Light-driven continuous rotating Möbius strip actuators

The study explores whether Möbius and twisted toroidal ribbons made from stimulus-responsive materials can achieve continuous, in situ rotation under external excitation. Classical Möbius strips possess a C2 symmetry axis and localize torsional strain at a twist locus. Prior theoretical and experimental work indicated that topological prestrain and rotational-symmetry breaking are required to access zero-elastic-energy modes enabling continuous motion; otherwise, stimuli often induce only uniform shrinkage without rotation. The objective is to design Möbius strip actuators that introduce controlled symmetry breaking via a photothermal contraction gradient, enabling redistribution of torsional energy along the strip and continuous rotation of the twist around the ring.

Previous work identified zero-energy deformation modes in prestrained toroidal polymer rods, showing that topological prestrain plus symmetry breaking (compression/tension imbalance) can yield continuous rotation (Baumann et al., Nat. Mater. 2018). A recent Möbius strip actuator under thermal stimulus exhibited only shrinkage due to identical inner and outer faces, lacking symmetry breaking (Davidson et al., Adv. Mater. 2020). Liquid crystal elastomers (LCEs) are well-established light/heat-responsive actuators capable of large, reversible deformations, with extensive work on photomobile polymers, alignment programming, and photothermal dyes enabling remote actuation. Elastic instability of higher-order Möbius strips leading to writhed loops has been theoretically described, setting expectations for shape transitions during actuation.

Materials and LCE synthesis: A polysiloxane LCE system was prepared using poly(methylhydrosiloxane) (PMHS) as the backbone and mesogenic monomer MBB. Crosslinkers 11UB and VBPB (varied ratio) tuned the LC–isotropic transition temperature. A croconaine NIR dye (YHD796, 0.07 wt%) provided photothermal conversion. Hydrosilylation (Karstedt’s catalyst) proceeded via two-step crosslinking. Two LCE types were made: LCE796 (mono-domain, uniaxially stretched during curing, dye-doped) and LCE0 (polydomain, unstretched, no dye). Actuator fabrication: Single-layer Möbius strips (S-Möbius[+1], S-Möbius[+2]) were formed by twisting LCE796 ribbons by 180° or 360° and gluing ends with silicone adhesive. Bilayered strips were made by laminating a pre-crosslinked, uniaxially stretched LCE796 strip onto a pre-crosslinked LCE0 film during the second crosslinking at 60 °C for 48 h, then cutting to 38×4 mm strips and twisting to form B-Möbius[+1], B-Möbius[+2], or B-Möbius[-2]. For continuous rotation of B-Möbius[+2], a diameter-matched inner cylinder (d=5 mm) was inserted to prevent transformation into writhed loops. Rolling robots were built by placing a hollow plastic tube in one of the holes of B-Möbius[±2]. Continuous, defect-free Möbius[±1] design (C-Möbius): A gradient bilayered LCE ribbon with homogeneous, one-sided photothermal contraction gradient was created. During the first curing, a PTFE mold on a 10° incline yielded a continuous thickness gradient along the ribbon. The pre-crosslinked film was uniaxially stretched (~140%), establishing an alignment gradient (thin end exhibits higher chain anisotropy). The film was cut longitudinally into two symmetric pieces and overlapped with thin-to-thick and thick-to-thin ends opposed, followed by the second crosslinking to bond into a bilayer (33×3 mm). The strip was twisted by 180° and ends joined such that thin-to-thin and thick-to-thick met, producing a one-sided, defect-free Möbius[+1] (and analogously [-1]). Characterization and actuation: NIR light (808±3 nm) provided photothermal stimulation; intensity varied from 0.03–1.1 W cm⁻². Surface temperature of the irradiated region was monitored; LC–isotropic transition for LCE796 was ~69.5 °C. 2D-WAXS quantified alignment at thin vs thick ends; order parameters S=0.70 (thin) and 0.55 (thick). Photothermal shrinkage ratios ΔL/L0 for thin vs thick ends were measured versus temperature, showing up to 47.3% vs 23.0% contraction (>100 °C). Rotation dynamics were recorded by video; positions of markers were analyzed (Tracker) to obtain rotation angle, speed, and trajectories of the twist locus and edge points. DMA measured elastic moduli of bilayered ribbons prepared with stretching ratios 105–150% (modulus 0.222–0.660 MPa). Parametric studies varied light intensity, scanning rate (2.4–30° s⁻¹), and ribbon modulus to assess their impact on continuous rotation.

• Single-layer LCE796 Möbius actuators (S-Möbius[+1], S-Möbius[+2]) under NIR illumination only shrank and wrinkled; no continuous rotation occurred due to uniform in-plane contraction.
• Bilayered actuators (B-Möbius[+1], B-Möbius[+2], B-Möbius[-2]) developed bending in the illuminated twist region from differential contraction (LCE796 vs LCE0), inducing rotation of the twist locus as the light spot scanned. B-Möbius[+1] rotation ceased at the glued junction (defect), while B-Möbius[+2] progressed until transforming into a helically writhed loop, halting rotation.
• Inserting a 5 mm inner cylinder into B-Möbius[+2] prevented writhed transition and enabled continuous rotation around the cylinder at ~16° s⁻¹ (clockwise). B-Möbius[-2] achieved anticlockwise rotation at ~11° s⁻¹. A light-fuelled B-Möbius[+2] tractor dragged a 183 mg load (robot mass 70 mg) at 0.51 mm s⁻¹.
• Authentic, one-sided, defect-free C-Möbius[±1] actuators with homogeneous homeotropic contraction gradient achieved in situ continuous rotation under NIR: C-Möbius[+1] rotated anticlockwise at ~4.9° s⁻¹; C-Möbius[-1] rotated clockwise at ~5.6° s⁻¹. A miniaturized C-Möbius[+1] (22×2 mm) reached ~5.1° s⁻¹.
• Optimal NIR intensity for continuous rotation was ~0.7 W cm⁻², maintaining the twist locus at ~83 °C; at 1.1 W cm⁻² speed increased to ~5.5° s⁻¹ but risked thermal damage. At 0.2–0.5 W cm⁻², rotation was discontinuous with 1.2–2.3° s⁻¹.
• Light scanning rate critically set local temperature; at 2.4° s⁻¹, the locus reached ~140 °C and deformed poorly; around 10° s⁻¹ yielded ~70 °C with slow/discontinuous motion; near 3–5° s⁻¹ produced ~105–83 °C with continuous rotation near 83 °C.
• Elastic modulus influenced capability: below ~0.35 MPa no continuous rotation; threshold ~0.45 MPa enabled ~1.1° s⁻¹; higher moduli 0.52–0.66 MPa gave 4.9–5.3° s⁻¹ (stretching ratio up to ~140–150%). Excessive stretching (~150%) risked ribbon breakage.
• Motion analysis: Twist locus traced a near-circle of radius ~3.7 mm (reduced from initial ~4.8 mm due to contraction). An edge point (mark C) executed a vertical flip on a circular path of radius ~1.5 mm, completing 360° after two 360° rotations of the twist locus (720° total). The flip angle vs twist rotation angle followed a sigmoidal (Boltzmann) relation with three stages of differing sensitivity.

Continuous rotation requires both topological prestrain and symmetry breaking. Single-layer LCE strips lack asymmetry and only contract uniformly, failing to delocalize energy and move the twist. Bilayers introduce homeotropic contraction gradients to change local curvature at the twist, initiating twist migration. However, structural defects (junctions) or elastic instabilities (writhed loop formation in higher-twist strips) interrupt motion. By engineering a one-sided, defect-free, homogeneous contraction gradient in a true Möbius[±1] geometry, continuous in situ rotation is achieved without external guides. The rotation dynamics scale with photothermal input and mechanical stiffness, consistent with expectations that sufficient heating above Tiso and adequate modulus are needed to sustain bending-driven propagation of the twist locus. The observed writhed transitions in B-Möbius[+2] align with elastic instability theory for higher-order Möbius strips. These findings demonstrate how geometric topology, material anisotropy, and spatially patterned stimuli can be combined to create continuous soft robotic motions, enabling rolling robots and load-carrying devices actuated by light.

The work introduces light-driven Möbius strip actuators based on liquid crystal elastomers that achieve continuous rotation. Bilayered B-Möbius[±2] can rotate when guided and, with an inner cylinder, perform sustained rotation and robotic rolling, including load transport. Most notably, authentic single-faced C-Möbius[±1] actuators with a continuous, defect-free, homogeneous homeotropic contraction gradient enable in situ continuous rotation under NIR irradiation. Rotation performance depends on light intensity, scanning rate, and ribbon modulus, with optimal conditions around 0.7 W cm⁻², ~83 °C at the twist locus, and modulus ≥0.45 MPa. This approach paves the way for actuators and shape-morphing systems exploiting continuous deformation rather than discrete state switching. Future efforts could focus on enhancing durability under thermal cycling, optimizing dye loading and photothermal efficiency, scaling down dimensions for higher speeds and energy efficiency, integrating autonomous light-guidance, and generalizing the topology-material design to other geometries and stimuli.

• High NIR intensities (~1.1 W cm⁻²) raise temperatures that risk thermal damage to LCEs; continuous operation favors moderate intensity (~0.7 W cm⁻²).
• B-Möbius[+1] rotation is halted by the end-to-end junction defect; B-Möbius[+2] tends to transform into writhed loops without an inner cylinder, interrupting rotation.
• Continuous rotation requires the twist locus to reach ~83 °C; at lower temperatures or overly fast scanning, motion is discontinuous or stalled.
• Mechanical constraints: achieving high modulus via high stretching ratios (~150%) can cause ribbon breakage; there is a trade-off between stiffness and robustness.
• Reported speeds are modest (few degrees per second) for C-Möbius[±1]; application throughput may be limited without further optimization.
• Dependence on NIR-absorbing dye and precise alignment gradients may affect manufacturability and long-term stability.