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AbstractWe develop a two-timing perturbation analysis to provide quantitative insights on the existence of temporal ratchets in an exemplary system of a particle moving in a tank of fluid in response to an external vibration of the tank. We consider two-mode vibrations with angular frequencies$$\omega $$ $\omega$and$$\alpha \omega $$ $\alpha \omega$, where$$\alpha $$ $\alpha$is a rational number. If$$\alpha $$ $\alpha$is a ratio of odd and even integers (e.g.,$$\tfrac{2}{1},\,\tfrac{3}{2},\,\tfrac{4}{3}$$ $\frac{2}{1},\frac{3}{2},\frac{4}{3}$), the system yields a net response: here, a nonzero time-average particle velocity. Our first-order perturbation solution predicts the existence of temporal ratchets for$$\alpha =2$$ $\alpha =2$. Furthermore, we demonstrate, for a reduced model, that the temporal ratcheting effect for$$\alpha =\tfrac{3}{2}$$ $\alpha =\frac{3}{2}$and$$\tfrac{4}{3}$$ $\frac{4}{3}$appears at the third-order perturbation solution. More importantly, we find closed-form formulas for the magnitude and direction of the induced net velocities for these$$\alpha $$ $\alpha$values. On a broader scale, our methodology offers a new mathematical approach to study the complicated nature of temporal ratchets in physical systems. Graphic abstract 

When applying an oscillatory electric potential to an electrolyte solution, it is commonly assumed that the choice of which electrode is grounded or powered does not matter because the time average of the electric potential is zero. Recent theoretical, numerical, and experimental work, however, has established that certain types of multimodal oscillatory potentials that are “nonantiperiodic” can induce a net steady field toward either the grounded or powered electrode [A. Hashemi et al., Phys. Rev. E 105, 065001 (2022)]. Here, we elaborate on the nature of these steady fields through numerical and theoretical analyses of the asymmetric rectified electric field (AREF). We demonstrate that AREFs induced by a nonantiperiodic electric potential, e.g., by a two-mode waveform with modes at 2 and 3Hz, invariably yields a steady field that is spatially dissymmetric between two parallel electrodes, such that swapping which electrode is powered changes the direction of the field. Furthermore, we show that, while the single-mode AREF occurs in asymmetric electrolytes, nonantiperiodic electric potentials create a steady field in electrolytes even if the cations and anions have the same mobilities. Additionally, using a perturbation expansion, we demonstrate that the dissymmetric AREF occurs due to odd nonlinear orders of the applied potential. We further generalize the theory by demonstrating that the dissymmetric field occurs for all classes of zero-time-average (no dc bias) periodic potentials, including triangular and rectangular pulses, and we discuss how these steady fields can tremendously change the interpretation, design, and applications of electrochemical and electrokinetic systems. 

We develop a linearly scaling variant of the force coupling method [K. Yeo and M. R. Maxey, J. Fluid Mech. 649, 205–231 (2010)] for computing hydrodynamic interactions among particles confined to a doubly periodic geometry with either a single bottom wall or two walls (slit channel) in the aperiodic direction. Our spectrally accurate Stokes solver uses the fast Fourier transform in the periodic xy plane and Chebyshev polynomials in the aperiodic z direction normal to the wall(s). We decompose the problem into two problems. The first is a doubly periodic subproblem in the presence of particles (source terms) with free-space boundary conditions in the z direction, which we solve by borrowing ideas from a recent method for rapid evaluation of electrostatic interactions in doubly periodic geometries [Maxian et al., J. Chem. Phys. 154, 204107 (2021)]. The second is a correction subproblem to impose the boundary conditions on the wall(s). Instead of the traditional Gaussian kernel, we use the exponential of a semicircle kernel to model the source terms (body force) due to the presence of particles and provide optimum values for the kernel parameters that ensure a given hydrodynamic radius with at least two digits of accuracy and rotational and translational invariance. The computation time of our solver, which is implemented in graphical processing units, scales linearly with the number of particles, and allows computations with about a million particles in less than a second for a sedimented layer of colloidal microrollers. We find that in a slit channel, a driven dense suspension of microrollers maintains the same two-layer structure as above a single wall, but moves at a substantially lower collective speed due to increased confinement. 

It is well established that application of an oscillatory excitation with zero time-average but temporal asymmetry can yield net drift. To date this temporal symmetry breaking and net drift has been explored primarily in the context of point particles, nonlinear optics, and quantum systems. Here, we present two new experimental systems where the impact of temporally asymmetric force excitations can be readily observed with mechanical motion of macroscopic objects: (1) solid centimeter-scale objects placed on a uniform flat surface made to vibrate laterally, and (2) charged colloidal particles in water placed between parallel electrodes with an applied oscillatory electric potential. In both cases, net motion is observed both experimentally and numerically with nonantiperiodic, two-mode, sinusoids where the frequency modes are the ratio of odd and even numbers (e.g., 2Hz and 3Hz). The observed direction of motion is always the same for the same applied waveform, and is readily reversed by changing the sign of the applied waveform, for example, by swapping which electrode is powered and grounded. We extend these results to other nonlinear mechanical systems, and we discuss the implications for facile control of object motion using tunable periodic driving forces.