Field synchronized bidirectional current in confined driven colloids

We investigate the collective colloidal current that emerges when strongly confined magnetic microspheres are subjected to a biased, but spatially uniform, precessing magnetic field. We observe a net bidirectional current composed of colloidal particles which periodically meet assembling into rotating dimers, and exchange their positions in a characteristic, “ceilidh”-like dance. We develop a theoretical model which explains the physics of the observed phenomena as dimer rupture and onset of current, showing agreement with Brownian dynamic simulations. By varying the tilt angle and the frequency of the applied field, we discover two separate transport mechanisms based on different ways the dimers break up during particle transport. Our results demonstrate an effective technique to drive microscale matter by using a combination of confinement and homogeneous field modulations, not based on any gradient of the applied field.

Figure 1

(a) Optical microscope images showing the exchange process between particles confined within a cell of thickness $h=4\mu \mathrm{m}$ and located close to the upper (dark) and lower (bright) plane. The applied precessing field has $B_0=7.2\mathrm{mT}$, $\theta =26.9^{\circ }$, and $f=6\mathrm{Hz}$. (b) Sequence of schematics corresponding to (a) and showing the exchanged-based transport mechanism. (c) Sketch of the biased precessing magnetic field with tilt angle $\delta$. (d) Experiments and (e) corresponding numerical simulations of a large system driven by the same field as (a) with $\delta \sim 7^{\circ }$. The particle trajectories are superimposed to the image in blue (dark) and red (bright) for up and down particles. Scale bar is $10\mu \mathrm{m}$. See corresponding Video 1 in the Supplemental Material [45].

Figure 2

(a)–(f) Normalized dipolar force $F_{\rho }$ (black curves) and driving frequency $f$ (red curves) vs the phase-lag difference $\mathrm{\Delta }={\varphi }_B-{\varphi }_p$ for a pair of paramagnetic colloids. The left column [(a), (c), (e)] refers to a system with thickness $h=3.9\mu \mathrm{m}$, and illustrates the breakage of a dimer after asynchronous slippage (mechanism M1). The right column [(b), (d), (f)] refers to a thickness $h=3.5\mu \mathrm{m}$ and corresponds to the direct dimer breaking (M2). The first row [(a), (b)] refers to the case of $\delta =0$ and ${\varphi }_B=0$, and the middle and bottom rows to $\delta =5^{\circ }$, ${\varphi }_B=0$ [(c), (d)] and ${\varphi }_B=\pi$ [(e), (f)].

Figure 3

Normalized current amplitude $I$ vs tilt angle $\delta$ for one driving frequencies. We define $I=2\left|\mathit{v}\right|/\left[fd\sqrt{\pi /\left(\sqrt{3}\mathrm{\Phi }\right)}\right]$, where $\mathrm{\Phi }$ is the normalized area fraction, and $\left|\mathit{v}\right|=\sum |{\mathit{v}}_i|$ the mean velocity of the particles. The velocity of a particle $i$ at an elevation $z_i$ from the middle plane is given by ${\mathit{v}}_i=\langle (v_{y,i}\mathit{x}-v_{x,i}\mathit{y})z_i/|z_i|\rangle$. The inset shows the corresponding particle velocities $v$ vs $\delta$ along the average direction of the current, indicated by a director $\mathit{n}$. Experimental images on the right are indicated by the corresponding shaded region in the graph. See corresponding Video 2 in the Supplemental Material [45]. (b) Theory: State diagram ($h,\delta ,f_c$) illustrating the regions related to the two different mechanisms of net colloidal current. (c) Diagrams in the $(\delta ,f)$ plane combining the simulation results (continuous color map) and the model (continuous thick line) for two different thickness corresponding to the two mechanisms.

Figure 4

Colloidal current $I$ vs normalized area fraction $\mathrm{\Phi }$ obtained from numerical simulation for a cell with $h=4\mu \mathrm{m}$, $B_0=7.2\mathrm{mT}$, $f=3\mathrm{Hz}$, $\delta =7^{\circ }$. The insets show experimental images at $\mathrm{\Phi }=0.23$ (left) and $\mathrm{\Phi }=0.77$ (right) with superimposed trajectories of up (down) particles in blue (red). See corresponding Video 3 in the Supplemental Material [45].