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Universal Dynamic Scaling for Sub-Yield Aging and Over-Yield Rejuvenation in A Cyclically Driven Metallic Glass

This study resolves the debate on the distinct origins of sub-yield aging and over-yield rejuvenation in cyclically driven metallic glasses by revealing a unified dynamic clock where over-yield rejuvenation is driven by macroscopic plastic shear work, while sub-yield aging is governed by second-order localized elastic stress fluctuations, thereby establishing a universal scaling identity that bridges non-linear rheology with stress fluctuation dynamics.

Original authors: Nannan Ren, Hongfei Zhang, Rui Wan, Pengfei Guan, Limin Wang, Baoshuang Shang

Published 2026-10-16

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CC BY 4.0

Original authors: Nannan Ren, Hongfei Zhang, Rui Wan, Pengfei Guan, Limin Wang, Baoshuang Shang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a material that looks and feels like a solid metal, yet its atoms are arranged with the chaotic disorder of a frozen liquid. This is a metallic glass. Unlike ordinary metals, which have neat, repeating crystal structures that allow them to bend and stretch without breaking, these glasses lack that internal order. Instead, their atoms are jammed together in a rugged, uneven landscape of energy. When you push or pull on them, the atoms must hop from one deep valley in this landscape to another to move. Scientists have long known that how they move depends on how hard you push. If you push gently, the material tends to settle deeper into its valleys, becoming harder and more stable, a process called aging. If you push hard enough, you shake the atoms out of their deep valleys and into shallower, higher-energy spots, making the material softer and more ductile, a process called rejuvenation. For years, researchers have debated whether these two opposite behaviors—settling down versus shaking up—stem from the same underlying physics or if they are driven by completely different microscopic rules.

A team of researchers has now used powerful computer simulations to settle this debate, revealing that both aging and rejuvenation are governed by a single, unified clock that dictates how the material navigates its energy landscape. By studying a model of a copper-zirconium metallic glass, they discovered that the difference between the two behaviors is not a change in the fundamental rules, but rather a shift in how energy is injected into the system. When the material is pushed beyond its yield point—the limit where it starts to flow permanently—the rejuvenation is driven directly by the large-scale work of the plastic deformation. However, when the material is pushed gently below that limit, the aging process is not driven by the overall push, but by tiny, localized fluctuations in stress that act like a mechanical catalyst, nudging atoms just enough to help them settle.

The researchers simulated the behavior of 16,000 atoms in a block of this glass, subjecting them to repeated cycles of stretching and squeezing. They varied the intensity of the stretch and the speed at which they applied the force. In the simulations, when the stretching was gentle, the material consistently lost energy, dropping into deeper, more stable states over time. This confirmed the aging effect. But when they increased the intensity of the stretch beyond a specific threshold, the behavior flipped. The material began to gain energy, climbing out of its deep valleys and settling into a higher-energy, more fluid state. The most intriguing results appeared at the boundary between these two worlds. At intermediate levels of stress, the material would initially gain energy, as if it were being rejuvenated, only to turn around and start losing energy and aging again after a certain number of cycles. This non-monotonic behavior, where the material first wakes up and then goes back to sleep, had been observed before but lacked a clear explanation.

By analyzing the data, the team found a precise mathematical relationship connecting the two regimes. They identified that the rate at which the material rejuvenates under heavy stress follows a specific power law, while the rate at which it ages under light stress follows a different power law. Crucially, they found that the exponent governing the aging process is exactly twice the exponent governing the rejuvenation process. This simple ratio revealed the hidden mechanism: the intense shaking of the material creates tiny, localized ripples in stress. Under heavy loads, these ripples are overwhelmed by the massive flow of the material. But under light loads, where the overall flow stops, these tiny ripples become the dominant force. They act as a source of mechanical noise, preventing the atoms from getting stuck in the deepest possible holes and slowing down the aging process. The faster the material is cycled, the stronger these ripples become, making it harder for the atoms to settle down.

This discovery provides a single framework to predict how the material will evolve. The researchers showed that the transition point between the initial rejuvenation and the subsequent aging can be predicted by balancing the energy injected by the external force against the energy dissipated by these internal stress fluctuations. It is as if the material has an internal timer that counts how many cycles of stress are needed before the tiny, local ripples take over and drive the system toward a stable state. This finding bridges the gap between the large-scale flow of the material and the microscopic jitters of its atoms. It suggests that by carefully controlling the speed and intensity of mechanical training, engineers could potentially tune the properties of metallic glasses, making them stronger or more ductile as needed. The work demonstrates that even in a material as disordered as a glass, the path from chaos to order is governed by a consistent, universal set of rules.

Technical Summary: Universal Dynamic Scaling for Sub-Yield Aging and Over-Yield Rejuvenation in a Cyclically Driven Metallic Glass

Problem Statement
The structural evolution of amorphous solids, specifically metallic glasses (MGs), under cyclic mechanical deformation presents a fundamental duality. Depending on the strain amplitude ( $\gamma_A$ ), cyclic driving induces either mechanical aging (sub-yield regime), where the system relaxes into deeper, more stable energy minima, or structural rejuvenation (over-yield regime), where the system is driven into shallower, higher-energy states. A longstanding unresolved debate concerns whether these opposing trajectories arise from distinct microscopic mechanisms or share a common physical origin. Furthermore, a unified quantitative framework connecting sub-yield relaxation kinetics to over-yield non-linear flow rheology has been conspicuously absent, particularly regarding the transition dynamics at intermediate strain amplitudes where non-monotonic energy evolution occurs.

Methodology
The authors employed extensive classical Molecular Dynamics (MD) simulations using the LAMMPS package on a prototypical binary metallic glass system, Cu $_{64}$ Zr $_{36}$ , comprising 16,000 atoms. Interatomic interactions were modeled using a well-validated Embedded-Atom Method (EAM) potential.

Simulation Protocol: Initial configurations were prepared by quenching the liquid phase from 2000 K to 300 K at 1 K/ps, followed by thermal relaxation.
Deformation: Cyclic uniaxial tension-compression was applied along the x-direction using a sinusoidal strain protocol, $\gamma(t) = \gamma_A \sin(2\pi t/t_p)$ , covering a wide range of strain amplitudes ( $\gamma_A$ ) and loading periods ( $t_p$ ).
Data Analysis: The potential energy change ( $\Delta U$ ) was tracked as a function of the loading cycle number ( $N_{cycle}$ ), recorded strictly at zero-strain states to eliminate transient elastic contributions. Statistical averages were computed over 10 independent configurations generated via the isoconfigurational method.

Key Contributions and Results

Identification of Distinct Scaling Regimes:
The study maps the dynamical response across the ( $\gamma_A$ , $t_p$ ) parameter space, identifying four operational domains:
- Region 1 (Sub-yield): Monotonic mechanically accelerated aging.
- Regions 2 & 3 (Crossover): A dynamic transition zone characterized by non-monotonic trajectories (initial rejuvenation followed by aging).
- Region 4 (Over-yield): Sustained mechanical rejuvenation reaching a steady-state energy plateau.
Universal Scaling Laws:
The authors uncover two distinct universal scaling exponents governing these regimes:
- Over-yield Rejuvenation (Region 4): The steady-state energy plateau ( $\Delta U_p$ ) scales with macroscopic plastic work derived from the Herschel-Bulkley (H-B) rheological model. The scaling relation is $\Delta U_p \propto \gamma_A \dot{\gamma}^\alpha$ , with a shear-thinning exponent $\alpha \approx 0.17$ .
- Sub-yield Aging (Region 1): The energy evolution follows a master curve when scaled by $\dot{\gamma}^\beta / \gamma_A$ , yielding an aging exponent $\beta \approx 0.34$ . The evolution is fitted by a stretched-exponential decay function.
The Unifying Scaling Identity ( $\beta \approx 2\alpha$ ):
A central finding is the quantitative relationship $\beta \approx 2\alpha$ (0.34 $\approx$ 2 $\times$ 0.17). The paper proposes a unified physical mechanism to explain this:
- First-Order Response (Over-yield): Rejuvenation is driven directly by macroscopic plastic shear work ( $\propto \dot{\gamma}^\alpha$ ).
- Second-Order Response (Sub-yield): In the absence of global plastic flow, structural aging is mediated by second-order local elastic stress fluctuations ( $\langle \delta\sigma^2_{local} \rangle$ ). These fluctuations, inheriting the stress scale of the driven matrix ( $\propto \dot{\gamma}^\alpha$ ), act as a mechanical noise source. The agitation energy scales as $\langle \delta\sigma^2_{local} \rangle \propto (\dot{\gamma}^\alpha)^2 = \dot{\gamma}^{2\alpha}$ .
- Mechanism: Intense high-rate agitation increases the probability of escaping deep potential energy minima, thereby suppressing the retention efficiency of structural relaxation. The authors establish a dynamic rationale demonstrating that $\beta \approx 2\alpha$ , suggesting a plausible physical origin connecting sub-yield relaxation to over-yield rheology.
Dynamics of the Crossover ( $N_c$ ):
For intermediate amplitudes (Regions 2 and 3), the system exhibits a transient crossover from rejuvenation to aging at a characteristic cycle number $N_c$ . The authors frame $N_c$ as a dynamical balance between first-order plastic work injection and second-order elastic fluctuation dissipation. They derive a universal scaling relation: $N_c \dot{\gamma}^{2\alpha} = G(\dot{\gamma}^\alpha / \gamma_A)$ , where $G$ is a universal scaling function. This "dynamic clock" dictates the transition from short-term excitation (rejuvenation) to long-term collective relaxation (aging).

Significance and Claims
The paper claims to resolve the duality between sub-yield aging and over-yield rejuvenation by establishing a unified dynamic clock that dictates energy landscape navigation across both regimes.

Theoretical Bridge: It directly bridges non-linear rheology (Herschel-Bulkley framework) with stress fluctuation dynamics, providing a single framework to predict structural evolution.
Mechanistic Insight: It redefines sub-yield aging not merely as passive relaxation but as a process governed by second-order local elastic stress fluctuations acting as a "mechanical catalyst" and noise source.
Predictive Capability: By identifying the scaling identity $\beta \approx 2\alpha$ and the master collapse of crossover boundaries, the work offers a predictive protocol for tailoring the structural state of amorphous materials through cyclic mechanical training.

The authors note that while the specific exponent values ( $\alpha \approx 0.17, \beta \approx 0.34$ ) are specific to the uniaxial NPT conditions of the Cu $_{64}$ Zr $_{36}$ model, the underlying fluctuation-driven framework and the scaling relation $\beta \approx 2\alpha$ represent a fundamental connection between stress fluctuations and relaxation dynamics that warrants further verification across diverse loading modes.

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