Figure 1

Molecular configuration in the $C$ phase. (a) Molecular orientation at one position in the $C$ phase. (b) Molecular configuration in one unit cell of the $C$ phase. At $z=0$ the molecule is oriented as in (a). At other positions the previous configuration precesses about the $z$ axis. All the molecules belonging to a single plane normal to $Oz$ are oriented in the same way. The helix drawn in the figure is a help for visualizing the precession of the molecular orientation but it has no direct physical meaning.Reuse & Permissions

Figure 2

Limit structures of the $R$ and EL phases arising when the induced density wave is sufficient to single out the molecular configurations located at density maximums. The density minimums are indicated by straight horizontal lines. (a) $R$ phase when the maximums of density coincide with the polarization maximums. The structure is mainly antiferroelectric. (b) $R$ phase when the density maximums coincide with the zeroes of the polarization wave. The structure is mainly anticlinic. (c) EL phase resulting from the deformation of the $R$ phase depicted in (a). (d) EL-phase resulting from deformation of (b). In this picture of the EL phase the polarization wave seems linearly polarized because the molecular orientations between the density maximums are omitted.Reuse & Permissions

Figure 3

Theoretical phase diagrams of the homogeneous vector-wave model. (a) For the free energy given by Eq. (7a). (b) For the free energy (7b). (c) For the free energy (7c).Reuse & Permissions

Figure 4

Structure of the incommensurate elliptic phase with a period $\lambda$. Locally the structure is elliptic while the axes of the ellipse precess with a period ${\lambda }_{\text{mod}}$. The precession is evident on the projection of the structure onto the $x-y$ plane. The projection has a flower-shape with a finite number of elliptic petals, because we have chosen for the picture a commensurate modulation wave vector. For general incommensurate structure the projection fills completely the disc surrounding the flower.Reuse & Permissions

Figure 5

Temperature $\left(T\right)$ concentration of a chiral doping (c) phase diagram. At zero concentration the circular phase is stable. For nonzero concentrations the isotropic liquid $\left(L\right)$ and the circular right-handed (for $c>0$) and left-handed (for $c<0$) phases are stabilized. Below the zero-concentration critical temperature $T_c$, the right and left circular phases can coexist. The corresponding demixion region is represented by the hatched area.Reuse & Permissions

Figure 6

Symmetries and molecular configurations corresponding to the flip process in the (a) zero-field $R$ phase, (b) $R1$ phase, (c) zero-field EL phase, (d) ${\text{EL}}_{∕∕}$ phase.Reuse & Permissions

Figure 7

Symmetries and molecular configurations corresponding to the flop process in (a) the $R2$ phase, (b) the ${\text{EL}}_{∕∕}$ phase. In the $R2$ phase the polarization directions in two adjacent smectic layers are opposite and their magnitudes are different. The molecular planes are not tilted so that two adjacent layers have the same thickness. In the ${\text{EL}}_{∕∕}$ phase the molecules are tilted. The tilt angle is different in two adjacent smectic layers, which consequently, have different thickness.Reuse & Permissions

Figure 8

(a) Zero-field minimal phase diagram. (b) Deformation of the minimal phase diagram under a low applied electric field $E$. (c) $b_1-E$ phase diagram with ${\mathrm{a}}_1$ constant and negative. The thermodynamic paths $\left(AB\right)$ and $\left(XY\right)$ represented by thin dashed straight lines in (a) and (b) are indicated. $R_{∕∕}$ represents either $R_1\left(\gamma >0\right)$ or $R_2\left(\gamma <0\right)$. Thick dashed curves and thick plain curves represent first-order and second-order transition lines, respectively.Reuse & Permissions

Figure 9

Temperature $\left(T\right)$ field $\left(E\right)$ phase diagrams resulting from the minimal zero-field diagram represented in Fig. 8a, for $b_1$ constant and ${\mathrm{a}}_1$ varying linearly with $T$. (a) $b_1>0$. The zero-field phase stable below $T_c$ is circular. It becomes elliptic parallel when the field is switched on and rectilinear parallel above a second-order transition line. The line becomes a first-order transition line at temperatures below the tricritical point $T_t$. Above $T_c$ the polarized liquid state is stable at low fields. Above a critical field the liquid state undergoes a field-induced transition toward the rectilinear parallel ($R_{∕∕}=R1$ or $R2$) phase. (b) $b_1<0$. The stable zero-field phases are $E$ and $R$. They transform into the $R_{∕∕}$ phase above a critical field. The three ordered structures merge at the triple point $T_T$. (c) Hysteresis curve (polarization $P$ vs field $E$) along the thermodynamic path $AB$ indicated by a dashed line in (a). (d)–(f) Modification of the phase diagram (a) resulting from higher degree coupling terms. The direct elliptic $\to$liquid first-order transition becomes possible (d). An isostructural transition within the elliptic stability domain separates the almost circular from the almost rectilinear elliptic states (e). The corresponding first-order transition line ends at the critical point $\left(T_R\right)$.Reuse & Permissions

Figure 10

Temperature-field phase diagrams resulting from the deformation of the nonminimal zero-field diagram (a). According to the degree of non-linearity and to the signs of the phenomenological coefficients in the free energy, various topologies may be predicted including various triple $\left(T_T\right)$ and tricritical $\left(T_t\right)$ points.Reuse & Permissions

Figure 11

Temperature-field phase diagrams where the coupling coefficients favor the liquid state. In contrast to the case of the diagrams shown in Fig. 9, the liquid state remains stable at high fields. The field destabilizes the ordered states, which undergo phase transformations toward the liquid phase above critical fields. The corresponding transition lines may be first or second order.Reuse & Permissions

Figure 12

Field-induced rotation of the extinction cross in a circular domain of the elliptic phase. (a) The smectic layers are normal to the plane of the figure and the wave vector $\mathbf{k}$ is radial. The local twofold symmetry axes are either normal to $\mathbf{k}$ ($C_{2x}$ and $C_{2y}$) or parallel $\left(C_{2z}\right)$. (b) Orientation of the principal axes of the optic tensor at various positions inside the circular domain. The axes are parallel to $C_{2x}$, $C_{2y}$, and $\mathbf{k}$. The extinction cross observed under polarized light with a polarizer parallel to $X$ and an analyzer parallel to $Y$ is indicated by dashed lines. (c) The application of an electric field $\mathbf{E}$ normal to the plane of the figure breaks the twofold axes $C_{2x}$ and $C_{2z}$. The optic axes within the plane $\left(x,\mathbf{k}\right)$ of the figure are no more locked and rotate in a sense which depends on the polarity of the field. The extinction cross rotates to the same angle.Reuse & Permissions