Phase transitions and self-organization of Janus disks in two dimensions studied by Monte Carlo simulations

The phase behavior of Janus disks on a square lattice is studied using the Monte Carlo method. A particle is composed of two different parts, $A$ and $B$. The interactions between neighboring particles depend on their orientations. To control the strength of the interactions, we use energy parameters characterizing $AA,BB$, and $AB$ contacts. The phase diagrams are estimated. We found two phase transitions, namely first-order transitions between colloidal-rich and colloidal-poor phases, and continuous order-disorder transitions to different ordered phases. A variety of ordered structures occurs depending on the relation between the energy parameters and on the fluid density. The influence of energy parameters on the phase diagram topology and critical parameters is shown. The competition between phase transitions and self-organization is discussed.

Figure 1

Orientations of the model Janus disks (a), and representative examples of pairs $x_{k1}$ (b) and $y_{k1}$ (c). For pairs $x_{k1}$ $\left(y_{k1}\right)$, the lines connecting the centers of the particles are aligned along the $x$ $\left(y\right)$ axis. The indexes indicate the orientations of particles located on sites with the successive $x$ $\left(y\right)$ coordinates.

Figure 2

Schematic representation of different possible ordered structures at a fully filled lattice and decomposition of a square lattice into four equivalent sublattices $a,b,c$, and $d$.

Figure 3

Ground-state diagram for the system with $u_{BB}=-1$ at a fully filled lattice. Vertical dashed line corresponds to the same $AA$ and $BB$ interactions $\left(u_{AA}=-1\right)$. The solid line $\left(\varepsilon =0\right)$ delimits the regions of the stability of the SAF and AF phases. Symbols show the points at which the calculations have been carried out.

Figure 4

Distributions of the component ${\Psi }_{\text{SAF}1}$ (the distributions of the component ${\Psi }_{\text{SAF}2}$ are the same) of the order parameter ${\Psi }_{\text{SAF}}$ for the system M1 obtained at $T^{*}=0.5$ and ${\mu }^{*}=-1.200$ (green line), ${\mu }^{*}=-1.225$ (red line), and $\mu =-1.250$ (black line). The size of the simulation cell $L=84$.

Figure 5

The density-temperature (a) and the chemical potential-temperature (b) projections of phase diagrams obtained for the following systems: M1 (black line), M2 (red line), and M3 (green line). Solid lines represent the first-order coexistence curves. The symbols are located on the $\lambda$ lines corresponding to the second-order transitions to the SAF phase.

Figure 6

The density-temperature (a) and the chemical potential-temperature (b) projections of phase diagrams obtained for the systems M4 (black lines) and M5 (red lines). Solid lines represent the first-order coexistence curves. The symbols are located on the $\lambda$ lines corresponding to the second-order transitions to the AF phase.

Figure 7

Fragments of snapshots of configurations recorded for M6 at $T^{*}=0.1$ and $\rho =0.4$ (a) and $\rho =0.7$ (b); $L=192$.

Figure 8

The results for the system M6. Changes of the order parameter ${\Psi }_{\text{SAF}}$ with the chemical potential (a,c) and the corresponding changes of the fourth-order cumulant $U_L\left({\Psi }_{\text{SAF}}\right)$(b,d) obtained at $T^{*}=0.25$ (a,b) and $T^{*}=0.2$ (c,d). The calculations have been carried out for the different sizes of the simulation cell, $L=60$ (black lines), 72 (red lines), 84 (green lines), 94 (blue lines), 120 (yellow lines), 140 (maroon lines), 144 (brown lines), and 168 (magenta lines).

Figure 9

The block density probability distribution for the system M6 at temperature $T^{*}=0.1$ and $\rho =0.6$ density. The system size $L=192$ and the block size $L_b=48$.

Figure 10

Schematic representation of possible ordered structured at low and medium densities in the monolayer: (a) dimers, (b) “micelles,” (c) zigzag structures at $\rho =2/3$, and (d) S-structure $\left(\rho =4/5\right)$.

Figure 11

Fragments of snapshots of configurations recorded for M7 at $T^{*}=0.025$ and $\rho =0.573$ (a) and for $T^{*}=0.1,\rho =0.763$ (b).