Mode-locking Steps Research Articles

Transport of a colloidal particle driven across a temporally oscillating optical potential energy landscape

A colloidal particle is driven across a temporally oscillating one-dimensional optical potential energy landscape and its particle motion is analysed. Different modes of dynamic mode locking are observed and are confirmed with the use of phase portraits. The effect of the oscillation frequency on the mode locked step width is addressed and the results are discussed in light of a high-frequency theory and compared to simulations. Furthermore, the influence of the coupling between the particle and the optical landscape on mode locking is probed by increasing the maximum depth of the optical landscape. Stronger coupling is seen to increase the width of mode locked steps. Finally, transport across the temporally oscillating landscape is studied by measuring the effective diffusion coefficient of a mobile particle, which is seen to be highly sensitive to the driving velocity and mode locking.

Directional mode-locking of driven two-dimensional active magnetized colloids with periodic pinning centers

We investigate the directional mode-locking of driven two-dimensional active magnetized colloids on the substrate with periodically distributed point-like pinning centers. The active magnetized colloids are modeled as Janus particles, which interact with each other through repulsive magnetic dipole and attractive electric dipole potentials. The movements of the depinning colloidal particles are found to be locked collectively in some symmetric directions of the substrate pinning potential. As a result, there form directional mode-locking steps in the driving force direction dependence of the average velocity of colloidal particles. The deviations of some colloidal particles from the mode-locking directions are also observed clearly. Our results provide guidance for continuous separation of different kinds of active microscopic particles, including macroscopic molecule and biological cells.

Dynamic mode locking in a driven colloidal system: experiments and theory

In this article we examine the dynamics of a colloidal particle driven by a modulated force over a sinusoidal optical potential energy landscape. Coupling between the competing frequencies of the modulated drive and that of particle motion over the periodic landscape leads to synchronisation of particle motion into discrete modes. This synchronisation manifests as steps in the average particle velocity, with mode locked steps covering a range of average driving velocities. The amplitude and frequency dependence of the steps are considered, and compared to results from analytic theory, Langevin dynamics simulations, and dynamic density functional theory. Furthermore, the critical driving velocity is studied, and simulation used to extend the range of conditions accessible in experiments alone. Finally, state diagrams from experiment, simulation, and theory are used to show the extent of the dynamically locked modes in two dimensions, as a function of both the amplitude and frequency of the modulated drive.

Interference mode-locking of 2D magnetized colloids driven by dc and ac forces in periodic pinning arrays

Using the Langevin simulation, we investigate the interference mode-locking of two-dimensional (2D) magnetized colloids on a substrate with periodic pinning centers. The colloidal particles are prepared initially in a 2D crystalline state and then driven simultaneously by direct-current (dc) and alternating-current (ac) forces. In the presence of a superimposed ac force, we find pronounced mode-locking steps in the characteristics of the averaged velocity versus dc force within a certain range of amplitude and frequency of the ac force. The mode-locking steps are attributed to an interference effect between the ac force and the modulation generated by the coherent motion of colloidal particles in a weak pinning potential. The step width Δfdc is found to oscillate in a Bessel function-like form with the amplitude of the ac force, in good agreement with previous results of vortex lattices. But, we find that Δfdc changes in an inverted parabola form with the frequency of the ac force and the substrate pinning strength as well as the interaction strength between colloidal particles. The averaged velocity at the step vstep is shown to increase linearly with the frequency of the ac force and the interaction strength between colloidal particles. The obtained results are helpful for fractionation of mesoscopic particles.

Directional Mode-Locking of Dynamics of Two-Dimensional Superparamagnetic Colloidal System in a Periodic Pinning Array

Using Langevin simulations, we study numerically the directional mode-locking of the dynamics of two-dimensional superparamagnetic colloidal system in a periodic pinning array. When the colloidal particles are initially commensurate with the pinning sites, there appear mode-locking steps in the average velocity of colloidal particles along certain directions of the external driving force. With an increase in the pinning strength, the width of the step increases linearly but the velocity at the step decreases parabolically.

Irrational Mode Locking in Quasiperiodic Systems

A model for ac-driven systems, based on the Tang-Wiesenfeld-Bak-Coppersmith-Littlewood automaton for an elastic medium, exhibits mode-locked steps with frequencies that are irrational multiples of the drive frequency, when the pinning is spatially quasiperiodic. Detailed numerical evidence is presented for the large-system-size convergence of such a mode-locked step. The irrational mode locking is stable to small thermal noise and weak disorder. Continuous-time models with irrational mode locking and possible experimental realizations are discussed.

Mode-Locking Behaviour in Driven Colloids with Random Pinning

We find mode-locking steps in simulated force-velocity characteristicsof external alternating-force (AF) driven colloids on a disorderedsubstrate. Studies of mode-locking patterns in systems show thatmode-locking steps are accompanied with the emergence of a dynamicsphase: transverse solid phase. We also study the influence oftemperature on the width of mode-locking steps. The mode-locked state isdestroyed by thermal fluctuation and the width of mode-locking stepsdecreases rapidly with increasing temperature. In high velocity and lowtemperature regimes, due to the appearance of transverse solid phase andmicroscopically periodic velocity modulation, the step width changeslittle as temperature is varied.

Mode-locking and dynamical correlation effects in a driven Josephson junction network

Mode-locking phenomena in a Josephson junction network (JJN) driven by both dc and ac currents are studied using a numerical simulation. We consider a ladder type of structure of the JJN, i.e., Josephson junction ladder (JJL). Under a random magnetic field, it is found that the mode locking exists in the present system, and mode-locked steps are observed in the dc current–voltage characteristics. Outside the steps, due to the effect of the random field, plastic deformation of phase–phase coupling occurs, and then there appears plastic flow. As the bias current is increased further, transition from the plastic to elastic flow state occurs. Around the transition point, a critical behavior of the voltage–voltage correlation is observed. We discuss the peculiar nature of the mode-locking phenomena in the JJL.

Mode-locking behaviour: ac response of a driven disordered vortex lattice

Using random pinning force simulations, we study numerically the alternating current (ac) response of the driven disordered vortex lattice. In the presence of a superimposed ac force, mode-locking steps are observed in the velocity–direct current (dc) force characteristics at high dc forces, where a smectic order occurs. The step width shows damped oscillations with the strength of interaction between vortices, in good agreement with recent experiments.

Collective Directional Transport of Coupled Oscillators in Symmetric Periodic Potentials

Collective directional transport of particles in symmetric periodic potentials is studied. An example is given to reveal the directed motion of a single particle in a symmetric periodic potential and subject to an asymmetric ac force with zero mean. It is then shown that the asymmetric coupling can give rise to a directed transport. The transport failure (pinning of the lattice) is related to the phase delocalization (crisis) in the circle map. For symmetric couplings, it is found that a train of plane wave can also lead to a directed transport. The mode-locking steps of the transport velocity is found and analyzed. The collective transport can be well optimized by adjusting parameters in the system.

Alternating Current Response of Two-Dimensional Vortex Lattice with Random Pinning**The project supported by the Ministry of Science and Technology of China (NKBRSF-G19990646), the Postdoctoral Science Foundation of China, and the Scientfic Research Foundation of State Education Ministry for Returned Overseas Chinese Scholars

Using a model of long-range interactions between vortices, we investigate numerically the alternating current (ac) response of two-dimensional vortex lattice with randomly distributed point-like pinning centers. Mode-locking steps are observed in the simulated current-voltages characteristics, and the number of steps increases with the superimposed ac amplitude and frequency. Our results are in good agreement with recent experiments.

Mode locking of vortex matter driven through mesoscopic channels.

We investigated the driven dynamics of vortices confined to mesoscopic flow channels by means of a dc-rf interference technique. The observed mode-locking steps in the IV curves provide detailed information on how both the number of vortex rows and the lattice structure in each flow channel change with magnetic field. Minima in flow stress occur when an integer number of rows is moving coherently, while maxima appear when the incoherent motion of mixed n and n+/-1 row configurations is predominant. Simulations show that the enhanced pinning at mismatch originates from quasistatic fault zones with misoriented edge dislocations induced by disorder in the channel edges.

Mode locking in driven vortex lattices with transverse ac drive and random pinning

We find mode-locking steps in simulated current-voltage characteristics of driven vortex lattices with {\it random} pinning when an applied ac-current is {\it perpendicular} to the dc-current. For low frequencies there is mode-locking only above a non-zero threshold ac force amplitude, while for large frequencies there is mode-locking for any small ac force. This is consistent with the nature of {\it transverse} temporal order in the different regimes in the absence of an applied ac-drive. For large frequencies the magnitude of the fundamental mode-locked step depends linearly with the ac force amplitude.

Mode locking in ac-driven vortex lattices with random pinning.

We find mode-locking steps in simulated current-voltage characteristics of ac-driven vortex lattices with random pinning. For low frequencies there is mode locking above a finite ac force amplitude, while for large frequencies there is mode locking for any small ac force. This is correlated with the nature of temporal order in the different regimes in the absence of ac drive. The mode-locked state is a frozen solid pinned in the moving reference of frame, and the depinning from the step shows plastic flow and hysteresis.

Thermal stability of mode-locking steps in charge density waves

We examine the stability of mode-locking steps in incommensurate charge density waves in the presence of finite temperature. The steps turn out to be quite stable by showing a series of definite plateaus in the current-voltage characteristics. We also study the dynamical phase transition from a mode-locked state to an unlocked state by assuming the fact that there exists a similar power-law scaling hypothesis as in a pinning-depinning phase transition. The critical exponents around both the edges of the mode-locking steps are found to have similar values for the depinning transition.

Mode-locking hysteresis in the globally coupled model of charge-density waves.

We study the response of a recently proposed global coupling model of charge-density waves to a joint ac+dc drive. The model is the standard Fukuyama-Lee-Rice model with an additional global coupling term to account for interaction with (uncondensed) quasi-particles. We find that traditional mode-locking is accompanied by hysteresis on mode-locked steps. The experimentally observed asymmetry in the size of hysteresis is reproduced and explained by drawing attention to the importance of the pinning force. For large values of global coupling, increased rigidity leads to vanishing of subharmonic steps consistent with experiments.

Dimensionality dependence of mode-locking dynamics in charge-density-wave transport.

We estimate the widths of the harmonic mode-locked steps observed on the dc I-V characteristic of charge-density-wave (CDW) conductors when ac and dc voltages are applied together using the classical deformable medium model. For weakly pinned CDW's in the high-velocity limit, \ensuremath{\delta}${\mathit{E}}_{\mathit{p}/1}$/${\mathit{E}}_{\mathit{T}}$ =2\ensuremath{\Vert}${\mathit{J}}_{\mathit{p}}$(${\mathrm{\ensuremath{\omega}}}_{\mathit{c}0}$${\mathit{E}}_{\mathit{a}\mathit{c}}$/${\mathrm{\ensuremath{\omega}}}_{\mathit{a}\mathit{c}}$${\mathit{E}}_{\mathit{T}}$)${\mathrm{\ensuremath{\Vert}}}^{4/(4\mathrm{\ensuremath{-}}\mathit{D})}$ where \ensuremath{\delta}${\mathit{E}}_{\mathit{p}/1}$ is the width in electric field of the p/1 step and D is the effective dimension of the pinning. An analytic argument suggests that the phase deformations are much larger in the mode-locked state than in the normal sliding state. They are largest for ac amplitudes that yield maxima in the step width. On the p/1 step the time-averaged phase-phase correlation length is predicted to vary as \ensuremath{\delta}${\mathit{E}}_{\mathit{p}/1}^{\mathrm{\ensuremath{-}}1/2}$. These analytic estimates are supported by numerical simulations. Measurements of the step width variation with ac amplitude and frequency for ${\mathrm{NbSe}}_{3}$ crystals whose static pinning is two dimensional (2D) are consistent with the 2D step width prediction.

Elastic properties of charge-density-wave conductors in applied electric fields.

The elastic properties of the charge-density-wave (CDW) conductors ${\mathrm{TaS}}_{3}$ and ${\mathrm{NbSe}}_{3}$ have been measured in the presence of ac, dc, and combined ac and dc longitudinal electric fields. For both ${\mathrm{TaS}}_{3}$ and ${\mathrm{NbSe}}_{3}$, application of ac fields ${E}_{\mathrm{ac}}$ in the MHz frequency range softens the crystal lattice (as indicated by a decrease in Young's modulus Y), even for values of ${E}_{\mathrm{ac}}$ well below the dc threshold field ${E}_{T}$ for CDW depinning. For applied dc fields ${E}_{\mathrm{dc}}$, lattice softening is observed only for ${E}_{\mathrm{dc}>{E}_{T}}$, consistent with previous studies. For combined ac and dc fields, anomalies in the electronic response (such as Shapiro step structure) are reflected also in Y and the internal friction \ensuremath{\delta}. On mode-locked steps, the elastic constants tend toward their pinned CDW values, despite the finite CDW drift velocity. It is suggested that these electroelastic experiments probe directly the internal degrees of freedom of the CDW, and our results are discussed in terms of applicable theoretical models.

Mode-locking and chaos in Rayleigh-Bénard convection

We have experimentally studied the transition to chaos in a thermally convecting dilute solution of 3He in superfluid 4He. Two natural oscillatory instabilities are found in our geometry. Mode-locking is observed over a range of Rayleigh and Prandtl number. The mode-locking steps have a measured fractal dimension in good agreement with theoretical predictions and the larger steps show Farey tree ordering. Far below chaos many features of the measurements are suggestive of the circle map. We produce a one-dimensional map showing a tangent bifurcation to a mode-locked state. An effective phase angle change across a mode-locked step is calculated from the data. Close to chaos, hysteresis and other effects are suggestive of a two-dimensional map. The fractal dimension of the attractor is measured over a range of Rayleigh number close to the chaotic onset. Our Poincaré sections show the fractalization of the torus and are characterized by extremely high resolution.