GLASS Flows: Mechanisms and Models

• GLASS Flows are frameworks describing glassy and jammed systems that exhibit yield stress, non-Newtonian flow, and slow structural relaxation.
• They integrate multiaxial flow experiments with continuum models, using Herschel–Bulkley and Von Mises criteria to quantify yield behavior and stress invariants.
• Emerging machine-learning methods, such as normalizing flows, accelerate sampling of complex energy landscapes, linking microscopic plastic events to macroscopic flow.

GLASS Flows refer to a broad set of phenomena and theoretical frameworks describing the flow, yielding, and rheological response of glassy or glass-forming systems—ranging from atomic and molecular glasses, colloidal suspensions, foams, and emulsions, to dense granular materials. These systems are characterized by slow structural relaxation, dynamic arrest, and the emergence of yield stress and non-Newtonian flow regimes. The central challenge is to relate macroscopic flow properties to the underlying microscopic or mesoscopic mechanisms, particularly under complex, multiaxial or heterogeneous flow fields.

1. Multiaxial Flow and Yielding Mechanisms in Soft Glasses

Experiments involving multiaxial flows, such as simultaneous shear and elongation, demonstrate that increasing stress along one direction enhances strain not only along that direction but also orthogonally. This coupling is generic in jammed, three-dimensional glassy networks. In thin films of soft glassy materials subjected to both rotational shear and normal (tensile) forces, the deformation in both rotational and elongational directions is accelerated as the invariant of the stress tensor increases. These effects are well described by the Herschel–Bulkley model, with the yield threshold captured by the Von Mises criterion:

• The combined deviatoric stress invariant is $J_2 = \frac{1}{2}(\tau_{rz}^2 + \tau_{r\theta}^2)$.
• Yielding occurs when $J_2 = \tau_Y^2$, or, equivalently, when the quadratic sum of the orthogonal components exceeds the material yield stress.

Strain–time curves for various stress combinations exhibit self-similarity, indicating the unjamming pathway is universal across loading configurations, with only the timescale changing. Beyond the yielding threshold, failure time (in, e.g., tensile plate separation) is inversely proportional to the invariant stress, $t_f \sim 1/\tau_\text{inv}$, and a yield-phase diagram can be constructed. Except for instabilities (such as Saffman–Taylor fingering in highly stretched films), the yielding boundary closely follows the predicted circular locus in the $(\tau_{rz}, \tau_{r\theta})$-plane (Shaukat et al., 2012).

2. Microscopic Origin of Flow: Dynamical Heterogeneity and Creep

The flow of glasses under stress is governed by the emergence and evolution of localized regions of enhanced mobility—dynamical heterogeneities. Below the yield stress $\sigma_Y$, deformation is dominated by sparse, isolated plastic events, leading to sub-linear "Andrade creep" ($\gamma(t) \sim t^a$, $a < 1$). As the stress approaches and exceeds $\sigma_Y$, these regions nucleate, grow, and coalesce into system-spanning structures, resulting in the onset of steady flow.

Crucially, both the mean-squared displacement (MSD) and the fraction of dynamically active regions scale linearly with the macroscopic strain over multiple regimes:

• $\langle \Delta y^2(t) \rangle \sim \gamma(t)$
• Fraction of active regions $\Phi_\text{active}(t) \sim \gamma(t)$

For $\sigma \gtrsim \sigma_Y$, the spatial correlation length of mobile regions grows as $\xi(t) \sim t^{2/3}$, whereas below yield it remains localized (Sentjabrskaja et al., 2015). This hierarchical percolation and growth of mobile clusters provide the missing microstructural ingredient for constitutive models of glass flow.

3. Flow Heterogeneities and Shear Banding under Shear

In supercooled liquids and glasses under imposed shear, transient, spatially localized bands of high mobility—shear bands—emerge during the transition from solid-like to fluid-like response:

• Supercooled liquids: transient vertical bands (perpendicular to flow) arise immediately following stress overshoot and decay within a narrow strain window ($\Delta\gamma \approx 0.1$), after which the system returns to homogeneous flow.
• Glassy state (deeply supercooled): Both short-lived vertical and long-lived horizontal bands (parallel to the flow) occur. Horizontal bands nucleate stochastically post-overshoot and correspond to rapid system-spanning fluidization. Their broadening is subdiffusive in strain, $\xi(\gamma) \propto \gamma^{0.32}$.

System size sharply affects the probability and sharpness of horizontal band formation but not the steady-state flow field (Golkia et al., 2020). These observations support the view that yielding and relaxation dynamics are governed by stochastic, spatially organized plastic events—consistent with mode-coupling and elastoplastic theoretical models.

4. Continuum Models: Nonlocal Fluidity and Cooperativity

The macroscopic rheology of soft glassy materials can often be described by continuum models incorporating nonlocal effects:

• The "fluidity" $f(y, t)$—the local rate of plastic rearrangement—evolves according to reaction–diffusion equations with a cooperativity length $\xi$:

$$\frac{\partial f}{\partial t} = f\big[\xi^2 \partial_{yy} f + m(\Sigma)f - f^{3/2}\big] + \eta(y,t)$$

where $m(\Sigma)$ encodes yielding behavior as a function of scaled stress $\Sigma = \sigma/\sigma_Y$.

• Under shear start-up, the model predicts stress overshoots followed by the growth of fluidized shear bands emanating from the boundaries. The band grows at a rate $dy^*/dt \sim \xi m^{5/2}$, and fluidization time scales as $T_f \sim 1/[\xi \dot\Gamma^{9/4}]$ in the small strain-rate limit, matching experimental data (Benzi et al., 2022).
• Incorporation of spatial noise yields avalanche-like events, connecting the model to observed intermittency in plastic flow.

This framework generalizes the local Herschel–Bulkley law by embedding cooperative, spatially extended yielding processes, crucial for capturing observed heterogeneity and geometry dependence in experimental flows.

5. Unified Rheological Models: Glass–Jamming Crossover

The rheology of dense suspensions and soft glasses is governed by both thermal (glass) and athermal (jamming) physics. The unified constitutive model decomposes the total steady-state stress

$$\sigma(\dot\gamma) = \sigma_G(\dot\gamma, \varphi, T) + \sigma_J(\dot\gamma, \varphi) + \eta_s \dot\gamma$$

with $\sigma_G$ describing thermal, glassy behavior (controlled by the colloidal glass transition packing fraction $\varphi_G$) and $\sigma_J$ the athermal, jamming sector (with jamming packing $\varphi_J$). Each sector features distinct scaling of viscosity and yield stress near their respective critical densities and stress/strain-rate scales (Ikeda et al., 2012, Ikeda et al., 2013). Experimental systems such as colloidal hard spheres, emulsions, and foams exhibit crossover behaviors that can be quantitatively collapsed within this additive framework.

6. Advances in Sampling and Simulation: GLASS Flows in Machine Learning

In machine-learning-based generative modeling for glass-forming systems, "GLASS Flows" (as an Editor's term for Generative models of GLASSy landscapes) embody the use of normalizing flows to sample complex, multimodal energy landscapes characteristic of glassy systems:

• Normalizing flows parameterize invertible transformations from tractable priors (e.g., high-temperature Boltzmann distributions) to target (low-temperature) equilibrium distributions. Architectures include continuous normalizing flows (CNFs) optimized as neural ODEs with equivariance to relevant symmetries (Jung et al., 2024).
• Training objectives involve minimizing forward and reverse KL divergences between the flow-induced and target Boltzmann distributions, using mixtures for optimal stability.
• Such approaches can dramatically accelerate sampling compared to molecular dynamics and annealing, though remain limited by expressivity, particularly when proposing large-scale rearrangements in deeply glassy regimes.

For generative modeling of discrete spin glasses, continuous-density formulations (e.g., Hubbard–Stratonovich transformations) enable training of invertible Real-NVP flows, which can efficiently capture multimodal steady-state distributions and reproduce glass-specific order parameters (e.g., bimodal overlap distributions, ultrametricity) if trained with forward-KL objectives and appropriately supervised MC-generated datasets (Hartnett et al., 2020).

Advancements such as the GLASS Flows algorithm ("GLASS Flows: Transition Sampling for Alignment of Flow and Diffusion Models" (Holderrieth et al., 29 Sep 2025)) introduce "flow-inside-a-flow" transition samplers, which enable efficient sampling of Markov transitions within reward-alignment frameworks, replacing expensive SDE integration with ODE-based solvers that preserve stochasticity and improve inference scalability for high-dimensional tasks such as text-to-image generation.

7. Experimental and Practical Implications

Experimental research using physical vapor deposition and broadband picosecond photoacoustics enables mechanical probing of glass flow in ultrastable glasses down to apparent viscosities of $\sim 10^{18}$ Pa·s ("exapoise"), extending well beyond conventional calorimetric timescales. The absence of a true dynamical divergence in viscosity, together with robust scaling between glass age, high-frequency elastic modulus, and effective fragility, challenges prevailing glass transition theories predicated on finite-temperature singularities (Pogna et al., 2015).

Flow studies of granular heaps, wet glass spheres, and soft emulsions under varying degrees of cohesion and confinement highlight transitions from steady to unsteady (avalanche/jump) regimes and the emergence of inhomogeneous packing, stratified by local flow and deposition events (Xiao et al., 2018). Lattice-Boltzmann and hydro-kinetic simulations demonstrate that the global rheology of these materials can be collapsed onto a universal Herschel–Bulkley form, provided that nonlocal cooperativity and geometry are properly accounted for (Benzi et al., 2013, Papenkort et al., 2013).

Taken together, GLASS Flows encompass a spectrum of phenomena unifying the microstructural basis of glass flow, advanced continuum and numerical models, recent generative machine-learning approaches, and emerging experimental techniques, offering a comprehensive framework for understanding and advancing the rheology and dynamics of amorphous solid-like materials.