Colloidal particle electrorotation in a nonuniform electric field

A model to study the dynamics of colloidal particles in nonuniform electric fields is proposed. For an isolated sphere, the conditions and threshold for sustained (Quincke) rotation in a linear direct current (dc) field are determined. Particle dynamics becomes more complex with increasing electric field strength, changing from steady spinning around the particle center to time-dependent orbiting motion around the minimum field location. Pairs of particles exhibit intricate trajectories, which are a combination of translation, due to dielectrophoresis, and rotation, due to the Quincke effect. Our model provides a basis to study the collective dynamics of many particles in a general electric field.

Figure 1

One particle trajectory starting at $x=2.5$, $z=6.0$. Initial perturbations at the magnitude of $O\left({10}^{-4}\right)$ are randomly generated. The red circle indicates the non-Quincke region satisfying Eq. (16). The markers p1 to p4 indicate four positions when $z$ first hits the value of 0.0,1.0,2.0,3.0. $G=0.4000$. The particle does not rotate in the equilibrium state.

Figure 2

The evolution of the multipole moments and rotation rate for the single particle motion shown in Fig 1. (a) Dipole moments. (b) Quadrupole moments.

Figure 3

Particle trajectories in different field gradient strength. Initial position $x=5.0$, $z=2.0$ and a random initial polarization perturbation at $O\left({10}^{-4}\right)$. $D=5.1520$, $D^{\prime }=5.6054$. (a) $G=1.0.$ (b) $G=2.3,$ inset shows steady state trajectory. (c) $G=3.0,$ inset shows the rotating elliptical trajectory. The final time interval $\tilde{t}\in (380,400)$ is indicated by red color. See Supplemental Material for movies [31].

Figure 4

Stationary trajectory shape vs $G$. $r_a$ and $r_b$ are the short and long radii of the steady orbit. When no steady orbit is observed, $r_a$ and $r_b$ are the minimum and maximum distances to the origin in a chosen time frame. $D^{\prime }=5.6054$, ${\epsilon }_{mw}^{\prime }=-0.0670$, ${\sigma }_{mw}^{\prime }=-0.3333$.

Figure 5

Particle dynamic patterns and their rotation rates as a function of time. The dashed (solid) line in the upper figure corresponds to the dashed (solid) line in the lower figure. Initial $\mathbf{P}$ is given randomly at $O\left({10}^{-4}\right)$. The nonelectrorotation region is indicated by the circle satisfying Eq. (16). $D=5.1520,D^{\prime }=5.6054$. (a) Corotating pair, $G=0.1$, (b) counter-rotating pair, $G=0.1$, (c) co-rotating pair, $G=1.0$, (d) counter-rotating pair, $G=1.0$. See Supplemental Material for movies [31].

Figure 6

Dynamics of 20 particles in a linear field with random initial positions. Chaining at $Y$ axis is observed. $G=1.0.D=5.1520$. See Supplemental Material for movies [31]. (a) $\tilde{t}=0$; (b) $\tilde{t}=20$; (c) $\tilde{t}=200$; (d) $\tilde{t}=2000$.

Figure 7

Dynamics of 60 particles in periodical field with random initial positions with $\tilde{x}\in [0,60]$. Clustering at $\tilde{z}=0$ plane is observed. Four chains are formed at $\tilde{x}=8,24,40,56.E_0/E_c=1.0.{\delta }^{\prime }=\frac{\pi }{16}$. $D=5.1520$. See Supplemental Material for movies [31]. (a) $\tilde{t}=0$; (b) $\tilde{t}=20$; (c) $\tilde{t}=200$; (d) $\tilde{t}=2000$.