Potential-Energy-Based Material Time in Glassy Aging

• Potential-energy-based material time is a formalism that defines an intrinsic clock for describing aging in glassy materials through potential energy and configuration-space metrics.
• It connects equilibrium fluctuations with nonlinear aging responses by mapping laboratory time into material time using distance-based relaxation functions.
• The approach enables master-curve collapse of aging data, highlighting dynamic rigidity percolation and the dominant role of slow particles in glass dynamics.

Potential-energy-based material time is a formalism that provides a unified framework for describing the physical aging of noncrystalline glassy materials. It models the evolution of glassy states via an intrinsic "material clock," whose progression is set by the evolving microscopic structure of the system, particularly through metrics quadratic in potential energy or configuration-space distances. This approach robustly connects nonlinear glass aging dynamics to equilibrium response functions and equilibrium fluctuation statistics.

1. Formal Definition and Material-Time Clock Rate

The central concept is the material time $\xi(t)$, a generalized clock variable defined by the integral

$\xi(t) = \int_0^t \gamma(t') \, dt'$

where $\gamma(t) = d\xi/dt$ is the instantaneous aging or clock rate. In equilibrium at temperature $T$, $$\gamma(t) = \gamma_\text{eq}(T) = 1/\tau_\text{eq}(T)$$ corresponds to the inverse (α-)relaxation time. Out of equilibrium, the clock rate depends on a single structural parameter $X(t)$, such as potential energy, density, or enthalpy:

$$\ln[\gamma(t)] - \ln[\gamma_\text{eq}(T)] = \Lambda [X(t) - X_\text{eq}(T)]$$

Thus,

$$\gamma(t) = \gamma_\text{eq}(T) \exp[\Lambda \cdot (X(t) - X_\text{eq}(T))]$$

where $\Lambda$ is a material-specific constant and $X_\text{eq}(T)$ is the equilibrium value of $X$ (Riechers et al., 2021).

2. Material-Time Response and Aging Convolution Formalism

For small perturbations, the linear aging response following an instantaneous temperature jump $\Delta T$ at $t=0$ is described by

$X(t) - X_\text{eq}(T_b) = \Delta X R_\text{lin}(t)$

where $$\Delta X \equiv X_\text{eq}(T_i) - X_\text{eq}(T_b)$$, and $R_\text{lin}(t)$ is the normalized relaxation function ($R_\text{lin}(0) = 1$, $R_\text{lin}(\infty) = 0$).

The Narayanaswamy-Kovacs material-time generalization provides that the same relaxation function governs nonlinear aging if laboratory time is replaced by material time. For $N$ instantaneous jumps with magnitudes $\Delta X_i$ at times $t_i$, the trajectory in material time is

$$X(\xi) = X_\text{eq}(T_N) + \sum_{i=1}^{N} \Delta X_i R_\text{lin}(\xi - \xi_i)$$

with $\xi_i = \xi(t_i)$, and the mapping between $\xi$ and $t$ is performed by integrating $\gamma(t)$ (Riechers et al., 2021).

3. Connection to Equilibrium Potential-Energy Fluctuations via FDT

The fluctuation-dissipation theorem (FDT) establishes that the linear relaxation function $R_\text{lin}(t)$ is directly computable from equilibrium autocorrelation functions. For observables $X$ with generalized enthalpy $U$ (i.e., potential energy),

$$R_\text{lin}(t) = \frac{\langle \delta U(0) \delta U(t) \rangle}{\langle \delta U(0)^2 \rangle}$$

where $\delta U = U - \langle U \rangle_\text{eq}$.

In the infinitesimal-jump (small $\Delta X$) regime, the response is linear, but for finite $\Delta X$, the rate $\gamma(t)$ evolves nonlinearly according to the instantaneous deviation $\Delta X(t)$, leading to "stretching" or "compression" of aging timescales (Riechers et al., 2021).

4. Distance-as-Time: Microscopic Interpretation and Geometric Properties

Recent developments have reinterpreted material time as a configuration-space distance. For an $N$-particle glass, considering the $3N$-dimensional position vector $\mathbf{R}(t)$ or its inherent-state analogue $\mathbf{I}(t)$ (potential-energy minima), one defines mean-square displacement (MSD) measures:

• Standard MSD: $$d_{12} = \langle |\mathbf{R}(t_2) - \mathbf{R}(t_1)|^2 \rangle / N$$
• Inherent MSD: $$d_{12}^I = \langle |\mathbf{I}(t_2) - \mathbf{I}(t_1)|^2 \rangle / N$$

Material time can be set proportional to such distance metrics:

$\xi(t) \equiv d(t_0, t)$

A particularly effective measure is the inherent harmonic MSD:

$$\langle \Delta^2 \rangle_\text{harm}(t) \equiv \left[\langle 1/\Delta_i^2(t) \rangle_i \right]^{-1}$$

where $$\Delta_i^2(t) = |\mathbf{r}^I_i(t) - \mathbf{r}^I_i(t_0)|^2$$, emphasizing the slowest particles and providing optimal data collapse of aging curves (Douglass et al., 2022).

Additivity and monotonicity of such distance-based material time yield two key geometric properties in configuration space:

• Unique-triangle property: for any triplet of times, the third distance is a function $F$ of the other two, i.e., $d_{13} = F(d_{12}, d_{23})$.
• Geometric reversibility: $F(x, y) = F(y, x)$, holding both in equilibrium and under aging (Douglass et al., 2022).

5. Dominance of Slow Particles and Dynamic Rigidity Percolation

Physical aging is predominantly controlled by the slowest particles in the system. Computer simulations on the Kob–Andersen binary Lennard-Jones mixture reveal that at low temperatures, a percolating network of the slowest $\sim$25% of particles persists for long times, with overall relaxation occurring only when this network finally breaks apart. Correspondingly, the materal time $\xi(t)$—when defined by harmonic MSD—collapses aging curves more effectively than metrics based on all particles (Douglass et al., 2022). This suggests a dynamic-rigidity-percolation perspective in which global aging is arrested or advanced by the collective motion of slow constituents.

6. Recovery of the Tool–Narayanaswamy Convolution Integral

The formalism of material time subsumes classical aging convolution models, notably the Tool–Narayanaswamy (TN) approach. The Stieltjes integral for linear response,

$$q_a(t) = \int_{-\infty}^t \phi_{ab}(t-t')\, \delta e_b(t')\, dt'$$

with $e_b$ an external field and $\phi_{ab}$ the response kernel, is reparametrized by distance-based material time as

$$q_a(t) = \int_{-\infty}^{\xi(t)} \psi_{ab}(\xi(t) - \xi')\, \delta e_b(\xi')\, d\xi'$$

where $\psi_{ab}$ is appropriately redefined. This construction recovers TN aging convolution, conditional on the assumption that the relationship between response and distance under small perturbations holds under moderate aging (Douglass et al., 2022).

7. Master-Curve Collapse and Quantitative Validation

Simulations of potential-energy relaxation after temperature jumps display master-curve collapse when relaxation is plotted against configuration-space material time. For example, in the Kob–Andersen system, plotting the normalized relaxation

$R(t) \equiv [U(t) - U(T)] / [U(T_i) - U(T)]$

against material time defined by inherent harmonic MSD aligns a wide range of aging curves (induced by various temperature jumps) onto a single function $J(\xi)$. This behavior strongly supports the material time approach's universality and predictive power in physical aging (Douglass et al., 2022).

Experimental and simulation results confirm that, for both molecular glasses and model binary mixtures, nonlinear aging predictions derived from equilibrium potential-energy fluctuations yield quantitatively accurate trajectories, even for substantial temperature jumps (~15%) (Riechers et al., 2021). This rigorously establishes that the nonlinearity in glass aging is effectively governed by the material-time clock rate controlled by single-parameter deviations and their connection to equilibrium fluctuations.

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Taken together, potential-energy-based material time unifies phenomenological aging models, equilibrium response theory, microscopic dynamics, and geometric properties of configuration space trajectories. Its application spans predictive aging in glassy systems, connects theory with experiment via equilibrium fluctuation measurements, and elucidates the dominant role of slow constituents and dynamic percolation phenomena.