Theoretical and experimental study of the static behaviour of a nematic liquid crystal near the Freedericksz transition

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Theoretical and experimental study of the static behaviour of a nematic liquid crystal near the

Freedericksz transition

C. Oldano, E. Miraldi, P. Taverna Valabrega

To cite this version:

C. Oldano, E. Miraldi, P. Taverna Valabrega. Theoretical and experimental study of the static be-

haviour of a nematic liquid crystal near the Freedericksz transition. Journal de Physique, 1984, 45

(4), pp.755-759. �10.1051/jphys:01984004504075500�. �jpa-00209807�

Theoretical and experimental study ^{of the} static behaviour of a nematic liquid crystal ^{near the} Freedericksz transition

C. Oldano (*), ^E. Miraldi and P. Taverna Valabrega

Dipartimento di Fisica del Politecnico, ¹⁰¹²⁹ Torino, Italy

(*) Gruppo Nazionale Struttura della Materia del C.N.R., ^U.R. 24, Italy

(Reçu ^{le 27} juillet ^1983, révisé le 20 octobre, accepté le 30 novembre 1983 )

Résumé.

Nous étudions la déformation statique ^{d’un film} nématique homéotrope

voisinage de la transition de Freedericksz en fonction de l’intensité et de la direction du champ magnétique. ^{Nous montrons} qu’une ^tech- nique interférométrique

^incidence oblique permet de contrôler l’orthogonalité ^du champ magnétique ^{et du}

directeur avec

précision ^de 0,01 degré ^et d’éliminer pratiquement ^{cette source d’erreur} dans les

de

champ critique.

Abstract.

We discuss the static distortion of

homeotropic nematic film in the neighbourhood ^{of the Free-}

dericksz transition as

function of the strength ^and of the direction of the magnetic field. We show that an inter- ferometric technique ^at oblique incidence allows the orthogonality between the magnetic field and the liquid crystal ^{director to} be checked with an accuracy of the order of 0.01 degrees, and allows this source of

in the measurements of the critical field to be practically eliminated.

Classification Physics ^Abstracts 61.30G

1. Introduction

In a preceding ^paper [1] ^an optical technique ^has

been described, which allows _very accurate measure- ments to be made of the director profile ^{of a nematic} liquid crystal ^{under the} distorting effects of externally applied ^{electric or} magnetic fields. The technique ^is

based on the interference effect between ordinary

and extraordinary rays propagating ^{in a} liquid crystal sample placed between crossed polarizers, making

an angle of x/4 ^with respect ^to the incidence plane.

The use of an oblique geometric arrangement provides

both an increase of the sensitivity ^{of the method and}

the possibility ^of evidencing the existence of statio- nary points ^{in the} light intensity ^{curves, which} permits

very accurate measurements of some physical quan- tities involved in the distortion process.

In the present paper, this technique ^{will be} applied

to the study of the Freedericksz transition in ^a nematic

liquid crystal.

This transition has been actually widely ^studied using several methods. It will be shown here that the interferometric method in oblique geometry, pre-

sently proposed, yields results which are much more

accurate and clear than the previously published ^ones.

In particular, it allows to check, ^with great accuracy, the orthogonality between the magnetic field and the

liquid crystal director, ^{which is a} common source

of error in the measurement of the critical field Hc.

In section 2 it is shown that the study of the direc- tor?s distortions in the proximity of the Freedericksz transition can be performed ^{on the basis of an} appro- ximate expression ^{similar to that used} by ^{Landau for}

a 2nd order phase transition in the presence of ^an external field This expression ^{is used to calculate}

the effect, ^on the evaluation of the critical field H,,

of a small mismatch angle between the magnetic ^field

and the normal to the crystal ^director.

In addition it is shown that an accurate measure- ment of Hc ^{can be obtained even} in the presence of ^a mismatch angle by ^{means of} measurements of ’ I vs.

H with a best fit proGedure.

Furthermore, ^{in the case of a} splay ^{+ bend defor-} mation, ^{it is} possible, by using ^a suitable tilted field,

to find both the elastic constants K11 ^and K33.

Finally, in section 3, ^we report experimental ^results relating ^to MBBA nematic liquid crystals ^homeotro- pically aligned, ^{which are in} extremely good agree- ment with the computed ^values.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01984004504075500

756

2. Theoretical discussion.

The free _energy for unit surface of an homeotropi- cally aligned liquid crystal in the presence of a magne- tic field H, making ^an angle OH

x/2 + EH ^with respect ^{to a z axis} perpendicular ^{to the} boundary surfaces, ^is given by

where D is the thickness of the sample, xa ^{is the magne-}

tic anisotropy, 0(z) ^{is the} angle between the liquid crystal director and the normal to the sample ^surfaces

in _any given point and q ^is equal ^to (K I I -K 3 3)/K 3 31 K11 ^and K33 being the elastic constants of splay ^and

bend respectively.

The case where OH

x/2 ^has already ^been widely

studied [2, 3]. ^{The usual} approximation ^{near the criti-}

cal field He consists in assuming ^that 0(z) ^{is a sinusoidal}

function. With this assumption (1) gives for the free energy ^an expression ^{similar to} the Landau _expan- sion in the neighbourhood ^{of a 2nd} order transition.

The amplitude Om of this sinusoidal function plays

the role of the order parameter.

A suitable extension of this method to the case

where BH ^is slightly different from x/2 consists in

assuming ^that

If strong anchoring ^is assumed, ^one _gets 00

EH, k

(x ^{+ 2} EH/Bm,)/D (see Appendix) ^{and :}

where

In these equations Hc

n(K33jXa)1/2/D ^{is the Free-}

dericksz transition critical field.

We recognize ⁱⁿ (3) ^{a Landau} expansion ^{in the}

presence of an external field [4]. ^The quantity ’EH

On - n/2 plays the role of the external field

The _accuracy obtained by using ^this approximation

is discussed in the Appendix.

By putting dF/d0m

^{0, we} get :

In figure ¹ the real solutions of (5) ^vs. H/Hc, for ^diffe-

rent sH values, ^are shown. If EH

0 one obtains the

curve corresponding ^to the Freedericksz transition : for H H,, 0.

^{0 is the} unique ^real solution, ^while for H > Hc, ^{there are two stable} solutions, symmetrical

relative to the abscissa axis. If SH =A ^0, that is if the field is tilted with respect ^to the normal to the undis-

Fig. ^1.

Plots of the distortion angle 6m

H/Hc. ^The

curves a, b, c, d, have been calculated with sH

0, n/1 000, n1180, n/90 respectively, ^{and with} KI1/Xa

^4.6, K33/Xa ^{= 7.3} (c.g.s. units). ^{The dashed}

f is the locus of the points

where the free energy

has

flexus with horizontal tangent. In the insert the Bm

HI Hc plots corresponding

to the upper

reported.

torted director, the Freedericksz transition does not occur.

Three curves corresponding ^to different values of SH ^are plotted. ^{Each curve} is constituted by ^{two bran-} ches, corresponding ^{to values of} Om having opposite sign. ^The upper branches, ^which correspond ^{to the}

absolute minimum of the free _energy, describes the behaviour of 6m ^{when H is increased without} changing

its direction. The second branches correspond ^{to the}

condition obtained when the field is initially ^tilted

in the opposite direction and afterwards rotated to obtain the same value of EH by keeping ^{its modulus}

constant. The dashed curve f separates ^{the two zones} where stable solutions of the second kind exist ^or do not exist, ^and corresponds ^to the locus of the points

where the free energy ^curve has a flexus with horizon- tal tangent.

The Freedericksz transition can be used to deter- mine the elastic constants through ^{a measure of} Hc.

If methods sensitive to 0’ ^{are used, as} for instance

capacitive ^{methods or} interferometric methods with

light ^at normal incidence [5], ^rather large ^{errors can}

be easily ^{made. In} the insert of figure 1, 02m ^values

vs. Hj Hc, for several values of SH ^are reported ^From

this insert the occurrence of such errors can be easily

understood, by taking ^{into account} the fact that the

behaviour of the 02 ^{vs. H curves for} :0 ^{0 can be}

distinguished, from that corresponding ^to EH

0, only through the deviations from linearity ^observed

at small Om values. Our calculations show that an error of about 0.5 deg. ^{in the field} alignment, ^{which is a} quite reasonable estimate for methods sensitive to

0’ [6], gives ^{rise to an error of about} 2 % ⁱⁿ H,. ^The

interferometric method in oblique geometry, proposed

in the present paper, allows a very good check of the conditions of orthogonality ^{between H} and,the liquid crystal ^{director to be made. Let us} consider the situa- tion described in [1] ^{where the} magnetic field H and the nematic crystal director lie in the light ^incidence plane. The relative light intensity Ij 10 transmitted

through ^the sample, held between two crossed pola-

rizers making ^an angle ^of n/4 ^with respect ^{to the inci-} dence plane, ^is given by :

where L1 is the phase difference between ordinary ^and extraordinary rays outside the sample. ^We recall that

where Ko

² nIA, O ^{is the incidence} lightangle ^and 00

is the ordinary refraction angle. ^The extraordinary

refraction angle Oe(z) ^is given by

where D 2(z)

^sin2 Ø(z)jn; ^{+ COS2} Ø(z)jnõ, and no

and ne ^{are the} ordinary ^and extraordinary ^refrac-

tion indices respectively.

The 0(z) ^function appearing ⁱⁿ (8) ^can be evaluated

through ^the approximate expression (5) only ^{for small}

values of both Om and sH5 i.e. in close proximity ^of

the critical point ^{However the} following plots ^of Ij 10 ^{vs. H are} generally ^extended beyond ^the region

of validity ^of approximation (5). ^{As a} consequence, in these _cases, 0(z) has been calculated numerically starting ^from (A . 6).

In figure 2a, 2b, ^{2c the curves} representing ^the intensity I/Io ^{vs. H} for different values of EH ^are report- ed These curves correspond ^{to the} upper ^ones of

figure ^1. They show how sensitive is this method in

checking ^the orthogonality condition : a deviation of 0.01 deg. ^{is in this case} easily ^evidenced Furthermore, contrary ^{to the} methods sensitive to 02 ^{it is now} possible ^to evidence also the sign of the deviation with respect ^to orthogonality, ^as figure 2d, having ^an

CH opposite ^{to that of} figure 2b, ^shows.

Incidentally ^{we notice} that the broad maximum with I/I0 ^1, occurring in all the curves corresponds

to an extremal point ^{of the} phase difference.

Fig. ^2.

Calculated intensity of the transmitted light 1110

^{H with}

^incidence angle 0; ^{= 80°, with}

^thickness

of the liquid crystal ^D

²⁰ gm, (ns - no)

0.225 and the

following ^values of EH : a) 7c/l 000, b) (n/2)

10-4, c) 0, d) - (n/2)

^10-4.

3. Experimental ^results.

The calculations of the transmitted light intensity, given ^{in the} preceding ^{section, have been} experimen- tally ^{verified on a} homeotropically aligned sample ^of

MBBA. The sample ^{had a thickness D}

^{109 ±} 1 gm and the measuring apparatus ^{was the same as des-} cribed in references [7, 8]. ^The sample ^was kept ^at

23 ± 0.5 °C. The rather large thickness of the sample

was chosen to reduce _errors, if strong anchoring

conditions are not met This also further enhances the sensitivity of the interferometric method.

In figures ^{3a and} 3b, experimental ^{results concern-} ing different values of 0 H ^are compared with the theo- retical curves calculated by using ^the K11, K33 ^and

An values obtained for the same specimen ^{in a} pre- vious experiment [1]. Namely ^On

0.225, KIIjXa

4.6, K33/Xa

^{7.3 (C.G.S. units). It is seen that the}

agreement ^is exceptionally good for the whole curve.

In figure ^{4 results} concerning geometrical ^condi-

tions _very near to orthogonality ^are reported By comparing these results with the theoretical curve, in which OH

89.99o with the other quantities ^the

same of figure 3, ^{it can} be, concluded that in the experi-

ment the deviation angle BH ^was smaller than 0.01 °.

The incidence angle O¡ ^{was chosen to} obtain the maximum sensitivity ^{in the curve} I/Io ^{vs. H for}

H Hc. The value of H, corresponds ^to the maximum

758

Fig. ^3.

Intensity ^{of the} transmitted light I/I0

H through

a

homeotropic ^MBBA sample ^for Oi

44° 20’ and D

109 pm. The values of EH

a) ^{1 °} 30’, b) ^{3°. The} points represent ^the experimental values, the full lines the

computed ^values.

Fig. ^4.

Experimental points ^of the relative light intensity III,

^{H for the same MBBA} sample

ⁱⁿ figure 3, with

oi

42° 45’. Full line represents the theoretical curve

calculated _{with c,}

(n/2)

^10-4. The values of the other parameters

^{the same of} figure ^3.

value of H, above which the light intensity ^{I becomes}

unstable because of the close packing of the inter- ference fringes.

We estimate for the reported KiilXa ^{values an error}

of the order of 2 : 3 %. ^{This error is} mostly ^{due to}

the uncertainty ^{in the} measurements of the sample

thickness.

4. Conclusions.

A new technique ^for studying the Freedericksz tran- sition by interferometric methods in oblique geome-

try ^{is described.}

The use of the Freedericksz transition to determine the elastic constants characterizing ^{a nematic} liquid crystal ^{is limited} by ^the difficulty ^of verifying ^the orthogonality ^between magnetic field and liquid crystal director in the undistorted conditions. Actually

small deviation angles give ⁱⁿ general ^rather large

errors in the evaluation of the critical field Hc. ^Con-

ventional methods, ^{based on} capacitance ^measure-

ments or interferometry with normal incident light,

which are sensitive to em, ^{easily give errors of the}

order of 0.5°, while with the technique proposed ⁱⁿ

the present paper ^a deviation angle ^{down to 0.01 °}

is easily detected. This allows to practically ^elimi-

nate this source of error.

Another important point ^{of this} technique ^{is the}

fact that, ^{in an} oblique geometry optical arrangement, stationary points appear in the Ij 10 ^{vs. H curves.}

As discussed in a separate paper, these points ^can

be related to the ratio K11/K33, and thus allow a

further elastic constant on the same specimen ^{to be}

measured with great accuracy.

Appendix

In this Appendix ^we discuss the problem of the best choice of the parameters 60, Om ^{and k} appearing ⁱⁿ (2),

under the boundary ^condition

Instead of using ^a variational method involving ^all

these three parameters (which gives ^{rise to rather} complicated expressions), ^we prefer ^{to derive an} approximate expression ^for 00 ^{and k on the basis}

of more intuitive arguments. ^A variational method is then applied ^{to obtain} only ^the quantity Om.

In performing ^these calculations, ^{we were} guided by ^a simple mechanical analogy. ^{In fact} when ’1

⁰

the expression within the integral appearing ⁱⁿ (1) corresponds ^{to the} Lagrangian function of a simple pendulum whose oscillation angle 0(t) ^{is identified}

with the quantity ² 9(z). Furthermore the term con-

taining q, which is of the order of 0’, ^{has a small}

influence near the critical point, where CH and 0(z)

are also small, and cannot greatly change ^the pendu-

lum-like solutions.

We first consider the case of EH

^{0, which has} already been discussed in [2], ^{where a sinusoidal}

solution is assumed, ^which corresponds ^to (2) ^with 00

^{0 and k}

7c/D. ^{This is not the best} assumption (in ^the class of the functions defined by (2), ^a purely

variational method gives, ^{for H} > Hc, ^{a value of} 00

which is zero only in the limit Om -+ 00) but is the

simplest one, and it is good enough ^to describe the

configuration ^{near to} Hc.

In fact with this assumption ^one easily ^{obtains :}

For il

^{0 this} equation corresponds, ⁱⁿ analogy ^with

the pendulum, ^to the relation between the oscilla- tion amplitude ^{and the} period, ^{and is correct} up to terms in 0m. ^{This means that the} 00 = 0 assumption gives ^not only ^{the correct} value of the critical field He

but also the exact slope ^{of the} 0’ ^{vs. H curve in the}

H = He point.

Next we notice that the Euler-Lagrange equation

reads

and that, for EH

0, the result 00

^{0 can be obtained} by requiring that the function 0(z) (given by Eq. (2))

satisfies this equation ^{in the} point ^where d20jdz2

^0.

The 00 and k values for BH =1= ⁰ have been obtained

on the basis of this same requirement ^{and of the} boundary ^condition given by (A.1). ^The validity ^of

the extension of this argument ^{to the case} of sH :A ⁰

lies in the fact that, ^{near to} the critical point, ^we certainly ^have 00 Om, ^as suggested by the mechani- cal analogy. ^{This can be also seen as a} logical ^conse-

quence of the assumption ^that 00

⁰ for EH

0, by considering ^{that, for H} > Hr, Om is different from

zero in the limit _EH

0, and that the Om ^vs. 00 ^curve

is continuous for _any value of _SH different from zero.

By inserting (2) ⁱⁿ (A. 2), ^and considering ^the point

where d’O/dZ2

^{0, one} easily ^{obtains, for small} 0.

and 00 ^{values :}

From equations (A.1) ^and (A. 4), ^and taking ^{in account}

that 00 0, ^{one then obtains :}

Once 00 ^{and k are known,} calculations proceed ⁱⁿ

a straight-forward ^{manner. We observe} only ^that (3),

which gives ^{the Landau} expansion of the free _energy, has been obtained from the defining (1) by perform- ing ^the following steps :

a) ^{the free} energy has been linearized with respect

to EH ;

b) the obtained expression ^{has been} expanded ^{in a}

power series of Om up ^to O;

c) ^{the term} containing SH 0§§, ^{which is} negligible

with respect ^to the other terms, has been omitted.

The consistency ^{of an} expansion ^{where the terms}

in 8£ ^{are omitted,} while those in 0’ ^{are retained, is}

justified by the fact that the (5), ^{for H}

Hc, gives :

03m

⁸ 6H/[7T(1 ⁺ q)]. This also confirms the ine-

quality 00 EH Om ^near H,,.

We finally ^{notice that the} Euler-Lagrange equa- tion admits the first integral

which has been used for the numerical calculations.

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