🔬 condensed matter

Criticality of the viscous to inertial transition near jamming in non-Brownian suspensions

Using Discrete Element Method simulations, this study demonstrates that the viscous-to-inertial shear thickening transition in dense frictionless non-Brownian suspensions near jamming is governed by a diverging microstructural length scale that causes the characteristic transition shear rate to vanish as the volume fraction approaches the jamming limit.

Original authors: Nishanth Murugan, Donald Koch, Sarah Hormozi

Published 2026-07-16

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CC BY 4.0

Original authors: Nishanth Murugan, Donald Koch, Sarah Hormozi

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 the rules of how things flow change depending on how fast you push them. This is the fascinating realm of fluid dynamics, specifically focusing on "suspensions"—mixtures where tiny solid particles are floating in a liquid, like sand in water or chocolate chips in milk. Usually, we think of fluids as either runny (like water) or thick (like honey), but these particle-filled mixtures can be tricky. Sometimes, if you stir them slowly, they act like a normal liquid. But if you stir them fast, they can suddenly turn stiff, almost like a solid. This phenomenon is called "shear thickening." Scientists are obsessed with understanding exactly when and why this switch happens because it affects everything from how we process cement and ceramics to how we predict dangerous natural events like mudslides and avalanches. The big mystery has been: what is the exact "speed limit" where the fluid decides to stop being runny and start acting stiff? Is it a fixed number, or does it change depending on how crowded the particles are?

In this study, researchers Nishanth Murugan, Donald Koch, and Sarah Hormozi from Cornell University used powerful computer simulations to solve a piece of this puzzle. They focused on a specific type of mixture: dense suspensions of particles that don't stick together (frictionless) and are too big to be jiggled around by heat (non-Brownian). Think of them as a crowded dance floor where the dancers are wearing slippery suits and can't bump into each other. The team wanted to find the precise moment the dance floor shifts from a smooth glide to a chaotic, jostling crowd.

Their simulations revealed that there isn't a single, universal speed limit for this transition. Instead, the "speed limit" depends entirely on how packed the dance floor is. As the particles get closer to the point where they are completely jammed and can't move at all (a state called "jamming"), the speed required to trigger the stiffening behavior drops to almost zero. In other words, the more crowded the system gets, the easier it is to turn the fluid into a solid.

The authors discovered that this happens because of a hidden, growing structure within the mixture. As the particles crowd together, they start forming giant, temporary clusters that act like a single, massive unit. The size of these clusters grows larger and larger as the system approaches jamming. The researchers found that the transition from "runny" to "stiff" is actually triggered by the inertia (the momentum) of these giant clusters, not just the individual particles. They proved this by showing that the size of these clusters and the speed at which the fluid stiffens follow the same mathematical pattern.

Interestingly, the paper argues against the idea that this transition is controlled by the size of a single particle or a fixed force. In systems where particles do have friction (like rough sand), the transition happens at a different, constant speed. But for these smooth, slippery particles, the transition is "critical," meaning it is deeply tied to the system's crowding. The study suggests that as you get closer to the jamming point, a "length scale" (a measure of how big these clusters get) grows infinitely large, making the system incredibly sensitive to even the tiniest push.

The researchers used a method called the Discrete Element Method (DEM), which is like a super-accurate video game where they calculate the forces between millions of particles. They simulated both 2D and 3D versions of these crowded systems. Their results showed that the "Stokes number" (a fancy way of measuring the balance between the fluid's stickiness and the particles' momentum) drops to zero as the volume fraction approaches the jamming limit. They found that the size of the clusters grows according to a specific power law, with an exponent of about 0.79 in 2D and 0.84 in 3D.

By testing how the results changed when they made the simulation box smaller or larger, they confirmed that these giant clusters really do exist and are the key to the transition. If the simulation box was smaller than the cluster, the results changed, proving the cluster was trying to "reach through" the walls. This confirmed that the transition is a non-local event, meaning what happens in one part of the fluid depends on what's happening far away, connected by these massive, invisible structures.

In short, the paper suggests that for smooth, crowded particles, the switch from liquid to solid isn't about the individual dancers; it's about the formation of a massive, invisible crowd that grows until it takes over the whole room. This insight helps scientists understand that near the point of jamming, the rules of flow become critical and depend on these growing structures, paving the way for better theories on how these complex materials behave.

Technical Summary: Criticality of the Viscous to Inertial Transition near Jamming in Non-Brownian Suspensions

Problem Statement
The rheology of dense non-Brownian suspensions is critical to understanding natural phenomena (e.g., landslides, mudslides) and industrial processes (e.g., cement slurry, ceramics). A central feature of these systems is the transition from a viscous regime, characterized by Newtonian rheology ( $\Sigma \propto \dot{\gamma}$ ), to an inertial regime, characterized by Bagnoldian rheology ( $\Sigma \propto \dot{\gamma}^2$ ). This transition is traditionally attributed to the increasing dominance of particle inertia over hydrodynamic forces.

However, the physical mechanism triggering this transition and the characteristic shear rate at which it occurs remain debated. Experimental and numerical literature presents conflicting results regarding the critical Stokes number ( $St_{v \to i}$ ), a dimensionless measure of the ratio of inertial to viscous stresses. In frictionless suspensions (e.g., polystyrene spheres with electrostatic repulsion), $St_{v \to i}$ has been observed to vary with the volume fraction ( $\phi$ ) and approach zero as the system nears the jamming volume fraction ( $\phi_m$ ). Conversely, in frictional suspensions (e.g., rough PMMA particles), $St_{v \to i}$ appears independent of $\phi$ and remains finite. The lack of consensus on the physical mechanism, particularly the role of microstructure in frictionless systems, motivates this study.

Methodology
The authors employ Discrete Element Method (DEM) simulations to investigate dense, non-Brownian, frictionless suspensions across a wide range of shear rates and volume fractions.

Force Model: The dynamics are governed by Newton's equations of motion, accounting for two primary forces: hydrodynamic lubrication forces ( $F^L_{ij}$ ) arising from relative motion in the inter-particle gap, and repulsive contact forces ( $F^R_{ij}$ ).
Frictionless Approximation: To model the electrostatic repulsive barrier found in experiments (which prevents direct contact), the repulsive force is modeled as a Hookean spring ( $F^R_{ij} = K\delta n$ ) that activates when the surface separation $h$ falls below a minimum threshold $h_{min}$ .
Simulation Setup: Simulations are conducted in both 2D and 3D domains using Lees-Edwards boundary conditions to simulate shear flow. The system uses a polydisperse particle distribution (16%) to prevent crystallization. The lubrication forces are calculated assuming a minimum separation of $h_{min}$ when particles attempt to get closer, effectively constraining normal motion.
Parameters: The study explores the parameter space defined by the volume fraction ( $\phi$ ), the Stokes number ( $St = 4\rho a^2 \dot{\gamma} / \eta_f$ ), and the dimensionless stiffness ratio ( $\Gamma$ ). The simulations are performed in a regime where $\Gamma \gg 1$ , ensuring the repulsive barrier dominates over lubrication forces at small separations.

Key Contributions and Results

Criticality of the Transition: The simulations confirm that for frictionless suspensions, the characteristic Stokes number for the transition to inertial shear thickening ( $St_{v \to i}$ ) is not constant. Instead, it decreases as the volume fraction increases, approaching zero as $\phi \to \phi_m$ . The relationship follows a power law: $St_{v \to i} \sim (\phi_m - \phi)^\beta$ .
Scaling Framework: To explain this behavior, the authors propose a scaling framework based on the hypothesis that a microstructural length scale diverges as the system approaches jamming. They introduce a modified Stokes number, $St_{Cluster} = B St (\phi_m - \phi)^{-\beta}$ , which scales the inertial effects relative to a growing cluster size $L_c \sim (\phi_m - \phi)^{-\beta/2}$ .
Data Collapse: Using this scaling framework, the rheological data (relative shear viscosity $\eta_r$ $η_{r}$ vs. $St$) for various volume fractions collapses onto a single master curve in both 2D and 3D simulations.
- In 2D, the exponent is $\beta \approx 0.79$ .
- In 3D, the exponent is $\beta \approx 0.84$ .
- These values align with experimental observations suggesting $St_{v \to i} \sim (\phi_m - \phi)^{-1}$ .
System Size Dependence: To validate the existence of the proposed diverging length scale, the authors performed system size dependence tests. They observed that when the simulation box size is smaller than the characteristic cluster size, the viscosity increases due to the cluster interacting with itself through periodic boundaries. By fitting the viscosity data to a power law dependent on system size, they successfully extracted a cluster length scale $L_c$ that diverges as $\phi \to \phi_m$ with an exponent consistent with the rheological scaling ( $L_c \sim (\phi_m - \phi)^{-\beta/2}$ ).

Significance and Claims
The paper claims that the transition from viscous to inertial shear thickening in frictionless suspensions is a critical phenomenon driven by a diverging microstructural length scale (clusters) as the system approaches jamming.

Mechanism: The criticality of the transition is attributed to the growth of these clusters, which alters the scale at which inertial effects become dominant. As the cluster size diverges, the characteristic shear rate required to trigger inertial effects vanishes.
Non-Local Rheology: The existence of a diverging length scale implies that the rheology of these suspensions is non-local; stress transmission depends on the formation and destruction of these large-scale clusters.
Contrast with Frictional Systems: The authors note that this critical behavior is specific to frictionless systems. In frictional suspensions, where particle contacts are governed by surface roughness rather than long-range repulsion, the transition appears independent of volume fraction, suggesting the microstructural mechanism differs fundamentally.
Future Direction: The work motivates the development of non-local theories that account for stress transmission through these growing clusters. It also highlights the need for further study on scaling laws for cluster sizes in frictional suspensions to establish a unified theory across different flow regimes.

Technical Summary: Criticality of the Viscous to Inertial Transition near Jamming in Non-Brownian Suspensions

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