Paramagnetic colloids: Chaotic routes to clusters and molecules

We present computer simulations and experiments on dilute suspensions of superparamagnetic particles subject to rotating magnetic fields. We focus on chains of four particles and their decay routes to stable structures. At low rates, the chains track the external field. At intermediate rates, the chains break up but perform a periodic (albeit complex) motion. At sufficiently high rates, the chains generally undergo chaotic motion at short times and decay to either closely packed clusters or more dispersed, colloidal molecules at long times. We show that the transition out of the chaotic states can be described as a Poisson process in both simulation and experiment.

Figure 1

(a) Schematic of the experimental setup. Two iron-core solenoids are used to create a magnetic field with constant magnitude of $H_0$, which rotates in the $xy$ plane with angular velocity of $\mathrm{\Omega }$ with an optical microscope imaging the $xy$ plane. (b) Snapshots from a typical trajectory in experiments at $\text{Mn}>{\text{Mn}}_2$. In this regime, the short-time behavior is chaotic motion, and the long-time behavior is a stable periodic orbit. In this case, the final stable periodic orbit is a rotating closely packed cluster.

Figure 2

Snapshots from typical trajectories in (left) experiments and (right) simulations. Particles are colored arbitrarily in the simulation to indicate identity. Time increases from left to right, and $\text{Mn}$ increases from top to bottom. Field direction is shown in the rightmost columns.

Figure 3

(a) Magnetostatic energy, $U_{MS}$, and (b) its power spectrum for a typical trajectory with ${\text{Mn}}_1<\text{Mn}<{\text{Mn}}_2$. (c) Magnetostatic energy, $U_{MS}$, and (d) its power spectrum for a typical trajectory with $\text{Mn}>{\text{Mn}}_2$. For the experimental curves, $U_{MS}$, is inferred as described in the text, and the Fourier analysis is done only over the chaotic part.

Figure 4

For $\text{Mn}=1.253$ with ${\varphi }_0=0$ (simulation result), (a) magnetostatic energy and (b) logarithm of the magnitude of the configuration space separation, $\text{log}\left(\right|\vec{\epsilon }\left|\right)$ vs time.

Figure 5

Long-time-averaged magnetostatic energy, $\langle U_{MS}\rangle$, vs $\text{Mn}$ (simulation result). ${\varphi }_0$ is the initial phase lag between the field direction and the chain orientation. ${\text{Mn}}_1$ and ${\text{Mn}}_2$ are the first and the second bifurcation points beyond which the rigid body rotation and the do-si-do motion become unstable, respectively. Above ${\text{Mn}}_2$, the typical dynamics consists of an episode of chaotic motion which eventually decays into a rotating colloidal cluster or molecule.

Figure 6

Fraction of trajectories in chaotic and nonchaotic states vs time. (a) $\text{Mn}=1.253$ (simulation) and (b) $\text{Mn}=1.06$ (simulation). (c) High $\text{Mn}$ limit (experiment). For (a) the dark blue curve which decreases monotonically corresponds to the fraction of chaotic states, the magenta to the fraction of molecules and the green to the fraction of clusters. For (b) the cyan curve corresponds to the fraction of trajectories in the second kind of special case orbits described in the text. In the experiment in (c) we observe escape only into clusters.