🔬 condensed matter

Unified Theory of Relaxation in Equilibrium and Nonequilibrium Glass-Forming Liquids

By demonstrating that steady shear suppresses equilibrium stringlike cooperative rearrangements and can be quantitatively described using a shear-dependent effective temperature, this study establishes a unified microscopic theory linking structural relaxation in glass-forming liquids across both equilibrium and nonequilibrium conditions without requiring additional fitting parameters.

Original authors: Zi-Long Wang, Qi-Lu Yuan, Yun-Jiang Wang, Jack F. Douglas, Matteo Baggioli, Zhao-Yan Sun, Wen-Sheng Xu

Published 2026-07-21

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CC BY 4.0

Original authors: Zi-Long Wang, Qi-Lu Yuan, Yun-Jiang Wang, Jack F. Douglas, Matteo Baggioli, Zhao-Yan Sun, Wen-Sheng Xu

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 crowded dance floor where everyone is trying to move, but the music is slowing down to a crawl. This is what happens to liquids when they get cold or get squeezed tight: they turn into glass. You might think glass is just a frozen solid, but to a physicist, it's a liquid that has forgotten how to flow. The big mystery in this field is why it gets so stuck. Is it because the molecules are getting stuck in a specific pattern? Or is it because they are trying to move together in a giant, coordinated group, like a conga line that gets too long to move? Scientists have long suspected that these "conga lines" of molecules, called cooperative motion, are the key to understanding why things slow down. But here's the twist: what happens when you push the glass? If you stir it or shear it (slide layers past each other), it suddenly becomes fluid again, moving fast. The big question has been: does this pushing create a brand new way of moving, or is it just the same old conga lines moving under different rules?

A team of scientists decided to find out by running a massive digital simulation of glass-forming liquids. They built a virtual world filled with tiny particles—some like simple balls and others like long chains of beads—and watched what happened when they cooled them down and then started "stirring" them with a steady shear force. They discovered that the secret to the glass's behavior lies in two things working together. First, the stirring breaks up those long molecular conga lines, making them shorter and less frequent. Second, and more surprisingly, the stirring makes the system act as if it is much hotter than it actually is. The researchers developed a new theory that combines these two effects. They found that if you account for the broken-up lines and this "fake" higher temperature (which they call an effective temperature), you can perfectly predict how fast the glass will relax, whether it's sitting still or being pushed hard. It turns out that the same basic mechanism controls the glass whether it's calm or chaotic; the only difference is that the push changes the "thermodynamic weather" inside the material.

The Story of the Glassy Dance Floor

To understand this discovery, we first need to picture what happens when a liquid turns into glass. Usually, when things cool down, they freeze into a neat, orderly crystal, like snowflakes. But glass is different. When you cool a liquid fast enough, the molecules get too sluggish to arrange themselves into a crystal. Instead, they get stuck in a messy, jumbled state, like a crowd of people frozen in mid-dance. This is the "glass transition."

For decades, scientists have been trying to figure out the rules of this frozen dance. One popular idea is the "Potential Energy Landscape." Imagine the molecules are hikers trying to find the lowest point in a valley. As the liquid cools, the hikers get stuck in deep, narrow valleys (metastable states) and can't climb out because the hills between them are too high. To move, they need to cooperate. This is where the "String Model" comes in. Think of the molecules not as individual dancers, but as people holding hands in a long, winding line—a "string." To move, the whole line has to shuffle together. The longer the line, the harder it is to move, and the slower the liquid flows. This model works great for explaining why glass slows down when it's just sitting there.

But what happens when you push the glass? If you take a jar of honey and stir it, it flows much easier. If you take a block of glass and shear it (slide the top layer sideways), it can suddenly start flowing like a liquid again. This is called "shear thinning" or "shear-induced acceleration." The big puzzle was: How does the push make it flow? Does it break the rules of the game, or does it just change the conditions?

The Virtual Experiment

The authors of this paper, a group of researchers from China and the United States, decided to test this using computer simulations. They didn't use real glass; they used two types of "model" liquids. The first was a mix of two types of particles (called the Kob–Andersen model), and the second was a model of a polymer melt, which is like a soup of long, tangled spaghetti strands. They simulated these systems at different temperatures and applied a steady "shear rate" (a measure of how fast they were sliding the layers past each other).

They watched two main things:

How fast the liquid relaxed: They measured the "structural relaxation time" ( $\tau_\alpha$ ), which is basically how long it takes for the molecules to forget their old positions and move to new ones.
How the "strings" behaved: They tracked the length and number of those cooperative "strings" of particles moving together.

The Findings: Breaking the Lines and Faking the Heat

The results were clear and dramatic. As they increased the shear rate (stirred faster), the relaxation time dropped massively. The liquid started moving orders of magnitude faster. This confirmed that the push definitely speeds things up.

But why? The researchers first checked the "strings." They found that the shear force was indeed breaking up the cooperative strings. The lines of particles moving together got shorter and less common. This made sense: if you break the conga line, it's easier to move. However, when they tried to use this fact alone to predict how fast the liquid would move, the math didn't work. Just knowing the strings got shorter wasn't enough to explain the huge speed-up they saw. There was something else going on.

This led them to a clever idea: Effective Temperature.

In a normal, calm liquid, the temperature tells you how much energy the molecules have to jump over barriers. But when you push a system out of equilibrium (like stirring it), the standard temperature of the room isn't the whole story. The system acts as if it is hotter. The researchers defined a new temperature, $T_{map}$ , which is the temperature an equilibrium system would need to have to move as fast as the pushed system.

They found that as they pushed harder, this "effective temperature" ( $T_{map}$ ) went up. It was as if the shear force was heating up the system from the inside, even though the actual temperature stayed the same.

The Unified Theory

The breakthrough came when they combined these two ideas into a single theory. They took the "String Model" (which works for calm glass) and modified it with two ingredients:

The broken strings: The shear force shortens the cooperative lines ( $z(T)$ ).
The fake heat: The shear force raises the effective temperature ( $T_{map}$ ).

When they plugged these two numbers into their equation, the prediction was perfect. The theory could accurately predict the relaxation time for all the different temperatures and shear rates they tested, without needing to add any new "fudge factors" or adjustable parameters.

This means that the same microscopic mechanism—the cooperative string motion—governs the glass whether it's sitting still or being pushed. The push doesn't create a new way of moving; it just changes the environment. It breaks the long lines and makes the system feel hotter, allowing the molecules to move faster.

What This Means

The paper rules out the idea that shear creates a completely different kind of motion. It also rules out the idea that just breaking the strings is enough to explain the speed-up. Instead, it suggests that the "effective temperature" is the missing link.

The researchers used a concept called the "fluctuation-dissipation relation" to prove that this effective temperature isn't just a mathematical trick. They showed that if you look at how the system responds to tiny nudges while it's being pushed, it behaves exactly like a hotter system would. This gives their theory a solid physical foundation.

In short, the study provides a unified picture: whether a glass-forming liquid is relaxing in peace or being dragged through a shear flow, it's the same dance. The only difference is that the music is faster and the dancers are more energetic because the "effective temperature" has risen. This helps scientists understand not just how glass forms, but how to control it when it's being processed in factories, where materials are constantly being stirred, stretched, and pushed. The paper suggests that by understanding this effective temperature, we might be able to predict and control the behavior of complex materials in ways we couldn't before.

Technical Summary: Unified Theory of Relaxation in Equilibrium and Nonequilibrium Glass-Forming Liquids

Problem Statement
The fundamental mechanism governing the dramatic slowdown of structural relaxation in glass-forming liquids as they transition from equilibrium to nonequilibrium states (such as those under steady shear) remains a central unresolved problem in condensed-matter physics. While the "String Model" successfully describes equilibrium dynamics by linking relaxation times to stringlike cooperative particle rearrangements, it is unclear whether this microscopic mechanism persists under external driving. Furthermore, mechanical deformation accelerates relaxation by orders of magnitude, yet no predictive framework has successfully connected the equilibrium cooperative motion to these accelerated nonequilibrium dynamics without relying on phenomenological fitting parameters.

Methodology
The authors employed molecular dynamics simulations of three representative glass-forming systems:

The Kob–Andersen binary Lennard–Jones (KABLJ) model.
A coarse-grained bead-spring polymer melt under constant volume (Poly-V).
The same polymer melt under constant pressure (Poly-P).

The study investigated these systems across a wide range of temperatures and steady shear rates ( $\dot{\gamma}$ ). Key quantities analyzed included:

Structural Relaxation Time ( $\tau_\alpha$ ): Extracted from the self-intermediate scattering function $F_s(q, t)$ .
Stringlike Cooperative Motion: Quantified via the number-averaged string length $\langle s(t) \rangle$ and the characteristic string length $L$ .
Effective Temperature ( $T_{eff}$ ): Derived independently using the fluctuation-dissipation relation (FDR). The authors compared three definitions: the asymptotic long-time limit ( $T_{slow}$ ), an average over the full FDR curve ( $T_{avg}$ ), and a phenomenological mapping temperature ( $T_{map}$ ) defined by equating nonequilibrium and equilibrium relaxation times.

Key Contributions and Results

Shear-Induced Acceleration and Suppression of Strings:
Simulations confirmed that increasing shear rates dramatically accelerate structural relaxation (reducing $\tau_\alpha$ by several orders of magnitude). Concurrently, steady shear progressively suppresses the size and prevalence of stringlike cooperative rearrangements. The characteristic string length $L$ decreases markedly with increasing shear rate, mirroring the reduction in $\tau_\alpha$ .
Limitations of the Equilibrium String Model under Shear:
The authors tested a minimal extension of the equilibrium String Model, assuming that shear only affects the characteristic string length $L$ while keeping other thermodynamic parameters (activation enthalpy $\Delta H_0$ and entropy $\Delta S_0$ ) at their equilibrium values. This approach failed to quantitatively predict the relaxation times under shear, demonstrating that the suppression of cooperative motion alone is insufficient to explain the accelerated dynamics.
The Generalized String Model with Effective Temperature:
To resolve the discrepancy, the authors proposed a unified theory where the nonequilibrium relaxation time is governed by the same String Model formalism but evaluated under an effective thermodynamic state. The model incorporates two shear-dependent quantities:
- The reduced characteristic string length $z(T) = L(T, \dot{\gamma})/L_A$ .
- An effective temperature $T_{map}$ , which characterizes the thermodynamic conditions of the driven state.
The relaxation time is predicted by:
$\tau_\alpha = \tau_0 \exp\left[ \frac{\Delta H_0 - T_{map} \Delta S_0}{k_B T_{map}} z(T) \right]$

Crucially, all parameters ( $\tau_0, \Delta H_0, \Delta S_0$ ) were fixed using only equilibrium simulation data. No additional nonequilibrium fitting parameters were introduced. This "parameter-free" Generalized String Model quantitatively collapsed the simulation data for all three systems across the full range of temperatures and shear rates.
Microscopic Origin of the Effective Temperature:
The study rigorously linked the phenomenological $T_{map}$ to a microscopic definition derived from the fluctuation-dissipation relation.
- The standard asymptotic effective temperature ( $T_{slow}$ ), derived from the long-time limit of the FDR, failed to match $T_{map}$ and could not predict $\tau_\alpha$ .
- However, the averaged effective temperature ( $T_{avg}$ ), defined by integrating the FDR over the full correlation range (Eq. 6 in the text), was found to be in excellent agreement with $T_{map}$ .
- This establishes that $T_{avg}$ , which captures the coupling of slow structural modes with faster cage-scale fluctuations induced by shear, is the relevant thermodynamic parameter governing activated dynamics in driven glass-formers.

Significance
The paper claims to provide a unified microscopic description of thermal and mechanically driven dynamics in glass-forming liquids. The primary significance lies in demonstrating that equilibrium and nonequilibrium relaxation are governed by the same underlying cooperative mechanism (stringlike motion), but they occur under different effective thermodynamic conditions.

Specifically, the work establishes that:

External driving does not introduce a fundamentally new relaxation mechanism but rather modifies the effective thermodynamic state (via $T_{map} \approx T_{avg}$ ) and suppresses the scale of cooperative motion.
The effective temperature derived from the fluctuation-dissipation relation is not merely a phenomenological fitting parameter but a physically grounded quantity that quantitatively predicts structural relaxation times without ad hoc adjustments.
This framework bridges the gap between equilibrium theories (like Adam-Gibbs and the String Model) and nonequilibrium rheology, offering a predictive tool for understanding glassy dynamics under processing conditions.

The authors conclude that this unified description holds for both simple liquids and polymer melts, suggesting a robust mechanism for driven amorphous materials, while noting that extension to other nonequilibrium conditions (e.g., physical aging) remains a subject for future investigation.

The Story of the Glassy Dance Floor

The Virtual Experiment

The Findings: Breaking the Lines and Faking the Heat

The Unified Theory

What This Means

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