Observation of Condensed Phases of Quasiplanar Core-Softened Colloids

We experimentally study the condensed phases of repelling core-softened spheres in two dimensions. The dipolar pair repulsion between superparamagnetic spheres trapped in a thin cell is induced by a transverse magnetic field and softened by suitably adjusting the cell thickness. We scan a broad density range and we materialize a large part of the theoretically predicted phases in systems of core-softened particles, including expanded and close-packed hexagonal, square, chainlike, stripe or labyrinthine, and honeycomb phase. Further insight into their structure is provided by Monte Carlo simulations.

Figure 1

Measured variability of the in-plane pair force profiles: $r^{-4}$ repulsion (thin cell, $h\ll h_m$; circles), softened repulsion ($h\approx h_m$; squares), and oversoftened interaction (thick cell, $h>h_m$; triangles) with the attractive part at small separations. For $h=h_m$, the in-plane hard-core diameter of the spheres is $\approx 0.89\mu \mathrm{m}$. The small-separation part of the force profile is not accessible with our method as the laser would act on both spheres, rendering the measurement impossible. In our experiment, $B$ is typically around 10 mT and the induced pair forces are of the order of 0.1 pN at separations comparable to sphere diameter. Solid lines are theoretical fits for $h=0$, $h=0.46\mu \mathrm{m}$, and $h=0.72\mu \mathrm{m}$, respectively. Inset: experimental geometry in cross section.

Figure 2

Micrographs of the representative mesophases induced by varying the density (top row; $B=12.5\mathrm{mT}$ which gives $k\approx 400$) and the simulation snapshots (bottom row) labeled by the corresponding 2D volume fractions of colloids. Because of optical image distortion, the colloids on the micrographs appear slightly larger than their true size; in some of the micrographs, patches of the underlying lattices are emphasized to guide the eye. The reduced interaction strength of $k=375$ was used in simulations at all but the largest two volume fractions where $k$ was set to 125 to improve convergence; the number of particles $N=1000$ except at $\eta =0.60$, where $N=200$.

Figure 3

Energy per particle of the ground states of hexagonal (broad shaded line; the line thickness represents the numerical inaccuracy), square (solid line), and stripe phases (dashed line) compared to the energies obtained by the Monte Carlo simulation (open circles with error bars). Also shown are the snapshots for $\eta =0.10$ (hexagonal lattice), $\eta =0.30$ (square lattice), and $\eta =0.52$ (labyrinthine structure) where the spheres’ vertical positions are encoded by shades of gray.