Phase diagrams of chiral nematic liquid crystals are studied within the framework of a generalized Landau-Ginzburg-de Gennes theory. Using the parametrization of Grebel, Hornreich, and Shtrikman for the tensor order parameter Q, all relevant elastic terms are included for the helicoidal phase and the blue phases of chiral nematic liquid crystals up to fourth order in Q and its gradient &#8706;Q. The influence of the additional elastic terms on the phase diagrams of the chiral nematic phases is then investigated. The theory correctly describes the variation of the pitch with temperature and the induced biaxiality of the cholesteric phase. The results resolve the discrepancies encountered by Hornreich and Shtrikman in the comparison of experiment and theory. New features in the topology of the phase diagrams of blue phases, like re-entrant phase transitions, are predicted

Longa, Lech

Monselesan, Didier

Trebin, Hans-Rainer

Online Publikationen der Universität Stuttgart

 
 
 
 
 
 
 
 
 
LIQUID CRYSTALS, 1989, VoL. 5, No. 3, 889-898 
Phase diagrams of cholesteric liquid crystals obtained with a 
generalized Landau-de Gennes theory 
by L. LONGA 
Jagellonian University, Department of Statistical Physics, Reymonta 4, 
Krak6w, Poland 
D. MONSELESAN and H.-R. TREBIN 
Institut fiir Theoretische und Angewandte Physik der Universitat Stuttgart, 
Pfaffenwaldring 57, D-7000 Stuttgart 80, F.R. Germany 
Phase diagrams of chiral nematic liquid crystals are studied within the frame-
work of a generalized Landau-Ginzburg-de Gennes theory. Using the parametriz-
ation of Grebe!, Hornreich, and Shtrikman for the tensor order parameter Q, all 
relevant elastic terms are included for the helicoidal phase and the blue phases of 
chiral nematic liquid crystals up to fourth order in Q and its gradient 8Q. The 
influence of the additional elastic terms on the phase diagrams of the chiral 
nematic phases is then investigated. The theory correctly describes the variation of 
the pitch with temperature and the induced biaxiality of the cholesteric phase. The 
results resolve the discrepancies encountered by Hornreich and Shtrikman in the 
comparison of experiment and theory. New features in the topology of the phase 
diagrams of blue phases, like re-entrant phase transitions, are predicted. 
1. Introduction 
Though the cholesteric and blue phases of chiral nematic liquid crystals have been 
intensively studied for more than a century [I, 2], only recently has it become evident 
that three different blue phases can be distinguished: blue phase I (BP I) with a 
body-centred structure, blue phase II (BP Il) with a simple cubic structure and the still 
mysterious amorphous blue phase Ill (BP Ill). The analysis of BP I and BP II by 
Landau-Ginzburg-de Gennes theory (3-6] has been very successful in explaining the 
basic features of the experimental observations. In particular, it has been shown that 
the properties of BP I and BP 11 are consistent with a space group assignment of 
body-centred cubic os (141 32) for BP I and simple cubic 0 2 (P4232) for BP 11. Up to 
three different modifications of the os phase have been predicted theoretically. The 
free energy difference between the different structures has been predicted to be 
extremely small, which indicates that higher order terms in the free energy density 
expansion [7] could play an important role, leading to qualitative changes in the 
topology of cholesteric phase diagrams. 
It is the purpose of this paper to study this suggestion theoretically. 
2. Landau-Ginzburg-de Gennes theory 
The fundamental ingredient of the Landau-Ginzburg-de Gennes theory is a 
second rank, symmetric and traceless tensor field, Q(r), with cartesian components 
Q"p(r) (a, f3 = x, y, z). The tensor vanishes in the isotropic phase and thus serves as 
an order parameter. Macroscopically, it may be associated with the anisotropic part 
0267-8292/S9 $3.00 © 1989 Taylor & Francis Ltd 
 
 
 
 
 
 
 
 
 
890 L. Longa et al. 
of the dielectric and diamagnetic susceptibilities. A spatial dependence of Q requires 
elastic terms, i.e. those proportional to 8Q (in components oa.Qpr) in a Landau-
Ginzburg expansion of the free energy. To the lowest orders in 8Q and in the absence 
of electric and magnetic fields, the free energy is 
F p2 + p3 + F4, 
~ 1 dvt[aQ f; + c1(o1Q;; )2 + c2o;Q;Jo1Q1J- 2de;11 Q;"o1 Q1"], 
- P! f dv Q;JQ11 Qii, V V (1) 
Here a is proportional to a reduced temperature, c1, c2 , d, fJ and y are temperature 
independent parameters, and e;J1 is the Levi-Civita tensor. The Einstein summation 
convention over repeated indices is applied. The term proportional to dviolates parity 
and is responsible for the formation of a helical ground state. 
The stability of expansion (I) requires that 
y ;;::: 0, (2) 
In order to study the minima ofF for periodic structures of the classical helicoidal 
cholesteric (C) phase, of BP I and BP 11, it is convenient to expand Q(r) in a Fourier 
series, 
Q(r) = L N.\12 ( L Q(k) exp (ik · r)), 
*k *k k <= *k 
(3 a) 
where 
Q(k) = { m~J2 n~~ekF e~~~} . (3 b) 
Here k is a reciprocal lattice vector, *k = {k'lk' = Sk, {Sit} E G} is the star of k, 
G denotes a cubic space group, N.k is the number of prongs of the star *k, n~~~~EkF 
are coefficients in the expansion and, finally, 
cl 0 0) el2l = S~" ~ -1 0 ' o. ~ 
0 2 
0 0 ±i) 
el2J ~E 0 0 I ' ± l . k 
±i 0 
~E H~ +i ~} el2J = -1 ± 2, k 0 
 
 
 
 
 
 
 
 
 
Phase diagrams of cholesterics 891 
are second rank tensors represented in an orthogonal, right handed coordinate system 
with the quantization axis parallel to the unit vector k. 
Writing 
(4a) 
the space group symmetry requires that amplitudes and phases of modes k and Sk 
belonging to the same star *k fulfil the conditions 
f n~yEkF y = f n~~s~<EpkFl = Qm(<i) ~ 0 (4h) 
and 
t/lm, k - t/lm.Sk = k · t - m4Js- I,I( (4c) 
where <i = (h2 + k 2 + /2 ) 112 is the star index and Q5 _ , I< is a phase, which is deter-
mined by the space group symmetry of the order parameter field (3 a). 
In order to make calculations feasible we must decide how many different sym-
metry allowed stars we have to retain in the expansion (3). We must also determine 
the relative phases k · t - mqy5 _,_1<· The information about the relevant k vectors 
may be partially deduced from experiment [1, 2]. In particular, for the usual cholester-
ic (C) phase the tensor order parameter takes the simple form of a transverse spiral 
along the z axis 
Qc(r) = - Qo(O) eb~Nt + 2~N2 Q2(1)[expi(kcz + qy) e~~lz + c.c.] (5) 
where c.c. denotes the complex conjugate, qy is an arbitrary phase expressing a 
freedom of choice of the reference frame and kc = 2njp for the wavevector associated 
with the cholesteric pitch p. Thus, only two amplitudes Q0(0) and Q2(1) must be 
determined in order to find the explicit form of Qc ( r). The twist induced simple spiral 
of the C phase causes the order parameter Q to be biaxial. A measure of the biaxiality, 
the so-called asymmetry parameter '1 = '1(r) (0 :::::; '1 :::::; 1) may be defined by diag-
onalizing the order parameter (5) (8] 
(
H -1 - 11) 
[Qcl""' = 6'1' [Q0(0) ~ 3"' Q, ( l)] ~ 
where 
0 
t( - 1 + '1) 
0 
(6a) 
(6b) 
In considering the blue phases, we shall concentrate on 0 2 , 0 5 and 0 8 cubic 
symmetries. Following the notation of Hornreich and Shtrikman we first introduce 
the reduced parameters 
p 
Qm(<i) = Y61/2 ~m EuFI 
d 
qc = -, 
c, 
k q = - = q[h, k, /], 
qc 
K = qC ~o · 
1. 3y 
41 = pz a, (7) 
 
 
 
 
 
 
 
 
 
892 L. Longa et al. 
10 
7.0 
t 
6.5 
6 
6.0 
5.5 
2.25 
,. 
2 
0 1 2 3 
Figure I. Phase diagrams predicted by Grebel et al. [5], shown with all relevant phases 
included. 
These definitions are introduced to eliminate the freedom of scale for F and Q in 
equation {1). The meaning of the parameters in equation (7) is explained in [5]. In 
particular, K is the chirality parameter and ~o is the correlation length at the isotropic-
racemic (qc = K = 0) phase boundary. 
All of the relevant features of BP I and BP II phase diagrams can now be 
reproduced by taking into account in the expansion (3 a) only the first two stars for 
the 0 2 , 0 5 and 0 8 space groups, i.e. [1, 0, 0], [1, 1, 0) for 0 2 , [1, 1, 0] for 0 5 and 
[1, I, 0), [2, 0, 0] for 0 8 • With these assumptions and definitions the free energy in 
equation (I) is expressed in terms of J1m(l1) (m = 0, ±I, ± 2; l1 = 1, 2, 4), K, q and 
t, where J.lm(l1) and q are variational parameters. Retaining the dominant m = 2 order 
parameter and minimizing F:ccd with respect to q and 112 ( l1) finally yields the phase 
diagrams presented in figure I (5]. 
3. Generalized Landau-Ginzburg-de Gennes theory 
We now study how elastic terms of higher order than in expansion (1) effect the 
topology of the phase diagram. 
The dominant higher order invariants are cubic terms of the form Q 8Q 8Q, where 
Q iJQ oQ denotes the class of all independent S0(3) invariants built up from the 
tensors n..Ipapny~ocrnpv· For the stabilization of the free energy expansion we must 
also retain QQ 8Q 8Q terms. Without proof we mention that the class of pseudo scalar 
invariants QQ oQ, QQQ 8Q and 8Q oQ oQ may safely be disregarded in the limit of 
weak chiral interactions, which are discussed here. Also the terms 8Q 8Q 8Q 8Q are 
not expected to be relevant for the type of long wavelength structural organization 
observed in C, BP I and BP IT. On the other hand, the phase transitions discussed here 
are first order, which means that the non-chiral elastic terms proportional to Q and 
QQ may be important. 
 
 
 
 
 
 
 
 
 
Phase diagrams of cholesterics 893 
These have been enumerated in our recent paper [7]. Here we showed that the free 
energy expansion up to order QQ 8Q 8Q contains, in addition to F 2 , F 3 and F4 terms 
(see Equation (1)) 19 more elastic terms, denoted in [7] as xi~Pl]I ex = 1, ... , 6 and 
xi~4F ]I {3 = 1, .. . , 13. Though three surface relations eliminate one invariant xi~P l ] 
and two invariants x£~4>] I studies of the free energy expansion with 16 free parameters 
appear hopeless. It is important, therefore, to identify among x£~>]I i = 3, 4 the 
relevant terms. Unique features of these terms come to light, if we realize that the 
dominant local distortion in C, 0 2 , os and os structures is the twist. Thus, we neglect 
those elastic terms, which in the mean field decomposition into splay, bend and twist 
(see table 4 of[7]) do not contain the twist component. The validity of this assumption 
is supported by explicit calculations, given in the Appendix, of the individual con-
tributions xi~l ]I i = 3, 4, to the free energies of C, 0 2 , os and os phases. Using the 
results given there we identify as dominant the elastic terms 
i~Pgn;N oknilok nN~> (8 a) 
i~4ln;N nuoknIm okntm• (8b) 
i~4l Q;k Q1k a, Q;m o, Q1m. (8 c) 
The stability of the corresponding extended Landau- Ginzburg-de Gennes free energy 
requires, in addition to conditions (2), that 
( £ (3) )2 i~4g ~ 0, (44J + i~4lF - _24 ~ 0. (9) 
cl 
Using the convenient notation 
11 
(a) 
7 
3 
u = !f._(£(4) + l £(4)) 2 2 2 7 , 
CIY 
(a) (K , A, u,v) 
(b) (K,A ,u,v) 
(c) (K,A,u,v) 
(K, O,O,O) 
(3, 0, 0.5, 0) 
(1,0, 1,0) 
-8 -4 ( t - tc) 
tc 
0 
Figure 2. Chirality q/qc versus temperature; tc denotes the cholesteric- isotropic 
transition temperature. 
 
 
 
 
 
 
 
 
 
894 
12 
(a) 
8 
4 
0 
0 
L. Longa et al. 
(d) 
\ 
\ 
' .......... 
(a) 
(b) 
(c) 
(d) 
(e) 
1 
---
(.A,u,v) 
(>.,u,v) 
(.A,u,v) 
(.A,u,v) 
(.A,u,v) 
2 
= (0,0,0) 
= (0, 1, 0) 
( - 2,1,0) 
(-1,1,0) 
( - 1,4,0) 
"' 
3 
Figure 3. Chirality q/qc versus chirality parameter " at the cholesteric-isotropic transition . 
10 Tf 
8 
4 
0 
(a) (.A,u ,v) 
(b) (.A,u,v) 
(c) (.A,u,v) 
0 
(0,0,0) 
( - 0.1,1,1) 
( - 0.1,2,-1) 
1 2 3 
Figure 4. Biaxiality versus chirality parameter 1<: at the cholesteric-isotropic transition. 
we can now minimize the free energies, listed in the Appendix, with respect to q and 
/12 (o-). 
The results, calculated at the low temperature phase transition to the C phase, are 
shown in figures 2-4, with chirality qfqc defined in close relation to experiment. As 
 
 
 
 
 
 
 
 
 
20 
(.\,u,v) -
I 
10 
0 
2 
4.5 
(.\,u,v) 
t 
3.5 
2.5 
!.5 
I.L 
Phase diagrams of cholesterics 
c 
3 
(a) 
{0.5,1,1) 
1.6 
(c) 
4 
2.1 
12 
t 
8 
4 
3 
t 
2 
1 
2 
1 
(A,u,v) = 
I 
3 
(b) 
(.\,u,v) 
1.5 
(d) 
895 
K, 4 
(0.1, 2, 1) 
2 
Figure 5. Possible modifications of phase diagrams given in figure 1, caused by the presence of 
higher order elastic terms. 
seen from the figures, qualitatively new results are obtained for both cholesteric and 
blue phases. In particular, we find that the theory yields a strong dependence of pitch 
and asymmetry parameter of the C phase on temperature and chirality (see figures 
2-4). The trends are consistent with experimental observations [8, 9] and in con-
trast with the predictions of Hornreich-Shtrikman theory [10] (see branch (a) in 
 
 
 
 
 
 
 
 
 
896 L. Longa et al. 
figures 2- 4). Only the asymmetry parameter Yf, calculated at the isotropic- cholesteric 
transition, follows the trends found in [10]. When normalized to its value at the 
cholesteric-isotropic transition, it appears to be a universal function of the reduced 
temperature for fixed K. The curves are essentially indistinguishable from those of 
Hornreich and Shtrikman (see figure 2, [10]). 
Another interesting aspect is the behaviour of blue phase diagrams, figures 5 (a)-
(d). The inclusion of higher order terms may dramatically alter the phase sequence of 
blue phases and may even lead to re-entrant phases. The reason is the strong non-
linear coupling between q and J.l2 ((J) . In this respect our re-entrant phases are very 
similar to those observed by Feldman et al. [11] under external, chiral strain. The 
important difference, however, is that in our model the chiral strain (i.e. the changes 
of q) is a thermodynamic variable, calculated at equilibrium. 
This work is partially supported by the Alexander von Humboldt Foundation, by 
the Polish Project C.P.B.P.01.03, and by Deutsche Forschungsgemeinschaft. 
Appendix 
The free energies of C, 0 2 , 0 5 and 0 8 phases calculated for all invariants xi~> ] [7]. 
For completeness the standard Landau-Ginzburg-de Gennes part (I ) is also included 
(see [5]). The terms 
2 
- (3) _ qc L(3) 
Lm p m ' m= 1, . . . , 7, 
2 
-(4) qc L(4) Ln = n • y n = 1, ... , 15, 
denote the higher order material parameters in the rescaled free energies. These 
energies read: 
Fe = itg.l~ElF + i[t - h2]g.l~ElF + ~EMF + g.l~ElF 
+ (- 3 + x~P>q2 + 2i}Pl qF~ElFg.l~ElF 
+ [2 + c£~4> + ;I£~4> - tlW)q2 + !l~~q + l~~ qlttR ClFtt~ClF 
+ [t + c£~4F + t£~4>Fq2 - tlWq + ly~qgg.l~ElFI 
Fos i[t - K2 + K2(q - NF2 ]tt~ E2F 
23(2)112 
+ 32 [- 1 + E N ·MMMMx~
P> - N·MMMMx~P> + N·MMMMl~P> Fq2 
- N·4N42i~P> q]g.l~ E2F 
+ 
499 [l + EN·TPPRx~4F - M·2SSR£~4F + 0·6944D4> - 0·3056D4> 384 7 8 
+ M·M4PN£~4> - o·M8S2iy~> + M·NPPP£y~ + O· t333LW)q2 
- M·4SMRl~~q + tJ4t42ly~qgttiE2FI 
 
 
 
 
 
 
 
 
 
Phase diagrams of cholesterics 
F0 z = ~ [t - K 2 + K2 ( 2i12 - 1 J] Iu~ElF + i([t - K2 + K 2(q - 1)2],uH2) 
+ 
23ji_112 [1 + (- N·MMMM£~> + N·MMMM£~P> - N·MMMMi~>Fq2 
+ N·4N42i~P}q]Iu~E2F 
+ 3[4 + 83(2)
112
] [- 1 + EM·SSSTx~PF - M·SM9Ri~P} + M·SM9Rf~PlFq2 
- N·NP8Ni~Plq]Iu~ElFgl2E2F 
+ 
499 [I + (1·7335£<4> - 0·2665£<4> + 0·6944£<4> - 0·3056£<4> 384 2 3 7 8 
+ o·M4PNi~4> - 0·0862iW + l·lPPPl~~ + 0·1333lW)q2 
- M·4SMRly~q + N·4N42i~~q]gl~E2F 
897 
+ 139 - 12(2)
112 
[1 + (1·2372£<4> - 0·2544£<4> + 0·6231£<4> - 0·0984£<4> 48 2 3 7 8 
- o·ooT2i~4>J o·o137lW + M·N2T2£~~ + 0·13t4lW)q2 
- M·RMSPl~~q + N·2MTfi~~q]gl~ENFIu2 E2F2 
+ 12 [1 + EM·92PNx~4} + M·MTS9x~4F + M·PMTT£~4} - M·N92P£~4F 
12 
+ o·MP8sl~4> - o·o769lW + o·oP8Rl~~ + o-03sslW)q2 
- M·2S92£y~ 1 + N·MMMMiy~q]Iu2 ENF4 I 
1 [ 2 2 ( 2q )
2
] 2 F0s = 4 t - K + K 2112 - 1 ,u2 (4) 
1 
+ - [t - K2 + K2(q - 1)2]JlH2) 4 
+ ~ [- 1 + EN·MMMMx~P> - N·MMMMx~P> + N·MMMMx~P>Fq2 - N·4N42l~P>q]gl~E2F 
+ 3[3 + 2(2Y12] [- 1 + EN·PPPPx~> - N·2N9Ml~P> + N·2N9Mf~>Fq2 
8 
- N·SM9Rx~P>q]Iu2E4FIu~E2F 
+ 449 [1 + (1·8508£<4> - 0·1492£<4> + 0·6537£<4> - 0·3463£<4> 384 2 3 7 8 
+ M·MS8Ml~4F - O·l359lW + M·MT4S£y~ + 0·0746LW)q2 
- MJ4N42iy~q + NJ4t42ltqlgl~E2F 
+ 24 + 17(2)
112 
[1 + (1·4411£<4>- 0·9731£<4> + 1·0221£<4>- 0·1850£<4> 48 2 3 7 8 
- M·MTR4£~4> + 0·1079£W+ 0-4865LW + 0·5294lW)q2 
 
 
 
 
 
 
 
 
 
898 L. Longa et al. 
- M·S9SPiy~q + 1·5607 iy~ q]gN2 E4F.u~E2F 
+ 151 H4~ 2E2F N N2 [1 + E2·R2P2x~4F - M·4MPPx~4F + N·M4P2x~4F - M·4MPT£~4F 
+ oJMRMR£~4> - 0·1419£\jl + M·2MNS£~~ + M·2P84iy~Fq2 
. - M·RTMT£y~q + 1·707tLWq]J12(4)2 NN~E2F 
+ .!2 [1 + (3·6923£<4> + 0·3077£<4) + 1·2308£<4> - 0·7692D4> 12 2 3 7 8 
+ M·NRP8£~4> - o-3077 LW + oJNRP8£y~ + o-t538LW)q2 
- M·RP84i~~q + 2Jooooi~~q]gN~E4F. 
References 
[1] BELYAKOV, V. A., and DMITRIENKO, V. E., 1985, Usp.fiz. Nauk, 146, 369 (1985, Soviet 
Phys. Usp., 27, 535). 
[2] STEGEMEYER, H ., BLiJMEL, TH., HILTROP, K., 0NUSSEIT, H., and PORSCH, F., 1986, Liq. 
Crystals, 1, 3 (erratum: 1986, Liq. Crystals, 1, 305). 
[3] HORNREICH, R. M., and SHTRIKMAN, S., 1981, Phys. Rev. A, 24, 635. 
(4] KLEINERT, H., and MAKI, K., 1981, Fortschr. Phys., 29, 219. 
[5) GREBEL, H., HORNREICH, R. M., and SHTRIKMAN, S., 1983, Phys. Rev. A, 28, 1114. 
(6] GREBEL, H., HORNREICH, R. M., and SHTRIKMAN, S., 1984, Phys. Rev. A, 30, 3264. 
(7] LONGA, L., MONSELESAN, D., and TREBIN, H.-R., 1987, Liq. Crystals, 2, 769. 
[8) YANIV, Z., CHIDICHTMO, G., and DOANE, J. W., 1983, Phys. Rev. A, 28, 3012. 
[9] YANIV, Z., NEUBERT, M. E., and DOANE, J. W., 1983, Liquid Crystals and Ordered Fluids, 
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Phase diagrams of cholesteric liquid crystals obtained with a generalized Landau-de Gennes theory

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Phase diagrams of cholesteric liquid crystals obtained with a generalized Landau-de Gennes theory

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