Uniform phases in fluids of hard isosceles triangles: One-component fluid and binary mixtures

We formulate the scaled particle theory for a general mixture of hard isosceles triangles and calculate different phase diagrams for the one-component fluid and for certain binary mixtures. The fluid of hard triangles exhibits a complex phase behavior: (i) the presence of a triatic phase with sixfold symmetry, (ii) the isotropic-uniaxial nematic transition is of first order for certain ranges of aspect ratios, and (iii) the one-component system exhibits nematic-nematic transitions ending in critical points. We found the triatic phase to be stable not only for equilateral triangles but also for triangles of similar aspect ratios. We focus the study of binary mixtures on the case of symmetric mixtures: equal particle areas with aspect ratios (${\kappa }_i$) symmetric with respect to the equilateral one, ${\kappa }_1{\kappa }_2=3$. For these mixtures we found, aside from first-order isotropic-nematic and nematic-nematic transitions (the latter ending in a critical point): (i) a region of triatic phase stability even for mixtures made of particles that do not form this phase at the one-component limit, and (ii) the presence of a Landau point at which two triatic-nematic first-order transitions and a nematic-nematic demixing transition coalesce. This phase behavior is analogous to that of a symmetric three-dimensional mixture of rods and plates.

Figure 1

Sketch of I (a), N (b), and TR (c) uniform phases of HT (the latter two exhibit perfect orientational ordering.

Figure 2

Sketch of the excluded area between two HT with relative angle between their axes equal to $\varphi$. The main parameters (base, height, angles) used to define the geometry of the triangle are labeled.

Figure 3

(a) The function $A^{\left(\mathrm{spt}\right)}\left(\varphi \right)/\left(2a\right)$ for triangles with $\kappa \ge \sqrt{3}$. Chosen values are $\kappa =\sqrt{3}$ (solid), $\sqrt{5+2\sqrt{5}}$ (dashed), 4 (dotted), and 6 (dot-dashed). (b) The same function for triangles with $\kappa \le \sqrt{3}$ and values: $\sqrt{3}$ (solid), 1 (dashed), 2/3 (dotted), and 1/2 (dot-dashed).

Figure 4

(a) Free-energy densities minus a straight line: ${\mathrm{\Phi }}^{*}\equiv \mathrm{\Phi }-a\eta -b$ (with $a=623.45$ and $b=-562.94$) corresponding to the I (dot-dashed), TR (solid), N (dashed), and ${\mathrm{TR}}^{*}$ (dotted) phases of HT with $\kappa =2.2$. The solid circle indicates the I-TR bifurcation, while open circles represent the TR-N coexistence (with ${\eta }_{\mathrm{TR}}=0.949157$ and ${\eta }_{\mathrm{N}}=0.949785$) and the square shows the ${\mathrm{N}\text{−}\mathrm{TR}}^{*}$ bifurcation. (b) Orientation distribution functions corresponding to TR (solid) at $\eta =0.949$, N (dashed) at $\eta =0.95$, both for stable phases of HT with $\kappa =2.2$. Dotted line is a distribution function corresponding to a ${\mathrm{TR}}^{*}$ metastable phase with $\eta =0.957$.

Figure 5

Phase diagram of HT in the packing-fraction ($\eta$)-aspect ratio ($\kappa$) plane. (a) $\eta \in [0.6,1]$ in logarithmic scale for $\kappa$. (b) Zoom of the phase diagram (with $\kappa$ in linear scale), showing the presence of N-N transitions ending in critical points (solid circles), the first-order character of the I-N transition for certain values of $\kappa$, and the region of stability of the TR phase. Dashed and solid lines show second- and first-order transitions, respectively. Regions of stability of different phases are correspondingly labeled.

Figure 6

(a) Order parameters $Q_k$ (solid for $k=2$ and dashed for $k=6$) as a function of the packing fraction $\eta$ for HT with $\kappa =1.215$, showing the ${\mathrm{N}}_1\text{−}{\mathrm{N}}_2$ first-order transition (shown with circles and squares, respectively). Inset: free-energy densities minus a straight line, ${\mathrm{\Phi }}^{*}=\mathrm{\Phi }-a\eta -b$ (with $a=-288.33$ and $b=334.6$), corresponding to the I (dot-dashed) and ${\mathrm{N}}_i$ (solid) phases as a function of $\eta$. The solid circles indicates the I-N bifurcation point, while the open circles show the N-N coexistence values. (b) Orientation distribution functions corresponding to the coexisting ${\mathrm{N}}_1$ (dashed) and ${\mathrm{N}}_2$ (solid) phases for the packing-fraction values ${\eta }_1=0.92680$ and ${\eta }_2=0.92776$, respectively.

Figure 7

(a) The function $A_{12}^{\left(\mathrm{spt}\right)}\left(\varphi \right)$ related to the excluded areas between two symmetric (${\kappa }_1{\kappa }_2=3$) triangles of equal areas ($a_1=a_2$) and $r=b_2/b_1=1.453$ (solid), and between the sublime and Gnomon (with a ratio between its base and the equally sized lengths equal to the golden number) triangles of equal areas (dashed). (b) Scaled Fourier coefficients of the pair of species corresponding to the solid line in (a). Open and solid circles correspond to positive and negative coefficients, respectively.

Figure 8

Phase diagram in the scaled pressure ($p^{*}=\beta p{a}_1$)-molar fraction ($x\equiv x_1$) plane of the HT with ${\kappa }_1=2.52$ (species 1) and its symmetric (${\kappa }_1{\kappa }_2=3$) counterpart, both of equal areas and with $r=b_2/b_1=1.453$. Regions of stability of different phases are correspondingly labeled. Dashed and solid lines represent the second-order transition curves and the binodals of first-order transitions, respectively. Coexistence regions are shaded in gray. The solid and open circles represent the critical and Landau points, respectively.

Figure 9

Distribution functions of the first (solid) and second (dashed) species of the same binary mixture as in Fig. 8 for a fixed scaled pressure $p^{*}=346.042$ and molar fractions $x=0.7$ (a) and $x=0.3$ (b). Dotted lines indicate the total orientation distribution function $f_{\mathrm{t}}\left(\varphi \right)$ (see text for definition). Insets are schematic particle configurations for binary mixtures corresponding to each of the subfigures (for simplicity, perfect orientational ordering was assumed with arrows indicating the orientations of particle axes).

Figure 10

Phase diagram of a binary mixture of the sublime triangle (species 1) and a triangle with ${\kappa }_2=5{\kappa }_1$, both having the same area.