Low-temperature ordering of the dimer phase of a two-dimensional model of core-softened particles

Purely pairwise interactions of the core-softened type, i.e., featuring a soft repulsion followed by a hard-core interaction at shorter distance, give rise to nontrivial equilibrium structures entirely different from the standard close packing of spheres. In particular, in a suitable low-temperature region of their phase diagram, such interactions are well known to favor a transition from a fluid to a cluster crystal. The residual mutual interaction between individual clusters can lead to the formation of patterns of their reciprocal orientations. In this work, we investigate two examples of such models in two dimensions, at the density most appropriate to the dimer phase, whereby clusters consist of just two particles, studying them with optimization techniques and Monte Carlo simulations. We focus on the dimer crystal, and unveil a second phase transition at extremely low temperature. This transition leads from a triangular dimer lattice with randomly disordered dimer orientations at high temperature to a reduced-symmetry ground state with nematic orientational order and a slightly distorted structure characterized by a centered-rectangular lattice at low temperature.

Figure 1

Hard-core plus GEM4 potential ${\varphi }_{\mathrm{HCGEM}4}\left(r\right)$ of Eq. (2) with hard-core diameter $\delta =0.05R$. The inset displays the Fourier transform of the GEM4 potential ${\tilde{w}}_{\mathrm{GEM}4}\left(k\right)$, exhibiting a negative minimum at $k_m\simeq 5.09618R^{-1}$.

Figure 2

Same as Fig. 1, but here for the smooth-core plus GL6 potential ${\varphi }_{\mathrm{SCGL}6}\left(r\right)$ of Eq. (4) with smooth-core repulsion strength $A=5\times {10}^{-5}$. The negative minimum of the Fourier transform of GL6 occurs at $k_m\simeq 4.82021R^{-1}$.

Figure 3

The six degenerate antinematic states of lowest energy at $T=0$ for a dimer crystal on a triangular lattice (black dots). The interaction is HCGEM4, and the density is $\rho =1.13971R^{-2}$. The dimer length is magnified by a factor 10 relative to the actual value $\delta =0.05R$ for ease of readability. The angles ${\vartheta }_1, {\vartheta }_2$ between the directions of the dimers and the horizontal axis are specified in each panel.

Figure 4

Examples of sampled MC configurations, for the relaxed states of dimers (represented as small green dashes) of particles interacting via a HCGEM4 potential at $T={10}^{-7}\varepsilon /k_{\mathrm{B}}$. In these simulations, the rectangular box aspect ratio is kept fixed to the value compatible with the triangular lattice. (a) A configuration of 968 particles. (b) A configuration of 1800 particles. Red and blue crosses mark the positions of the topological defects with charge $W=\pm 1/2$ induced by the periodic boundary conditions incompatible with the actual nematic ground state.

Figure 5

Three orientations of the nematic ground state for a dimer crystal at $\rho =1.13971R^{-2}$. The dimer size is magnified by a factor 10 as in Fig. 3. In each panel, $\alpha$ is the angle between the primitive vectors ${\mathbf{e}}_1, {\mathbf{e}}_2$, and $\vartheta$ is the angle between the direction of the dimers and the horizontal axis. The insets give a qualitative representation of the lattice vectors (not to scale). The arrows represent the primitive vectors ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$. The ticks mark the sides of equal length of the isosceles triangles. The dashed symmetry axis marks the direction of the nematic alignment.

Figure 6

Two examples of metastable configurations representing local minima of the total energy $U$ of a dimer crystal at $\rho =1.13971R^{-2}$. In both cases, the dimer length is magnified by a factor 10 as in Fig. 3, and the unit cell is replicated several times in order to illustrate the periodicity. The energy per particle amounts to (a) $E=0.68577624\varepsilon$ and (b) $E=0.75980667\varepsilon$, both well above the nematic ground state.

Figure 7

Simulated potential energy per particle $E$ in $\varepsilon$ units (black dots) at $T={10}^{-9}\varepsilon /k_{\mathrm{B}}$ as a function of the number of particles $N$, in the presence of four vortices generated by the rectangular box aspect ratio kept fixed to the value compatible with the triangular lattice, as in Fig. 4. Error bars are smaller than the symbol size. The energy of the nematic ground state (Fig. 5) obtained by gradient minimization is shown as a blue dashed line.

Figure 8

$S\left(q_x,0\right)$ (blue circles) and $S\left(0,q_y\right)$ (gray diamonds), obtained by $T={10}^{-7}\varepsilon /k_{\mathrm{B}}$ MC simulations for $N=800$ particles in the nematic state sketched in Fig. 5. Red line: $N\times f\left(q_y\right)$, Eq. (32) ($T=0$ theoretical prediction).

Figure 9

Scaling of the first Bragg peak intensity $S_1$ as a function of $N$, where the data points are taken from simulations of $S(q_x,0)$ carried out in supercells compatible with the centered-rectangular lattice, and ${\alpha }_0=59.{835}^{\circ }$ at temperatures $T={10}^{-7}\varepsilon /k_{\text{B}}$ (blue circles) and $T={10}^{-3}\varepsilon /k_{\text{B}}$ (red squares). The dashed lines are linear fits of the simulated data. Green diamonds signify twice the intensity of the corresponding Bragg peak $S_{\text{CM}}(8.8426R^{-1},0)$ in the structure factor of the dimer centers of mass at $T={10}^{-3}\varepsilon /k_{\text{B}}$, which is only sensitive to phonon excitations, not to thermal fluctuations in the dimers orientation.

Figure 10

The temperature dependence of the average energy per particle $E$, obtained by means of MC simulations in a supercell containing $N=800$ particles. For $T>T_{\times }\simeq 2.25\times {10}^{-5}\varepsilon /k_B$ (marked by a red arrow), the energy corresponding to the triangular lattice (blue circles) turns lower than that of the centered-rectangular nematic state (orange squares), which prevails instead at low temperature. Error bars are smaller than the symbol size. The black dashed line marks the energy of the nematic ground state.

Figure 11

The dimer-dimer angular correlation function ${\langle {\sigma }_{\alpha }{\sigma }_{{\alpha }^{\prime }}\rangle }^x$, as obtained in simulations carried out at temperatures $T={10}^{-7},{10}^{-5},{10}^{-3}\varepsilon /k_{\mathrm{B}}$. The correlation function is expressed as a function of the dimer-dimer $x$ separation $|\alpha -{\alpha }^{\prime }|$ expressed in units of the lattice spacing $d^{\prime \prime }$. Error bars are smaller than the symbol size. Inset: zoom on the $T={10}^{-3}\varepsilon /k_{\mathrm{B}}$ data points.