Forces that control self-organization of chemically-propelled Janus tori

The study addresses how anisotropic, chemically self-propelled Janus microtori behave and self-organize near confining walls—a key question in active matter with implications for microrobotics, drug delivery, and precision medicine. While synthetic active particles exhibit propulsion, chemotaxis, and rheotaxis, most realizations have simple shapes (spheres, rods), limiting functionality. Recent additive manufacturing enables complex shapes such as tori that, in experiments, sediment and hover near walls, spontaneously tilt, translate parallel to the wall, and form rotating/translating bound states without applied magnetic fields. The research aims to identify and quantify the forces and interactions—gravitational, hydrodynamic, diffusiophoretic, electrostatic, and steric—governing hovering, tilt-induced translation, and the emergence of bound states of one or multiple tori near a wall, and to reconcile simulations with experimental observations.

Prior work showed active colloids near walls can hover or slide at constant height and tilt depending on surface mobility and wall effects (e.g., diffusiophoretic Janus spheres) and explored dynamics of more complex swimmers (linked rotating beads, conformational swimmers). For toroidal geometries, analytical and numerical studies examined inactive tori and various active mechanisms (rotation, linked rotating beads, diffusiophoresis). These studies established wall-modulated concentration and flow fields as central to near-wall dynamics. However, a comprehensive account combining self-diffusiophoresis, hydrodynamics, gravity, electrostatics, and sterics for toroidal Janus motors near walls remained lacking, motivating the present simulations.

The authors use multiparticle collision dynamics (MPCD), a particle-based mesoscale simulation method that incorporates thermal fluctuations, hydrodynamic interactions, chemical reactions, and solid inclusions. Geometry: slab with top/bottom walls parallel to (x,y), separation Lz; periodic boundaries in x,y. Tori (centerline radius a=5) are constructed from NA=39 overlapping spherical beads linked by harmonic springs (k=500); beads are Janus with catalytic (C) and noncatalytic (N) hemispheres. Solvent consists of A (fuel) and B (product) particles with reactions A→B on catalytic faces; a bulk reaction maintains nonequilibrium. Diffusiophoresis arises from asymmetric A/B distributions and bead–solute interactions; propulsion direction defines forward- vs backward-moving tori relative to the torus symmetry vector û (from catalytic to noncatalytic side). Walls: bounce-back for fluid; short-range repulsive Lennard-Jones wall potential; optional uniformly charged wall interacting with bead-centered point dipoles distributed around the torus with sinusoidally varying sign/magnitude. Gravity: a downward force parameterized by strength g; beads also experience short-range bead–solvent Lennard-Jones interactions, and steric repulsion between beads on different tori prevents overlap. Quantities measured include the probability distribution of center-of-mass (c.o.m.) height P(z), tilt-angle distribution p(θ) with θ from wall normal (cos θ = uz), inter-torus distance distributions P(D), and concentration (CB) and velocity fields v(r) in planes cutting through the torus. Simulations are performed for single tori (hovering, tilt, translation), pairs of tori (formation of stacked or side-by-side bound states under varying g and charge–dipole strength εcd), and four-tori systems (cluster states). A phenomenological Langevin/Fokker–Planck theory models the tilt-angle distribution p(θ), first without charge–dipole interactions (leading to p(θ) ∝ e^{βγ cos θ}) and then including charge–dipole torque (yielding p(θ) ∝ e^{β(γ1 cos θ − (γ2/2) cos^2 θ)} that captures symmetry breaking and nonzero tilt). Simulation parameters (dimensionless): MD timestep dt=0.001; MPCD collision time τ=0.5; temperature kBT=0.2; solvent density ρ0=10; bead–solute potentials with εA, εB=0.16 (interchanged for forward/backward modes as specified); wall potential εw=3.0; charge–dipole potential strength εd varied (εcd in figures); bulk reaction rate k=0.001; typical box sizes 40×40×40 for single torus and 60×60×40 for multi-tori. Derived transport coefficients: dynamic viscosity η≈1.43, diffusion D≈0.117, kinematic viscosity ν≈0.143; dimensionless numbers for a representative speed V=0.006 include Re≈0.2, Pe≈0.25, Da≈1.2, Sc≈1.2.

• Hovering due to coupled gravity and self-diffusiophoresis/hydrodynamics: For backward-moving tori with catalytic side facing the wall, P(z) shows stable confinement near the wall for g in 0.1–0.3, with mean height increasing as g decreases; for g ≤ 0.15 the beads do not contact the wall, indicating hovering arises from phoretic/hydrodynamic effects rather than direct repulsion. Tilt-angle distributions p(θ) peak at θ=0 and broaden as g decreases; strong confinement (g=0.3) yields a very narrow peak. Concentration and flow fields reveal elevated product concentration within the torus hole and characteristic near-field recirculations and wall-limited suction/flows. Forward-moving tori exhibit inverted near-interface flow direction relative to backward-moving tori, and wall-distorted concentration gradients modify propulsion.
• Electrostatic charge–dipole interactions induce symmetry breaking, tilt, and translation: Including bead–wall charge–dipole interactions with strength εcd causes spontaneous tilt of a hovering torus. The peak tilt angle θmax transitions from ~0 to finite values beyond a critical εcd ≈ 0.12 (for g=0.15), consistent with a bifurcation smoothed by thermal noise. The mean propulsion speed parallel to the wall V increases with θmax; even near θmax≈0, thermal fluctuations yield nonzero V. A simple Langevin/Fokker–Planck model with p(θ) ∝ e^{β(γ1 cos θ − (γ2/2) cos^2 θ)} fits simulation data well over εcd values (fit parameters reported in the paper for εcd ∈ {0,0.1,0.15,0.2}).
• Pair interactions and bound states: Two backward-moving tori (g=0.15, εcd=0.2) hover independently without forming long-lived bound states; c.o.m. separations D mainly 18–28 (a=5), with similar P(z) and p(θ) for both—indicating minimal clustering propensity. Two forward-moving tori form long-lived metastable bound states whose structure depends on g and εcd: for εcd=0.2, g=0.07, D concentrates between ~7–10 (≈ stacked configuration; since a=5), with unequal heights and larger tilt on the upper torus; for εcd=0.2, g=0.09, a predominantly side-by-side bound pair forms parallel to the wall with D ≈ 14, similar heights and small tilt (p(θ) peaked near 0). For εcd=0.1, g=0.07 dynamics fluctuates between stacked and side-by-side; for εcd=0.1, g=0.09 side-by-side dominates. In bulk (no gravity), two forward-moving tori also form stacked metastable states: either co-aligned catalytic sides that co-propagate along their common axis, or opposing catalytic faces that tumble and execute active Brownian rotation without net propagation.
• Four-torus clusters: For forward-moving tori with εcd=0.2, g=0.07, clusters predominantly contain a stacked pair modified by interactions with additional tori, with some tori flipping to have catalytic sides toward the wall; P(z) is double-peaked and p(θ) extends to large angles (including a minor peak near ~150°). For εcd=0.2, g=0.09, clusters are dominated by two staggered interacting side-by-side pairs; P(z) and p(θ) resemble the side-by-side two-torus case.
• Control parameters and transitions: Gravitational strength g suppresses orientational fluctuations and favors side-by-side configurations; stronger εcd induces tilt and stacked configurations. The study predicts a transition from stacked to side-by-side bound states of forward-moving tori as g increases (or εcd decreases), and that backward-moving tori do not form bound pairs near walls under comparable conditions.

The simulations demonstrate that the near-wall dynamics of Janus microtori emerges from an interplay of gravity, self-diffusiophoretic concentration and flow fields, electrostatic charge–dipole interactions with the wall, and steric effects. Gravity confines the torus near the wall while diffusiophoresis and hydrodynamics generate a hovering state with suppressed large tilts. Introducing charge–dipole interactions breaks the symmetry of the horizontal hovering state, producing a preferred finite tilt that, in turn, drives translation parallel to the wall—qualitatively consistent with experiments on 3D-printed Pt/Ni-coated microtori. The phenomenological Langevin/Fokker–Planck description captures the tilt-angle distributions and their εcd dependence, linking microscopic torques to macroscopic orientation statistics. For multiple tori, the geometry-dependent concentration and flow fields mediate effective interactions that stabilize distinct bound states: stacked (vertically offset) versus side-by-side (parallel) configurations, with transitions controlled by g and εcd. Forward-moving tori exhibit significant binding propensity, while backward-moving tori behave largely independently, mirroring observations for other active colloids. These insights clarify how tuning particle mass density (e.g., catalyst loading or hollow printing) and electrostatic interactions can program collective behaviors and bound-state architectures near walls.

The study identifies and quantifies the combined roles of gravitational, hydrodynamic, diffusiophoretic, electrostatic, and steric interactions in governing the self-organization of chemically propelled Janus tori near walls. Key contributions include: (i) explanation of torus hovering without direct contact via diffusiophoresis/hydrodynamics under gravity; (ii) demonstration that charge–dipole interactions with a charged wall induce spontaneous tilt and near-wall translation, matching experimental phenomenology; (iii) prediction that backward-moving tori do not form bound states near walls, whereas forward-moving tori form long-lived stacked or side-by-side pairs with a gravity-controlled transition between them; and (iv) identification of diverse four-torus cluster states dependent on g and εcd. These findings suggest design routes to control collective states by tuning effective mass density (e.g., via Pt deposition or hollow structures) and electrostatic coupling; future work could extend the algorithms to other shapes (propellers, helices), more complex reaction schemes, and include magnetic interactions to explore richer programmable collective dynamics.

The model assumes an effective charge–dipole interaction between the torus and wall rather than explicitly resolving the full electrochemistry and electrohydrodynamics; direct experimental measurements of the dipole are not yet available. Surface mobilities are treated uniformly across catalytic and noncatalytic faces, and some material-specific interactions are simplified. Simulations are in dimensionless units with finite system sizes and times; metastable states and rare escape events may be under-sampled. The phenomenological tilt theory captures statistics but uses fitted parameters (γ1, γ2) rather than deriving them from first principles. Wall charging and environmental ionic strength effects are simplified and not varied systematically.