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PHASE DIAGRAMS OF CHOLESTERICS IN APPLIED FIELDS : THE TWO-DIMENSIONAL HEXAGONAL PHASE

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APPLIED FIELDS : THE TWO-DIMENSIONAL HEXAGONAL PHASE

R. Hornreich, M. Kugler, S. Shtrikman

To cite this version:

R. Hornreich, M. Kugler, S. Shtrikman. PHASE DIAGRAMS OF CHOLESTERICS IN APPLIED FIELDS : THE TWO-DIMENSIONAL HEXAGONAL PHASE. Journal de Physique Colloques, 1985, 46 (C3), pp.C3-47-C3-60. �10.1051/jphyscol:1985305�. �jpa-00224622�

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JOURNAL DE PHYSIQUE

Colloque C3, supplément au n°3, Tome *6, mars 1985 page C3-^7

PHASE DIAGRAMS OF CHOLESTERICS IN APPLIED FIELDS : THE TWO-DIMENSIONAL HEXAGONAL PHASE

R.M. H o r n r e i c h + * 3 . K u g l e r * * and S. S h t r i k m a n + + * * *

*Department of Mathematics, Imperial College of Science and Technology, London SW7 2BT and Department of Theoretical Physics, Oxford University, Oxford 0X1 3NP, U.K.

**Physics Department, Weizmann Institute of Science, Rehovot, Israel

***Department of Electrical Engineering and Computer Science, University of California, La Jolla, CA 92093, U.S.A.

Résumé- Nous construisons le diagramme de phase (à la Landau) des cristaux liq- uides cholestériques placés dans un faible champ électrique ou magnétique. Quand le champ excède une valeur seuil, une nouvelle phase, à symétrie hexagonale, de- vient stable. Ceci se produit pour des valeurs accessibles de la chiralité; les champs- seuils sont de l'ordre de 70/(Ae)ï[kV/cm] ou H = 60/(Ax)s[Oe], [où Ae(Ax) est l'anisotropie diélectrique (diamagnétique)]. A champ relativement bas, la phase nouvelle doit apparaître entre la phase désordonnée (isotrope) et la phase cu- bique bleue. Pour des champs plus élevés, elle vient s'intercaler entre la phase désordonnée et la phase hélicoïdale. La nouvelle phase peut être visualisée en tant que réseau triangulaire de structures cylindriques parallèles. Au sein de chaque cylindre, la directrice présente une torsion radiale.

Abstract- Thermodynamic phase diagrams of cholesteric liquid crystals in relatively weak electric or magnetic fields are calculated approximately in a Landau approach.

It is found that a new phase, having a planar hexagonal structure, can become ther- modynamically stable at physically realizable chiralities when the field exceeds a threshold value. The threshold fields are approximately E = 70/(Ae)2[kV/cm]

and H — 60/(Ax)^\Oe] for the usual cholesteric systems, with Ae(A%) the di- electric (diamagnetic) anisotropy. The new phase is predicted to appear between the disordered (isotropic) and cubic blue phases at low fields and between the dis- ordered and helicoidal phases at higher fields. The new phase xan be envisioned as an array of parallel cylindrical structures on a triangular lattice. Within each cylinder the cholesteric director configuration is described by a curling mode.

I. Introduction

Intensive experimental and theoretical investigations on cholesteric blue phases have been carried out during the past few years.1 These studies have now definitely established that these phases (with the exception of the apparently amorphous "gray" or "fog" phase) are characterized by cubic orientational order. Although a two-dimensional hexagonal structure was suggested as a possible blue phase model by Brazovslcii and Dmitriev2, no

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985305

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such structure has been observed and later calculations1 showed that either cubic ordering or the usual cholesteric spiral is the thermodynamically stable, spatially-periodic, state below the clearing point.

Recently, several groups3-6 have begun to study experimentally the effect of applied electric fields on cholesteric blue phases. In general, these studies have concentrated on the disappearance of the cubic phases in sufficiently strong fields. This is a rather complex problem and we shall therefore restrict ourselves here to a simpler although related one;

the theoretical phase diagram of cholesteric systems in relatively & fields, where the cholesteric order-parameter can be taken to have the same form as in the zero-field case.

This simplifies the calculations considerably.

11. Landau Theory

To study electric field effects in thermotropic liquid crystals, an appropriate choice for the order-parameter is the anisotropic part of the dielectric tensor

(To study magnetic field effects, an analogous order-parameter, based upon the magnetic susceptibility tensor, would be used. We shall discuss this case later.) For cholesteric systems, the zero-field average Landau free energy density is given in terms of by1,217

Here a is proportional t o a reduced temperature, cl, c2,d, and 7 are temperature- independent parameters, ~ z ~ ~ , ~ ~ ~ E ~ ~ / L ' z ~ , and we sum on repeated indices. For thermody- .namic stability, it is necessary that1r7 cl, 7, and cl

+

;c2 all be positive.

In the presence of an applied electric field

E,

equation (2) must be supplemented by

in order to obtain the total average free energy density. Note that E is in units of stat- volts/cm. We restrict ourselves to the case of uniform applied fields, in which case (3a) becomes

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For periodic structures, it will be convenient t o expand tij(3) in Fourier components1

From (3b) and (4), two conclusions can immediately be reached: (a) only the h = k = l = O Fourier component of tij(3) contributes to (3b), and (b) for structures having symmetry, where Q~(O,O,O) vanishes, Fe = 0. Thus khe free energy of cubic cholesteric blue phases is unaffected by an applied electric field. Of course, in a sufficiently strong field, the initially cubic structure will be distorted by the field5 so as to reduce the system free energy. However,this high field region is not within the framework of the calculations being presented here.

For the usual helicoidal cholesteric phase (C), the zero-field tensor order-parameter takes the exact form

where c.c. denotes complex conjugate. In the weak field regime we expect [EC] to have, to a first approximation, the same form with the Fourier coefficients E,, 62 now &dependent.

From (3), we have

Clearly, the minimum value of (Fe)c is dependent upon the algebraic sign of co. Defining

€11 and t l as the dielectric constants parallel and perpendicular to the nematic axis in a fully aligned specimen(i.e., a racmic mixture), then (ell

+

E*) and € 1 are the dielectric constants perpendicular and parallel to the cholesteric axis and

We shall be interested particularly in materials in which A t

>

0 and therefore take E to point in the X direction; E -+ = E i,. Then

It is now obvious that the free energy of the helicoidal phase is lowered by a weak electric

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field whereas that of the cubic phase remains unchanged. Thus the cubic phase becomes unstable with respect to the helicoidal one even in relatively weak fields.

Upon substituting (5) into (2) and minimizing F with respect to qc, we obtain q c = d / c l and

This can be put into a more convenient form by introducing the reduced quantities1

E i = spa'

S = --

P

f i r

f = ~ s - ~ ~

Substituting (9) into (8) gives

The C-phase equilibrium values po(t, K, e2), p2(t, K , e2) are obtained by setting a ( f ~ ) c / a p o = a(fT)C/ap2 = 0.

We next consider the two-dimensional hexagonal structure8. As introduced by Bra- zovskii and Dmitriev, the order-parameter describing this phase was composed entirely of Fourier components with associated wave vectors of magnitude qc. In this case, of course, Pc = 0. However, it has been pointed outlyg that the symmetry of the planar hexagonal structure allows a zero wave vector nematic-like Fourier component when the associated tensor amplitude has its unique axis perpendicular to the hexagonal plane. In accordance with our basic assumption we take the form of the modified hexagonal (Hm) phase order- parameter in an applied field to be the same as in the zero-field case. Then, in reduced units, (2) become:;')

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For this configuration F, is minimized for Ac

>

0 when the field is parallel to the unique axis and

( f ~ ) f l m = fHm

-

2poe 2 (12)

Comparing (10) and (12) we note that the H, phase has a term -2p0e2 in contrast with -poe2 for the helicoidal phase. This results in the H , phase becoming stable in a limited range of external fields.

As before, the H,-phase equilibrium values of po(t, K, e2) and p2(t, K, e2) are found from (12) by setting ~ 9 ( f ~ ) ~ , / a p ~ = a(fT)Hm/ap2 = 0.

Since within the framework of our calculations the free energies of cubic blue phases are field-independent, we can simply use the results obtained ~ l s e w h e r e ~ ~ ' ~ for the 0 ~ ( P 4 ~ 3 2 ) and 0~(14~32) stl-uctures.

After solving numerically the various expressions for the equilibrium values of the order-parameter and calculating the free energies, we obtained the (K, t , e2) phase dia- grams shown in Figs. l and 2. The first figure shows, for various values of e2, the ther- modynamical phases in the ( ~ , t ) plane. The most significant result is the appearance of a thermodynamically stable planar hexagonal structure in a region of the phase diagram above a threshold e;h(n = 1.3) = -028. This is seen from a different aspect in Fig. 2, where we show the (t,e2) plane for K = 1 and K = 1.3. Note that n is related to

1FR,

the C-phase Bragg back-reflection wavelength, by K = 4?rfztR/hFR. Typical values"~12 are n=1.6 and = 25nrn, so n = l corresponds to $IR = 500nm and the region of physical interest is roughly 0.6

5

n

5

1.4.

Rather than minimizing the free energies for each value of e2 separately, we could have used a lowest order perturbation method, taking p. and p2 from the e=O case and substituting those values into eqs. (10) and (12). This cruder method also yields a stable hexagonal phase at somewhat different values of K and t.

From figures (1) and (2) we see that there is a non-zero threshold electric field eth, for the appearance of the hexagonal phase. Since our calculations assume explicitly that the C, H,, and cubic structures are essentially undistorted by the applied field, the question naturally arises as to whether this field strength is within the range of validity of our model.

An answer t o this question can be found by calculating the field e% in which a spatially periodic phase undergoes a phase transition and the system becomes nematic (N). Near this field strength distortions in the periodic structure can not be neglected, as can be seen from the rigorous low-temperature solution for the C-N transition found by de ~ e n n e s l ~ and ~ e ~ e r ' ~ . We therefore require that e;h be significantly less than e& and thus in a field region in which the zero-field structures are still essentially undistorted. Unfortunately, the exact de Gennes-Meyer solution cannot be used as it is' obtained by assuming the magnitude of the order-parameter to be both temperature and field independent. This is,

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of course, & valid in the region near the clearing point which is of interest to us here.

We can, however, obtain rigorous lower-bounds on e&(t, re), which will be sufficient for our purposes.

The procedure is as follows: (1) We solve rigorously for the N-phase free energy minimum. (2) We obtain, by convenient approximations, the free energies of periodic phases in the field region of interest. Three cases will be considered; (a) a distorted C phase in which only a single 180° twist or domain wall exists, (b) an undistorted Hm phase, and (c)an undistorted helicoidal phase. (3) Since the latter minimization procedure is an approximate one, the resulting expressions for the free energies must be everywhere greater than the true one. (4) Therefore, upon equating t o the exact N-phase free energy expression, the fields 8% obtained will in all cases be than e s . For the C-N transition

2% -t

e L

asymptotically from below as t -+ -00.

The N-phase free energy density in an applied field is, in reduced units1,

Setting a(jT)N/3p = 0 gives

In general, the reduced C-phase order-parameter has the form

We shall approximate Lhc <:-phase free energy by coristrai~lilig the order-parameter (15) to satisfy p = 2po = 2p2/

4,

with p position-indepe~~de~it and given by (14). Using (2), (3)) arid (9) tile free energy density is then found to be

with z = qC( and $'rrlpl/d(. 'J'his is a static version of the sine-Gordon equatio~i. Tlie 12ulcr-Lagrange equation obtained by setting = 0 is exactly that considered by de eennes13 and ~ e ~ e r l * and it is straightforward t o show that ( f T ) ~ = fN when

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Fig. 1: Thermodynamic phase diagram (chirality-temperature plane) of cholesteric liquid crystals for applied fields (a)e2 = 0 (b)e2 = 0.018 (c)e2 = 0.04. Units are de- fined in the text.

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Equations (14) and (17) determine ?%c at any point (t, K). Solving, we find that the true transition field e& must satisfy

Note that (18) p r o d e s a lower bound only for t

5

1 = (9

+

h r 2 ~ 2 ) / 8 . This can be greater or less than the zero-field I-C transition which (in the absence of blue phases) occurs at1

For the H, and C phases, we obtain approximate high field energies directly from (10), (11) and (12) by substituting the equilibrium values of p and p2. Setting ( f T ) ~ , =

f~ then determines

zsHm

at any point ( t , K). Similarly the bound

z % ~ ~ ~

is calculated. Of course, the true transition field satisfies

e~ 2

>

and e&

> i?XHeI

The lower bounds i?&& and i?hl are shown in Fig. 2. We see that e:h/eL

<

0.27. The experimental result~~-~indicate that at such fields the cubic phases are not distorted, this supports our basic assumption that the periodic structures are essentially undistorted at the hexagonal phase threshold field.

Turning t o the case of an applied magnetic rather than electric field, the appropriate order-parameter is xijrather than eij in (2) and (5). Then (3) is replaced by (in gaussian units)

Equations (10) and (12) are then obtained by using (9) and

xi

= spi and e = (3r2//?3)4~. (9') Of course, the phenomenological Landau coefficients appearing in-(2) change when a different order-parameter is used (by the appropriate power of eij/xij). Thus, for a given value of e2, the electric and magnetic energy densities are simply related by E , E ~ / ~ T =

X o ~ 2 , as would be expected. In analogy with (7) we have

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l I K.1.3 3 -

NEM.

2- NEM. -

Fig. 2: Thcrmodynamic phase di- '\.

-.+..., agram (applied Geld-temperature plane) L '... ....,.

t \ \ .'...

for chirality parameter rc = 1 and c = \ , '..., ...

\ \ ...

1.3. The lower bounds for stability as de- \

,

2. '...

\ 'i.

e2 111. Alternate Description of the Hexagonal Phase scribed in the text are: z & ~

. . . . .

.; 1.

-0 ehHrn -

. - .

- .; - - - - -

One of the factors entering into our Landau theory calculation in Sec. I1 was the choice of the zero-field Hm-phase order-parameter. Whereas the corresponding C-phase order-parameter is exact and those for the cubic phases included up t o four different wave vector magnitudesl~10 the Hm-phase order-parameter was restricted t o a single non-zero spatial frequency. In principle, including higher harmonics in c i j alters the equilibrium value of ,uo(t, K , e 2 ) and thus the field-induced change in the H,-phase free energy. To show that such an effect is likely to be small and t o give also a more physical picture of the H,,,, phase, we consider an alternate point of view in which this phase is regarded as an assembly of parallel infinite cylinders whose axes lie.on the sites of a planar triangular lattice (see Fig. 3).

- \ \ '.. '..

\

\

\

I J

The equilibrium configuration and free energy of a single such cyliqder, for the case in which ~ i j is constrained t o be locally uniaxial, have been found rigorously15. The requirement that c i j be locally uniaxial is

.05 .K)

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Fig. 3: The two-dimensional hexagonal phase as an assembly of cylindrical elements.

In each element the director describes a curling mode configuration as described in Ref.

15.

with it (?) a unit vector. The applied field term in the normalized free energy density takes the form

For the isolated cylinder, ii is given in cylindrical coordinates by1'

A = B einw

+

2 coew, (22)

and w,p are functions of the radial coordinate r only.

Decomposing e into two components: eZ parallel to the cylinder axis and e l perpen- dicular to the axis and integrating over B gives

where A, the area per cylinder in the planar structure, is related to the lattice spacing l by ~ = & l ~ / 2 . We define a normalized length scale p = r / t R with p C d / 2 t R . For p, sufficiently large (see below) the cylinders become essentially independent and the results obtained for a single isolated cylinder can be used as a first approximation. Further, the exact zero-field single cylinder solution15 for w(p), p(p) can, t o a very good approximation, be parametrized in the form

We assume, as before, that this configuratioc is undistorted by the field. From (23) and (24) we obtain

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The parameters 4 , j i and A are determined, in principle, by minimizing the total averaged free energy density, (fT)S = fs

+

(fe)S. Substituting (20) and (22) into (2) gives15

with fip = dpldp and wp = dwldp. Since we are in any case ignoring overlap effects and thus not determining pc by rigorous energy minimization, we simplify the estimate of

(fTIs

by minimizing fs rather than ( f T ) ~ . This is, of course, also in accord with our assumption that the director configuration is not changed by the field.

Substitutirlg (24) into (26)) f s is easily minimized numerically for given (t, c ) and some typical parameter values are given in Table I. For definiteness we have chosen the temper- atures tIs(n) at which the isotropic to single cylinder phase transition occurs15, i.e., when (fs),;, = 0. To an excellent approximation, we see that p (tIS) is kindependent while

4

and A scale as n and /c2, respectively. Thus I = I ( ~ ~ / A ) is essentially c-independent for t = tIs(rc) and, from (25b), we obtain I(tIs) = 1.148. This is less than

4

so, from (25a), (fe)S is minimized when e l = 0 and 2 = eiZ, as would be expected from the Landau theory calculation.

Finally, pc is required in order to obtain a numerical value for ( f , ) ~ . A reasonable esti- mate can be reached by noting that in the H, structure, the distance between lattice sites

l = 2tR/pe would be equal to 2a/qe. However, from cubic blue phase c a l c u l a t i ~ n s ~ * ~ ~ ~ ~ ~ and also e ~ ~ e r i m e n t a l l ~ ~ ~ ~ ~ ' , it is known that harmonics of the basic spatial frequency re-

sult in a "red shift", i.e., an expansion of the unit cell. For BPI, which would be analogous to the planar phase, the observed shift is 25 to 40 %and the calculated value is 32 %.

Taking p, = 1 . 3 2 ~ / ~ , we find from Table I that ezp(-Ap:) = 0.167. This indicates that using the results of an isolated cylinder calculation should give a reasonable estimate for the applied field energy. From (25)) we obtain

(fe)s = -0.187e2 for tls, (274

which can be compared with

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( f e ) ~ , = -0.149e2 for t = tIHrn(s = 1.3),

= -0.182e2 for t = tIHrn(~ = l),

= -0.214e2 for t = tIHrn(~ = 0.75), (276) Table I: Calculated values of the isolated cylindrical curling mode config- uration transition temperatures t f ~ and optimum parameters p , $, A as a function of the chirality parameter K.

All the values in (27) are for e2 -+ 0. These results, which are based on quite distinct descriptions of the two-dimensional structure, differ by less than 20 %in the region of inter- est. This indicates that our prediction, based upon Landau theory, of a stable hexagonal phase in cholesterics in an applied field is not simply an anomaly associated with the model or order-parameter employed.

IV. Discussion

In this paper, we haw calculated the phase diagram of cholesteric liquid crystals in weak electric or magnetic fields. We find that a new planar structure should be thermody- namically stable in such fields above a threshold value a t physically realizable chiralities.

Experimentally, the relevant quantity when electric fields are employed is

Thus the physical electric field E required to reach a given point in the normalized phase diagram is inversely proportional to ( A C ) ~ / ~

.

In many cholesteric systems Ac E 0.1;

however, this can vary considerably and values as much as one-to-two orders-of-magnitude larger have been observed. When magnetic fields are used, the relevant expression (in Gaussian units) is

( a x ) n 2 = (p4/3r3)p0e2. (28b)

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For most materials AX N 1 0 - ~ . The numerical coeilicients in (28) can be estimated by noting that the elastic energy density term

i c l ~ $ , e

in (2) appears in the reduced free energy density of a uniaxial nematic or racemic mixture as

$

[ i p 2 ( n i , t ) 2 . Noting that p = 112 at the transition, we have for the corresponding Pranlr. elastic constant

and with (28a) wc get

Using K N 4 X IO-? dynes, = 25 X 10-? cm, e & ( ~ . = 1) = 0.018, and

= 0.11, (28) becomes

Thus, for AE = 0.1 and AX = 1 0 - ~ fields of approximately E=210 kv/cm and B = 19 X l o 4 0 e would be required to reach the lower threshold of the planar phase stabi1it.x region for a cholesteric system whose C-phase Bragg bsck-reflecton wavelength is at 500nm. These field values are experimentally accessible.

To summarize, we have pointed out that a new phase should be observable in cholester- ics in the presence of an applied electric or magnetic field. Verification of this prediction and studies of this planar structure could lead to an improved understanding of the com- plex behavior of cholesterics below their clearing point.

Acknowledgement

We are grateful to H. Grebe1 for numerical assistance and t o H. Stegemeyer for useful discussions and correspondence. This work was supported in part by a grant from the U.S.-Israel ~inational Science Foundation (BSF), Jerusalem, Israel.

REFERENCES

+SERC Senior Visiting Fellow. Permanent Address: Department of Electronics, Weizmann Institute of Science, Rehovot, Israel.

++The Samuel Sebba Professor of Applied Physics. Permanent Address: Department of Electronics, Weizmann Institute of Science, Rehovot, Israel.

1. GREBEL H., HORNREICH R. M., and SHTRIKMAN S., Phys. Rev. m ( 1 9 8 3 ) 1114. This paper contains an extensive list of references on cholesteric blue phases.

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BRAZOVSKII S. A., and DMITRIEV S. G., Zh. Eksp. Teor. Fiz. @(1975)979 [Sov. Phys. JETP 42(1976)497].

ARMITAGE D., and COX R. J., Mol. Cryst. Liq. Cryst. Lett. 64(1980) 41.

FINN P. L., and CLADIS P., Mol. Cryst. Liq. Cryst. 84(1982)159.

HEPPKE G., KRUMREY M., and OESTREICHER F., Proc.Ninth Intern. Conf.

Liq. Cryst., Mol. Cryst. Liq. Cryst. @(1983)99.

PORSCH F., and STEGEMEYER H.,

-

private communication. PORSCH F., STEGEMEYER H., and HILTROP K., Z. Naturforsch. In press.

DE GENNES P. G.

,

Mol. Cryst. Liq. Cryst. l2(1971)193.

Actually, the two-dimensional Bravais lattice of this structure is triangular; since this structure is generally referred to as hexagonal, we do so here also.

HORNREICH R. M., and SHTRIKMAN S., J. Phys. (Paris) 4l(1980)335 and Errata 42(1981)367.

-

GREBEL H., HORNREICH R. M., and SHTRIKMAN S.,

-

Phys. Rev. A. in press.

JOHNSON D. L., FLACK J. H., and CROOKER P. P.,

,

Phys. Rev. Lett.

45(1980)641.

-

YANG C. C., Phys. Rev. Lett. 28(1972)955.

DE GENNES P. G., Sol. State Commun. 8(1968)163.

MEYER R. B., Appl. Phys. Lett. l4(1968)208.

HORNREICH R. M., KUGLER M., and SHTRIKMAN S., Pbys. Rev. Lett.

48(1982)1404.

-

MEIROOM S., SAMMON M., and BRINKMAN W. F., Phys. Rev. m ( 1 9 8 3 ) 438.

MEIBOOM S., and SAMMON M., Phys. Rev. Lett. 44(1980)882; Phys. Rev.

&(1981)468.

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