Theory of flow alignment in biaxial nematics and nematic discotics

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Theory of flow alignment in biaxial nematics and nematic discotics

H. Brand, H. Pleiner

To cite this version:

H. Brand, H. Pleiner. Theory of flow alignment in biaxial nematics and nematic discotics. Journal de

Physique, 1982, 43 (6), pp.853-858. �10.1051/jphys:01982004306085300�. �jpa-00209462�

Theory ^{of flow} alignment ⁱⁿ biaxial nematics and nematic discotics

H. Brand (*) ^{and H. Pleiner}

FB 7, Universität Essen, ^D-4300 Essen, W.-Germany (Reçu ^{le 21} janvier ^1982, accepté ^le 16 fevrier 1982)

Résumé.

Nous présentons ^{une théorie} microscopique de l’orientation des cristaux liquides nématiques (et nématiques discotiques) biaxes dans une expérience d’écoulement. Nous proposons trois configurations géomé- triques ^très simples pour ^mesurer les trois paramètres réversibles qui ^sont affectés. Le cas particulier ^des nématiques discotiques ^{uniaxes est discuté} séparément. ^{En outre, nous} discutons d’effets similaires dans les smectiques ^{C et} cholesteriques.

Abstract

We present ^a microscopic theory for the flow alignment in biaxial nematics and nematic discotics and we propose three simple geometrical configurations ^{to measure} the three reversible parameters ^involved.

The uniaxial special ^case for nematic discotics is discussed separately. In addition we discuss similar effects in smectics C and cholesterics.

Classification

Physics ^Abstracts

61.30

03.40G - 47.90

1. Introduction.

The interest in the theoretical

description of biaxial nematics increased considera-

bly since the first (lyotropic) biaxial nematic liquid crystal ^{has been} produced ^last year [1]. ^{In the} present letter we will discuss in some detail one aspect ^{of the}

hydrodynamic description [2-5] of biaxial nematics

(and ^nematic discotics) namely ^{the flow} alignment

under an external shear flow. This experimental ^arran-

gement is well known for uniaxial nematics [6] ^and

serves there (via determination of the alignment angle) ^to determine the only nontrivial (i.e. ^different

from 1 or 1/2) reversible transport parameter ^of uniaxial nematics. On the other hand this coefficient

can be calculated by ^means of Poisson brackets [7].

In section 2 we will _carry out the same procedure ^for

biaxial nematics (and ^nematic discotics) and, ^{as it}

turns out, the three reversible transport parameters

can be determined separately by ^the study ^{of three} simple geometric orientations of the preferred ^direc-

tions n and m of biaxial nematics. These coefficients

(at ^least the instantaneous parts ^of them) ^are expressed by ^{the four} microscopic ^{« order} parameters >> ^which

are necessary for a consistent description ^of the sys- tem [8, 9]. ^Of course, the uniaxial situation is always

contained as a special ^{case in our} presentation. ^In

section 4 we will discuss separately ^{the flow} alignment

in uniaxial nematic discotics exploiting previous ^work

by D. Forster [7] ^{on the usual} (rod-like) type of uniaxial nematics. Discotic liquid crystals ^{have been} produced

for the first time in 1977 by S. Chandrasekhar et al. [10]

and since then columnar phases ^of hexagonal ^and rectangular symmetry ^{as well as} nematic like and cholesteric like discotic liquid crystals ^{have been} produced [11-15]. ^{In section 5 we} briefly investigate

a shear flow experiment ^{which can be used to deter-}

mine one of the two reversible parameters ^{which enter} the reversible currents [16, 17] of smectics C and we

clarify ^the microscopic ^nature of the variable charac-

terizing ^the spontaneously broken rotational symme- try ⁱⁿ smectics C. In addition we calculate in section 6 the instantaneous part of the coefficients _gkl entering

the hydrodynamic description of cholesterics. These terms, which couple ^the displacement ^{field to the} symmetrized gradients ^{of the} velocity field, ^{have been}

introduced by the authors recently [17].

2. Flow alignment in biaxial nematics.

The non-

linear hydrodynamic equations for biaxial nematics have been derived for the first time by ^the present authors _very recently ^{in detail} [2]. ^{As has been} pointed

out, there, the reversible currents for the three variables

characterizing ^the spontaneously broken rotational symmetry ^{in real} space take the form

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004306085300

854

with

In the following ^we only ^need equations (2.1)

and (2.2). ^For the other reversible ^currents as well

as for the irreversible and static contributions we refer to reference [2]. ^{As it turns out it} proves ^to be conve-

nient to introduce Eulerian angles ^to characterize the flow alignment angles. Using standard notation of textbooks on mechanics we have

We consider in the following ^a stationary ^shear flow,

which we can assume without loss of generality ^{to lie}

in the y - ^z plane (i. e. VyVz # 0). Using (2 .1 )-(2 . 5)

we then find for _{al, Ct.2} and _as (in ^{order to} guarantee

stationarity)

Since the general expressions (2.6)-(2.8) ^{are not}

very illuminating ^{we discuss three} simple geometries :

where the equations ^for a2 and _as are satisfied identi-

cally.

The geometrical picture corresponding ^{to case} (i) ^{has been} plotted ⁱⁿ figure ^1.

(ii) (p = 0 = 00 :

The equations ^for al and _as are again satisfied identi-

cally ^{and we} get the situation shown graphically ^on figure 2.

Fig. ^1.

^Flow alignment angle 0, ^{which measures the}

coefficient _al (2.9); ^the preferred direction n lies in the shear plane, ^{while m is} orthogonal ^{to it.}

The direction m is orthogonal ^{to the shear} plane

and the vector n is titled by ^the angle 0, ^{which is} determined by the coefficient _al (2. 9).

Fig. ^2.

^Flow alignment angle 9 ^{which measures the}

coefficient _a2 (2. 10) ; ^both preferred directions, n and m,

lie in the shear plane.

As is immediately ^{seen from} figure ^{2 both} preferred

directions m and n lie in the shear plane ^{and the} corresponding alignment angle ⁰ is determined by ^the

coefficient _a2 (2. 10).

In the third case the equations ^for al and _a2 are

satisfied identically ^whereas a5 is determined by

The corresponding geometrical situation is plotted

in figure ^3.

Fig. ^3.

^Flow alignment angle T which measures the coefficient as (2.11); ^the preferred ^{direction m} lies in the shear plane, ^{while n is} orthogonal ^{to it.}

In the third case n points ^{out of the shear} plane and,

m is oriented in the plane according ^to (2 .11 ).

On the other hand, ^{it is} possible by measuring ^the alignment angles ⁱⁿ the three different geometries

to obtain measured values of the reversible transport parameters ^{ai, a2 and} a5 by ^{means of} equations (2. 9)- (2 .11 ).

3. Microscopic description.

^{In the next} step ^we relate the transport parameters al, a2 and _as to the

microscopic ^order parameters using ^a technique ^which

is quite ^{similar to that} proposed for uniaxial nematics

by ^{D. Forster} [7]. ^The hydrodynamic variables cha-

racterizing the broken rotational symmetry ^{can be} expressed by the elements of the quadrupole ^tensor Rij ^{in the following way} (n°/%Z == ê3, mO/êy- == ^ê2)

where Qij = Rij > ^{and where} Rij ^{is the operator for}

the quadrupole ^tensor [7]. Using ^these definitions for the variables bni, 6m; ^{we have the} commutation relations (or rather Poisson brackets) ^{with the} opera- tor for the angular ^{momentum L}

Generally, the coefficients _{cxl, Ct.2} and _{a 5} contain two contributions : an instantaneous one arising

from the frequency matrix and a collision dominated one, which arises from the non-hermitian part ^{of the}

memory matrix [7, 18]. ^{Then we} find for the instan- taneous contributions to the coefficients _{«1, a2} and _as

where _g is the linear momentum density and Q ^{is the}

sum of the moments of inertia Ii.

As for uniaxial nematics [7] the non-instantaneous collision-dominated contributions to al, a2 and _a5

are expected ^{to be} very small. In a last step ^{we connect} the quantities Qij ^{to the four} microscopic ^order parameters (S, A, q, ^and il’) ^{which are} necessary ^to guarantee ^an appropriate microscopic description

of biaxial nematics [8, 9].

Using the work of R. G. Priest and T. C. Lubensky

we have in detail

with

Taking ^{into account} the relation [7]

/

we have B ’e

856

Combining equations (3.3)-(3.6) ^we obtain the final result (expressing ^the phenomenological ^coeffi-

cients _{al, a2} and a5 by ^the microscopic ^order para- meters S, L1, r¡ ^and r¡’)

with

Thus we have outlined in this section ^on biaxial nema-

tics (or ^nematic discotics) ^a possibility ^to get ^from flow alignment experiments informations about the

microscopic ^order parameters ^{and vice versa.}

4. Flow alignment ^{in uniaxial} nematic discotics.

Recently nematic discotics have been synthesized

and flow alignment measurements are possible [19].

It is the _purpose of the present ^{section to show that} the theory ^{of flow} alignment for uniaxial nematic discotics is implicitly ^{contained in the} previous ^cal-

culations of D. Forster for uniaxial rod-like nematics

[7]. Furthermore, ^{we discuss an} experimental ^scheme appropriate for uniaxial nematic discotics. For the

flow-alignment angle D. Forster derived the result

with

This expression ^is also contained in our general ^for-

mulas (3. 7) ^and (2 . 9)-(2 .11 ), if, there, the biaxia-

lity is switched off, ^i.e. al

a2 = - 2 A, as = 0,

L1 = q = 11’ = 0.

For extreme rod-like molecules (1 - 0)

À

3 S/(2 ⁺ S) and the flow alignment angle ⁰

varies between 7r/4 (for S

0, ^no ordering) ^{and zero} (for S

1, ^full ordering), ^{In the case of extreme disc-} shaped ^molecules (I, ^--+ 0) ^{one obtains A}

³ S/(4 - S)

and the flow-alignment angle varies between 7r/4 (for S

0) ^and 7r/2 (for ^S

1). Thus, ^{for discotics}

the flow-alignment angle increases for increasing ordering ⁱⁿ contrast to (rod-like) nematics. Of _course, for full ordering ^S

¹ the molecules are horizontal,

i.e. 0

900 for discotics and 0

00 for (rod-like)

nematics (cf Fig. 4). The results for S

1 (e.g. À = - 1

for extreme discotics) ^were previously given by

Fig. ^4a.

^Flow alignment angle ^for rod-like molecules

(IIi It ^{N 5, S z} 0.8).

Fig. ^4b.

^Flow alignment angle ^for disc-shaped ^molecules (It/ II ^N 5, ^{S =} 0.8).

Volovik [20]. However, ^S

^{1 is never reached in}

nature. In figure ^{4 we} sketch the experimental ^situa-

tion for realistic values of S.

5. Flow alignment in smectics C.

As is well known [6, 16] smectics C has two variables charac-

terizing spontaneously broken continuous _symme- tries : one characterizing the broken translational symmetry along ^the layer ^normal (called 3-axis in the

following) ^of the smectic layers (like in smectics A)

and one characterizing ^a rotational symmetry ^about this 3-axis.

If the molecules are assumed to lie in the 1-3 plane

in equilibrium, the rotational degree of freedom is described by fluctuations of n which are perpendicu-

lar to the 1-3 plane. The reversible part ^{of the} dynamic

equation ^{for that} variable reads [16, 17.]

(In ^reference [17] the coefficients )t(1), ^{A(3) are called}

-

2 _{a2, -} 2 al, respectively,)

By inspection ^of equation (5.1) it becomes obvious

immediately ^{that a} stationary ^flow alignment experi-

ment is possible only ^for V 1 V 2 =1= ^{0 or} V 2 V 1 =1= 0;

thus only ^)t(3) (but ^not )t(1») ^is measurable by ^means

of a flow alignment angle 0, ^{and we find}

In order to relate ;}1) and A(3) to the microscopic

order parameters ^{one must first} express the hydro- dynamic ^variable bn2 ^{in terms of the} operators Rij.

Thereby ^{one has to take care} of the fact that

( Rij > ^_-- Qij ^{is not diagonal} in smectics C [6].

Using the relation

one finds that

(with ç

^{Q13(Q22 -} Q33)-1) ^{is the} appropriate expression because it leads to the canonic commu-

tation relation

The existence of Q13 =A ⁰ is characteristic for smec-

tics C (compared ^{to nematics or smectics} A).

Using (5.4) ^we find for the instantaneous parts ^of A (1) and A(3) the relations

and

Thus it becomes possible ^to give ^a connection between the flow alignment parameter  (3), ^{which is acces-}

sible to experiment, ^{and the} quantities Qij (espe- cially Q13 !) ^{which can} be calculated by fully ^micro- scopic ^theories.

6. Remark on a reversible parameter ^{in choleste-}

rics.

Recently ^the present authors have shown [17]

that in the hydrodynamic equation for the variable Ri characterizing ^{the broken} symmetry ^of cholesterics, symmetry arguments ^{allow for two} reversible trans-

port coefficients _gi and _g2 which were not taken into account previously [16, 21], ^i.e.

(p - pitch ^{axis; 2} trio

pitch).

Although ^the phenomenological parameters g 1 and g2 ^{are not} measurable by ^flow alignment experiments [22], they show ^some similarities with the reversible transport parameters in nematics and smectics C described above. We will derive a microscopic expres- sion for _gl and _g2 in the same spirit ^{as we have done}

above for the A’s and a’s in nematics (discotics) ^and

smectics C. The spontaneously ^broken symmetry ⁱⁿ cholesterics is the translational symmetry along ^the pitch axis p, ^{which can} equally well be described as a rotation about this axis. The variable Ri has, ^there- fore, the Poisson bracket relations

(Pi, Li ^are total linear momentum and total angular

momentum, respectively). ^The frequency matrix ele- ment

describes the instantaneous reversible dynamic ^cou- pling ^{of R and the momentum} density g (or ^{the velo-} city V) given by equation (6.1). Using ^{a coarse} grained

version of R(r) namely

in order to allow Fourier transformation (cf ^Ref [23]),

we obtain

Equation (6.2) reproduces ^{the structure of} equa- tion (6 .1 ) and relates (the instantaneous part of) ^the

coefficients _{g 1} and _g2 to the microscopic ^order para-

meter Am > Pm. The difference _{gl - g2} is non zero,

therefore, only ^{due to} the collision dominated parts of _{g 1} and _g2, which we have neglected ^above.

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