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ON THE DYNAMICS OF HELICOIDAL FERROELECTRIC SMECTIC [MATH] LIQUID
CRYSTALS
B. Žekš, A. Levstik, R. Blinc
To cite this version:
B. Žekš, A. Levstik, R. Blinc. ON THE DYNAMICS OF HELICOIDAL FERROELECTRIC SMEC- TIC [MATH] LIQUID CRYSTALS. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-409-C3-412.
�10.1051/jphyscol:1979382�. �jpa-00218778�
JOURNAL DE PHYSIQUE Colloque C3, supplkment au no 4, Tome 40, Avril 1979, page C3-409
ON THE DYNAMICS OF HELICOIDAL FERROELECTRIC SMECTIC LIQUID CRYSTALS
B.
ZEKS,
A. LEVSTIK and R. BLINCJ. Stefan Institute, University of Ljubljana, Ljubljana, Yugoslavia
R6sum6. - On calcule le spectre des fluctuations de la polarisation, a basse frtquence (vecteur d'onde q = 0) d'un systkme chiral ayant un changement de phase Sm A -t Sm C ferrotlectrique htlicoidal, au-dessus et au-dessous de Tc. Au-dessus de Tc le spectre a le caractere d'un mode mou tandis qu'au-dessous de Tc, il consiste en un mode mou et un mode de Goldstone. En l'absence de couplage flexotlectrique entre la polarisation et l'angle de basculement des moltcules, la frtquence et l'intensitt ditlectrique du mode de Goldstone ne dtpendent pas de la temptrature. Le couplage flexotlectrique modifie la dtpendance en temptrature de telle sorte que l'intensite ditlectrique du
mode de Goldstone tende vers ztro a T,.
Abstract. - The low frequency polarization_fluctuation spectrum of a chiral system undergoing a helicoidal ferroelectric smectic A + smectic C phase transition is evaluated for the wave vector q = 0 both above and below T,. It has a soft mode character above To, while it consists of the soft mode part and Goldstone mode part below T,. In the absence of the flexoelectric coupling between the polarization and the tilt of molecules both the frequency and the dielectric strength of the Gold- stone mode do not depend on temperature. The flexoelectric coupling influences the temperature dependence so that the Goldstone mode dielectric strength goes to zero at T,.
1. Introduction. - The smectic C liquid crystalline phase is ferroelectric if the molecules are chiral (non-centrosymmetrical) and have a permanent dipole moment transverse to their long molecular axes [l].
In the high temperature smectic A phase the long axes of the molecules are oriented perpendicular to the smectic layers. The point symmetry of each layer corresponds to the group D,. The transition to the ferroelectric smectic phase is induced [2] by the two dimensional representation E, and the point symmetry of the layers is reduced to C,. The order parameters of the transition are the components of the in-plane spontaneous polarization P, and P,
- describing the ordering of dipoles transverse to the long molecular axes - or the quadratic combi- nations
5 ,
= n, nx and5 ,
= n, n, of the components of the molecular director n (describing the orientation of the long molecular axis). In view of the smallness of the observed [3, 41 in-plane spontaneous polariza- tion and the small difference in the smectic A + smec- ticc
transition temperatures between chiral and non-chiral modifications of the same compound, it is clear that the tilt of the long molecular axes with respect to the layer normals (which is a consequence of a non-zero value of n, n, and n, n,) is the primary order parameter and the polarization only a secondary order parameter. Thus, the spontaneous polarization is induced by the molecular tilt, and the ferroelectricsmectic liquid crystals are improper ferroelec- trics [5].
Indenbom, Pikin and Loginov [2] have shown that there are no third order invariants in the expansion of the free energy density in terms of the order para- meters so that the transition may be of second order.
However, there is a Lifshitz term [5, 21 producing a helicoidal distribution of molecular tilt and the spontaneous polarization as one goes from one smectic layer to another.
The symmetry properties of the high temperature phase allow for two types of bilinear coupling terms [2]
between the molecular tilt and the dipolar ordering :
A piezoelectric coupling [l, 2, 6 , 101
between the polarization and the tilt, and a flexo- electric term [2]
Previously [l l] we discussed the static and dynamic properties of a helicoidal smectic ferroelectric with both the piezoelectric and flexoelectric coupling terms present in the free energy density expansion.
We have shown that the flexoelectric coupling term qualitatively changes neither the static properties
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979382
C3-410 B. ZEKS, A. LEVSTIK AND R. BLINC
of the system nor the behaviour of the fluctuations with the critical wave vector q,.
The aim of this paper is to analyse the low frequency dynamical properties of the system for the non-critical wave vector q = 0 corresponding to the dielectric measurements. We will show that q = 0 relaxational eigenfrequencies and dielectric strengths for T T , depend strongly on the flexoelectric coupling. Because of this we believe that the analysis of dielectric data could give us the relative strengths of piezoelectric and flexoelectric couplings in ferroelectric liquid crystals.
2. Static properties. - The non-equilibrium free energy density can be written in the vicinity of the phase transition between a smectic A -+ smectic phase as [2J
Here g, is the free energy density of the smectic A phase in the absence of smectic
21
fluctuations, K,, is an elastic constanta = a(T - To) , h = const. > 0
and all other coefficients are assumed to be constant.
The static properties of the model can be obtain- ed [ I l l by minimizing the free energy
The spontaneous tilt and the spontaneous polarization can be expressed as
cl
= 8, cos q, z, t2
= 8, sin q, z ( 4 4 P, = - P, sin q, z , P, = P, cos q, z (4h) where 8, and P, are the absolute values of the spon- taneous tilt and polarization which are given byfor T < T,. Above T, in the smectic A phase, P, and 8, are equal to zero. The transition temperature
Tc is obtained from
and the critical wave vector which determines the pitch of the helix in the smectic
c
phase is given byand is temperature independent in this approximation.
3. Dynamic properties for T > Tc. - Neglecting inertial terms, we get the Landau-Khalatnikw equa- tions of motion as
where F is the free energy and the kinetic coefficients for tilt ( r , ) and for polarization (T,) vary only slightly with temperature. We assume that the pola- rization frequencies are much higher than the tilt frequencies. Introducing the Fourier transform
5 -
=C
(68 l q COS qz - BeZq sin qz)4
( 9 4
t2
=C
(801q sin qz+
dozq cos qz)4
(9h) P, =
C
(- 6Plq sin qz - 6P2, cos qz) (9c)4
P, = (6P1, cos qz - 6P2, sin qz) ( 9 4
4
and linearizing the equations of motion (8) we obtain for each wave vector, q, two - doubly degenerate -
solutions for relaxational frequencies ( l / z ) , of the above system, which describes the exponential approach of the fluctuations in the tilt angle and in the polarization to the equilibrium [ l l j . For the critical wave vector q = qo and T -+
c
we find that 112- oc ( T - T,) and 1'7, = const. Thus, 117- is the doubly degenerate soft and l / z , the doubly degenerate hard mode in the smectic A phase.For the non-critical wave vector q = 0 the relaxati'o- nal frequency l / z - decreases when approaching Tc from above but has a finite value at T,. The soft mode frequency 117- is an even function of (q - q,) in the absence of the flexoelectric coupling ( p = 0 ) and the frequency for q = 0 is the same as the one for q = 2 q,. For p # 0, this symmetry does not exist. It is possible to show that in this case 1 ! ~ - , ~ = , is always bigger than l / z - , , =,,,.
The dielectric response can be calculated adding
- EP, exp(- jot) term to the free energy density (eq. (3)). Here E and CO are the amplitude and the frequency of the homogeneous external electric field applied in the X-direction. Only q = 0 fluctuations contribute to the dielectric susceptibility, which can be expressed as a sum of two Debye terms with characteristic relaxation frequencies l / z - ,q=O and l/z+,,=,. The dielectric strength of the low frequency
HELICOIDAL FERROELECTRIC SMECTIC C LIQUID CRYSTALS C3-411 part (l/z_,*=,) has a soft mode behaviour. It is
increasing when approaching T, from above but does not diverge at T,, because q = 0 is not the critical wave vector. If the piezoelectric coupling is not present (C = O), the soft mode for q = 0 does not have any dielectric strength. The flexoelectric coupling does not essentially influence the dielectric response for T > T,.
4. Dynamic properties for T < T,. - In the low temperature ferroelectric smectic
c
phase we trans- form at first the coordinatescl,
r,, P, and P, into the new coordinate system rotating with the helix t l ( z ) = [8,+
601(z)] cos q, z - 68,(z) sin q, z( 1 0 4 t 2 ( z ) = [00
+
SO1(z)] sin q, z+
602(z) cos q, z( 1 0b) P,(z) = - [P,
+
SP,(z)] sin q, z-
6P2(z) cos q, z( 1 Oc) P,(z) = [ P ,
+
6Pl(z)] cos qo z - 6P2(z) sin q, z.
( 1 0 4 Here 60, and SP, represent the amplitude fluctuations while 68, and 6P2 describe the orientational fluctua- tions. After linearizing the equations of motion (8) and after introducing the Fourier transforms, we obtain a system of coupled equations for the fluctua- tions with the wave vector k and with the wave vector (- k ) in the rotating frame. In the laboratory coor- dinate system this corresponds to the coupling bet- ween the fluctuations with the wave vector ( k
+
q,)and the ones with (- k
+
q,).For the wave vector k = 0, which corresponds to the critical wave vector q, in the laboratory frame, the equations of motion separate into the amplitude fluctuations part and the orientational fluctuations part. The two eigenfrequencies of the amplitude fluctuations are given by the same expression as
l I Z ,,q=,, for T > T, only replacing a(T - Tc) by
2 a(Tc - T ) . The out of phase amplitude fluctuation mode is the hard mode, and the in phase amplitude fluctuation mode is the soft mode which vanishes at T,. The in phase orientational fluctuations represent the Goldstone mode of the transition. The eigenvector of this mode describes the rotation of the helix, and its frequency identically equals zero.
Since we are interested in dielectric response, we have to analyse the equations of motion in the rotating frame for k = q,. This corresponds in the laboratory system to the coupled fluctuations with the wave vectors q = 0 and q = 2 q,. Only these degrees of freedom contribute to the dielectric response.
For p = 0 the equations of motion for k = q, separate again into amplitude and orientational part.
The orientational part is temperature independent and consists of the hard orientational mode and of
the low frequency Goldstone mode with temperature independent' frequency l ! z l which is finite because of the non-critical wave vector. The dielectric strength of the Goldstone mode is also temperature indepen- dent. The low frequency part of the amplitude fluc- tuations has a soft mode behaviour. Its relaxational frequency decreases and dielectric strength increases when approaching Tc from below. Both stay finite at T, because of the non-critical wave vector. At the transition temperature we have
Figure 1 shows the temperature dependence of the Goldstone (2,) and soft mode (2,) frequencies in units of r , ( z = (l ! z ) / T 1 ) for T < T,. The parameters of the model have been chosen as
FIG. 1. - Temperature dependence of the relaxational frequencies in units of
r ,
for Goldstone mode (z,) and for soft mode (z,) forp = 0.
and determined from the approximate values for the pitch of the helix (5 X 10F6 m) and for Tc
-
T o (-- 1 K ) takingr,/r,
E = l0 K,, q t . In figure 2 the corresponding dielectric strengths in units of E ( I l and I,) are plotted as functions of tem- perature. The Goldstone mode dielectric strength is temperature independent, while the soft mode strength increases with temperature to a finite maximal value at T,. Only very close to T, I , > I,.For p # 0 the equations of motion for k = q, do not separate into amplitude and orientational part. This means that in contrast to p = 0 case the elementary excitations have a mixed character. Figu- res 3 and 4 show the temperature dependence of frequencies and dielectric strengths for p # 0, where all other parameters are determined in the same
C3-412 B. ZEKS. A. LEVSTIK AND R. BLINC
0.20 0 15 0.10 0 05 0
T t - T ["Cl
FIG. 2. - Temperature dependence of the Goldstone mode (I,) FIG. 4. - Temperature dependence of the Goldstone mode (I,) and soft mode (I,) dielectric strengths for p = 0. and soft mode (I,) dielectric strengths for p # 0.
FIG. 3. - Temperature dependence of the relaxational frequencies in units of
r ,
for Goldstone mode (2,) and for soft mode (2,) forp # 0.
- BOO
S2
= 0.8K 33 - 700
way as in figures 1 and 2. Now, the Goldstone mode frequency is only approximately temperature inde- pendent with a small decrease close to T,. The two frequencies are not equal at T, and the following relation is valid
21
-
1 1
- ( T = 71 T L ) = - - ( T = z - T,+, q = 2 q,) <
ZOO
The dielectric strength of the Goldstone mode (I,) and its temperature dependence have changed more than its frequency. Now the intensity is no longer temperature independent as in p = 0 case, but it shows a critical behaviour going to zero at T,.
Since the experimental results (ref. [l31 and A. Lev- stik et al., this conference DP 23) show this kind of behaviour, we can conclude that the flexoelectric coupling is essential for the understanding o f the dielectric response of ferroelectric liquid crystals.
\
References
[l] MEYER, R. B., LIEBERT, L., STRZELECKI, L. and KELLER, P., J. Physique Lett. 30 (1975) 69.
[2] INDENBOM, V. L., PIKIN, S. A. and LOGINOV, E. B., Kristallo- grafiya 21 (1976) 1093.
[3] OSTROVSKI, B. I., RABINOVICZ, A. S., SONIN, A. S., STRU-
KOV, B. A. and CZERNOVA, N. I., JETP-Pisma 25 (1977) 80.
[4] MARTINOT-LAGARDE, Ph., J. Physique Lett. 38 (1977) 17.
[5] D V O ~ A K , V., Ferroelectrics 7 (1974) 1.
[6] BLINC, R., Phys. Status Solidi (b) 70 (1975) K29, Ferroelectrics 14 (1976) 603.
[7] BLINC, R. and ZEKS, B., ~erroe1ect;ics and Antiferroelectrics : Lattice Dynamics, MIR, Moscow (1975).
[8] MICHELSON, A., BENGUIGUI, L. and CABIB, D., Phys. Rev. A 16 (1977) 394.
[9] MICHELSON, A., CABIB, D. and BENGUIGUI, L., J. Physique 38 (1977) 961.
[l01 CABIB, D . and BENGUIGUI, L., J. Physique 38 (1977) 419.
[l11 BLINC, R. and ~ E K S , B., Phys. Rev. A 18 (1978) 740.
[l21 GAROFF, S. and MEYER, R. B., Phys. Rev. Lett. 38 (1977) 848.
[l31 HOPFMANN, J., KUCZYNSKI, W. and MALECKI, J., Mol. Cryst.
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