Controlled structuring of binary hard-disk mixtures via a periodic, external potential

Ordering phenomena on surfaces or in monolayers can be successfully studied by model systems as binary hard-disk mixtures, the influence of a substrate being modeled by an external potential. For the field-free case the thermodynamic stability of space-filling lattice structures for binary hard-disk mixtures is studied by Monte Carlo computer simulations. As these structures prove to be thermodynamically stable only in high pressure environments, the phase behavior of an equimolar binary mixture with a diameter ratio of ${\sigma }_B/{\sigma }_A=0.414$ exposed to an external, one-dimensional, periodic potential is analyzed in detail. The underlying ordering mechanisms and the resulting order differ considerably, depending on which components of the mixture interact with the external potential. The simulations show that slight deviations in the concentration of large particles $x_A$ or the diameter ratio ${\sigma }_B/{\sigma }_A$ have no impact on the occurrence of the various field-induced phenomena as long as the mixture stays in the relevant regime of the packing fraction $\eta$. Furthermore the importance of the commensurability of the external potential to the $S_1\left(AB\right)$ square lattice for the occurrence of the induced ordering is discussed.

Figure 1

(Color online) Schematic drawing of the energy landscape for particles interacting with the external field, which is commensurate to the $S_1\left(AB\right)$ square lattice with a commensurability ratio $p=2$, for an equimolar, binary mixture with diameter ratio ${\sigma }_B/{\sigma }_A=0.414$.

Figure 2

(Color online) The probability distribution of the shape factor $P\left(\zeta \right)$ for an equimolar binary mixture with diameter ratio ${\sigma }_B/{\sigma }_A=0.414$ in the fluid phase at ${\varrho }^{\ast }=1.6$ and $V_0^{\ast }=0$. The inset shows a typical Voronoi diagram of a fluid configuration. Voronoi cells are colored according to their structure: rectangular black, pentagonal dark gray (blue) and hexagonal light gray (orange). All other structures are plotted in white.

Figure 3

(Color online) The packing fraction $\eta$ as a function of decreasing pressure $p^{\ast }$ in an equimolar binary mixture. Two different lattice structures are tested for their stability: (a) the $S_1\left(AB\right)$ square lattice for various mixtures with ${\sigma }_B/{\sigma }_A∊\left[0.392,0.414\right]$. (b) The $H_2\left(AB\right)$ lattice for mixtures with ${\sigma }_B/{\sigma }_A∊\left[0.627,0.646\right]$.

Figure 4

(Color online) Overview of analyzed lattice structures for binary hard-disk mixtures. Structures for which a pressure regime, in which the structures stay stable, could be identified are marked by crosses (green). For the mixture marked by a triangle (blue) the lattice structure was not stable within the simulated pressure regime of $p^{\ast }\le 200$. Also given are the packing fraction $\eta$ and the pressure $p^{\ast }$ above which all of the simulated mixtures for the given lattice structure were found to be stable.

Figure 5

(Color online) Comparison of the total pair correlation functions (a) $h_{AA}\left(r\right)$ and (b) $h_{BB}\left(r\right)$ of an equimolar mixture with ${\sigma }_B/{\sigma }_A=0.414$ exposed to an external, periodic potential at ${\varrho }^{\ast }=1.60$. Shown is the case when both components of the mixture interact with the external potential.

Figure 6

(Color online) Comparison of the total pair correlation functions (a) $h_{AA}\left(r\right)$ and (b) $h_{BB}\left(r\right)$ of an equimolar mixture with ${\sigma }_B/{\sigma }_A=0.414$ exposed to an external, periodic potential at ${\varrho }^{\ast }=1.60$. Shown is the case when only the large component of the mixture interacts with the external potential.

Figure 7

(Color online) The change in the probability distribution of the shape factor $\Delta P\left(\zeta \right)$ for the different scenarios of coupling to the external field of the components of the mixture at ${\varrho }^{\ast }=1.71$. The coupling of a component to the external field is indicated in the schematic inset by a filled circle, while an open circle denotes no coupling to the external field of the component concerned. (a) All three cases are in the demixing regime for $V_0^{\ast }=0.9$. (b) At $V_0^{\ast }=2.1$ systems with a coupling of the small components to the external field have switched into the coexistence regime of the square lattice with the modulated liquid.

Figure 8

(Color online) Detail of configurations containing a fissure in the $S_1\left(AB\right)$ square lattice, which forms in the presence of an external, one-dimensional modulated potential at ${\varrho }^{\ast }=1.73$ and $V_0^{\ast }=5.0$. (a) A typical fissure for the case, that only the small component interacts with the external potential. Small particles filling the fissure form dimers that orient along the minima of the external potential. (b) The situation for the case that only the large component of the mixture interacts with the external field. The placement of the small particles within the fissure is arbitrary.

Figure 9

(Color online) The phase diagram as obtained from the analysis of the order parameters $S_B$ and $S_A$ and the shape factor $\zeta$ for an equimolar binary mixture with diameter ratio ${\sigma }_B/{\sigma }_A=0.414$ exposed to an external periodic potential, commensurate to the $S_1\left(AB\right)$ square lattice for the case that only the small component of the mixture interacts with the external potential. Lines are a guide for the eyes. Figure reprinted from Franzrahe et al. [30], Fig. 4.

Figure 10

(Color online) The phase diagram as obtained from the analysis of the order parameters $S_B$ and $S_A$ and the shape factor $\zeta$ for an equimolar binary mixture with diameter ratio ${\sigma }_B/{\sigma }_A=0.414$ exposed to an external periodic potential, commensurate to the $S_1\left(AB\right)$ square lattice for the case that both components of the mixture interact with the external potential. For this case of coupling to the external potential no laser induced coexistence occurs, but the fissuring regime and the $S_1\left(AB\right)$ lattice with defects are stabilized down to far lower packing fractions $\eta$ as compared to the case, when only the small component interacts with the external potential (Fig. 9). Lines are a guide for the eyes.

Figure 11

(Color online) Probability distributions of the rotational order parameter ${\Psi }_8$ for various simulated strengths of coupling to the external field for the large component of the binary mixture at ${\varrho }^{\ast }=1.68$ and $V_{B,0}^{\ast }=V_0^{\ast }=2.1$.

Figure 12

(Color online) The probability distributions of the rotational order parameter ${\Psi }_6$ evaluated only with respect to the positions of the large particles show a substructure due to the grain boundaries and defects in the irregular monodisperse, rhombic crystal, but no tendency of the system to show a laser induced freezing transition into the commensurate $S_1\left(AB\right)$ square lattice locked floating solid.

Figure 13

Pair correlation functions $g_{BB}\left(\vec{r}\right)$ and $g_{AA}\left(\vec{r}\right)$ for a binary mixture exposed to an incommensurate, periodic external potential with $\lambda =0.546$ and $V_0^{\ast }=5.0$ at two different number densities ${\varrho }^{\ast }$.

Figure 14

(Color online) The phase diagrams as obtained from the analysis of the order parameters $S_B$ and $S_A$ and the shape factor $\zeta$ for an equimolar binary mixture with diameter ratio ${\sigma }_B/{\sigma }_A=0.414$ exposed to external periodic potentials, incommensurate to the $S_1\left(AB\right)$ square lattice for the case that only the small component of the mixture interacts with the external potential. Lines are a guide for the eyes.

Figure 15

(Color online) The change in the probability distributions of the shape factor $\zeta$ at various amplitudes of the external potential $V_0^{\ast }$ with respect to the probability distribution at $V_0^{\ast }=0.0$ for a binary mixtures with diameter ratio ${\sigma }_B/{\sigma }_A=0.414$ with exclusive coupling of the small particles to the external potential is plotted for mixtures with various concentrations of large particles $x_A$. (a) At $x_A=46.9\%$ and a number density of ${\varrho }^{\ast }=1.71$ the mixture has a packing fraction of $\eta =75.2\%$. (b) The reference mixture with $x_A=50.0\%$ at ${\varrho }^{\ast }=1.71$ has a packing fraction of $\eta =78.7\%$ as has the mixture with $x_A=53.1\%$ in (c) at ${\varrho }^{\ast }=1.64$.

Figure 16

(Color online) The change in the probability distributions of the shape factor $\zeta$ at various amplitudes of the external potential $V_0^{\ast }$ with respect to the probability distribution at $V_0^{\ast }=0.0$ for equimolar, binary mixtures with exclusive coupling of the small particles to the external potential is plotted for mixtures with various diameter ratios ${\sigma }_B/{\sigma }_A$. Shown are the distributions in the demixing regime of the reference mixture at (a) $V_0^{\ast }=0.3$ and (b) $V_0^{\ast }=0.9$ and in the coexistence regime of the square lattice with the modulated liquid of the reference mixture at (c) $V_0^{\ast }=2.1$.