Determination of tilt angle and its behavior in chiral smectic phases by exploring molecular conformations using complementary methods

Density functional theory (DFT) calculations and x-ray diffraction techniques were employed to evaluate the value of the tilt angle in ferroelectric smectic $C^{*}$ and antiferroelectric smectic $C_A^{*}$ phases. Five homologues from the chiral series denoted as $3\mathrm{F}m\mathrm{HPhF}6(m=2,4,5,6,7)$, based on 4-(1-methylheptyloxycarbonyl) phenyl 4'-octyloxybiphenyl-4-carboxylate (MHPOBC), were studied. Two types of conformations for the nonchiral terminal chain (fully extended and $gauche$) and three types of deviation from the rodlike shape of the molecules (hockey stick, zigzag, and $\mathsf{C}$ shape) were computationally considered. The nonlinear shape of the molecules was accounted for by introducing a shape parameter $\delta \mathrm{\Theta }$. We observe that calculations of the tilt angle which consider the $\mathsf{C}$-shaped structures, in both the fully extended or $gauche$ conformations, lead to good agreement with the values of the tilt angle obtained from electro-optical measurements below the saturation temperature. The results allow us to conclude that such structures are adopted by molecules in the examined series of smectogens. Additionally, this study proves the presence of the standard orthogonal $\mathrm{Sm}{A}^{*}$ phase for the homologues with $m=6$, 7, and the de Vries $\mathrm{Sm}{A}^{*}$ phase for $m=5$.

Figure 1

Ordering of molecules in the ferroelectric $\mathrm{Sm}{C}^{*}$ (a) and antiferroelectric $\mathrm{Sm}{C}_A^{*}$ (b) phases along with a side view of an electro-optic cell with a smectic phase in a bookshelf geometry (c) and switching of the molecules by an electric field (d). Perpendicular arrows in the bottom of (d) visualize the optic axes of the polarizers. Vector $\widehat{z}$ is the smectic layer's normal, $\widehat{n}$ is the director, $\vec{E}$ denotes the external electric field, $\mathrm{\Theta }$ is the tilt angle, and $d$ is the smectic layer spacing. Both $\widehat{z}$ and $\widehat{n}$ vectors lay in the tilt plane, parallel to the plane of the electro-optic cell.

Figure 2

Molecular formula of the $3\mathrm{F}m\mathrm{HPhF}6$ series (a) and schematic representation of exemplary conformations possible for a molecule with a rigid core and two terminal chains (b). The dashed frame in (a) indicates the molecular core which was used in calculations of the shape parameter $\delta \mathrm{\Theta }$ (details in the text). In (b), $d$ is the smectic layer spacing, $l$ is the molecular length, and ${\mathrm{\Theta }}_{\mathrm{POM}}$ and ${\mathrm{\Theta }}_{\mathrm{XRD}}$ are the optical and steric tilt angle, respectively.

Figure 3

Visualization of the considered conformations for the 3F5HPhF6 molecule optimized with DFT-BLYP/SVP, representative for the remaining $3\mathrm{F}m\mathrm{HPhF}6$ homologues. The upper row shows four extended $max$ conformations. The bottom row shows four representative $gauche$ conformations obtained from $max1$. Definition of the shape parameter $\delta \mathrm{\Theta }$ is presented for $max3$.

Figure 4

Potential energy profiles for rotational scans performed for the isolated 3F5HPhF6 molecule with the DFT-BLYP/SVP method. In each panel, $\mathrm{\Delta }E$ is given relative to the most energetically favorable conformation. The ${\phi }_1, {\phi }_2, {\phi }_3, {\phi }_4$ torsional angles are defined in the main text and in Fig. 3. For other investigated $3\mathrm{F}m\mathrm{HPhF}6$ compounds, DFT geometry optimizations were performed only for the local minima analogous to those identified for 3F5HPhF6.

Figure 5

Boltzmann-averaged values of the molecular length (a) and the shape parameter (b) calculated for $3\mathrm{F}m\mathrm{HPhF}6$ molecules at $80{}^{\circ }\mathrm{C}$ in $max$ and $gauche$ conformations, separately for the hockey-stick shape $\left(1+4\right), \mathsf{C}$ shape $\left(2\right)$, and zigzag shape $\left(3\right)$.

Figure 6

Smectic layer spacing $d$ in the $3\mathrm{F}m\mathrm{HPhF}6$ series determined based on low-angle XRD patterns collected on cooling: in a wide temperature range (a) and in the vicinity of the phase transitions for $m=5$, 6, 7 (b). The inset in (a) shows the results for 3F2HPhF6 about the $\mathrm{Sm}{C}^{*}\to \mathrm{Sm}{C}_A^{*}$ transition. The dashed and dash-dotted lines mark, respectively, the $\mathrm{Sm}{C}^{*}\to \mathrm{Sm}{C}_A^{*}$ and the $\mathrm{Sm}{A}^{*}\to \mathrm{Sm}{C}^{*}$ transition. For each compound, one representative uncertainty bar is shown. The layer spacing for $m=7$ was taken from [22], after correction of the systematic error determined on the basis of the wide-angle XRD pattern.

Figure 7

Tilt angle in the $\mathrm{Sm}{C}^{*}$ and $\mathrm{Sm}{C}_A^{*}$ phases of $3\mathrm{F}m\mathrm{HPhF}6$ homologues: $m=2$ (a), $m=4$ (b), $m=5$ (c), $m=6$ (d), and $m=7$ (e) determined from the XRD and DFT results combined according to Eq. (1). The tilt angle determined by the electro-optic method [25, 26] is shown for a comparison. The average tilt angle values below $50{}^{\circ }\mathrm{C}$ for each studied homologue are presented in (f).