🔬 condensed matter

Emergent odd viscoelasticity in chiral soft glassy materials

This paper introduces a chiral soft glassy rheology model of actively rotating inclusions in a glassy matrix to demonstrate how odd viscoelastic responses emerge, revealing that odd viscosity increases as active rotation slows and that oscillatory shear induces a complex spectrum combining resonance effects with glassy power laws.

Original authors: Debarghya Banerjee, Peter Sollich

Published 2026-08-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Debarghya Banerjee, Peter Sollich

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 world where materials don't just sit there or flow like water, but remember their past and react with a bit of a twist. This is the realm of rheology, the science of how things flow and deform. Think of it as the study of the "personality" of matter: is it a stiff solid that snaps back, a runny liquid that slides, or something in between, like toothpaste or silly putty? Scientists have long known that many complex materials, from biological tissues to industrial paints, are "glassy." This doesn't mean they are made of windows, but that their internal parts are jammed together like a crowded dance floor, moving so slowly they seem frozen, yet they can eventually flow if pushed hard enough.

Recently, researchers discovered something strange about "active" materials—things that generate their own energy, like tiny robots or living cells. These materials can be "chiral," meaning they have a handedness, like a left or right hand, and they often spin or rotate on their own. When these spinning, jammed-up materials are pushed, they don't just resist; they push back in a sideways, "odd" way that breaks the usual rules of symmetry. While scientists have studied how these materials act as simple liquids or simple solids, there was a big mystery: what happens when they are in that tricky middle ground, acting like a glassy, spinning, memory-having fluid? Understanding this is crucial because it could explain how living tissues move, heal, or even how we might build new, smart materials that twist and turn on command.

The Spinning Jam: A New Model for Twisted Materials

In this paper, Debarghya Banerjee and Peter Sollich introduce a new way to think about these mysterious materials. They build a "chiral Soft Glassy Rheology" (SGR) model. To visualize this, imagine a giant bowl of thick, sticky jam (the glassy matrix). Now, sprinkle in thousands of tiny, motorized gears (the inclusions) that are constantly spinning in circles. In the real world, these could be tiny bacteria or synthetic particles that burn fuel to rotate. The authors ask: if you stir this spinning jam, how does it push back?

The team creates a mathematical story for this system. They treat each tiny gear and the jam immediately around it as a single unit. These units have a "yield energy," which is like a threshold of stress they can take before they snap, relax, and reset. The gears spin at a specific frequency, which the authors call $\Omega$ . The model tracks how the stress builds up in the jam and the gears as they are sheared (stirred) by an outside force.

The Surprising Twist: Slower Spin, Stronger Push

The most exciting discovery comes when the researchers look at what happens under a steady, constant stir. They find that the material develops an "odd viscosity." In normal fluids, viscosity is just a measure of how thick or sticky the fluid is. But "odd" viscosity is weird: it creates a force that is perpendicular to the direction of the flow, a bit like how a spinning top might wobble sideways instead of just moving forward.

Here is the counter-intuitive part: the authors find that the slower the gears spin (as the frequency $\Omega$ decreases), the stronger this odd viscosity becomes. It sounds like magic, but the math explains it. The glassy jam has a wide range of relaxation times—some parts relax quickly, others take forever. When the gears spin slowly, they allow the "slow" parts of the jam to stay stressed for longer, building up a massive sideways push. As the spin rate drops, this effect grows, following a specific power law where the viscosity scales as $\Omega^{X-2}$ (where $X$ is a parameter describing how crowded the jam is). This is a non-trivial result: usually, you'd expect less activity to mean less effect, but here, slowing down the spin actually amplifies the weird sideways force.

The Resonance: When the Stirring Matches the Spin

The story gets even more interesting when the researchers test the material with an oscillating stir, like wiggling a spoon back and forth at different speeds. They discover a "resonance" effect. Imagine the gears spinning at a rate $\Omega$ . If you wiggle the spoon back and forth at exactly twice that speed ( $\omega = 2\Omega$ ), something special happens.

The authors explain this with a vivid picture: every time the spoon completes a half-cycle of its wiggle, the gears have rotated exactly 90 degrees. This means the direction the gear is pushing aligns perfectly with the new direction the spoon is pushing. It's like a child on a swing; if you push at just the right moment in the swing's arc, the swing goes higher and higher. In the material, this synchronization causes a massive spike in the response, a "resonance-like peak" in the viscosity spectrum. This peak appears clearly in their calculations at the frequency $\omega = 2\Omega$ .

What the Model Predicts and What It Doesn't

The authors are careful to state that their findings come from a theoretical model and simulations, not a physical experiment in a lab. They explicitly rule out the idea that this behavior is just a simple extension of known liquid or solid physics; the "odd" terms are a unique feature of the combination of chirality (spinning) and glassy dynamics (jamming).

They also note that their model assumes the spinning gears are distributed randomly, like a disordered crowd, and that the deformations are small. If the spins were arranged in a perfect pattern, or if the material was stretched too far, the rules might change. Furthermore, they point out that at very high frequencies (much faster than the spin rate), the material behaves more like a standard glass, and the weird odd effects fade away.

Why This Matters

While this is a theoretical paper, it provides a roadmap for understanding real-world systems. The authors suggest that this model could help explain the behavior of biological tissues where cells rotate, or "starfish embryos" that have been observed to exhibit odd elasticity. By showing how a simple combination of spinning parts and a jammed environment creates these complex, twisting forces, the paper offers a new lens for scientists to look at active matter. It suggests that the "odd" mechanical properties we see in nature might not be accidents, but a natural consequence of how spinning, jammed systems respond to the world around them. The paper doesn't claim to have solved the mystery of all active materials, but it offers a compelling, mathematically grounded story for how chirality and glassiness dance together to create something truly unique.

Technical Summary: Emergent Odd Viscoelasticity in Chiral Soft Glassy Materials

Problem Statement
While significant progress has been made in understanding "odd" mechanical responses—specifically odd viscosity in fluids and odd elasticity in solids—in active chiral systems, the field lacks a comprehensive understanding of odd viscoelasticity. This is particularly relevant for amorphous, glassy materials where structural disorder generates a broad spectrum of relaxation times. Existing models for active matter often neglect the interplay between structural disorder (glassy dynamics) and chirality (active rotation), leaving the rheological response of chiral soft glasses underexplored.

Methodology
The authors propose a Chiral Soft Glassy Rheology (SGR) model to address this gap. The methodology builds upon the original SGR framework, which describes passive glassy materials as an ensemble of mesoscopic elements characterized by local strains and yield energies. The chiral adaptation introduces the following key features:

Mesoscopic Structure: Each element consists of an actively rotating "inclusion" (angular frequency $\Omega$ ) embedded within a glassy "matrix."
Elastic Energy: The model defines a quadratic elastic energy coupling the strain tensors of the inclusion ( $l^I_{ij}$ ) and the matrix ( $l^M_{ij}$ ), including a coupling term ( $k_3$ ) representing strain mismatch.
Dynamics: The evolution of the system is governed by a master equation tracking the distribution of yield energies ( $E$ ) and deformation tensors. Plastic yield events occur stochastically with rates determined by mechanical noise (effective temperature $X$ ) and the energy barrier between the current elastic energy and the yield energy.
Active Rotation: Unlike the passive SGR model, the inclusion deformation tensor evolves with an additional term representing active rotation ( $-\Omega \epsilon_{ik}$ ), while the matrix follows standard flow gradients.
Derivation: The authors employ a pre-averaging approximation (exact for linear response) to derive closed hydrodynamic equations for the stresses of the inclusion and matrix. They then analyze the linear stress response to both steady and oscillatory shear flows, integrating over the distribution of yield energies (assumed to be exponential, $\rho(E) = e^{-E}$ ).

Key Contributions and Results
The paper derives the linear viscoelastic spectrum for the chiral SGR model, revealing emergent odd terms in the mechanical response:

Steady Shear Flow:
- The model predicts an odd viscosity term in the stress response.
- A non-trivial result is that the odd viscosity increases as the active rotation frequency $\Omega$ decreases. This scaling follows $\eta_{odd} \sim \Omega^{X-2}$ (where $1 < X < 2$ is the SGR exponent), diverging as $\Omega \to 0$ .
- This behavior arises because the power-law spectrum of relaxation times in glassy materials causes the viscosity to diverge at zero frequency; the active rotation $\Omega$ acts as an effective upper cutoff for these relaxation times. As $\Omega$ decreases, longer relaxation times contribute, enhancing the odd viscosity.
Oscillatory Shear Flow:
- The model yields a frequency-dependent odd viscoelastic spectrum containing both odd viscous (real part) and odd elastic (imaginary part) components.
- High Frequency ( $\omega \gg \Omega$ ): The odd viscosity scales as $\eta_{odd} \sim \Omega \omega^{X-3}$ . In this regime, the odd response vanishes as $\Omega \to 0$ , recovering the passive limit.
- Low Frequency ( $\omega \ll \Omega$ ): The odd viscosity approaches a plateau scaling as $\Omega^{X-2}$ .
- Resonance: A distinct, non-smooth crossover occurs at $\omega = 2\Omega$ . The authors identify this as a resonance-like peak where the period of the external shear flow matches the time required for the active inclusion to rotate by $\pi/2$ . This synchronization reinforces extensional and compressional deformations accumulated in previous half-cycles.
Shear Viscoelasticity:
- The shear response of the inclusions also exhibits a low-frequency plateau and a resonance peak at $\omega = 2\Omega$ .
- At high frequencies ( $\omega \gg \Omega$ ), the active rotation becomes negligible, and the shear response reverts to the standard passive SGR scaling ( $\sim \omega^{X-2}$ ).

Significance and Claims
The authors claim that this work provides a theoretical basis for understanding odd viscoelasticity in disordered, active materials. The primary significance lies in demonstrating how the combination of chirality and glassy dynamics generates unique mechanical responses that are not present in either purely active fluids or passive glasses alone.

Specifically, the paper highlights:

The emergence of a non-monotonic dependence of odd viscosity on activity, where slower rotation leads to stronger odd viscous effects in the steady limit.
The existence of a resonance mechanism at $\omega = 2\Omega$ driven by the interplay between external driving and internal active rotation.
The potential relevance of these findings to biological systems where active rotation occurs within viscoelastic environments, such as rotating cell nuclei, starfish embryos, and epithelial tissues.

The paper concludes that the Chiral SGR model offers a paradigmatic framework for exploring odd mechanical effects in experimental systems ranging from dense mixtures of passive particles and active spinners to biological tissues, bridging the gap between active matter physics and soft glassy rheology.

The Spinning Jam: A New Model for Twisted Materials

The Surprising Twist: Slower Spin, Stronger Push

The Resonance: When the Stirring Matches the Spin

What the Model Predicts and What It Doesn't

Why This Matters

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