🔬 materials science

Thermal softening of the yield stress in glasses with different stability

Using molecular dynamics simulations, this study demonstrates that while thermal fluctuations soften the yield stress of glasses across all stability levels following a universal $T^{2/3}$ law, the magnitude of this softening at shear start-up is strongly dependent on the glass's stability, with highly stable glasses exhibiting significantly less sensitivity to temperature than poorly annealed ones.

Original authors: Tianze Wan, Ludovic Berthier

Published 2026-10-14

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CC BY 4.0

Original authors: Tianze Wan, Ludovic Berthier

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

Amorphous solids are materials that look like solids but lack the orderly, repeating crystal structure found in metals or salt. Think of window glass, certain plastics, or even foams; they are rigid enough to hold their shape under small pushes, yet they can flow like thick liquids if pushed hard enough. When these materials are stressed beyond a certain point, they undergo a dramatic change called "yielding," where they stop behaving like a solid and begin to flow. Scientists have long known that how a glass is prepared matters immensely. A glass that has been cooled slowly and carefully, known as a well-annealed glass, tends to break suddenly and sharply when stressed. In contrast, a glass that was cooled quickly or prepared with less care, a poorly annealed glass, tends to deform smoothly and gradually before flowing.

For decades, researchers studied this breaking point in a theoretical world where temperature plays no role, focusing only on the mechanical force applied. However, in the real world, heat is always present. Tiny, random jiggles of atoms caused by temperature can help a material overcome the barriers that keep it rigid, effectively softening it. While scientists have a good grasp of how heat affects flowing liquids, it has been much harder to understand how thermal energy influences the precise moment a solid glass decides to break, especially when that glass has different levels of internal stability. The question remains: does the history of how a glass was made change the way heat helps it yield?

To answer this, researchers at ESPCI Paris turned to a powerful computer simulation to watch a model glass deform under heat and pressure. They created a virtual container filled with 1,500 particles of varying sizes, designed to mimic the messy, non-crystalline structure of real amorphous solids. Using a sophisticated algorithm that allows particles to swap places to find their most stable arrangements, they prepared these virtual glasses at different "parent" temperatures. Some were cooled from very high temperatures to create unstable, poorly prepared glasses, while others were cooled from much lower temperatures to create highly stable, well-prepared glasses. They then subjected these digital materials to a shearing force, sliding layers of the material past one another, while carefully controlling the temperature of the environment.

The team observed two distinct scenarios. First, they looked at what happens when the material is already flowing steadily. They found that as the temperature rose, the force required to keep the material flowing decreased. This softening effect followed a predictable pattern: the yield stress dropped in proportion to the temperature raised to the power of two-thirds. This mathematical relationship held true regardless of how stable the glass was or how fast it was being sheared. When they mathematically removed the temperature from their equations, the results perfectly matched the known strength of these materials in a cold, heat-free world. This confirmed that for a material already in motion, the way heat softens it is universal and does not depend on its preparation history.

The story changed, however, when they examined the moment the material first started to flow, known as the start-up phase. Here, the initial stability of the glass became the deciding factor. When they pulled on the highly stable glasses, the material held its ground firmly, and the rising temperature had very little effect on its strength. These stable materials were remarkably resistant to thermal softening. In stark contrast, the poorly annealed, unstable glasses were much more sensitive. For these materials, even a small increase in temperature caused a significant drop in the force needed to make them yield. The researchers found that while the mathematical rule for how the strength dropped with temperature remained the same, the size of the drop was entirely controlled by the glass's stability. The unstable glasses softened dramatically, while the stable ones barely budged.

This discovery reveals a crucial nuance in the physics of breaking materials. It shows that while the fundamental mechanism of thermal softening is the same for all glasses, the magnitude of the effect is not. A material's history—specifically how stable its internal structure is—acts as a filter that determines how much the ambient heat will weaken it. For a stable glass, the internal barriers to flow are so high that thermal jiggles cannot easily push the material over the edge. For an unstable glass, those barriers are lower, allowing heat to do the heavy lifting of breaking the structure. Consequently, one cannot simply look at how a flowing liquid softens with heat and assume the same rules apply to the moment a solid first breaks. The stability of the glass must be accounted for explicitly, as it dictates whether the material will be a tough, heat-resistant solid or a fragile one that yields easily to the warmth of its environment.

Technical Summary: Thermal Softening of the Yield Stress in Glasses with Different Stability

Problem Statement
The yielding behavior of amorphous solids is well-characterized in the athermal quasi-static (AQS) limit, where preparation history (specifically glass stability) dictates whether yielding is brittle or ductile. However, the influence of thermal fluctuations on yielding remains less understood, particularly regarding how the stability of the glass affects thermal softening. While previous studies have identified a universal $T_s^{2/3}$ reduction in yield stress for metallic glasses (attributed to thermal activation near saddle-node instabilities), it is unclear whether this exponent and the amplitude of softening depend on the initial stability of the glass or the shear rate. Crucially, existing numerical tests have been limited to conventionally prepared glasses with narrow stability ranges, failing to systematically decouple the effects of the deformation temperature ( $T_s$ ) from the parent preparation temperature ( $T_p$ ).

Methodology
The authors employ molecular dynamics simulations of a three-dimensional size-polydisperse soft-sphere model glass ( $N=1500$ particles) to investigate yielding under finite temperature ( $T_s$ ) and finite shear rate ( $\dot{\gamma}$ ).

Glass Preparation: To access a broad range of glass stabilities, the authors utilize the swap Monte Carlo algorithm to equilibrate the system at various parent temperatures ( $T_p$ ) ranging from 0.062 (highly stable) to 0.125 (poorly annealed). This allows for the preparation of glasses far below the conventional glass-transition regime.
Simulation Protocols:
- Steady Flow: Equilibrium configurations are sheared at constant $\dot{\gamma}$ and $T_s$ to determine the steady-state stress $\sigma_{ss}$ .
- Shear Start-up: Equilibrium configurations are quenched to their inherent structures, assigned Maxwell–Boltzmann velocities at $T_s$ , and then sheared. This protocol isolates the response of the initial glass structure before structural relaxation occurs.
Thermal Control: A Nosé–Hoover thermostat is applied to the kinetic temperature with a short damping time (5 time steps) to prevent oscillations and ensure the thermostat removes shear heating without interfering with the driven dynamics.
Analysis: The study analyzes flow curves $\sigma_{ss}(\dot{\gamma})$ and start-up stress-strain curves to extract the dynamic yield stress ( $\sigma_y$ ) and the start-up yield stress ( $\sigma_{max}$ ), respectively.

Key Contributions and Results

Universal Thermal Softening Law:
The study confirms that thermal fluctuations soften both the dynamic yield stress (steady state) and the start-up yield stress. In all cases, the temperature dependence of the yield stress reduction follows the $T_s^{2/3}$ law expected for thermally activated yielding near a saddle-node instability:
$\sigma(T_s) = \sigma_{ath} - A T_s^{2/3}$
The exponent $\mu = 2/3$ is found to be robust, showing no systematic dependence on glass stability ( $T_p$ ) or shear rate ( $\dot{\gamma}$ ). Extrapolations to zero temperature successfully recover the known athermal yield stresses ( $\sigma_{AQS}$ ).
Stability-Dependent Softening Amplitude:
While the exponent is universal, the amplitude of the thermal softening ( $A$ ) depends strongly on the glass stability.
- Steady Flow: The dynamic yield stress softening amplitude ( $A_{ss}$ ) is independent of preparation history, as the system loses memory of its initial state in the steady flow.
- Shear Start-up: The start-up yield stress softening amplitude ( $A_{max}$ ) varies significantly with $T_p$ . Highly stable glasses (low $T_p$ ) exhibit a much smaller softening amplitude and are less sensitive to temperature than poorly annealed glasses (high $T_p$ ).
- Quantitative Comparison: For stable glasses, the start-up softening is weaker than the steady-flow softening ( $A_{max} < A_{ss}$ ). Conversely, for poorly annealed glasses, the start-up softening is stronger ( $A_{max} > A_{ss}$ ). The crossover occurs around $T_p \approx 0.08$ .
Mechanical Response Characteristics:
The study reproduces the known dichotomy in yielding behavior at finite temperature: stable glasses exhibit brittle yielding (sharp stress drop, shear banding) even with thermal fluctuations, while poorly annealed glasses yield in a ductile manner (smooth stress evolution). The thermal softening preserves this distinction, with the amplitude of softening scaling with the initial stability.

Significance and Claims
The paper claims that a complete description of yielding at finite temperature must explicitly account for the stability of the glass. The authors demonstrate that while the mechanism of thermal softening (governed by the $T_s^{2/3}$ exponent) appears universal, the magnitude of the effect is not. Consequently, thermal sensitivity measured in steady flow cannot be used to predict the thermal sensitivity of the start-up yield stress, as the latter can be either larger or smaller depending on the preparation history.

The results suggest that the distribution of activation barriers in the initial glass state is preparation-dependent, with stable glasses possessing fewer low-barrier regions or larger typical barriers, making them less susceptible to thermal activation prior to mechanical instability. The authors conclude that these findings necessitate theoretical descriptions of thermal softening that incorporate glass stability, potentially within frameworks such as elastoplastic models, shear transformation zone theory, or random first-order transition theory.

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