A self-consistent approach for the SmA–SmC* phase transition

Abstract

By using an analogy with the Maier–Saupe theory of the nematic phase, a mean field theory of the chiral smectic-C (SmC*) phase is constructed. As an order parameter of the SmC* phase, the tilt angle θ is selected, and the feedback effect of θ is introduced into the hindered rotational potential about the molecular long axis. By solving the self-consistent equation for the order parameter θ, the second-order phase transition appears. Also, to describe the SmA–SmC* phase transition under an electric field, a coupling term between a molecular dipole moment and an external electric field is introduced to the potential function. The electroclinic effect in the SmA phase and the hysteresis effect in the SmC* phase, which were described phenomenologically in the past, are found directly from the microscopic theory.

Acknowledgments

This research was partially supported by the Grants-in-Aid for Scientific Research of Young Scientists (B) (No. 17760124) and Priority Area (No. 18048042) from the Ministry of Education, Science, Sports and Culture of Japan.

Notes

‡However, by later experiments, it was clarified that the rotation about the molecule long axis for the achiral SmC phase was almost free. See, e.g. 2.

†Maier and Saupe constructed the mean field theory of nematics by using the analogy with the molecular field theory of ferromagnetics 9.

‡The nematic order parameter S can be expressed by using an arbitrary macroscopic anisotropy; e.g. by using the magnetic anisotropy Δη, S is rewritten as represents the equilibrium value of Δη as T→0. This representation is called Tsvetkov's representation 10. Therefore, V is also rewritten as .

†In general, the solution of the Žekš theory is obtained by using an approximate free energy 11. Besides, the semi-empirical approach using the numerical simulation and the experimental data is also proposed 12. On the other hand, in this paper, we propose quite a different approach using a self-consistent equation. Though the use of a hindered rotational potential about the molecule long axis is similar to the semi-empirical approach, it is very new to obtain the solution without the approximation as shown in the final part of section 4.

‡Though the spontaneous polarization P_s is defined by , the primary order parameter of the SmC* phase is not P_s but θ. N represents the number density of the molecules. For the nematic phase, it is necessary to consider three-dimensional integration (see equation (2)). However, for the SmC* phase, it is enough to consider two-dimensional integration.

†This means the first term of the potential described by Žekš et al. in 11,

was adopted. For the SmC* phase with complex behaviour, e.g. the sign reversal phenomenon of the spontaneous polarization, the second term of equation (A) is necessary with an increase in θ. The use of a simple cosine form is valid because equation (4) satisfies a symmetrical requirement for ferroelectric LCs. With the decrease of T, many materials with SmC* phases change to the solid phases (or smectic phases with lower symmetry) in the region of rad (≈57^○). Therefore, because the second term of equation (A) is very small compared with the first term, the neglect of the second term like equation (4) is valid in the simple SmC* phases.

†Here, we consider the validity of the approximation of equation (11). Near the phase transition point , because is satisfied, equation (9) can be written as . Moreover, near the phase transition point, because and are satisfied, equation (10) can be written as . Therefore, by substituting these into equation (8), equation (11) is obtained. Actually, when equation (11) and equation (8) are plotted and compared, both are very consistent with each other at .

†As known well, the electroclinic effect of SmA phases is very small . Therefore, since is very small, the approximation of equation (14) is permitted. Of course, as E become larger, the effect of higher-order terms of the Taylor expansion appears (see section 5).

‡In equation (15), the macroscopic tilt θ diverges as T tends to T_c , for non-zero E. The problem of the divergence might be solved by retaining higher-order terms. But, we wanted to show here that our approach led to an expression equivalent to the classical Landau approach. Therefore, we just want to retain equation (15) as future work without particularly mentioning this problem.

†At the Curie temperature (T=T _c ), equation (22) simplifies to .