Remarks on the flow alignment of disc like nematics

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Remarks on the flow alignment of disc like nematics

T. Carlsson

To cite this version:

T. Carlsson. Remarks on the flow alignment of disc like nematics. Journal de Physique, 1983, 44 (8),

pp.909-911. �10.1051/jphys:01983004408090900�. �jpa-00209673�

909

Remarks on the flow alignment of disc like nematics

T. Carlsson

Institute of Theoretical Physics, ^Chalmers University ^of Technology,

S-412 96 Göteborg, ^Sweden

(Reçu ^le 22 fevrier 1983, révisé le 11 avril, accepté le 1 S avril 1983)

Résumé. 2014 Dans le cadre de la théorie hydrodynamique ^de Leslie, ^{Ericksen et} Parodi,

examine la nature de l’orientation des cristaux liquides nématiques ^dans

expérience d’écoulement On montre que l’alignement

ne

peut

produire que dans

intervalle d’angle ^limité. L’application

^des nématiques discotiques ^montre

que seule l’une des deux théories récemment discutées

l’origine ^de l’alignement d’écoulement [2-4] ^est

^bon

accord

la théorie de l’hydrodynamique ^des nématiques.

Abstract.

The nature of the flow alignment of nematic liquid crystals ^is discussed within the hydrodynamic theory ^of Leslie, Ericksen and Parodi. It is shown that there is

limited range of angles ^{for which flow} alignment

can occur.

With this

starting point ^{the flow} alignment ^{of disc like} nematics is discussed. It is proved ^that only

one

out of the two models of flow alignment recently discussed in the literature [2-4] ^is consistent with the hydro- dynamic theory of nematics.

J. Physique ⁴⁴ (1983) ^{909-911 AOÛT} 1983,

Classification

Physics ^Abstracts

61. 30G

1. Introduction.

Since the existence of discotic liquid crystals ^was

first reported by Chandrasekhar et al. [1] ^{there have} appeared ^an increasing number of _papers, both theoretical and experimental, concerning ^these sys- tems. Although ^{there have been no} papers reporting

an experimental study of the flow properties ^{of the}

discotics (at ^{least to the} knowledge ^{of this} author)

two theoretical papers have recently appeared [2-4], concerning ^flow alignment ^{of nematic} discotics under shear. The flow alignment predicted by ^Carlsson [2]

is shown in figure 1 a and the prediction by ^Brand

and Pleiner [3] is shown in figure ^{lb. In an erratum}

to their paper [4] ^{Brand and Pleiner however} modify

their prediction ^to be consistent with that of figure ^la.

It is the aim of this paper ^to show that it is only ^the

situation pictured ⁱⁿ figure 1 a that is consistent with the hydrodynamic theory of nematic liquid crystals developed by ^{Leslie, Ericksen} and Parodi (the ^LEP- theory). It is also shown that although ^{the flow} align-

ment pictured ⁱⁿ figure ^{1 b} represents ^an equilibrium

solution of the hydrodynamic equations, this solution is not physically feasible, because it describes an

equilibrium ^{which is} unstable, ^{so a} system ^{with this} configuration would relax to the situation pictured ⁱⁿ figure ^{la due to} fluctuations.

Fig. ^1.

Predictions of the flow alignment ^{of nematic}

discotics which have been discussed in the literature [2-4].

We show in this paper that figure ^{1 a} represents

stable equilibrium ^while figure ^{1 b} represents

^unstable

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01983004408090900

910

2. Flow alignment

In this section we discuss the flow alignment ^of

nematics in the spirit ^{of the} LEP-Theory. The results

displayed ^{below were} originally ^discussed by ^Erick-

sen [5] ^and recently ^reviewed by ^Leslie [6]. ^{We intro-}

duce coordinates as shown in figure ^{2. We} study

a nematic under shear. The nematic is confined between two parallel glass plates, ^the upper plate moving in the x-direction with

velocity

Assuming

the director (denoted by n) ^to remain in the plane

of shear

introduce a coordinate 0 to describe it, where 0 is the angle between the director and the normal to the glass plates ^{and is counted as} positive

for a clockwise rotation. At the present stage ^{we have}

not restricted ourselves to the study of either rod- like or disc-like nematics. The meaning of the director in these two different cases is shown in figure ^{2. The} equivalence ^{of n with -}

^{allows us to} study ⁰ only

in the interval [ - 900, 900].

Fig. ^2.

Definition of coordinates in the present ^work.

The lower plate ^{is at} rest, ^while the _uppear

is moved in the x-direction with

velocity

The z-direction is normal to the plates ^{and the} y-axis ^is pointing inwards. The angle

which the director makes with the z-axis is denoted by 6, t4king 0

positive ^for

clockwise rotation. The direction of the y-axis ^and the reference direction of 0

chosen in such

way that

positive torque ry ^{would tend to rotate}

the director in the positive 0-direction.

When we study ^flow alignment ^{we must consider}

two of the six Leslie viscosities; a2 and (X3. It is impor-

tant to notice the following inequality [7] ^{which must}

be fulfilled by ^{these :}

The viscous torque, Fy, ^{exerted on} the director in the stationary ^state (i.e. 010t

0) ^is given by

where u’ is the local shear. Flow alignment ^is pos- sible when r) ^{is zero} giving ^the following ^solution

for the flow alignment angle 0.

We can divide the solutions of equation (3) ^into two different ^cases. Either both a2 and a3 ^are negative

(case a)

^both

positive (case b). ^{The two} signs

in the solution of equation (3) represents ^one stable and one unstable solution. It is _easy to show (see ^for

instance Ref. [2]) that when a2 and 01533 ^are negative

it is the + sign ^which gives the stable solution. When a2 and a3 ^are positive ^however it is the - sign ^which

has to be chosen to give the stable solution.

Case a : 01532 and a3 ^are negative.

^{In this case the} inequality (1) implies that I 01532 I 1> I a3 1. ^{As is men-}

tioned earlier the stable solution of equation (3) ^is

the one given by ^{the +} sign. ^{This means that the}

allowed values of tan On will be in the interval [1, oo[

and we thus get :

Case b : a2 ^and a3

positive.

^{In this case the} inequality (1) implies that a3 > a2. The stable solu- tion of equation (3) ^{is now} given by ^{the -} sign. ^The

allowed values of tan 8n ^{will now be in to} the interval

[ - 1, 0] ^{and we thus} get :

We thus draw the conclusion that flow alignment

is possible either with a value of 9n ^{between - 45°}

and 0° (when a2 and a3 ^are both positive) ^{or between}

45° and 90° (when _{a2 and a3} ^{are both} negative). ^This

is illustrated in figure ^3.

Fig. ^3.

^Possible values of the flow alignment angle ^within

the LEP-theory (shaded areas). ^{A flow} alignment angle

between - 45° and 0° is consistent with positive ^{values of}

a2 and a3, while

flow alignment angle ^{between 450 and}

900 is consistent with negative ^values of a2 and _01523. The

areas

not shaded in the figure represent ^the unstable solu- tions of equation (3) ^and

^not physically feasible. The definition of 9 is shown in the upper left part ^{of the} figure.

3. Flow alignment ^{of disc-like nematics.}

Recently ^two papers have appeared predicting ^the

nature of flow alignment ^{of disc-like} nematics. The

predictions given ^{in these two papers are} illustrated in figure ^1. By ^{the aid of the} discussion in section 2

(the ^{results are} summarized ⁱⁿ Fig. 3), ^we immediately

see that the situation pictured ⁱⁿ figure ^{1 b} represents

an unstable equilibrium while that of figure 1 a repre-

sents a stable one. We also ^see that disc-like nematics

911

demand positive ^values of the Leslie viscosities _a2 and a3 (in ^{the case} where flow alignment occurs).

The reasoning ^{which led to the} proposal ^of figure ^la

was based on

calculation by ^Volovic [8], ^{and is}

discussed in reference [2]. ^Here

will however mention

calculation of the flow alignment angle

for uniaxial nematics by ^Forster [9] and show that the prediction ^of figure ^{1 a} is consistent also with this. With the choice of 0 made in this work, ^the calculations by Forster establish that the flow align-

ment angle ^is given by

S is the order parameter ^{of the} system and I, and It

are the components of the molecular moments of inertia evaluated in the system ^of principal ^molecular

axes as defined by Forster. We can distinguish ^two special types ^of molecules : rod-like molecules where

It

0 and disc-like molecules where ( - ^0.

Rod-like molecules (It

0).

Putting It

^{0 in equa-}

tion (6) ^we get

By letting ^S increase from 0 to 1

see that tan2 On

lies in to the interval [1, oo]. ^This corresponds ^to

case a mentioned in section 2 and rod-like nema-

tics should be described by negative values of _a2 and a3 and have ^a flow alignment angle 0. ^which

lies between 450 and 900, ^{where 450} corresponds ^to complete ^disorder (S

0) ^{and 900} corresponds ^to

full ordering (S

1). ^For realistic ^{values of} S, On ^lies

close to 90° which implies I (X2 1 » I (X3 according ^to equations (3) ^and (7). This is in accordance with all

experiments reported ^{on the flow} alignment ^{of rod-}

like nematics. (We ^{do not} here consider the rather

exceptional ^cases of rod like nematics where flow

alignment ^{does not} occur.)

Disc-like molecules (II

0).

^{If we} put II

^{0 in} equation (6)

get

By letting ^S increase from 0 to 1

see that tan2 On

lies in the interval [1, 0]. ^{This is case} b discussed in section 2. Disc-like nematics therefore should be described by positive ^{values of} a2 and a3 and the

flow alignment angle should lie between - 450 and 00 depending ^{on the order} parameter. ^{For values} of S close to one, 8" is small and negative ^which implies a3 > IX2 according ^to equations (3) ^and (8).

(We only discuss discotics showing ^flow alignment

and do not pay attention to the possibility ^{of tum-} bling, which resembles the situation that occurs for

some of rod-like nematics.)

4. Conclusions.

Two models of the nature of flow alignment ^of

disc-like nematics have recently been discussed in the literature [2-4]. We have shown that the model

suggested ⁱⁿ figure ^{1 a} represents

^stable equilibrium

while that of figure ^{1 b} represents ^{an unstable one.}

We have also shown (as ^far

^flow alignment ^is concerned) ^{that as a} consequence of Forster’s cal- culations [9] ^{rod-like nematics must} be described by negative values of a2 and a3 while the disc-like ^nema- tics must be assigned positive values of both these constants. This is in accordance with the earlier

predictions by ^Carlsson [2].

In figure ⁴

have shown the flow alignment ^of

disc-like and rod-like nematics, ^{and we see that the} physical situations in the two cases are quite ^similar.

In figure ^{4a we have rods} lying ^almost parallel ^{to the}

flow direction while in figure ^{4b we} have discs which

are oriented with the disc planes ^almost parallel ^to

the same direction (this is the situation occurring ^for

a

value of the order parameter ^{which is not} very much less than one). ^The similarity ^{of the two cases}

should not be surprising ^{if one} imagines ^{some kind}

of steric force to be responsible ^{for the flow} alignment.

Fig. ^4.

^Flow alignment angle, tan On

a2 ja3, ^for

rod-like molecules. Both oe2 and a3

negative ^and I r:J.2 1 ’> a3 I (left). Proposed ^flow alignment angle, tan 0.

a2 ja3, for disc like molecules. Both a2 and a3

positive and 0(3 > OC2 (right).

References,

[1] CHANDRASEKHAR, S., SADHASHIVA, B. K. and SURESH, K. A., Pramana 9 (1977) ^471.

[2] ^{CARLSSON, T., Mol.} Cryst. Liq. Cryst. ⁸⁹ (1982) ^57.

[3] ^{BRAND, H. and} PLEINER, H., ^J. Physique ⁴³ (1982) ^853.

[4] ^{BRAND, H. and} PLEINER, H., ^{Erratum to} reference 3.

J. Physique (in press).

[5] ERICKSEN, ^J. L., Kolloid-Z. 173 (1960) ^117.

[6] LESLIE, ^F. M., ^Mol. Cryst. Liq. Cryst. ⁶³ (1981) ^111.

[7] LESLIE, ^F. M., Quart. ^{J. Mech.} Appl. ^{Math. 19} (1966) ^357.

[8] VOLOVIC, ^G. E., JETP Lett. 31 (1980) ^273.

[9] FORSTER, D., Phys. Rev. Lett. 32 (1974) ^1161.