Nonequilibrium Glassy State Research Articles

Mixed equilibrium/nonequilibrium effects govern surface mobility in polymer glasses

Using angle-resolved X-ray photoelectron spectroscopy, sum-frequency generation vibrational spectroscopy, contact angle measurements, and molecular dynamics simulations, we verify that the glass transition temperature (Tg) of polymer glass is lower near the free surface. However, the experimental Tg-gradients showed a linear variation with depth (z) from the free surface, while the simulated equilibrium Tg-gradients exhibited a double exponential z-dependence. In typical simulations, Tg is determined based on the relaxation time of the system reaching a prescribed threshold value at equilibrium. Conversely, the experiments determined Tg by observing the unfreezing of molecular mobility during heating from a kinetically arrested, nonequilibrium glassy state. To investigate the impact of nonequilibrium effects on the Tg-gradient, we reduced the thermal annealing time in simulations, allowing the system to fall out of equilibrium. We observe a decrease in the relaxation time and the emergence of a modified z-dependence consistent with a linear Tg-gradient near the free surface. We further validate the impact of nonequilibrium effects by studying the dependence of the Tg on the heating/cooling rate for polymer films of varying thickness (h). Our experimental results reveal significant variations in the Tg-heating/cooling rate dependence with h below the bulk Tg, which are also observed in simulation when the simulated system is not equilibrated. We explain our findings by the reduction in mass density within the inner region of the system under nonequilibrium conditions, as observed in simulation, and recent research indicating a decrease in the local Tg of a polymer when placed next to a softer material.

Entropy Scaling of Molecular Dynamics in a Prototypical Anisotropic Model near the Glass Transition.

Dynamics and thermodynamics of molecular systems in the vicinity of the boundary between thermodynamically nonequilibrium glassy and metastable supercooled liquid states are still incompletely explored and their theoretical and simulation models are imperfect despite many previous efforts. Among them, the role of total system entropy, configurational entropy, and excess entropy in the temperature-pressure or temperature-density dependence of global molecular dynamics (MD) timescale relevant to the glass transition is an essential topic intensively studied for over half of a century. By exploiting a well-known simple ellipsoidal model recently successfully applied to simulate the supercooled liquid state and the glass transition, we gain a new insight into the different views on the relationship between entropy and relaxation dynamics of glass formers, showing the molecular grounds for the entropy scaling of global MD timescale. Our simulations in the anisotropic model of supercooled liquid, which involves only translational and rotational degrees of freedom, give evidence that the total system entropy is sufficient to scale global MD timescale. It complies with the scaling effect on relaxation dynamics exerted by the configurational entropy defined as the total entropy diminished by vibrational contributions, which was earlier discovered for measurement data collected near the glass transition. Moreover, we argue that such a scaling behavior is not possible to achieve by using the excess entropy that is in excess of the ideal gas entropy, which is contrary to the results earlier suggested within the framework of simple isotropic models of supercooled liquids. Thus, our findings also warn against an excessive reliance on isotropic models in theoretical interpretations of molecular phenomena, despite their simplicity and popularity, because they may reflect improperly various physicochemical properties of glass formers.

Universal features of annealing and aging in compaction of granular piles

This paper links the nonequilibrium glassy relaxation behavior of otherwise athermal granular materials to those of thermally activated glasses. Thus, it demonstrates a much wider universality among complex glassy materials out of equilibrium. Our three-dimensional molecular dynamics simulations, fully incorporating friction and inelastic collisions, are designed to reproduce experimental behavior of tapped granular piles. A simple theory based on a dynamics of records explains why the typical phenomenology of annealing and aging after a quench should extend to such granular matter, activated by taps, beyond the more familiar realm of polymers, colloids, and magnetic materials that all exhibit thermal fluctuations.

Reconciling computational and experimental trends in the temperature dependence of the interfacial mobility of polymer films.

Many measurements have indicated that thin polymer films in their glass state exhibit a mobile interfacial layer that grows in thickness upon heating, while some measurements indicate the opposite trend. Moreover, simulations and limited measurements on glass-forming liquids at temperatures above the glass transition temperature Tg exhibit a growing interfacial mobility scale ξ upon cooling. To better understand these seemingly contradictory trends, we perform molecular dynamics simulations over a temperature regime for which our simulated polymer film enters a non-equilibrium glassy state and find that the relaxation time τα within the film interior, relative to the polymer-air interfacial layer, exhibits a maximum near the computational Tg. Correspondingly, we also observe that the interfacial mobility length scale exhibits a maximum near Tg, explaining the apparent reversal in the temperature dependence of this scale between the glass and liquid states. We show that the non-monotonic variation of ξ and the relative interfacial mobility to the film interior arise qualitatively from a non-monotonic variation of the gradient of the effective activation free energy of the film; we then obtain a quantitative description of this phenomenon by introducing a phenomenological model that describes the relaxation time layer-by-layer in the film for a temperature range both above and below Tg of the film as a whole. This analysis reveals that the non-monotonic trend in the relative interfacial mobility and ξ both arise primarily from the distinctive temperature dependence of relaxation in the interfacial layer, which apparently remains in local equilibrium over the whole temperature range investigated.

Electron doping induced magnetic glassy state in phase separated YBaCo2O5.5-δ

The structural, magnetic and transport properties of the layered RBaCo$_2$O$_{5.5}$ cobaltites are sensitive to the oxygen stoichiometry. In this present study, we report the presence of a low-temperature magnetic glassy state in electron-doped polycrystalline YBaCo$_2$O$_{5.5}$ cobaltite. The ac magnetization studies show the absence of conventional spin glass features, while the various dc magnetization studies demonstrate the presence of non-equilibrium magnetic glassy state at low temperature. The magnetic glassy state of this sample results from the kinetic arrest of the first order ferro (or ferri) to antiferromagnetic transition. The role of electron doping in the occurrence of magnetic glassy phenomena is discussed in terms of magnetic phase separation involving the Co3+/Co2+ clusters of the ferrimagnetic phase in the Co3+/Co3+ antiferromagnetic matrix.

Statistical mechanics of topological fluctuations in glass-forming liquids

All liquids are topologically disordered materials, yet the degree of disorder can vary as a result of internal fluctuations in structure and topology. These fluctuations depend on both the composition and temperature of the system. Most prior work has considered the mean values of liquid or glass properties, such as the average number of topological degrees of freedom per atom; however, the localized fluctuations in properties also play a key role in governing the macroscopic characteristics. This paper proposes a generalized approach for modeling topological fluctuations in glass-forming liquids by linking the statistical mechanics of the disordered structure to topological constraint theory. In doing so we introduce the contributions of localized fluctuations into the calculation of the topological degrees of freedoms in the network. With this approach the full distribution of properties in the disordered network can be calculated as an arbitrary function of composition, temperature, and thermal history (for the nonequilibrium glassy state). The scope of the current investigation focuses on describing topological fluctuations in liquids, concentrating on composition and temperature effects.

Changes of the Molecular Mobility of Poly(ε-caprolactone) upon Drawing, Studied by Dielectric Relaxation Spectroscopy

Dielectric relaxation spectroscopy (DRS) of poly(e-caprolactone) with different draw ratios showed that the mobility of polymer chains in the amorphous part decreases as the draw ratio increases. The activation energy of the α process, which corresponds to the dynamic glass transition, increases upon drawing. The enlarged gap between the activation energies of the α process and the β process results in a change of continuity at the crossover between the high temperature a process and the α and β processes. At low drawing ratios the a process connects with the β process, while at the highest drawing ratio in our measurements, the a process is continuous with the α process. This is consistent with X-ray diffraction results that indicate that upon drawing the polymer chains in the amorphous part align and densify upon drawing. As the draw ratio increases, the α relaxation broadens and decreases its intensity, indicating an increasing heterogeneity. We observed slope changes in the α traces, when the temperature decreases below that at which τ α ≈ 1 s. This may indicate the glass transition from the ‘rubbery’ state to the non-equilibrium glassy state.

Characterization of the kinetic arrest of martensitic transformation in Ni45Co5Mn36.8In13.2 melt-spun ribbons

The kinetic arrest (KA) of martensitic transformation (MT) observed in Ni45Co5Mn36.8In13.2 melt-spun ribbons has been studied. These alloy ribbons show an ordered columnar-like grain microstructure with the longer grain axis growing perpendicular to ribbon plane and transform martensitically from a single austenitic (AST) parent phase with the L21-type crystal structure to a monoclinic incommensurate 6M modulated martensite (MST). Results show that the volume fraction of austenite frozen into the martensitic matrix is proportional to the applied magnetic field. A fully arrest of the structural transition is found for a magnetic field of 7T. The metastable character of the non-equilibrium field-cooled glassy state was characterized by introducing thermal and magnetic field fluctuations or measuring the relaxation of magnetization. The relaxation of magnetization from a field-cooled kinetically arrested state at 5 and 7T follows the Kohlrausch–Williams–Watts (KWW) stretched exponential function with a β exponent around 0.95 indicating the weak metastable nature of the system under the strong magnetic fields. The relationship between the occurrence of exchange bias and the frozen fraction of AST into the MST matrix was studied.

Relaxation dynamics of glasses along a wide stability and temperature range.

While lots of measurements describe the relaxation dynamics of the liquid state, experimental data of the glass dynamics at high temperatures are much scarcer. We use ultrafast scanning calorimetry to expand the timescales of the glass to much shorter values than previously achieved. Our data show that the relaxation time of glasses follows a super-Arrhenius behaviour in the high-temperature regime above the conventional devitrification temperature heating at 10 K/min. The liquid and glass states can be described by a common VFT-like expression that solely depends on temperature and limiting fictive temperature. We apply this common description to nearly-isotropic glasses of indomethacin, toluene and to recent data on metallic glasses. We also show that the dynamics of indomethacin glasses obey density scaling laws originally derived for the liquid. This work provides a strong connection between the dynamics of the equilibrium supercooled liquid and non-equilibrium glassy states.

Negligible ageing in poly(ether-ether-ketone) membranes widens application range for solvent processing

Organic solvent nanofiltration membranes have been prepared from poly(ether-ether-ketone) (PEEK) by phase inversion. Remarkably, these membranes undergo negligible ageing, even under extreme conditions of high temperature air annealing, and high temperature solute filtration with aggressive solvents. This negligible ageing of PEEK membranes is contrasted with substantial ageing of crosslinked polybenzimidazole (PBI) and polyimide (PI) membranes. After air annealing at 120°C, PBI and PI membranes become brittle and lose all permeance, whereas PEEK membranes remain flexible and retain a constant permeance of ~0.2Lh−1m−2bar−1 for tetrahydrofuran (THF). The structural change in PBI and PI membranes is attributed to polymer transition from a non-equilibrium glassy state towards an equilibrium state at which chain packing precludes permeation of solvent. High temperature filtrations in DMF up to 140°C for the three polymeric membranes showed PEEK performance to be stable and consistent with negligible ageing, whereas PBI and PI rapidly loose stability.

Modeling of carbon dioxide and water sorption in glassy polymers through PC-SAFT and NET PC-SAFT

In this work, perturbed-chain statistical associating fluid theory (PC-SAFT) and its combination with the NET-GP (non-equilibrium thermodynamics of glassy polymer) model were both used to better understand both CO2 and water sorption in glassy polymers. For CO2 sorption in poly(methyl methacrylate) (PMMA) the NET PC-SAFT model provides excellent predictions, reflective of the glassy state of this polymer. A poor result is obtained by the PC-SAFT model, which does not account for the non-equilibrium nature of such systems. However, it is a different case for water sorption in polylactic acid (PLA) and PMMA. The PC-SAFT model successfully fitted the water sorption isotherms at three different temperatures with only one fitting parameter, although both polymers were in a non-equilibrium glassy state. This is because the impact of excess free volume on water sorption is small. This PC-SAFT model is able to account for induced-association i.e. the hydrogen bonding between water and polymers that have no self-association. Satisfactory results could not be achieved if only self-association of water molecules was considered. On the other hand, the NET PC-SAFT model was unable to provide as good a fit to experimental data from the literature for water sorption in PMMA, even when two fitting parameters were employed. In this case, the final result was strongly dependent upon the experimental value of the polymer density used.

Dynamical heterogeneity of the glassy state.

We compare dynamical heterogeneities in equilibrated supercooled liquids and in the nonequilibrium glassy state within the framework of the random first order transition theory. Fluctuating mobility generation and transport in the glass are treated by numerically solving stochastic continuum equations for mobility and fictive temperature fields that arise from an extended mode coupling theory containing activated events. Fluctuating spatiotemporal structures in aging and rejuvenating glasses lead to dynamical heterogeneity in glasses with characteristics distinct from those found in the equilibrium supercooled liquid. The non-Gaussian distribution of activation free energies, the stretching exponent β, and the growth of characteristic lengths are studied along with the four-point dynamical correlation function. Asymmetric thermodynamic responses upon heating and cooling are predicted to be the result of the heterogeneity and the out-of-equilibrium behavior of glasses below Tg. Our numerical results agree with experimental calorimetry. We numerically confirm the prediction of Lubchenko and Wolynes in the glass that the dynamical heterogeneity can lead to noticeably bimodal distributions of local fictive temperatures during some histories of preparation which explains in a unified way recent experimental observations that have been interpreted as coming from there being two distinct equilibration mechanisms in glasses.

Comment on “Residual entropy and structural disorder in glass: A two level model and a review of spatial and ensemble vs. temporal sampling” by A. Takada, R. Conradt, and P. Richet [J. Non-Cryst. Solids, 360, 13 (2013)

We show that, for a two level system, the non-equilibrium (glassy state) entropy and the statistical (Gibbs) entropy – as defined by Takada et al. – are mathematically equivalent and therefore the equality between these two entropies shown by Takada et al. using simulations does not imply any new physics. In addition, we point out that the free energy of a glass, in the Takada et al. treatment, becomes less than the free energy of the liquid for all temperatures below the fictive temperature of the glass, a result that is clearly unphysical.

Water Sorption Thermodynamics in Glassy and Rubbery Polymers: Modeling the Interactional Issues Emerging from FTIR Spectroscopy

Water sorption thermodynamics has been effectively investigated in rubbery and glassy polymers using, respectively, an equilibrium lattice fluid model, originally introduced by Panayiotou et al. [Panayiotou, C.; Tsivintzelis, I.; Economou, I.G. Ind. Eng. Chem. Res.2007, 46, 2628], accounting for hydrogen bond (HB) interactions in the system (i.e., the nonrandom hydrogen bonding, NRHB, model), and an extension of this model to a nonequilibrium glassy state (i.e., nonequilibrium thermodynamics for glassy polymers (NETGP)) that follows the same line of thought adopted originally by Doghieri and Sarti [Doghieri, F.; Sarti, G.C. Macromolecules1996, 29, 7885] to develop the NETGP approach. NRHB and NETGP-NRHB models have been used to interpret water sorption thermodynamics respectively in polycaprolactone and in polyimides. Model predictions in terms of self- and cross-HB established in the system are compared with quantitative information gathered from in situ infrared spectroscopy experiments, exploiting the wealth of information provided by proper elaboration of spectroscopy data by means of 2D correlation techniques.

Effect of fragility on relaxation of density fluctuations in glass

Density fluctuations play a crucial role in governing the optical, mechanical, thermodynamic, and kinetic properties of glass. The relaxation of density fluctuations displays an inherently nonmonotonic behavior, yielding a minimum in fluctuations following a quench and isothermal hold. Here we investigate the impact of liquid fragility on the relaxation of density fluctuations in the nonequilibrium glassy state. While fragility has a direct impact on the kinetics of the relaxation process, the minimum level of density fluctuations is unaffected by changes in fragility alone. The magnitude of density fluctuations can be minimized by tailoring the thermal history of the glass.

Fictive Temperature and the Glassy State

Within the glass science community, it is common to describe the state of a glass in terms of a fictive temperature or a distribution of fictive temperatures. However, a number of different definitions of fictive temperature are being practiced in the field. Based on the previous literature, at least three definitions are possible: (a) microscopic, which entails a mapping of the nonequilibrium glassy structure to comparable equilibrium liquid structures; (b) macroscopic, by representing the property values of a glass in terms of equilibrium states with equivalent configurational property values; and (c) kinetic, where the fictive temperatures are used to represent the various relaxation modes within the glass. Of these, the first, microscopic definition offers the possibility of writing a simplified statistical mechanical model of the nonequilibrium glassy state in terms of a linear combination of equilibrium liquid states. However, in this paper, we show that the microscopic physics of a glass cannot, in general, be described in this manner. The fictive temperature description of the glassy state is rigorous only under special circumstances, such as the case of an infinitely fast quench through the glass transition regime. Nevertheless, the microscopic definition of fictive temperature can provide a reasonable description of ensemble‐averaged state properties such as enthalpy and molar volume, where there is a cancellation of errors. The concept of fictive temperature distribution cannot, however, capture accurately the fluctuations in enthalpy and molar volume in the glassy state. We also show that fictive temperature mapping does not provide an accurate description of the low‐temperature dynamics of glass. An alternative description of the glassy state in terms of an enthalpy landscape can account for the true statistical mechanics of the glassy state without relying on any notion of fictive temperature.

Mysteries of the glass transition

As someone who has long been interested in the glass transition and glassy-state kinetics, I would like to comment on some of the issues raised by James Langer in his Reference Frame column.I affirm Langer’s statement about healthy contentiousness. Whether or not the glass transition has thermodynamic roots definitely makes for exciting science. The reason some of us think thermodynamics is important is that we find it difficult to dismiss as co-incidences the similarities in the values of the kinetic temperature T 0 and the thermodynamic Kauzmann temperature T K. One common objection to the Kauzmann analysis, that an amorphous solid should not have zero entropy, can be assuaged by noting that the entropy at T K does not have to be zero, just very small.Langer briefly mentions the success of the simplistic Adam–Gibbs (AG) model in describing the dynamics of supercooled liquids. Its nonlinear extension into the glass-transition region and glassy state (NLAG) is also surprisingly successful. 1 1. G. W. Scherer,J. Amer. Ceram. Soc. 67, 504 (1984); I. M. Hodge, Macromolecules 20, 2897 (1987). https://doi.org/10.1021/ma00177a044 That extension is based on concepts introduced by several researchers over several decades: Simon Rekhson in 1994, George Scherer in 1984, Cornelius Moynihan in 1976, O. S. Narayanaswamy in 1971, and others. The successes of the NLAG model go far beyond expectations, and raise issues of their own. The resolution of these issues might provide important clues to a theoretical understanding of the glass transition. ▸As noted by Langer, the experimentally observed effective activation energy E(T) increases rapidly with decreasing temperature down to the glass-transition temperature T g, but it then decreases through the T g range until it reaches a constant value E(T g), so that glassy-state relaxation exhibits Arrhenius behavior. The singularity at T 0 noted by Langer only occurs in the equilibrium supercooled liquid state and not in the experimentally observed nonequilibrium glassy state. The change from non-Arrhenius to Arrhenius behavior at T g is well described by the NLAG model and its precursor, the Tool-Narayanaswamy-Moynihan model. ▸NLAG predicts a simple relation between the ratio T K/T g and an empirical constant that parameterizes the nonlinearity of the glass transition and glassy-state kinetics. This intriguing prediction needs to be independently confirmed or unambiguously refuted.▸The NLAG model, together with the plausible assumption that smaller localized activation energies Δµ enable the kinetic T g to get closer to the thermodynamic T K, generates many of the correlations captured by Angell’s fragility. In fact, the ratio T K/T g is an excellent metric that allows fragility to be applied to the glassy state.▸Estimated values of Δµ for canonical glasses are often comparable with rotational energy barriers in polymers, and ionic, covalent, and hydrogen bond strengths. In these cases the NLAG model is almost quantitatively accurate.▸Incorporation of a distribution in Δµ yields a respectable account 2,3 2. G. Sartor, E. Mayer, G. P. Johari, Biophys. J. 66, 249 (1994). https://doi.org/10.1016/S0006-3495(94)80774-X 3. I. M. Hodge, Biophys. J. 91, 993 (2006). https://doi.org/10.1529/biophysj.106.080796 of thermal manifestations of motions in hydrated proteins and B-DNA. 2,4 2. G. Sartor, E. Mayer, G. P. Johari, Biophys. J. 66, 249 (1994). https://doi.org/10.1016/S0006-3495(94)80774-X 4. J. L. Green, J. Fan, C. A. Angell, J. Phys. Chem. 98, 13780 (1994). https://doi.org/10.1021/j100102a052 The mean value for Δµ is comparable with hydrogen bond strengths, albeit with a large uncertainty, and the large standard deviation—30% of the average—is consistent with the insightful but qualitative analysis of Jennifer Green and coworkers. 4 4. J. L. Green, J. Fan, C. A. Angell, J. Phys. Chem. 98, 13780 (1994). https://doi.org/10.1021/j100102a052 The fact that NLAG gives a decent account of annealing in hydrated proteins and B-DNA strongly supports Austen Angell’s suggestion that the glass transition and protein dynamics have much in common. 5 5. C. A. Angell, Science 267, 1924 (1995). https://doi.org/10.1126/science.267.5206.1924 I share Langer’s belief that short-range interactions are probably the key. Since the current models accommodate a wide range of interactions, such as covalent, hydrogen, and ionic bonding, the glass-transition phenomenon is evidently insensitive to the details of those interactions. This generality is missing from too many theoretical attempts at explaining the problem. Perhaps the averaging of details is why the simplistic NLAG model is so successful.REFERENCESSection:ChooseTop of pageREFERENCES <<CITING ARTICLES1. G. W. Scherer,J. Amer. Ceram. Soc. 67, 504 (1984); Google Scholar I. M. Hodge, Macromolecules 20, 2897 (1987). https://doi.org/10.1021/ma00177a044 , Google ScholarCrossref, ISI, CAS2. G. Sartor, E. Mayer, G. P. Johari, Biophys. J. 66, 249 (1994). https://doi.org/10.1016/S0006-3495(94)80774-X , Google ScholarCrossref, CAS3. I. M. Hodge, Biophys. J. 91, 993 (2006). https://doi.org/10.1529/biophysj.106.080796 , Google ScholarCrossref, CAS4. J. L. Green, J. Fan, C. A. Angell, J. Phys. Chem. 98, 13780 (1994). https://doi.org/10.1021/j100102a052 , Google ScholarCrossref, CAS5. C. A. Angell, Science 267, 1924 (1995). https://doi.org/10.1126/science.267.5206.1924 , Google ScholarCrossref, ISI, CAS© 2008 American Institute of Physics.

Glassy aging with modified Kohlrausch-Williams-Watts form

In this paper, we address the question of whether aging in the nonequilibrium glassy state is controlled by the equilibrium alpha-relaxation process, which occurs at temperatures above T(g) . Recently, Lunkenheimer [Phys. Rev. Lett. 95, 055702 (2005)] proposed a model for the glassy aging data of dielectric relaxation using a modified Kohlrausch-Williams-Watts form exp[-(t(age)/tau(age)(beta(age))] . The aging time t(age) dependence of the relaxation time tau(age) is defined by these authors through a functional relation involving the corresponding frequency nu(t(age))=1/(2pi tau(age)) , but the stretching exponent beta(age) is the same as beta(alpha), the alpha-relaxation stretching exponent. We present here an alternative functional form for tau(age)(t(age)) directly involving the relaxation time itself. The proposed model fits the data of Lunkenheimer perfectly with a stretching exponent beta(age) different from beta(alpha) .

Change in the temperature dependence of segmental dynamics in deeply supercooled polycarbonate

The equilibrium and nonequilibrium segmental dynamics of bisphenol-A-polycarbonate (PC) have been investigated near the glass transition by studying the rotational and translational diffusion of probe molecules in the polymer matrix. The temperature dependence of equilibrium dynamics was studied from 149 to 124 °C. We report a change in the temperature dependence of equilibrium molecular motion near 134 °C. Below this temperature, the activation energy of both the translational diffusion coefficient DT and the rotational correlation time τc decreases by nearly a factor of 3. This change is not caused by the transition to the nonequilibrium glassy state, but occurs in equilibrium. Additionally, we present evidence that the equilibrium dynamics of PC are spatially heterogeneous. Translational diffusion of rubrene shows an enhancement of 2.5 decades over what would be expected in a homogeneous system at Tg and the mean-square displacement of tetracene translation is not linear with time at low temperatures. In the nonequilibrium glassy state, differences in the physical aging behavior of rotational and translational observables suggest that physical aging dynamics are spatially heterogeneous. During isothermal aging, regions of faster mobility initially age towards equilibrium faster than regions of slower mobility.

On the application of the Adam–Gibbs equation to the non-equilibrium glassy state

A new extension of the Adam–Gibbs (AG) equation to the non-equilibrium glassy state is presented, whereby the configurational entropy is a function of both fictive temperature and thermodynamic temperature. The relative contributions of each are determined by an entropic non-linearity parameter x s analogous to the non-linearity parameter x of the Tool–Narayanaswamy–Moynihan (TNM) equation. This new form of the AG equation resolves many inconsistencies in the TNM approach: for example, the dependence of x on annealing temperature and time observed by many workers, as well as often unreasonable values for x and for T 2. Furthermore, it can rationalise the frequently reported correlation between x and β, the KWW non-exponentiality parameter. A physical interpretation of x s is made in terms of the Gibbs–DiMarzio theory and the relative importance of the freezing-in of either holes or of high energy bond conformations.