Magnetic cilia carpets with programmable metachronal waves

The study investigates how to experimentally realize and program metachronal waves—phase-shifted collective beating patterns—in large carpets of artificial cilia to emulate and study biological transport phenomena. Natural metachronal waves in motile cilia enhance fluid transport at low Reynolds numbers and are central to processes such as mucus clearance, egg transport, and feeding. While prior work largely relies on numerical simulations and observations of natural organisms, there is a need for an experimental platform with both large cilia numbers and programmable beating patterns to probe emergent behaviors. The authors aim to create a soft magnetic cilia carpet with encoded magnetization patterns that yield programmable metachronal waves under rotating magnetic fields, and to assess its capabilities in fluid pumping and surface locomotion, thereby contributing to both fundamental biophysical understanding and cilia-inspired robotic applications.

Metachronal coordination has been extensively studied via simulations and in natural systems, showing advantages for transport efficiency over in-phase beating. Natural studies are constrained by limited tunability of cilium properties and sensitivity to imaging conditions. Artificial cilia have been driven by pneumatic, optical, acoustic, electric, and magnetic stimuli and used for pumping and mixing. Magnetic artificial cilia demonstrations include lines with length-dependent thresholds producing metachrony in uniform fields and assemblies achieving 2D metachronal waves. However, prior systems typically either feature many cilia beating in phase or only a few cilia with limited programmability. The present work addresses this gap by enabling large arrays with programmable, spatially varying magnetization to produce desired metachronal waveforms.

Fabrication: Cilia carpets were produced by a two-step molding process using 3D-printed polymer molds (Vero-Clear). Molds were cleaned, UV-cured, and coated with a thin insulating/smoothing resin layer (XTC-3D). Step 1: a magnetic composite (NdFeB microparticles MQP-S-11-9 and Ecoflex 00-30, 1:1 wt) was mixed, degassed, and pressed into 0.8 mm-diameter, 4 mm-deep cylindrical cavities to form cilia hairs. Step 2: pure Ecoflex 00-30 was poured to form a soft, stretchable substrate. Cure: 65 °C for 8 h; carpets were then demolded. Magnetization: Non-magnetized NdFeB particles were magnetized post-fabrication. Carpets were stretched/wrapped onto 3D templates of prescribed curvature to encode spatially varying magnetization, then pulsed in an impulse magnetizer (IM-10, ASC Scientific) at up to 1.2 T, imprinting magnetization vectors across the array. Actuation: An 8-coil electromagnetic navigation system (CardioMag) generated uniform rotating fields (typically 80 mT at 30°/s) in specified planes (e.g., x–z). Single-cilium dynamics were characterized under slowly rotating fields; motion comprised synchronized and asynchronized phases analogous to power and recovery strokes. Modeling: A quasi-static magneto-elastic model based on Cosserat rod theory simulated cilium deformation under gravity and magnetic loading, with internal elastic forces/torques (linearly elastic, isotropic material; permanent, homogeneous magnetization). State variables included position, orientation (quaternion), internal force, and torque along arc length. The external field from the 8-coil system was modeled via a calibrated linear multi-dipole matrix; desired rotating fields were synthesized with currents computed from the Moore–Penrose pseudoinverse of the calibration matrix. The system was discretized into 20 segments and solved using numerical continuation (MATCONT) to follow equilibrium branches. Parameters (from Table 1): magnetization magnitude 3.82×10⁻⁵ A·m², E = 1.85×10⁵ Pa, G = 6.16×10⁴ Pa, ρ = 2.39×10³ kg/m³, length 4 mm, diameter 0.8 mm. Fluid transport experiments: Carpets (36×36 mm) were placed in an acrylic tank (160×110×45 mm) filled with 99% glycerol (η ≈ 1.15 Pa·s) seeded with 200 µm fluorescent particles. A POM baffle was positioned to define the observation region. A uniform 80 mT rotating field at 30°/s actuated the cilia. Particle trajectories were imaged (Fujifilm X‑T20) and analyzed with custom Matlab scripts to compute displacement over one actuation period (12 s), mapping displacement fields (x–z) and vertical profiles. Average displacement within 0–4 mm above the carpet quantified pumping performance; various metachronal wavelengths and cilia spacings were tested. Locomotion experiments: Cilia carpets (hairs: 4 mm length, 0.8 mm diameter, 4 mm pitch; substrate: 1 mm Ecoflex) were inverted onto a smooth, dry POM surface in the 8-coil system. Baby powder reduced adhesion. A clockwise rotating field (x–z plane) at 30°/s was applied. Locomotion modes (crawling vs rolling) were assessed as a function of field magnitude; long carpets (20×5 cilia) with encoded antiplectic waves (e.g., λ = 6d) emulated millipede-like gaits. Locomotion speeds were measured from the geometric center under different wave vectors (antiplectic, symplectic, no wave).

• Programmable metachronal waves: By magnetizing stretched carpets on curved templates, spatial magnetization patterns were encoded, yielding symplectic, antiplectic, and diaplectic (standing) metachronal waves with controllable wavelengths across large arrays (e.g., 8×8).
• Single-cilium dynamics and model validation: Cilia exhibited synchronized and asynchronized phases under slow rotating fields. Reciprocal motion at 10–40 mT transitioned to non-reciprocal, biomimetic 3D trajectories at 50–80 mT, crucial for low-Re pumping. Trajectories depended on magnetization orientation (e.g., z-axis magnetization produced D-shaped tip paths; transverse magnetization produced figure‑8 paths). Simulations (Cosserat rod model) matched experimental trajectories.
• Fluid transport enhancement: Metachronal waves increased tracer displacement. Carpets with long metachronal wavelengths (e.g., XL) showed approximately threefold higher average particle displacement per period compared to an infinite-wavelength (identically beating) carpet, and produced coherent flow structures over the carpet.
• Density dependence: Varying inter-cilium spacing d from 4 to 10 mm showed average particle displacement scaling approximately as d^a with a ≈ 1.5, consistent with numerical predictions (a ≈ 1.4–1.6). At very small spacings, magnetic/mechanical interactions impeded full cilium trajectories, reducing performance due to crowding.
• Multimodal locomotion: Two modes observed depending on field strength. Crawling via travelling metachronal waves occurred at low fields (~40 mT). Above ~60 mT, magnetic torque curled the carpet, inducing rolling locomotion that was faster than crawling for 8×8 carpets.
• Wave-vector effect on crawling: Millipede-inspired long carpets (20×5) with antiplectic waves (λ = 6d) achieved efficient crawling. Antiplectic waves yielded higher locomotion speeds than symplectic waves or no wave. Mechanism: during recovery stroke, antiplectic waves bulged the substrate (reducing drag), whereas symplectic waves dented it (increasing friction), hindering leg recovery.
• Design/fabrication: Demonstrated scalable, assembly-free fabrication of >200-cilium carpets with customizable geometry (length gradients, density) and high substrate stretchability enabling complex magnetization patterns.

The work addresses the challenge of experimentally studying emergent metachronal phenomena with both large cilia counts and programmable beat phase patterns. By encoding magnetization distributions via template-assisted magnetization, the authors decoupled wave formation from hydrodynamic self-organization, enabling systematic exploration of wave types and wavelengths. The validated magneto-elastic model explains how magnetization direction and field strength shape non-reciprocal trajectories crucial for pumping at low Reynolds numbers. Fluid experiments corroborate key numerical predictions: metachronal coordination significantly enhances transport over in-phase beating and exhibits a density-dependent scaling of performance. Locomotion tests further reveal how the direction of wave propagation relative to the power stroke influences gait efficiency through substrate curvature–friction interactions, favoring antiplectic coordination. Collectively, the findings demonstrate a versatile platform to probe cilia–fluid and cilia–surface interactions, bridging simulation and biology, and informing the design of cilia-inspired soft robotic systems for transport and locomotion.

This study introduces a scalable, highly customizable soft magnetic cilia carpet that encodes programmable metachronal waves via template-assisted magnetization. The platform reproduces and extends key phenomena: non-reciprocal cilium trajectories, wavelength- and density-dependent fluid transport gains (up to threefold over in-phase carpets), and multimodal locomotion with superior antiplectic crawling. A validated Cosserat-rod-based model provides predictive capability for cilium trajectories under varied magnetization and fields. The system offers an experimental testbed for studying collective cilia behavior and active matter, and a foundation for cilia-inspired soft robots with potential biomedical utility. Future work could downscale cilia dimensions, broaden material/magnetization design spaces, integrate faster and more compact actuation systems, explore 3D flow fields and transport of biologically relevant cargo, and investigate feedback-controlled or self-organized coordination.

• Fabrication resolution: Mould dimensions are limited by the 3D printer resolution, constraining minimum cilium size and inter-cilium spacing.
• Material processing: Higher NdFeB loading increases composite viscosity, complicating molding and potentially limiting magnetization magnitude and uniformity.
• Actuation hardware: The 8-coil CardioMag system is bulky with slow dynamic response, limiting actuation bandwidth and portability.
• Crowding effects: At high cilia densities, strong magnetic and mechanical interactions prevent full trajectories, degrading pumping performance.
• Scale mismatch: Experiments are at millimeter scale in high-viscosity fluids to approximate low-Re conditions; direct microscale biological equivalence may require microfabrication and different materials.
• Single-material assumptions: Modeling assumes homogeneous, permanent magnetization and linear elastic isotropy, which may not capture all material behaviors.