Variety of subphase emerging sequences, the frustration of three main phases, $\mathrm{Sm}{C}_A^{*}$, $\mathrm{Sm}{C}^{*}$, and $\mathrm{Sm}A$, and the long-range interlayer interactions

Prompted by the existence of biaxial subphases 1/4, 2/5, and 3/7 [Phys. Rev. E 96, 012701 (2017)], we reconsidered the three-phase frustration and the resulting degeneracy lifting by combining the phase diagram of $\mathrm{Sm}{C}_A^{*}$, $\mathrm{Sm}{C}^{*}$, and $\mathrm{Sm}A$ with the discrete flexoelectric effect. We systematically calculated the phase diagrams and tried to understand the overall picture of the phenomena by means of a simple and intuitively clear way in terms of minimal number of parameters. The treatment naturally explains the highly distorted helical structures of the biaxial subphases as well as the microscopic helical short-pitch of $\mathrm{Sm}{C}_{\alpha }^{*}$ which increases or decreases accordingly with rising temperature. The regular subphase emerging sequence is $\mathrm{Sm}{C}_A^{*}(\mathrm{Sm}{C}_{\alpha }^{*}$)–1/4–$\mathbf{1}/\mathbf{3}$–2/5–3/7–$\mathbf{1}/\mathbf{2}$–$\mathrm{Sm}{C}^{*}(\mathrm{Sm}{C}_{\alpha }^{*}$), where the subphases other than 1/3 and 1/2 may or may not emerge. At the same time, we can see a variety of irregular sequences; in particular, any one of the biaxial subphases may singly emerge between $\mathrm{Sm}{C}_A^{*}(\mathrm{Sm}{C}_{\alpha }^{*}$) and ($\mathrm{Sm}{C}^{*})\mathrm{Sm}{C}_{\alpha }^{*}$. Moreover, the experimentally confirmed extraordinary subphase emerging sequence $\mathrm{Sm}{C}^{*}$–$\mathbf{1}/\mathbf{2}$–$\mathrm{Sm}{C}_{\alpha }^{*}$ appears for particular parameter values. Contrastingly to these affirmative aspects, some calculated results are contradictory to the previously reported experimental results: the change from $\mathrm{Sm}{C}_A^{*}$ and $\mathrm{Sm}{C}^{*}$ to $\mathrm{Sm}{C}_{\alpha }^{*}$ is always continuous, the 6-layer 2/3 subphase is not stabilized, and the subphase emerging sequence $\mathrm{Sm}{C}_A^{*}$–1/3–$\mathrm{Sm}{C}^{*}$ does not appear. The causes of inconsistency and how to resolve them were discussed in comparisons with experimental findings.

Figure 1

Calculated $g-\tilde{T}$ phase diagram with $\chi c_{\mathrm{p}}{c}_{\mathrm{f}}/\tilde{B}$ = $-0.12$ and $c_{\mathrm{f}}{c}_{\mathrm{p}}=-1.0$ by taking account of superlattice structures consisting of up to 10 smectic layers.

Figure 2

List of all the conceivable $q_T$ numbers for the subphases with unit cells of up to 10 smectic layers. Numbers in parentheses cannot be realized in unit cells of up to 10 smectic layers. Bold-faced numbers actually emerge in Figs. 3, 4, 5, 6.

Figure 3

The $V_{1,\mathrm{eff}}$ dependence of the $r$ – $\mathrm{\Theta }$ phase diagram at fixed values of $\mathcal{R}$ = $-0.1$, $g=0.2$, and $c_{\mathrm{f}}/c_{\mathrm{p}}$ = $-1$.

Figure 4

The $\mathcal{R}$ dependence of the $r$ – $\mathrm{\Theta }$ phase diagram at fixed values of $g=0.2$ and $c_{\mathrm{f}}/c_{\mathrm{p}}$ = $-1$. See also Figs. 3, 3, and 3 for $\mathcal{R}$ = $-0.1$.

Figure 5

The $g$ dependence of the $r$ – $\mathrm{\Theta }$ phase diagram at fixed values of $\mathcal{R}$ = $-0.1$ and $c_{\mathrm{f}}/c_{\mathrm{p}}$ = $-1$. See also Figs. 3, 3, and 3 for $g=0.2$.

Figure 6

The $c_{\mathrm{f}}/c_{\mathrm{p}}$ dependence of the $r$ – $\mathrm{\Theta }$ phase diagram at fixed values of $\mathcal{R}$ = $-0.1$ and $g=0.2$. See also Figs. 3, 3, and 3 for $c_{\mathrm{f}}/c_{\mathrm{p}}$ = $-1$.

Figure 7

The expanded characteristic head part of the $r$ – $\mathrm{\Theta }$ phase diagram at $V_{1,\mathrm{eff}}$ = $-20$, $\mathcal{R}$ = $-0.2$, $g=0.3$, and $c_{\mathrm{f}}/c_{\mathrm{p}}$ = $-1$. Here we can see the extraordinary subphase emerging sequences (iii) and (iv) as well as the single emergence of 1/4 in between $\mathrm{Sm}{C}_{\alpha }^{*}$ and the subphase emerging sequence (i). For further details, see text.

Figure 8

The expanded head parts of the $r$ – $\mathrm{\Theta }$ phase diagrams at fixed values of $\mathcal{R}$ = $-0.02$, $g=0.2$, and $c_{\mathrm{f}}/c_{\mathrm{p}}$ = $-1$. Here we can see an evolution of subphase emerging sequence (ii). For further details, see text.

Figure 9

The $\mathrm{Sm}{C}_{\alpha }^{*}$ helical pitch as a function of $\mathrm{\Theta }$ for various values of $r=0.2\sim 1.2$ at fixed values of $\mathcal{R}$ = $-0.1$, $g=0.2$, and $c_{\mathrm{f}}/c_{\mathrm{p}}$ = $-1$. Notice that helical pitches are drawn even in the area of the $r$ – $\mathrm{\Theta }$ phase diagram where $\mathrm{Sm}{C}_{\alpha }^{*}$ does not have the global lowest free energy. For further details, see text.

Figure 10

Highly distorted helical structures of 10-layer 2/5 subphases: (a) antiferroelectric ($FAFAAFAFAA$) and (b) ferrielectric ($FAFAFAAFAA$). Distortion angles used are given in Table 4. For further details, see text.