Hybrid finite-amplitude periodic modes for two uniformly magnetized spheres.

We analyze a system of two uniformly magnetized spheres, one fixed and the other free to slide in frictionless contact with the surface of the first. The centers of the two magnets, and their magnetic moments, are restricted to a plane. We search for sets of initial conditions that yield finite-amplitude oscillatory periodic solutions. We extend two small-amplitude base modes, one with orbital and spin motions that are in phase and the other out of phase, to finite amplitudes and show that the motion for arbitrary oscillatory solutions can be considered to be a nonlinear superposition of these base modes. Some solutions are pure periodic finite-amplitude extensions of one base mode, while others are hybrid finite-amplitude superpositions of the two modes. Hybrid modes with rational frequency ratios are periodic and come in families defined by their frequency ratios. We further characterize hybrid periodic modes by identifying two symmetry classes that describe their relative phases. We see continuous transitions between one finite-amplitude base mode and the other, with one mode gradually transforming into the other. We also calculate frequency spectra of nonperiodic modes, show that the two base modes have well-defined frequencies even for nonperiodic states, and show that periodic solutions can give clues about the behavior of nearby nonperiodic solutions. In the limit of small amplitudes, we confirm that the computed frequencies of these modes agree with small-amplitude analytical results. We also generate a Lyapunov exponent heatmap that reflects periodic and nonperiodic regions of state space.

Measuring the electrophoretic mobility and size of single particles using microfluidic transverse AC electrophoresis (TrACE).

The ability to measure the charge and size of single particles is essential to understanding particle adhesion and interaction with their environment. Characterizing the physical properties of biological particles, like cells, can be a powerful tool in studying the association between the changes in physical properties and disease development. Currently, measuring charge via the electrophoretic mobility (µep) of individual particles remains challenging, and there is only one prior report of simultaneously measuring µep and size. We introduce microfluidic transverse AC electrophoresis (TrACE), a novel technique that combines particle tracking velocimetry (PTV) and AC electrophoresis. In TrACE, electric waves with 0.75 to 1.5 V amplitude are applied transversely to the bulk flow and cause the particles to oscillate. PTV records the particles' oscillating trajectories as pressure drives bulk flow through the microchannel. A simple quasi-equilibrium model agrees well with experimental measurements of frequency, amplitude, and phase, indicating that particle motion is largely described by DC electrophoresis. The measured µep of polystyrene particles (0.53, 0.84, 1, and 2 µm diameter) are consistent with ELS measurements, and precision is enhanced by averaging ∼100 measurements per particle. Particle size is simultaneously measured from Brownian motion quantified from the trajectory for particles <2 µm or image analysis for particles ≥2 µm. Lastly, the ability to analyze intact mammalian cells is demonstrated with B cells. TrACE systems are expected to be highly suitable as fieldable tools to measure the µep and size of a broad range of individual particles.

Correction: Measuring the electrophoretic mobility and size of single particles using microfluidic transverse AC electrophoresis (TrACE).

Correction for 'Measuring the electrophoretic mobility and size of single particles using microfluidic transverse AC electrophoresis (TrACE)' by M. Hannah Choi et al., Lab Chip, 2023, https://doi.org/10.1039/D3LC00413A.

Inertial motion on the earth's spheroidal surface.

As seen by an observer in the rotating frame, the earth's small spheroidal deformations neutralize the centrifugal force, leaving only the smaller Coriolis force to govern the "inertial" motion of objects that move on its surface, assumed smooth and frictionless. Previous studies of inertial motion employ weakly spheroidal equations of motion that ignore the influence of the centrifugal force and yet treat the earth as a sphere. The latitude dependence of these equations renders them strongly nonlinear. We derive and justify these equations and use them to identify, classify, name, describe, and illustrate all possible classes of inertial motion, including a new class of motion called circumpolar waves, which encircle both poles during each cycle of the motion. We illustrate these classes using CorioVis, our freely available Coriolis visualization software. We identify a rotational/time-reversal symmetry for motion on the earth's surface and use this symmetry to develop and validate closed-form small-amplitude approximations for the four main classes and one degenerate class of inertial motion. For these five classes, we supply calculations of experimentally relevant frequencies, zonal drifts, and latitude ranges.

Normal-mode oscillations for the circular and dipolar states of a filled hexagonal magnetic dipole cluster.

We analyze the rotational dynamics of six magnetic dipoles of identical strength at the vertices of a regular hexagon with a variable-strength dipole in the center. The seven dipoles spin freely about fixed axes that are perpendicular to the plane of the hexagon, with their dipole moments directed parallel to the plane. Equilibrium dipole orientations are calculated as a function of the relative strength of the central dipole. Small-amplitude perturbations about these equilibrium states are calculated in the absence of friction and are compared with analytical results in the limit of zero and infinite central dipole strength. Normal modes and frequencies are presented. Bifurcations are seen at two critical values of the central dipole strength, with bistability between these values.

Periodic bouncing modes for two uniformly magnetized spheres. II. Scaling.

A uniformly magnetized sphere moves without friction in a plane in response to the field of a second, identical, fixed sphere and makes elastic hard-sphere collisions with this sphere. Numerical simulations of the threshold energies and periods of periodic finite-amplitude nonlinear bouncing modes agree with small-amplitude closed-form mathematical results, which are used to identify scaling parameters that govern the entire amplitude range, including power-law scaling at large amplitudes. Scaling parameters are combinations of the bouncing number, the rocking number, the phase, and numerical factors. Discontinuities in the scaling functions are found when viewing the threshold energy and period as separate functions of the scaling parameters, for which large-amplitude scaling exponents are obtained from fits to the data. These discontinuities disappear when the threshold energy is viewed as a function of the threshold period, for which the large-amplitude scaling exponent is obtained analytically and for which scaling applies to both in-phase and out-of-phase modes.

Periodic bouncing modes for two uniformly magnetized spheres. I. Trajectories.

We consider a uniformly magnetized sphere that moves without friction in a plane in response to the field of a second, identical, fixed sphere, making elastic hard-sphere collisions with this sphere. We seek periodic solutions to the associated nonlinear equations of motion. We find closed-form mathematical solutions for small-amplitude modes and use these to characterize and validate our large-amplitude modes, which we find numerically. Our Runge-Kutta integration approach allows us to find 1243 distinct periodic modes with the free sphere located initially at its stable equilibrium position. Each of these modes bifurcates from the finite-amplitude radial bouncing mode with infinitesimal-amplitude angular motion and supports a family of states with increasing amounts of angular motion. These states offer a rich variety of behaviors and beautiful, symmetric trajectories, including states with up to 157 collisions and 580 angular oscillations per period.

Periodic nonlinear sliding modes for two uniformly magnetized spheres.

A uniformly magnetized sphere slides without friction along the surface of a second, identical sphere that is held fixed in space, subject to the magnetic force and torque of the fixed sphere and the normal force. The free sphere has two stable equilibrium positions and two unstable equilibrium positions. Two small-amplitude oscillatory modes describe the sliding motion of the free sphere near each stable equilibrium, and an unstable oscillatory mode describes the motion near each unstable equilibrium. The three oscillatory modes remain periodic at finite amplitudes, one bifurcating into mixed modes and circumnavigating the free sphere at large energies. For small energies, the free sphere is confined to one of the two discontiguous domains, each surrounding a stable equilibrium position. At large energies, these domains merge and the free sphere may visit both positions. The critical energy at which these domains merge coincides with the cumulation point of an infinite cascade of mixed-mode bifurcations. These findings exploit the equivalence of the force and torque between two uniformly magnetized spheres and the force and torque between two equivalent point dipoles, and offer clues to the rich nonlinear dynamics of this system. Online MagPhyx visualizations illustrate the dynamics.

Diffusion layer formation drives zone migration in travelling wave electrophoresis.

COMSOL finite element modeling software is used to simulate 2D traveling-wave electrophoresis for microfluidic separations and sample concentration. A four-phase AC potential is applied to a periodic interdigitated four-electrode array to produce a longitudinal electric wave that travels through the channel. Charged particles are carried along with the electric wave or left behind, depending on their mobilities. A simplified model of asymmetric electrode reactions resolves the issue of electric double layer shielding at the electrodes. Selective reactions allow for the formation of diffusion layers of charged particles which follow the traveling electric wave. These diffusion layers determine the transport of charged species through the system. Our model reproduces experimental separations of charged species based on mobility. With easy control over the frequency and direction, one may employ this method for concentrating and/or separating charged particles.

Velocity plateaus in traveling-wave electrophoresis.

One-dimensional models are used to study traveling-wave electrophoresis, a tunable method for separating charged analytes. A traveling-electrode model reveals the mechanism for longitudinal oscillations. A stationary-electrode model explains the origin of mode-locked plateaus in the average velocity, predicts devil's staircases with nested Farey sequences, and reduces to a continuum sinusoidal model in the high electrode-density limit.

Fabrication and performance of a microfluidic traveling-wave electrophoresis system.

A microfluidic traveling-wave electrophoresis (TWE) system is reported that uses a locally defined traveling electric field wave within a microfluidic channel to achieve band transport and separation. Low voltages, over a range of -0.5 to +0.5 V, are used to avoid electrolysis and other detrimental redox reactions while the short distance between electrodes, ∼25 µm, provides high electric fields of ∼200 V cm(-1). It is expected that the low voltage requirements will simplify the future development of smaller portable devices. The TWE device uses four interdigitated electrode arrays: one interdigitated electrode array pair is on the top of the microchannel and the other interdigitated electrode array pair is on the microchannel bottom. The top and bottom substrates are joined by a PDMS spacer that has a nominal height of 15 µm. A pinched injection scheme is used to define a narrow sample band within an injection cross either electrokinetically or hydrodynamically. Separation of two dyes, fluorescein and FLCA, with baseline resolution is achieved in less than 3 min and separation of two proteins, insulin and casein is demonstrated. Investigation of band broadening with fluorescein reveals that sample band widths equivalent to the diffusion limit can be achieved within the microfluidic channel, yielding highly efficient separations. This low level of band broadening can be achieved with capillary electrophoresis, but is not routinely observed in microchannel electrophoresis. Sample enrichment can be achieved very easily with TWE using a device with converging electric field waves controlled by two sets of independently controlled interdigitated electrodes arrays positioned serially along the microchannel. Sample enrichment of 40-fold is achieved without heterogeneous buffer/solvent systems, sorptive, or permselective materials. While there is much room for improvement in device fabrication, and many capabilities are yet to be demonstrated, it is anticipated that the capabilities and performance demonstrated herein will enable new lab-on-a-chip processes and systems.

Simultaneous separation and detection of cations and anions on a microfluidic device with suppressed electroosmotic flow and a single injection point.

A rapid and simultaneous separation of cationic and anionic peptides and proteins in a glass microfluidic device that has been covalently modified with a neutral poly(ethylene glycol) (PEG) coating to minimize protein adsorption is presented. The features of the device allow samples that contain both anions and cations to be introduced from a central flow stream and separated in different channels with different outlets-all in the presence of low electroosmotic flow (EOF) imparted by the PEG coating. The analytes are electrophoretically extracted from a central hydrodynamic stream and electrophoretically separated in two different channels, in which pressure driven flow has been suppressed through the use of hydrodynamic restrictors. Having different outlets for the electrophoretic separation channels that are spatially separated from the injection enables coupling with further downstream functionalities or off-chip detection, such as mass spectrometry. A plug of charged analyte is hydrodynamically pumped to the sampling intersection and anions from the plug migrate electrophoretically toward the anode in one channel while cations migrate toward the cathode in the other channel due to suppressed EOF from the PEG coating. The separations presented here required less than a minute to complete and produced average separation efficiencies of up to about 3,500 plates from a separation length of 2 cm. The extraction efficiency of both cations and anions from the hydrodynamic stream is determined experimentally and compared with a previously reported model that was used to determine anion extraction efficiency. The extraction efficiency is determined to be 87% and 98% for the two sample mixtures analyzed, and the values predicted by the model are within 3.5% of the experimental data. It is anticipated that this basic approach for simultaneous separation of anions and cations with reduced EOF will be integrated into larger microfluidic systems because the design provides separate outlets that can feed downstream processes or linked to off-chip detection.

Self-similar nested sequences on a chaotic attractor for traveling-wave electrophoresis.

Oscillating electric potentials are applied to interdigitated arrays of cylindrical electrodes above and below a stationary conducting viscous fluid. The phases of these potentials are chosen to produce a longitudinal traveling wave that traps high-mobility ions and partially traps intermediate-mobility ions in periodic and narrowband chaotic attractors with average velocities that are commensurate with the wave speed. Stable periodic attractors have periods up to 101 times the wave period. Incommensurate broadband chaotic attractors are described by one-dimensional iterated contact-angle return maps, which feature self-similar nested sequences that converge geometrically at unstable trapped orbits. Sequences of singular angles and sequences of step transitions are characterized by distinct convergence factors. A criterion for allowed interelectrode orbits is developed. Experiments are suggested to evaluate the applicability of the theory to microfluidic separations.

A theoretical and experimental study of the electrophoretic extraction of ions from a pressure driven flow in a microfluidic device.

The electrophoretic extraction of ions from a hydrodynamic flow stream is investigated at an intersection between two microfluidic channels. A pressure gradient is used to drive samples through the main channel, while ions are electrophoretically extracted into the side channels. Hydrodynamic restrictors and a neutral coating are used to suppress bulk flow through the side channels. A theoretical model that assumes Poiseuille flow in the main channel and neglects molecular diffusion is used to calculate the extraction efficiency, eta, as a function of the ratio, R, of the average hydrodynamic velocity to the electrophoretic velocity. The model predicts complete extraction of ions (eta=1) for R<2/3 and a monotonic decrease in eta as R becomes greater than 2/3, which agrees well with the experimental results. Additionally, the model predicts that the aspect ratio of the microfluidic channel has little effect on the extraction efficiency. It is anticipated that this device can be used for on-line process monitoring, sample injection, and 2D separations for proteomics and other fields.

Traveling-wave electrophoresis for microfluidic separations.

Models and microfluidic experiments are presented of an electrophoretic separation technique in which charged particles whose mobilities exceed a tunable threshold are trapped between the crests of a longitudinal electric wave traveling through a stationary viscous fluid. The wave is created by applying periodic potentials to electrode arrays above and below a microchannel. Predicted average velocities agree with experiments and feature chaotic attractors for intermediate mobilities.

Propagation velocities of chemical reaction fronts advected by Poiseuille flow.

Poiseuille flow between parallel plates advects chemical reaction fronts, distorting them and altering their propagation velocities. Analytical solutions of the cubic reaction-diffusion-advection equation resolve the chemical concentration for narrow gaps, wide gaps, and small-amplitude flow. Numerical solutions supply a general description for fluid flow in the direction of propagation of the chemical reaction front, and for flow in the opposite direction. Empirical relations for the velocity agree with numerical solutions to within a few percent, and agree exactly with the analytical limits. Applications to nonlinear fingering are discussed.

Poiseuille advection of chemical reaction fronts.

Poiseuille flow between parallel plates alters the shapes and velocities of chemical reaction fronts. In the narrow-gap limit, the cubic reaction-diffusion-advection equation predicts a front-velocity correction equal to the gap-averaged fluid velocity epsilon. In the singular wide-gap limit, the correction equals the midgap fluid velocity 3epsilon/2 when the flow is in the direction of propagation of the reaction front, and equals zero for adverse flow of any amplitude for which the front has a midgap cusp. Stationary fronts are possible only for adverse flow and finite gaps. Experiments are suggested.

River meandering dynamics.

The Ikeda, Parker, and Sawai river meandering model is reexamined using a physical approach employing an explicit equation of motion. For periodic river shapes as seen from above, a cross-stream surface elevation gradient creates a velocity shear that is responsible for the decay of small-wavelength meander bends, whereas secondary currents in the plane perpendicular to the downstream direction are responsible for the growth of large-wavelength bends. A decay length D=H/2C(f) involving the river depth H and the friction coefficient C(f) sets the scale for meandering, giving the downstream distance required for the fluid velocity profile to recover from changes in the channel curvature. Using this length scale and a time scale T, we explicitly trace the observed length scale invariance to the equations of motion, and predict similar time and velocity scale invariances. A general time-dependent nonlinear modal analysis for periodic rivers reveals that modes higher than the third mode are needed to describe upstream migration of bend apexes just before oxbow cutoff, and are important to accurate calculations of the time and sinuosity at cutoff.