Controlled propulsion in viscous fluids of magnetically actuated colloidal doublets

We study the propulsion of a micron-size paramagnetic colloidal doublet dispersed in water and driven above a surface by an external precessing magnetic field. The applied field forces the doublet to precess around an axis parallel to the plane of motion and the rotation of the colloidal assembly is rectified into translation due to a periodic asymmetry in dissipation close to the bounding plate. These recent experimental findings [P. Tierno, R. Golestanian, I. Pagonabarraga, and F. Sagués, Phys. Rev. Lett. 101, 218304 (2008)] are complemented here with a theoretical analysis of the system and extended to more complex magnetic modulations such as elliptical driving fields. Experimental results show a good agreement with numerical simulations with the aim to find the best conditions toward the optimization of propulsion speed and swimming efficiency.

Figure 1

(Color online) (a) Scheme showing a paramagnetic doublet floating above a glass plate (distance from center of mass $h$) and subjected to an external magnetic field $H$ precessing around the $y$ axis with frequency $\Omega$. $\theta$ denotes the angle that the doublet director makes with the $y$ axis and $H_0,H_1$ the field components along the $y$ and $z$ axes. (b) Microscope images taken at different times of a doublet traveling with a speed of $\sim 3\mu \text{m}/\text{s}$. Superimposed is the particle trajectory after 7.8 and 16.0 s. In [3] the corresponding video file is deposited.

Figure 2

(Color online) Scaled doublet velocity $V/V_0$ as a function of the normalized frequency $\Omega /{\Omega }_B$ for $H_0/H_1=0.27$ and $h=2.05\mu \text{m}$. The experimental data (open circles) are plotted together with a nonlinear curve fit (continuous line) and Eqs. (29, 30) for the limits $\Omega /{\Omega }_B\to 0$ and $\Omega /{\Omega }_B\to \infty$ (dashed lines), respectively.

Figure 3

(Color online) Scaled doublet velocity $V/V_0$ as a function of the normalized frequency $\Omega /{\Omega }_B$ (a) for different precession ratios $H_0/H_1$ with $h=2.05\mu \text{m}$ and (b) for different distances from the wall $h$ with $H_0/H_1=0.27$.

Figure 4

(Color online) (a) Representation in the three-dimensional space of the director tip trajectories for different ellipticity parameters, $\beta =-0.7$, 0, and 0.7. (b) Scaled doublet velocity as a function of $\Omega /{\Omega }_B$ for different values of $\beta$. The upper inset shows an enlargement of the graphs close to the velocity maximum with $H_0/H_1=0.27$ and $h=2.05\mu \text{m}$.

Figure 5

(Color online) Scaled peak velocity $V_p$ of the doublet as a function of the ellipticity parameter for different precession ratios $H_0/H_1$ with $h=2.05\mu \text{m}$. The experimental points (closed circles) refer to a ratio $H_0/H_1=0.25$.