Reciprocal swimming in viscoelastic granular hydrogels

This study demonstrates that a reciprocally flapping swimmer achieves significant locomotion in cohesive granular hydrogels specifically when its flapping frequency matches the material's inverse relaxation time, a phenomenon driven by hysteresis in drag and propulsion forces caused by low-density zones and inertia, which results in motion opposite to that observed in cohesion-free granular media.

The Gist

Imagine you are trying to swim through a bucket filled with wet, squishy jelly balls (hydrogel spheres). Now, imagine you are a tiny robot shaped like a scallop, with two wings that flap open and shut.

In a normal, sticky liquid like honey, or in a bucket of dry, hard plastic beads, physics has a strict rule: if you flap your wings open and shut in the exact same way (a "reciprocal" motion), you won't go anywhere. You'll just wiggle in place. This is known as the "Scallop Theorem."

However, this paper describes a surprising experiment where that rule gets broken, but only under very specific conditions. Here is what the researchers found, explained simply:

The Setup: A Squishy Bucket

The researchers built a small robot with two square wings. They placed it inside a box filled with hydrogel spheres. These are tiny, water-filled balls that are:

1. Very slippery (almost no friction).
2. Very squishy (they can stretch and bounce back).
3. Slightly sticky to each other because of the water between them.

They made the robot flap its wings open and shut at different speeds and watched what happened.

The Surprise: Going Backward

When they tested the robot in a bucket of hard, dry plastic beads, it moved forward, just like you might expect.

But in the squishy hydrogel balls, something weird happened:

• Too Slow: If the robot flapped very slowly, it didn't move. It just wiggled back and forth.
• Too Fast: If it flapped very quickly, it also didn't move.
• Just Right: At a specific "Goldilocks" speed (about 1 flap per second), the robot started moving!

The most shocking part? The robot moved in the opposite direction compared to how it moved in the hard plastic beads. In the squishy jelly, it swam backward.

Why Did This Happen? (The Three Ingredients)

The paper explains that this backward motion is a magic trick created by mixing three things together: Inertia, Squishiness, and Time.

1. The Heavy Bucket (Inertia)

Usually, we think of the swimmer having weight. But in this experiment, the robot was fixed in place, and the entire bucket of jelly balls was sitting on air cushions so it could slide freely.

• The Analogy: Imagine standing on a skateboard (the bucket) while holding a heavy spring (the robot). When you push the spring, the skateboard moves.
• Because the bucket of jelly is heavy, it has inertia. It doesn't want to start or stop moving instantly. When the robot's wings flap, the bucket lags behind. This delay creates a "push" that helps the robot move.

2. The Memory of the Jelly (Viscoelasticity)

The hydrogel balls aren't just solid; they are like a memory foam that takes time to settle.

• The Analogy: Think of a crowded dance floor. If someone suddenly pushes through, they leave an empty space (a void) behind them. If they stop, the crowd slowly shuffles to fill that gap.
• When the robot's wings flap, they push the jelly balls apart, creating low-density "voids" or empty pockets.
• The Timing:
  • Too Fast: The wings flap so fast the jelly balls can't move out of the way or fill the gaps. The robot just flaps in a solid block.
  • Too Slow: The jelly balls have plenty of time to shuffle back and fill the gaps perfectly. The robot just flaps in a fluid.
  • Just Right: The wings flap at a speed that matches how fast the jelly balls can rearrange themselves. The robot creates a gap, and the jelly balls start to fill it, but not quite in time. This creates a "lag" or hysteresis.

3. The Perfect Mismatch (Resonance)

The magic happens when the speed of the flapping matches the speed at which the jelly balls relax and rearrange.

• Because of the inertia (the heavy bucket lagging) and the viscoelasticity (the jelly taking time to fill gaps), the forces acting on the robot change depending on when the wings are moving.
• For a brief moment during the flap, the resistance pushes the robot in one direction, and then the "springiness" of the jelly pushes it further in that same direction before the wings even change direction.
• This creates a net push in the backward direction, effectively breaking the "Scallop Theorem."

The Takeaway

The paper concludes that you can make a simple, symmetric flapping robot move in a complex, squishy material, but only if you hit the perfect rhythm. It's like pushing a child on a swing: if you push at the wrong time, they stop. If you push at the exact right moment (matching the swing's natural rhythm), they go higher and faster.

In this case, the "swing" is the squishy jelly, and the "push" is the robot's flapping wings. When the timing is perfect, the robot surfs the lag between the jelly's movement and its own, propelling itself backward through the granular goo.

Technical Summary

1. Problem Statement

The study addresses the challenge of active locomotion in complex, non-Newtonian granular media. Specifically, it investigates how a "scallop-like" swimmer (which undergoes reciprocal, symmetric shape changes) can achieve net propulsion in a granular hydrogel.

• Theoretical Context: In low-Reynolds-number Newtonian fluids, Purcell's Scallop Theorem dictates that reciprocal motion cannot generate net propulsion due to time-reversibility. However, in non-Newtonian or granular media, this theorem can be bypassed.
• The Gap: While locomotion in rigid, frictional granular media (like plastic beads) is well-studied, the behavior of swimmers in cohesive, nearly frictionless, viscoelastic granular hydrogels remains poorly understood. These materials exhibit unique properties (low friction, high elasticity, liquid capillary bridges) that differ significantly from dry granular matter or standard polymer solutions.

2. Methodology

The authors employed a combination of experimental mechanics, high-speed imaging, and X-ray radiography.

• Experimental Setup:
  • Swimmer: A 3D-printed scallop-inspired device with two counter-rotating square wings (15 mm × 15 mm) driven by a servo motor and gears.
  • Medium: A bed of hydrogel spheres (swollen from dry powder to 1–2 mm diameter). These particles are nearly frictionless ($\mu \approx 0.01$) and cohesive due to liquid capillary bridges.
  • Comparison: Experiments were also conducted in polystyrene granules (rigid, frictional) to contrast behaviors.
  • Container: The granular bed was contained in a box resting on air bearings, allowing the entire container to move frictionlessly in the swimming direction ($y$). The swimmer was fixed relative to the lab frame, meaning the observed motion is the reaction of the container/swimmer system.
• Measurement Techniques:
  • Kinematics: Linear Hall encoders tracked the displacement of the container ($y_s$).
  • Forces: A load cell measured the driving force ($f_y$) on the swimmer.
  • Imaging: X-ray radiography (110 kV, 400 µA) was used to visualize density variations and void formation within the granular bed during flapping cycles.
• Variables: The flapping frequency ($1/T$) was varied systematically to observe transitions in locomotion regimes.

3. Key Results

A. Directional Reversal and Frequency Dependence

• Polystyrene (Rigid/Frictional): The swimmer moved forward ($+y$) with net displacement that was largely rate-independent, consistent with previous findings where particle jamming drives propulsion.
• Hydrogel (Viscoelastic/Cohesive):
  • Low Frequency ($1/T < 0.5$ Hz): No net locomotion. The swimmer returns to its starting position after each cycle.
  • Intermediate Frequency ($1/T \approx 1.0$ Hz): Significant backward locomotion ($-y$ direction), opposite to the motion in rigid media. This is the peak efficiency regime.
  • High Frequency ($1/T > 2.0$ Hz): Locomotion ceases again; the swimmer oscillates in place.

B. Mechanism of Propulsion

The study identifies a unique mechanism driven by the interplay of inertia and viscoelastic relaxation:

1. Inertia and Phase Shift:

   • At the optimal frequency ($1 \approx 1$ Hz), the swimmer's inertia (coupled with the massive container) creates a phase shift between the wing motion and the container's displacement.
   • The swimmer continues moving in the direction of the previous stroke even after the wings reverse, creating a period where the driving force and velocity align ($\dot{y}_s f_y > 0$), generating net work.
   • This resembles a resonant oscillation of a driven damped oscillator.

2. Viscoelastic Relaxation and Hysteresis:

   • Density Variation: X-ray radiograms revealed that wing flapping creates low-density zones (voids) in the hydrogel.
   • Time Scales: The hydrogel has two relaxation times: a fast viscoelastic time ($\tau_1 \approx 0.3$ s) and a slow particle rearrangement time ($\tau_2 \approx 4.9$ s).
   • Hysteresis: At the optimal frequency ($T \sim \tau_2$), the voids created during the "opening" stroke are only partially refilled during the "closing" stroke. This asymmetry in density (and thus drag/propulsion forces) breaks the time-reversal symmetry required by the Scallop Theorem, allowing net propulsion.
   • Extreme Frequencies:
     • Too Slow: Particles rearrange fully between strokes; symmetry is restored; no net motion.
     • Too Fast: Voids persist but the medium behaves purely elastically (no relaxation); the system oscillates without net drift.

4. Key Contributions

• Discovery of Backward Locomotion: The paper demonstrates that in cohesive, viscoelastic granular media, reciprocal swimmers can move in the opposite direction compared to their motion in rigid, frictional granular media.
• Identification of a Resonance Regime: It establishes that maximum propulsion occurs when the flapping frequency matches the inverse of the material's relaxation time, creating a resonant interaction between the swimmer's inertia and the medium's viscoelasticity.
• Mechanistic Insight: The study provides experimental evidence (via X-ray) that hysteresis in granular density (void formation and incomplete refilling) is the primary driver for breaking time-reversal symmetry in this specific medium.
• Theoretical Framework: The authors propose a simplified dynamical equation (Eq. 2) incorporating mass, drag, elasticity, and driving force to model the system, highlighting the necessity of all three terms (inertia, viscosity, elasticity) for locomotion.

5. Significance

• Fundamental Physics: This work challenges the application of the Scallop Theorem in complex granular environments, showing that viscoelasticity and inertia can combine to enable propulsion even with symmetric strokes. It bridges the gap between fluid mechanics (viscoelastic fluids) and granular physics (jamming/rearrangement).
• Biological and Engineering Applications:
  • Microrobotics: The findings are crucial for designing microrobots intended to navigate through biological tissues (e.g., blood clots, mucus, or soft tissues) which often exhibit viscoelastic and granular properties similar to hydrogels.
  • Soft Matter: It offers a new model for understanding how organisms (like worms or larvae) might move through cohesive, wet soils or sediments where friction is low but elasticity is high.

In conclusion, the paper reveals that reciprocal swimming in viscoelastic granular hydrogels is a resonant phenomenon dependent on the precise matching of actuation frequency to the material's relaxation timescales, resulting in a unique backward propulsion mechanism driven by density hysteresis and inertial phase shifts.