🔬 condensed matter

Exotic rheology of materials with active rearrangements

This paper introduces a mean-field elasto-plastic model demonstrating that active T1 transitions in biological tissues can generate exotic rheological behaviors, including negative viscosity, non-monotonic flow curves, and hysteresis, by incorporating active elements that respond oppositely to applied stress.

Original authors: Aondoyima Ioratim-Uba, Tanniemola B. Liverpool, Silke Henkes

Published 2026-07-08

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CC BY 4.0

Original authors: Aondoyima Ioratim-Uba, Tanniemola B. Liverpool, Silke Henkes

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 crowd of people in a busy hallway. Usually, if you push the crowd from one side, they move in the direction of the push. This is how normal materials, like honey or silly putty, behave. But what if some people in that crowd were secretly programmed to run away from the push?

That is the core idea behind this paper. The researchers are studying biological tissues (like the skin of an embryo) not just as a pile of cells, but as a complex, "active" material where the cells themselves generate their own energy and movement. They created a mathematical model to understand how these tissues flow and change shape during development.

Here is a breakdown of their findings using simple analogies:

1. The Two Types of "Dancers"

The researchers imagined the tissue as a mixture of two types of dancers:

The Passive Dancers (Blue): These are normal cells. If you push them, they move with the push and build up tension, just like a spring being stretched.
The Active Dancers (Red): These are special cells that generate their own internal energy. Crucially, when they are pushed, they react in the opposite direction. If you push them right, they push back left.

The paper asks: What happens if you mix these two groups together and try to flow the whole crowd?

2. The "Exotic" Flow Curve

In normal materials, if you push harder (increase the strain rate), the resistance (stress) goes up steadily. It's a straight line.

However, the researchers found that when they mixed enough "Active Dancers" into the crowd, the flow curve became non-monotonic. Imagine a rollercoaster:

At slow speeds: The crowd behaves strangely. Even though you are pushing them forward, the "Active Dancers" push back so hard that the whole group actually moves backward or resists your push. It's like trying to walk forward on a treadmill that is suddenly speeding up against you.
At fast speeds: If you push the crowd very quickly, the "Active Dancers" can't keep up with their weird backward logic. The crowd finally gives in and starts moving in the direction you pushed, behaving like a normal fluid again.

This "backward then forward" behavior is exactly what happens in convergence-extension, a process where tissues get longer in one direction and thinner in another during embryonic development.

3. The "Yield Stress" Surprise

Materials usually have a "yield stress"—a minimum amount of force needed to get them to start flowing (like the force needed to get ketchup out of a bottle).

Positive Yield Stress: You have to push hard to get it moving.
Negative Yield Stress: This is the paper's "exotic" discovery. The researchers found that under certain conditions, the material actually wants to move on its own. It's as if the ketchup bottle is sitting on a table, but the ketchup starts flowing before you even touch it, or flows in the opposite direction of your hand.

They discovered that by adjusting the ratio of "Active" to "Passive" dancers and how quickly the stress spreads through the crowd, they could create materials that are:

Solid but flow backward.
Fluid but flow backward.
Solid that flows forward only when pushed hard.

4. The "Residual Stress" (The Hysteresis Effect)

The researchers also added a twist: What if the "Active Dancers" don't fully relax after they move? What if they stay slightly tense?

They found this creates a hysteresis effect. Think of it like a door with a sticky hinge. If you push the door open, it stays open a bit. If you pull it closed, it stays closed a bit.
In the tissue, this means the material remembers its history. If you push it one way and then the other, the path it takes isn't the same. It creates a loop, making the material's behavior depend on which way it was last moving.

5. Why This Matters (According to the Paper)

The paper compares these biological tissues to metamaterials. Metamaterials are man-made structures designed to have weird properties (like bending light in impossible ways).

The Analogy: Nature has been building these "metamaterials" for millions of years using cells.
The Claim: By understanding these "Active Dancers," we can see how biology uses disorder and internal energy to create complex flows. The paper suggests that if engineers could build materials that rearrange themselves and rebuild their structure (like the tissue does), they could create "liquid engines" that generate their own movement, mimicking how embryos grow.

Summary

The paper presents a model showing that when you mix normal cells with "rebellious" active cells that push back against stress, you get a material with exotic rheology (weird flow rules). This material can flow backward when pushed slowly, have negative resistance, and remember its past movements. This explains how biological tissues can perform complex shape-shifting tasks during development that normal, passive materials simply cannot do.

Technical Summary: Exotic Rheology of Materials with Active Rearrangements

Problem Statement
Biological tissues exhibit complex rheological properties driven by internal active stresses, particularly during developmental processes like convergence-extension (CE). In CE, epithelial tissues elongate in one direction while contracting in another, a process governed by active T1 transitions where cell neighbor exchanges occur opposite to the applied stress direction. This results in "exotic" rheology, including emergent negative shear modulus and viscosity. While theoretical models have captured passive yield stress behavior and fluid-solid transitions, the mechanics of feedback-driven active T1 transitions—where yielding occurs opposite to the polarization axis—remain unexplored within a mean-field framework. Existing models often lack a unified description of how these active rearrangements couple with passive elements to generate non-monotonic flow curves and negative stresses.

Methodology
The authors introduce a mean-field elasto-plastic model based on the Hébraud-Lequeux (HL) framework, extended to include a two-component mixture of passive and active elements.

Model Structure: The system consists of a fraction $\phi$ of active blocks and $(1-\phi)$ of passive blocks. Both are subject to an external shear strain rate $\dot{\gamma}$ .
Stress-Strain Relations: Passive blocks follow a standard elastic response ( $\sigma = G_0 \gamma$ ) up to a yield stress $\sigma_{0,c}$ . Active blocks exhibit an opposite stress response ( $\sigma = G_1 \gamma$ with $G_1 < 0$ ) up to a yield stress $\sigma_{1,c}$ , modeling the active T1 transition where junctions contract against the stress.
Dynamics: The evolution of the stress probability distributions $P_k(\sigma, t)$ (for $k=0$ passive, $k=1$ active) is governed by a Fokker-Planck equation. This includes advection due to elastic loading, a diffusion term $D(t)$ representing mechanical noise from stress redistribution, and a yield term where blocks exceeding their local yield stress are removed and reset.
Self-Consistency: The diffusion coefficient is coupled to the plastic activity $\Gamma(t)$ via $D(t) = \alpha \Gamma(t)$ , where $\alpha$ is a dimensionless parameter controlling the transition from yield-stress materials to fluids. The model also explores the effect of a residual active stress $\sigma_R$ remaining after yielding.
Solution: The authors seek stationary solutions ( $\partial_t P_k = 0$ ) to derive closed-form expressions for macroscopic stress $\langle \sigma \rangle$ and yield stress as functions of strain rate, active fraction, and stress diffusion parameters.

Key Contributions and Results

Non-Monotonic Flow Curves: The introduction of active elements with opposite stress responses fundamentally alters the rheology. The model predicts non-monotonic flow curves where the macroscopic stress opposes the strain rate at low $\dot{\gamma}$ but aligns with it at high $\dot{\gamma}$ . This crossover behavior is characteristic of convergence-extension.
Negative Yield Stress and Viscosity: Beyond a threshold fraction of active elements, the system exhibits negative yield stresses and negative initial effective viscosity ( $\eta_0$ ). This occurs when the competition between the active fraction and the difference in yield stresses ( $\sigma_{1,c} - \sigma_{0,c}$ ) favors the active response at low strain rates.
Phase Diagram: The authors map the $(\phi, \alpha)$ $(ϕ, α)$ parameter space into seven distinct phases:
- Passive (P): Monotonic positive stress for all strain rates.
- Yield Stress (Y) vs. Fluid (F): Separated by a critical $\alpha_c$ .
- Turnover (T) Phases: Regions exhibiting non-monotonic behavior (YT+, YT-, FT) where stress or viscosity changes sign.
- Backwards (B) Phases: Mostly active systems ( $\phi > 0.5$ ) where the stress and gradient remain negative across all strain rates.
Residual Active Stresses: Incorporating a non-zero residual stress $\sigma_R$ $σ_{R}$ breaks the symmetry of the stress-strain rate relationship. This leads to:
- A lowered effective yield threshold, promoting fluidization.
- Finite macroscopic stress at zero strain rate in the fluid phase.
- Asymmetry in the flow curves (hysteresis-like behavior) where the stress and viscosity depend on the direction of the strain rate limit ( $0^+$ vs $0^-$ ).

Significance and Claims
The paper claims that this two-component mean-field model successfully reproduces the exotic rheological features observed in biological convergence-extension, specifically the ability of tissues to yield opposite to applied stress at low rates while recovering passive behavior at high rates. The authors highlight that these features are characteristic of metamaterials, yet biological systems uniquely possess the ability to rearrange and rebuild their mechanical structure dynamically.

The study suggests that active rheology, driven by the interplay of active fractions and stress diffusion, could serve as a novel driver for developmental patterning, such as primitive streak formation or spatially controlled jamming/unjamming transitions. The authors position their work as a foundational step toward understanding the rich phenomenology of active elements, noting that while the model is deliberately simple, it provides a roadmap for engineering "build your own" rheological responses, potentially leading to autonomous reconfiguring liquid engines that mimic tissue mechanics. The paper does not claim to directly map to specific biological experiments but rather offers a theoretical framework where key parameters (moduli, yield stresses, time scales) can be mapped to biological scales (e.g., $\sim 1$ kPa stress, $\sim 15$ min T1 timescales).