Frequency-dependent Specific Heat Research Articles

Revisiting impulsive stimulated thermal scattering in supercooled liquids: Relaxation of specific heat and thermal expansion.

Impulsive stimulated thermal scattering (ISTS) allows one to access the structural relaxation dynamics in supercooled molecular liquids on a time scale ranging from nanoseconds to milliseconds. Till now, a heuristic semi-empirical model has been commonly adopted to account for the ISTS signals. This model implicitly assumes that the relaxation of specific heat, C, and thermal expansion coefficient, γ, occur on the same time scale and accounts for them via a single stretched exponential. This work proposes two models that assume disentangled relaxations, respectively, based on the Debye and Havriliak-Negami assumptions for the relaxation spectrum and explicitly accounting for the relaxation of C and γ separately in the ISTS response. A theoretical analysis was conducted to test and compare the disentangled relaxation models against the stretched exponential. The former models were applied to rationalize the experimental ISTS signals acquired on supercooled glycerol. This allows us to simultaneously retrieve the frequency-dependent specific heat and thermal expansion up to the sub-100MHz frequency range and further to compare the fragility and time scale probed by thermal, mechanical, and dielectric susceptibilities.

Frequency-dependent specific heat in quantum supercooled liquids: A mode-coupling study.

Frequency-dependence of specific heat in supercooled hard sphere liquid is computed using quantum mode-coupling theory (QMCT). Mode-coupling equations are solved using a recently proposed perturbative method that allows us to study relaxation in the moderate quantum regime where quantum effects assist liquid to glass transition. Zwanzig's formulation is used to compute the frequency-dependent specific heat in the supercooled state using dynamical information from QMCT. Specific heat shows strong variation as the quantumness of the liquid is changed, which becomes more significant as density is increased. It is found that, near the transition point, different dynamical modes contribute to specific heat in classical and quantum liquids.

Capturing molecular multimode relaxation processes in excitable gases based on decomposition of acoustic relaxation spectra

Existing two-frequency reconstructive methods can only capture primary (single) molecular relaxation processes in excitable gases. In this paper, we present a reconstructive method based on the novel decomposition of frequency-dependent acoustic relaxation spectra to capture the entire molecular multimode relaxation process. This decomposition of acoustic relaxation spectra is developed from the frequency-dependent effective specific heat, indicating that a multi-relaxation process is the sum of the interior single-relaxation processes. Based on this decomposition, we can reconstruct the entire multi-relaxation process by capturing the relaxation times and relaxation strengths of N interior single-relaxation processes, using the measurements of acoustic absorption and sound speed at 2N frequencies. Experimental data for the gas mixtures CO2–N2 and CO2–O2 validate our decomposition and reconstruction approach.

Communication: High pressure specific heat spectroscopy reveals simple relaxation behavior of glass forming molecular liquid.

The frequency dependent specific heat has been measured under pressure for the molecular glass forming liquid 5-polyphenyl-4-ether in the viscous regime close to the glass transition. The temperature and pressure dependences of the characteristic time scale associated with the specific heat is compared to the equivalent time scale from dielectric spectroscopy performed under identical conditions. It is shown that the ratio between the two time scales is independent of both temperature and pressure. This observation is non-trivial and demonstrates the existence of specially simple molecular liquids in which different physical relaxation processes are both as function of temperature and pressure/density governed by the same underlying "inner clock." Furthermore, the results are discussed in terms of the recent conjecture that van der Waals liquids, like the measured liquid, comply to the isomorph theory.

Frequency dependence of specific heat in supercooled liquid water and emergence of correlated dynamics

Molecular origin of the well-known specific heat anomaly in supercooled liquid water is investigated here by using extensive computer simulations and theoretical analyses. A rather sharp increase in the values of isobaric specific heat with lowering temperature and the weak temperature dependence of isochoric specific heat in the same range are reproduced in simulations. We calculated the spatio-temporal correlation among temperature fluctuations and examined the frequency dependent specific heat. The latter shows a rapid growth in the low frequency regime as temperature is cooled below 270 K. In order to understand the microscopic basis of this increase, we have performed a shellwise decomposition of contributions of distant molecules to the temperature fluctuations in a central molecule. This decomposition reveals the emergence, at low temperatures, of temporally slow, spatially long ranged large temperature fluctuations. The temperature fluctuation time correlation function (TFCF) can be fitted to a William-Watts stretched exponential form with the stretching parameter close to 0.6 at low temperatures, indicating highly non-exponential relaxation. Temperature dependence of the relaxation time of the correlation function can be fitted to Vogel-Fulcher-Tamermann expression which provides a quantitative measure of the fragility of the liquid. Interestingly, we find that the rapid growth in the relaxation time of TFCF with lowering temperature undergoes a sharp crossover from a markedly fragile state to a weakly fragile state around 220 K.

Does the Viscosity Exponent Derive from Ultrasonic Attenuation Spectra?

Based on a representation of the sound velocity of critical liquids in terms of a frequency-dependent complex specific heat at constant pressure, a simple relation between the low-frequency normalized sonic attenuation coefficient and the correlation length of fluctuations is derived. This relation provides a promising alternative for the determination of the dynamics exponent and thus the critical exponent of the shear viscosity. Sonic attenuation data from the literature, measured at frequencies down to 50 kHz, are re-evaluated with a view of the viscosity exponent determination. It is found that only in a small temperature range, the major requirement of the approach is fulfilled with the available data. Close to the critical temperature, the frequencies of measurement are still insufficiently small as compared to the inverse relaxation time of order parameter fluctuations. Criteria for future experiments are discussed briefly.

Fragility of supercooled liquids from differential scanning calorimetry traces: Theory and experiment

Starting from the Debye model for frequency-dependent specific heat and the Vogel-Fulcher-Tammann (VFT) model for its relaxation time, an analytic expression is presented for the heat capacity versus temperature trace for differential scanning calorimetry (DSC) of glass transitions, suggesting a novel definition of the glass transition temperature based on a dimensionless criterion. An explicit expression is presented for the transition temperature as a function of the VFT parameters and the cooling rate, and for the slope as a function of fragility. Also a generalization of the results to non-VFT and non-Debye relaxation is given. Two unique ways are proposed to tackle the inverse problem, i.e., to extract the fragility from an experimental DSC trace. Good agreement is found between theoretically predicted DSC traces and experimental DSC traces for glycerol for different cooling rates.

Analytical model for acoustic multi-relaxation spectrum in gas mixtures

To identify the correlation between sound propagation and molecular multimode vibrational relaxation in polyatomic gas mixture, an analytical model that constructs acoustic multi-relaxation spectrum is presented. The frequency-dependent effective specific heat of gas is formulated from the micro view of vibrational mode energy transfer as well as the macro view of relaxation process due to vibrational-vibrational mode energy coupling. With the aid of the general relaxation equations of multimode vibrational energy transfer, the analytical expressions to calculate acoustic relaxation absorption and dispersion, which reflect both primary and secondary relaxation processes, are developed from the effective specific heat. The constructed absorption spectra of various gas mixtures, consisting of carbon dioxide, methane, nitrogen, and oxygen, accord with the experimental data very well. Especially, the peak errors of those results are less than 1%. Moreover, the simulation results illustrate that less than two single processes with higher strength appear generally in a multi-relaxation absorption spectrum. Compared with the existing models, the analytical model can directly obtain the analytical expressions of characteristic points in the relaxation spectrum of gas mixtures, which makes it advantageous to analyze the spectral characteristics qualitatively and quantitatively. Consequently, the model provides an effective approach to analyzing the relationship between sound propagation and molecular vibrational relaxation of gas mixtures.

Frequency-dependent specific heat from thermal effusion in spherical geometry

We present a method of measuring the frequency-dependent specific heat at the glass transition applied to 5-polyphenyl-4-ether. The method employs thermal waves effusing radially out from the surface of a spherical thermistor that acts as both a heat generator and a thermometer. It is a merit of the method compared to planar effusion methods that the influence of the mechanical boundary conditions is analytically known. This implies that it is the longitudinal rather than the isobaric specific heat that is measured. As another merit the thermal conductivity and specific heat can be found independently. The method has highest sensitivity at a frequency where the thermal diffusion length is comparable to the radius of the heat generator. This limits in practice the frequency range to 2-3 decades. An account of the 3ω technique used including higher-order terms in the temperature dependence of the thermistor and in the power generated is furthermore given.

Coupling effects in dynamic calorimetry: Frequency-dependent relations for specific heat and thermomechanical responses — A one-dimensional approach based on thermodynamics with internal state variables

As shown by Christensen et al. [2], temperature-induced stresses or strains can have a strong influence on the frequency-dependent specific heat, especially of thin layers of viscoelastic materials near the glass transition. Thus, both the mathematical representation and the physical understanding of these couplings are essential for the interpretation of temperature-modulated DSC data. The approach provided in this paper is based on thermodynamics with internal state variables. It thus differs from the transfer-matrix method which has been applied in Refs. [2,29] and constitutes a thermodynamic basis from a different point of view for the interpretation of the results obtained in Refs. [2,29,32]. Furthermore, although not the object of this paper, time-domain simulations can also be done with the model presented thus allowing for the calculation of temperature-ramping experiments and effects observed therein [42]. The approach in this paper is restricted to one-dimensional states of stress and strain to focus on the main idea and keep the mathematical formalism to a minimum. The Gibbs free energy is chosen as thermodynamic potential and the primary variables – the stress and the temperature – are supplemented by a set of internal state variables which is introduced to include history-dependent and hence viscoelastic effects. The Gibbs free energy is approximated up to second order terms in the vicinity of a reference state. Employing the Legendre transform, a corresponding expression for the Helmhotz free energy is obtained. Evaluating the laws of thermodynamics, explicit frequency-dependent expressions for the specific heat under constant stress or strain, the thermal expansion behaviour as well as the mechanical response functions are obtained. Recently published formulations of the Prigogine–Defay ratio can also be derived from the proposed constitutive model.

On the frequency-dependent specific heat and TMDSC: Constitutive modelling based on thermodynamics with internal state variables

To develop constitutive models to represent the thermomechanically chemically coupled behaviour of curing resins, vulcanizing elastomers or melting and crystallizing polymers the technique of DSC is extremely helpful. In the present study, the method of TMDSC is interpreted and evaluated in the context of thermodynamics with internal state variables. The balance equation of energy and the dissipation principle in the form of the Clausius–Duhem inequality form the theoretical basis of our study. Since the pressure and the temperature are the external variables in DSC, the specific Gibbs free energy is used as thermodynamic potential. It depends on temperature, stress and a set of internal state variables to represent the microstructure of the material on a phenomenological basis. The temperature- and internal variable-induced changes in the Gibbs free energy are approximated by a Taylor series up to second order terms. As a substantial result of this work, closed-form expressions for the dynamic calorimetric response due to harmonic temperature perturbations and the frequency-dependent complex heat capacity are derived. The theory allows a physical interpretation of the complex heat capacity and its underlying phenomena and is in accordance with experimental observations from literature.

Theory of specific heat in glass-forming systems

Experimental measurements of the specific heat in glass-forming systems are obtained from the linear response to either slow cooling (or heating) or to oscillatory perturbations with a given frequency about a constant temperature. The latter method gives rise to a complex specific heat with the constraint that the zero frequency limit of the real part should be identified with thermodynamic measurements. Such measurements reveal anomalies in the temperature dependence of the specific heat, including the so called "specific heat peak" in the vicinity of the glass transition. The aim of this paper is to provide theoretical explanations of these anomalies in general and a quantitative theory in the case of a simple model of glass formation. We first present interesting simulation results for the specific heat in a classical model of a binary mixture glass former. We show that in addition to the formerly observed specific heat peak there is a second peak at lower temperatures which was not observable in earlier simulations. Second, we present a general relation between the specific heat, a caloric quantity, and the bulk modulus of the material, a mechanical quantity, and thus offer a smooth connection between the liquid and amorphous solid states. The central result of this paper is a connection between the micromelting of clusters in the system and the appearance of specific heat peaks; we explain the appearance of two peaks by the micromelting of two types of clusters. We relate the two peaks to changes in the bulk and shear moduli. We propose that the phenomenon of glass formation is accompanied by a fast change in the bulk and the shear moduli, but these fast changes occur in different ranges of the temperature. Last, we demonstrate how to construct a theory of the frequency dependent complex specific heat, expected from heterogeneous clustering in the liquid state of glass formers. A specific example is provided in the context of our model for the dynamics of glycerol. We show that the frequency dependence is determined by the same alpha -relaxation mechanism that operates when measuring the viscosity or the dielectric relaxation spectrum. The theoretical frequency dependent specific heat agrees well with experimental measurements on glycerol. We conclude the paper by stating that there is nothing universal about the temperature dependence of the specific heat in glass formers-unfortunately, one needs to understand each case by itself.

Solution of the spherically symmetric linear thermoviscoelastic problem in the inertia-free limit

The coupling between mechanical and thermal properties due to thermal expansion complicates the problem of measuring frequency-dependent thermoviscoelastic properties, in particular for highly viscous liquids. A simplification arises if there is spherical symmetry where--as detailed in the present paper--the thermoviscoelastic problem may be solved analytically in the inertia-free limit, i.e., the limit where the sample is much smaller than the wavelength of sound waves at the frequencies of interest. As for the one-dimensional thermoviscoelastic problem [Christensen et al., Phys. Rev. E 75, 041502 (2007)], the solution is conveniently formulated in terms of the so-called transfer matrix, which directly links to the boundary conditions that can be experimentally controlled. Once the transfer matrix has been calculated, it is fairly easy to deduce the equations describing various experimentally relevant special cases (boundary conditions that are adiabatic, isothermal, isochoric, etc.). In most situations the relevant frequency-dependent specific heat is the longitudinal specific heat, a quantity that is in between the isochoric and isobaric frequency-dependent specific heats.

Conventional methods fail to measurecp(ω)of glass-forming liquids

The specific heat is frequency dependent in highly viscous liquids. By solving the full one-dimensional thermoviscoelastic problem analytically it is shown that, because of thermal expansion and the fact that mechanical stresses relax on the same time scale as the enthalpy relaxes, the plane thermal-wave method does not measure the isobaric frequency-dependent specific heat c{p}(omega) . This method rather measures a "longitudinal" frequency-dependent specific heat, a quantity defined and detailed here that is in between c{p}(omega) and c{V}(omega) . This result means that no reliable wide-frequency measurements of c{p}(omega) on liquids approaching the calorimetric glass transition exist. We briefly discuss consequences for experiment.

Single-order-parameter description of glass-forming liquids: A one-frequency test

Thermoviscoelastic linear-response functions are calculated from the master equation describing viscous liquid inherent dynamics. From the imaginary parts of the frequency-dependent isobaric specific heat, isothermal compressibility, and isobaric thermal expansion coefficient, we define a "linear dynamic Prigogine-Defay ratio" LambdaTp(omega) with the property that if LambdaTp(omega)=1 at one frequency, then LambdaTp(omega) is unity at all frequencies. This happens if and only if there is a single-order-parameter description of the thermoviscoelastic linear responses via an order parameter (which may be nonexponential in time). Generalizations to other cases of thermodynamic control parameters than temperature and pressure are also presented.

Inelastic-neutron-scattering study of excited spin multiplets and low-energy phonons in theFe8nanomagnet: Implications for relaxation

Inelastic neutron scattering is used to study excited spin multiplets and low-energy phonons in the ${\mathrm{Fe}}_{8}$ nanomagnet. The energies of the three lowest-lying excited multiplets are determined. In addition, we find weakly dispersive optical phonons with energies lower than the anisotropy splitting of the ground $S=10$ spin multiplet. We show that these optical phonons may produce sharp minima in the field dependence of the relaxation time. The positions of the calculated minima agree with those that had been inferred from frequency-dependent specific heat and ac-susceptibility measurements.

Frequency dependent heat capacity within a kinetic model of glassy dynamics

There has been renewed interest in the frequency dependent specific heat of supercooled liquids in recent years with computer simulation studies exploring the whole frequency range of relaxation. The simulation studies can thus supplement the existing experimental results to provide an insight into the energy landscape dynamics. We here investigate a kinetic model of cooperative dynamics within the landscape paradigm for the dynamic heat capacity C(omega,T) behavior. In this picture, the beta-process is modeled as a thermally activated event in a two-level system and the alpha-process is described as a beta-relaxation mediated cooperative transition in a double well. The model resembles a landscape picture, apparently first conceived by Stillinger [Science 267, 1935 (1995)], where an alpha-process is assumed to involve a concerted series of beta-processes. The model provides a description of the activated hopping in the energy landscape in close relation with the cooperative nature of the hopping event. For suitable choice of parameters, the model predicts a frequency dependent heat capacity that reflects the two-step relaxation behavior. The separation between the two peaks grows as the temperature drops, indicating the stringent constraint on the alpha-process due to the cooperativity requirement. The temperature dependence of the position of the low-frequency peak, due to the alpha-relaxation, shows a non-Arrhenius behavior as observed experimentally. The shape of the alpha-peak is, however, found to be temperature independent. The high-frequency peak appears with considerably larger amplitude than the alpha-peak. We attempt a plausible reason for this observation that is in contrast with the general feature revealed by the dielectric spectroscopy. The relative amplitudes of the beta- and alpha-peaks in the present framework are found to depend on several characteristic features of the energy landscape, including the extent of cooperativity requirement for the alpha-relaxation and the asymmetry of the double well.

Frequency dependence and equilibration of the specific heat of glass-forming liquids.

We have performed molecular dynamics simulations on a glass-forming liquid consisting of a three-dimensional binary mixture of soft spheres. We show that a peak in the specific heat versus temperature can occur because a glassy system that shows no signs of aging progresses so slowly through the energy landscape that the minimum sampling time needed to obtain accurate thermodynamic averages exceeds the observation time. We develop a systematic technique to determine the equilibrium value of the specific heat and the minimum sampling time. Below the temperature of the specific heat peak, the minimum sampling time is orders of magnitude longer than the alpha relaxation time. We find that an equilibrium system that is not undergoing structural relaxation or aging has a frequency dependent specific heat that rises as the frequency decreases. The rise occurs at frequencies corresponding to periods that are long enough for the system to sample statistically independent energies. When the period is comparable to the minimum sampling time, the frequency dependent specific heat reaches a plateau. As a result, the specific heat has a frequency dependence at frequencies orders of magnitude lower than is implied by the inverse alpha relaxation time.

Magnetic field dependent thermodynamics of Fe 8 single crystal in the thermally activated regime

Quantum tunneling of the magnetization in the thermally activated regime has been studied on Fe 8 single crystals by frequency dependent specific heat measurements as a function of applied magnetic field. Curves are very well described by a microscopic theory based on phonon assisted spin tunneling. However, an additional peak appears that can be due to transitions in higher multiplets.

Structural relaxation and frequency-dependent specific heat in a supercooled liquid.

We have studied the relation between the structural relaxation and the frequency-dependent thermal response or the specific heat, c(p)(omega), in a supercooled liquid. The mode coupling theory (MCT) results are used to obtain c(p)(omega) corresponding to different wave vectors. Due to the two-step relaxation process present in the MCT, an extra peak, in addition to the low-frequency peak, is predicted in specific heat at high frequency.