Light-steerable locomotion using zero-elastic-energy modes

Biology inspires synthetic materials that emulate adaptive, dynamic behaviours of living systems; a key to lifelike functionality is to drive materials out of equilibrium, enabling autonomous, adaptive, and interactive responses. In soft actuators and soft robotics, self-regulating, feedback-driven deformation coupled with energy dissipation can yield self-sustained periodic motions under constant energy input from light, heat, or chemistry, enabling locomotion, transport, homeostasis, and signal transduction. Zero-elastic-energy modes (ZEEMs) are particularly intriguing non-equilibrium mechanisms arising in mechanically frustrated, prestrained systems that can undergo continuous motion without elastic energy cost when driven. ZEEM-based motions have been shown in diverse materials, but achieving controllable, dynamically steerable, self-sustained locomotion remains challenging because direction and velocity are often poorly controlled far from equilibrium; unidirectional motion typically requires built-in asymmetries that are hard to tune post-fabrication. This study addresses these challenges by introducing a light- or heat-fuelled liquid crystal elastomer (LCE) torus that exhibits self-sustained autorotation due to ZEEMs, and demonstrates optical steering of locomotion via controlled friction or viscous drag across terrestrial, confined, and fluidic environments, including fully steerable swimming in the Stokes regime.

The paper situates ZEEMs within broader nonequilibrium materials research, referencing prior demonstrations of self-sustained motions and oscillators in soft systems driven by light, heat, or chemical reactions. ZEEMs, rooted in zero-frequency hydrodynamic states of symmetry-broken systems, have manifested in contexts such as quantum phenomena, DNA topology, knots, and soft matter structures. Previous ZEEM-related demonstrations include nylon, PDMS, hydrogels, fibreboids, LCEs, and rotating Möbius strips, highlighting the potential for autonomous soft robotics. However, prior systems often lacked dynamic steerability and relied on fixed geometric or mechanical asymmetries. The authors build on theoretical work proposing toroidal swimming at low Reynolds numbers and on prior observations of thermally driven toroidal self-rotation, aiming to realize Purcell’s envisioned toroidal swimmer with optical steerability and to overcome control limitations by leveraging friction and viscous drag asymmetries induced by photothermal effects.

Materials and fibre fabrication: Light/thermally responsive LCE fibres were synthesized from a monomer mixture comprising RM82 (LC crosslinker), amine chain extenders (8-amino-1-octanol and dodecylamine), non-polymerizable mesogen 5CB (50 wt% to reduce viscosity), and photoinitiator (2,2-dimethoxy-2-phenylacetophenone). The mixture (80 °C) was injected into 8-mm-diameter elastic silicone tubes, held for 10 min, then cooled to 45 °C (1 °C min−1). Oligomerization proceeded via aza-Michael addition (acrylate:amine molar ratio 1.1:1) over 24 h at 45 °C.

Alignment programming and polymerization: Mechanical forces were applied before UV photopolymerization (365 nm, 180 mW cm−2, 30 min) to program LC alignment and thus the axial thermal expansion coefficient α. Uniaxial stretching yielded monodomain alignment along the fibre axis with negative thermal expansion (α < 0), producing reversible contraction (~35%) upon heating. Twisting the tube 3 turns per cm during polymerization induced alignment perpendicular to the fibre axis (assisted by homeotropic anchoring on the tube wall), generating positive thermal expansion (α > 0) with reversible ~35% axial expansion upon heating. After polymerization, tubes were removed in isopropanol, and fibres were post-cured on a 120 °C hot plate to complete curing and evaporate 5CB. To impart photothermal sensitivity, fibres were dyed by immersion in Disperse Red 1 (DR1) in isopropanol; DR1 diffusion provided efficient visible-light absorption without significantly altering thermal deformability. Estimated work capacity was ~5 J g−1, supporting repeated deformations.

Torus construction: Fibres were glued end-to-end to form closed loops (tori). The torus is characterized by torus radius R, fibre radius r, and slenderness ε = r/R. The sign of α determines rotation sense: α < 0 leads to inversion; α > 0 to eversion.

Experimental setups: Autorotation was triggered either by uniform heating on a hot plate or by photothermal heating with a continuous-wave 532 nm laser. Illumination intensities used included thresholds near 1.5 W cm−2 for α < 0 autorotation on glycerol, 1.8–1.9 W cm−2 for terrestrial motion demonstrations, and up to ~7.3 W cm−2 for viscous liquid and Stokes-regime swimming (non-polymerized PDMS, viscosity 5.5 Pa·s). Infrared thermography (FLIR T420BX) monitored temperature fields; imaging (Canon 5D Mark III) recorded motion. Dynamic mechanical analysis (Anton Paar MCR-702e) characterized viscoelastic properties and loss modulus.

Mechanistic modeling: ZEEM formation in a closed elastic loop arises from topological prestrain generating a static strain field across the cross-section. A thermal gradient induces a dynamic strain field; their interplay generates a torque Ma that can overcome internal and external losses (M1) to produce autorotation. For photothermal activation, the maximum driving torque is modeled as Mα = παβτε I, with E Young’s modulus, α thermal expansion coefficient, r fibre radius, ε slenderness, β a fitting parameter, τ the thermal relaxation time, and I the light intensity. Thresholds and rotation speeds follow ω² = A I − B, where A = ε α β and B = 1/τ (with internal viscoelastic losses dominating resistance torque). Temperature oscillations during rotation arise from alternating exposure to and shadowing from the light as segments circulate.

Locomotion modalities and force analyses: Terrestrial steering relies on light-induced softening and increased friction coefficient μ on the illuminated side, creating asymmetric friction drags f1 and f2 that bias translation away from the light for α < 0 and towards the light for α > 0. On a suspended thread, the single contact point supplies static friction for directed motion under lateral illumination. In viscous liquids (glycerol, PDMS), photothermal effects can generate bubbles that adhere at interfaces (in glycerol), modifying motion (e.g., stepwise climbing along a pipette). In confined conduits (glass tube), α > 0 tori maintain outer surface contact and move towards the light via frictional drag. For Stokes-regime swimming (PDMS, Re ~ 0.0001), viscous drag asymmetry between outer and inner torus surfaces under eversion/inversion produces net propulsion; direction reverses with the sign of α. The relation between translational displacement s and rotational angle θ is linear, matching theoretical predictions for toroidal swimmers with limited fluid sliding (sliding coefficient ζ ≈ 0.36).

Data acquisition and analysis: Motion trajectories and angular kinematics were extracted from videos (Tracker). Flow fields were analyzed by particle image velocimetry (PIVlab in Matlab). Absorption spectra were measured with a UV–vis spectrophotometer (Cary 60).

• Programmable thermal expansion in LCE fibres: uniaxial alignment (α < 0) yields reversible axial contraction ~35%; twisted alignment (α > 0) yields reversible axial expansion ~35% upon heating.
• DR1-doped LCE fibres exhibit efficient photothermal response with comparable thermally driven deformability and approximate work capacity ~5 J g−1, enabling repeated deformation and autorotation under visible light (532 nm).
• Photothermal activation threshold: for α < 0 tori on glycerol, autorotation initiates at I ≈ 1.5 W cm−2 with ΔT ≈ 80 °C; during rotation, ΔT exhibits sawtooth-like oscillations with ~20 °C amplitude due to periodic exposure and cooling.
• Rotation kinetics and scaling: ω increases with I and obeys ω² = A I − B, consistent with theory accounting for slenderness ε, thermal expansion α, fitting factor β, and thermal relaxation 1/τ; internal viscoelastic losses dominate resistance torque. Thinner fibres have higher activation thresholds but rotate faster once activated (tested radii ~260, 290, 325 μm).
• Light-controlled friction modulation: illumination softens LCE and increases friction coefficient μ by ~3× at 1.3 W cm−2, producing asymmetric drags (f1, f2) under oblique illumination that steer terrestrial translation. For α < 0, motion is away from the light; for α > 0, towards the light.
• Multirealm locomotion: Directed motion demonstrated on dry surfaces, on a 50 μm thread (single contact point providing static friction), and in liquids. In glycerol, photothermally generated bubbles at interfaces cause stepwise adhesion-assisted climbing along a pipette wall. In a 5 mm glass conduit filled with glycerol, α > 0 tori translate towards the light via outer-surface friction.
• Transition from friction-dominated rolling to drag-dominated self-propulsion observed on tapered pipettes in viscous liquids, rectifying translation as the local confinement increases.
• Stokes-regime swimming: In non-polymerized PDMS (viscosity 5.5 Pa·s), Re ≈ 0.0001. The torus swims in arbitrary directions (up, down, left, right) under light steering; left/right velocities are similar, upward swimming is faster than downward due to convective heat effects along the z-axis. Translational displacement s scales linearly with rotational angle θ, matching theoretical prediction for toroidal swimmers with sliding coefficient ζ ≈ 0.36.
• 3D steerability and reorientation: Under inhomogeneous light, the torus inclines to equalize excitation, with the brighter side acting as a stator. Optical reorientation rates of ~7° s−1 enable dynamically programmable 3D trajectories by repositioning the excitation beam.
• Direction reversibility: Changing illumination direction or switching the sign of α reverses rotation sense and swimming direction under identical conditions.

The study demonstrates that continuous spatiotemporal symmetry breaking via ZEEMs in a toroidal LCE enables net propulsion even at very low Reynolds numbers, addressing a long-standing challenge highlighted by Purcell’s scallop theorem. By programming the sign of the axial thermal expansion coefficient (through alignment control) and using photothermal gradients, the torus exhibits self-sustained, unidirectional rotation that, when coupled to asymmetric friction or viscous drag, converts into controlled translation. This approach circumvents the backflow losses typical of reciprocal swimmers because the rotation is unidirectional and the potential energy change during rotational deformation is minimized. The ability to steer direction dynamically with light across dry, confined, and viscous environments—including fully steerable Stokes-regime swimming and 3D path planning—demonstrates a versatile, untethered platform for active soft matter and microrobotics. The observed dependence of swimming performance on orientation relative to convection (upward faster than downward) and the predicted reversal of swimming direction with α corroborate the mechanistic model. Overall, the findings validate ZEEM-driven toroidal swimmers as promising, efficient low-Re locomotors with externally programmable navigation.

A light-fuelled, millimetre-scale LCE torus was engineered to exhibit self-sustained autorotation under constant illumination via global ZEEMs, with rotation direction set by the programmed sign of the axial thermal expansion. By harnessing light-induced friction and drag asymmetries, the system achieves robust, steerable locomotion across multiple realms (terrestrial, confined, and fluidic). Crucially, the toroidal swimmer operates in the Stokes regime (Re ≈ 0.0001) with fully controllable 3D navigation, experimentally realizing a concept envisioned for efficient low-Re swimming. The work establishes prestrained topological structures and ZEEMs as a general design principle for out-of-equilibrium soft robots. Future research could explore geometry generalizations (e.g., Möbius strips, supercoils), alternative responsive materials (hydrogels, shape memory, piezoelectrics), miniaturization and integration with sensing/feedback, and optimization of efficiency and control strategies under diverse fluidic conditions.

• Performance depends strongly on light intensity and thermal gradients; relatively high intensities (≈1.5–7.3 W cm−2) are required for activation and swimming in viscous media.
• Thermal effects introduce environmental dependencies: convection along the vertical axis aids upward swimming but hampers downward motion, leading to direction-dependent speeds.
• In glycerol, photothermally generated bubbles accumulate at interfaces, causing adhesion and stepwise motion that can perturb smooth translation.
• Rotation thresholds and speeds are sensitive to fibre geometry: thinner fibres require higher activation intensity though they rotate faster once active, implying trade-offs in design.
• Internal viscoelastic losses constitute a primary resistance torque, which may limit efficiency; the model indicates material losses dominate during steady rotation.
• Directionality depends on the sign of α and on precise light-field alignment; inhomogeneous illumination can cause facet reorientation before steady swimming, adding control complexity.