Figure 1

Molecular structures of the one-dimensional circular (C), linear $(R=\mathrm{Sm}CP)$ and elliptic $(\mathrm{EL}=B2)$ phases. All the molecules lying in one plane normal to $O_z$ have the same orientation. One bent-core molecule is represented with one vector and one two-end arrow. The mean polarization $\vec{P}\left(\vec{r}\right)$ is parallel to the molecular polarization shown in the figures by a one-end arrow. The tilt vector $\vec{A}\left(\vec{r}\right)$ is parallel to the rotation axis turning the molecular plane from its reference orientation (parallel to the vector wave $\vec{k}$ and the polarization) to its actual configuration shown in the figure. One sees that in C and EL $\vec{A}\left(\vec{r}\right)$ is parallel to $\vec{P}\left(\vec{r}\right)$ and cancels in $R$ (at least where the molecules are represented). In $R$ and EL, only the molecules located at the maxima of the density wave (i.e., in the smectic planes) are represented. We have pictured the antiferroelectric configuration of $R$, in which the molecules lying in the smectic planes are not tilted (whereas molecules located between the planes are tilted). In the anticlinic configuration (not represented), the molecules in the smectic planes are not polarized on the average (due to statistical jumps between opposite values of the polarization), but they are tilted.Reuse & Permissions

Figure 2

Phase diagrams of the $R\left[\mathrm{Sm}CP\right]\to A_1^x\left[B1\right]$ transition. This diagram holds also for the phases $A_y^1$ $A_2^{x,y}$, $B_1^{x,y}$, and $B_2^{x,y}$ (see Tables ). Continuous and broken lines indicate second- and first-order transitions, respectively. (a) When the free energy [Eq. (7)] is truncated up to the fourth degree. (b) When the free energy is truncated up to the sixth degree. The axes represent phenomenological coefficients defined in Eq. (7), which are combinations of two external parameters such as temperature and concentration. Varying only the temperature amounts to a move along straight lines in the diagrams.Reuse & Permissions

Figure 3

(a) Molecular structures in three variants of the phase $A_1^x$. (b) Waves and molecular structures in one unit cell of the phases $A_1^x$, $A_2^x$, $B_1^x$, and $B_2^x$, (see Table ). The wave components normal to $O_y$ are represented by in-plane arrows, whereas the parallel component is represented by circles. First column: Tilt fields $\vec{A}(x,z)$. Second column: Polarization fields $\vec{P}(x,z)$. Third column: Molecular configurations. The representation of single molecules is explained in Fig. 1.Reuse & Permissions

Figure 4

Columnar subphases of $R$ when $\vec{K}=K{\vec{e}}_y$. (a) Scheme of bent-core molecules seen from within their own plane. (b) Molecular arrangements in one unit cell of the phases $R$, $A_1^y$, $A_2^y$, $B_2^y$, and $B_2^y$ (see Table ). The two columns display two variants of the same phase corresponding to two possible locations of the density maxima. The positions of the symmetry elements in the unit cell are the same in the two columns. In $A_1^y$ (first column) and $B_1^y$ (second column) the molecular polarizations are normal to $O_z$.Reuse & Permissions

Figure 5

Phase diagrams of the transitions $R\left[\mathrm{Sm}CP\right]\to {\mathrm{I}}_{\pm },{\mathrm{II}}_{\pm },{\mathrm{III}}_{\pm }$ (see Table ). The axes represent phenomenological coefficients, which are linear combinations of two external parameters such as temperature and concentration.Reuse & Permissions

Figure 6

Molecular structures and group-subgroup relationships between the phases ${\mathrm{I}}_{\pm }$, ${\mathrm{II}}_{\pm }$, ${\mathrm{III}}_{\pm }$ (Table ). The conventional orthorhombic unit cells depicted for the 3D phases ${\mathrm{II}}_{\pm }$ and ${\mathrm{III}}_{\pm }$ contain two primitive cells. In ${\mathrm{II}}_{+}$ the molecular polarizations are parallel to $O_x$ and the molecular planes are in the $x\text{−}z$ plane. In ${\mathrm{III}}_{+}$ the polarizations are rotated around $O_x$ in the $x\text{−}y$ plane. In ${\mathrm{II}}_{-}$ and ${\mathrm{II}}_{-}$ only the molecules lying within the visible sides of the cell are represented. The molecular planes are parallel to $x\text{−}z$. In ${\mathrm{I}}_{-}$ and ${\mathrm{III}}_{-}$ the orientation of the molecule (a) is not locked along symmetric directions and it can change with temperature. The orientation of (b) is obtained from that of (a) after a twofold rotation around $O_z$. The orientation of (c) is obtained after a mirror symmetry transformation with respect to the $x\text{−}y$ plane. The orientation of (d) is obtained from (a) by space inversion.Reuse & Permissions

Figure 7

Unit cell in subphases (Tables ) of $\mathrm{EL}\left[B2\right]$. In $B^x$ the molecular polarizations ${\vec{p}}_{q=a,b,c,d}$ are in fact out of the $x\text{−}z$ plane. The orientations of the molecules $\left(b\right)$, $\left(c\right)$, and $\left(d\right)$ are obtained from that of $\left(a\right)$ by applying the twofold rotations $C_{2x}$, $C_{2y}$, and $C_{2z}$, respectively: ${\vec{p}}_a=(p_x,p_y,p_z)$, ${\vec{p}}_b=(p_x,-p_y,-p_z)$, ${\vec{p}}_c=(-p_x,p_y,-p_z)$, ${\vec{p}}_d=(-p_x,-p_y,p_z)$. In ${\mathrm{I}}_E$, ${\mathrm{II}}_E$, and ${\mathrm{III}}_E$, the molecules $\left(b\right)$ are turned by $C_{2z}$ with respect to the molecules $\left(a\right)$. In ${\mathrm{II}}_E$ the molecular polarizations are along $Ox$ whereas in ${\mathrm{I}}_E$ and ${\mathrm{III}}_E$ they have three independent components: ${\vec{p}}_a=(p_x,p_y,p_z)$, ${\vec{p}}_b=(-p_x,-p_y,p_z)$.Reuse & Permissions

Figure 8

$AB$-type subphases (Table ) of $\mathrm{EL}\left[B2\right]$, stabilized when the waves $A\left({\vec{e}}_y\right)$ and $B\left({\vec{e}}_y\right)$ condense simultaneously. The direction of the macroscopic polarization is indicated for the monoclinic $A{B}^y$, $AB_Z{}^y$, and triclinic $AB_O{}^y$ phases. In $AB_Z{}^y$ and $AB_O{}^y$ in-plane components of the molecular polarizations are omitted for clarity.Reuse & Permissions

Figure 9

Phase diagrams of the transition $\mathrm{EL}\left[B2\right]\to \{A^x,B^x\}\to {AB}^x$ (or equivalently $\mathrm{EL}\to \{A^y,B^y\}\to A{B}^y$).Reuse & Permissions

Figure 10

Group-subgroup relationships. Elliptic, rectangular, and thick rectangular boxes indicate 1D, 2D, and 3D structures, respectively. The phases inside gray boxes are achiral. (a) When the wave vector is parallel to ${\vec{e}}_x$, the symbols $A_i$, $B_i$, $A^{\prime }{}_i$, and $B^{\prime }{}_i$, $A$ and $B$ represent $A_1^x$, $A_2^x$, etc. When the wave vector is parallel to ${\vec{e}}_y$, $A_i$, $B_i$, $A^{\prime }{}_I$, and $B^{\prime }{}_i$, $A$ and $B$ stand for the phases $A_1^y$, $A_2^y$, etc. (b) Phases induced from $R$ $\left(\mathrm{Sm}CP\right)$ or EL $\left(B2\right)$ when the wave vector lies in the general direction of the smectic planes.Reuse & Permissions

Figure 11

Molecular structures of the phases $R$ and $A_1^x$ when $k=\pi ∕L$ (first row) and $k=2\pi ∕L$ (second row). In the first row $R$ represents the phase $\mathrm{Sm}CP$ whereas the $A_1^x$ structure has not been experimentally reported. In the second row, $R$ represents the intercalated structure $B6$, and $A_1^x$ the phase $B1$. The model by Roy et al. involves the nematic phase, $\mathrm{Sm}CP$ (or $B2$), $B1$, and $B6$. The vector-wave theory relates the isotropic liquid (or the nematic) to $\mathrm{Sm}CP$ and $B2$, on the one hand, and the isotropic phase (or nematic) to $B1$ and $B6$, on the other hand.Reuse & Permissions