Magneto-capillary particle dynamics at curved interfaces: inference and criticism of dynamical models.

Time-varying fields drive the motion of magnetic particles adsorbed on liquid drops due to interfacial constraints that couple magnetic torques to capillary forces. Such magneto-capillary particle dynamics and the associated fluid flows are potentially useful for propelling drop motion, mixing drop contents, and enhancing mass transfer between phases. The design of such functions benefits from the development and validation of predictive models. Here, we apply methods of Bayesian data analysis to identify and validate a dynamical model that accurately predicts the field-driven motion of a magnetic particle adsorbed at the interface of a spherical droplet. Building on previous work, we consider candidate models that describe particle tilting at the interface, field-dependent contributions to the magnetic moment, gravitational forces, and their combinations. The analysis of each candidate is informed by particle tracking data for a magnetic Janus sphere moving in a precessing field at different frequencies and angles. We infer the uncertain parameters of each model, criticize their ability to describe and predict experimental data, and select the most probable candidate, which accounts for gravitational forces and the tilting of the Janus sphere at the interface. We show how this favored model can predict complex particle trajectories with micron-level accuracy across the range of driving fields considered. We discuss how knowledge of this "best" model can be used to design experiments that inform accurate parameter estimates or achieve desired particle trajectories.

Synchronization and alignment of model oscillators based on Quincke rotation.

Colloidal spheres in weakly conductive fluids roll back and forth across the surface of a plane electrode when subject to strong electric fields. The so-called Quincke oscillators provide a basis for active matter based on self-oscillating units that can move, align, and synchronize within dynamic particle assemblies. Here, we develop a dynamical model for oscillations of a spherical particle and investigate the coupled dynamics of two such oscillators in the plane normal to the field. Building on existing descriptions of Quincke rotation, the model describes the dynamics of the charge, dipole, and quadrupole moments due to charge accumulation at the particle-fluid interface and particle rotation in the external field. The dynamics of the charge moments are coupled by the addition of a conductivity gradient, which describes asymmetries in the rates of charging near the electrode. We study the behavior of this model as a function of the field strength and gradient magnitude to identify the conditions required for sustained oscillations. We investigate the dynamics of two neighboring oscillators coupled by far field electric and hydrodynamic interactions in an unbounded fluid. Particles prefer to align and synchronize their rotary oscillations along the line of centers. The numerical results are reproduced and explained by accurate low-order approximations of the system dynamics based on weakly coupled oscillator theory. The coarse-grained dynamics of the oscillator phase and angle can be used to investigate collective behaviors within ensembles of many self-oscillating colloids.

Designing negative feedback loops in enzymatic coacervate droplets.

Membraneless organelles within the living cell use phase separation of biomolecules coupled with enzymatic reactions to regulate cellular processes. The diverse functions of these biomolecular condensates motivate the pursuit of simpler in vitro models that exhibit primitive forms of self-regulation based on internal feedback mechanisms. Here, we investigate one such model based on complex coacervation of the enzyme catalase with an oppositely charge polyelectrolyte DEAE-dextran to form pH-responsive catalytic droplets. Upon addition of hydrogen peroxide "fuel", enzyme activity localized within the droplets causes a rapid increase in the pH. Under appropriate conditions, this reaction-induced pH change triggers coacervate dissolution owing to its pH-responsive phase behavior. Notably, this destabilizing effect of the enzymatic reaction on phase separation depends on droplet size owing to the diffusive delivery and removal of reaction components. Reaction-diffusion models informed by the experimental data show that larger drops support larger changes in the local pH thereby enhancing their dissolution relative to smaller droplets. Together, these results provide a basis for achieving droplet size control based on negative feedback between pH-dependent phase separation and pH-changing enzymatic reactions.

Accelerating the Design of Self-Guided Microrobots in Time-Varying Magnetic Fields.

Mobile robots combine sensory information with mechanical actuation to move autonomously through structured environments and perform specific tasks. The miniaturization of such robots to the size of living cells is actively pursued for applications in biomedicine, materials science, and environmental sustainability. Existing microrobots based on field-driven particles rely on knowledge of the particle position and the target destination to control particle motion through fluid environments. Often, however, these external control strategies are challenged by limited information and global actuation where a common field directs multiple robots with unknown positions. In this Perspective, we discuss how time-varying magnetic fields can be used to encode the self-guided behaviors of magnetic particles conditioned on local environmental cues. Programming these behaviors is framed as a design problem: we seek to identify the design variables (e.g., particle shape, magnetization, elasticity, stimuli-response) that achieve the desired performance in a given environment. We discuss strategies for accelerating the design process using automated experiments, computational models, statistical inference, and machine learning approaches. Based on the current understanding of field-driven particle dynamics and existing capabilities for particle fabrication and actuation, we argue that self-guided microrobots with potentially transformative capabilities are close at hand.

Active Colloids as Models, Materials, and Machines.

Active colloids use energy input at the particle level to propel persistent motion and direct dynamic assemblies. We consider three types of colloids animated by chemical reactions, time-varying magnetic fields, and electric currents. For each type, we review the basic propulsion mechanisms at the particle level and discuss their consequences for collective behaviors in particle ensembles. These microscopic systems provide useful experimental models of nonequilibrium many-body physics in which dissipative currents break time-reversal symmetry. Freed from the constraints of thermodynamic equilibrium, active colloids assemble to form materials that move, reconfigure, heal, and adapt. Colloidal machines based on engineered particles and their assemblies provide a basis for mobile robots with increasing levels of autonomy. This review provides a conceptual framework for understanding and applying active colloids to create material systems that mimic the functions of living matter. We highlight opportunities for chemical engineers to contribute to this growing field.

Convection Confounds Measurements of Osmophoresis for Lipid Vesicles in Solute Gradients.

Lipid vesicles immersed in solute gradients are predicted to migrate from regions of high to low solute concentration due to osmotic flows induced across their semipermeable membranes. This process─known as osmophoresis─is potentially relevant to biological processes such as vesicle trafficking and cell migration; however, there exist significant discrepancies (several orders of magnitude) between experimental observations and theoretical predictions for the vesicle speed. Here, we seek to reconcile predictions of osmophoresis with observations of vesicle motion in osmotic gradients. We prepare quasi-steady solute gradients in a microfluidic chamber using density-matched solutions of sucrose and glucose to eliminate buoyancy-driven flows. We quantify the motions of giant DLPC vesicles and Brownian tracer particles in such gradients using Bayesian analysis of particle tracking data. Despite efforts to mitigate convective flows, we observe directed motion of both lipid vesicles and tracer particles in a common direction at comparable speeds of order 10 nm/s. These observations are not inconsistent with models of osmophoresis, which predict slower motion at ca. 1 nm/s; however, experimental uncertainty and the confounding effects of fluid convection prohibit a quantitative comparison. In contrast to previous reports, we find no evidence for anomalously fast osmophoresis of lipid vesicles when fluid convection is mitigated and quantified. We discuss strategies for enhancing the speed of osmophoresis using high permeability membranes and geometric confinement.

Insights into Chemically Fueled Supramolecular Polymers.

Supramolecular polymerization can be controlled in space and time by chemical fuels. A nonassembled monomer is activated by the fuel and subsequently self-assembles into a polymer. Deactivation of the molecule either in solution or inside the polymer leads to disassembly. Whereas biology has already mastered this approach, fully artificial examples have only appeared in the past decade. Here, we map the available literature examples into four distinct regimes depending on their activation/deactivation rates and the equivalents of deactivating fuel. We present increasingly complex mathematical models, first considering only the chemical activation/deactivation rates (i.e., transient activation) and later including the full details of the isodesmic or cooperative supramolecular processes (i.e., transient self-assembly). We finish by showing that sustained oscillations are possible in chemically fueled cooperative supramolecular polymerization and provide mechanistic insights. We hope our models encourage the quantification of activation, deactivation, assembly, and disassembly kinetics in future studies.

Automating Bayesian inference and design to quantify acoustic particle levitation.

Self-propulsion of micro- and nanoparticles powered by ultrasound provides an attractive strategy for the remote manipulation of colloidal matter using biocompatible energy inputs. Quantitative understanding of particle motion and its dependence on size, shape, and composition requires accurate characterization of the acoustic field, which depends sensitively on the experimental setup. Here, we show how automated experiments based on Bayesian inference and design can accurately and efficiently characterize the acoustic field within resonant chambers used to propel acoustic nanomotors. Repeated cycles of observation, inference, and design (OID) are guided by a physical model that describes the rate at which levitating particles approach the nodal plane. Using video microscopy, we observe the relaxation of tracer particles to this plane following the application of the acoustic field. We use sequential Monte Carlo methods to infer model parameters such as the amplitude and frequency of the resonant chamber while accounting for particle-level measurement noise and population-level heterogeneity in the field. Guided by simulated outcomes, we select the optimal design for the next experiment as to maximize the information gain in the relevant parameters. We show how this iterative process serves to discriminate between competing hypotheses and efficiently converges to accurate parameter estimates using only few automated experiments. We discuss the need for model criticism to ensure the validity of the guiding model throughout automated cycles of observation, inference, and design. This work demonstrates how Bayesian methods can learn the parameters of nonlinear, hierarchical models used to describe video microscopy data of active colloids.

Quincke Oscillations of Colloids at Planar Electrodes.

Dielectric particles in weakly conducting fluids rotate spontaneously when subject to strong electric fields. Such Quincke rotation near a plane electrode leads to particle translation that enables physical models of active matter. In this Letter, we show that Quincke rollers can also exhibit oscillatory dynamics, whereby particles move back and forth about a fixed location. We explain how oscillations arise for micron-scale particles commensurate with the thickness of a field-induced boundary layer in the nonpolar electrolyte. This work enables the design of colloidal oscillators.

Fabrication and Electric Field-Driven Active Propulsion of Patchy Microellipsoids.

Active colloids are a synthetic analogue of biological microorganisms that consume external energy to swim through viscous fluids. Such motion requires breaking the symmetry of the fluid flow in the vicinity of a particle; however, it is challenging to understand how surface and shape anisotropies of the colloid lead to a particular trajectory. Here, we attempt to deconvolute the effects of particle shape and surface anisotropy on the propulsion of model ellipsoids in alternating current (AC) electric fields. We first introduce a simple process for depositing metal patches of various shapes on the surfaces of ellipsoidal particles. We show that the shape of the metal patch is governed by the assembled structure of the ellipsoids on the substrate used for physical vapor deposition. Under high-frequency AC electric field, ellipsoids dispersed in water show linear, circular, and helical trajectories which depend on the shapes of the surface patches. We demonstrate that features of the helical trajectories such as the pitch and diameter can be tuned by varying the degree of patch asymmetry along the two primary axes of the ellipsoids, namely longitudinal and transverse. Our study reveals the role of patch shape on the trajectory of ellipsoidal particles propelled by induced charge electrophoresis. We develop heuristics based on patch asymmetries that can be used to design patchy particles with specified nonlinear trajectories.

Programmable topotaxis of magnetic rollers in time-varying fields.

We describe how spatially uniform, time-periodic magnetic fields can be designed to power and direct the migration of ferromagnetic spheres up (or down) local gradients in the topography of a solid substrate. Our results are based on a dynamical model that considers the time-varying magnetic torques on the particle and its motion through the fluid at low Reynolds number. We use both analytical theory and numerical simulation to design magnetic fields that maximize the migration velocity up (or down) an inclined plane. We show how "topotaxis" of spherical particles relies on differences in the hydrodynamic resistance to rotation about axes parallel and perpendicular to the plane. Importantly, the designed fields can drive multiple independent particles to move simultaneously in different directions as determined by gradients in their respective environments. Experiments on ferromagnetic spheres provide evidence for topotactic motions up inclined substrates. The ability to program the autonomous navigation of driven particles within anisotropic environments is relevant to the design of colloidal robots.

A perturbation solution to the full Poisson-Nernst-Planck equations yields an asymmetric rectified electric field.

We derive a perturbation solution to the one-dimensional Poisson-Nernst-Planck (PNP) equations between parallel electrodes under oscillatory polarization for arbitrary ionic mobilities and valences. Treating the applied potential as the perturbation parameter, we show that the second-order solution yields a nonzero time-average electric field at large distances from the electrodes, corroborating the recent discovery of Asymmetric Rectified Electric Fields (AREFs) via numerical solution to the full nonlinear PNP equations [Hashemi Amrei et al., Phys. Rev. Lett., 2018, 121, 185504]. Importantly, the first-order solution is analytic, while the second-order AREF is semi-analytic and obtained by numerically solving a single linear ordinary differential equation, obviating the need for full numerical solutions to the PNP equations. We demonstrate that at sufficiently high frequencies and electrode spacings the semi-analytical AREF accurately captures both the complicated shape and the magnitude of the AREF, even at large applied potentials.

Swelling Cholesteric Liquid Crystal Shells to Direct the Assembly of Particles at the Interface.

Cholesteric liquid crystals can exhibit spatial patterns in molecular alignment at interfaces that can be exploited for particle assembly. These patterns emerge from the competition between bulk and surface energies, tunable with the system geometry. In this work, we use the osmotic swelling of cholesteric double emulsions to assemble colloidal particles through a pathway-dependent process. Particles can be repositioned from a surface-mediated to an elasticity-mediated state through dynamically thinning the cholesteric shell at a rate comparable to that of colloidal adsorption. By tuning the balance between surface and bulk energies with the system geometry, colloidal assemblies on the cholesteric interface can be molded by the underlying elastic field to form linear aggregates. The transition of adsorbed particles from surface regions with homeotropic anchoring to defect regions is accompanied by a reduction in particle mobility. The arrested assemblies subsequently map out and stabilize topological defects. These results demonstrate the kinetic arrest of interfacial particles within definable patterns by regulating the energetic frustration within cholesterics. This work highlights the importance of kinetic pathways for particle assembly in liquid crystals, of relevance to optical and energy applications.

Magneto-Capillary Particle Dynamics at Curved Interfaces: Time-Varying Fields and Drop Mixing.

Spatially uniform magnetic fields induce nonzero forces on magnetic particles adsorbed at curved liquid interfaces thereby driving their motion. Such motions, prohibited in bulk fluids, arise due to interfacial constraints that couple magnetic torques to capillary forces at curved interfaces. Here, we show that time-varying (spatially uniform) magnetic fields can be used to drive a variety of steady particle motions on water drops in decane. Upon application of a precessing field, magnetic Janus particles with amphiphilic surface chemistry move either along circular orbits at the drop poles or along zigzag paths at the drop equator. The different magneto-capillary motions depend on the frequency and precession angle of the field as well as the initial position of the particle on the drop surface. Our experimental observations are reproduced and explained by a mathematical model that accounts for the relevant magnetic, capillary, and hydrodynamic forces and torques that contribute to particle motion. In addition to ferromagnetic Janus particles, we show that similar dynamics can be achieved using superparamagnetic carbonyl iron particles, which are manufactured on industrial scales and respond to even weaker magnetic fields. We demonstrate how the field-driven motion of such particles at the drop interface can induce fluid flows that effectively mix the drop interior. These results suggest that magneto-capillary particle motions could be used to enhance mass transfer within emulsions stabilized by magnetic particles.

Learning Retrosynthetic Planning through Simulated Experience.

The problem of retrosynthetic planning can be framed as a one-player game, in which the chemist (or a computer program) works backward from a molecular target to simpler starting materials through a series of choices regarding which reactions to perform. This game is challenging as the combinatorial space of possible choices is astronomical, and the value of each choice remains uncertain until the synthesis plan is completed and its cost evaluated. Here, we address this search problem using deep reinforcement learning to identify policies that make (near) optimal reaction choices during each step of retrosynthetic planning according to a user-defined cost metric. Using a simulated experience, we train a neural network to estimate the expected synthesis cost or value of any given molecule based on a representation of its molecular structure. We show that learned policies based on this value network can outperform a heuristic approach that favors symmetric disconnections when synthesizing unfamiliar molecules from available starting materials using the fewest number of reactions. We discuss how the learned policies described here can be incorporated into existing synthesis planning tools and how they can be adapted to changes in the synthesis cost objective or material availability.

Directed propulsion of spherical particles along three dimensional helical trajectories.

Active colloids are a class of microparticles that 'swim' through fluids by breaking the symmetry of the force distribution on their surfaces. Our ability to direct these particles along complex trajectories in three-dimensional (3D) space requires strategies to encode the desired forces and torques at the single particle level. Here, we show that spherical colloids with metal patches of low symmetry self-propel along non-linear 3D trajectories when powered remotely by an alternating current (AC) electric field. In particular, particles with triangular patches of approximate mirror symmetry trace helical paths along the axis of the field. We demonstrate that the speed and shape of the particle's trajectory can be tuned by the applied field strength and the patch geometry. We show that helical motion can enhance particle transport through porous materials with implications for the design of microrobots that can navigate complex environments.

Shape-directed rotation of homogeneous micromotors via catalytic self-electrophoresis.

The pursuit of chemically-powered colloidal machines requires individual components that perform different motions within a common environment. Such motions can be tailored by controlling the shape and/or composition of catalytic microparticles; however, the ability to design particle motions remains limited by incomplete understanding of the relevant propulsion mechanism(s). Here, we demonstrate that platinum microparticles move spontaneously in solutions of hydrogen peroxide and that their motions can be rationally designed by controlling particle shape. Nanofabricated particles with n-fold rotational symmetry rotate steadily with speed and direction specified by the type and extent of shape asymmetry. The observed relationships between particle shape and motion provide evidence for a self-electrophoretic propulsion mechanism, whereby anodic oxidation and cathodic reduction occur at different rates at different locations on the particle surface. We develop a mathematical model that explains how particle shape impacts the relevant electrocatalytic reactions and the resulting electrokinetic flows that drive particle motion.

Measurement and mitigation of free convection in microfluidic gradient generators.

Microfluidic gradient generators are used to study the movement of living cells, lipid vesicles, and colloidal particles in response to spatial variations in their local chemical environment. Such gradient driven motions are often slow (less than 1 µm s-1) and therefore influenced or disrupted by fluid flows accompanying the formation and maintenance of the applied gradient. Even when external flows are carefully eliminated, the solute gradient itself can drive fluid motions due to combinations of gravitational body forces and diffusioosmotic surface forces. Here, we develop a microfluid gradient generator based on the in situ formation of biopolymer membranes and quantify the fluid flows induced by steady solute gradients. The measured velocity profiles agree quantitatively with those predicted by analytical approximations of relevant hydrodynamic models. We discuss how the speed of gradient-driven flows depends on system parameters such as the gradient magnitude, the fluid viscosity, the channel dimensions, and the solute type. These results are useful in identifying and mitigating undesired flows within microfluidic gradient systems.