Phase Diagram of Dipolar Hard and Soft Spheres: Manipulation of Colloidal Crystal Structures by an External Field

Phase diagrams of hard and soft spheres with a fixed dipole moment are determined by calculating the Helmholtz free energy using simulations. The pair potential is given by a dipole-dipole interaction plus a hard-core and a repulsive Yukawa potential for soft spheres. Our system models colloids in an external electric or magnetic field, with hard spheres corresponding to uncharged and soft spheres to charged colloids. The phase diagram of dipolar hard spheres shows fluid, face-centered-cubic (fcc), hexagonal-close-packed (hcp), and body-centered-tetragonal (bct) phases. The phase diagram of dipolar soft spheres exhibits, in addition to the above mentioned phases, a body-centered-orthorhombic (bco) phase, and it agrees well with the experimental phase diagram [Nature (London) 421, 513 (2003)]. Our results show that bulk hcp, bct, and bco crystals can be realized experimentally by applying an external field.

Figure 1

(a) Body-centered structure in three dimensions, whose conventional unit cell is $a\times b\times c$. The field is along the $z$ axis. The white arrows show the direction of the field-induced dipole moments. The bct structure corresponds to $a=b\ne c$, and the bco to $a\ne b,c\ne a$, and $c\ne b$. (b) Top view of the body-centered structure, that can be constructed by placing strings of particles shifted by $c/2$ into two interpenetrating rectangular lattices. (c),(d) The hcp and fcc structures shown in side and top views. The hcp structure has $AB$ stacking of the hexagonal planes, and the fcc $ABC$.

Figure 2

Phase diagram of dipolar hard spheres in the dipole moment strength $\gamma$-packing fraction $\eta$ representation. The circles denote points where the phase boundary was determined and the gray areas denote coexistence regions (where the tie lines are vertical). Snapshots are taken (a) in the string fluid phase ($\gamma =8.0$, $\eta =0.01$), and (b),(c) in the fluid-bct coexistence region at ($\gamma =13.1$, $\eta =0.4$) and at ($\gamma =26.1$, $\eta =0.3$), respectively.

Figure 3

Phase diagram of dipolar soft spheres with parameters $\kappa \sigma =10.0$ and $ϵ=12.54$ in the (dimensionless dipole moment strength $\gamma$, packing fraction $\eta$) representation. The circles denote points where the phase boundary was determined and the gray area denotes the coexistence region (where the tie lines are vertical).

Figure 4

Change in the Madelung energy $\Delta U_{\mathrm{M}}(a/b)=U_{\mathrm{M}}(a/b)-U_{\mathrm{M}}(1)$ of a bco crystal when $a/b$ is increased from one. The results are for dipolar soft spheres with $\kappa \sigma =10.0$, $ϵ=12.54$, and $\gamma =37.5$, at packing fractions $\eta =0.27$, 0.4, and 0.5. The insets (a) and (b) show the soft and dipolar parts of the Madelung energy change, respectively.