Chiral Soft Glassy Rheology Model

• Chiral soft glassy rheology is a model that couples active inclusion rotations with glassy yield dynamics to produce emergent odd viscoelastic responses.
• It employs a mean-field, tensorial approach with linear elasticity and activated yielding to capture a broad spectrum of relaxation times.
• The model predicts key regimes including a low-frequency odd viscosity plateau and a resonance-like peak at ω=2Ω under oscillatory shear.

Chiral soft glassy rheology is a generalization of the soft glassy rheology model that incorporates actively rotating inclusions within a glassy matrix to describe emergent odd viscoelastic responses in a class of amorphous solids. The model was introduced to address the lack of understanding of odd viscoelastic responses, using a mean-field, tensorial approach suitable for two-dimensional systems and focusing on the linear stress response to steady and oscillatory shear flows. In steady shear it predicts an odd viscosity that, non-trivially, grows as the active rotation frequency $\Omega$ decreases. In oscillatory shear it predicts an odd viscoelastic spectrum with a non-trivial dependence on the driving frequency $\omega$, combining resonance effects around $\omega = 2\Omega$ with glassy power laws at larger $\omega$ (Banerjee et al., 4 Sep 2025).

1. Soft-glassy background and conceptual lineage

The chiral model inherits the central structure of soft glassy rheology. In the standard SGR framework, a sample is represented as an ensemble of mesoscopic elements or clusters, each associated with a local shear strain $l$, a local stress $kl$, and an energy trap of depth $E$. Between yielding events, element strains evolve affinely with the imposed macroscopic flow, $\dot{l} = \dot{\gamma}$. Yielding is treated as an activated process with rate

$$\tau^{-1}(E,l)=\tau_0^{-1}\exp\left[-\frac{E-\frac{1}{2}kl^2}{x}\right],$$

where $x$ is an effective noise temperature encoding mechanical noise from elsewhere in the system. After a hop, the element is reset to $\omega$0 and assigned a new trap depth drawn from $\omega$1 (Fielding, 2013).

This trap-based construction was developed to capture structural disorder, metastability, ageing, and yield-stress behavior in soft glassy materials. For $\omega$2, it predicts glassy nonergodicity and ageing; for small $\omega$3, the flow curve acquires a finite yield stress, with

$\omega$4

A central consequence is that SGR provides a broad relaxation spectrum rather than a single relaxation time. The chiral extension preserves this glassy backbone while adding a chiral, active degree of freedom. This places it within a broader family of SGR-based constitutive theories in which mesoscopic disorder and activated relaxation remain the organizing principles, but the local state variables and couplings are enlarged.

2. Mesoscopic construction of the chiral model

In the chiral soft glassy rheology model, each mesoscopic element consists of an active, chiral inclusion that rotates at frequency $\omega$5 and a passive matrix that remains glassy and is characterized by a distribution of yield energies $\omega$6. The key parameters are $\omega$7, the externally imposed oscillation frequency $\omega$8, the effective noise temperature $\omega$9, the local yield energy $\omega = 2\Omega$0, and the microscopic attempt time $\omega = 2\Omega$1 (Banerjee et al., 4 Sep 2025).

The model assumes linear elasticity, activated Arrhenius-like yielding, and a mean-field description in which spatial correlations are neglected except through average mechanical noise. It also employs a pre-averaging approximation applicable for small strains. These assumptions distinguish it from spatially resolved SGR variants developed for shear banding and related inhomogeneous flows. In the chiral setting, the objective is instead to isolate how active rotation modifies the linear viscoelastic response of a glassy medium.

The elastic energy combines inclusion and matrix strains. Denoting the corresponding strain tensors by $\omega = 2\Omega$2 and $\omega = 2\Omega$3, the element-level elastic energy is

$\omega = 2\Omega$4

This form introduces elastic self-energies for the inclusion and matrix together with a coupling term between them. The physical picture is therefore not a purely passive trap model with an added antisymmetric transport coefficient, but an explicitly composite mesoscopic element in which activity is localized in the inclusion and glassiness in the surrounding matrix.

3. Governing dynamics and constitutive structure

The local deformation dynamics separate the actively rotating inclusion from the passive matrix. If $\omega = 2\Omega$5 and $\omega = 2\Omega$6 denote the corresponding deformation tensors, their evolution is

$\omega = 2\Omega$7

where $\omega = 2\Omega$8 is the two-dimensional Levi-Civita symbol and $\omega = 2\Omega$9 is the velocity gradient. The inclusion therefore experiences both imposed deformation and active rotation, whereas the matrix is purely advected by the imposed flow (Banerjee et al., 4 Sep 2025).

Plastic events are activated by the stored elastic energy. The local yielding rate is

$\omega$0

After yield, both the inclusion and matrix strains are reset to a residual distribution, often centered at zero. At the population level, the probability density $\omega$1 obeys a master equation of gain–loss form: $\omega$2

In the linear and small-strain regime, this description is recast into pre-averaged continuum equations for the inclusion and matrix stresses: $\omega$3 Here $\omega$4, and $\omega$5 is the advective derivative. The antisymmetric $\omega$6-dependent term is the explicit source of chiral stress dynamics. In this sense, odd response is not imposed phenomenologically but emerges from the coupling of active rotation to a glassy spectrum of relaxation times.

4. Odd viscosity and frequency-dependent response

The odd, or chiral, component of the viscoelastic response arises from the active rotation of inclusions, specifically through the antisymmetric tensor $\omega$7 in the inclusion dynamics and stress equation. In isotropic two-dimensional systems this produces stress components perpendicular to the strain-rate direction. Under an imposed oscillatory shear $\omega$8, the inclusion stress takes the form

$\omega$9

with

$l$0

The corresponding shear and odd viscosities are

$l$1

After integrating over the distribution of yield energies, the total complex viscosities become

$l$2

$l$3

where $l$4 is the complex shear viscosity of the original non-chiral SGR model (Banerjee et al., 4 Sep 2025).

The resulting spectrum has several distinct regimes. For $l$5, $l$6 scales as $l$7 at low frequencies close to the glass transition. In steady shear, the odd viscosity scales as

$l$8

so the odd viscosity increases as $l$9 is decreased. At high frequencies, $kl$0,

$kl$1

For $kl$2, both shear and odd viscosities exhibit a low-frequency plateau scaling as $kl$3. The model also predicts a resonance-like peak at $kl$4. The stated mechanism is that oscillatory shear aligns maximally with the active rotation of the inclusion, reinforcing distortions each half-period and producing a large stress response. The paper describes this as a hallmark emerging from the coupling of activity and glassiness (Banerjee et al., 4 Sep 2025).

5. Relation to ageing, banding, and other SGR-based rheologies

The chiral model is part of a broader SGR literature in which ageing and flow rejuvenation play a central role. Spatially resolved SGR studies of large-amplitude oscillatory shear showed that shear banding can be strong over an extensive range of amplitudes and frequencies, even in materials that do not shear band under steady imposed shear. Highly counterintuitively, this banding persists in the limit $kl$5, and the stated explanation is an alternating competition within each cycle between glassy ageing and flow rejuvenation. Related work on LAOStrain and LAOStress likewise found shear banding to be important over a broad range of amplitudes and frequencies, with pronounced consequences for Lissajous–Bowditch curves. Numerical studies of strain-rate sweeps within SGR and fluidity models further connected stress overshoot, transient banding, and rheological hysteresis in simple yield stress fluids (Radhakrishnan et al., 2016, Radhakrishnan et al., 2017, Radhakrishnan et al., 2016).

These results matter for interpreting the chiral model because they show that monotonic steady constitutive behavior does not guarantee homogeneous response in time-dependent protocols. A common misconception in soft-glassy rheology is to identify monotonicity of $kl$6 with the absence of banding under all driving conditions. The SGR literature explicitly rejects that inference for ageing materials. The chiral model, by contrast, is introduced as a linear-response theory with a mean-field, tensorial structure and small-strain pre-averaging. This suggests a distinction between odd linear viscoelasticity on the one hand and spatially heterogeneous nonlinear flow phenomena on the other. It also connects to earlier remarks that the SGR framework is extensible: by changing the local variables and their evolution equations, one could include a handedness to local rearrangements or chiral couplings between elements or between stress and local structural order parameters (Radhakrishnan et al., 2017).

6. Experimental relevance, scope, and theoretical limits

The chiral model proposes several candidate realizations. The listed systems include dense suspensions of active spinners, chiral active matter experiments such as spinning colloidal magnets, active glassy tissues, bacterial or cell monolayers with rotation, chiral particle assemblies, actomyosin gels, chiral cytoskeletal networks, and embryonic tissues. The corresponding observable signatures are an odd viscosity plateau at low shear rates or frequencies that increases as the spinner rotation frequency $kl$7 is reduced, a resonance in the loss modulus upon sweeping oscillatory shear frequency with a signature at $kl$8, and glassy scaling exponents in the complex moduli that reflect the underlying relaxation spectrum (Banerjee et al., 4 Sep 2025).

Its scope is nevertheless constrained by its assumptions. The model is formulated for linear elasticity, small deformations, activated yielding, and a mean-field treatment in which spatial correlations are neglected except through average mechanical noise. In the wider SGR literature, several theoretical limits of such constructions are already clear. One strand of work on delayed solidification in Laponite found that an SGR-type model appeared quantitatively successful at first sight, yet its core assumptions could not be reconciled with the extremely large strain amplitudes in the experiments; the paper concluded that semi-quantitative agreement was deceptive and that the quantitative explanation remained an open theoretical challenge. More generally, the broader SGR literature identifies wall slip and boundary effects, nonlocal effects, and bridging scales between mean-field stochastic descriptions and microstructural dynamics as open problems (Joshi et al., 2012, Fielding, 2013).

Within that landscape, the chiral soft glassy rheology model is best understood as a minimal constitutive theory for emergent odd viscoelasticity in chiral, glassy matter. It combines active chiral drive and a glassy distribution of yielding times in a single framework, while leaving spatial heterogeneity, strong-deformation regimes, and nonlocal couplings to future extensions.