Minimal model for Beta relaxation in viscous liquids

Image of the solid-state rotary motion encoded in the dielectric response

The future development of advanced molecular systems with controlled rotation requires the development of an effective methodology for assessing the rotational performance of artificial machine components. We identified two patterns of the dielectric behavior for polar rotators in a static non-polar framework of sizable crystal showing relations between the spectral and molecular-level features of solid-state rotary motion. Various functionalization of phenylene rotors with a fluorine atom(s) changed rotational performance from high to low with rotational barriers ranging from 6.06 to 11.84 kcal mol-1. The meta-F-substitution favored rotator-rotator contacts allowing for the implementation of fast rotary motion. Contrary, the presence of rotator-stator contacts inhibited independent rotator dynamics leading to opposite spectral behavior in terms of temperature evolution of loss peak amplitude. Our observations, supported by an analysis based on an asymmetric double well-potential model, show that easily noticeable spectral differences encoded some molecular-level information important for the implementation of rotary motion.

Tracing the slow Arrhenius process deep in the glassy state-quantitative evaluation of the dielectric relaxation of bulk samples and thin polymer films in the temperature domain

The slow Arrhenius process (SAP) is a dielectric mode connected to thermally activated equilibration mechanisms, allowing for a fast reduction in free energy in liquids and glasses. The SAP, however, is still poorly understood, and so far, this process has mainly been investigated at temperatures above the glass transition. By employing a combination of methods to analyze dielectric measurements under both isochronal and isothermal conditions, we were able to quantitatively reproduce the dielectric response of the SAP of different polymers and to expand the experimental regime over which this process can be observed down to lower temperatures, up to 70 K below the glass transition. Employing thin films of thicknesses varying between 10 and 800 nm, we further verified that the peak shape and activation energy of the SAP of poly(4-bromostyrene) are not sensitive to temperature, nor do they vary upon confinement at the nanoscale level. These observations confirm the preliminary trends reported for other polymers. We find that one single set of parameters-meaning the activation barrier and the pre-exponential factor, respectively, linked to the enthalpic and entropic components of the process-can describe the dynamics of the SAP in both the supercooled liquid and glassy states, in bulk and thin films. These results are discussed in terms of possible molecular origins of the slow Arrhenius process in polymers.

Nonlinear response theory for Markov processes. IV. The asymmetric double-well potential model revisited

The dielectric response of noninteracting dipoles is discussed in the framework of the classical model of stochastic reorientations in an asymmetric double-well potential (ADWP). In the nonlinear regime, this model exhibits some pecularities in the static response. We find that the saturation behavior of the symmetric double-well potential model does not follow the Langevin function and only in the linear regime are the standard results recovered. If a finite asymmetry is assumed, then the nonlinear susceptibilities are found to change the sign at a number of characteristic temperatures that depend on the magnitude of the asymmetry, as has been observed earlier for the third-order and fifth-order responses. If the kinetics of the barrier crossing in the ADWP model is described as a two-state model, then we can give analytical expressions for the values of the characteristic temperatures. The results for the response obtained from a (numerical) solution of the Fokker-Planck equation for the Brownian motion in a model ADWP behaves very similarly to the two-state model for high barriers. For small barriers no clear-cut timescale separation between the barrier crossing process and the intrawell relaxation exists and the model exhibits a number of timescales. In this case, the frequency-dependent linear susceptibility at low temperatures is dominated by the fast intrawell transitions and at higher temperatures by the barrier crossing kinetics. We find that for nonlinear susceptibilities the latter process appears to be more important and the intrawell transitions play only a role at the lowest temperatures.

1D-Confinement Inhibits the Anomaly in Secondary Relaxation of a Fluorinated Polymer

We present an experimental study of the dynamics of a well-pronounced secondary relaxation observed in bulk and ultrathin films of the fluorinated copolymer poly(vinylidene fluoride-co-hexafluoropropylene) (PVdF-HFP). In proximity to the glass transition, an anomalous phenomenon is observed: the β-relaxation slows down upon heating. Measurements as a function of the film thickness show that this exceptional behavior gradually vanishes upon confinement at the nanoscale level. Regardless of sample size, the relaxation dynamics could be described in terms of the Minimal Model via an asymmetric double well potential. Supported by a structural investigation of surfaces and interfaces, our results reveal that the presence of adsorbing walls induces an increase in glass transition temperature, which counterbalances the asymmetry in the double well potential responsible for molecular motion.

Single-parameter aging in a binary Lennard-Jones system

This paper studies physical aging by computer simulations of a 2:1 Kob-Andersen binary Lennard-Jones mixture, a system that is less prone to crystallization than the standard 4:1 composition. Starting from thermal-equilibrium states, the time evolution of the following four quantities is monitored by following up and down jumps in temperature: potential energy, virial, average squared force, and the Laplacian of the potential energy. Despite the fact that significantly larger temperature jumps are studied here than in typical similar experiments, to a good approximation, all four quantities conform to the single-parameter-aging scenario derived and validated for small jumps in experiments [T. Hecksher, N. B. Olsen, and J. C. Dyre, J. Chem. Phys. 142, 241103 (2015)]. As a further confirmation of single-parameter aging with a common material time for the four different quantities monitored, their relaxing parts are found to be almost identical for all temperature jumps.

Thermodynamic Ultrastability of a Polymer Glass Confined at the Micrometer Length Scale

We employ fast scanning calorimetry to assess the thermodynamic state attained after a given cooling rate and the molecular mobility of glassy poly(4-tert-butylstyrene) confined at the micrometer length scale. We show that, for such a large confinement length scale, thermodynamic states with a fictive temperature (T_{f}) 80 K below the polymer glass transition temperature (T_{g}) are attained, which allows to bypass the geological timescales required for bulk glasses. Access to such states is promoted by a fast mechanism of equilibration. Importantly, the tremendous T_{f} decrease takes place while the molecular mobility remains bulklike, indicating marked decoupling between vitrification kinetics and molecular mobility.

Structural rearrangements governing Johari-Goldstein relaxations in metallic glasses

The Johari-Goldstein secondary (β) relaxations are an intrinsic feature of supercooled liquids and glasses. They are crucial to many properties of glassy materials, but the underlying mechanisms are still not established. In a model metallic glass, we study the atomic rearrangements by molecular dynamics simulations at time scales of up to microseconds. We find that the distributions of single-particle displacements exhibit multiple peaks, whose positions quantitatively match the pair distribution function. These are identified as the structural signature of cooperative string-like excitations. Furthermore, the most probable time of the string-like motions coincides with the β-relaxation time as probed by dynamical mechanical simulations over a wide temperature range and is consistent with a theoretical model. Our results provide insights into the long-standing puzzle regarding the structural origin of β relaxations in glassy metallic materials.

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.

Heterogeneous Viscoelasticity: A Combined Theory of Dynamic and Elastic Heterogeneity

We present a heterogeneous version of Maxwell's theory of viscoelasticity based on the assumption of spatially fluctuating local viscoelastic coefficients. The model is solved in coherent-potential approximation. The theory predicts an Arrhenius-type temperature dependence of the viscosity in the vanishing-frequency limit, independent of the distribution of the activation energies. It is shown that this activation energy is generally different from that of a diffusing particle with the same barrier-height distribution, which explains the violation of the Stokes-Einstein relation observed frequently in glasses. At finite but low frequencies, the theory describes low-temperature asymmetric alpha relaxation. As examples, we report the good agreement obtained for selected inorganic, metallic, and organic glasses. At high frequencies, the theory reduces to heterogeneous elasticity theory, which explains the occurrence of the boson peak and related vibrational anomalies.

Probing cooperative liquid dynamics with the mean square displacement

Literature data for picosecond mean square displacements show that the anharmonicity explains only about half of the fragility (with different fractions for different glass formers). The other half must be ascribed to the Adam-Gibbs mechanism of a growing cooperatively rearranging region. One can measure both influences separately by a simultaneous measurement of liquid and crystal in the coexistence region.

The β-relaxation in metallic glasses

Focusing on metallic glasses as model systems, we review the features and mechanisms of the β-relaxations, which are intrinsic and universal to supercooled liquids and glasses, and demonstrate their importance in understanding many crucial unresolved issues in glassy physics and materials science, including glass transition phenomena, mechanical properties, shear-banding dynamics and deformation mechanisms, diffusion and the breakdown of the Stokes–Einstein relation as well as crystallization and stability of glasses. We illustrate that it is an attractive prospect to incorporate these insights into the design of new glassy materials with extraordinary properties. We also outline important questions regarding the nature of β-relaxations and highlight some emerging research directions in this still-evolving field.

Picosecond dynamic heterogeneity, hopping, and Johari-Goldstein relaxation in glass-forming liquids

We show that incoherent quasielastic neutron scattering from molecular liquids reveals a two-state dynamic heterogeneity on a 1 ps time scale, where molecules are either highly confined or are free to undergo relatively large excursions. Data ranging from deep in the glassy state to well above the melting point allows us to observe temperature-dependent population levels and exchange between these two states. A simple physical picture emerges from this data, combined with published work, that provides a mechanism for hopping and for the Johari-Goldstein (β_{JG}) relaxation, and allows us to accurately calculate the diffusion coefficient, D_{T}, and characteristic times for α, and β_{JG} relaxations from ps time scale neutron data.

Characteristics of the Johari-Goldstein process in rigid asymmetric molecules

Molecular dynamics simulations were carried out on a Lennard-Jones binary mixture of rigid (fixed bond length) diatomic molecules. The translational and rotational correlation functions, and the corresponding susceptibilities, exhibit two relaxation processes: the slow structural relaxation (α dynamics) and a higher frequency secondary relaxation. The latter is a Johari-Goldstein (JG) process, by its definition of involving all parts of the molecule. It shows several properties characteristic of the JG relaxation: (1) merging with the α relaxation at high temperature; (2) a change in temperature dependence of its relaxation strength on vitrification; (3) a separation in frequency from the α peak that correlates with the breadth of the α dispersion; and (4) sensitivity to volume, pressure, and physical aging. These properties can be used to determine whether a secondary relaxation in a real material is an authentic JG process, rather than more trivial motion involving intramolecular degrees of freedom. The latter has no connection to the glass transition, whereas the JG relaxation is closely related to structural relaxation, and thus can provide new insights into the phenomenon.

Dynamics of glass relaxation at room temperature

The problem of glass relaxation under ambient conditions has intrigued scientists and the general public for centuries, most notably in the legend of flowing cathedral glass windows. Here we report quantitative measurement of glass relaxation at room temperature. We find that Corning® Gorilla® Glass shows measurable and reproducible relaxation at room temperature. Remarkably, this relaxation follows a stretched exponential decay rather than simple exponential relaxation, and the value of the stretching exponent (β=3/7) follows a theoretical prediction made by Phillips for homogeneous glasses.

Aging of CKN: modulus versus conductivity analysis

It was recently reported that the electrical modulus peaks narrows upon annealing of the ionic system CKN [Paluch et al., Phys. Rev. Lett. 110, 015702 (2013)], which was interpreted as providing evidence of dynamic heterogeneity of this glass. The present analysis of the same data in terms of the ac conductivity shows no shape changes, however. We discuss the relation between both findings and show further that the ac conductivity conforms to the prediction of the random barrier model at all times during the annealing.

Glassy Interfacial Dynamics of Ni Nanoparticles: Part II Discrete Breathers as an Explanation of Two-Level Energy Fluctuations

Recent studies of the dynamics of diverse condensed amorphous materials have indicated significant heterogeneity in the local mobility and a progressive increase in collective particle motion upon cooling that takes the form of string-like particle rearrangements. In a previous paper (Part I), we examined the possibility that fluctuations in potential energy E and particle mobility μ associated with this 'dynamic heterogeneity' might offer information about the scale of collective motion in glassy materials based on molecular dynamics simulations of the glassy interfacial region of Ni nanoparticles (NPs) at elevated temperatures. We found that the noise exponent associated with fluctuations in the Debye-Waller factor, a mobility related quantity, was directly proportional to the scale of collective motion L under a broad range of conditions, but the noise exponent associated with E(t) fluctuations was seemingly unrelated to L. In the present work, we focus on this unanticipated difference between potential energy and mobility fluctuations by examining these quantities at an atomic scale. We find that the string atoms exhibit a jump-like motion between two well-separated bands of energy states and the rate at which these jumps occur seems to be consistent with the phenomenology of the 'slow-beta' relaxation process of glass-forming liquids. Concurrently with these local E(t) jumps, we also find 'quake-like' particle displacements having a power-law distribution in magnitude so that particle displacement fluctuations within the strings are strikingly different from local E(t) fluctuations. An analysis of these E(t) fluctuations suggests that we are dealing with 'discrete breather' excitations in which large energy fluctuations develop in arrays of non-linear oscillators by virtue of large anharmonicity in the interparticle interactions and discreteness effects associated with particle packing. We quantify string collective motions on a fast caging times scale (picoseconds) and explore the significance of these collective motions for understanding the Boson peak of glass-forming materials.

Liquid-liquid transition without macroscopic phase separation in a water-glycerol mixture

The existence of more than two liquid states in a single-component substance and the ensuing liquid-liquid transitions (LLTs) has attracted considerable attention because of its counterintuitive nature and its importance in the fundamental understanding of the liquid state. Here we report direct experimental evidence for a genuine (isocompositional) LLT without macroscopic phase separation in an aqueous solution of glycerol. We show that liquid I transforms into liquid II by way of two types of kinetics: nucleation and growth, and spinodal decomposition. Although liquid II is metastable against crystallization, we could access both its static and dynamical properties experimentally. We find that liquids I and II differ in density, refractive index, structure, hydrogen bonding state, glass transition temperature and fragility, and that the transition between the two liquids is mainly driven by the local structuring of water rather than of glycerol, suggesting a link to a plausible LLT in pure water.

On the mechanism of the highly viscous flow

The asymmetry model for the highly viscous flow postulates thermally activated jumps from a practically undistorted ground state to strongly distorted, but stable structures, with a pronounced Eshelby backstress from the distorted surroundings. The viscosity is ascribed to those stable distorted structures which do not jump back, but relax by the relaxation of the surrounding viscoelastic matrix. It is shown that this mechanism implies a description in terms of the shear compliance, with a viscosity which can be calculated from the cutoff of the retardation spectrum. Consistency requires that this cutoff lies close to the Maxwell time. The improved asymmetry model compares well with experiment.

Effects of water contamination on the supercooled dynamics of a hydrogen-bonded model glass former

Broad-band dielectric spectroscopy is a commonly used tool in the study of glass-forming liquids. The high sensitivity of the technique together with the wide range of probed time scales makes it a powerful method for investigating the relaxation spectra of liquids. One particularly important class of glass-forming liquids that is often studied using this technique consists of liquids dominated by hydrogen (H) bond interactions. When investigating such liquids, particular caution has to be taken during sample preparation due to their often highly hygroscopic nature. Water can easily be absorbed from the atmosphere, and dielectric spectroscopy is a very sensitive probe of such contamination due to the large dipole moment of water. Our knowledge concerning the effects of small quantities of water on the dielectric properties of these commonly investigated liquids is limited. We here demonstrate the effects due to the presence of small amounts of water on the dielectric response of a typical H-bonded model glass former, tripropylene glycol. We show how the relaxation processes present in the pure liquid are affected by addition of water, and we find that a characteristic water induced relaxation response is observed for water contents as low as 0.15 wt%. We stress the importance of careful purification of hygroscopic liquids before experiments and quantify what the effects are if such procedures are not undertaken.

Configurational and residual entropies of nonergodic crystals and the entropy's behavior on glass formation

We use thermodynamics of lattice vacancies to test the merits of the view that (i) statistical entropy, k(B) ln Omega, vanishes on vitrification of a liquid and hence there is no residual entropy and (ii) k(B) ln Omega of a nonergodic state would increase with time t as its structure relaxes. We argue that this view conflicts with the precepts of the configurational entropy of a crystal, -R[x ln x+(1-x)ln(1-x)], where x is the fractional population of vacancies, and with the observed decrease in x with t on structural relaxation. The issue of whether the entropy of a kinetically arrested crystal state is equal to k(B) ln Omega or equal to -R[x ln x+(1-x)ln(1-x)] can be resolved by measuring the vapor pressure, the emf of an electrolytic cell, and by scanning calorimetry. We also consider how the energy landscapes of a crystal and liquid differ, and point out that since crystals are in a nonequilibrium state, their thermodynamic data are inappropriate for testing the validity of the third law.

Broadband dielectric spectroscopy on benzophenone: alpha relaxation, beta relaxation, and mode coupling theory

We have performed a detailed dielectric investigation of the relaxational dynamics of glass-forming benzophenone. Our measurements cover a broad frequency range of 0.1 Hz to 120 GHz and temperatures from far below the glass temperature well up into the region of the small-viscosity liquid. With respect to the alpha relaxation this material can be characterized as a typical molecular glass former with rather high fragility. A good agreement of the alpha relaxation behavior with the predictions of the mode coupling theory of the glass transition is stated. In addition, at temperatures below and in the vicinity of T(g) we detect a well-pronounced beta relaxation of Johari-Goldstein type, which with increasing temperature develops into an excess wing. We compare our results to literature data from optical Kerr effect and depolarized light scattering experiments, where an excess-wing-like feature was observed in the 1-100 GHz region. We address the question if the Cole-Cole peak, which was invoked to describe the optical Kerr effect data within the framework of the mode coupling theory, has any relation to the canonical beta relaxation detected by dielectric spectroscopy.

Feasibility of a single-parameter description of equilibrium viscous liquid dynamics

Molecular dynamics results for the dynamic Prigogine-Defay ratio are presented for two glass-forming liquids, thus evaluating the experimentally relevant quantity for testing whether metastable-equilibrium liquid dynamics is described by a single parameter to a good approximation. For the Kob-Andersen binary Lennard-Jones mixture as well as for an asymmetric dumbbell model liquid, a single-parameter description works quite well. This is confirmed by time-domain results where it is found that energy and pressure fluctuations are strongly correlated on the alpha time scale in the constant-volume, constant-temperature ensemble; similarly, energy and volume fluctuations correlate strongly in the constant-pressure, constant-temperature ensemble.

Effect of chain length on fragility and thermodynamic scaling of the local segmental dynamics in poly(methylmethacrylate)

Local segmental relaxation properties of poly(methylmethacrylate) (PMMA) of varying molecular weight are measured by dielectric spectroscopy and analyzed in combination with the equation of state obtained from PVT measurements. Significant variations of glass transition temperature and fragility with molecular weight are observed. In accord with the general properties of glass-forming materials, single molecular weight dependent scaling exponent gamma is sufficient to define the mean segmental relaxation time taualpha and its distribution. This exponent can be connected to the Gruneisen parameter and related thermodynamic quantities, thus demonstrating the interrelationship between dynamics and thermodynamics in PMMA. Changes in the relaxation properties ("dynamic crossover") are observed as a function of both temperature and pressure, with taualpha serving as the control parameter for the crossover. At longer taualpha another change in the dynamics is apparent, associated with a decoupling of the local segmental process from ionic conductivity.

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.

Changes of relaxation dynamics of a hydrogen-bonded glass former after removal of the hydrogen bonds

Dielectric relaxation spectra of two closely related glass formers, dipropylene glycol [H-(C3H6O)2-OH] and dipropylene glycol dimethyl ether [CH3-O-(C3H6O)2-CH3], were measured at ambient and elevated pressures in the supercooled and the glassy states are presented. Hydrogen bonds formed in dipropylene glycol are removed when its ends are replaced by two methyl groups to become dipropylene glycol dimethyl ether. In the process, the primary relaxation, the excess wing, and the resolved secondary relaxation of dipropylene glycol are all modified when the structure is transformed to become dipropylene glycol dimethyl ether. The modifications include the pressure and temperature dependences of these relaxation processes and their interrelations. Thus, by comparing the dielectric spectra of these two closely related glass formers at ambient and elevated pressures, the differences in the relaxation dynamics and properties in the presence and absence of hydrogen bonding are identified.

Dielectric secondary relaxations in polypropylene glycols

Broadband dielectric measurements of polypropylene glycol of molecular weight M(w)=400 g / mol (PPG 400) were carried out at ambient pressure over the wide temperature range from 123 to 353 K. Three relaxation processes were observed. Besides the structural alpha relaxation, two secondary relaxations, beta and gamma, were found. The beta process was identified as the true Johari-Goldstein relaxation by using a criterion based on the coupling model prediction. The faster gamma relaxation, well separated from the primary process, undoubtedly exhibits the anomalous behavior near the glass transition temperature (T(g)) which is reflected in the presence of a minimum of the temperature dependence of the gamma-relaxation time. We successfully applied the minimal model [Dyre and Olsen, Phys. Rev. Lett. 91, 155703 (2003)] to describe the entire temperature dependence of the gamma-relaxation time. The asymmetric double-well potential parameters obtained by Dyre and Olsen for the secondary relaxation of tripropylene glycol at ambient pressure were modified by fitting to the minimal model at lower temperatures. Moreover, we showed that the effect of the molecular weight of polypropylene glycol on the minimal model parameters is significantly larger than that of the high pressure. Such results can be explained by the smaller degree of hydrogen bonds formed by longer chain molecules of PPG at ambient pressure than that created by shorter chains of PPG at high pressure.