TEMPERATURE DEPENDENCE OF THE PITCH IN CHOLESTERIC LIQUID CRYSTALS : A MOLECULAR STATISTICAL THEORY

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TEMPERATURE DEPENDENCE OF THE PITCH IN CHOLESTERIC LIQUID CRYSTALS : A

MOLECULAR STATISTICAL THEORY

W. Goossens

To cite this version:

W. Goossens. TEMPERATURE DEPENDENCE OF THE PITCH IN CHOLESTERIC LIQUID CRYSTALS : A MOLECULAR STATISTICAL THEORY. Journal de Physique Colloques, 1979, 40 (C3), pp.C3-158-C3-163. �10.1051/jphyscol:1979332�. �jpa-00218728�

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TEMPERATURE DEPENDENCE OF THE PITCH IN CHOLESTERIC LIQUID CRYSTALS

:

A MOLECULAR STATISTICAL THEORY

W. J. A. GOOSSENS

Philips Research Laboratories, Eindhoven, The Netherlands

RBsumB. - Une thkorie molkculaire statistique est prksentk qui dkcrit le comportement du pas en fonction de la tempkrature dans les cristaux liquides cholestkriques. Cette thkorie est formulke en termes de ceux paramktres d'ordre qui sont pertinents pour la description de 1'Ctat nkmatique correspondante.

Abstract. ^-A molecular statistical theory is given which describes the temperature dependence of the pitch in cholesteric liquid crystals in terms of all order parameters relevant to the untwisted nematic phase.

1. Introduction. - Some years ago a molecular statistical theory was proposed which explained the helical structure of the cholesteric phase, however with a temperature independent pitch [I]. The model has been extended to account for the temperature dependence of the pitch in terms of hindered rotation of the molecules around the long principal axes [2, 31.

The temperature dependence is then determined by an orde; parameter-which describes biaxiality on a microscopic scale [3]. The distribution of the long molecular axes themselves with respect to the local axis of alignment, i.e. the director, was considered to be uniaxial. Consequently any influence of hindered rotation of the long molecular axes around the director was disregarded. This hindered rotation, generated by the above mentioned microscopic or molecular biaxiality, give rise in a nematic liquid crystal to a macroscopic or phase biaxiality. In cholesteric liquid crystal, where the director rotates smoothly around the helix axis, this biaxiality should

Priest and Lubenski [4]. They introduced a model Hamiltonian in terms of the order parameters of a biaxial nematic phase, having the proper symmetry for producing a twisted nematic phase. General symmetry arguments then allowed them to arrive at similar conclusions, albeit with no physical back- ground and interpretation as to the molecular aspects and origin of the twist.

2. General formulation. - The theory of the cholesteric phase presented in reference [l] can be viewed as an extension of the Maier-Saupe theory for the nematic phase. The dispersion energy between two anisotropic, optically active molecules was calculated, taking into account not only the dipole-dipole interaction, but also the dipole-quadrupole interaction.

The latter, which is specific for optically active molecules, acts as a rather small perturbation. The dispersion energy thus obtained for two molecules i and j can be written as,

manifest itself on a local scale. i.e. intermediate

vij

⁼

vij

⁺

vij

⁺

vji

P P P 9 P9

between molecular and pitch dimensions, the helix (1)

axis still being an axis of uniaxial symmetry. where V,$ is the second order perturbation energy It is the purpose of this paper to review and extend of the dipole-dipole interaction, i.e. the London- the above mentioned theory [I] and to show how local Van der Waals energy, and

v$~

is the second order biaxiality may play a significant role in the tempera- perturbation energy due to the combination of the ture dependence of the pitch. A different, rather dipole-dipole interaction and the dipole-quadrupole phenomenological model has been proposed by interaction. These respective energies are defined as

v,v'

(2) ( P ; / P ~ ) v ( P : , I ~ ~ , , ) v ,

'2, DgBr

(p;/pi), ( p ~ , / q ~ , j , ) v , C2,(rg/rij)

(,. 9 7 +

1

^(,.ij)7

+

^C.C.

v,v' v,v'

(3)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979332

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TEMPERATURE DEPENDENCE OF THE PITCH IN LC C3-159

where @ ~ I ~ b ) , , = ( O l p i I v ) ( v l p ~ I 0 ) and (P:

I

q;,), =

<

⁰

^I

^pi

^I

^v⁾

<

^v

^I

^q;,

^{l o}

⁾are products of molecular matrix elements of the dipole moment operator p and quadrupole moment operator q.

Repeated indices a,

P,

y , etc., which refer to the coor- dinates X, Y , Z of a macroscopic, fixed, Cartesian coordinate system, indicate a summation over the corresponding components. Besides one has to sum over all excited states i1 and 1:' with energies Ev and Ev, such that E::,

,,

⁼^(E:

+

^E;.)^-^(EA

+

E i ) # 0 .

The scalar r i j is the absolute value of the vector rij = (r;, r;, r:), pointing from the centre of mass of molecule i to that of molecule j. The quantities C$

and D:~, are tensor elements depending only on the relative positions of the molecules, not on their orientations ; they are defined by :

The matrix elements for the electric dipole moment operator p and electric quadrupole moment operator q transorm under rotations like the corresponding components of p and q respectively. For convenience we will therefore use the following abbreviations :

The above pair potential V i j will be used for an internal field approximation in which the precise interaction V i = Vij of one molecule with all the

J

others will be approximated by a suitable averaged interaction. In order to do so we will use the following

symmetry arguments.

a) The helix axis, which we identify with the macroscopic y-axis, is an axis of twofold rotational symmetry. Therefore only those dipole-dipole transitions contribute to the interaction energy that transform like XX, Y Y , Z Z and X Z . With regard to the dipole-quadrupole transitions only those contribute to the interaction energy that transform like X X Y ,

Y Y Y , Z Z Y and X Y Z .

b) The helix axis is an axis of screw symmetry.

There is thus a correlation between orientation and translation along the y-axis. The easiest way to account for this correlation is to first average over the positions of the molecules with regard to their rx and r, dependence, keeping r, fixed ; we then assume that all molecules with the same r, have on the average the same orientation. The distribution of the centres of mass of the molecules with regard to the helix axis can be taken as having rotational symmetry. Then only those combinations C Ca,Bj, 9 CGIB DarSPy and Cap r, have an averaged value different from zero that are not odd in rx and r,, i.e. Cxx C x , = 0 , Cxx Dxrz = 0 ,

@a

1

pb)v

(A 1

q1,$v, - (ap)i

P,

y ) j

C

^-

v.v' E:!,,oo ⁽⁷⁾ r , # 0, C,, C , # 0,

ex,

^Dxzz^#0 etc. Taking

these arguments into account the dipole-dipole and which clearly reflect the symmetry properties. Besides dipole-quadrupole interaction averaged over

r2

^and

r2

(ED),

⁼(pa), , (a,

P Y ) ~

⁼(a, Y P ) ~ , etc. can then be written as

v:;

⁼v i j ( 4 ( x x

+

^{Y Y}

+

Z Z ) , ( Y Y ) j

+

4 ( Y Y ) i ( X X

+

^{Y Y}

+

^{z Z ) j}

+

2(XX

+

Z Z ) , ( X X

+

^{Z z ) j}

+

^{( X X}^-^Zz)i^{( X X}^-zZ)j

+

4 ( X Z ) , ( X Z ) , )

.

(8) 2 V'j

v z

=

7

(2(XX

+

^{2 Y Y}

+

Z Z ) , ((Z'+' X, XY),.

+

( Y , YY),.)

+

r~

where

( P ' ) X, X Y ) , = ( X , X Y -

:

Y , X X ) ,

+

^(2,^{Z Y}^-

⁴

^{Y , ZZ)i}

(z(-'

x , X Y ) , = ( X , X Y -

:

Y , X X ) , - ( Z , Z Y -

:

Y , Z Z ) , ( 1 0 ) ( Z X , Y Z ) , = ( X , Y Z - Y ,

zx +

Z , X Y ) ,

.

The common factor Vij = 3/(b 4(ry)4) arises from to the fixed macroscopic coordinate system X, Y , Z.

the averaging over the various combinations of the In order tcvdefine these quantities in terms of mole- tensor elements Cap and DaB,. cular properties and of the order parameters describing the phase, we have to project the tensor components pa, qp, upon a molecular fixed coordinate 3. Order parameters and molecular properties. -

system X , Y , Z . Quite generally we may write We shall now consider the molecular aspects in

more detail. According to equations ( 6 ) and (7) (XX),, ( X , Y Z ) , etc. refer to products of matrix ele- ments for the electric dipole and electric quadrupole moments. These moments were defined with respect

(4)

where x, = X, = a, a,, x, = Y, = a, a, etc. Here to have an explicit representation of these order a,, a,, a, etc. are unit vectors along the corresponding parameters in terms of Euler angles. Therefore we coordinate axes. The order parameters then appear use the following representation for A, which spe- as the averaged values (

c, ^lp

^- d Z p dC5 ) ; cifies the orientation of the molecular coordinate

5, 5

= x, y, z and a,

p

⁼X, Y, 2, whereas the brac- system ^X,y, z with respect to the macroscopic coor- kets denote an ensemble average. It is convenient dinate system X, Y , Z [ 5 ] ,

cos $ cos cp cos 0 - sin $ sin cp, - sin $ cos cp cos 0 - cos $ sin cp, cos cp sin 0 cos $ sin cp cos 0

+

^{sin $ cos}cp, - sin $ sin cp cos 0

+

cos $ cos cp, sin cp sin 0

- cos $ sin 0, sin $ sin 0, cos 8

Here 9 is an angle that describes a rotation around the macroscopic Z-axis, 0 an angle that describes a rotation around an axis perpendicular to the Z-axis, i.e. the angle between the molecular z-axis and the macroscopic Z-axis, and $ describes a rotation around the molecular z-axjs.

In addition to the macroscopic symmetry elements already' mentioned and used, there is also twofold rotational symmetry around the molecular z-axis (the long axis) and around any axis perpendicular to that axis. The order parameters describing the nematic phase can then be defined as the ensemble averages of the following invariants [6] (in fact these invariants appear in this form quite natural$, as will be seen below) which are normalized to unity :

F, = $(z; - $) = ;(cos2

e

^-+) I;, = x i - yg = cos 2 $ sin2 0

I; = z 2 - z 2

,

^-^-^{cos 2}^cp^{sin2 0} ^{(1 3)}

F - ' . 2

4 - J.XX - x;) - q(Y; - y;) =

4

cos 2 cp cos 2 @(I

+

^{c0s2 0).}

The ensemble average ( F, ) is the basic order parameter to describe the nematic phase ; the ordering corresponds to uniaxial alignment of the molecufar z-axes along the macroscopic Z-axis. In this uniaxial phase there may be biaxial order on a molecular scale described by ( F2 ). When both ( F, ) and ( F, ) are non- zero, for which it is necessary that both ( F, ) and ( F2 ) are non-zero, the nematic ordering is biaxial [6].

This phase biaxiality is due to molecular biaxiality. Doing the transformations for (XX),, (YY), and (ZZ), one finds :

The quantities 6 and E are molecular anisotropies formally introduced as 6, = i((zz) - $(xx

+

yy)),

and ^{E ,}= $(XX

+

y ~ ) ~ , reflecting their symmetry properties. Analogous to equation (6) the precise definition is

The quantity

Ji'

⁼^i(xx

+

yy

+

zz), is defined similarly. Next we calculate the transforms of (XZ), and of the dipole-quadrupole transition moments (X, Y Z ) , etc. After some lengthy calculations one finds

(XZ)i = '(3 6 - s cos 2 ^$)icos cp, sin 2 0,

(z(+)

X, XY), = - $(y, yy), = -

(r ⁺ *

[)i cos 2 $i sin2 qoi cos cpi sin 2 0,

(C(-) X, XY), = :(2 y

+

(c -

( E +

c) sin2 cp) cos 2 +), cos cp, sin 2 11 7) (CX, Yz)i = 2 ~i(F1 -

3

F3)i - (I - 5)i (F2)i

+

(CF," - 5F:)i

where F; = cos 2 $ cos 2 9 cos2 0, Fi = cos 2 $ cos 2 cp sin2 0 and 2

Fl +

F; ⁼2 F4 as defined in equation (13). The molecular quantities y, I; and

l

are introduced for their symmetry properties as

7, = (x, YZ - y, Z X ) , / ~ ,

li

⁼^(z,^xy), ^and

5,

⁼(x-, YZ

+

Y, zx-),/2

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TEMPERATURE DEPENDENCE OF THE PITCH IN LC C3-161

The precise definition is analogous to equation (7) given by :

and similar equations for

C

and

5

in combination with 6 and E. The molecular quantities y,

c

^and

⁵

are only non-zero in the case of molecules possessing neither a plane nor a centre of symmetry, i.e. optically active molecules.

The important thing to observe is the appearance of some new invariants all proportional to

cos cp sin 2 8 = 2 z, z, .

These clearly demonstrate the overall planar symmetry of the cholesteric phase. Twofold rotational symmetry around the fixed macroscopic X- or Z-axis destroys the asymmetric interaction which, as will be shown below, gives rise to the twist.

4. Calculation of the twist. - In order to illustrate the physical meaning of the odd terms in the dipole- dipole and dipole-quadrupole interaction we first consider the rather simple case of a purely planar structure in which the molecular z-axes are confined to the X-Z planes [I, 71 ; then z, = 0, i.e. cp = 0.

Consequently we take

II/

⁼0

+

^7c. One easily finds that (XX

+

Z Z ) , and (YY), are constants, that (Y, YY), = 0 and

(ZZ - XX), = (3 6, - E,) cos 2 Oi 2(XZ), = (3 Si - E,) sin 2 0,

(CX, YZ), = (2 y,

+

ci) cos 2 8, (1 9 ) ( c ( - ) X, X Y ) , = (2 yi

+

Ci) sin 2 8,

leading to

2 vij

v:~

⁼ ^{(3 6,}^-^{ci) (2 yj}

+

^{Cj) sin 2}^€Iij

.

(21)

t~

The twist angle, which can be defined as the average rotation ( a ) of the projection of the z-axis upon the X-Z plane (see Fig. 1) is then given by :

where a is a molecular dimension. The order of magnitude of ( cc ) has been discussed in reference [I].

Now we return to the general case. From figure 1

FIG. 1. - Orientation of the molecular z-axis, i.e. the long molecular axis, with respect of the macroscopic coordinate system X, Y, Z in terms of the angles 0 and cp and in terms of the angle cc = arctg (z,/z,) and the scalar z,, where z i

+

^z;

+

^z$⁼¹^;

Y denotes a rotation around the molecular z-axis.

we see that now a is not identical with 8 but determined by

sin a = z,/(z;

+

~ 3 ' / ~ , where z i

+

z; = 1 - r;.

So we find

sin 2 a = 2 z, z,/(l - 22,) =

= cos cp sin 2 8/(cos2 cp sin2 8

+

cos2 8)

.

If we introduce instead of 8 and ^cpthe new variables a

and 2 ; defined by

cos cp sin 2 8 = (1 - 22,) sin 2 a

=

2 z, z,

cos28 - cos2 cpsin28 = (1 - Z $ ) C O S ~ ~

=

zz - z$ = F 1 - i F j

(6)

where

the dipole-dipole and dipole-quadrupole interaction can be written as

m

Vpq =

C

^V:($,^,^:^r r,) sin 2 ncc

.

I ] = 1 (2 5 )

The potential in this form is similar to the potential proposed in reference [7] for a purely planar configuration in which only the variables y

=

cc and r ,

-

^{r ,}enter and where

_-;

⁼0. In view of the limit discussed above for a purely planar configuration, this model seems rather artificial as the appearance of the higher harmonics is

h

fact due to the three-

dimensionality of the problem ; we find that (V,),

, ,

^cc^r;. Using equation (24) and (25) all moments of sin 2 cc can be calculated from which ( a ) can be obtained. This method is rather cum- bersome and tedious.

It is far more practical to use the fact that the cholesteric can be considered as a continuous rotating nematic phase [3, 4, 71. The corresponding rotating coordinate system X', Y', 2' can be defined by :

X' = X cos Qr,

+

Z sin Qr,

Y' = Y (26)

Z' = - X sin Qr,

+

Z cos Qr,

where the y-axis is the helix axis and Q = 2 n/p, with p being the pitch of the helix. We now apply

this transformation to the matrix elements ( X X ) , ( X , Y Z ) etc. occurring in equation (8) and equation (9).

Since these elements transform like the corresponding coordinates, one finds :

( X X - Z Z ) i = ( X X - Z Z ) ; cos 2 Qr: - 2 ( X Z ) ; sin 2 Qri 2 ( X Z ) , = ( X X

-

22); sin 2 Qr:

+

2(XZ)l cos 2 Qri ( E ( - ) X, X Y ) , = ( F - ) X, X Y ) ; cos 2 Qr: - (EX, Y Z ) ; sin 2 Qr:

(EX, Y Z ) , = (8'-I X, X Y ) , sin 2 Qr:

+

(EX, Y Z ) f cos 2 Qri all other terms being independent of Q.

We now substitute these expressions for the corresponding matrix elements in equations (8) and (9) for V:, and V$ respectively. For the Q dependent part of V2P one finds :

V f p ( Q ) = ( ( Z Z - XX ); ( Z Z

-

XX); $. 4 ( X ~ ) i ( X Z ) j)

vij

cos 2 Qr?

+

( ( Z Z - X X ) ; (2 XZ)j

-

(2 X Z ) ; ( Z Z - XX);) V" sin 2 ~ r :

.

₍₂₈₎

Here the primed quantities are defined with respect to the rotating coordinate system. The local equilibrium configuration with respect to this coordinate system corresponds to nematic ordering, for which

( ( Z Z - X x ) ; ) = ( ( Z Z -

xx);

) ,

independently of position. In this case one also may demand that the averaged value of the odd terms vanish identically, i.e. ( (XZ)" ) = ( ( X Z ) ; )

=

0. Hence the averaged value of V:',(Q) in a molecular field approxi- mation can be defined by :

( V p p ( Q ) ) = ( ( Z Z - XX)' )'.

S'

dry V ( r y ) cos 2 Qr,

The prime on the integral restricts r, to values not running over the mutually excluded volume of two molecules. Proceeding in the same way we can define the averaged value of V p q = V:,

+

^V&^by^:

( V p q ( Q ) ) = 4 ( ( Z Z - X X ) ' ) ( ( Z X ,

YZ)')j'dry-•

^{V ( r}⁾ sin 2 ~ r ,

.

Y Y

Since in a molecular field approximation the entropy does not depend on the spatial variation of the direc- tion of alignment [4, 71, the value of Q that minimizes the free energy can be found by minimizing the average potential energy. Expanding the sine and cosine up to order Q 2 one finds :

Q

⁼ ( ( Z X , Y Z ) ' )

-

1 ( ( Z Z

-

XX)' ) a2

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TEMPERATURE DEPENDENCE OF THE PITCH IN LC

where a is a molecular dimension defined by

Using equations (14) and (17), the equation for Q can finally be written as :

5. Discussion. -Our final result is an equation for Q = 2 n/p, where p is the pitch which is charac- teristic of the cholesteric or twisted nematic phase, in terms of molecular quantities and of the order parameters ( Fi ) which are characteristic of the corresponding untwisted nematic phase. For a purely planar configuration the above equation reduces to equation (22) with ( Bij ) = 2 aQ. When both ( F, ) and ( F4 ) are zero the result of reference [3] is retained, which accounts for biaxiality on a molecular scale described by ( F, ). When the local nematic ordering is biaxial we have to use the full expression given by equation (32). To our knowledge no macroscopic biaxiality in nematic phases has been reported.

Local biaxiality in cholesteric phases is rather difficult to observe directly because of the overall uniaxial symmetry. Nevertheless cholesteric phases are biaxial on a local scale, at least of the order of (Qa)', due to the dipole-quadrupole interaction [4]. It seems quite plausible that this biaxiality can be reinforced by the dipole-dipole interaction since the constituting molecules can be considered to be planar, i.e. E f 0.

Since cholesterics do not have a corresponding untwisted nematic phase, it seems worthwhile to make a kind of corresponding untwisted nematic

phase by making a compensated mixture of two natural cholesteric compounds. Whether phase biaxiality exists or not can then be detected directly. This point is not only interesting in itself but can be important for the understanding of the temperature dependence of the pitch. As to the magnitude of the order parameters ( F3 ) and ( F4 ) nothing is known except for the results of the model calculations given in reference

[q.

There it is was found that when there is biaxiality, ( F4 ) and therefore also ( Fi ) are large compared to both ( F, ) and ( F, ) and consequently also

<

^{F i ).}The temperature dependence of the pitch should then be governed by the ratios ( F,

)/(

F , ) and ( F:

)I(

F,

).

However since we have no knowledge of the relative magnitude and sign of the molecular quantities y,

5

and it seems almost impossible to make any prediction. As a function of temperature the pitch may increase, decrease or do both successi- vely, depending on the relative magnitude of the various biaxial order parameters. We hope that systematic experiments on the temperature depen dence of the pitch will give more conclusive results that will enable us to interprete and test the theoretical description.

References

[I] GOOSSENS, W. J. A., Mol. Cryst. Liqu. Cryst. 12 (1971) 237. [5] SLATER, J. C., Quantum Theory of Molecules and Solids (McGraw [2] STEGEMEYER, H. and FINKELMAN, H., Naturwissenschaften 62 Hill, Book Company, New York) 1965, vol. 11.

(1975) 436. [6] STRALEY, J. P., Phys. Rev. A 10 (1974) 1801.

[3] VAN DER MEER, B. W. and VERTOGEN, G., Phys. Lett. 59A [7] LIN-LRJ, Y. R., Yu MING, SHIH, CHIA-WEI WOO and TAN, H. T.,

(1976) 279. Phys. Rev. A 14 (1976) 445.

[4] PRIEST, R. G . and LUBENSKI, T. C., Phys. Rev. A 9 (1974) 893.

TEMPERATURE DEPENDENCE OF THE PITCH IN CHOLESTERIC LIQUID CRYSTALS : A MOLECULAR STATISTICAL THEORY

HAL Id: jpa-00218728

https://hal.archives-ouvertes.fr/jpa-00218728