🔬 condensed matter

Pressure sensitivity in non-local flow behaviour of dense hydrogel particle suspensions

This study demonstrates that pressure sensitivity is a critical component of non-local flow rules for dense hydrogel particle suspensions, as experiments and simulations reveal that shear band widths vary dramatically from broad to narrow depending on confining stress, a behavior successfully captured by a pressure-sensitive non-local granular fluidity model.

Original authors: Zohreh Farmani, Nazanin Ghods, Harkirat Singh, Jing Wang, Ralf Stannarius, Stefan Radl, David L. Henann, Joshua A. Dijksman

Published 2026-08-24

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CC BY 4.0

Original authors: Zohreh Farmani, Nazanin Ghods, Harkirat Singh, Jing Wang, Ralf Stannarius, Stefan Radl, David L. Henann, Joshua A. Dijksman

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 sand, sugar, or even tiny beads can behave like a solid block one moment and a flowing liquid the next. This is the realm of granular materials, a field of physics that studies how collections of small, solid particles move and interact. These materials are everywhere, from the sand in a desert dune to the grains of coffee in a machine, and understanding how they flow is crucial for industries ranging from construction to food production. For decades, scientists have known that when you push or pull these materials, they do not always move uniformly. Instead, they often concentrate their movement into narrow, twisting lanes called shear bands, while the rest of the material sits still. Predicting exactly where these bands form and how wide they are has been a persistent puzzle, especially because the behavior seems to depend on how much pressure is pressing down on the particles. While some theories suggest that the width of these moving lanes is a fixed property of the material itself, others argue that the local pressure changes everything, acting like a dial that widens or narrows the flow.

To settle this debate, a team of researchers turned to a unique and delicate system: a suspension of soft, water-filled gel spheres. Unlike the hard, rigid glass beads often used in previous studies, these hydrogel particles are squishy and nearly frictionless, floating in water where they are almost perfectly matched in density to the liquid around them. This setup allowed the scientists to create a scenario where the internal pressure of the material is incredibly low, effectively removing the weight of the particles as a driving force. They placed these soft spheres into a specialized container known as a split-bottom shear cell, a device that rotates a disk at the bottom to stir the mixture. By using a powerful medical imaging technique called magnetic resonance imaging, they could peer inside the container and watch the particles move in three dimensions without disturbing them. They then compared these real-world observations with computer simulations and a mathematical model designed to predict how granular materials flow.

The results revealed a striking sensitivity to pressure. When the researchers let the soft gel spheres flow without any extra weight pressing down on them, the shear bands became incredibly wide. Instead of a narrow lane of movement, the entire top layer of the material seemed to participate in the flow, spreading out broadly across the container. This happened because the low internal pressure allowed the "fluidity" of the material to spread far and wide. However, when the team applied a gentle external squeeze, pressing down on the top of the mixture with a force equivalent to just a few hundred pascals, the behavior changed dramatically. The wide, diffuse flow collapsed into a much narrower, more defined band, similar to what is seen in rigid, dry sand. This confirmed that the pressure acting on the particles is a critical factor in determining how they flow, validating a specific class of theories that had previously been tested mostly on hard, dry grains.

The researchers also explored how the speed of the flow affected the material. In the slow, steady movements they observed with the imaging technique, the flow behaved in a predictable, steady manner that did not depend heavily on how fast they turned the disk. However, when they pushed the system to move faster or applied higher levels of compression, the behavior shifted. Under high pressure, the material began to show a different kind of resistance that did not fit the standard rules for slow-moving granular flows. The team suspects this change is due to the slow, elastic nature of the gel particles themselves, which take time to relax and reshape after being squeezed, adding a layer of complexity that simple models of rigid particles cannot capture.

Ultimately, this work demonstrates that the rules governing how soft, wet particles flow are deeply connected to the pressure they feel. The study confirms that mathematical models which account for this pressure sensitivity can successfully predict the flow of these soft materials, bridging a gap between the physics of dry sand and wet, squishy gels. By showing that pressure acts as a control knob for the width of the flow, the research provides a clearer picture of how to model complex mixtures found in nature and industry. It suggests that to truly understand how granular materials move, one must always consider the invisible hand of pressure shaping the path of every particle.

Technical Summary: Pressure Sensitivity in Non-Local Flow Behaviour of Dense Hydrogel Particle Suspensions

Problem Statement
Predicting the flow behavior of granular materials is a fundamental challenge in soft matter physics and engineering. While non-local continuum models, specifically the Non-Local Granular Fluidity (NGF) model, have successfully described the quasi-static flow of rigid, dry granular materials (e.g., glass beads), their applicability to other dense particulate systems remains unproven. A critical gap exists in understanding how stress gradients, particle softness, and interstitial fluids influence non-local flow rules. Specifically, it is unclear if the NGF model's pressure sensitivity—where the width of shear bands depends on the local distance to the yield point—holds for dense suspensions of soft, nearly frictionless particles where particle deformation and buoyancy effects are significant.

Methodology
The authors employed a tripartite approach combining experimental observation, discrete numerical simulation, and continuum modeling to investigate dense hydrogel particle suspensions in a Split-Bottom Shear Cell (SBSC).

Experimental Setup: The study utilized 2–3 mm soft, slippery hydrogel spheres submerged in water. The system was designed to be nearly density-matched ( $\Delta\rho \approx 0$ ), minimizing hydrostatic pressure gradients and allowing external confining stresses to dominate. Experiments were conducted using Magnetic Resonance Imaging (MRI) coupled with Particle Image Velocimetry (PIV) to measure 3D flow profiles. A unique protocol involved stepwise shear and compression to capture quasi-static flow fields, as the MRI scanner requires static snapshots. Confining pressures were tuned from near-zero up to $\approx 250$ Pa, reaching approximately 1% of the particle Young's modulus ( $E_p \approx 10$ kPa).
Discrete Element Method (DEM): Simulations replicated the experimental geometry and conditions. The model incorporated Hertzian contact laws for elastic deformation, viscous lubrication forces for the interstitial fluid, and buoyancy. Parameters, including the particle Young's modulus and density mismatch, were calibrated against experimental torque and rheology data.
Continuum Modeling (NGF): The Non-Local Granular Fluidity model was adapted for these soft, frictionless suspensions. The model relates the local stress state to the strain rate via a non-local length scale (cooperativity length, $\xi$ ) that depends on the stress field. Parameters ( $\mu_s, A$ ) were fitted to unconfined experimental data and then tested against confined scenarios without further adjustment.

Key Results

Shear Band Width and Pressure Sensitivity: In unconfined conditions (where pressure is dominated by the negligible hydrostatic gradient), the hydrogel suspensions exhibited extremely broad shear bands. As external confining pressure was applied, the shear bands collapsed into narrow zones. This transition was qualitatively and quantitatively captured by both the DEM simulations and the NGF continuum model, confirming that the flow behavior is highly sensitive to the local stress field.
Model Validation: The NGF model, originally developed for stiff, frictional grains, successfully predicted the bulk flow profiles of soft, nearly frictionless hydrogel suspensions. The fitted parameters ( $\mu_s \approx 0.1, A \approx 0.4$ ) were consistent with values for frictionless systems, suggesting the model's robustness across different particle stiffnesses.
Rate Dependence:
- Quasi-Static Regime: In the slow flow regime probed by MRI, hydrodynamic forces (lubrication) were found to be negligible compared to contact forces, validating the quasi-static assumption.
- Rheological Behavior: Unconfined suspensions exhibited Herschel-Bulkley (HB) type rheology with a flow index $n \approx 0.5$ . However, under elevated confinement, the flow deviated from HB behavior, transitioning to a power-law dependence with a lower exponent ( $n \approx 0.2$ ).
- DEM Limitations: While DEM matched the quasi-static flow profiles well, it failed to reproduce the power-law flow behavior observed in confined rheological experiments, suggesting the standard contact models lack the necessary time-dependent physics (e.g., viscoelastic relaxation) to capture this regime.

Significance and Claims
The paper claims that the NGF model is a generalizable framework for dense granular flows, extending beyond rigid, dry systems to include soft, nearly frictionless suspensions. The primary contribution is the explicit demonstration that pressure sensitivity is a fundamental component of non-local flow rules, even when particle softness and fluid buoyancy are present. The results indicate that the width of shear bands is governed by the local stress state relative to the yield point, a mechanism that holds true for hydrogel suspensions despite their distinct micro-physics.

The authors modestly note that while the NGF model captures the qualitative trends of shear band narrowing under pressure, it does not yet account for the specific effects of particle softness ( $P/E_p$ ) on the yield stress or the viscoelastic relaxation timescales that appear to drive the anomalous rate dependence in highly confined flows. Consequently, the work establishes the necessity of stress-dependent closure equations in non-local models but identifies particle softness and time-dependent contact mechanics as critical areas for future refinement to achieve fully quantitative agreement in all flow regimes.

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