Degeneracy and hysteresis in a bidisperse colloidal ice

We use numerical simulations to investigate the low-energy states of a bidisperse colloidal ice, realized by confining two types of magnetic particles into double wells of different lengths. For this system, theoretical calculations predict a highly degenerate ground state where all the vertices with zero topological charge have equal energy. When raising the applied field, we find a re-entrant transition where the system passes from the initial disordered state to a low-energy one and then back to disorder for large interaction strengths. The transition is due to the particle localization on top of the central hill of the double wells, as revealed from the position distributions. When we decrease the applied field, the system displays hysteresis in the fraction of low-energy vertices, and a small return point memory by cycling the applied field.

Figure 1

(a) Schematic (not on a scale) showing a colloidal ice made of a square lattice of double wells, each filled with one paramagnetic colloidal particle. The vectors within the particles denote the magnetic moments $\mathit{m}$ induced by the external field $\mathit{B}$. (b) Different types of vertices in the square colloidal ice with their associated topological charge $Q$. The ratio between the magnetostatic energy of one vertex $u$ and that of the type 3 ($u_3$) is illustrated at the bottom. (c) Average fraction of vertices $\varphi$ for the square colloidal ice vs the applied magnetic field $\mathit{B}$.

Figure 2

(a) Comparison between one vertex in the monodisperse (left) and in the bidisperse (right) colloidal ice. In the latter case, the particles have different magnetic susceptibilities (${\chi }_1>{\chi }_2$) and distances ($d_1>d_2$) at the vertex. (b) Energetic splitting of different types of vertices in the bidisperse ice. (c) Vertex energy $\overline{u}$ vs magnetic susceptibility ${\chi }_2$ for the bidisperse system with particles subjected to an applied field of amplitude $B=13\mathrm{mT}$. For ${\chi }_2=0.067$, the energies ${\overline{u}}_{5a}\approx {\overline{u}}_{5b}, {\overline{u}}_{5a}/{\overline{u}}_{5b}=0.99$ and ${\overline{u}}_{3a}={\overline{u}}_{3b}={\overline{u}}_4$. (d) Mean vertex energy $\langle \overline{u}\rangle /N$ vs total number of vertices $N$ for a system composed of type 3 (squares) and type 4 (circles). Inset shows the energy difference $(\langle {\overline{u}}_4\rangle -\langle {\overline{u}}_3\rangle )/N$ vs $N$.

Figure 3

Average fraction of vertices for different field values for the bidisperse square ice. The results from the simulations are averaged over 10 realizations.

Figure 4

(a) Snapshots of the numerical simulations of the bidisperse ice taken at $B=0$ (first image), 85 (second) 120 (third), and $200\mathrm{mT}$ (fourth). The particles are indicated by the black disks on the square lattice, while the vertex defects are large blue (type 1, $Q=-4$), small blue (type 2, $Q=-2$), small red (type 5, $Q=+2$), and large red (type 6, $Q=+4$) disks. The green disks denote vertices which fulfill the GS ($Q=0$). (b) Corresponding histograms of the particle positions in long horizontal traps (first row) and short vertical traps (second row) for the three field values.

Figure 5

(a) Average fraction $\varphi$ of type 3 vertices vs applied field $B$ obtained up to $B=200\mathrm{mT}$ (red and blue circles), and lowering back to $B=0\mathrm{mT}$ (green and purple triangles) at two different rates. (b) Hysteresis area $A$ vs rate ${\alpha }_B$ of the applied field. Inset shows the same plot in the log-log scale. (c) Spin overlap order parameter $q_s$ vs number of cycles calculated from a minor loop centered at the peak of the type 3 vertex fraction.