Anomalous Dynamics of Superparamagnetic Colloidal Microrobots with Tailored Statistics

Abstract

Living organisms have developed advanced motion strategies for efficient space exploration, serving as inspiration for the movements of microrobots. These real-life strategies often involve anomalous dynamics displaying random movement patterns that deviate from Brownian motion. Despite their biological inspiration, autonomous stochastic navigation strategies of current microrobots remain much less versatile than those of their living counterparts. Supported by theoretical reasoning, this work demonstrates superparamagnetic colloidal microrobots with fully customizable stochastic dynamics displaying the entire spectrum of anomalous diffusion, from subdiffusion to superdiffusion, across statistically significant spatial and temporal scales (covering at least two decades). By simultaneously tuning microrobots' step-length distribution and, critically, their velocity autocorrelation function with magnetic fields, fundamental anomalous dynamics are reproduced with tailored properties mimicking Lévy walks and fractional Brownian motion. These findings pave the way for programmable microrobotic systems that replicate optimal stochastic navigation strategies found in nature for applications in medical robotics and environmental remediation.

Figures (15)

• Figure 1: Anomalous diffusion of superparamagnetic colloidal microrobots in the comoving frame. (A) Two concentric Halbach cylinders (inner dipole; outer quadrupole) generate a linear magnetic field $\mathbf{B}$ (color gradient) and a constant force $\mathbf{F_\mathrm{B}}$ (black arrow) on a sample of colloidal microrobots (black rectangle). The cylinders’ magnetic north (N, red) and south (S, blue) are shown. Rotating the quadrupole (purple arrows) around the fixed dipole (orange arrow) reorients $\mathbf{F}_\mathrm{B}$ and the microrobots' direction. Two consecutive rotations define their turning angle $\varphi$ as twice the quadrupole rotation angle. (B) Trajectory of a colloidal microrobot moving at approximately constant speed ($4.6~\pm0.8$ µm $\rm s^{-1}$) in a comoving frame defined by its velocity vector $\mathbf{v}$ and $\varphi$. Scale bar: 25 µm. (C-D) Bespoke sequences $\varphi_n$ in time $t$ of uniformly distributed $\varphi$ in $[-\pi, \pi)$, obtained by sampling the quadrupole rotation time $\tau_n$ (solid lines) from (C) a half-Gaussian and (D) a power-law (anomalous exponent $\mu = 1.5$) distribution. (E) Time-averaged mean squared displacements (${\rm MSD}$, symbols) and trajectories (inset) of microrobots yielding long-time normal diffusion ($\mu = 1$) and superdiffusion ($\mu = 1.5$) as confirmed by a logarithmic curve fit (dashed lines). Diffusive ($\propto \Delta t$) and ballistic ($\propto \Delta t^2$) slopes shown for reference. Scale bar: 1 mm.
• Figure 2:
• Figure 3: Tailoring microrobots' superdiffusion by controlling step-length distributions. (A) Normalized time-averaged mean squared displacements (${\rm MSD}$, dots) yielding long-time superdiffusion for microrobots' trajectories (inset) generated according to Eqs. \ref{['eq:overd1d']} and \ref{['eq:overd2d2']} by sampling the turning angle $\varphi_n$ from the uniform distribution on the circle and the flight time $\tau_n$ from power-law distributions of varying exponent $\alpha = 3 - \mu$ between the normal diffusive ($\mu = 1$) and ballistic ($\mu = 2$) limits. Case for $\mu = 1.5$ as in Fig. \ref{['fig:Fig1']}E. Fit lines (dashed lines) confirm the different superdiffusive regimes (Table \ref{['tab_Stats_Fig3']}). The MSDs are normalized to the square of each microrobot's short-term drift distance $L$ in the driving magnetic field. Scale bar: 5 mm. (B-E) Probability distribution functions (PDF, dots) of experimental step lengths $\hat{\ell}_n$ for (B) $\mu = 1$, (C) $\mu = 1.25$, (D) $\mu = 1.75$ and (E) $\mu = 2$. ${\rm PDF} (\hat{\ell}_n) \sim {\rm PDF} (\langle \hat{v} \rangle\,\hat{\tau}_n)$ (thick background lines). Case for $\mu = 1.5$ in Fig. \ref{['fig:Fig2']}B. Fit lines (dashed lines) show power-law scalings ($\sim \hat{\ell}_n^{\hat{\mu}-4}$) consistent with the desired ground-truth values of $\mu$ (Table \ref{['tab_Stats_Fig3']}). Diffusive (A: $\propto \Delta t$; B-E: $\propto \ell_n^{-3}$) and ballistic (A: $\propto \Delta t^2$; B-E: $\propto \ell^{-2}_n$) limits shown for reference.
• Figure 4: Tailoring anomalous diffusion by controlling the microrobots' velocity autocorrelation function. (A) Normalized time-averaged mean squared displacements (${\rm MSD}$, dots) yielding different anomalous dynamics at long times, from subdiffusion ($\mu < 1$) to superdiffusion ($\mu > 1$) through normal diffusion ($\mu = 1$), for microrobots' trajectories (inset) consistent with fractional Brownian motion (Experimental Section, Supplementary Text). Fit lines (dashed lines) confirm the different anomalous diffusion regimes (Table \ref{['tab_Stats_Fig4']}). The MSDs are normalized to the square of each microrobot's short-term drift distance $L$ in the driving magnetic field. Diffusive ($\propto \Delta t$) and ballistic ($\propto \Delta t^2$) limits shown for reference. Scale bar: 100 µm. (B) Respective normalized time-averaged velocity autocorrelation functions $C_v$ (dots) as a function of lag time $\Delta t$ calculated from the trajectories in (A) for different ground-truth values of $\mu$. Fitting the tail of the data with a quadratic polynomial scaled by a power law (dashed lines) confirms the asymptotic scaling characteristic of fractional Brownian motion at different values of $\mu$ ($\sim \hat{\mu}(\hat{\mu}-1)\Delta t^{\hat{\mu}-2}$, Table \ref{['tab_Stats_Fig4']}).
• Figure S1: Magnetic fields of the Halbach cylinders. (A-B) In-scale schematics of a cylindrical Halbach (A) dipole and (B) quadrupole, built with cubic permanent magnets (gray squares with black arrows pointing to their north poles). The cylinders' magnetic north (N, red) and south (S, blue) poles as well as flux lines are shown. The strong dipolar field $\mathbf{B}^{\rm D}=B_0 \mathbf{e}_y$ is homogeneous and directed along the $y$-axis (unit vector, $\mathbf{e}_y$). The weaker quadrupolar field $\mathbf{B}^{\rm Q}$ consist of two orthogonal linear components. (C) The cubic magnets (side length, $a_{\rm m}$) are arranged in a circular pattern in the $xy$ plane, with the cylinders' axis along $z$ (Experimental Section, table \ref{['tab_magnets']}). The inner radius $r_\mathrm{in}$ (from the center to the magnets’ inner edge) defines the experimental area. The outer radius $r_\mathrm{out}$ (from center to the magnets' outer edge) delimits the size of the cylinder. (D) Measured dipolar magnetic field intensity $B_0$ with (E) $x$- and (F) $y$-components, $B^\mathrm{D}_x$ and $B^\mathrm{D}_y$. $B^\mathrm{D}_x$ is one order of magnitude smaller than $B^\mathrm{D}_y$. (G) Measured quadrupolar magnetic field intensity $B_{\rm Q}$ with (H) $x$- and (I) $y$-components, $B^\mathrm{Q}_x$ and $B^\mathrm{Q}_y$, which are orthogonal and linear in space.