Primary and secondary relaxations in bis-5-hydroxypentylphthalate revisited

The molecular structure of bis-5-hydroxypentylphthalate (BHPP) is like dihexyl phthalate but having appended to it two hydroxyl end groups, which contribute additional dipole moments and capacity for hydrogen-bond formation. In a previously published dielectric study of the primary and secondary relaxations of BHPP, it was found that all the dynamic properties are normal except for the anomalously large width of the primary relaxation loss peak. There are two secondary relaxations, the relaxation time of the slower one increases with increasing pressure, whereas that of the faster one is practically insensitive to pressure. Hence, the slower secondary relaxation is the "universal" Johari-Goldstein (JG) [J. Chem. Phys. 53, 2372 (1970); 55, 4245 (1971)] relaxation in BHPP. All is well except if the observed large width of the primary relaxation were an indication of a corresponding large coupling parameter n=0.45 in the coupling model. Then the predicted relations between the primary relaxation time tau(alpha) and the JG relaxation time tau(JG) found previously to hold in many glass formers would be violated. It was recognized that this singular behavior of BHPP is likely due to broadening of the primary loss peak by the overlapping contributions of two independent dipole moments present in BHPP, and the actual coupling parameter is smaller. However, at the time of publication of the previous work there were not enough data to support this explanation. By making broadband dielectric measurements of dibutyl phthalate (DBP) and dioctyl phthalate (DOP) that have chemical structures closely related to BHPP but with only one dipole moment, we show that all their dynamic properties are almost the same as BHPP but the widths of their primary relaxation loss peaks are significantly narrower corresponding to a smaller coupling parameter n=0.34. The new data presented here indicate that the coupling parameter of BHPP is about the same as DBP and DOP, and the predicted relations between tau(alpha) and tau(JG) of BHPP are brought back in agreement with the experimental data.     

Two secondary modes in decahydroisoquinoline: which one is the true Johari Goldstein process?

Broadband dielectric measurements were carried out at isobaric and isothermal conditions up to 1.75 GPa for reconsidering the relaxation dynamics of decahydroisoquinoline, previously investigated by Richert et al. [R. Richert, K. Duvvuri, and L.-T. Duong, J. Chem. Phys. 118, 1828 (2003)] at atmospheric pressure. The relaxation time of the intense secondary relaxation tau(beta) seems to be insensitive to applied pressure, contrary to the alpha-relaxation times tau(alpha). Moreover, the separation of the alpha- and beta-relaxation times lacks correlation between shapes of the alpha-process and beta-relaxation times, predicted by the coupling model [see for example, K. L. Ngai, J. Phys.: Condens. Matter 15, S1107 (2003)], suggesting that the beta process is not a true Johari-Goldstein (JG) relaxation. From the other side, by performing measurements under favorable conditions, we are able to reveal a new secondary relaxation process, otherwise suppressed by the intense beta process, and to determine the temperature dependence of its relaxation times, which is in agreement with that of the JG relaxation.     

Effect of physical aging on the Johari-Goldstein and alpha relaxations of D-sorbitol: a study by thermally stimulated depolarization currents

The relaxations in amorphous D-sorbitol have been studied by thermally stimulated depolarization currents during annealing at 255 K, which is 17 K below its calorimetric glass transition temperature Tg=272 K. As the glass structurally relaxes on aging, the features of the alpha relaxation and of the Johari-Goldstein (JG) relaxation change with time. For the alpha relaxation (i) the dielectric strength decreases; (ii) the activation energy decreases; and (iii) the relaxation time increases. For the JG relaxation the dielectric strength also decreases but with a different time dependence, and there is no evidence for any modification of the kinetic features of the mobility. The amplitude of response to aging is higher for the higher temperature motional components of the Johari-Goldstein relaxation compared with the lower temperature ones.     

Primary and secondary relaxations in bis-5-hydroxypentylphthalate

Broadband dielectric spectroscopy was used to study the relaxation dynamics in bis-5-hydroxypentylphthalate (BHPP) under both isobaric and isothermal conditions. The relaxation dynamics exhibit complex behavior, arising from hydrogen bonding in the BHPP. At ambient pressure above the glass transition temperature T(g), the dielectric spectrum shows a broad structural relaxation peak with a prominent excess wing toward higher frequencies. As temperature is decreased below T(g), the excess wing transforms into two distinct peaks, both having Arrhenius behavior with activation energies equal to 58.8 and 32.6 kJmol for slower (beta) and faster (gamma) processes, respectively. Furthermore, the relaxation times for the beta process increase with increasing pressure, whereas the faster gamma relaxation is practically insensitive to pressure changes. Analysis of the properties of these secondary relaxations suggests that the beta peak can be identified as an intermolecular Johari-Goldstein (JG) process. However, its separation in frequency from the alpha relaxation, and both its activation energy and activation volume, differ substantially from values calculated from the breadth of the structural relaxation peak. Thus, the dynamics of BHPP appear to be an exception to the usual correlation between the respective properties of the structural and the JG secondary relaxations.     

Secondary relaxation behavior in a strong glass

In the present paper we study the enthalpy relaxation behavior of the hyperquenched GeO(2) (HQGeO(2)) glass, one of the strongest glass systems. By applying the hyperquenching-annealing-calorimetry approach, we have found that unlike fragile glasses the strong HQGeO(2) glass relaxes in a manner that all the secondary relaxation units contribute to the primary relaxation. By analyzing dynamic properties of the secondary relaxation, we have identified two typical features of the Johari-Goldstein relaxation in the HQGeO(2) glass. First, the quantitative relationship observed here between E(beta) and T(g) agrees well with the empirical relation of the JG relaxation. Second, the characteristic relaxation time of the GeO(2) glass at T(g) is found to be about 10 s, larger than that of relatively fragile glasses. These results verify that the JG peak in strong glasses is hidden by the alpha peak in the dielectric loss curves.     

Relation between the alpha-relaxation and Johari-Goldstein beta-relaxation of a component in binary miscible mixtures of glass-formers

The coupling model was applied to describe the alpha-relaxation dynamics of each component in perfectly miscible mixtures A(1-x)B(x) of two different glass-formers A and B. An important element of the model is the change of the coupling parameter of each component with the composition, x, of the mixture. However, this change cannot be determined directly from the frequency dispersion of the alpha-relaxation of each component because of the broadening caused by concentration fluctuations in the mixture, except in the limits of low concentrations of either component, x --> 0 and x --> 1. Fortunately, the coupling model has another prediction. The coupling parameter of a component, say A, in the mixture determines tau(alpha)/tau(JG), the ratio of the alpha-relaxation time, tau(alpha), to the Johari-Goldstein (JG) secondary relaxation time, tau(JG), of the same component A. This prediction enables us to obtain the coupling parameter, n(A), of component A from the isothermal frequency spectrum of the mixture that shows both the alpha-relaxation and the JG beta-relaxation of component A. We put this extra prediction into practice by calculating n(A) of 2-picoline in binary mixtures with either tri-styrene or o-terphenyl from recently published broadband dielectric relaxation data of the alpha-relaxation and the JG beta-relaxation of 2-picoline. The results of n(A) obtained from the experimental data show its change with composition, x, follows the same pattern as assumed in previous works that address only the alpha-relaxation dynamics of a component in binary mixtures based on the coupling model. There is an alternative view of the thrust of the present work. If the change of n(A) with composition, x, in considering the alpha-relaxation of component A is justified by other means, the theoretical part of the present work gives a prediction of how the ratio tau(alpha)/tau(JG) of component A changes with composition, x. The data of tau(alpha) and tau(JG) of 2-picoline mixed with tri-styrene or o-terphenyl provide experimental support for the prediction.     

Emergence of the genuine Johari-Goldstein secondary relaxation in m-fluoroaniline after suppression of hydrogen-bond-induced clusters by elevating temperature and pressure

The dielectric spectra of the glass former, m-fluoroaniline (m-FA), at ambient pressure show the presence of a secondary relaxation, which was identified in the literature as the universal Johari-Goldstein (JG) beta relaxation. However, published elastic neutron scattering and simulation data [D. Morineau, C. Alba-Simionesco, M. C. Bellisent-Funel, and M. F. Lauthie, Europhys. Lett. 43, 195 (1998); D. Morineau and C. Alba-Simionesco, J. Chem. Phys. 109, 8494 (1998)] showed the presence of hydrogen-bond-induced clusters of limited size in m-FA at ambient pressure and temperature of the dielectric measurements. The observed secondary relaxation may originate from the hydrogen-bond-induced clusters. If so, it should not be identified with the JG beta relaxation that involves essentially all parts of the molecule and has certain characteristics [K. L. Ngai and M. Paluch, J. Chem. Phys. 120, 857 (2004)], but then arises the question of where is the supposedly universal JG beta relaxation in m-FA. To gain a better understanding and resolving the problem, we perform dielectric measurements at elevated pressures and temperatures to suppress the hydrogen-bond-induced clusters and find significant changes in the dielectric spectra. The secondary relaxation observed at ambient pressure in m-FA is suppressed, indicating that indeed it originates from the hydrogen-bond-induced clusters. The spectra of m-FA are transformed at high temperature and pressure to become similar to that of toluene. The new secondary relaxation that emerges in the spectra has properties of a genuine JG relaxation like in toluene.     

The nearly constant loss,Johari‐Goldstein β‐relaxation,and α‐relaxation of 1,4‐polybutadiene

From high‐resolution dielectric spectroscopy measurements on 1,4‐polybutadiene (1,4‐PB), we show that in addition to the structural α‐relaxation and higher frequency secondary relaxations in the spectra, a nearly constant loss (NCL) is observed at shorter times/lower temperatures. The properties of this NCL are compared to those of another chemically similar polymer, 1,4‐polyisoprene. The secondary relaxations in 1,4‐PB include the well‐known Johari‐Goldstein (JG) β‐relaxation and two other higher‐frequency peaks. One of these, referred to as the γ‐relaxation, falls between the JG‐relaxation and the NCL. Seen previously by others, this γ‐relaxation in 1,4‐PB is not the JG‐process and bears no relation to the glass transition. At very low temperatures (<15 K), we confirm the existence of a very fast secondary relaxation, having a weak dielectric strength and an almost temperature‐invariant relaxation time. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 342–348, 2007     

Study of dielectric relaxations of anhydrous trehalose and maltose glasses

We investigated the frequency dependent dielectric relaxation behaviors of anhydrous trehalose and maltose glasses in the temperature range which covers a supercooled and glassy states. In addition to the α-, Johari-Goldstein (JG) β-, and γ-relaxations in a typical glass forming system, we observed an extra relaxation process between JG β- and γ-relaxations in the dielectric loss spectra. We found that the unknown extra relaxation is a unique property of disaccharide which might originate from the intramolecular motion of flexible glycosidic bond. We also found that the temperature dependence of the JG β-relaxation time changes at 0.95T(g) and it might be universal.     

Dynamics of supercooled and glassy dipropyleneglycol dibenzoate as functions of temperature and aging: Interpretation within the coupling model framework

Dielectric relaxation measurements of a typical small molecular glassformer, dipropyleneglycol dibenzoate show the presence of two secondary relaxations. Their dynamic properties differ in the equilibrium liquid and glassy states, as well as the changes during structural recovery after rapid quenching the liquid to form a glass. These differences enable us to identify the slower secondary relaxation as the genuine Johari-Goldstein (JG) beta-relaxation, acting as the precursor of the primary alpha-relaxation. Agreement between the JG beta-relaxation time and the independent relaxation time of the coupling model leads to predicted quantitative relations between the JG beta-relaxation and the alpha-relaxation that are supported by the experimental data.     

Johari-Goldstein relaxation and crystallization of sorbitol to ordered and disordered phases

The equilibrium permittivity epsilon(s) and the dielectric relaxation spectra of supercooled liquid D-sorbitol were measured during its crystallization to orientationally disordered or ordered phases depending on the sample preparation procedure at several fixed temperatures up to a period of 6 days. The epsilon(s) measurements showed that when the sample was contaminated by a minute amount of crystals, it crystallized to an ordered phase. When the liquid was not contaminated, the sample crystallized to an orientationally disordered phase. When supercooled D-sorbitol was kept close to its T(g), its dielectric spectra did not change over a period of 138.5 h. It was found that the Johari-Goldstein (JG) relaxation rate of the orientationally disordered crystalline phase is higher in comparison with that of the supercooled liquid, the spectrum broader, and the relaxation strength lower. Its glasslike transition temperature is higher than T(g) of the liquid. The results on crystallization showed that the structural changes occurring at a temperature where the alpha relaxation emerges from the JG relaxation affects the crystallization kinetics of the liquid.     

Effect of high hydrostatic pressure on the dielectric relaxation in a non-crystallizable monohydroxy alcohol in its supercooled liquid and glassy states

The complex relative permittivity of a non-crystallizable secondary alcohol, 5-methyl-2-hexanol, is measured over a wide range of temperatures and pressures up to 1750 MPa (17.5 kbar). The data at atmospheric pressure (P = 0.101 MPa) are analyzed in terms of three processes, and the results are in complete agreement with that of O. E. Kalinovskaya and J. K. Vij [J. Chem. Phys. 112, 3262 (2000)]. Process I is of the Debye type and process II is of the Davidson-Cole type, whereas process III is identified as the Johari-Goldstein relaxation process. For pressures of ～500 MPa and higher, processes I and II are seen to merge into each other to form a single dominant process which unambiguously cannot be resolved into more than one process. The dielectric relaxation strength of process I decreases slightly initially with pressure and when the two processes have merged at elevated pressures, the total relaxation strength increases with increase in pressure. Process III is better resolvable at higher pressures especially above T(g) in the supercooled liquid state for the reason that the separation in the time scales between the dominant and the JG relaxation process increases at elevated pressures. Surprisingly we find a change in the slope in the plot of log τ(JG) vs. 1/T for P = 1750 MPa. The results for the relaxation time of alcohols are compared with the Kirkwood correlation factor, g, and it is found that higher is the g, lower is the relaxation time for process I, and it is more of the Debye type. On a reduction in g brought about by an increase in pressure at lower temperatures, the dominant process becomes non-Debye though extensive hydrogen bonding is still present. The dielectric strength of the merged processes increases with increase in pressure. The values of the steepness index, m = |d log τ/d(T(g)/T)|(T = Tg) for processes I and II are different for P = 0.1 MPa. However the value of m, for the composite process, which is a merger of processes I and II, for P = 1750 MPa is almost the same for process II at P = 0.1 MPa. From the results of the activation volume, activation enthalpy, and a comparison of the relaxation times with the g factor, we conclude that both processes I and II are significantly affected by hydrogen bonding and both contribute to the structural relaxation.     

Interdependence of primary and Johari-Goldstein secondary relaxations in glass-forming systems

We report evidence from broadband dielectric spectroscopy that the dynamics of the primary alpha- and secondary Johari-Goldstein (JG) beta-processes are strongly correlated in different glass-forming systems over a wide temperature T and pressure P range, in contrast with the widespread opinion of statistical independence of these processes. The alpha-beta mutual dependence is quantitatively confirmed by (a) the overall superposition of spectra measured at different T-P combinations but with an invariant alpha-relaxation time; (b) the contemporary scaling of the isothermal-pressure and isobaric-temperature dependences of the alpha-and beta-relaxation times as plotted versus the reduced variable Tg(P)/T where Tg is the glass transition temperature. These novel and model-independent evidences indicate the relevance of the JG relaxation phenomenon in glass transition, often overlooked by most current theories.     

Fundamentals of ionic conductivity relaxation gained from study of procaine hydrochloride and procainamide hydrochloride at ambient and elevated pressure

The pharmaceuticals, procaine hydrochloride and procainamide hydrochloride, are glass-forming as well as ionically conducting materials. We have made dielectric measurements at ambient and elevated pressures to characterize the dynamics of the ion conductivity relaxation in these pharmaceuticals, and calorimetric measurements for the structural relaxation. Perhaps due to their special chemical and physical structures, novel features are found in the ionic conductivity relaxation of these pharmaceuticals. Data of conductivity relaxation in most ionic conductors when represented by the electric loss modulus usually show a single resolved peak in the electric modulus loss M(")(f) spectra. However, in procaine hydrochloride and procainamide hydrochloride we find in addition another resolved loss peak at higher frequencies over a temperature range spanning across T(g). The situation is analogous to many non-ionic glass-formers showing the presence of the structural α-relaxation together with the Johari-Goldstein (JG) β-relaxation. Naturally the analogy leads us to name the slower and faster processes resolved in procaine hydrochloride and procainamide hydrochloride as the primary α-conductivity relaxation and the secondary β-conductivity relaxation, respectively. The analogy of the β-conductivity relaxation in procaine HCl and procainamide HCl with JG β-relaxation in non-ionic glass-formers goes further by the finding that the β-conductivity is strongly related to the α-conductivity relaxation at temperatures above and below T(g). At elevated pressure but compensated by raising temperature to maintain α-conductivity relaxation time constant, the data show invariance of the ratio between the β- and the α-conductivity relaxation times to changes of thermodynamic condition. This property indicates that the β-conductivity relaxation has fundamental importance and is indispensable as the precursor of the α-conductivity relaxation, analogous to the relation found between the Johari-Goldstein β-relaxation and the structural α-relaxation in non-ionic glass-forming systems. The novel features of the ionic conductivity relaxation are brought out by presenting the measurements in terms of the electric modulus or permittivity. If presented in terms of conductivity, the novel features are lost. This warns against insisting that a log-log plot of conductivity vs. frequency is optimal to reveal and interpret the dynamics of ionic conductors.     

Pressure effects on the alpha and alpha' relaxations in polymethylphenylsiloxane

In some polymers, in addition to the usual structural alpha relaxation, a slower alpha' relaxation is observed with a non-Arrhenius temperature dependence. In order to understand better the molecular origin of this alpha' relaxation in poly(methylphenylsiloxane) (PMPS) we have studied, for the first time, the pressure dependence of its relaxation time, together with the usual temperature dependence, by means of dynamic light scattering (DLS). For the same material the alpha relaxation was also studied by means of DLS and dielectric spectroscopy (DS) in broad temperature and pressure ranges. We find that the temperature dependence of both alpha and alpha' relaxation times, at all pressures studied, can be described by a double Vogel-Fulcher-Tammann (VFT) law. The pressure dependence of the characteristic temperatures Tg (glass transition temperature) and T0 (Vogel temperature) as well as the activation volumes for both alpha and alpha' processes are very similar, indicating, that both relaxation processes originate from similar local molecular dynamics. Additionally, for both alpha and alpha' relaxations the combined temperature and pressure dependences of the relaxation times can be described using a parameter Gamma=rhon/T with the same value of the exponent n.     

Kinetics of spontaneous change in the localized motions of D-sorbitol glass

The dielectric relaxation spectra of D-sorbitol glass have been studied in real time during annealing at 221.1 K, which is 47 K below its T(g) of 268 K. As the glass structurally relaxes during annealing, features of the Johari-Goldstein (JG) relaxation change with time: (i) the relaxation strength decreases, (ii) the relaxation peak at 48 Hz shifts to a higher frequency, and (iii) the relaxation spectra become narrower. All seem to follow the relation p proportional, variant exp[-(kt)(n)], where p is the magnitude of a property, k the rate constant, and t the time. The parameter n may well be less than 1, but this could not be ascertained. It is proposed that shift of the relaxation peak to a higher frequency and narrowing of the relaxation spectra occur when local, loosely packed regions of molecules in the glass structure collapse nonuniformly and the relaxation time of some of the molecules in the collapsed state becomes too long to contribute to the JG-relaxation spectra. Consequently, the half width of the spectra decreases, and the relaxation peak shifts to a higher frequency. Molecules whose diffusion becomes too slow after the local regions' collapse would contribute to the alpha-relaxation spectra and thus the net relaxation strength would increase on structural relaxation. It is argued that these findings conflict with the NMR-based conclusions that motion of all molecules in the glass and supercooled liquid contributes to the faster relaxation process.     

Aging of the Johari-Goldstein relaxation in the glass-forming liquids sorbitol and xylitol

Employing frequency-dependent dielectric susceptibility we characterize the aging in two supercooled liquids, sorbitol and xylitol, below their calorimetric glass transition temperatures. In addition to the alpha relaxation that tracks the structural dynamics, the susceptibility of both liquids possesses a secondary Johari-Goldstein relaxation at higher frequencies. Following a quench through the glass transition, the susceptibility slowly approaches the equilibrium behavior. For both liquids, the magnitude of the Johari-Goldstein relaxation displays a dependence on the time since the quench, or aging time, that is quantitatively very similar to the age dependence of the alpha peak frequency. The Johari-Goldstein relaxation time remains constant during aging for sorbitol while it decreases slightly with age for xylitol. Hence, one cannot sensibly assign a fictive temperature to the Johari-Goldstein relaxation. This behavior contrasts with that of liquids lacking distinct Johari-Goldstein peaks for which the excess wing of the alpha peak tracks the main part of the peak during aging, enabling the assignment of a single fictive temperature to the entire spectrum. The aging behavior of the Johari-Goldstein relaxation time further calls into question the possibility that the relaxation time possesses stronger temperature dependence in equilibrium than is observed in the out-of-equilibrium state below the glass transition.     

Primary and secondary relaxations in supercooled eugenol and isoeugenol at ambient and elevated pressures: dependence on chemical microstructure

Dielectric loss spectra of two glass-forming isomers, eugenol and isoeugenol, measured at ambient and elevated pressures in the normal liquid, supercooled, and glassy states are presented. The isomeric chemical compounds studied differ only by the location of the double bond in the alkyl chain. Above the glass transition temperature T(g), the dielectric loss spectra of both isomers exhibit an excess wing on the high frequency flank of the loss peak of the alpha relaxation and an additional faster gamma process at the megahertz frequency range. By decreasing temperature below T(g) at ambient pressure or by elevating pressure above P(g), the glass transition pressure, at constant temperature, the excess wing of isoeugenol shifts to lower frequencies and is transformed into a secondary beta-loss peak, while in eugenol it becomes a shoulder. These spectral features enable the beta-relaxation time tau(beta) to be determined in the glassy state. These changes indicate that the excess wings in isoeugenol and eugenol are similar and both are secondary beta relaxations that are not resolved in the liquid state. While in both isoeugenol and eugenol the loss peak of the beta relaxation in the glassy state and the corresponding excess wing in the liquid state shifts to lower frequencies on elevating pressure, the locations of their gamma relaxation show little change with increasing pressure. The different pressure sensitivities of the excess wing and gamma relaxation are further demonstrated by the nearly perfect superposition of the alpha-loss peak together with excess wing from the data taken at ambient pressure and at elevated pressure (and higher temperature so as to have the same alpha-peak frequency), but not the gamma-loss peak in both isoeugenol and eugenol. On physical aging isoeugenol, the beta-loss peak shifts to lower frequencies, but not the gamma relaxation. Basing on these experimental facts, the faster gamma relaxation is a local intramolecular process involving a side group and the slower beta relaxation mimics the structural alpha relaxation in behavior, involves the entire molecule and satisfies the criteria for being the Johari-Goldstein beta relaxation. Analysis and interpretation of the spectra utilizing the coupling model further demonstrate that the excess wings seen in the equilibrium liquid states of these two isomers are their genuine Johari-Goldstein beta relaxation.     

Activated hopping and dynamical fluctuation effects in hard sphere suspensions and fluids

Single particle Brownian dynamics simulation methods are employed to establish the full trajectory level predictions of our nonlinear stochastic Langevin equation theory of activated hopping dynamics in glassy hard sphere suspensions and fluids. The consequences of thermal noise driven mobility fluctuations associated with the barrier hopping process are determined for various ensemble-averaged properties and their distributions. The predicted mean square displacements show classic signatures of transient trapping and anomalous diffusion on intermediate time and length scales. A crossover to a stronger volume fraction dependence of the apparent nondiffusive exponent occurs when the entropic barrier is of order the thermal energy. The volume fraction dependences of various mean relaxation times and rates can be fitted by empirical critical power laws with parameters consistent with ideal mode-coupling theory. However, the results of our divergence-free theory are largely a consequence of activated dynamics. The experimentally measurable alpha relaxation time is found to be very similar to the theoretically defined mean reaction time for escape from the barrier-dominated regime. Various measures of decoupling have been studied. For fluid states with small or nonexistent barriers, relaxation times obey a simple log-normal distribution, while for high volume fractions the relaxation time distributions become Poissonian. The product of the self-diffusion constant and mean alpha relaxation time increases roughly as a logarithmic function of the alpha relaxation time. The cage scale incoherent dynamic structure factor exhibits nonexponential decay with a modest degree of stretching. A nearly universal collapse of the different volume fraction results occurs if time is scaled by the mean alpha relaxation time. Hence, time-volume fraction superposition holds quite well, despite the presence of stretching and volume fraction dependent decoupling associated with the stochastic barrier hopping process. The relevance of other origins of dynamic heterogeneity (e.g., mesoscopic domains), and comparison of our results with experiments, simulations, and alternative theories, is discussed.