Landau theory for smectic-$A$–hexatic-$B$ coexistence in smectic films

We explain theoretical peculiarities of the smectic-$A$–hexatic-$B$ equilibrium phase coexistence in a finite-temperature range recently observed experimentally in free-standing smectic films [I. A. Zaluzhnyy et al., Phys. Rev. E 98, 052703 (2018)]. We quantitatively describe this unexpected phenomenon within Landau phase transitions theory assuming that the film state is close to a tricritical point. We found that the surface hexatic order diminishes the phase coexistence range as the film thickness decreases, shrinking it to zero at some minimal film thickness $L_c$, of the order of a few hexatic correlation length. We established universal laws for the temperature width of the phase coexistence range in terms of the reduced variables. Our theory is in agreement with the existing experimental data.

Figure 1

Found numerically by solving Eqs. (29) and (30) values of the dimensionless value of the hexatic order parameter ${\psi }^2/{\psi }_0^2$ at $z=0$ in the coexistence regime (sea-green circles) as a function of the dimensionless temperature deviation $(a-a_0)/a_0$. The dark green solid line shows corresponding analytic result [Eq. (7)], and the blue vertical dashed line shows the limit of the existence for this local minimum $[(a-a_0)/a_0\le 1/3]$.

Figure 2

Numerically found by solving Eqs. (29) and (30), using Eqs. (19) and (A2), values of ${\gamma }_A$ for the Sm-$A$ phase (dark blue circles) and ${\gamma }_H$ for Hex-$B$ phase (green circles) as a function of $(a-a_0)/a_0$. Light green dashed line represents the analytical dependence $g\left({\psi }_m\right)/{({\psi }_0/\xi )}^2$. Using Eqs. (B6) we fit the numerical data near the point $\gamma =g_c$. Dark blue solid lines are for the Sm-$A$ phase ($\gamma \le g_c$) and dark green for the Hex-$B$ phase ($\gamma \ge g_c$).

Figure 3

Found numerically by solving Eqs. (29), (30), and (24) values of the dimensionless film thickness $L/\xi$ in the coexistence regime (lilac circles) as a function of the dimensionless temperature deviation $(a-a_0)/a_0$. The azure dashed line in the left shows the corresponding analytic result [Eq. (A12)]. The green solid line [for $(a-a_0)/a_0<(a_c-a_0)/a_0$] is obtained by fitting with Eq. (B5) the result near the point $\gamma =g_c$.

Figure 4

Numerical values [found by solving Eqs. (29) and (30) with Eq. (24) and using Eqs. (19) and (A2)] for the dimensionless film thickness $L/\xi$ in the coexistence regime of Sm-$A$ (dark blue circles) and Hex-$B$ phase (green circles) presented as a function of the dimensionless variable $\gamma /{({\psi }_0/\xi )}^2$, in accordance with Fig. 2. The last right numerical point corresponds to the condition $\partial \gamma /\partial L=0$. Corresponding analytical description according to the Eq. (A12), using Eq. (A2), is shown by the light green dashed line. Fitting [see Eqs. (B5) and (B6)] in the vicinity of the point $\gamma =g_c$ is shown by the blue solid line (for the Sm-$A$ phase, $\gamma \le g_c$) and by the dark green line (for the Hex-$B$ phase, $\gamma \ge g_c$).

Figure 5

The temperature interval of the phase coexistence as a function of the dimensionless film thickness $L/\xi$, computed by solving numerically Eqs. (29), (30), and (33) (dark blue points). Our theory result for the thick films [Eq. (A17)] is shown by the azure dashed line. The violet solid line (started from the point $L_c/\xi$) denotes the analytic solution to Eqs. (34), (46), (47), and (B5) in the vicinity of the point $a=a_c$.

Figure 6

Computed from Eq. (23) for $a/a_0=1.1$ (i.e., for the case $1<a/a_0<4/3$) function $\mathrm{\Xi }/\xi$ versus $\gamma {({\psi }_0^2/{\xi }^2)}^{-1}$.

Figure 7

Comparison of the numeric solution of Eq. (23) for $\mathrm{\Xi }/\xi$ as a function of $(\gamma -g_c){({\psi }_0^2/{\xi }^2)}^{-1}$ with our theory analysis [Eq. (B1)] (thin azure solid line). $a/a_0=1.57$ and $a/a_0=1.572$ (i.e., for the case $a/a_0>4/3$).

Figure 8

Comparison of numerically found by solving Eq. (23) $\mathrm{\Xi }/\xi$ as a function of $(\gamma -g_c){({\psi }_0^2/{\xi }^2)}^{-1}$ with analytical approximation Eq. (B1) (thin azure solid line). $a/a_0=1.5773$ (i.e., for the case $a/a_0>4/3$) and $a/a_0=a_c/a_0=1.57741$.

Figure 9

Comparison of our numeric results [solution to Eq. (28) for $\partial \mathrm{\Delta }\mathrm{\Phi }/\partial a$ versus $(a_c-a)/a_0$] in the coexistence regime (dark blue circles) with analytic theory [Eq. (47)] near the point $a=a_c$ (blue solid line).

Figure 10

Experimental data, borrowed from Ref. [3], on the temperature range $\mathrm{\Delta }T$ of the Sm-$A$ and Hex-$B$ phase coexistence as a function of the dimensionless film thickness $L/\xi$ (on cooling, shown by dark blue circles). Our analytical description of the data by Eq. (A17) for the thick films is shown by the azure dashed line. In numeric computation we used the following fitting parameters: $\xi =3.5\times {10}^{-7}$ m and $\beta {\psi }_0^2/\mathrm{\Gamma }=1.6$. Our results near the point $a=a_c$ (the violet solid line) are plotted by solving Eqs. (34), (46), (47), and (B5).

Figure 11

Comparison of the numeric results for $L/\xi$ (light green circles) versus $a_c-a$ with their analytic counterparts given by Eq. (B5) near the point $a=a_c$. Branch corresponding to the Eq. (B5) is shown by the dark green solid line.

Figure 12

Comparison of the numeric results for $y_H-y_c$ (light green circles) and $-(y_A-y_c)$ (blue circles) versus $a_c-a$ with analytic ones given by Eqs. (B6) near the point $a=a_c$. Curves corresponding to the Eqs. (B6) are shown by the solid lines (upper dark green for Hex-$B$ phase and bottom blue for Sm-$A$).