Figure 1

Schematic representation of the distribution function $P$ [Eq. (2)] in the order-parameter space of the conventional biaxial phase. The main fluctuations of the stick like molecules are schematized by a cone. For ${\eta }_2={\eta }_0=0$ the phase is isotropic and the cone is a sphere (same probability in all directions). In the uniaxial calamitic phase [$C\left(x\right),C\left(y\right)$, and $C\left(z\right)$] the molecule is preferentially oriented along one direction ($x,y$ or $z$) and fluctuates isotropically around it (symmetric cone). In the biaxial phase ($B1$ to $B6$) the fluctuations around the preferred axis are anisotropic and lead to a flat cone. The anisotropy and the amplitude of the fluctuations increase when going from calamitic to diskotic. In the uniaxial diskotic phase [$D\left(xy\right),D\left(xz\right)$ and $D\left(yz\right)$] the molecular orientation is completely disordered in one plane [e.g., $Oxy$ for $D\left(xy\right)$] and the cone becomes a disk. For instance in the sector $B1$ the preferred direction is $Oz$ and the fluctuations are in the $Oxz$ plane whereas in $B6$ the fluctuations are in $Oyz$. All the nonequivalent states lie within the gray sector $B1$. The other sectors correspond to domains of the same states: $B3$ and $B5$ are obtained from $B1$ by $2\pi ∕3$ and $4\pi ∕3$ external rotations around the direction (111) whereas $B6,B4$, and $B2$ are obtained from $B1,B3$, and $B5$, respectively, after fourfold rotations around $Oz$.Reuse & Permissions

Figure 2

(a) The three main axies $X,Y,Z$ of one bent-core molecule. The molecular polarization $p$ lies along the two fold symmetry axis $Z$. (b) Euler angles $\alpha ,\beta ,\gamma$ characterizing the orientation of one molecule in the laboratory frame $Ox,y,z$.Reuse & Permissions

Figure 3

Action of external and internal symmetries upon the statistical distribution $P$. (a) $P$ is represented by a set of molecules oriented in various directions. Two of them are represented in the figure. The figure can also be understood as a distribution containing only two possible orientations ($A$ and $B$) with probabilities $1∕2$. $x,y,z$ are the fixed axes of the laboratory frame and $X,Y,Z$ represent the frames rigidly tied to the molecules. (b) Under an external two fold rotation around $Oy$ the orientation $A$ is reversed whereas $B$ is unchanged. The new distribution $P^{\prime }$ gives probabilities $1∕2$ for the two new orientations and is equivalent to $P$ seen by an observer rotated in the laboratory frame. (c) Under an internal two fold rotation around $OY$ the configuration $A$ rotates around $Oy$ and is still reversed whereas the configuration $B$ is rotated around $Oz$ and is also reversed. The corresponding distribution $P^{″}$ is different from $P^{\prime }$. This example illustrates the difference between $C_{2y}^{\mathrm{int}}$ and $C_{2y}^{\mathrm{ext}}$. $P^{″}$ is not equivalent to $P$ seen by any rotated observer.Reuse & Permissions

Figure 4

Schematic representation of the equilibrium phases. (a) Conventional nematics. The single tensor order parameter $\eta$ is represented in the biaxial state by its three proper directions. In the uniaxial states two directions become equivalent and the tensor is invariant under rotation around the third direction. The plane containing the two degenerate directions is represented by a disk. (b) Bent-core nematics described by two tensors $\eta$ and ${\eta }^{\prime }$. The external symmetry groups of the corresponding phase are given by the intersection of the two tensor groups. When $\eta$ and ${\eta }^{\prime }$ are uniaxial and their axes are parallel, the phase is uniaxial since the continuous rotation axes of the two tensors coincide. When the two tensors are parallel and at least one of them is biaxial, the phase is biaxial. When they share only one common axis the symmetry is $C_{2h}$ and it is $C_i$ when they have not any common axis.Reuse & Permissions

Figure 5

(a) Three-branch star associated with the distribution $P(\alpha ,\beta ,\gamma )$ [Eq. (5)]. Each branch describes the fluctuations of $X,Y$, or $Z$. The projections ${\eta }_Z={\eta }_0,{\eta }_Y=-(1∕2){\eta }_0-(\sqrt{3}∕2){{\eta }^{\prime }}_0$ and ${\eta }_X=-(1∕2){\eta }_0+(\sqrt{3}∕2){\eta }_0^{\prime }$ of each branch onto the horizontal axis coincide with the amplitudes of the partial orientation distributions of the corresponding axes [Eq. (6)]. (b) Positions, in the order-parameter space $({\eta }_0,{\eta }_0^{\prime })$, of the six uniaxial configurations. On the middle of each region (indicated by dashed lines) a special state, occurring when one branch of the star becomes horizontal, is stabilized.Reuse & Permissions

Figure 6

(a) subunit shapes in the uniaxial special states. The subunits are generated by isotropic spinning of one bent-shape molecule (thick lines) around $X,Y$, or $Z$ and, in state (ii), by up-down jumps between the two cones. (b) Corresponding subunits in the general states obtained by completely freezing the statistical internal spinning of molecules presented in (a). The three subunits have then the same form and the same orthorhombic symmetry.Reuse & Permissions

Figure 7

Most likely orientations of the subunit in the unconventional uniaxial states. On the three calamitic special states (along the dashed lines) the subunit is cylindrically symmetric due to internal fluctuations. The subunit axis fluctuates and is on average oriented along the macroscopic optic axis. In the intermediate sectors $A,B$, and so on the subunit is orthorhombic and its external fluctuations create a macroscopic cylindrically symmetric state.Reuse & Permissions

Figure 8

(a) Star representing an unconventional biaxial state in the order-parameter space. Each branch describes one of the molecular axes $X,Y$, and $Z$. The statistical meaning of each branch is given in Fig. 1. (b) The six domains of the state (1) depicted in (a) Domains 2 and 3 result from 1 after three fold rotations around the direction (111) (i.e., permutations $zxy$ and $yzx$ of the laboratory axes). Domains 4, 5, and 6 result from mirror symmetries ($yxz$, etc.). (c) Three stars associated with the uniaxial phase.Reuse & Permissions

Figure 9

Classes of distinct unconventional biaxial states.Reuse & Permissions

Figure 10

Phase diagrams resulting from the minimization of $F_u$ [Eq. (11)] for various values of the coefficients $\epsilon$ and $\delta$. The continuous lines represent the symmetry breaking first-order transition from the isotropic liquid to the nematic phase. Dashed lines represent isostructural first-order transitions. $T_1,T_2,T_p$, and $T_p^{\prime }$ are triple points. $T_c$ and $T_c^{\prime }$ are critical points occurring at the end of isostructural transition lines. $N,N^{\prime }$, and Ne are different states of the uniaxial nematic phase. All the diagrams [except (f)] correspond to small values of the coefficients $f,g,h,i$ in Eq. (11) and almost unit values of the asymmetric ratios.Reuse & Permissions

Figure 11

Theoretical phase diagram obtained for systems invariant under the permutation of the molecular axes $X,Y,Z$. $a$ and $b$ are the phenomenological coefficients below the second- and third-order invariants in the free energy. I, II, and III are the stability domains of the three uniaxial phases which differ by their internal symmetries. I and II have cylindrical internal symmetry groups and coincide with the special states CDC,DCC,CCD and DDC,DCD, CDD, respectively. III has the orthorhombic internal symmetry $D_{2h}$. The four phases merge into a single Landau point for $a=b=0$. This diagram is the internal analog of the conventional phase diagram associated with the isotropic→uniaxial $(\mathrm{I}=\text{calamitic},\mathrm{II}=\text{diskotic})$→biaxial (III) transitions.Reuse & Permissions