Two-Phase Crystallization in a Carpet of Inertial Spinners

We study the dynamics of torque driven spherical spinners settled on a surface, and demonstrate that hydrodynamic interactions at finite Reynolds numbers can lead to a concentration dependent and nonuniform crystallization. At semidilute concentrations, we observe a rapid formation of a uniform hexagonal structure in the spinner monolayer. We attribute this to repulsive hydrodynamic interactions created by the secondary flow of the spinning particles. Increasing the surface coverage leads to a state with two coexisting spinner densities. The uniform hexagonal structure deviates into a high density crystalline structure surrounded by a continuous lower density hexatically ordered state. We show that this phase separation occurs due to a nonmonotonic hydrodynamic repulsion, arising from a concentration dependent spinning frequency.

Figure 1

(a) A schematic showing the simulation system. A constant torque $T$ is subjected to each particle sedimented on a flat wall. (b) The streamlines showing the flow field created by a rotating spherical particle with $\mathrm{Re}\approx 10$. (c) A hexagonal structure of the particles for an area fraction ${\varphi }_0\approx 32\%$. (d) Phase separation to particle dense and dilute regions for ${\varphi }_0\approx 37\%$. The color maps show the local area fractions. The insets in (c) and (d) show the structure factors with the characteristics of a hexatic order in both cases.

Figure 2

(a) The probability distributions of the local area fractions $P(\varphi )$ are shown for different global area fractions ${\varphi }_0$ marked by the vertical dashed line for a $\mathrm{Re}\approx 10$ sample. In the case of a single phase ($\varphi <35\%$) the peak corresponds to ${\varphi }_0$. The $P(\varphi )$ becomes bimodal for ${\varphi }_0>35\%$. (b) The averaged local hexatic order as a function of overall area fraction. (c) Density profile $\varphi (r)$ as a function of the radial distance $r$ from the center of the dense region.

Figure 3

The time evolution of (a) the local hexatic order parameter ${\mathrm{\Psi }}_6$ and (b) the domain length scale $L$. When a constant torque was applied on each particle, the domain length shows a condensation of the particles for ${\varphi }_0\approx 37\%$ ($\mathrm{Re}\approx 10$). After the system reaches a steady two density state, the constant torque $T$ is replaced by a constant frequency $\omega$ corresponding to $\mathrm{Re}\approx 8$. (c)–(e) The snapshots show particle positions and are color coded by the spinning frequencies $\omega$ of the particles.

Figure 4

(a),(b) The hydrodynamic repulsion between a pair of spinners confined to move only along one direction ($Y$). (a) The repulsion force shows a $F\sim {\omega }^2$ scaling and (b) a $F\sim r^{-2.7}$ decay. (c) The spinning frequency $\omega$ as a function of overall area fraction for a constant torque for a small sample (open circles) ($L_X=L_Y=L_Z=20R$ and $\mathrm{Re}\approx 10$). The black curve is a fit to polynomial function $\omega /{\omega }_0=0.308{\varphi }^3-1.291{\varphi }^2+0.122\varphi +0.821$. The green shadow shows the correlation between the frequency and the local area fraction calculated from the two-phase state of Fig. 1. (d) A schematic showing the repulsion force as a function of the overall area fraction when combing the measurements from (a)–(c). (e) A snapshot showing the measurement of the expansion force of a small spinner cluster ($N=80$). (f) The measured effective repulsion force of a spinner cluster as a function of the area fraction.

Figure 5

(a) The probability distributions of the local area fractions $P(\varphi )$ are shown for different Reynolds numbers for a global area fraction ${\varphi }_0\approx 39\%$ (marked by the vertical dashed line). (b) The averaged local hexagonal order parameter ${\mathrm{\Psi }}_6$ as a function of the Reynolds number.