Mean-field model of boomerang nematic liquid crystals with diminished coupling of molecular uniaxial and biaxial susceptibilities

The mean-field theory approach has been applied to the boomerang type particles from P. I. C. Teixeira, A. Masters, and B. Mulder [Mol. Cryst. Liq. Cryst. 323, 167 (1998)] but with diminished strength of the interaction coefficient responsible for the coupling between molecular uniaxial and biaxial susceptibilities. For the rodlike particles, when the apex boomerang angle is larger than $107.35{}^{\circ }$, the stable uniaxial rodlike phase occurs. For smaller angles, beyond the point where the transition is of the second order (the Landau point) and for diminished parameter of molecular biaxial-uniaxial coupling, a biaxial phase is observed with the transition undergoing directly from the isotropic phase. According to the order parameters the character of this transition is of the first order. Such behavior is in accordance with the Sonnet-Durand-Virga model of the biaxial phases. The change in the type of the phase transition order is also illustrated by the changes in the equations of state and the changes in second and third derivatives of the free energy. The possibilities to tailor interaction coefficients of real molecules to obtain such a phase transition scenario are discussed.

Figure 1

Hard boomerang. The boomerang angle $\mathrm{\Psi }$ is defined here as the angle between the $Z$ axis and the boomerang arms, when it is positioned symmetrically, and hence the apex angle between the boomerang arms is $\kappa =\pi -2\mathrm{\Psi }$. The length of the arm is $L$ and the width is $D$.

Figure 2

Order parameters $s_1,s_2,s_3$, and $s_4$ for the phases of the boomerang particles of different boomerang angle $\mathrm{\Psi }$, where the transition to the uniaxial phase is of the first order. The closest to the Landau point ($\mathrm{\Psi }=0.2017835\pi$) results, where the transition to the uniaxial phase becomes of the second order, are for ($\mathrm{\Psi }=0.20\pi$). Panel (a) shows rodlike boomerangs case and panel (b) shows boardlike case.

Figure 3

Derivative of the order parameter $s_1=\langle P_2\rangle$ versus temperature. In the biaxial phase its linear character shows that $s_1$ is of the square type for any boomerang angle in the case when the particles are rodlike.

Figure 4

$P–T$ equation of state. (The excess pressure $\beta P$ versus temperature.) The curve with squares is the closest to the Landau point, which occurs for $\mathrm{\Psi }=0.2017835\pi$. Note the smoothness of this curve in the mesogenic region in comparison to the other profiles. The negative values come here from the fact that the presented results are for the excess pressure relative to the isotropic phase.

Figure 5

The specific heat according to (19) for the standard boomerang case. LP denotes the boomerang angle corresponding to the Landau point, for which the specific heat is given also in the inset.

Figure 6

$P–T$ equation of state for different $\gamma$ values and the boomerang angle $\mathrm{\Psi }=0.19\pi$.

Figure 7

The specific heat according to (19) for different $\gamma$ values and the boomerang angle $\mathrm{\Psi }=0.19\pi$.

Figure 8

Order parameters $s_1,s_2,s_3$, and $s_4$ versus the reduced temperature for the boomerang angle $\mathrm{\Psi }=0.201783\pi$, (the Landau point) (a) and the magnified view (b) for different values of $\gamma$.

Figure 9

$P–T$ equation of state for the boomerang angle corresponding to the Landau point for perfect boomerang and by adding influence of the $\gamma$ coefficient

Figure 10

The specific heat according to (19) for different $\gamma$ values and the boomerang angle $0.20178\pi$. The visible change of the slope character at NU-NB transition indicates that this phase transition becomes weakly of the first order for $\gamma <0.8$.

Figure 11

Order parameters $s_1,s_2,s_3$, and $s_4$ for the boomerang angle $\mathrm{\Psi }=0.22\pi$ (the apex angle $100.8{}^{\circ }$). (a) order parameters for larger values of $\gamma \text{"s}$ where the type of ordering is like for $\gamma =1$ (b) order parameters for smaller values of $\gamma \text{"s}$ with a new type of ordering for particles arrangement as proposed in Fig. 12.

Figure 12

An idea for the possible arrangement of boomerangs in the planar phase. Long axes are distributed in the plane, whereas the short axes can placed in two possible ways leading to the change of their order parameters sign.

Figure 13

$P–T$ equation of state for the boomerang angle $0.215\pi$ with different $\gamma$ coefficients.

Figure 14

The specific heat according to (19) for different $\gamma$ values and the boomerang angle $0.215\pi$.