Macroscopic and molecular symmetries of unconventional nematic phases

The effects of the molecular symmetry on macroscopic properties of conventional and unconventional nematic phases are investigated theoretically. These effects concern: (1) the form of the molecular orientation fluctuations, (2) the thermodynamic behavior and the list of stable ordered phases, (3) the effective symmetry of the molecules in the ordered phases. The order-parameter consists of tensors forming the main harmonics of the fluctuation distribution. In conventional models (valid for uniaxial molecules) a single tensor is sufficient while unconventional models (valid for less symmetric molecules) need several tensors with the same rank. We analyze the qualitative differences arising when the number of equivalent tensors varies. We show how to work out complete models in the general case, and to calculate the sequences of stable phases and the corresponding effective molecular symmetries. This yields, for each molecular group and each tensor rank, a complete classification and a deep insight into the structure of thermotropic nematics. This work generalizes the approach we have applied to polar nematics recently observed in polyester compounds and to unconventional uniaxial and biaxial phases of bent-core materials.

Figure 1

(a) Action of an internal twofold rotation $R_{00\pi }^{\mathrm{int}}$ around $\mathbf{K}$ on a statistical set composed of two molecules. The initial distribution $P$ is transformed into $P^{\prime }=R_{00\pi }^{\mathrm{int}}$ $P$. Since the sets associated with $P$ and $P^{\prime }$ are different, $P$ is not invariant under $R_{00\pi }^{\mathrm{int}}$. (b) Action of an internal rotation $R_{\Omega }^{\mathrm{int}}$ onto a frame $\Sigma$ oriented by the rotation $R_{\omega }$ with respect to the laboratory frame ${\Sigma }_0$. In step 1, $\Sigma$ is put parallel to ${\Sigma }_0$ (or equivalently the laboratory frame is turned parallel to $\Sigma$). In step 2, $\Sigma$ is turned by the external rotation $R_{\Omega }$ (or equivalently by an internal rotation in the turned laboratory frame). Step 3 reverses step 1 (or equivalently the laboratory frame is back to its initial orientation). The final internally-rotated state is thus oriented with respect to ${\Sigma }_0$ by the rotation $R_{\omega }{R}_{\Omega }$ after the internal rotation $R_{\Omega }^{\mathrm{int}}$, while it is oriented by $R_{\omega }{R}_{\Omega }$ after the external rotation $R_{\Omega }^{\mathrm{ext}}$.

Figure 2

Action of internal and external selections (projections) on the matrix of fluctuation coefficients $P_{m{m}^{\prime }}^{L,+}$ associated with a single tensor rank $L$ and parity, say +. Internal projection cancels a number of columns whereas external projection cancels a number of rows in the matrix. More precisely, the projections make linearly dependent a number of rows or of columns. After a linear transformation independent of the coefficients, the fluctuation matrix takes the form (with zero rows and columns) indicated in the figure.

Figure 3

Symmetry groups of the polar vector model. When the model contains a single vector, or several parallel vectors (first row) the macroscopic group $G_{\mathrm{Nem}}$ is $C_{\infty v}$ (achiral case) or $C_{\infty }$ (chiral case). When the model contains two vectors, or many vectors in a single plane (second row), the symmetry of the achiral case is reduced to one mirror plane $\left(C_S\right)$, or it becomes triclinic $\left(C_1\right)$ in the chiral case. With three vectors (third row), the symmetry is triclinic in both cases.

Figure 4

(a) Classes of subgroups of $\mathrm{O}\left(3\right)$ when the order parameter is an even second-rank tensor. Some group-subgroup relationships are indicated by broken lines. The five groups of the maximal SG sequence [$\mathrm{O}\left(3\right)$, $D_{\infty h}$, $D_{2h}$, $C_{2h}$, and $C_i$] are inside ovals. The groups forming the five corresponding classes are surrounded by thick lines. When the symmetry of one molecule belongs to the isotropic class, no standard nematic can be stabilized. Molecules belonging to the class of uniaxial molecules can stabilize the uniaxial and biaxial conventional phases. Molecules belonging to the orthorhombic, monoclinic and triclinic classes can stabilize all the ordered standard phases [$\mathrm{O}{\left(3\right)}^{\mathrm{ext}}$, $D_{\infty h}^{\mathrm{ext}}$, $D_{2h}^{\mathrm{ext}}$, $C_{2h}^{\mathrm{ext}}$, and $C_i^{\mathrm{ext}}$]. The noncrystallographic groups that have not been indicated on the figure all belong to the isotropic class. (b) Effective molecular groups associated with the molecules of each class. When the phase is impossible with the molecules of one class, the class is hatched in gray.

Figure 5

Molecular classes when the order parameter is an axial $\left({\Gamma }_1^{+}\right)$ or a polar $\left({\Gamma }_1^{-}\right)$ vector. The classes are surrounded by thick lines. The maximal element of each class lies inside an oval. The value of $n_0^{\pm }$ (or of $n_0^{\prime \pm }$) common to all the groups in one class is indicated beside the class. Some group-subgroup relationships are indicated by broken lines. The maximal SG sequence is presented on the right. Each group in this sequence is represented with its group-parametricity, $\overline{a}$. The order parameter multiplicity is ${\mathrm{n}}_{\Gamma }=\overline{a}\left(C_1\right)$ or $\overline{a}\left(C_i\right)$. Straight lines represent the group-subgroup relationships between the elements of the maximal SG sequence. The successive nonmaximal SG sequences are formed with all the groups having $\overline{a}⩽0,1,2,\dots ,n_{\Gamma }$.

Figure 6

Molecular classes and maximal SG-sequence when the order parameter is an even $\left({\Gamma }_2^{+}\right)$ or an odd $\left({\Gamma }_2^{-}\right)$ second-rank tensor.

Figure 7

Molecular classes and maximal SG sequence when the order parameter is an even $\left({\Gamma }_3^{+}\right)$ or an odd $\left({\Gamma }_3^{-}\right)$ third-rank tensor.

Figure 8

Molecular classes and maximal SG sequence when the order parameter is an even $\left({\Gamma }_4^{+}\right)$ or an odd $\left({\Gamma }_4^{-}\right)$ fourth-rank tensor.