation enthalpies E(i)=R[(d ln tau/dT(-1))]i, at constant volume V and constant pressure P(E(V)/E(P)), is frequently determined. We present arguments for necessity to substitute E(V) by E(Vf), the activation enthalpy at constant (hole) free volume, and show that E(Vf)/E(P) changes as expected: it increases with increasing free volume, i.e., with increasing temperature, decreasing pressure, and decreasing molecular weight. E(Vf)/E(P) exhibits smaller values than E(V)/E(P), which leads to the general inference that the free volume plays a larger role in dynamics than concluded from E(V)/E(P). The same conclusion is obtained when scaling tau to T(-1)V(f)(-gamma) instead of to T(-1)V(-gamma), where both gamma's are material constants.

Thermal conductivity of a polymerizing liquid

Thermal conductivity kappa of seven polymerizing liquids has been measured in real time at different temperatures, and calorimetry and dielectric spectroscopy of one liquid are performed to help interpret the results. As a covalently bonded linear chain or a network structure in the liquid grows, kappa of the Debye equation initially increases with the polymerization time t(polym) as the molecular weight, density, and sound velocity increase, as on cooling a liquid. The measured kappa reaches a maximum and then decreases, thus showing a peak at a certain t(polym) and finally becomes constant, which is not the true behavior of steady state kappa. The dielectric relaxation time of the covalently bonded structure at the t(polym) for the kappa peak is less than 5 s and the extent of polymerization is below the vitrification plateau value. The peak height increases when the pulse time for kappa measurement is increased. An increase in the liquid's temperature shifts the kappa peak to a shorter t(polym). Liquid compositions polymerizing rapidly show a similar shift, and those polymerizing slowly or whose viscosity does not reach a high enough value show a small kappa peak or none. The kappa peak may be an artifact of the time dependence of heat capacity during the pulse time used for the kappa measurement, as proposed for glasses and supercooled liquids, similar to the changes in other properties observed as an artifact of kinetic freezing/unfreezing. For a polymerizing liquid, the peak may additionally arise when the rate of increase in the elastic modulus becomes equal to the rate of decrease in equilibrium Cp. In either case, its appearance does not distinguish the Brownian motions' slowing on polymerization from that on cooling or compressing a liquid.

Structural relaxation of acetaminophen glass

PURPOSE

The aim is to determine the structural stability of acetaminophen glass with time and temperature change, and to examine the merits of adapting the structural relaxation models of the glassy state for pharmaceuticals.

METHODS

Differential scanning calorimetry technique has been used to study the acetaminophen glass after keeping the samples for various periods at fixed temperatures and after keeping at various temperatures for fixed periods.

RESULTS

A general formalism for thermodynamic changes during storage in a temperature fluctuating environment is given and the kinetics of the enthalpy and entropy decrease determined. At a fixed temperature, the decrease occurs according to a non-exponential kinetics. For the same storage time, but at different temperatures, the enthalpy and entropy decrease rises to a maximum value at a certain temperature and then declines. The peak appears at the temperature at which the internally equilibrated state of the sample is reached for a fixed storage time. The change in the normalized heat capacity during the heating of acetaminophen has been analysed in terms of a non-exponential, non-linear enthalpy relaxation model.

CONCLUSION

A single set of parameters that fit the data for unannealed acetaminophen glass does not fit the calorimetric data for annealed glass. Since acetaminophen molecules form intermolecular hydrogen-bonds in the crystal state and likely to form such bonds more easily in the disordered state, effect of such bonds on structural relaxation is likely to be significant.

Dynamic Heat Capacity and Relaxation Time of Ultraviscous Melt and Glassy Acetaminophen

The real and imaginary components, C'(p) and C''(p), of the complex heat capacity, C*(p)=C'(p)-iC"(p) of supercooled, ultraviscous melt of acetaminophen have been measured at different temperatures during cooling through its vitrification range and during heating through its glass-softening range by using a modulation frequency of 3.3 mHz. From these data, the distribution of relaxation time parameter, beta, and a characteristic (calorimetric or configurational) relaxation time, tau(cal), have been determined. A constant value of 0.65 for beta fits the data, and tau(cal) varies with the temperature according to the Vogel-Fulcher-Tammann equation, tau(cal) = 10(-12.95) exp[1813/(T - 240.5)]. This relation differs significantly from the one deduced by others in which the configurational entropy theory was used to deduce tau(cal). The C'(p) and C''(p) values measured during the cooling of its ultraviscous melt and during the heating of its glassy state show a small hysteresis only at low temperatures. These investigations also provide a comparison of calorimetric and dielectric relaxation times in ultraviscous acetaminophen and highlight the role of faster modes of relaxation at low temperatures in organic, molecular glasses that can help in a better understanding of the crystal nucleation process in glasses at T below their T(g).

Heat capacity of tetrahydrofuran clathrate hydrate and of its components, and the clathrate formation from supercooled melt

We report a thermodynamic study of the formation of tetrahydrofuran clathrate hydrate by explosive crystallization of water-deficient, near stoichiometric, and water-rich solutions, as well as of the heat capacity, C(p), of (i) supercooled tetrahydrofuran-H2O solutions and of the clathrate hydrate, (ii) tetrathydrofuran (THF) liquid, and (iii) supercooled water and the ice formed on its explosive crystallization. In explosive freezing of supercooled solutions at a temperature below 257 K, THF clathrate hydrate formed first. The nucleation temperature depends on the cooling rate, and excess water freezes on further cooling. The clathrate hydrate melts reversibly at 277 K and C(p) increases by 770 J/mol K on melting. The enthalpy of melting is 99.5 kJ/mol and entropy is 358 J/mol K. Molar C(p) of the empty host lattice is less than that of the ice, which is inconsistent with the known lower phonon frequency of H2O in the clathrate lattice. Analysis shows that C(p) of THF and ice are not additive in the clathrate. C(p) of the supercooled THF-H2O solutions is the same as that of water at 247 K, but less at lower temperatures and more at higher temperatures. The difference tends to become constant at 283 K. The results are discussed in terms of the hydrogen-bonding changes between THF and H2O.

Dielectric Studies of Molecular Motions in Amorphous Solid and Ultraviscous Acetaminophen

The dielectric permittivity and loss spectra of glassy and ultraviscous states of acetaminophen have been measured over the frequency range 10 Hz-0.4 MHz. The relaxation spectra show an asymmetric distribution of times expressed in terms of the Kohlrausch exponent, beta, which remains constant at 0.79+/-0.02 over the 305-341 K range. The dielectric relaxation time increases on cooling according to the Vogel-Fulcher-Tammann equation. However, the values of the parameters are considerably different from the values deduced from earlier work by other researchers using the heat capacity of ultraviscous acetaminophen and relating it to its molecular mobility. The calorimetric glass softening temperature of 296 K obtained from differential scanning calorimetry is close to the value measured from dielectric relaxation. The equilibrium permittivity of ultraviscous acetaminophen decreases on heating like that of a normal dipolar liquid, as anticipated from the Curie law. But, its value decreases rapidly with time when it begins to crystallize. The equilibrium permittivity of this crystal phase is approximately 3.1 at 300 K and increases with temperature, which indicates a partial, orientational-disordering of its structure. The results show limitations of the procedures used in the modeling of the kinetics of molecular motions, that is, estimating physical stability, using thermodynamic considerations based on thermal analyses of the amorphous solid phase of acetaminophen.

Thermodynamic−Kinetic Correlations in Supercooled Liquids: A Critical Survey of Experimental Data and Predictions of the Random First-Order Transition Theory of Glasses

Thermodynamics and kinetics are thought to be linked in glass transitions. The quantitative predictions of alpha-relaxation activation barriers provided by the theory of glasses based on random first-order transitions are compared with the experimental results for 44 substances. The agreement found between the predicted activation energies near T(g) and experiment is excellent. These predictions depend on the configurational heat capacity change on vitrification and the entropy of melting the crystal which are experimental inputs.

Configurational entropy and its crisis in metastable states: ideal glass transition in a dimer model as a paragidm of a molecular glass

We discuss the need for discretization to evaluate the configurational entropy in a general model. We also discuss the prescription using restricted partition function formalism to study the stationary limit of metastable states where a more stable equilibrium state exists. We introduce a lattice model of dimers as a paradigm of molecular fluid and study stationary metastability in it to investigate the root cause of glassy behavior. We demonstrate the existence of entropy crisis in metastable states, from which it follows that the entropy crisis is the root cause underlying the ideal glass transition in systems with particles of all sizes. The orientational interactions in the model control the nature of the liquid-liquid transition observed in recent years in molecular glasses.

Entropy crisis, ideal glass transition, and polymer melting: exact solution on a Husimi cactus

We investigate an extension of the lattice model of melting of semiflexible polymers originally proposed by Flory. Along with a bending penalty epsilon, present in the original model and involving three sites of the lattice, we introduce an interaction energy epsilon (p), corresponding to the presence of a pair of parallel bonds and an interaction energy epsilon (h), associated with a hairpin turn. Both these new terms represent four-site interactions. The model is solved exactly on a Husimi cactus, which approximates a square lattice. We study the phase diagram of the system as a function of the energies. For a proper choice of the interaction energies, the model exhibits a first-order melting transition between a liquid and a crystalline phase at a temperature T(M). The continuation of the liquid phase below T(M) gives rise to a supercooled liquid, which turns continuously into a new low-temperature phase, called metastable liquid, at T(MC)<T(M). This liquid-liquid transition seems to have some features that are characteristic of the critical transition predicted by the mode-coupling theory. The exact calculation provides a thermodynamic justification for the entropy crisis (entropy becoming negative), generally known as the Kauzmann paradox, caused by the rapid drop of the entropy near the Kauzmann temperature. It occurs not in the supercooled liquid, but in the metastable liquid phase since its Helmholtz free energy equals the absolute zero equilibrium free energy at a positive temperature. A continuous ideal glass transition occurs to avoid the crisis when the metastable liquid entropy, and not the excess entropy, goes to zero. The melting transition in the original Flory model, corresponding to the vanishing of the four-site interactions, appears as a tricritical point of the model.

Possible resolution of the Kauzmann paradox in supercooled liquids

Generally, the entropy of the supercooled liquid decreases more rapidly than that of the crystal. Thus, the former, if we extrapolate it smoothly below the glass-transition temperature T(g), becomes equal to the latter at the so-called Kauzmann temperature T(K). Further extrapolation below T(K) leads to the unphysical situation that the entropy of disordered liquid is lower than the ordered crystal, which results in the violation of the third law of thermodynamics. This is known as the "Kauzmann paradox" which has been the key problem of liquid-glass transition for a long time. Here we propose a simple resolution of the Kauzmann paradox by answering a fundamental question of how deeply we can supercool a liquid. We argue that the lower metastable limit T(LML), below which a liquid should crystallize before its structural relaxation, is located above the Kauzmann temperature T(K). Thus, the entropy crisis at T(K) is naturally avoided by crystallization. We suggest that it is dynamic heterogeneity that destabilizes a deeply supercooled "equilibrium" liquid state as well as a glassy state against crystallization. This may have a significant implication on the stability of a glassy state, which is of industrial importance in relation to the storage of glassy material.