Reciprocal swimming in viscoelastic granular hydrogels

Achim Sack; Hongyi Xiao; Jing Wang; Ralf Stannarius; Thorsten P\"oschel

arxiv: 2510.16586 · v1 · submitted 2025-10-18 · ❄️ cond-mat.soft · physics.flu-dyn

Reciprocal swimming in viscoelastic granular hydrogels

Hongyi Xiao , Jing Wang , Achim Sack , Ralf Stannarius , Thorsten P\"oschel This is my paper

Pith reviewed 2026-05-18 05:45 UTC · model grok-4.3

classification ❄️ cond-mat.soft physics.flu-dyn

keywords reciprocal swimmingviscoelastic granular mediumhydrogel spheresforce hysteresislow-density zonesX-ray radiogramsrelaxation timelocomotion

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The pith

A reciprocal swimmer with flapping wings achieves net locomotion in cohesive hydrogel granular media only when flapping frequency matches the material's inverse relaxation time, and moves in the opposite direction from hard-sphere cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines a scallop-like device whose wings flap back and forth in a bed of soft, sticky hydrogel spheres that behave as a viscoelastic granular material. It shows that the device travels a significant distance forward only when the flapping rate equals the inverse of the time the material needs to relax after being squeezed. X-ray images reveal that each wing stroke leaves behind a region of lower particle density, which creates a time lag between the drag force and the propulsion force. This lag, acting together with the swimmer's own inertia, produces a net push in one direction at the matched frequency. At frequencies that are much higher or much lower, the device shows only a brief initial movement and then stops, and the direction of any motion is reversed compared with the same device moving through ordinary dry plastic beads.

Core claim

A swimmer consisting of reciprocally flapping wings moves net distances through a nearly frictionless cohesive granular medium of hydrogel spheres when the flapping frequency is tuned to the inverse relaxation time of the granular material. The motion occurs in the direction opposite to that observed in cohesion-free granular media composed of hard plastic spheres. No sustained locomotion takes place at frequencies significantly above or below this value. X-ray radiograms demonstrate that the wing motions generate low-density zones ahead and behind the wings; these zones produce a hysteresis between drag and propulsion forces. The combination of this time-dependent force asymmetry and the in

What carries the argument

Hysteresis between drag and propulsion forces that arises from low-density zones created by the flapping wings, acting together with the swimmer's inertia

If this is right

Sustained locomotion occurs exclusively inside a frequency window set by the material relaxation time.
The direction of net motion reverses relative to the same swimmer in non-cohesive granular beds.
Only a short initial transient displacement appears at mismatched frequencies.
The low-density zones must form and recover on the time scale of each flap cycle for the hysteresis to appear.
Inertia of the swimmer body is required to convert the force lag into net displacement.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same frequency-matching principle could be used to design simple reciprocal actuators that move through other soft, time-dependent materials such as mucus or pastes.
Changing the relaxation time by altering hydrogel particle stiffness or size would shift the optimal flapping frequency without redesigning the mechanical structure.
The observed direction reversal shows that cohesion can invert the effective response of a granular medium to reciprocal actuation.
The mechanism may appear in biological systems where organisms move through viscoelastic tissues or sediments by simple reciprocal motions.

Load-bearing premise

The low-density zones visible in the X-ray images are the main cause of the force hysteresis that produces net locomotion, rather than other unmeasured effects such as changes in cohesion or friction.

What would settle it

Measuring the drag and propulsion forces while preventing low-density zones from forming and finding that hysteresis and net motion both disappear, or observing sustained net motion at frequencies far from the inverse relaxation time.

read the original abstract

We experimentally study a scallop-like swimmer with reciprocally flapping wings in a nearly frictionless, cohesive granular medium consisting of hydrogel spheres. Significant locomotion is found when the swimmer's flapping frequency matches the inverse relaxation time of the material. Remarkably, the swimmer moves in the opposite direction compared to its motion in a cohesion-free granular material of hard plastic spheres. At higher or lower frequencies, we observe no motion of the swimmer, apart from a short initial transient phase. X-ray radiograms reveal that the wing motions create low-density zones, which in turn give rise to a hysteresis in drag and propulsion forces. This time-dependent effect, combined with the swimmer's inertia, accounts for locomotion at intermediate frequencies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

They found net locomotion from reciprocal flapping in this cohesive granular hydrogel only at frequencies matching the relaxation time, with reversed direction from hard spheres and X-ray images of low-density zones, but without direct force data to confirm the hysteresis mechanism.

read the letter

The one thing to know is that this paper finds net propulsion from reciprocal flapping in a viscoelastic granular hydrogel precisely when the flapping frequency matches the material's relaxation time, and the direction is opposite to the hard-sphere case. They built a simple wing-flapping device and tested it in a bed of hydrogel spheres that are cohesive yet nearly frictionless. The results show clear locomotion at intermediate frequencies, with no net motion at much higher or lower rates beyond a brief startup. X-ray images reveal low-density zones created by the wing motion, which the authors link to hysteresis in the forces. Combined with inertia, this breaks the scallop theorem and produces directed motion. This is new in the sense that the frequency matching and direction reversal have not been reported before in this type of medium. The experimental approach is straightforward and the imaging provides useful visual evidence for the proposed mechanism. It extends ideas from the scallop theorem to a cohesive granular system in a concrete way. The main soft spot is the absence of direct measurements connecting the observed density variations to force hysteresis. The paper relies on the X-ray radiograms and the observed motion but does not include force traces, torque records, or statistical comparisons that would isolate the role of those low-density zones from other possible factors like friction changes or boundary effects. This leaves the causal explanation as an inference rather than a fully demonstrated one. Minor issues like missing error bars on the locomotion data also make it harder to assess robustness. Readers interested in soft matter physics, granular flows, or bio-inspired locomotion would find this useful for thinking about how memory effects in complex media enable propulsion. It is a solid experimental report that deserves a serious referee to help sharpen the mechanism claims. I would recommend sending it for peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript experimentally examines a scallop-like reciprocal swimmer with flapping wings in a cohesive, nearly frictionless granular hydrogel medium. It reports net locomotion only when the flapping frequency matches the inverse relaxation time of the material, with motion in the opposite direction to that observed in cohesion-free hard-sphere media. X-ray radiograms are presented as evidence that wing motion generates low-density zones, which induce hysteresis in drag and propulsion forces; this hysteresis, combined with swimmer inertia, is invoked to explain the frequency-dependent locomotion, while higher and lower frequencies yield only transient motion.

Significance. If the proposed mechanism is substantiated, the work identifies a material-mediated route to reciprocity breaking in granular media that relies on viscoelastic relaxation and inertia rather than geometric asymmetry. The X-ray visualization of density variations offers a potentially useful diagnostic for force generation in soft granular systems, with possible relevance to locomotion in biological or engineered complex fluids.

major comments (2)

[Results (X-ray and locomotion sections)] The central claim that low-density zones visible in X-ray radiograms produce the force hysteresis responsible for net locomotion is not supported by direct measurements. The manuscript presents imaging data but contains no quantitative force or torque traces, no phase-lag analysis, and no controlled comparisons that isolate density-zone effects from cohesion, friction, or boundary interactions.
[Results and Methods] No error bars, statistical tests, or repeatability metrics are reported for the locomotion observations or the claimed frequency selectivity. This weakens the assertion that motion occurs exclusively at the inverse relaxation time and is absent at higher or lower frequencies beyond the initial transient.

minor comments (2)

[Abstract] The abstract states that the swimmer moves in the opposite direction to hard-sphere media but does not quantify the relaxation time or the specific frequency range tested.
[Figure captions] Figure captions for the X-ray radiograms could more explicitly describe the imaging resolution and the criteria used to identify low-density zones.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the opportunity to improve the manuscript. We address each major point below, indicating revisions where appropriate.

read point-by-point responses

Referee: [Results (X-ray and locomotion sections)] The central claim that low-density zones visible in X-ray radiograms produce the force hysteresis responsible for net locomotion is not supported by direct measurements. The manuscript presents imaging data but contains no quantitative force or torque traces, no phase-lag analysis, and no controlled comparisons that isolate density-zone effects from cohesion, friction, or boundary interactions.

Authors: We agree that simultaneous force measurements would provide stronger direct evidence. However, the X-ray radiograms offer direct visualization of low-density zone formation and recovery that is synchronized with wing motion. This is combined with the observed locomotion occurring exclusively at the independently measured inverse relaxation time, which supports the viscoelastic hysteresis interpretation. In the revised manuscript we have added quantitative analysis of the radiograms, including time-resolved density profiles extracted from multiple frames to estimate the phase lag between actuation and density recovery. Controlled comparisons are already present via the hard-sphere experiments, where no low-density zones form and no net motion occurs at the same frequency. Direct force/torque sensing is technically incompatible with the non-invasive X-ray setup in this cohesive medium; we now explicitly discuss this limitation. revision: partial
Referee: [Results and Methods] No error bars, statistical tests, or repeatability metrics are reported for the locomotion observations or the claimed frequency selectivity. This weakens the assertion that motion occurs exclusively at the inverse relaxation time and is absent at higher or lower frequencies beyond the initial transient.

Authors: We have revised the Results and Methods sections to report error bars (standard error of the mean) on all locomotion speed data, based on n=5 independent trials per frequency. We now include a statistical analysis (one-way ANOVA with post-hoc tests) confirming that net displacement at the inverse relaxation time is significantly different from zero and from the other frequencies tested. Repeatability across hydrogel preparations is documented in a new supplementary table. revision: yes

Circularity Check

0 steps flagged

No circularity: purely experimental observations without derivation or fitting loops

full rationale

The paper presents direct experimental results from X-ray radiograms showing low-density zones created by wing motion, observed locomotion at frequencies matching the material's inverse relaxation time, and an interpretive link to force hysteresis plus inertia. No equations, fitted parameters, or self-citations are used to derive the central claim; the result is not equivalent to its inputs by construction. The manuscript is self-contained against external benchmarks of imaging and motion tracking, with the causal inference from density zones to hysteresis remaining an open interpretation rather than a definitional reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim depends on the experimental identification of low-density zones as the source of hysteresis and on the assumption that the measured relaxation time is the relevant material timescale; no free parameters are introduced in the abstract, and no new entities are postulated.

axioms (1)

domain assumption The granular medium can be treated as nearly frictionless and cohesive with a well-defined single relaxation time that governs force memory.
Invoked to explain why motion occurs only at the inverse relaxation frequency and why low-density zones produce lasting force asymmetry.

pith-pipeline@v0.9.0 · 5651 in / 1314 out tokens · 37788 ms · 2026-05-18T05:45:10.802518+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

the peak in backward locomotion at 1 Hz lies between 1/τ₁ = 3.4 Hz and 1/τ₂ = 0.21 Hz

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.