Anchoring strength for twist deformation at a nematic liquid crystal-wall interface

HAL Id: jpa-00210335

https://hal.archives-ouvertes.fr/jpa-00210335

Submitted on 1 Jan 1986

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Anchoring strength for twist deformation at a nematic liquid crystal-wall interface

G. Barbero, E. Miraldi, C. Oldano, M.L. Rastello, P. Taverna Valabrega

To cite this version:

G. Barbero, E. Miraldi, C. Oldano, M.L. Rastello, P. Taverna Valabrega. Anchoring strength for twist deformation at a nematic liquid crystal-wall interface. Journal de Physique, 1986, 47 (8), pp.1411-1416.

�10.1051/jphys:019860047080141100�. �jpa-00210335�

Anchoring strength ^{for twist} deformation at a nematic liquid crystal-wall ^interface

G. Barbero (1,2), ^{E. Miraldi} (1), ^{C. Oldano} (1,3), ^{M. L. Rastello} (4) and P. Taverna Valabrega (1) (1) Dipartimento di Fisica del Politecnico, Torino, Italy

(2) GNSM, ^{Unità di} Cosenza, Dipartimento ^di Fisica, Università della Calabria, Arcavacata di Rende, Italy (3) ^{GNSM U.R.} 24, Torino, Italy

(4) I.E.N. G. Ferraris di Torino, Torino, Italy

(Reçu le 10 décembre 1985, révisé le 7 avril 1986, accepté le 9 avril 1986)

Résumé.

Nous avons mesuré la rotation de torsion près de l’interface entre un cristal liquide nématique ^{et une} paroi pour des échantillons de MBBA ^et de ZLI 1738 placés ^{dans un} champ magnétique qui ^{induit une} déformation de torsion pure. Nous ^avons obtenu des conditions aux limites homogènes ^{en frottant avec} du téflon des lames de

verre couvertes de silanes. Dans les échantillons _que nous avons examinés, ^un champ magnétique de 5 kOe pro- duisait une rotation de 10°. Ceci correspond ^{à une} longueur d’extrapolation de l’ordre de 1 _03BCm et à une constante

d’ancrage de l’ordre de 0,005 dyne/cm. ^{Les mesures ont} été faites par analyse de la lumière transmise par les échan- tillons. Nous montrons que, dans ^ces expériences, ^{le théorème} adiabatique n’est pas applicable : ^en effet, ^{des dévia-}

tions même petites par rapport ^à l’adiabaticité jouent ^un rôle déterminant pour l’intensité ^et l’état de polarisation

de la lumière transmise.

Abstract.

We measured the twist rotation at the wall-nematic liquid crystal interface in MBBA and ZLI 1738

samples placed ^{in a} magnetic ^field giving ^a pure twist deformation. Homogeneous boundary conditions were

obtained by rubbing ^two silane coated glasses softly with teflon. In the samples examined, ^{a rotation} angle ^{of the}

order of 10° was obtained with a magnetic field of 5 kOe. This corresponds ^{to an} extrapolation length ^{and an} anchoring ^constant of the order of 1 03BCm and 0.005 dyn/cm respectively. Measurements were made by analysing

the light transmitted through ^the samples. It is shown that in such experiments ^{the adiabatic theorem cannot be}

considered valid, ^{since even} very small deviations from adiabaticity play a leading ^{role in} determining ^the intensity

and the polarization ^state of the transmitted beam.

Classification Physics ^Abstracts

61. 30G

1. Introduction.

The anisotropic part of the interactions between a

liquid crystal ^{and a} substrate is generally ^described by introducing ^a surface free _energy term (anchoring energy) ^which depends ^{on the} angles defining ^the

orientation of the director at the wall. By ^{means of a}

suitable treatment of the substrate liquid crystal interface, ^{one can} impose ^a specific easy axis (which

defines the director orientation corresponding ^{to the}

minimum value of the surface free energy), ^{and the} strength of the forces arising ^from every change ^{in the}

director orientation. The study of such forces is of great interest from both the fundamental and the

practical point ^{of view,} but their description ^and comprehension is far from being complete. ^A great ^deal of work has recently ^{been done on the} dependence ^of

surface _energy on the angle ^between the director and the normal to the sample, i.e. the forces arising ^when

this angle ^is changed by ^a splay ⁺ bend deformation.

The present aim of the authors is to consider

homogeneous boundary conditions (easy axis in the

plane ^{of the} wall) ^{and to measure the} director rotation at the surface, ^{in the} plane ^{of the wall, induced} by ^a magnetic field such to give ^a pure twist deformation.

The experiment ^can be carried out by analysing ^the light reflected from the front surface of the sample ^or

the light transmitted through ^it.

As far as we know, ^the anchoring strength ^{for twist}

deformation has been measured only in references [1, 2]

by transmitted light techniques (1).

Here we report ^some preliminary ^{results, obtained}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047080141100

1412

by this rather simple method, by ^which measurements can be made on the samples normally ^{used for} practical

purposes.

We have used a distorting magnetic ^{field H 7 k0e.}

For such a low field, the adiabatic theorem is generally

considered valid, ^{since the} magnetic ^coherence length

is at least one order of magnitude greater ^{than the}

light wavelength [3].

We have carefully tested the validity of the adia- batic theorem with ^a computer simulation. Our conclusions are that even very small deviations from

adiabaticity play a leading role in this type ^{of mea-}

surement and cannot be neglected

An accurate analysis ^{of our} experimental ^results

shows that the anchoring strength ^for samples ^treated

with suitable surfactant and then rubbed, ^is of the order of 5 x 10-3 dyne/cm.

2. Theory.

Let us consider (see Fig. 1) ^an homogeneously aligned liquid crystal ^cell orthogonal ^{to the} z-axis, held between two polarizers P 1 and P2 ^{in a} magnetic ^{field H which}

lies in the x, y plane. ^In these ideal conditions, ^one

obtains a pure twist deformation. Let 9(z) ^{be the} angle defining the director orientation with respect ^{to a} given axis, 9. ,, and qJs2 the values of 9 ^{at the} boundary planes,

(Pr. I and CPe2 the easy axis directions in the same planes,

91 and (P2 the angles defining the orientations of the

polarizer P1 and of the analyser P2 respectively.

Let us first assume that the adiabatic theorem is valid This means that within the sample ^a light ^beam travelling parallel ^to the z-axis can be considered as a

superposition ^{of an} extraordinary ray, with the electric field E everywhere parallel ^{to n, and of an} ordinary ^one,

with E 1 n. Under this assumption ^the amplitude ^{of the} ordinary ^and extraordinary rays, beyond ^the analyser,

is proportional ^{to cos} ((p,

9rl) ^cos (qJ2 - qJs2) ^and

-

sin (qJl - qJsl) ^sin (qJ2 - 9,,2) respectively. ^{The in-} tensity ^{of the} light ^{at the} photodiode is therefore given by :

where al = (PI - (P.11 a2 = lfJ2 - lfJs2’ and b =

(ne - no) ² nd/A ^{is the} phase difference between the

extraordinary ^and ordinary rays and d the sample

thickness. Io is the maximum intensity ^{which can be}

obtained by rotating P1 ^and P2 ^and corresponds ^to

ai

a2

0 or n/2.

In order to determine the anchoring strength torque

one must measure lfJsl and qJs2 ^{as a} function of H. To this end, ^two different methods are suggested by equation (1).

(A) It will be noticed that I = 0 when al ⁼ 0 and a2 ⁼ n/2 (or al ⁼ n/2 ^and a2 = 0), ^{i.e. when} P1 ^is

rotated in such a way ^{as to} excite only ^{the extraordi-}

nary (ordinary) ^{ray within the} sample ^and P2 stops ^it.

Fig. ^1.

^Schematic representation ^{of the} experimental set-up. P1 ^and P2 ^{are the} polarizer ^and analyser respectively,

S is the sample. In the actual experiment ^a beam-splitter ^is placed ^between P1 ^{and the} sample, ^{in order to obtain a}

reference light ^beam.

For 6 different from zero and from n, these are the

only positions ^of P1 ^and P2 giving ^{I = 0. In this} way gr ,, and lfJs2 ^can be measured as a function of H.

(B) ^{If the two} surfaces of the sample ^are equally

oriented and equally treated, i.e. if lfJs2 ⁼ Pst, and

a crossed polarizer arrangement ^{is used} (a2 = ai + 7r/2), equation (1) ^{reads :}

Moreover, by assuming ({Jl - ({Jel = n/4, ^i.e. al = n/4

for H ⁼ 0, equation (2) gives :

«(fJs1 - (fJe1) ^can be determined by equation (3) ^{as a}

function of H, by simply measuring ^a light intensity

ratio.

In order to evaluate the effects of a non perfect validity of the adiabatic theorem, ^{let us} perform ^a computer simulation of the experience described in

(B), ^{for a} sample ⁵⁰ J.1m thick, ^{in the} range H ⁼ 0-5.Hc,

where Hr ^{is the} critical field for a twist Freedericksz transition.

The light propagation through ^the distorted nematic is evaluated by ^means of Berreman’s 4 x 4 ^matrix

method [4]. ^{It allows to} obtain the matrix T defined

by ^the equation :

where Pi ^and Pf ^are two vectors defining amplitude

and polarization ^{state of the} light coming ^{in and}

going ^{out of the} sample.

For a given ^director configuration, the matrix T is expected ^to depend ^on the actual values of ne ^and no.

Therefore Ir ^{vs. H curves} for different values of ne and no ^{have been} computed ^and plotted ⁱⁿ figure ^2.

For strong anchoring (({Jst

({Jet) ^{all these curves are} practically coincident, ^and collapse ^{in a} straight ^line (Fig. 2a). ^The light intensity is therefore independent

of ne, no and ^H (i.e. of the director field distortion).

Fig. ^2.

Computed light intensity through ^a planar sample placed ^{between crossed} polarizers ^{at 450 with} respect ^{to the} easy axes, in a magnetic ^field giving ^a pure twist deformation.

The values of the parameters ^{are :} xa

10-’ _c.g.s. units;

K22

^{6 x 10-’} dyne; d(sample thickness)

⁵⁰ um; angle

between H and the easy axis 45° ; ws

^oo in a), ^0.003 dyn/cm

in b) ^{and 0.001} dyn/cm ⁱⁿ c). ^In figures b) ^and c) ^{the curves} 1, 2, 3, 4, 5, 6, ⁷ correspond ^to phase differences of 10°, 300, 90°, 180°, 2700, 3300, ³⁵⁰⁰ respectively. The dashed lines give

the intensities obtained by assuming the adiabatic theorem

to hold.

Notice that this result is precisely ^that expected ^when

the adiabatic theorem is assumed to hold Apparently,

the possible deviations from adiabaticity ^are negligible

up ^to H ⁼ 5 He (this may ^{not seem too} surprising,

since the minimum magnetic ^coherence length ^is always ^{about one order of} magnitude greater ^than A).

The dashed ^curves of figures ^{2b and 2c are the} plots

of equation (3) ^{for two} different values of the anchoring

constant, i.e. the ^curves obtained by assuming ^the

adiabatic theorem to .hold ^even in the case of weak

anchoring. ^We have assumed that the elastic free energy per unit area is given by ^the simple expression

where ws is the surface stiffness constant.

Full line curves are the actual ones for different values of 6. They show that the non validity ^{of the}

adiabatic theorem can play a fundamental role in this

type of measurement, depending ^{on the} phase ^dif- ference 6. We explicitly observe that the parameter of interest is 6 ⁼ 2 n(ne - no) d/ À (and ^not n,, no

separately). ^/

In order to understand why ^{the non} validity ^{of the}

adiabatic theorem plays ^{no role in} figure 2a, ^while it has striking effects in figures 2b, 2c, ^we have searched for analytical approximations of the matrix T. For any practical purpose ^a suitable first order pertur- bation approximation of Berreman’s 4 x 4 matrix is

good enough in all the H range examinated Jones’s matrices notation has been used, ^{in a coordinate} system with the x-axis coincident with the director orientation at the surfaces. This means that Pi ^and Pf

are vectors of a bidimensional space, whose compo- nents are the complex amplitudes ^{of the} extraordinary

and ordinary rays, respectively. ^{In this} notation, the

vector Pi ^is given by

where ai is the ^amplitude of the incident beam, ^and

the matrix T is given by (2)

where

The meaning ^of equation (7) ^is quite ^{clear :} light entering ^the sample ^{as an} extraordinary (ordinary)

ray gives ^{rise to an} ordinary (extraordinary) ^{one of} complex amplitude ^{it. If} cp(z) ^{is an even} function of z, t is real according ^to equation (8).

The complex amplitude a ^{of the} light ^{beam after}

the analyser ^is given by ^the projection ^{of the vector} Pf ⁼ TP; ^{on the vector} P2 ^of components ^cos (a2)

and sin(a2), respectively. Since a2 ⁼ al + n/2, ^we

obtain

The four terms in equation (9) derive from the four elements of the matrix T. Notice that the t-dependent

terms, ^which represent the first order correction to the adiabatic theorem, exactly cancel each other for al ⁼ 7r/4.

In the experimental conditions of case (A) ^{one must}

search for the polarizer ^and analyser setting giving

I ⁼ 0. By assuming CfJsI ⁼ CfJs2 ^{once more} this condi-

(2) ^{The deduction of} equations (7), (8) ^is quite complicated

and will be given elsewhere, being beyond the scope of this paper. We point ^{out that the} adiabatic theorem holds good

in the limits within which t may be neglected ^with respect

to the unity. According ^to equation (8) ^a rough ^estimation

of a superior ^limit for t is given by the maximum variation of the angle ⁹ (expressed ⁱⁿ radiants), through ^a layer ^of

thickness A’

A12(n,. - no), ^{or over the} sample thickness d

for d A’ (if w changes monotonically ^{over a thickness} A’).

1414

tion writes

After straightforward calculations one obtains

The zeros of intensity ^are found with crossed pola-

rizers (as found in the hypothesis ^of validity ^{of the}

adiabatic theorem), ^{but no} longer ^for al ⁼ PI - (psi 0.

In figure 3 the behaviour of t and al ^{as a} function of 6

are plotted for different H-values. They show that the

angle ai, which is ^{a measure} of the deviation from

adiabaticity, ^becomes very great for 6 z 0. Slightly

more complicated expressions for ai and a2 ^are found when the hypothesis of identical boundary ^conditions

is dropped

3. Experimental set-up ^{and results.}

The experimental set-up is shown in figure ^{1. One of}

the two beams is sent to the sample, ^{the other one to a}

second photodiode, ^{in order to obtain a reference} signal ^for the correction of incident light fluctuations.

A quarter-wave quartz plate is inserted between the

sample ^{and the} analyser before and after _any experi-

ment, ^to determine the sign ^{of the} phase difference 6 between the extraordinary ray and the ordinary ^one.

The quantity ^{cos 6 is more} easily ^obtained by ^measur- ing the modulation of the light intensity ^induced by ^a

rotation of the analyser. ^The measurement of 6 is essential for two reasons. First because it determines the parameter t ^defined by equation (8), giving ^the

deviation from adiabaticity. ^Besides which, ^{the measu-}

rement of 6 is the best test to detect the possible splay

distortion induced by ^the magnetic field, ^which may be due to a misalignment of the field or to a possible pretilt angle ^{at the} boundary ^{surfaces. In all the}

measurements reported ^{below no} correction was

Fig. ^3.

Plots of the parameters t (defined by Eq. (8)) ^and

al (defined by Eq. (11)) ^{vs. the} phase diflerence 6, ^{for the}

same sample ^of figure 2, ^{and for H}

^1.5 He 1, ³ He ^{2 and}

5 H,, ³ respectively. Notice that t is a periodic function of 6 with a period ^{of 4} n, while al (which gives the first order corrections due to the nonvalidity ^{of the adiabatic theorem}

in a type (A) experiment) ^{has a} period ^{of 2 n, and is} nearly equal ^{to zero when 6 is} nearly equal ^{to a} semi-integer ^mul- tiple ^{of 2 n.}

needed for this source of error. The samples ^{are 25}

and _{50 ym} films of MBBA and ZLI 1738, placed

between glass plates. Homogeneous alignment ^is

achieved by performing ^the following ^{surface treat-}

ment : a) ^the glasses ^{are rinsed in} chromic mixture and washed in bidistilled water; b) ^their coating ^is

obtained dipping ^{in an} ethanole solution of silane

(EMAP) ^and drying ^at 120 °C for half an hour;

c) finally they ^are softly rubbed with teflon. The spacers used are mylar ^or teflon films. The two glasses

are kept together ^{with a} small mechanical pressure, in order to avoid birefringence ^induced by ^stresses.

Two types ^of measurements have been performed a) ^The angles qJsl and qJs2, defining the director orientations at the boundary surfaces, ^{have been} measured with method (A) of section 2, ^{for different} field strengths. ^Two slightly ^different angles (of ^88.50

and 91 .5° respectively) between the field and the _easy axis have been used, ⁱⁿ order to give opposite ^dis-

tortions of the director field and opposite ^rotations

at the surfaces.

The values of qJsl and qJs2 for ^two different field

strengths ^{and for a} 50 pm sample ^{of ZLI 1738 are} reported ⁱⁿ figure ^4. Since the values of H are large

with respect ^to the critical field H,, for twist Freede- ricksz transition, the director in the middle of the

sample, ^where dqJ/dz ⁼ 0, ^is practically parallel ^{to H.}

In these conditions the following simple equation

can be written (see ^{Ref. [3], p.} 83)

where _CPH gives the field direction.

The surface torques zsi ^at the two surfaces are :

and are therefore proportional ^{to the} quantities H sin (qJsi - qJH)’

Fig. ^4.

^Director orientations T., and lfJs2 at ^{the two boun-} dary ^{surfaces of a 50 um thick} sample ^{of ZLI} 1738, ^{as a} function of the quantity H sin «({Jsi - ({JH). ^The reported

errors are computed according ^to equation (14).

The phase difference 6 is n (this ^{value was obtained} by ^{means of} temperature control). ^{In these} conditions, the director orientation at the sample ^{surfaces are} given by the directions of the polarizer ^and analyser,

without _any correction for ^non adiabaticity (see Fig. 3). Notice that for the polarizer ^and analyser setting considered in the experiment, ^the intensity

never goes ^{to zero} but reaches a minimum value Im;n . Presumably this is due to the fact that within the illuminated area ( ~ 1 mm2), ^the alignment ^{of the}

director at the surfaces is not exactly ^{the same. Let}

us assume that 1) the surface angles lfJsl and lfJs2

change randomly ^from point ^to point, ^{with mean}

values CPsl’ CPs2 ^and standard deviations (âlfJsl)2, (âlfJs2)2 respectively, ^and 2) the overall transmitted

intensity ^{is obtained} by averaging equation (3) ^with

respect ^to 9. ,, and lfJs2’ Under these assumptions ^the

minimum intensity ^is obtained for lfJl = CPsl (polarizer parallel ^{to the} average director direction), lfJ2 ⁼ CPs2 ⁺ n/2, ^{and is} given by

Equation (13) ^is easily obtained from equation (3) by considering ^that ai ⁼ (Psl - ^gsl, sin al x al,

cos ai = ¹ and that ai has ^mean value equal ^{to zero} (analogous, ^{but not} identical, ^{relations are obtained}

for a2). ^{The errors} reported ⁱⁿ figure ^{4 have been}

Fig. ^5.

^Relative intensity Ir ^{vs. H} plots, ^{in the} experimen-

tal conditions of a Freedericksz twist transition, ^{for the same} sample ^of figure 4, ^{for two} different values of 6. The full lines

are the theoretical curves. The sample ^is placed ^between

crossed polarizers. ^The angle al between the polarizer ^and

the easy axis is 220 30’ (where ^{the function} I( 0153l) ^{has its}

maximum derivative).

evaluated by ^the expression :

obtained from equation (13) by assuming A(p,,, ⁼ A9.2 ⁼ AT.. ^This quantity ^is supposed ^to represent the intrinsic uncertainty of the surface alignment.

The actual uncertainty ^is perhaps ^{lower. In fact other}

sources of error may contribute to Imin, ^as for instance the depolarized ^forward light scattering ^{from thermal}

fluctuations of the director field

Moreover, according ^{to our} evaluation, ^{the sources}

of error related to a possible splay deformation and to the deviation from adiabaticity ^are (in ^the experi-

mental condition of Fig. 4) smaller than that given by equation (14).

In order to obtain the extrapolation lengths Li ⁼ K22/Wsi at ^{the two surfaces, we} have measured the critical field He with the standard method of Freedericksz transition (see Fig. 5). By considering

that d > Li, ^the following simple expression ^for He

can be used :

From equations (15) ^and (12) ^one easily ^{obtains :}

From the data plotted ⁱⁿ figure 4, ^{one obtains} L1 ⁼ ( 1.1 ± 0.3) pm and L2 ⁼ (1 ± 0.2) um.

b) ^It should be noticed that the previously ^described

measurements give ^{all the} quantities Tsl, ({Js2, ({Je1’ 9e2

(the ^{latter are} coincident with ({Js1 and 9s2 ^{at zero} field strength), ^{so that one can test} whether the two surfaces are equally ^{treated. For some} samples, ^and

for some areas of other samples, ^the easy ^axes and the anchoring strengths ^{are found to} be rather dif- ferent. The measurements of type (B), ^{on the} contrary, do not separate the effects of the two boundary

surfaces. The experimental data have therefore been

compared ^{with the} theory by assuming ^identical extrapolation lengths ^at the surfaces.

We have measured the light intensity ^{as a function}

of H, ^{with the} sample between crossed polarizers ^at

± ^{450 with} respect ^{to the} easy direction (method (B)

of Sect. 2).

An Ir ^{vs. H} plot obtained with the same sample ^but

not exactly ^{with the same} illuminated area as figure ⁴

is shown in figure 6. The full line is the computed ^one,

with L1 ⁼ L2

^{1.35 um. It} only ^{fits the} points corresponding ^{to the} higher ^field strengths very well.

The other experimental points ^are fitted within the

experimental ^{errors, but} they ^are systematically ^lower.

Similar behaviour is found in all the curves obtained with different samples ^or with different zones of the

same sample. This experiment ^{seems to} suggest ^{a non} linear dependence of the surface twist angles «({Js

({Je)

on the torques ^even for I (Pr,

9e I 100, and ^more

1416

Fig. ^6.

^Relative intensity Ir ^{vs. H} plots ^{for the same} sample

of figure 4, placed between crossed polarizers ^{at 450 with}

respect ^to the easy ^axes. The angle ^{between H} and the easy

axes is 45°. The full line is the theoretical intensity computed

with an extrapolation length ^L

^{1.35 pm and 6}

^2250.

precisely ^{that w. is an} increasing function of i. However, the previous analysis suggests ^{an extreme caution in} the interpretation ^{of the} experimental ^{data. We cannot}

exclude a possible experimental effect which might

account for the intensity behaviour without assuming

a variable _wr.

4. Conclusions.

This _paper shows that a rather weak anchoring ^can

be obtained for twist deformations in planar samples,

and gives ^a simple experimental method for the

measurement of the anchoring ^{constant. In all the} experiments ^{carried out} the surface deformation is

fully reversible, within the experimental ^{errors. The}

same curves are obtained by increasing ^or decreasing

the field strength. Furtherly they ^are independent ^of

the time taken to perform ^the experiment (from ^a

few minutes to a few hours). Furthermore the same

order of magnitude for the twist stiffness constant is obtained for all the samples. The results can be summarized as follows : a surface twist deformation of the order of magnitude of 100 is obtained by ^a magnetic field of the order of 5 k0153. This corresponds

to an extrapolation length of the order of _{1 um,} and to a stiffness constant of the order of 0.005 dyn/cm.

Other _very important ^related problems ^{are under} study by ^{our research} group, in particular ^{the deve-} lopment ^{of an} experimental technique ^{able to ensure}

a reasonably reproducible ^weak anchoring, ^{and the}

theoretical interpretation ^{of the} anchoring ^{forces in}

terms of molecular interactions. Concerning ^{the first} point, ^no procedure ^{are known at the} present [5].

With regard ^to the second point ^{we observe as follows.}

If the measured anchoring ^{forces are} interpreted ⁱⁿ

the framework of the Berreman _grooves model [6],

the conclusion is that the surface is quite ^{flat, i.e. that}

the _grooves are of a very small amplitude. ^{This is not}

unreasonable since the surfaces have been rubbed

softly ^with teflon, ^{but the} applicability ^{of this} simple

model is questionable. Other forces _may play a

dominant role, e.g. those due to : i) ^surface irregu- larities, ii) electrostatic interactions between the polar

groups of the liquid crystal ^{and the} substrate, iii) ^che-

mical binding between the molecules of the surfactant and those of the liquid crystal [7]. The conclusion is that other experiments ^are required ^to give ^a really satisfactory theoretical explanation ^{of the} anchoring

energy.

References

[1] SICART, J., ^J. Physique ^{Lett. 37} (1976) ^L-25.

[2] ^VAN SPRANG, ^H. A., ^J. Physique ⁴⁴ (1983) ^421.

[3] ^DE GENNES, ^P. G., ^The Physics of Liquid Crystals (Clarendon Press, Oxford) ^1974.

[4] BERREMAN, ^D. W., ^J. Opt. Soc. Am. 62 (1971) ^502.

[5] BECKER, ^M. E., NEHRINGS, J., SHEFFER, ^T. J., ^J. Appl.

Phys. ⁵⁷ (1985) ^4539.

[6] BERREMAN, ^D. W., Phys. Rev. Lett. 28 (1972) ^{1683 and}

Mol. Cryst. Liq. Cryst. ²³ (1973) ^215.

[7] OKANO, K., MATSUURA, N., KOBAYASHY, S., Japan ^J.

Appl. Phys. ²¹ (1982) ^L-109.