Anomalous structural transition of confined hard squares

Structural transitions are examined in quasi-one-dimensional systems of freely rotating hard squares, which are confined between two parallel walls. We find two competing phases: one is a fluid where the squares have two sides parallel to the walls, while the second one is a solidlike structure with a zigzag arrangement of the squares. Using transfer matrix method we show that the configuration space consists of subspaces of fluidlike and solidlike phases, which are connected with low probability microstates of mixed structures. The existence of these connecting states makes the thermodynamic quantities continuous and precludes the possibility of a true phase transition. However, thermodynamic functions indicate strong tendency for the phase transition and our replica exchange Monte Carlo simulation study detects several important markers of the first order phase transition. The distinction of a phase transition from a structural change is practically impossible with simulations and experiments in such systems like the confined hard squares.

Figure 1

Hard squares between two parallel walls, where σ is the side length and H is the pore width. The arrow at the leftmost square of each panel indicates the square's orientation. The longitudinal and transversal directions are denoted as $x$ and $y$. Three configurations are shown with seven squares at $H^{*}=H/\sigma =1.9$: the close packing in the parallel arrangement (upper panel), five squares are placed in closely packed zigzag order with two ends left in parallel arrangement (middle panel), and the close packed structure in zigzag order (lower panel). ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ are the loss and the gain in space as compared to the parallel configuration, respectively.

Figure 2

Equation of state (a), order parameter (b), isothermal compressibility (c), and heat capacity (d) of freely rotating squares for different values of $H^{*}$. The inset of (a) shows the transversal pressure as a function of longitudinal pressure. Diamonds are REMC data, while the continuous curves are TMM results. The employed dimensionless quantities are $P_x^{*}=\beta P_x{\sigma }^2$, $P_y^{*}=\beta P_y{\sigma }^2$, ${\chi }_Y^{*}=k_{B}T{\chi }_T/{\sigma }^2$, and $C_P^{*}=C_P/N{k}_B$.

Figure 3

Fluctuations of $x$, $\left|y\right|$, and φ (accounting for the fourfold symmetry) variables as a function of packing fraction at $H^{*}=1.9$. Here $x$ means the longitudinal distance between the neighboring particles. The fluctuations are defined as ${\sigma }_{\xi }^2=\langle {\left(\xi -\langle \xi \rangle \right)}^2\rangle$, where ξ can be $x$, $\left|y\right|$, and φ.

Figure 4

(a) Equation of state of the restricted orientation model using the exact TMM (continuous curve) and the approximate TMM with level-crossing approximation (short dashed curve) for various pore widths. The inset zooms in the transition regime for $H^{*}=1.9$. (b) Phase diagram of the restricted orientational model using TMM with level-crossing approximation. Coexisting packing fractions are shown as a function of pore width. The inset shows the transition pressure.