Rheological model for the alpha relaxation of glass-forming liquids and its comparison to data for DC704 and DC705.

Dynamic shear-modulus data are presented for two silicone oils DC704 and DC705 for frequencies between 1 mHz and 10 kHz at temperatures covering more than five decades of relaxation-time variation. Data are fitted to the alpha part of a phenomenological model previously shown to describe well the dynamic shear modulus of squalane, which has a large beta process [Hecksher et al., J. Chem. Phys. 146, 154504 (2017)]. That model is characterized by additivity of the alpha and beta shear compliance and by a high-frequency decay of the alpha process in proportion to ω-1/2, where ω is the angular frequency. The fits of the alpha part of this model to the DC704 and DC705 data are compared to fits by a Havriliak-Negami type model, a Barlow-Erginsav-Lamb model, and a Cole-Davidson type model. At all temperatures, the best fit is obtained by the alpha part of the squalane model. This strengthens the conjecture that so-called t-relaxation, leading to high-frequency loss decays proportional to ω-1/2, is generic to the alpha relaxation of supercooled liquids [J. C. Dyre, Phys. Rev. E 74, 021502 (2006); Nielsen et al., J. Chem. Phys. 130, 154508 (2009); and Pabst et al., J. Phys. Chem. Lett. 12, 3685-3690 (2021)].

Fast contribution to the activation energy of a glass-forming liquid.

This paper presents physical-aging data for the silicone oil tetramethyl-tetraphenyl trisiloxane. The density and the high-frequency plateau shear modulus [Formula: see text] were monitored following temperature jumps starting from fully equilibrated conditions. Both quantities exhibit a fast change immediately after a temperature jump. Adopting the material-time formalism of Narayanaswamy, we determine from the dielectric loss at 0.178 Hz the time evolution of the aging-rate activation energy. The relative magnitude of the fast change of the activation energy differs from that of the density, but is identical to that of [Formula: see text] In fact, the activation energy is proportional to [Formula: see text] throughout the aging process, with minor deviations at the shortest times. This shows that for the silicone oil in question the dynamics are determined by [Formula: see text] in-as well as out of-equilibrium.

Model for the alpha and beta shear-mechanical properties of supercooled liquids and its comparison to squalane data.

This paper presents data for supercooled squalane's frequency-dependent shear modulus covering frequencies from 10 mHz to 30 kHz and temperatures from 168 K to 190 K; measurements are also reported for the glass phase down to 146 K. The data reveal a strong mechanical beta process. A model is proposed for the shear response of the metastable equilibrium liquid phase of supercooled liquids. The model is an electrical equivalent-circuit characterized by additivity of the dynamic shear compliances of the alpha and beta processes. The nontrivial parts of the alpha and beta processes are each represented by a "Cole-Cole retardation element" defined as a series connection of a capacitor and a constant-phase element, resulting in the Cole-Cole compliance function well-known from dielectrics. The model, which assumes that the high-frequency decay of the alpha shear compliance loss varies with the angular frequency as ω-1/2, has seven parameters. Assuming time-temperature superposition for the alpha and beta processes separately, the number of parameters varying with temperature is reduced to four. The model provides a better fit to the data than an equally parametrized Havriliak-Negami type model. From the temperature dependence of the best-fit model parameters, the following conclusions are drawn: (1) the alpha relaxation time conforms to the shoving model; (2) the beta relaxation loss-peak frequency is almost temperature independent; (3) the alpha compliance magnitude, which in the model equals the inverse of the instantaneous shear modulus, is only weakly temperature dependent; (4) the beta compliance magnitude decreases by a factor of three upon cooling in the temperature range studied. The final part of the paper briefly presents measurements of the dynamic adiabatic bulk modulus covering frequencies from 10 mHz to 10 kHz in the temperature range from 172 K to 200 K. The data are qualitatively similar to the shear modulus data by having a significant beta process. A single-order-parameter framework is suggested to rationalize these similarities.

Communication: Direct tests of single-parameter aging.

This paper presents accurate data for the physical aging of organic glasses just below the glass transition probed by monitoring the following quantities after temperature up and down jumps: the shear-mechanical resonance frequency (∼360 kHz), the dielectric loss at 1 Hz, the real part of the dielectric constant at 10 kHz, and the loss-peak frequency of the dielectric beta process (∼10 kHz). The setup used allows for keeping temperature constant within 100 µK and for thermal equilibration within a few seconds after a temperature jump. The data conform to a new simplified version of the classical Tool-Narayanaswamy aging formalism, which makes it possible to calculate one relaxation curve directly from another without any fitting to analytical functions.

Mechanical spectra of glass-forming liquids. I. Low-frequency bulk and shear moduli of DC704 and 5-PPE measured by piezoceramic transducers.

We present dynamic shear and bulk modulus measurements of supercooled tetraphenyl-tetramethyl-trisiloxane (DC704) and 5-phenyl-4-ether over a range of temperatures close to their glass transition. The data are analyzed and compared in terms of time-temperature superposition (TTS), the relaxation time, and the spectral shape parameters. We conclude that TTS is obeyed to a good approximation for both the bulk and shear moduli. The loss-peak shapes are nearly identical, while the shear modulus relaxes faster than the bulk modulus. The temperature dependence of this decoupling of time scales is constant over the temperature range explored here. In addition, we demonstrate how one can measure reliably the DC shear viscosity over ten orders of magnitude by using the two measuring techniques in combination.

Shear and dielectric responses of propylene carbonate, tripropylene glycol, and a mixture of two secondary amides.

Propylene carbonate and a mixture of two secondary amides, N-methylformamide and N-ethylacetamide, are investigated by means of broadband dielectric and mechanical shear spectroscopy. The similarities between the rheological and the dielectric responses of these liquids and of the previously investigated tripropylene glycol are discussed within a simple approach that employs an electrical circuit for describing the frequency-dependent behavior of viscous materials. The circuit is equivalent to the Gemant-DiMarzio-Bishop model, but allows for a negative capacitive element. The circuit can be used to calculate the dielectric from the mechanical response and vice versa. Using a single parameter for a given system, good agreement between model calculations and experimental data is achieved for the entire relaxation spectra, including secondary relaxations and the Debye-like dielectric peak in the secondary amides. In addition, the predictions of the shoving model are confirmed for the investigated liquids.

Communication: Identical temperature dependence of the time scales of several linear-response functions of two glass-forming liquids.

The frequency-dependent dielectric constant, shear and adiabatic bulk moduli, longitudinal thermal expansion coefficient, and longitudinal specific heat have been measured for two van der Waals glass-forming liquids, tetramethyl-tetraphenyl-trisiloxane (DC704) and 5-polyphenyl-4-ether. Within the experimental uncertainties the loss-peak frequencies of the measured response functions have identical temperature dependence over a range of temperatures, for which the Maxwell relaxation time varies more than nine orders of magnitude. The time scales are ordered from fastest to slowest as follows: Shear modulus, adiabatic bulk modulus, dielectric constant, longitudinal thermal expansion coefficient, and longitudinal specific heat. The ordering is discussed in light of the recent conjecture that van der Waals liquids are strongly correlating, i.e., approximate single-parameter liquids.

Physical aging of molecular glasses studied by a device allowing for rapid thermal equilibration.

Aging to the equilibrium liquid state of organic glasses is studied. The glasses were prepared by cooling the liquid to temperatures just below the glass transition. Aging following a temperature jump was studied by measuring the dielectric loss at a fixed frequency using a microregulator in which temperature is controlled by means of a Peltier element. Compared to conventional equipment, the new device adds almost two orders of magnitude to the span of observable aging times. Data for the following five glass-forming liquids are presented: dibutyl phthalate, diethyl phthalate, 2,3-epoxy propyl-phenyl-ether, 5-polyphenyl-ether, and triphenyl phosphite. The aging data were analyzed using the Tool-Narayanaswamy formalism. The following features are found for all five liquids: (1) The liquid has an "internal clock," a fact that is established by showing that aging is controlled by the same material time that controls the dielectric properties. (2) There are no so-called expansion gaps between the long-time limits of the relaxation rates following up and down jumps to the same temperature. (3) At long times, the structural relaxation appears to follow a simple exponential decay. (4) For small temperature steps, the rate of the long-time exponential structural relaxation is identical to that of the long-time decay of the dipole autocorrelation function.

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.

An electrical circuit model of the alpha-beta merging seen in dielectric relaxation of ultraviscous liquids.

We present a new model for dielectric data in the alpha-beta merging region. The model is constructed using electrical circuit analogies. It leads to an interpretation of the merging region as one where the total relaxation upon cooling separates in two relaxation processes, consistent with a view where the relaxing entities involved are the same for the two processes. We use this alpha-beta model to fit dielectric data in the merging region of two different molecular liquids. These fits are performed under the assumption that the intrinsic high-frequency behavior of the alpha relaxation is a -1/2 power law and that both the alpha and the beta process separately obey time temperature superposition. We get good quality fits in the entire frequency and temperature range studied. This supports the view that alpha relaxation high-frequency slopes that are found to be numerically smaller than 1/2 can be attributed to the influence of the beta relaxation.

Prevalence of approximate square root(t) relaxation for the dielectric alpha process in viscous organic liquids.

This paper presents dielectric relaxation data for organic glass-forming liquids compiled from different groups and supplemented by new measurements. The main quantity of interest is the "minimum slope" of the alpha dielectric loss plotted as a function of frequency in a log-log plot, i.e., the numerically largest slope above the loss peak frequency. The data consisting of 347 spectra for 53 liquids show prevalence of minimum slopes close to -1/2, corresponding to approximate square root(t) dependence of the dielectric relaxation function at short times. The paper studies possible correlations between minimum slopes and (1) temperature (quantified via the loss peak frequency); (2) how well an inverse power-law fits data above the loss peak; (3) degree of time-temperature superposition; (4) loss peak half width; (5) deviation from non-Arrhenius behavior; (6) loss strength. For the first three points we find correlations that show a special status of liquids with minimum slopes close to -1/2. For the last three points only fairly insignificant correlations are found, with the exception of large-loss liquids that have minimum slopes that are numerically significantly larger than 1/2. We conclude that--excluding large-loss liquids--approximate square root(t) relaxation appears to be a generic property of the alpha relaxation of organic glass formers.

A cryostat and temperature control system optimized for measuring relaxations of glass-forming liquids.

An experimental setup, including a cryostat and a temperature control system, has been constructed to meet the demands of measuring linear and nonlinear macroscopic relaxation properties of glass-forming liquids in the extremely viscous state approaching the glass transition. In order to be able to measure such frequency-dependent response functions accurately (including dielectric permittivity, specific heat, thermal expansivity, and shear and bulk moduli), as well as nonlinear relaxations following a temperature jump, one must have the ability to hold temperatures of liquids steady over the span of several days or even several weeks. To maximize temperature stability, special care is taken to thermally isolate the sample chamber of the cryostat. The main temperature control system is capable of maintaining temperatures within a few millikelvins. If liquid is deposited into a special transducer assembly that includes a subcryostat unit, the temperature of liquids can be maintained even more precisely, within a few tenths of a millikelvin. This subcryostat unit is more responsive to temperature changes because (i) it is equipped with a Peltier element that provides secondary heating and cooling, (ii) the transducer contains a layer of liquid that is only 50 micfom thick, and (iii) feedback proportional-integral-derivative temperature control is implemented by a fully analog circuit. The subcryostat permits us to change and stabilize temperatures quickly; it takes only 10 s to stabilize the temperature within tenths of a millikelvin after a jump of 1 K, for example, a capability that is highly advantageous for accurately observing relaxation processes following a temperature step.

An impedance-measurement setup optimized for measuring relaxations of glass-forming liquids.

An electronics system has been assembled to measure frequency-dependent response functions of glass-forming liquids in the extremely viscous state approaching the glass transition. We determine response functions such as dielectric permittivity and shear and bulk moduli by measuring electrical impedances of liquid-filled transducers, and this technique requires frequency generators capable of producing signals that are reproducible over the span of several days or even several weeks. To this end, we have constructed a frequency generator that produces low-frequency (1 mHz-100 Hz) sinusoidal signals with voltages that are reproducible within 10 ppm. Two factors that partly account for this precision are that signals originate from voltages stored in a look-up table and that only coil-less filters are used in this unit, which significantly reduces fluctuations of output caused by changes of temperatures of circuits. This generator also includes a special triggering facility that makes it possible to measure up to 512 voltages per cycle that are spaced apart at uniform phase intervals. Fourier transformations of such data yield precise determinations of complex amplitudes of voltages and currents applied to a transducer, which ultimately allows us to determine electrical impedances of transducers with a reproducibility error that is only a few parts per hundred thousand. This equipment is used in tandem with a commercial LCR meter and/or impedance analyzer that give(s) impedance measurements at higher frequencies, up to 1 MHz. The experimental setup allows measurements of the transducer impedance over nine decades of frequency within a single run.

Supercooled liquid dynamics studied via shear-mechanical spectroscopy.

We report dynamical shear-modulus measurements for five glass-forming liquids (pentaphenyltrimethyltrisiloxane, diethyl phthalate, dibutyl phthalate, 1,2-propanediol, and m-touluidine). The shear-mechanical spectra are obtained by the piezoelectric shear-modulus gauge (PSG) method. This technique allows one to measure the shear modulus (10(5)-10(10) Pa) of the liquid within a frequency range from 1 mHz to 10 kHz. We analyze the frequency-dependent response functions to investigate whether time-temperature superposition (TTS) is obeyed. We also study the shear-modulus loss-peak position and its high-frequency part. It has been suggested that when TTS applies, the high-frequency side of the imaginary part of the dielectric response decreases like a power law of the frequency with an exponent -1/2. This conjecture is analyzed on the basis of the shear mechanical data. We find that TTS is obeyed for pentaphenyltrimethyltrisiloxane and in 1,2-propanediol while in the remaining liquids evidence of a mechanical beta process is found. Although the high-frequency power law behavior w(-alpha) of the shear loss may approach a limiting value of alpha = 0.5 when lowering the temperature, we find that the exponent lies systematically above this value (around 0.4). For the two liquids without beta relaxation (pentaphenyltrimethyltrisiloxane and 1,2-propanediol) we also test the shoving model prediction, according to which the relaxation time activation energy is proportional to the instantaneous shear modulus. We find that the data are well described by this model.

Conventional methods fail to measure cp (omega) 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.

Dielectric and shear mechanical relaxations in glass-forming liquids: a test of the Gemant-DiMarzio-Bishop model.

The Gemant-DiMarzio-Bishop model, which connects the frequency-dependent shear modulus to the frequency-dependent dielectric constant, is reviewed and a new consistent macroscopic formulation is derived. It is moreover shown that this version of the model can be tested without fitting parameters. The reformulated version of the model is analyzed and experimentally tested. It is demonstrated that the model has several nontrivial qualitative predictions: the existence of an elastic contribution to the high-frequency limit of the dielectric constant, a shift of the shear modulus loss peak frequency to higher frequencies compared with the loss peak frequency of the dielectric constant, a broader alpha peak, and a more pronounced beta peak in the shear modulus when compared with the dielectric constant. It is shown that these predictions generally agree with experimental findings and it is therefore suggested that the Gemant-DiMarzio-Bishop model is correct on a qualitative level. The quantitative agreement between the model and the data is on the other hand moderate to poor. It is discussed if a model-free comparison between the dielectric and shear mechanical relaxations is relevant, and it is concluded that the shear modulus should be compared with the rotational dielectric modulus, 1(epsilon(omega)-n2), which is extracted from the Gemant-DiMarzio-Bishop model, rather than to the dielectric susceptibility or the conventional dielectric modulus M=1epsilon(omega).

Dielectric and shear mechanical alpha and beta relaxations in seven glass-forming liquids.

We present shear mechanical and dielectric measurements taken on seven liquids: triphenylethylene, tetramethyltetra-phenyltrisiloxane (Dow Corning 704 diffusion pump fluid), polyphenyl ether (Santovac 5 vacuum pump fluid), perhydrosqualene, polybutadiene, decahydroisoquinoline (DHIQ), and tripropylene glycol. The shear mechanical and dielectric measurements are for each liquid performed under identical thermal conditions close to the glass transition temperature. The liquids span four orders of magnitude in dielectric relaxation strength and include liquids with and without Johari-Goldstein beta relaxation. The shear mechanical data are obtained by the piezoelectric shear modulus gauge method giving a large frequency span (10(-3)-10(4.5) Hz). This allows us to resolve the shear mechanical Johari-Goldstein beta peak in the equilibrium DHIQ liquid. We moreover report a signature (a pronounced rise in the shear mechanical loss at frequencies above the alpha relaxation) of a Johari-Goldstein beta relaxation in the shear mechanical spectra for all the liquids which show a beta relaxation in the dielectric spectrum. It is found that both the alpha and beta loss peaks are shifted to higher frequencies in the shear mechanical spectrum compared to the dielectric spectrum. It is in both the shear and dielectric responses found that liquids obeying time-temperature superposition also have a high-frequency power law with exponent close to -12. It is moreover seen that the less temperature dependent the spectral shape is, the closer it is to the universal -12 power-law behavior. The deviation from this universal power-law behavior and the temperature dependencies of the spectral shape are rationalized as coming from interactions between the alpha and beta relaxations.

Landscape equivalent of the shoving model.

It is shown that the shoving model expression for the average relaxation time of viscous liquids, according to which the activation energy is proportional to the instantaneous shear modulus, follows largely from a classical "landscape" estimation of barrier heights from curvature at energy minima. Although the activation energy in this reasoning involves both instantaneous bulk and shear moduli, the bulk modulus contributes less than 8% to the temperature dependence of the activation energy. This reflects the fact that the physics of the two models are closely related.

Minimal model for Beta relaxation in viscous liquids.

Contrasts between beta relaxation in equilibrium viscous liquids and glasses are rationalized in terms of a double-well potential model with structure-dependent asymmetry, assuming structure is described by a single order parameter. The model is tested for tripropylene glycol where it accounts for the hysteresis of the dielectric beta loss peak frequency and magnitude during cooling and reheating through the glass transition.