Ultra-slow aging dynamics of glass and its application to geological dating

Domain: physics
Published: 2025-11-26
Editorial outcome: minor_revision

Abstract
The ultra-slow relaxation dynamics of glasses at ambient temperature provide a promising alternative for dating glasses with extremely low isotopic content that cannot be dated using traditional radiometric methods. However, these ultra-slow, nonlinear aging dynamics remain poorly understood due to the lack of accurate theoretical models and long-term experimental validation. Existing equilibrium-based dynamics models substantially overestimate relaxation times at temperatures far below the glass transition temperature, making it difficult to model and quantify non-equilibrium aging over geological timescales. We address this challenge by formulating an empirical equation that quantifies the non-equilibrium effective relaxation time (τ_eff) for various glasses, including metallic glasses, organic amber, and lunar glasses. Our findings demonstrate a universal nonlinear aging dynamics governed by a single τ_eff, which follows a robust empirical relation parameterized by aging temperature and material-specific fragility. Employing this equation, we propose a universal glass kinetics dating method, conceptually analogous to radioactive decay, where τ_eff serves as a material-specific decay constant. This approach enables dating of glassy materials over timescales spanning decades to billions of years. This work bridges a fundamental gap in glass aging theory and establishes a practical framework for dating geological and planetary glasses.

Introduction
Glasses, although macroscopically solid and apparently static, are fundamentally out-of-equilibrium systems that evolve continuously over time1–3. This structural evolution, known as physical aging, arises from an extremely slow relaxation of the disordered structure toward lower energy configurations. At temperatures near the glass transition temperature (Tg), the aging process has been extensively studied, with clear manifestations such as enthalpy recovery4,5, volume relaxation6, and mechanical stiffening7,8. However, at temperatures far below Tg—such as ambient or geological conditions—the dynamics of aging slow so dramatically that relaxation becomes effectively undetectable within conventional experimental timeframes. Yet mounting evidence challenges the long-held assumption that glasses are entirely “frozen” in this regime. Observable aging effects have been reported over timescales ranging from decades to billions of years, in systems including metallic glass (MG)5, polymer9–11, and silicate glasses12,13. This recognition raises a central question in non-equilibrium physics: how can we quantitatively describe and predict the dynamics of disordered systems evolving over geological timescales?

One of the most remarkable features of glass aging dynamics is its inherent nonlinearity8,14,15. In equilibrium supercooled liquids, the relaxation behavior in response to perturbations (e.g., subtle temperature jump) can be well described by a stretched exponential Kohlrausch-Williams-Watts (KWW) function16,17: ϕ(t)=exp[−(t/τ_eq)^β], where ϕ(t)=[p(t)−p(∞)]/(p(0)−p(∞)) is the normalized relaxation function, and p(t) is the measured quantity (e.g., enthalpy, density, modulus). And τ_eq here is the equilibrium relaxation time which does not depend on aging time. Unlike equilibrium supercooled liquids, non-equilibrium glasses exhibit progressively slower and history-dependent relaxation. To describe this nonlinear aging dynamics, the concept of material time is introduced14,15,18,19, and a nonlinear integral function is required8:

ϕ(t)=exp[−ξ^β]=exp[−(∫_0^t dt'/τ_neq(t'))^β]

where ξ(t)=∫_0^t dt'/τ_neq(t') is the dimensionless material time, and τ_neq is the instantaneous relaxation time which is not fixed but evolves continuously with aging time. This nonlinear integral form introduces substantial complexity, as the time-dependent nature of τ_neq(t) prevents a closed-form analytical solution, requiring numerical integration or approximation methods.

This mathematical complexity reflects a deeper conceptual challenge: accurately capturing the nonlinear, time-dependent nature of τ_neq(t) in theoretical models. However, most widely used models—including the Arrhenius, Vogel-Fulcher-Tammann (VFT)20–22, Parabolic23, and MYEGA relations24—were developed to capture equilibrium or near-equilibrium dynamics. When extrapolated to deep sub-Tg regimes, they often yield vastly overestimated relaxation times, in some cases even exceeding the age of the universe. Even frameworks explicitly designed for non-equilibrium behavior, such as the Narayanaswamy-Moynihan (NM) model18,25, treat relaxation as quasi-equilibrium and fail to fully account for the nonlinear growth of τ_neq(t). As a result, theoretical predictions consistently overestimate effective relaxation times by several orders of magnitude compared to experimental observations. For example, long-term studies on MGs, polymers, and oxide glasses—including those by Duan et al.26, Zhao et al.5, Perez-De Eulate et al.9, and Corning Incorporated12—have shown that the measured relaxation times are several orders of magnitude shorter than NM-based predictions. These discrepancies highlight a critical limitation in current theoretical limitation: the failure to completely incorporate nonlinear effects that dominate ultra-slow aging behavior.

This fundamental knowledge gap in glass physics has profound implications for Earth and planetary sciences. Natural glasses, such as obsidian, tektites, organic amber, and lunar glasses, are widespread in geological and extraterrestrial environments. They are often the only available amorphous records of thermal, impact, or volcanic events. However, some lunar glasses contain K or Pb at concentrations too low to produce measurable isotope signals27–30, while amber contains 14C but its short half-life (~5730 years)31 restricts dating to at most several tens of thousands of years, far shorter than the million of years typically required. As a result, glasses with extremely low isotope abundances cannot be reliably dated using traditional radiometric techniques, which calls for the development of alternative dating methods. The continuous evolution of physical properties in these aging glasses suggests an alternative: a kinetics-based dating approach that estimates age through intrinsic structural relaxation dynamics rather than isotopic decay. Such a method could unlock the chronology of otherwise undatable glassy materials, providing new tools for reconstructing geological history, tracing planetary surface processes, and interpreting extraterrestrial sample returns.

In this study, we address these challenges by formulating an empirical equation that quantifies the effective relaxation time (τ_eff) of glasses in non-equilibrium aging regimes at deep sub-Tg temperatures. Through systematic analysis of diverse glassy materials—including Ce-based MG, fossilized amber, and lunar glasses—we demonstrate that their nonlinear relaxation dynamics can be unified through a single τ_eff, governed by aging temperature and material-specific fragility. The resulting empirical relation captures essential features of ultra-slow aging across multiple classes of glasses and reveals a universal behavior that serves as a kinetic fingerprint. Based on this framework, we propose a glass kinetics dating method, conceptually analogous to radioactive decay, in which τ_eff serves as a material-specific decay constant. This approach provides a quantitative chronometer for glassy materials, applicable across timescales spanning decades to billions of years.

Results
Glass kinetics decay method: nonlinear features of aging
The introduction of material time is crucial for understanding and analyzing the aging dynamics of glass14,15,18,19. In the special case where τ_neq(t) is constant, the material time ξ(t) scales linearly with the real time t, i.e., ξ(t)∝t. However, during aging process, τ_neq(t) typically increases with t, hence ξ(t) is nonlinear with t. Struik32 systematically studied the change of creep relaxation time τ_neq(t) with aging time t in glasses, and observed a power-law relationship between them, i.e., τ_neq(t)∝t^μ, where μ is the Struik shift factor. In this case, the material time is related to the aging time by a power law, i.e., ξ(t)∝t^(1−μ). And the overall relaxation function still remains a KWW form, i.e.,

ϕ(t)=exp[−ξ^β]=exp[−(t/τ_eff)^β_eff]

where β_eff=β(1−μ) is the apparent stretching exponent, τ_eff corresponds to the time when ϕ(t) decays to e−1, which represents a characteristic timescale of glass aging. In such case, the material time can be reconstructed as ξ(t)=t/τ_eff^(1−μ), and the Struik relation can be deduced as τ_neq(t)=t^μ×τ_eff^(1−μ)/(1−μ). The Eq. (2) suggests that even in nonlinear aging regimes, if the Struik relation is valid, the relaxation dynamics can still be described using a single effective timescale and a modified exponent. Even in systems where the Struik power-law relation does not strictly apply, the Eq. (2) can still provide an empirical approximation over finite time windows33–35. Consequently, the KWW function is generally applicable to relaxation in diverse glass systems, and even in other many-body interacting systems36,37.

The broad validation of the KWW equation in modeling the aging dynamics of various glasses can be evidenced from the data collected in Fig. 1a. Metallic glasses such as Ce-based5, Pd-based6, and Zr-based7 glasses, as well as polymer glasses like poly(vinyl-chloride) (PVC)8, polystyrene (PS85K)38, polycarbonate (PC)38, and poly(methyl methacrylate) (PMMA)38, and silicate glasses12,39 have been shown to follow the KWW equation closely during their aging processes. In all cases, the initial glassy states were prepared by quenching from their supercooled liquids, and the as-cast samples were subsequently annealed isothermally below Tg. The temporal evolution of their physical properties, such as recovered enthalpy, hardness, creep modulus, fictive temperature, and strain, was then monitored during the approach toward equilibrium. Despite the different thermal histories and the diversity of measured properties, the KWW equation can accurately capture the kinetics of these changes, allowing for the determination of the non-equilibrium effective relaxation time τ_eff and the stretching exponent β_eff. In Fig. 1b, by normalizing the aging time with τ_eff and β_eff, all the decay curves can collapse into a single master curve. This collapse strongly supports the universality of the KWW equation in modeling the non-equilibrium aging kinetics of a wide range of glassy systems, irrespective of their composition or physical properties.

Empirical equation for the decay of non-equilibrium glasses
To characterize the temperature dependence of the effective relaxation time (τ_eff) in aging glasses, we begin by reviewing existing theoretical models. Numerous phenomenological equations have been proposed to describe the evolution of equilibrium relaxation time τ_eq with temperature in supercooled liquids. The simplest is the Arrhenius equation τ_eq=τ_0 exp(E_a/RT), where τ_0 is the prefactor and E_a is the apparent activation energy. The Arrhenius equation only works for several strong glass-forming systems (like silica), but fails for most other systems where apparent activation energy E_a varies with temperature. To address this, various non-Arrhenius models, such as the VFT (E_a=BRT/(T−T_0))20–22, parabolic (E_a=CRT(T_0/T−1)^2)23, and MYEGA (E_a=KRexp(C/T))24 have been developed. However, these models20–24 are only valid for equilibrium supercooled liquids and breakdown in the non-equilibrium glass state.

A more general framework for modeling non-equilibrium glass involves the concept of fictive temperature (Tf), defined as the effective thermodynamic temperature at which a glass’s instantaneous properties match those of the equilibrium liquid40. During aging, Tf continuously decreases from the temperature that glass is frozen (Tg) toward the ambient temperature (T). The Narayanaswamy–Moynihan (NM) model18,25 relates the growth of the instantaneous non-equilibrium relaxation time τ_neq(t) to the evolution of Tf(t), as:

τ_neq(t)=τ_0 exp[xE_g/RT+(1−x)E_g/RT_f(t)]

where Eg is the activation energy around Tg, x is the weighting factor. And the τ_neq(t) in turn determines the decaying rate of fictive temperature Tf(t) and related physical properties p(t), i.e.,

ϕ(t)=(T_f(t)−T)/(T_g−T)=(p(t)−p_eq)/(p_0−p_eq)=exp[−(∫_0^t dt'/τ_neq(t'))^β]

Due to the inherent complexity of nonlinearity, obtaining an exact analytical solution for these equations is challenging. To make analytical progress, we provide an approximating solution under two assumptions: 1) the τ_eff is defined as the time when relaxation function decays to e−1 (i.e., ϕ(τ_eff)=e−1); 2) a simple KWW function can be used to approximate the relaxation function with β_eff=β(1−μ). Consequently, the non-equilibrium relaxation time τ_neq(t) near t=τ_eff could be deduced as a power law relationship τ_neq(t)=t^μ×τ_eff^(1−μ)/(1−μ). Combining these simplified equations with Eq. (3) and Eq. (4) yields the following equation:

τ_eff=t^μ×τ_eff^(1−μ)/(1−μ)=τ_0 exp[xE_g/RT+(1−x)E_g/RT_f(t)]

where Tf(t)=T+(Tg−T)exp[−(t/τ_eff)^β_eff]. In such case, an analytical solution for τ_eff can be deduced as τ_eff=(1−μ)τ_0 exp[xE_g/RT+(1−x)E_g/RT_n], where Tn=Tf(τ_eff)=T+(Tg−T)e−1 is the fictive temperature at t=τ_eff, and the parameter μ=(1+(1−x)y_T)−1 with y_T=βE_g(Tn−T)/RT_n^2. Considering that Eg is connected to fragility index m (i.e., Eg=mln(10)RTg) and that τ_g=τ_0 exp(E_g/RT_g), the non-equilibrium effective relaxation time of glass τ_eff can be reconstructed as

log(τ_eff/τ_g)=m[xT_g/T+(1−x)T_g/T_n]−log(1+(1−x)y_T)

with Tn=T+(Tg−T)e−1 and y_T=βmln(10)T_g(Tn−T)/T_n^2. The upper limit of the τ_eff is related to the case of x=1, i.e., log(τ_eff/τ_g)=m(T_g/T−1), which is a simple Arrhenius equation. Whereas the lower limit of τ_eff is related to the case of x=0, i.e., log(τ_eff/τ_g)=m(T_g/T_n−1)−log(1+y_T). The lower limit of τ_eff has a limited value even at zero temperature, i.e., log(τ_eff(T=0)/τ_g)=m(e−1)−log(1+e^βmln(10)) < m(e−1), indicating that glass aging could take place even at temperature far below Tg. More importantly, the fragility index m is crucial here, and it controls how fast the relaxation time increase as temperature decreases.

As an illustration, the evolution of relaxation time for these different models is compared in a Ce-based MG with a fragility of 40.9 (see Supplementary Fig. 1) and a Tg of 410 K. As shown in Fig. 2, the lower limit of the NM model, which accounts for the nonlinear effect, predicts a room-temperature relaxation time of 958 years. In contrast, the upper limit of the NM model, which neglects the nonlinear effect, predicts a room-temperature relaxation time of 50.3 billion years—longer than the age of the universe (13.8 billion years). Moreover, the coupling model (CM)33,41 predicts a room-temperature relaxation time of 1.28 million years, which is much longer than the prediction of the lower limit of the NM model. For the non-Arrhenius models (VFT, MYEGA, and Parabolic), the predicted relaxation time is even longer. These results highlight the significance of nonlinearity in understanding the non-equilibrium relaxation process, as it can effectively reduce the relaxation time of glass by tens of orders of magnitude. However, by observing the distinct enthalpy decrease during a 10.5-year aging process in experiments, the relaxation time of Ce-based MG at room temperature is estimated to be on the order of 10^1 years, which is much shorter than the predictions of all these models. Therefore, these existing theories are still inadequate for effectively evaluating relaxation time of glass.

To address this issue, an empirical formula was developed to effectively describe the τ_eff below Tg42: ln(τ_eff/τ_g)=A(Tg−T)/T, where τ_g is the relaxation time at Tg, and A is an unknown constant without any given physical meaning. For better comparison among different glass systems, the temperature can be normalized with Tg, and a new empirical formula can be developed:

ln(τ_eff/τ_g)=w(Tg−T)/Tg

where w is a dimensionless constant. Actually, Hecksher and coworkers43,44 have proposed a single-parameter aging (SPA) ansatz, suggesting that the difference between logarithm of the aging rates in equilibrium and out-of-equilibrium is proportional to the temperature difference, which corresponds well to Eq. (7). As shown in Fig. 3a, the relationship between ln(τ_eff/τ_g) and (Tg−T)/Tg below Tg for different types of glasses is linear, with the slope varying depending on the type of glass. By multiplying the slope w by (Tg−T)/Tg, it is found that the τ_eff of all different types of glasses can be normalized onto a single master curve, as shown in Fig. 3b. This demonstrates the effectiveness of Eq. (7) in describing the τ_eff below Tg for different types of glasses. Furthermore, as shown in Fig. 3c, the slope w and the fragility m of various glasses show a good fit with the equation y=x, with a fitting degree of 94.7%. This indicates that the slope w in Eq. (7) can be approximated as the fragility value of the glass (w~m). Notably, as shown in Fig. 2, applying Eq. (7) to the Ce-based MG would produce a τ_eff of 1.14 years at ambient temperature. This value of τ_eff is of the same order of magnitude as the time showing distinct aging effects, more reasonable than the predictions by all other models. It indicates that the Eq. (7) can be effective and reliable even at ambient temperature.

Comparing the predicted age and the real age
To evaluate the validity of our model in describing long-term aging dynamics at ambient temperatures, we compare the predicted ages of the glasses with their known or independently estimated real ages. When using the equation ϕ(t)=exp[−(t/τ_eff)^β_eff] to assess the age of glass at ambient temperature, the parameters β_eff and τ_eff are essential as they determine the decay rate of the aging dynamics. At temperatures far below Tg, β_eff has been consistently found to be approximately a constant value of 3/712,45. The ambient temperature τ_eff can be calculated using the empirical relation ln(τ_eff/τ_g)=w(Tg−T)/Tg, where w is approximately equal to the kinetic fragility m. By analyzing the variation of Tg under different heating rates, the fragility values of Ce65Ga8Cu22Nb5 MG, lunar glass, and amber were determined to be 40.9±1.4, 45.8±3.1, and 98.2±4.6, respectively (see Supplementary Fig. 1). These values and their uncertainties are consistent with previously reported fragility measurements for MGs46, silicate glasses47, and polymers48.

The average annual ambient temperature at the storage location of Ce-based MG is ~288 K. Using this temperature, τ_eff at ambient temperature is estimated to be 17.4 seconds, or 1.14 years, as shown in Fig. 3d. Therefore, the relaxation function of Ce65Ga8Cu22Nb5 MG at ambient temperature is expressed as ϕ(t)=exp[−(t/1.14)^3/7]. From the observed changes in the physical properties of Ce-based MG during aging, its corresponding age t can be determined. As shown in Fig. 4a, compared to the initial sample, the 10.5 years’ aged Ce65Ga8Cu22Nb5 MG exhibits a more pronounced endothermic overshoot during the glass transition process, with the recovery enthalpy increasing from 1.1 J g−1 of the rejuvenated sample to 9.7 J g−1 of the aged sample. The recovery enthalpy of the aged sample is very close to the equilibrium value of recovery enthalpy. This yields a normalized relaxation value of ϕ ≈ 0.11. Substituting this into the relaxation function provides an estimated age of ~7.17 years (black solid square in Fig. 5a), which is reasonably close to the actual age of 10.5 years, confirming the model’s accuracy under ambient conditions.

According to the official climate report issued by the Institute of Meteorology and Water Management – National Research Institute (Poland), the national mean annual temperature in Poland is ~282.7 K. At this ambient temperature, the τ_eff of amber is e34.2 seconds, or about 21.6 million years (Fig. 3d). The corresponding relaxation function for amber at ambient temperature can be determined: ϕ(t)=exp[−(t/21.6)^3/7]. As shown in Fig. 4c, the recovery enthalpy of the amber sample during the glass transition process increases from 3.02 J g−1 of the rejuvenated sample to 12.4 J g−1 of the hyper-aged sample. After approximation, a ϕ value of ~0.24 can be obtained. Substituting into the relaxation equation yields a predicted age of ~49.5 million years (red solid circle in Fig. 5b), which falls within the known age range of Baltic amber49,50. This consistency indicates that the long-term aging dynamics of amber confirm well to the kinetic model we established even at geological timescale.

At the Chang’E-5 landing site (~43°N, 52°W)51, the lunar surface temperature exhibits strong diurnal variations52. The temperature profile at this site is shown in Supplementary Fig. 2 (data from ref. 52): nighttime temperatures are ~97 K with small fluctuations, whereas daytime temperatures rise to a peak of ~350 K. Using a trapezoidal approximation to the daytime segment, the average daytime temperature is ~286 K. Aging dynamics are exponentially sensitive to temperature. At the Chang’E-5 landing site, the average daytime temperature is ~189 K higher than the nighttime value. Therefore, the aging dynamics are expected to be dominated by daytime conditions, whereas the nighttime contribution can be neglected. At 286 K, τ_eff for the lunar glass during daytime is e36.9 seconds, or roughly 331 million years (Fig. 3d). The daytime aging dynamics function is therefore ϕ(t)=exp[−(t/331)^3/7]. From this relation, the age of lunar glass is estimated to be approximately twice the daytime aging duration. According to ref. 13, the reduced modulus of two lunar glass samples increased from 48.5 and 56.8 GPa in their rejuvenated states to 84.2 and 85.6 GPa in their hyperaged states, respectively. From these modulus changes, relaxation function values of approximately ϕ ≈ 0.58 and ϕ ≈ 0.66 were obtained. Substituting into the kinetic model yields estimated ages of ~161 million and 86.0 million years, respectively. These values fall within the range of 2 million to 2 billion years as determined by isotopic dating methods30. The agreement between our predictions and radiometric age estimates suggests that the glass kinetics dating method provides a reliable alternative and is capable of cross-validation with traditional techniques.

Methods
Glass samples
We investigated three distinct types of glasses that had aged for different durations under ambient conditions: a Ce-based MG stored for ~10.5 years, Baltic amber aged for tens of millions of years, and lunar glass samples aged for hundreds of millions of years, returned by China’s Chang’E-5 mission. Specifically, the Ce65Ga8Cu22Nb5 MG was produced in 2013 and stored in a sealed plastic container under ambient conditions for 10.5 years. The inset of Fig. 4b shows a photograph of this Ce-based MG after 10.5 years. The sample developed a dark brown oxide layer on its surface, which was removed with sandpaper to reveal a shiny metallic luster. The inset of Fig. 4d shows the amber sample, which originates from Poland. The age of this Baltic amber is estimated to be between 30 and 50 million years49,50.

Thermal and structural characterization
To assess the degree of aging in each glass sample, the aged state was compared to its original state (corresponding to the case of t=0). This can be achieved by heating the glass above the Tg, where it transitions into the supercooled liquid state, and then rapidly quenching it to ambient temperature10,11. This process rejuvenates the glass and recovers it to its initial state5. The aged Ce65Ga8Cu22Nb5 MG and amber samples were heated in a high-vacuum annealing furnace to a temperature 5 K above the end of the glass transition temperature (Tg) at a rate of 20 K min−1, held for 60 sec, and then furnace-cooled to room temperature to obtain rejuvenated samples (in initial state) with identical initial thermal histories. The thermodynamics of glassy samples was performed using PerkinElmer DSC8000 at a heating rate of 20 K min−1 under a purified argon atmosphere. The heat capacities (Cp) of amber and Ce65Ga8Cu22Nb5 MG were obtained by comparing with Cp of a sapphire standard sample. The amorphous structure of amber and Ce65Ga8Cu22Nb5 samples with different aging histories was confirmed by X-ray diffraction (XRD) using a D8/ADVANCE diffractometer with Cu Kα radiation at 40 kV. The kinetic fragility (m) of different types of glass was measured using DSC to analyze the relationship between Tg and various heating rates. Samples were subjected to heating rates ranging from 5 to 150 K min−1, and the corresponding Tg values were recorded. By plotting the logarithm of the heating rates against Tg, the fragility parameter (m) for each type of glass was determined61,62.

Chang’E-5 lunar glass
The lunar regolith was scooped from the lunar surface in Chang’E-5 lunar exploration program. The lunar regolith sample numbered CE5C0400 was applied from China National Space Administration. The lunar regolith from the Chang’E-5 mission contains ~15 wt.% of glassy materials63, which formed due to the rapid cooling of basalt melted by meteorite impacts64. The Tg of lunar glass is taken from ref. 13. According to our previous results13, the glass obtained by cooling at 240 K s−1 approximates the glass initially formed by impact on the Moon and is regarded as the rejuvenated state of the lunar glass. The samples aged on the lunar surface over geological timescales are considered to be in a hyperaged state, and the reduced modulus values for rejuvenated and hyperaged states are measured in ref. 13.

Data availability
All data in this study are included in the manuscript, the Supplementary Information/Source Data file, or can be obtained from the corresponding author upon request. Source data are provided with this paper.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by Zhao and coworkers represents a significant and timely contribution to the field of glass dynamics. As is well known, understanding the relaxation behavior of glasses—particularly at temperatures far below the glass transition temperature (Tg)—remains one of the central challenges in glass physics and materials science. In recent years, this topic has attracted growing interest, as mounting experimental evidence [PRL 110, 265901 (2013); PRL 112, 165901 (2014); Sci. Adv. 8, eabn3623 (2022); Sci. Adv. 9, eadi6086 (2023); PRL 132, 056101 (2024)] suggests that glasses continue to evolve structurally even in deep non-equilibrium regimes, far below Tg, and over timescales spanning from years to geological epochs. While existing models have proven effective in describing equilibrium or near-equilibrium relaxation, they largely fail to capture this ultra-slow, nonlinear aging behavior. This work effectively addresses and fills this critical gap.\nThrough a combination of theoretical modeling and experimental validation, the authors propose a general empirical framework that quantitatively captures the aging behavior of diverse glasses at ambient temperatures. Building on this understanding, they introduce an elegant and novel dating approach based on intrinsic relaxation dynamics—conceptually analogous to radiometric dating, but grounded in the physics of aging rather than isotopic decay. The method allows one to infer the age of glass samples by measuring the extent of property decay caused by structural relaxation.\nTo support their model, the authors conduct a systematic experimental study on three representative glass systems: a Ce-based metallic glass aged naturally for over 10 years, Baltic amber aged tens of millions of years, and lunar glass from China’s Chang’E-5 mission aged over hundreds of millions of years. These carefully selected materials span an extraordinary range of timescales and compositions, providing compelling evidence that the proposed framework can universally describe glass aging from laboratory to geological time domains.\nThe manuscript is clearly written, logically organized, and technically rigorous. The integration of theoretical analysis with experimental results from multiple glass types strengthens the validity of the proposed model. The innovative dating approach based on τeff not only fills a theoretical void but also opens up new opportunities in planetary science, archaeology, and materials reliability. The results are both novel and impressive, and the conceptual impact of treating glass aging as a chronometric process is profound.\nWhile the manuscript is strong overall, the following suggestions could enhance its clarity and impact:\n1) Discuss uncertainties in fragility (m): Since fragility plays a central role in determining τeff, it would be valuable to briefly comment on the sensitivity of τeff predictions to uncertainty in m—especially for ancient or natural glasses where m is inferred rather than directly measured.\n2) Highlight broader applications: While the primary focus is on geological and planetary dating, the proposed kinetics-based approach could also be relevant to other areas, such as long-term reliability of amorphous materials in engineering systems, optical drift in devices, or the preservation of historical glass artifacts. A sentence or two expanding on these possibilities would help readers appreciate the wider implications.\n3) A minor tip, adding the Ta/Tg value (where Ta is the annealing temperature) as a mark for each type of glass in Figure 1a would make it easier for readers to directly compare aging processes across different glasses.\nIn conclusion, this is an important and well-executed study that makes a meaningful advance in our understanding of glass aging and opens new directions for its application. I strongly support its publication in Nature Communications.", "reviewer_2": "Re: ”Ultra-slow aging dynamics of glass….” By Zhao et al.\nI read this paper with great interest, which addresses the important topic of predicting aging of systems taking place far below the glass transition temperature. From the formalism developed one can determine the age of a sample, i.e., the time that has passed since the sample left thermodynamic equilibrium and became a glass. The authors’ new approach to doing this is validated from data of metal-glass aging (for which the age is known), millions of year old amber, and billions of year old glass from the Moon.\nI quite like the fundamental idea of relating Struik’s old power-law ansatz with the TNM material-time formalism, which in conjunction with the assumption of a stretched exponential leads to testable predictions.\nThe paper is well written with few (if any) typos or other trivial errors. I recommend the paper is accepted by NC once the authors have had a chance to reflect on the below comments / suggestions.\n1. Assuming the paper is written for a general audience, not only aging experts, I think there could be more details given in the text. For instance, the relaxation function phi(t) is discussed but not defined (before page 8 in Eq. (4)). Likewise the concept of an “instantaneous relaxation time” (p. 2) is not defined (to me it is the aging rate of the TNM formalism but the reader may not know this).\n2. It is not clear whether the paper assumes the existence of a material time (like in Eq. (1)), but I assume this is the case.\n3. I could not find any reference to the works of Greg McKenna and co-workers on the aging of amber (published in NC, as I recall), which should be relevant to cite.\n4. On p. 3, some references refer to first names (Eloi and Daniele are not surnames).\n5. On the same page, what is “nonlinear cumulative effects”?\n6. P. 3: I am not an isotope expert, but am surprised the concentration may be so low it cannot be measured by modern mass spec. Could the authors give a reference to this claim? Maybe the problem is that there are no relevant isotopes to detect (?).\n7. On p. 5 it is not quite clear to me what the exact definition of tau_eff is.\n8. Also on p. 5 is mentioned a Taylor expansion, but of what?\n9. Figure 1(b) looks very convincing, but it should be mentioned to the reader that this is just a restatement of the fact that the full curves fit data in Fig. 1(a) (assuming the full curves are the stretched exp predictions which is not stated explicitly). Thus Fig. 1(b) provides no new information. I also think more details should be given – for instance it is not clear whether the data relate to relaxation after temperature jumps. And why not give the actually measured quantity for the benefit of the reader?\n10. Is beta_eff a fit parameter in this figure?\n11. On p. 7 it is mentioned that “these models .. break down in the non-equilibrium state..” but no reference is given to this claim.\n12. On p. 8 is mentioned that “Substituting these simplified equations into the NM equation yields…” but isn’t it the case that one also needs to assume Eq. (4).\n13. The basic equation of the paper, Eq. (7), is not entirely new but in fact equivalent to the so-called “single-parameter aging” (SPA) ansatz of Hecksher et al [J. Chem. Phys. 142, 241103 (2015)] (this equivalence with Eq. (7) only comes about because of the Struik power law assumption made, a most interesting observation). It was thought that SPA only works for relatively small temperature changes, but SPA has recently been shown to work for also large temperature jumps [J. Chem. Phys. 161, 194504 (2024)]. SPA postulates that the logarithm of the aging rates in equilibrium and out-of-equilibrium is proportional to the temperature difference, which is precisely Eq. (7).\n14. On p. 12 the word “naturally aged” appears a bit awkward to me.\n15. Also on p. 12, it seems to me that the authors’ approach of heating the sample to Tg is similar to that of McKenna et al. in their studies of amber aging.\n16. P. 13: How well can one estimate the aging temperature of a sample on the Moon where temperature varies very much? Likewise (p. 15) I am surprised that 288 is the average temperature in Poland – a reference to this should be provided.\n17. On p. 15: The numbers in the exponentials have too many decimals – we usually teach students only to write the numbers that are to be trusted.\n18. On p. 19, it is mentioned that “These results suggest that glass can never be completely frozen”. But is that not what Simon discussed at length in 1931?\n19. Also on p. 19: What is the MAP model?\n20. It would be nice in the final part of the paper to summarize the basic assumptions made: 1) The material time formalism with single-parameter aging, 2) A stretched exponential relaxation function, 3) the Struik power law relation between time and aging time."}, "author_response": "Thanks to the reviewers for the valuable comments and suggestions. The concerns of the reviewers are indeed crucial to the validity of our work, and all the comments, criticisms and suggestions have been carefully considered. We accept all these comments or suggestions and revise accordingly in the revised version. We have marked the changes in blue color in our revised manuscript to make the checking process of the manuscript more convenient. In the following response letter, we replied to the comments one by one. For convenience, the original comments and criticisms are reproduced in bold.\n--------------------------------------------------------------------------------------\nResponse to Comments of Referee #1-- NCOMMS-25-49362-T/Zhao\n--------------------------------------------------------------------------------------\n(From referee #1) The manuscript by Zhao and coworkers represents a significant and timely contribution to the field of glass dynamics. As is well known, understanding the relaxation behavior of glasses—particularly at temperatures far below the glass transition temperature (T )—remains one of the central challenges in glass physics and materials science. In recent years, this topic has attracted growing interest, as mounting experimental evidence [PRL 110, 265901 (2013); PRL 112, 165901 (2014); Sci. Adv. 8, eabn3623 (2022); Sci. Adv. 9, eadi6086 (2023); PRL 132, 056101 (2024)] suggests that glasses continue to evolve structurally even in deep non-equilibrium regimes, far below T , and over timescales spanning from years to geological epochs. While existing models have proven effective in describing equilibrium or near-equilibrium relaxation, they largely fail to capture this ultra-slow, nonlinear aging behavior. This work effectively addresses and fills this critical gap.\nThrough a combination of theoretical modeling and experimental validation, the authors propose a general empirical framework that quantitatively captures the aging behavior of diverse glasses at ambient temperatures. Building on this understanding, they introduce an elegant and novel dating approach based on intrinsic relaxation dynamics—conceptually analogous to radiometric dating, but grounded in the physics of aging rather than isotopic decay. The method allows one to infer the age of glass samples by measuring the extent of property decay caused by structural relaxation.\nTo support their model, the authors conduct a systematic experimental study on three representative glass systems: a Ce-based metallic glass aged naturally for over 10 years, Baltic amber aged tens of millions of years, and lunar glass from China’s Chang’E-5 mission aged over hundreds of millions of years. These carefully selected materials span an extraordinary range of timescales and compositions, providing compelling evidence that the proposed framework can universally describe glass aging from laboratory to geological time domains.\nThe manuscript is clearly written, logically organized, and technically rigorous. The integration of theoretical analysis with experimental results from multiple glass types strengthens the validity of the proposed model. The innovative dating approach based on τ eff not only fills a theoretical void but also opens up new opportunities in planetary science, archaeology, and materials reliability. The results are both novel and impressive, and the conceptual impact of treating glass aging as a chronometric process is profound.\nReply: We thank Reviewer #1 for the positive and insightful comments. We are pleased that the reviewer recognized the significance of our work in addressing the challenge of ultra-slow glass aging far below T , which has long been a central problem in glass physics. We also appreciate the recognition of the conceptual contribution of introducing a relaxation-based dating framework, supported by experimental validation across vastly different timescales. Below we respond to each suggestion in detail.\n(From referee #1) While the manuscript is strong overall, the following suggestions could enhance its clarity and impact:\n1) Discuss uncertainties in fragility (m): Since fragility plays a central role in determining τ eff, it would be valuable to briefly comment on the sensitivity of τ eff predictions to uncertainty in m—especially for ancient or natural glasses where m is inferred rather than directly measured.\nReply: We are grateful for the Reviewer #1’s insightful suggestion. Below we first briefly introduce the method used to determine the fragility m, and then discuss the sensitivity of effective relaxation time τ eff to its uncertainty.\n1. The fragility index (m) is a key parameter in glass physics, which is proportional to the apparent activation energy (E ) of α-relaxation in equilibrium supercooled liquid near T , i.e., . In our study, we directly determined E and m values of all three glass systems—Ce-based metallic glass (MG), Baltic amber, and lunar glass—via differential scanning calorimetry (DSC) measurements conducted over a broad range of heating rates (5-150 K min-1). The measured T shifts were analyzed using the Kissinger method to obtain E , based on which m can be calculated. This approach yielded fragility values of 40.9±1.4, 45.8±3.1, and 98.2±4.6 for Ce-based metallic glass, lunar glass, and amber, respectively (see Supplementary Fig. S1). These values and their uncertainties are consistent with previously reported fragility measurements for MGs [Mater. Futures 3, 025002 (2024)], silicate glasses [Nat. Commun. 14, 13 (2023)], and polymers [Macromolecules 41, 7232(2008)].\n2. In this paper, we reveal that fragility index m also controls the how fast the effective relaxation time (τ eff) of physical aging changes with temperature in glass, i.e., lnτ eff-lnτ g ≈m T -T T .According to the equation, the uncertainty of m will transform linearly to that of lnτ eff, and consequently to the estimated age t. To quantitatively assess the sensitivity of lnτ eff to m, we have performed an error propagation analysis. In our dating framework, the primary concern lies in the order of magnitude of the estimated age of the glass, which is governed by lnτ eff. The relative uncertainties obtained from our measurements of m are 3.4%, 4.7%, and 6.9% for Ce-based MG, amber, and lunar glass, respectively. With this level of uncertainty, the corresponding uncertainties in lnτ eff are ±0.4 for Ce-based MG, ±1.4 for amber, and ±2.2 for lunar glass, respectively. Propagating these uncertainties to the estimated ages gives an uncertainty of less than 50% for the Ce-based MG, while even for the more fragile amber and for lunar glass aged at lower temperature, the predicted ages remain within one order of magnitude. This demonstrates that the uncertainty in fragility does not affect the validity of our dating approach.\nWe have now added a brief discussion of this issue in the revised manuscript (page 13, line 7-9; page 18, lines 20-28; page 19, lines 1-5) for clarity.\n2) Highlight broader applications: While the primary focus is on geological and planetary dating, the proposed kinetics-based approach could also be relevant to other areas, such as long-term reliability of amorphous materials in engineering systems, optical drift in devices, or the preservation of historical glass artifacts. A sentence or two expanding on these possibilities would help readers appreciate the wider implications.\nReply: We appreciate the suggestion to highlight broader applications. In the revised manuscript, we have expanded the discussion following the example of glassy silk dating to include representative cross-disciplinary applications. Specifically, we now note that beyond cultural heritage cases, the same framework can assess the long-term stability of optical coatings, where refractive index drift affects performance, and monitor resistance drift in amorphous films for memory and sensing devices. We have added the related discussion on page 20, lines 11-14.\n3) A minor tip, adding the T a/T g value (where T a is the annealing temperature) as a mark for each type of glass in Figure 1a would make it easier for readers to directly compare aging processes across different glasses.\nReply: In the revised version of Fig. 1a on page 6, we have added the corresponding T a/T g values for each glass system in the legend.\nIn conclusion, this is an important and well-executed study that makes a meaningful advance in our understanding of glass aging and opens new directions for its application. I strongly support its publication in Nature Communications.\nReply: We sincerely thank the reviewer for the encouraging and positive evaluation of our study.\n--------------------------------------------------------------------------------------\nResponse to Comments of Referee #2-- NCOMMS-25-49362-T/Zhao\n--------------------------------------------------------------------------------------\n(From referee #2) I read this paper with great interest, which addresses the important topic of predicting aging of systems taking place far below the glass transition temperature. From the formalism developed one can determine the age of a sample, i.e., the time that has passed since the sample left thermodynamic equilibrium and became a glass. The authors’ new approach to doing this is validated from data of metal-glass aging (for which the age is known), millions of years old amber, and billions of years old glass from the Moon.\nI quite like the fundamental idea of relating Struik’s old power-law ansatz with the TNM material-time formalism, which in conjunction with the assumption of a stretched exponential leads to testable predictions.\nThe paper is well written with few (if any) typos or other trivial errors. I recommend the paper is accepted by NC once the authors have had a chance to reflect on the below comments / suggestions.\nReply: We sincerely thank Reviewer #2 for the thoughtful and encouraging feedback. We appreciate your recognition of the connection between Struik’s power-law ansatz and the TNM material-time formalism. We are also grateful for your support of the manuscript’s publication, and we have carefully addressed all of your specific comments and suggestions as detailed below.\n(From referee #2)\n1. Assuming the paper is written for a general audience, not only aging experts, I think there could be more details given in the text. For instance, the relaxation function phi(t) is discussed but not defined (before page 8 in Eq. (4)). Likewise the concept of an “instantaneous relaxation time” (p. 2) is not defined (to me it is the aging rate of the TNM formalism but the reader may not know this).\nReply: Thanks for your comments. We fully agree that providing clear definitions and explanations is crucial for ensuring the accessibility of our paper to a general audience. Following your suggestions, we have made the following revisions in the revised manuscript (page 2, lines 18-28; page 3, lines 1-3):\nIn equilibrium supercooled liquids, the relaxation behavior in response to perturbations (e.g., subtle temperature jump) can be well described by a stretched exponential Kohlrausch-Williams-Watts (KWW)16,17 function: , where is the normalized relaxation function, and p (t) is the measured quantity (e.g., recovered enthalpy, density, modulus). And τ eq here is the equilibrium relaxation time which does not depend on aging time. Unlike equilibrium supercooled liquids, non-equilibrium glasses exhibit progressively slower and history-dependent relaxation. To describe this nonlinear aging dynamics, the concept of material time is introduced14,15,18,19, and a nonlinear integral function is required8: , (1) where is the dimensionless material time, and τ neq is the instantaneous relaxation time which is not fixed but evolves continuously with aging time.\n2. It is not clear whether the paper assumes the existence of a material time (like in Eq. (1)), but I assume this is the case.\nReply: Indeed, our analysis is based on the existence of a material time. To avoid ambiguity, we have reconstructed Equation (1) in the revised manuscript (page 2, lines 18-28; page 3, lines 1-3; page 4, lines 26-27).\n3. I could not find any reference to the works of Greg McKenna and co-workers on the aging of amber (published in NC, as I recall), which should be relevant to cite.\nReply: We appreciate the reviewer’s suggestion regarding the works of Gregory B. McKenna and co-workers on the aging of amber. These studies have been cited in the revised manuscript as Refs. 10 and 11, both in the introduction (page 2, line 13) and the discussion (page 18, line 4). Ref. 10 is the work by Zhao et al. in collaboration with G. B. McKenna on using 20-million-year-old amber to test the super-Arrhenius behaviour of glass-forming systems [Nat. Commun. 4, 1783 (2013)]. Ref. 11 reports on the stability of two-level systems and the boson peak in 110-million-year-old amber glass [Phys. Rev. Lett. 112, 165901 (2014)].\n4. On p. 3, some references refer to first names (Eloi and Daniele are not surnames).\nReply: We have corrected the text to ensure that all authors are referred to by the surname of the first author rather than by the name of the corresponding authors. In the original text, “Eloi” and “Daniele” referred to the corresponding authors Eloi Pineda and Daniele Cangialosi, whereas the correct first authors are Duan (Y. J. Duan, Ref. 26) and Perez-De Eulate (N. G. Perez-De Eulate, Ref. 9), respectively. The sentence has been revised to: “… including those by Duan et al.26, Zhao et al.5, Perez-De Eulate et al.9, and Corning Incorporated12 ….”. This change ensures consistency with standard academic practice of citing by the first author’s surname. The revised sentence appears on page 3, lines 18-19 of the revised manuscript.\n5. On the same page, what is “nonlinear cumulative effects”?\nReply: The “nonlinear cumulative effects” here refers to the nonlinear integral function of Equation (1). The nonlinear aging dynamics here is represented by the accumulation of time-dependent growth of 1/τ neq(t) during aging. For clarification, we have simply replaced it with “nonlinear effects” in the revised manuscript (page 3, line 22).\n6. P. 3: I am not an isotope expert, but am surprised the concentration may be so low it cannot be measured by modern mass spec. Could the authors give a reference to this claim? Maybe the problem is that there are no relevant isotopes to detect (?).\nReply: We thank the reviewer for this comment. Below, we discuss present situations in lunar glass and amber glass, respectively.\n1. Lunar glass ages are indeed mainly obtained by isotope methods, particularly 40Ar/39Ar and U-Th-Pb dating [J. Geophys. Res. Planets 124, 2686 (2019)]. A main limitation is that many individual glasses contain K or Pb at concentrations too low to produce measurable signals, even with modern mass spectrometry. For instance, in Apollo 14 [J. Geophys. Res. Planets 107, 12 (2002)], Apollo 16 [Meteorit. Planet. Sci. 42, 993 (2007).], and Apollo 17 [Meteorit. Planet. Sci. 44, 839 (2009)] soils, 549 glass spheres were characterized, but only 68 yielded usable 40Ar/39Ar ages because of low K contents and diffusive Ar loss [Geochim. Cosmochim. Acta 73, 4590 (2009)]. Likewise, in the Chang’E-5 collection, 215 glass beads were analyzed, but only 166 provided U-Pb ages, while the rest had Pb signals below detection limits (206Pb < 1 cps) [Sci. Adv. 8, eabq2542 (2022)]. In addition to concentration, size is another critical factor: Ar-Ar studies of Apollo glasses generally selected beads >100 μm, while U-Pb dating of Chang’E-5 glasses focused on beads >50 μm to maximize the probability of retaining and detecting sufficient isotopes. Thus, the main challenge for lunar glass dating is their very low abundance and limited retention, particularly in small glass beads.\n2. In the case of amber, its age is determined in a different way. Unlike lunar glasses, amber contains measurable amounts of 14C. However, the half-life of 14C is only about 5730 years [Radiocarbon Dating: An Archaeological Perspective (Routledge, 2016)], which limits radiocarbon dating to timescales of at most several tens of thousands of years. Moreover, amber lacks long-lived isotopic systems such as U-Pb that can be used to establish ages of millions to billions of years. Consequently, the age of amber is usually constrained indirectly, most often through stratigraphic methods, that is by dating the geological layers in which the amber is preserved and using those ages to infer the age of the amber itself [Minerals 13, 948 (2023)].\nWe have provided more detailed explanations and appropriate references regarding the reasons why isotope methods are often difficult to apply to both lunar glasses and amber in the revised manuscript (page 3, lines 27-28; page 4, lines 1-5).\n7. On p. 5 it is not quite clear to me what the exact definition of τ eff is.\nReply: In this work, we have showed that, if the Struik’s relation ( ) is valid, the overall relaxation function (nonlinear integral function of Eq (1) ) still remains a KWW form, i.e., . In this case, τ eff corresponds to the time when (t) decays to e-1. Even if Struik’s relation is invalid and the overall relaxation function is much complicated. Mathematically, one can always obtain τ eff by fitting (t) with a KWW function. Or, one can simply define τ eff as the time when (t) decays to e-1.\nIn our views, τ eff represents a characteristic time scale of glass aging. If one wants to observe a significant aging effect experimentally, the aging time should roughly be of the same order of magnitude as τ eff. If the aging time is much shorter than τ eff, it will be difficult to observe the aging effect experimentally. As an example, the evaluated τ eff value for lunar glass is 331 million years, making it almost impossible to directly observed the aging effect at room temperature within laboratory time scale. We have revised the manuscript to clarify this definition on page 5, lines 8-9.\n8. Also on p. 5 is mentioned a Taylor expansion, but of what?\nReply: Here we mean Taylor expansion of the relaxation function. As as is mentioned above, if Struik’s relation is invalid and the overall relaxation function (t) is not a perfect KWW function. Then, we can still define τ eff as the time when (t) decays to e-1, and use a KWW function to approximate the real relaxation function (t) over finite time windows.\nMathematically, if (t) follows the KWW form, i.e., , then one can get ln . Therefore, one would get a straight line when plot ln as a function of time t in a double logarithmic coordinate graph. If the overall relaxation function (t) is not a perfect KWW function. We can always use a straight line to approximate the data in the double logarithmic coordinate graph (see Figure R1). To avoid ambiguity, we have removed the description of Taylor expansion in the revised manuscript.\nFig. R1. Linear approximation (first order Taylor expansion) of ln ϕ t function in double logarithmic coordinate graph.\n9. Figure 1(b) looks very convincing, but it should be mentioned to the reader that this is just a restatement of the fact that the full curves fit data in Fig. 1(a) (assuming the full curves are the stretched exp predictions which is not stated explicitly). Thus Fig. 1(b) provides no new information. I also think more details should be given – for instance it is not clear whether the data relate to relaxation after temperature jumps. And why not give the actually measured quantity for the benefit of the reader?\nReply: In the revised manuscript, we have modified Fig. 1a to explicitly present the absolute values of the measured physical quantities (e.g., recovered enthalpy, hardness, creep modulus, fictive temperature, and strain) for different types of glasses. The corresponding normalized relaxation functions are shown in Fig. 1b, providing a unified description of these diverse relaxation behaviors through the KWW function. For clarity, the revised figure (Fig. R2) is also provided below.\nIn addition, we clarify the experimental protocol. In all cases, the initial states of the glasses were obtained by quenching from the supercooled liquid. These as-cast glasses were subsequently annealed isothermally below T g, and the temporal evolution of their physical properties was monitored during the approach toward equilibrium. Thus, the observed aging dynamics represent non-equilibrium relaxation toward equilibrium, rather than relaxation directly induced by temperature jumps.\nThe revised Fig. 1 is presented on page 6, and the description of the aging protocol for the glass samples can be found on page 5, lines 24-28 and page 6, lines 1-2.\nFig. R2 Universality of the KWW function in describing the relaxation dynamics of various glassy materials. (a) Relaxation behaviors of various glassy systems are well captured by the KWW equation. These systems include metallic glasses (Ce-based5, Pd-based6, Zr-based7), polymer glasses (PVC8, PS85K38, PC38, PMMA38), and inorganic silica glasses12,39. The relaxation dynamics are extracted from diverse physical quantities, including recovered enthalpy, hardness, creep modulus, fictive temperature, and strain, measured under different aging protocols (the fitted values of τ eff and β eff are listed in Supplementary Table S1). (b) By reducing the aging time with the non-equilibrium effective relaxation time τ eff and the β eff index obtained in (a), the variations in different physical properties of various types of glass can be unified into a single master curve, demonstrating the effectiveness of the KWW empirical formula in describing the kinetic decay of relaxation.\n10. Is beta_eff a fit parameter in this figure?\nReply: Yes, β eff is indeed a fit parameter. In Fig. 1a, we used KWW function to fit the absolute values of diverse physical quantities, including recovered enthalpy, hardness, creep modulus, fictive temperature, and strain. These fitted values were then used to normalize the data in Fig. 1b. To clarify this, we have updated the figure caption accordingly and provided the fitting parameters in Supplementary Table S1.\n11. On p. 7 it is mentioned that “these models break down in the non-equilibrium state.” but no reference is given to this claim.\nReply: We have added the corresponding references for the VFT, parabolic, and MYEGA models (Refs. 20-24) to support the statement that these models break down in the non-equilibrium state. This change is on page 7, line 22 of the revised manuscript.\n12. On p. 8 is mentioned that “Substituting these simplified equations into the NM equation yields…” but isn’t it the case that one also needs to assume Eq. (4).\nReply: Yes, Equation (4) is the basic assumption ahead of all others. To avoid ambiguity, we have changed this sentence to “Combining these simplified equations with Equation (3) and Equation (4) yields…” on page 8, lines 16-17.\n13. The basic equation of the paper, Eq. (7), is not entirely new but in fact equivalent to the so-called “single-parameter aging” (SPA) ansatz of Hecksher et al [J. Chem. Phys. 142, 241103 (2015)] (this equivalence with Eq. (7) only comes about because of the Struik power law assumption made a most interesting observation). It was thought that SPA only works for relatively small temperature changes, but SPA has recently been shown to work for also large temperature jumps [J. Chem. Phys. 161, 194504 (2024)]. SPA postulates that the logarithm of the aging rates in equilibrium and out-of-equilibrium is proportional to the temperature difference, which is precisely Eq. (7).\nReply: Thank you for pointing out the connection between Equation (7) and single-parameter aging (SPA) model. After carefully reading the two papers by Hecksher et al [J. Chem. Phys. 142, 241103 (2015), J. Chem. Phys. 161, 194504 (2024)], we learned that SPA model suggests that the difference between logarithm of the aging rates in equilibrium and out-of-equilibrium is proportional to the temperature difference, i.e., , (R1) where ΔT is the temperature difference, (t) is the normalized relaxation function, and c is material-specific constant. By applying the Struik’s power law assumption together with a stretched exponential relaxation function for (t), the equation can be simplified as . (R2) Indeed, the right side of Equation (R2) corresponds exactly to that of Equation (7).\nWe have revised the manuscript (page 10, line 16; page 11, lines 1-3) to clarify the connection between SPA and Equation (7):\n“Actually, Hecksher and coworkers43,44 have proposed a single-parameter aging (SPA) ansatz, suggesting that the difference between logarithm of the aging rates in equilibrium and out-of-equilibrium is proportional to the temperature difference, which corresponds well to Equation (7).”\n14. On p. 12 the word “naturally aged” appears a bit awkward to me.\nReply: To improve clarity and consistency, we have replaced all occurrences of “naturally aged” in the manuscript with “stored or aged under ambient conditions”, which more accurately describes the storage/aging environments of the samples. The corresponding revisions can be found on page 20, lines 23-24 and page 21, lines 8-13.\n15. Also on p. 12, it seems to me that the authors’ approach of heating the sample to T g is similar to that of McKenna et al. in their studies of amber aging.\nReply: We agree with the reviewer’s observation. In the work by McKenna et al. [Nat. Commun. 4, 1783 (2013)] on 20-million-year-old Dominican amber, pristine samples were heated above T g into the equilibrium supercooled liquid region to erase the thermal history (thermal rejuvenation) before subsequent measurements. A similar thermal rejuvenation procedure was also applied in the work of Miguel A. Ramos et al. [Phys. Rev. Lett. 112, 165901 (2014)] on 110-million-year-old Spanish amber. In our experiments, we used the same approach by heating above T g to remove the prior thermal history of long-term aged glasses, which enabled a direct comparison of their properties before and after thermal rejuvenation. These references are cited in the Methods section as Refs. 10 and 11 (page 21, lines 21-23).\n16. P. 13: How well can one estimate the aging temperature of a sample on the Moon where temperature varies very much? Likewise (p. 15) I am surprised that 288 is the average temperature in Poland – a reference to this should be provided.\nReply: We thank the reviewer for this important comment. We have verified the ambient temperatures for both the Chang’E-5 lunar glass and Baltic amber from Poland, as detailed below.\n1. Ambient temperature for Chang’E-5 lunar glass\nIn the previous manuscript, we quoted an average lunar surface temperature of ~253 K, which corresponds to the equatorial mean value taken from Wikipedia and NASA (Christie R J, Plachta D W, Yasan M M. Transient thermal model and analysis of the lunar surface and regolith for cryogenic fluid storage[R]. 2008.). We have now carefully re-examined the actual conditions at the Chang’E-5 landing site (~43°N, 52°W) [Natl. Sci. Rev. 9, nwab188 (2022)].\nAs shown in Fig. R3a (data from [Icarus 283, 300 (2017)]), the surface temperature on the Moon varies strongly with both local time and latitude. At ~43°N, corresponding to the Chang’E-5 landing site (Fig. R3b), the average nighttime temperature of ~97 K with only small fluctuations, whereas daytime temperatures rise from ~97 K to a maximum of ~350 K before returning to ~97 K. By applying a trapezoidal approximation to the daytime temperature curve, the average daytime temperature is calculated to be ~286 K in Fig. R3b.\nAging dynamics are exponentially sensitive to temperature. At the Chang’E-5 landing site, the average daytime temperature is ~189 K higher than the nighttime value. Therefore, the aging dynamics are expected to be dominated by daytime conditions, whereas the nighttime contribution can be neglected. Using the average daytime temperature of 286 K, we revised the effective relaxation time (τ eff) of lunar glasses to ~331 million years (Fig. 3d). Based on the daytime aging dynamics equation ((t) = exp[-(t/331)3/7]), the age of lunar glass corresponds to approximately twice the daytime aging duration. On this basis, the two dated glass beads yield ages of ~161 Ma and ~86 Ma, respectively (Fig. 4).\nFig. R3 (a) Mean bolometric surface temperatures of the Moon as a function of local time for latitude bands at 0°, 30°, 45°, 60°, 70°, 80°, and 85° [Icarus 283, 300 (2017)]. (b) Surface temperature [Icarus 283, 300 (2017)] variation with local time at the Chang’E-5 landing site (~43°N, 52°W). The average nighttime temperature is ~97 K, while the average daytime temperature is ~286 K.\n2. Ambient temperature for Baltic amber from Poland\nRegarding the average annual temperature in Poland, we agree that it is not 288 K. After re-checking the data, we found that the national mean annual temperature in Poland was 282.7 K, as reported in the official Polish climate report issued by the Institute of Meteorology and Water Management - National Research Institute, Poland (www.imgw.pl). Based on this ambient temperature, we recalculated the effective relaxation time of amber to be ~21.6 Ma (Fig. 3d), and its age was accordingly revised to ~49.5 Ma (Fig. 5b).\nIn the revised manuscript, we have re-examined the ambient temperatures at both the Chang’E-5 landing site and in Poland, updated the corresponding effective relaxation times and ages, and added the relevant references. The corresponding revision can be found on page 14, lines 10-15 and page 15, lines 7-24.\n17. On p. 15: The numbers in the exponentials have too many decimals – we usually teach students only to write the numbers that are to be trusted.\nReply: We agree with this point and have revised the manuscript by retaining three significant figures for both the relaxation times in the exponential equations and the glass age data.\n18. On p. 19, it is mentioned that “These results suggest that glass can never be completely frozen”. But is that not what Simon discussed at length in 1931?\nReply: We thank Reviewer #2 for pointing this out. Simon’s paper [Z. Anorg. Allg. Chem. 203, 219-226 (1931)] indeed discussed in detail that molecular motion and entropy changes still occur in glassy materials at low temperatures. Our discussion of “glass can never be completely frozen” is consistent with Simon’s discussion. We have cited Simon’s paper and revised the sentence (page 18, lines 14-15) as:\n“These results are consistent with Simon’s discussion65 in 1931 that glass can never be completely frozen.”\n19. Also on p. 19: What is the MAP model?\nReply: Thank you for pointing this out. The “MAP” model refers to the Mauro-Allan-Potuzak model for glass viscosity. That is, , where A, B, C, K, ΔH are constants for a given glass composition. This phenomenological model was developed by Mauro, Allan, and Potuzak [Phys. Rev. B 80, 094204 (2009)].\nWe copied this abbreviation “MAP” from the paper by Welch et al. [Phys. Rev. Lett. 110, 265901 (2013)]. The relevant excerpt from the paper is: “The measured viscosity data are fit using the MYEGA [14] and MAP [15] expressions in the equilibrium and nonequilibrium regimes, respectively, and lead to an estimated room temperature glassy-state viscosity of 1022.25 Pa s. When this value of viscosity is divided by the shear modulus of the glass (29.5 GPa) following Maxwell’s relation, an estimated structural relaxation time on the order of 19000 yr is obtained.”\nTo ensure clarity for the reader, we have now added the appropriate citation and spelled out the name in the revised manuscript (page 19, lines 12-15).\n“In contrast, using the Mauro-Allan-Potuzak model66, Welch et al. estimated a room temperature viscosity of about 1022.25 Pa∙s for Gorilla glass, corresponding to a relaxation time of 19000 years12.”\n20. It would be nice in the final part of the paper to summarize the basic assumptions made: 1) The material time formalism with single-parameter aging, 2) A stretched exponential relaxation function, 3) the Struik power law relation between time and aging time.\nReply: We appreciate this constructive suggestion. Following your suggestions, we have added a brief summary in the final part of the paper as:\n“Based on the following assumptions: 1) the material time formalism with single-parameter aging, 2) a stretched exponential relaxation function for aging dynamics, and 3) the Struik’s power law relation between instantaneous relaxation time and aging time, we demonstrate that...”\nFinally, we sincerely thank Reviewer #2 again for all the constructive comments and suggestions. These insights have effectively improved our understanding of the issues and enhanced the clarity and readability of the manuscript. We hope that our revisions and detailed responses will meet with the reviewer’s approval."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have revised the manuscript based on the previous comments, now, it can be published in Nature Communications.", "reviewer_2": "The authors have thoroughly addressed all my questions and concerns, and I can now strongly recommend publication in NC of this very nice paper."}, "author_response": ""}]

Decision letter