🔬 condensed matter

A quantitative approach to flowing supercooled liquids: From microscopic heterogeneities to rheology

This paper presents a quantitative theoretical model for supercooled liquids that, using only equilibrium parameters, successfully links microscopic structural heterogeneities to macroscopic rheological behaviors by conceptualizing flow as a dynamic coexistence of solid-like and liquid-like regions.

Original authors: Dong-Xu Yu, Zhe Wang

Published 2026-06-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Dong-Xu Yu, Zhe Wang

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 you have a jar of honey that has been left in the freezer. It's not frozen solid like ice, but it's incredibly thick and sluggish. This is what scientists call a supercooled liquid. It's a strange state of matter: it looks like a liquid, but it acts like a solid in many ways.

For a long time, scientists have struggled to predict exactly how these "sticky" liquids will flow when you stir them, squeeze them, or pull them. This paper by Dong-Xu Yu and Zhe Wang offers a new, simpler way to understand this behavior by treating the liquid not as a uniform soup, but as a mixture of two different worlds happening at the same time.

Here is the breakdown of their discovery, using everyday analogies:

1. The Two-Worlds Idea: The "Solid Islands" in a Liquid Sea

Usually, we think of a liquid as one big, uniform puddle. But the authors suggest that when you start moving a supercooled liquid, it actually splits into two types of regions:

The Liquid-like Regions: These are the "normal" parts that flow easily, like water.
The Solid-like Regions (LERs): These are tiny, temporary islands of rigidity. Think of them as solid icebergs floating in a warm ocean.

Even though the whole jar is technically a liquid, these "icebergs" form because the molecules get stuck in a tight, cage-like arrangement. They can't move easily, so they act like a solid for a brief moment. They build up stress (like a rubber band being stretched) until they suddenly snap or "yield," turning back into liquid.

2. The Secret Ingredient: The "Social Network" of Molecules

The paper introduces two key concepts that make their model work:

The Correlation Length (The "Neighborhood" Effect):
Imagine a neighborhood where if one house gets a new roof, the neighbors are likely to get one too, not just because they want to, but because they are connected. In the liquid, the "solid islands" don't form randomly. They cluster together based on how tightly the molecules are packed. The authors found a specific "neighborhood size" (a correlation length) that determines how far this packing influence reaches. This size is fixed by the liquid's natural state before you even start stirring it.
Localized Elasticity (The "Walled Garden"):
This is the most crucial part. In a normal solid (like a metal beam), if you push one end, the whole beam feels the pressure instantly. But in these "solid islands" inside the liquid, the pressure stays local.
- Analogy: Imagine a crowd of people. In a solid, if you push the person at the front, the whole line shoves forward. In this supercooled liquid, if you push a group of people (a solid island), they push back only against each other. The people outside that specific group don't feel the push immediately. The "solid" behavior is trapped inside its own little garden.

3. What Happens When You Stir? (The Flow)

When you start stirring the liquid (applying shear):

At slow speeds: Most of the liquid is still "liquid-like." It flows easily.
As you speed up: You force more and more molecules to move faster than they naturally want to. This traps them in those "solid islands." The liquid starts to act more like a solid, resisting your stir.
The Result: This explains shear thinning. It sounds counterintuitive, but as you stir faster, the liquid actually gets thinner (flows easier) eventually. Why? Because the "solid islands" keep getting broken down and reformed faster than they can build up enough stress to stop you.

4. The "Start-Up" Surprise (The Stress Overshoot)

If you suddenly start stirring a thick liquid, you often feel a big "jolt" or peak in resistance before it settles into a smooth flow. This is called stress overshoot.

The Paper's Explanation: When you first start, the "solid islands" are fresh and strong. They fight back hard, creating a peak in resistance. But as you keep stirring, these islands get battered, break apart, and rearrange. Once they break, the resistance drops, and the liquid flows smoothly.
The Success: The authors' model predicted this "jolt" perfectly, matching computer simulations without needing to guess any numbers. They only used data from the liquid when it was sitting still (equilibrium) to predict how it would behave when moving.

5. The Big Picture: Why This Matters

The authors show that you don't need to invent new laws of physics to explain these liquids. You just need to realize that:

The liquid is a mix of solid-like and liquid-like patches.
These patches are connected by a specific "neighborhood" size determined by how the molecules pack together.
The "solid" parts can only push back on their immediate neighbors, not the whole system.

By combining these ideas, they created a mathematical recipe that accurately predicts how these tricky liquids will flow, how thick they will get, and how they will react to sudden movements. It bridges the gap between the microscopic world (how individual molecules move) and the macroscopic world (how the whole jar of honey flows).

In short: The paper says that flowing supercooled liquids are like a chaotic dance floor where some dancers suddenly freeze into rigid poses (solid islands) while others keep dancing (liquid). The way these frozen groups form, interact with their neighbors, and eventually break apart explains all the complex flow behaviors we see.

Technical Summary: A Quantitative Approach to Flowing Supercooled Liquids

Problem Statement
Supercooled liquids exhibit complex flow behaviors across macroscopic and molecular scales, including shear thinning, stress overshoots during start-up shear, and dynamic heterogeneity (DH). Current theoretical frameworks struggle to provide a unified description that simultaneously captures steady-state rheology, transient mechanical responses, and the evolution of microscopic heterogeneities. Existing approaches often focus either on the liquid-state physics of shear thinning (e.g., bond-breaking clusters, cooperative relaxations) or the solid-state physics of amorphous yielding (e.g., cage-scale dynamics), but fail to bridge these regimes or account for the transient response. Furthermore, many models rely on fitting parameters to flow data rather than deriving predictions from equilibrium properties.

Methodology
The authors propose a quantitative theoretical model based on a two-state paradigm, conceptualizing the flow of supercooled liquids as a dynamic coexistence of transient solid-like and liquid-like regions. The model is built upon the concept of Localized Elastic Regions (LERs)—mesoscopic regions that deform elastically before yielding.

The model integrates two critical physical ingredients:

Structural Correlation Length ( $\xi$ ): A length scale characterizing medium-range structural order and the spatial clustering of local energy barriers ( $E_b$ ). This is derived from equilibrium configurations by maximizing the correlation between coarse-grained structural parameters (local free volume) and dynamic parameters (particle-wise local configurational relaxation time, $\tau_{LC}$ ).
Localization of Elasticity-Mediated Interaction: Unlike standard elastoplastic models where stress redistribution is global, this model posits that long-range anisotropic elastic interactions (Eshelby-type) are restricted to solid-like regions (LERs). Liquid-like regions exhibit short-ranged, uncorrelated flow events.

The model operates on a spatially discretized grid of "blocks," each characterized by an energy barrier $E_b$ . The evolution of the system involves:

Shear-facilitated activation: Blocks activate when the accumulated elastic energy overcomes the barrier.
Block Renewal: After yielding, a block's energy barrier is renewed via a combination of thermal randomness, weighted averaging of neighboring barriers (governed by $\xi$ ), and a drift term to prevent spatial stagnation.
Two Moduli: The model assigns distinct shear moduli to solid-like ( $\mu_S$ ) and liquid-like ( $\mu_L$ ) blocks, reflecting the transient nature of elasticity in supercooled liquids.
Equation of Motion: A constitutive equation (Eq. 11) governs stress evolution, incorporating external loading, local relaxation, and inter-block stress redistribution (active only when a connected path of solid-like blocks exists).

Crucially, all model parameters are determined solely from the equilibrium state (quiescent properties), allowing for predictions without fitting to flow data.

Key Results
The model was validated against Molecular Dynamics (MD) simulations of the Kob-Andersen (KA) Lennard-Jones mixture across a broad range of shear rates and temperatures.

Macroscopic Rheology:
- The model quantitatively reproduces the steady-state viscosity curve, capturing the Newtonian regime, the crossover, and the power-law shear thinning ( $\eta \sim \dot{\gamma}^{-0.65}$ ).
- It accurately predicts transient stress overshoots in start-up shear, including both the peak strain ( $\gamma_{os}$ ) and the overshoot magnitude ( $M$ ).
- The overshoot magnitude follows a master curve when scaled by a characteristic shear rate $\dot{\gamma}_n$ , linking the onset of overshoot to the departure from Newtonian behavior.
Microscopic Dynamics:
- The model captures the evolution of Dynamic Heterogeneity (DH), quantified by the four-point susceptibility ( $\chi_{4}$ ).
- It reproduces the crossover in the peak susceptibility ( $\chi_{4}^{pk}$ ) from a plateau at low shear rates to a power-law shrinkage ( $\chi_{4}^{pk} \sim \dot{\gamma}^{-0.4}$ ) at high shear rates.
- The crossover shear rate coincides with the onset of power-law thinning ( $\dot{\gamma}_{pl}$ ).
Mechanism Validation:
- "Incomplete" models lacking the correlation length ( $\xi$ ) or the localization of elasticity ( $C_{ij}$ ) failed to reproduce key features. Removing $\xi$ led to underestimation of stress in the crossover regime, while removing localization ( $C_{ij}$ ) caused overestimation of stress at low shear rates and eliminated the crossover in DH. This confirms that the interplay between structural clustering and localized elasticity is essential.

Significance and Claims
The paper claims that this two-state model provides a unified picture linking the flow behaviors of supercooled liquids directly to their equilibrium properties. By treating the flow as a competition between intrinsic local mobility and external shear rates, the model explains how shear does not merely "fluidize" the system but rather recruits particles into solid-like, elastically interacting regions (LERs).

The authors highlight several specific contributions:

Dual Success: The model simultaneously describes macroscopic rheology (viscosity, stress overshoot) and microscopic spatiotemporal fluctuations (DH evolution) without empirical fitting.
Structural-Dynamic Decoupling: The model reveals a decoupling between structural correlation length (which remains constant and is determined by equilibrium packing) and dynamic correlation length (which shrinks with increasing shear rate). This contrasts with the quiescent glass transition where both lengths grow upon cooling.
Reframing Shear: The work suggests that shear highlights the "glassy" features of supercooled liquids by creating localized amorphous solids (LERs) within the flow, rather than simply acting as an effective temperature that enhances diffusion.
Theoretical Bridge: The model offers a perspective where LERs serve as a mesoscopic substrate for dynamic facilitation, potentially bridging competing views on the glass transition (dynamic facilitation vs. structural order vs. elastic correlations).

The authors conclude that while the model is currently applied to the KA system, its framework offers a route toward a broader mesoscopic description of disordered systems, including potential extensions to oscillatory shear and surface flows.