Tool-Narayanaswamy-Moynihan Research Articles

A novel approach for Tool-Narayanaswamy-Moynihan model parameter extraction using multi-scale neural model

The accurate determination of parameters in the Tool-Narayanaswamy-Moynihan (TNM) model, which describes the viscoelastic behavior of glass-forming materials, is crucial for predicting material responses through various thermal histories. Traditional methods rely heavily on curve-fitting techniques; however, these often fail due to noise in the data. Furthermore, traditional methods are computationally intensive and prone to inaccuracies, particularly when dealing with complex datasets or when the initial parameter guesses are far from optimal; also, they require a skilled personnel.In this study, we propose the application of a multi-scale convolutional neural network (MCNN) as a machine learning approach to address these challenges. The MCNN model is trained on a comprehensive simulated dataset encompassing a wide range of TNM parameters, allowing it to learn intricate patterns and dependencies within the data that are difficult to capture with conventional methods. Our results show that the MCNN significantly improves the accuracy of the parameter estimations for β and x across the entire spectrum of tested conditions, achieving performance that is not only comparable to, but often surpasses, traditional curve-fitting methods. Furthermore, the MCNN demonstrates superior robustness when initial parameter estimates are suboptimal or when the dataset exhibits significant noise. Although the prediction accuracy for the activation energy Δh∗ and the pre-exponential factor log(A) was somewhat lower, the method still provides valuable estimates that can be refined with supplementary techniques.This work highlights the potential of machine learning approaches like MCNN to revolutionize the parameter extraction process in complex physical models, reducing the reliance on manual curve-fitting and providing a more automated, scalable solution. We also analyze the primary sources of prediction errors in the MCNN outputs and offer insights into future improvements, including model architecture refinements and the integration of additional physical constraints. Our findings suggest that this approach can be extended to other domains where similar models are employed, paving the way for broader applications of machine learning in materials science.

Remarkable difference in structural relaxation dynamics of conventionally prepared bulk glass and vapor-deposited thin films.

The structural relaxation dynamics of conventionally prepared bulk glass of N,N'-bis(3-methylphenyl)-N,N'-diphenyl-benzidine (TPD) was measured by differential scanning calorimetry. The calorimetric data were quantitatively described in terms of the Tool-Narayanaswamy-Moynihan (TNM) model. The TNM parameters were evaluated using a combination of linearization and non-linear optimization methodologies: h*/R = 109.5 kK, ln(A/s) = -321, β = 0.37, x = 0.64. In addition, the TNM phenomenology was used to describe recently reported normalized thickness data of stable and aged thin TPD films measured by ellipsometry. The structural relaxation was found to proceed at a markedly higher rate in these thin films prepared by the physical vapor deposition compared to that of conventional bulk glass. This feature appears to be associated with the significantly narrower distribution of relaxation times (β ≅ 0.8) observed for stable thin film in "as-deposited" form with uniquely dense molecular packing. Interestingly, very similar attributes of the relaxation kinetics were also found in the aged thin film with a previously erased thermal history associated with the deposition.

Structural interpretation of the viscous flow and relaxation kinetics in the As-Se and Ge-Se chalcogenide systems

This study investigates the behaviour of As₂₀Se₈₀ and Ge₂₅Se₇₅ chalcogenide glasses near the glass transition temperature using differential scanning calorimetry (DSC) and thermomechanical analysis (TMA), addressing a gap in the literature for these compositions. Precise TMA measurements of viscosity in the range of 10⁶ – 10¹³ Pa·s clarified existing data and improved the quality of fits describing the temperature-dependent viscosity for both compositions. The DSC method enabled a comprehensive analysis of structural relaxation behaviours, which were described using the Tool-Narayanaswamy-Moynihan (TNM) model with the determination of its four parameters. The study discusses the viscosity and structural relaxation behaviours of both glass-formers in relation to their structure, referencing previous results for other binary chalcogenide glasses in the As-Se and Ge-Se systems. The findings indicate significant changes in viscosity and activation energy of structural relaxation with the addition of As or Ge, while the non-linearity and non-exponentiality parameters remain largely unaffected.

A model of heterogeneous undercooled liquid and glass accounting for temperature-dependent nonexponentiality and enthalpy fluctuation.

Dynamic heterogeneity is a fundamental characteristic of glasses and undercooled liquids. The heterogeneous nature causes some of the key features of systems' dynamics such as the temperature dependence of nonexponentiality and spatial enthalpy fluctuations. Commonly used phenomenological models such as Tool-Narayanaswamy-Moynihan (TNM) and Kovacs-Aklonis-Hutchinson-Ramos fail to fully capture this phenomenon. Here we propose a model that can predict the temperature-dependent nonexponential behavior observed in glass-forming liquids and glasses by fitting standard differential scanning calorimetry curves. This model extends the TNM framework of structural relaxation by introducing a distribution of equilibrium fictive temperature (Tfe) that accounts for heterogeneity in the undercooled liquid. This distribution is then frozen at the glass transition to account for the heterogeneous nature of the glass dynamics. The nonexponentiality parameter βKWW is obtained as a function of temperature by fitting the Kohlrauch-Williams-Watts (KWW) equation to the calculated relaxation function for various organic and inorganic undercooled liquids and glasses. The calculated temperature dependent βKWW shows good agreement with the experimental ones. We successfully model the relaxation dynamics far from equilibrium for two silicate systems that the TNM model fails to describe, confirming that temperature dependent nonexponentiality is necessary to fully describe these dynamics. The model also simulates the fluctuation of fictive temperature δTf during isothermal annealing with good qualitative agreement with the evolution of enthalpy fluctuation reported in the literature. We find that the evolution of enthalpy fluctuation during isothermal annealing heavily depends on the cooling rate, a dependence that was not previously emphasized.

Kinetics of physical aging of a silicate glass following temperature up- and down-jumps.

In this article, we investigate the structural relaxation of lithium silicate glass during isothermal physical aging by monitoring the temporal evolution of its refractive index and enthalpy following relatively large (10-40 °C) up- and down-jumps in temperature. The Kohlrausch-Williams-Watts function aptly describes the up- and down-jump data when analyzed separately. For temperature down-jumps, the glass exhibits a typical stretched exponential kinetic behavior with the non-exponentiality parameter β < 1, whereas up-jumps show a compressed exponential behavior (β > 1). We analyzed these datasets using the non-exponential and non-linear Tool-Narayanaswamy-Moynihan (TNM) model, aiming to provide a comprehensive description of the primary or α-relaxation of the glass. This model described both up- and down-jump datasets using a single value of β ≤ 1. However, the standard TNM model exhibited a progressively reduced capacity to describe the data for larger temperature jumps, which is likely a manifestation of the temperature dependence of the non-exponentiality or non-linearity of the relaxation process. We hypothesize that the compressed exponential relaxation kinetics observed for temperature up-jumps stems from a nucleation-growth-percolation-based evolution on the dynamically mobile regions within the structure, leading to a self-acceleration of the dynamics. On the other hand, temperature down-jumps result in self-retardation, as the slow-relaxing denser regions percolate in the structure to give rise to a stretched exponential behavior.

Kinetic criterion for triple-shape memory effect in amorphous polymers undergoing heterogeneous glass transitions

The triple-shape memory effect (triple-SME) in amorphous polymers arises from the heterogeneous glass transitions of different components. However, due to the complex composition and structural relaxation of triple-shape memory polymers (triple-SMPs), existing models struggle to characterize the shape memory temperature range for each component and its dependence on thermal history. This presents a challenge in developing a kinetic criterion to judge the occurrence of the triple-SME. To address this issue, we propose a kinetic phase transition model to predict the shape memory temperature range for each transition phase in triple-SMPs under different thermal histories. By combining the Tool-Narayanaswamy-Moynihan (TNM) model with the Adam-Gibbs theory, the effect of cooling rate during programming on subsequent triple-SME is investigated. Subsequently, the effectiveness of the proposed model is evaluated by applying it to predict the shape memory performance and thermomechanical behavior of SMPs with dual- and triple- SMEs under different thermal histories. Using this modeling strategy enables the calculation of the critical isothermal shape recovery time for each transition phase under different thermal histories, effectively preventing shape recovery overlap. Consequently, the proposed model is expected to provide theoretical guidance for designing SMPs with triple-SME and promoting their application in engineering.

A Phenomenological Model for Enthalpy Recovery in Polystyrene Using Dynamic Mechanical Spectra.

This paper studies the effects of annealing time on the specific heat enthalpy of polystyrene above the glass transition temperature. We extend the Tool-Narayanaswamy-Moynihan (TNM) model to describe the endothermic overshoot peaks through the dynamic mechanical spectra. In this work, we accept the viewpoint that the enthalpy recovery behavior of glassy polystyrene (PS) has a common structural relaxation mode with linear viscoelastic behavior. As a consequence, the retardation spectrum evaluated from the dynamic mechanical spectra around the primary Tg peak is used as the recovery function of the endothermic overshoot of specific heat. In addition, the sub-Tg shoulder peak around the Tg peak is found to be related to the structural relaxation estimated from light scattering measurements. The enthalpy recovery of annealed PS is quantitatively described using retardation spectra of the primary Tg, as well as the kinetic process of the sub-Tg relaxation process.

Non-linear physical aging of supercooled glycerol induced by large upward ideal temperature steps monitored through cooling experiments.

The physical aging of supercooled glycerol induced by upward temperature steps of amplitude reaching 45K was studied by a new method consisting in heating a micrometer-thick liquid film at a rate of up to 60 000 K/s, holding it at a constant high temperature for a controlled duration before letting it quickly cool down to the initial temperature. By monitoring the final slow relaxation of the dielectric loss, we were able to obtain quantitative information on the liquid response to the initial upward step. The so-called TNM (Tool-Narayanaswamy-Moynihan) formalism provided a good description of our observations despite the large distance from equilibrium, provided that different values of the nonlinearity parameter were used for the cooling phase and for the (much further from equilibrium) heating phase. In this form, it allowed to precisely quantify how to design an ideal temperature step, i.e., where no relaxation occurs during the heating phase. It helped bringing a clear physical understanding of how the (kilosecond long) final relaxation is related to the (millisecond long) liquid response to the upward step. Finally, it made possible the reconstruction of the fictive temperature evolution immediately following a step, evidencing the highly non-linear character of the liquid response to such large amplitude temperature steps. This work illustrates both the strengths and limitations of the TNM approach. This new experimental device offers a promising tool to study far-from-equilibrium supercooled liquids through their dielectric response.

The peak‐shift method for the description of structural relaxation in glassy materials: a critical review

AbstractNowadays, the glass transition kinetics is most commonly described in terms of the Tool–Narayanaswamy–Moynihan (TNM) model. Evaluation of one of the most prominent features of the structural relaxation motions—the relaxation nonlinearity—was traditionally done using the peak‐shift (PS) method. This paper introduces, for the first time, extensive testing of the PS method by means of theoretical simulations for all practically observable types of structural relaxation behavior and all types of glassy/amorphous materials (tested range of the glass transition temperatures: −55 to 1000°C; tested range of the relaxation activation energies: 300–1300 kJ⋅mol−1). For the majority of types of structural relaxation behavior, the PS method tends to slightly overestimate the value of the TNM nonlinearity parameter x (by ∼ 0.05–0.10). In the specific cases of the highly linear behavior (↑ x) combined with a broad distribution of relaxation times, the PS method systematically vastly underestimates the value of x. A new, improved temperature program was proposed for the PS method, eliminating the major intrinsic drawback of the originally proposed version of the PS methodology. In addition, based on the comparison between the theoretically simulated and real‐life experimental data, a new approach (based on the characteristic PS dependence shape) was introduced to estimate the width of the relaxation times distribution.

Structural relaxation in chalcogenide glasses

AbstractThe structural relaxation of chalcogenide glasses is discussed within Tool–Narayanaswamy–Moynihan (TNM) formalism. The TNM parameters for more than 70 different glassy compositions are compared on the basis of the relaxation rate defined as Rf(ΔT) = −(dTf/dlogt)i at the inflection point of the isothermal relaxation curve plotted on a logarithmic timescale. The Rf(ΔT) depends on the TNM parameter ß and the parameter σ, combining the nonlinearity parameter x, the effective activation energy h* or the fragility m. It is shown that Rf(10) estimated at 10 K below Tg is useful for the prediction of structural relaxation kinetics in different amorphous materials. The chalcogenide glasses are, for example, compared with oxide glasses and organic polymers. For all these materials, the Rf(10) versus σ plot shows a well‐defined pattern that is thoroughly discussed.

Physical Aging of an Epoxy Resin for Civil Engineering Applications from Fast Scanning Calorimetry Investigations

AbstractThis works deals with the structural relaxation of an epoxy resin based on bisphenol A diglycidyl ether polymer (DGEBA) and 4,7,10‐Trioxa‐1, 13‐tridecanediamine (TTDA) hardener. The physical aging experiments are performed in situ using fast scanning calorimetry (FSC). The profile of the relaxation enthalpy with temperature is obtained close to the glass transition by annealing the resin during 10 min at various aging temperatures. From this profile, the structural relaxation is probed at various distances from the glass transition temperature, with aging times ranging from 0 to 1000 min. The Tool–Narayanaswamy–Moynihan (TNM) model is employed to analyze the relaxation data.

Effect of lithium doping on the glass transition behavior of the Bioglass 45S5

Differential scanning calorimetry, thermomechanical analysis and Raman spectroscopy were used to study the role of the Li2O and P2O5 oxides on the structural relaxation phenomena in the Li-doped 45S5 Bioglass. The glass transition behavior was found to be similar for the enthalpy and volume manifestations of the relaxation motions. The Tool-Narayanaswamy-Moynihan (TNM) model was applied. An improved simulation-comparative methodology was used to determine the model relaxation parameters. The TNM model parameters and the selected glass transition characteristics (the glass transition temperature and the coefficients of thermal expansion) showed no statistically significant correlation with the particular elements present in the synthesized glasses. Hence, the information about the glasses composition was used to calculate the theoretical amounts of the QnSi species in the Li-doped 45S5 glasses. It was found that practically all explored physico-chemical quantities related to the glass transition kinetics show a correlation with the content of Q2Si species that are responsible for the formation of the chain-like structures in the present glasses.

Viscosity and structural relaxation of silver-doped (GeS2)50(Sb2S3)50

The viscosities of Agx[(GeS2)50(Sb2S3)50]100-x glasses and undercooled melts for x = 5, 7.5, and 15 are studied in the range of 106–1013 Pa·s. The measurements are recorded using a thermomechanical analyzer and two experimental setups, the parallel-plate and penetration methods. The structural relaxation of the composition Ag10[(GeS2)50(Sb2S3)50]90 is studied using differential scanning calorimetry and thermomechanical analysis. The enthalpy and volume relaxation are characterized and described using the Tool-Narayanaswamy-Moynihan (TNM) model. It is verified that both the enthalpy and volume relaxation can be described using similar values of the TNM model parameters. The parameters characterizing the viscous flow and structural relaxation, and their relationship, are discussed.

Anomalous structural recovery in the near glass transition range in a polymer glass: Data revisited in light of temperature variability in vacuum oven‐based experiments*

AbstractThere is considerable interest in data reported by Cangialosi et al. [Cangialosi, D., Boucher, V. M., Alegría, A., &amp; Colmenero, J. (2013). Physical Review Letters, 111(9), 095701] that purportedly showed unusual irregular responses in the isothermal structural recovery behavior of glasses because the observation challenges the general view of the dynamics being smooth functions of time as observed in extensive dilatometric experiments by Kovacs [Kovacs, A. J. (1964). In Fortschritte der Hochpolymeren‐Forschung, 3, 394–507. Springer, Berlin, Heidelberg.] in the 1960s. The reported data show an apparent two mechanism structural (enthalpy) recovery at aging temperatures ranging from 6 to 17 K below which is also controversial as it could not be reproduced in work from Koh and Simon [Koh, Y. P., &amp; Simon, S. L. (2013). Macromolecules, 46(14), 5815–5821.] where a second plateau observed in the enthalpy loss curve was not reproduced in nearly 1‐year‐long experiments at 15 K below . Therefore, it is important to determine what might be possible reasons for the apparent non‐smooth relaxations. Here, we examine the possibility that the poor temperature control of a typical vacuum oven could explain the anomalous results. We used the Tool–Narayanaswamy–Moynihan (TNM) model of structural recovery to calculate the effect of typical vacuum oven temperature variation on the structural recovery of polystyrene and find that we can reproduce the reported experimental results. The issue of temperature control using a vacuum oven is discussed, and data from thermocouple measurements of temperature at various locations inside a vacuum oven are shown.

Numerical simulation of the structural relaxation characteristics of K-LCV161 glass

Only a few glass structure relaxation parameters are applicable for finite element simulation. The structural relaxation characteristics of K-LCV161 glass were described using the Tool–Narayanaswamy–Moynihan (TNM) model in multiple differential scanning calorimetry(DSC) experiments. First, the proposed method of narrowing the fitting parameter range was based on the influence of TNM model parameters on DSC curve. The position of the DSC curve peak is affected by changing Δh*/R and A, the height of the DSC curve peak is affected by changing x and β. The least squares fitting method to solve the TNM model parameters was then systematically presented systematically presented, and the following results were obtained: ∆h*/R = 52,394.91 K, A = 6.15e−31 s, x = 0.77, β=0.70. Finally, the volume change of K-LCV161 glass during cooling was predicted by using the thermal expansion and structural relaxation models, and the difference between these two models was approximately 23%.

Physical Aging of Zr65Cu17.5Ni10Al7.5 Glassy Alloy

Physical aging of Zr65Cu17.5Ni10Al7.5 glassy alloy has been studied after holding its samples with variation of time and temperature of annealing. The results showed that the enthalpy and entropy decreased with time according to a non-exponential kinetics. The Differential Scanning Calorimetry (DSC) heating scan after annealing was analyzed using Tool-Narayanaswamy-Moynihan (TNM) model. The analysis showed that a set of parameters that fitted the heating scan data for one annealing time did not fit the heating scan data for another annealing time. This is explained due to approximations used in the model and the effect of Johari-Goldstein (JG) relaxation to the physical aging.

Correlation between the structure and relaxation dynamics of (GeS2)y(Sb2S3)1 − y glassy matrices

Differential scanning calorimetry was used to study enthalpy relaxation kinetics of the (GeS2)y(Sb2S3)1−y glasses in the y=0.1–0.9 compositional range. The relaxation behavior was described in terms of the phenomenological Tool-Narayanaswamy-Moynihan (TNM) model. Non-linear optimization procedure was used to determine unique set of relaxation parameters for each studied glass. With increasing GeS2 content the activation energy of enthalpy relaxation was found to steeply decrease whereas the glass transition temperature increased significantly. Both non-linearity and non-exponentiality features were not affected by compositional changes. Raman spectra have shown that the Ge-S structural units, as opposed to the Sb-S units, are significantly interconnected and create a 3-dimensional network capable of solely carrying the relaxation motions, which is in agreement with the interpretation of compositional evolution of relaxation parameters. Moreover, based on the present data the controversy regarding the (GeS2)y(Sb2S3)1−y fragilities found in literature was clarified, and the free-volume concept employing the Avrami growth theory was disputed.

Thermal characterization of (As2Se3)0.5(As2Te3)0.5 infrared glass

The structural relaxation and crystallization kinetics of the (As2Se3)0.5(As2Te3)0.5 chalcogenide glass was studied by means of differential scanning calorimetry (DSC) and the influence of experimental conditions (particle size, heating rate) was observed. The Tool-Narayanaswamy-Moynihan (TNM) model was used for description of structural relaxation data. The structure of the studied glass was found to be similar to its purely selenide analogue (As2Se3), but the Te atoms appear to randomly incorporate into the short Se-As-Te chains; also they coexist in AsTe3/2 pyramidal units alongside the AsSe3/2units. In case of the description of the crystallization kinetics, the nucleation-growth Johnson-Mehl-Avrami (JMA) model and the semi-empirical autocatalytic AC(M,N) model were tested on experimental DSC data. The suitability of JMA model was not confirmed due to the complexity of crystallization, therefore the AC(M,N) model was applied. These results were supported by the structural information from XRD analysis, Raman spectroscopy and infrared (IR) microscopy. The presence of As-Te, As-Se and As2Se(Te)3 phases was confirmed and the occurrence of tellurium in the structure has a huge influence on overall structure and subsequent thermal behavior of the (As2Se3)0.5(As2Te3)0.5 glassy material.

Revising the formulization of the Narayanaswamy equation using the Adam-Gibbs theory

Hodge's formulization of the Tool-Narayanaswamy-Moynihan (TNM) equation from the Adam-Gibbs (AG) theory was re-evaluated. The non-linearity parameter (x) and the apparent activation energy (Δh*) of the TMN equation were re-derived. Compared with the original derivations, the revised values of x and Δh* were better correlated with experimental results. Of particular importance, the revised theoretical x correctly predicts the experimental trend of x for the cooling rate of polystyrene, whereas the original derivation predicted an opposite trend. Furthermore, the new derivation establishes a strong relationship between the fragility coefficient (m) and the x of a material. The relationship correlates well with the experimental values for a wide range of materials. Ultimately, the revised equations for x and Δh* more precisely reveal the theoretical foundation of the phenomenological TNM equation, as it relates to structural relaxation of polymeric materials.

Stress Relaxation and Refractive Index Change of As2S3 in Compression Molding

As2S3 is one of the chalcogenide glasses that have attracted increasing interests for compression molding applications. This article aimed to evaluate the stress relaxation behavior of As2S3 above its glass transition temperature and calculate its refractive index change during cooling. First, creep tests were conducted with cylinder glass specimens at three different temperatures, in order to deduce the shear stress relaxation function by using the relationship with creep compliance function. In addition, the shift factor for thermo‐rheological simplicity using Williams–Landel–Ferry equation was obtained. Then, finite element simulation was implemented to verify the calculated shear stress relaxation function. The acquired shear stress relaxation function needs to be modified to compensate the influence of friction on the thickness change in the experiments, so that the simulation results using the modified shear stress relaxation would match the experiments better. Finally, the refractive index changes of As2S3 at different cooling rates were modeled by using the Tool–Narayanaswamy–Moynihan (TNM) model for structural relaxation behavior. It is confirmed that the slower the cooling rate is, the less the refractive index drop will be. It was also demonstrated that the refractive index drop is strictly dependent on the cooling rate logarithmically by using TNM model. In summary, the results presented in this article can provide reliable references for viscoelastic characterization of As2S3 glass, crucial for compression molding or similar applications.