The Scaling of Relaxation Processes

Glassy dynamics covers the extraordinary spectral range from 10⁺¹³ to 10⁻³ Hz and below. In this broad frequency window, four different dynamic processes take place: (i) the primary or α-relaxation, (ii) (slow) secondary relaxations (β-relaxations), (iii) fast absorption processes in the GHz and (iv) the boson-peak in the THz range. The dynamic glass transition is assigned to fluctuations between structural substates and scales well with the calorimetric glass transition temperature. It shows a similar temperature dependence as the viscosity and fluctuations of the density or heat capacity. The temperature dependence of the mean relaxation rate of the dynamic glass transition follows at first glance the empirical Vogel–Fulcher–Tammann law, albeit a further analysis unravels clear-cut deviations. The (slow) secondary relaxations are assigned to librational relaxations of molecular subgroups hence having a straightforward molecular assignment. They may also show up as a wing on the high-frequency side of the dynamic glass transition. The fast absorption processes at GHz frequencies can formally be described within the framework of the mode-coupling theory (MCT). The boson-peak resembles the Poley absorption and originates from overdamped oscillations. In this chapter, especially the first three contributions will be discussed in detail and compared with existing theoretical models.

In this article, we review broadband dielectric spectroscopy in supercooled liquids, in many cases covering more than 15 decades in frequency and a wide range of temperatures from the low-viscosity liquid to the rigid sub-T_g glass. The access to this extremely broad frequency window allows a detailed study of the complexity of glassy freezing and glassy dynamics in a large variety of materials. Dielectric spectroscopy not only documents the enormous slowing down of the structural relaxation when approaching the glass transition, but also reveals a variety of further relaxation processes, which are important to understand the physics of the transition from a supercooled liquid into a rigid glass. After a short introduction, mainly focusing on long-term experiments on glasses and on the classification of glass formers into strong and fragile, we shortly discuss some basics of relaxation and conductivity contributions when viewed via dielectric spectroscopy. We provide some prototypical examples of dielectric loss spectra covering a large frequency and temperature regime. The glass formers shown can be categorized into two classes, type A and type B. The latter reveal well-defined Johari–Goldstein secondary relaxations, which lead to peaks in the dielectric loss at least at low temperatures. The former exhibit an excess wing, showing only a change of slope of the high-frequency flank of the structural-relaxation loss peaks. Then, we exemplify the phenomenology of glassy dynamics as revealed by these broadband spectra: The structural relaxation, the Johari–Goldstein relaxation, the appearance of a fast process as proposed by the mode-coupling theory, and the boson peak, a well-defined feature in the dielectric loss at THz frequencies, are discussed in detail. In a further chapter, we focus on the importance of sub-T_g experiments: Aging experiments and a possible experimental evidence of the Gardner transition are discussed. Finally, we summarize the experimental dielectric results documenting the universality of glassy freezing, which can be directly derived from these measurements.

The inter- and intra-molecular interactions of low molecular weight and polymeric glass-forming model systems are studied by broadband dielectric (BDS) spectroscopy and Fourier-transform infrared (FTIR) spectroscopy. Analyzing the temperature dependence of specific IR absorption bands, reflecting the intra-molecular potentials of dedicated molecular moieties, enables one to unravel on an intra-molecular scale the process of glass formation and to compare it with the dielectrically determined primarily inter-molecular dynamics. Molecular systems to be studied are typical glass formers as the polyalcohols glycerol, threitol, xylitol, and sorbitol, as salol and three poly(ethylene-glycol) derivatives, namely poly(ethylene-glycol)methyl-ether-acrylate, poly(ethylene-glycol)phenyl-ether-acrylate, and poly(ethylene-glycol)-dibenzoate. Within this experimental framework, a wealth of novel information is obtained proving that the different molecular moieties of a glass former show characteristic features in the course of glassy solidification. This demonstrates the fundamental importance of intra-molecular dynamics for the dynamic glass transition, providing refined insights into the underlying interactions beyond coarse-grained models, approximating, for instance, glass-forming molecules as hard spheres.

Investigations of the sought after complete and commonly accepted theory of the glass transition and related phenomena have recently gained an essential support from a very promising idea of the density scaling of molecular dynamics in viscous liquids. This idea, often known as the thermodynamic scaling concept, has been initiated by many phenomenological observations, which have shown that dynamic quantities (e.g., viscosity, structural relaxation time, or segmental relaxation time in case of polymers) measured in different thermodynamic conditions (e.g., along different isobars and isotherms) can be scaled onto one master curve well described by a function of the single variable that is a product of the inverse temperature and the density power with the scaling exponent considered as a material constant independent of thermodynamic conditions. However, a crucial advantage of the phenomenological description has become its theoretical grounds relied on an effective short-range intermolecular potential, which has been derived from the well-known Lennard-Jones potential and satisfactorily verified by computer simulations. A relation suggested between the scaling exponent and the exponent of the dominant repulsive part of the effective intermolecular potential gives a tempting opportunity to study the macroscopic properties of materials by using the underlying intermolecular potential and vice versa to determine the intermolecular potential parameters based on measurements of macroscopic quantities. It opens new perspectives for our better understanding of complex physicochemical phenomena occurring near the glass transition. In this chapter, we present the density scaling concept as the idea that bears hallmarks of universality in case of both various materials and different quantities. We show that the density scaling law may concern not only dynamic but also thermodynamic quantities, constituting a convenient tool to explore relationships between molecular dynamics and thermodynamics based on the effective short-range intermolecular potential. We demonstrate predictive capabilities of the density scaling law that implies several rules for activation quantities and fragility parameters defined in different thermodynamic conditions, which enable to discover and verify physically well-defined invariants. We also discuss some nontrivial cases of the thermodynamic scaling for which the power density scaling law with a constant scaling exponent is not sufficient, but we can find density or timescale-dependent counterparts of the exponent. The exceptions to the standard power density scaling law delimit further challenges in making progress toward the development of the density scaling idea and its applicability range.

Supramolecular relaxations of the Debye or near-Debye type are featured by monohydroxy alcohols, water, and several other liquids. Mainly focusing on results from broadband dielectric spectroscopy, shear rheology, X-ray diffraction, and near-infrared absorption, scaling properties of chain-forming and ring-forming monohydroxy alcohols are examined. Deviations from ideal-mixing behavior in binary solutions involving these liquids in their supercooled state are given particular attention. The present survey is selective rather than comprehensive with a focus on exciting recent developments in this scientific area. Although most of the research summarized in this chapter is based on experiments and analyses carried out under linear-response and ambient-pressure conditions, phenomena emerging beyond these regimes are briefly touched upon as well. Finally, aiming at a faithful representation of the molecular dynamics taking place in these liquids at the microscopic level, overarching aspects arising from the complementary application of experimental techniques as well as perspectives for future developments are discussed.

Starting with an overview of major results of the main (α) and secondary (β) relaxation in neat glass formers as compiled by dielectric and nuclear magnetic resonance (NMR) spectroscopy as well as by light scattering, the contribution deals with elucidating the component dynamics in binary glass formers. Dynamically asymmetric mixtures with high-T_g contrast of their components are in focus. In addition to polymers, specially synthesized non-polymeric glass formers are considered as high-T_g component and mixed with a low-T_g simple liquid. While the high-T_g component in the mixtures shows relaxation features similar to that of neat glass formers, the low-T_g component displays significantly faster dynamics and pronounced dynamic heterogeneities, i.e., an extremely broad distribution of correlation times G(lnτ), which may lead to quasi-logarithmic correlation functions. Two glass transition temperatures with non-trivial concentration dependences are identified. The dynamic heterogeneities are transient in nature as proven by 2D exchange NMR. Thus, liquid-like (isotropic) reorientation of the low-T_g additive as well as exchange within its distribution G(lnτ) is observed in an essentially rigid high-T_g matrix. The results show similarity with those collected for glass formers in confining geometries, suggesting that in asymmetric binary glass formers (intrinsic) confinement effects may control the dynamics either. We also investigate the β-process in the mixed glasses introduced by the low-T_g additive. It is rediscovered for all concentrations with virtually unchanged time constants. NMR identifies the β-relaxations as being similar to those of neat glasses. A spatially highly restricted motion with an angular displacement below ±10° encompasses all molecules. Very similar spectral features are observed for the high-T_g component in NMR. Apparently, the (small) additive molecules “enslave” the large molecules to perform a common hindered reorientation. At lowest additive concentrations, one finds indications that the β-process starts to disintegrate. We conclude that the β-process is a cooperative process.

Broadband dielectric spectroscopy (BDS) can be considered the standard and most widespread method to experimentally access molecular reorientation in supercooled liquids, as it covers a range of time constants from sub picoseconds corresponding to the highly fluid liquid to several thousand seconds below the glass transition temperature. In a similar fashion, depolarized dynamic light scattering (DLS) is able to probe molecular reorientation. A comparable range of time scales is covered by combining Tandem Fabry Perot Interferometry (TFPI) and Photon Correlation Spectroscopy (PCS) with recent multispeckle techniques allowing to access even the non-ergodic regime below \(T_g\). Thus, DLS represents an alternative route to cover the full range of glassy dynamics. Moreover, due to the fact that both methods couple to different molecular properties, extra information in particular on the motional mechanism behind a certain dynamic process can be obtained by comparing experimental data from both techniques. In the present work we explore this approach for several examples, including ionic liquids and monohydroxy alcohols, and discuss the implications for different relaxation processes. For instance in the case of supercooled ionic liquids, i.e., molten salts, which are liquid at room temperature, the combination of both techniques allows to unambiguously disentangle the contribution of molecular reorientation from other polarization features that often mask reorientation in the dielectric spectra, and a detailed analysis reveals indications for a crossover in the motional mechanism involved in the \(\alpha \)-relaxation. In monohydroxy alcohols we discuss the appearance of the Johari-Goldstein \(\beta \)-process in both techniques and what the observations imply for the underlying motional mechansim. Furthermore, we consider the Debye relaxation, which is frequently observed in the dielectric spectra of monoalcohols and is usually ascribed to transient supramolecular structures. Here, such a comparison of data reveals molecular details about the conditions under which the supramolecular structures are formed.

In this chapter, we first introduce the main concepts related to quasielastic neutron scattering (QENS) techniques and the way they can be connected to dielectric spectroscopy (DS). This is not obvious, because they access different correlation functions. The dielectric permittivity measured by DS reflects the orientational dynamics of the molecular dipoles in a very broad temperature/frequency range, while, thanks to the transfer of energy (\(\hbar \omega \)) and momentum (\(\hbar Q\)) dependence of the measured intensities, QENS provides information about nuclear positions with space/time resolution. In particular, QENS on protonated samples follows the self-correlation function of the hydrogens. Next, we describe the general findings from both techniques relative to the \(\alpha \)-relaxation in glass-forming systems. From the comparison of the results, we define a Q-value (\(Q^\star \)) at which the timescale of the \(\alpha \)-process measured by QENS and DS become similar and compile its values from the literature for diverse systems ranging from polymers and low-molecular weight glass-forming systems to water and water solutions. The results are discussed in a phenomenological way in terms of structural and dynamic parameters. Thereafter, we show that in the case of a simple diffusive process, a simple approach based on molecular hydrodynamics and a molecular treatment of DS allows expressing \(Q^\star \) in terms of a many-body magnitude—a generalized Kirkwood parameter —and a single-molecule magnitude—the hydrodynamic radius. The application of these ideas to liquid water and water solutions is presented. Finally, we explore the possibility of extending this kind of treatment to the more complex subdiffusive case.

A combination of different complementary methods is employed to investigate scaling of the molecular dynamics of two different liquid crystals. Each method is sensitive to different kind of fluctuations and provides therefore a different window to look at the molecular dynamics. In detail, broadband dielectric spectroscopy is combined with specific heat spectroscopy and neutron scattering. As systems, the nematic liquid crystal E7 and a discotic liquid crystalline pyrene are considered. First of all, it was proven that both systems show all peculiarities which are characteristic for glassy dynamics and the glassy state. Especially for the nematic liquid crystal E7, it could be unambiguously shown by a combination of dielectric and specific heat spectroscopy that the tumbling mode is the underlying motional process responsible for glassy dynamics. Dielectric investigations on the discotic liquid crystalline pyrene reveal that at the phase transition from the plastic crystalline to the hexagonal columnar liquid crystalline phase, the molecular dynamics changes from a more strong to fragile temperature dependence of the relaxation rates. Moreover, a combination of results obtained by specific heat spectroscopy with data from structural methods allows an estimation of the length scale relevant for the glass transition.

The glass transition at common laboratory scan rates (K/min) has been a highly debated topic for the last decade. The continuous increase in the variety of available glass-forming materials and methods to characterize them maintains a research interest, as well as opens new perspective applications. In parallel, many different theoretical methods aimed at describing the glass transition have been proposed in the last 70 years. A general theory has yet to be developed and carefully tested. In the present chapter, we describe the results of theoretical and experimental investigations of the glass transition of a model polymer—polystyrene. State-of-the-art scanning calorimetry allows for measuring the temperature dependence of the isobaric heat capacity in an exceedingly wide range of cooling rates. Besides providing novel data on the glass transition of polymers at fast cooling rates, this allows for one to test the capabilities of convenient theoretical methods in modelling the kinetics of the glass transition under very different vitrifying conditions. The glass transition of atactic polystyrene was investigated at different cooling rates in the range of q_c = 10⁻⁶–10⁴ K/s. Dependencies of the glass transition temperature, T_g, and the shape of heat capacity, C_p, curves on q_c were obtained. Furthermore, we have applied a number of different theoretical methods to test their capability to model the glass transition kinetics for such a wide range of control parameter q_c. The list of investigated theoretical methods consists of the Tool–Narayanaswamy–Moynihan approach, Adam–Gibbs theory, an irreversible thermodynamics-based approach and some of their modern modifications. As a first step, we show that most of these methods are capable of fitting the cooling rate dependencies of the glass transition parameters (T_g and others). The model parameters in this case are close to literature data. Furthermore, we show that while fitting the C_p(T) curves for a single cooling–heating experiment bears acceptable results, the parameters have to be changed with respect to q_c, with their difference becoming significant for very slow or very fast cooling rates. Thus, none of the methods can be applied successfully to model and predict the kinetics of glass transition in a wide range of q. We compare the results of different methods and propose an expression for the relaxation time dependence on model parameters within an irreversible thermodynamics approach. Thus, we extend the experimental results for polystyrene and state that the presently applied theoretical methods are incapable of accurately describing the heat capacity temperature curves C_p(T) for a wide range of cooling/heating rates, q. The present methods and expressions for relaxation time τ do not account for a certain additional effect spanning over different rates of temperature change, which has yet to be discovered.

Broadband dielectric spectroscopy is able to follow the time correlation of its observable, the polarization of a sample, over an unprecedented range of frequency or timescales, respectively. Features in the dielectric susceptibility as a function of frequency are assigned to different molecular motions, which, for polymers in the melt, range from high-frequency vibrations over a possible Johari-Goldstein \(\beta \)-relaxation, the segmental or \(\alpha \)-relaxation, a possible normal mode for chains with a net dipole moment along the end-to-end vector of the chain to conductivity contributions on the low-frequency side. This assignment is a statement about correlations between the local dipole moments making up the sample polarization and their behavior across length and timescales. Such correlations can depend on temperature and on density as well as on confinement effects. A molecular dynamics simulation of a chemically realistic model of 1,4-polybutadiene confined between graphite walls allows for a study of such correlations over a broad temperature range in the bulk-like center of the film, as well as in the wall regions where confinement -induced correlations are present. We have already shown that confinement induces a new type of normal mode into this polymer which does not possess such a mode in the bulk. Here, we discuss the scaling of the dielectric relaxation process in this system and additionally analyze the temperature and confinement dependence of high-frequency vibrations.

In this chapter, we discuss coarse-grained and atomistic molecular-dynamics simulation studies of the rheological properties of bulk polymer systems and polymer nanocomposites. Both systems contain monodispersed and non-crosslinked chain molecules. A multiscale strategy is applied to characterize the rheological behavior on different length scales of the systems structural organization. Fully atomistic simulations provide insights in rheological properties on smaller length scales than those accessible through coarse-grained simulations. Different approaches are utilized to obtain rheological moduli at these different length scales. At both levels of description, cyclic shear deformation is performed to characterize macroscopic properties of the systems before and after filler insertion. In the fully atomistic simulations of polyimide R-BAPB, passive microrheology approach is employed in addition to active rheology. To this end, a probe particle is immersed into the atomistic polymer matrix. Then, local rheological properties on the length scales at and beyond the Kuhn length are estimated. Results are compared with macroscopic rheological properties obtained by shear deformation. Additionally, the influence of the strain amplitude on the resulting rheological properties is examined. The reported coarse-grained simulations show a strong decrease of the nanocomposites storage modulus with increasing strain amplitude, which is accompanied by a maximum in the loss modulus (the so-called Payne effect); the onset of the softening is observed in the linear regime of deformation at strain amplitude of about 0.01. Moreover, the dependence of the storage modulus on the instantaneous strain exhibits both softening and hardening regimes, in agreement with recently reported [22] Large Amplitude Oscillatory Shear (LAOS) experiments. The simulations suggest that the observed hardening is caused by the shear-induced decrease of the non-affine diffusion of the polymer segments due to filler particles acting as effective crosslinks between polymeric chains and, hence, hindering diffusion. Moreover, the formation of “glassy” immobile layers at the nanoparticle interface strongly increases the storage modulus at low strain amplitudes. The strain softening with increasing strain amplitude is connected to the mobilization of these glassy layers and an increase in the dynamic heterogeneity of the polymer matrix. A breakup of the network structure plays a role as well.