Ellipsometric studies of the nematic-substrate interface

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Ellipsometric studies of the nematic-substrate interface

J.P. Nicholson

To cite this version:

J.P. Nicholson. Ellipsometric studies of the nematic-substrate interface. Journal de Physique, 1988,

49 (12), pp.2111-2118. �10.1051/jphys:0198800490120211100�. �jpa-00210893�

Ellipsometric studies of the nematic-substrate interface

J. P. Nicholson

Dept. ^of Physics ^and Applied Physics, University ^of Strathclyde, Glasgow ^G4 ONG, Scotland, ^{U. K.}

(Requ le 16 mai 1988, révisé le 18 août, accepté le 31 août 1988)

Résumé.

L’interface entre la phase nématique ^{du 7CB et une} frontière solide a été étudiée par ellipsométie optique. ^La différence de phase ^entre les rayons ordinaire ^et extraordinaire, ⁰³⁹⁴

03B4p - ^{03B4S, et l’angle}

03C8

tan-1 |rp/rs| ^ont été mesurés en fonction de l’angle d’incidence et de la température. ^La modélisation des données expérimentales exige ^{une zone de} séparation exponentielle ^{avec une densité} 0,6 à 1,5 ^% plus grande ^et

un paramètre d’ordre 17 a 40 % plus grand que ^{ceux en masse.} L’épaisseur ^{de la} région interfaciale varie de 135 Å à 250 Å.

Abstract.

The interface of a nematic 7CB - solid substrate has been studied by optical ellipsometry. ^The phase difference between 0 and E rays, ⁰³⁹⁴

03B4p - ^{03B4s, and the} angle 03C8

tan-1 | rp/rs ^{| are measured as a}

function of incidence angle ^and temperature. ^{To fit the} experimental data, ^an exponential interfacial region having ^a density rising ^{to ~} 0.6-1.5 % and an order parameter ^{17-40 %} greater than bulk is required. ^The skindepth of the interfacial region varies from 135 A to 250 Å.

Classification

Physics ^Abstracts

61.30

42.80

68.00

1. Introduction.

There is considerable interest, both theoretical [1-6]

and experimental [7-13] ^{in the} interfacial region

between a nematic liquid crystal ^{and its} enclosing boundary ^wall. Apart from the well known alignment

effect of a microgrooved [18] ^or obliquely ^coated [19]

substrate which is of practical application ^{to L. C.}

electro-optic devices, ^{there exists also an effect on}

the order parameter ^{in the} vicinity of the substrate.

Thus the order parameter S (o ) ^{at the substrate} boundary ^is expected ^to differ from the bulk value

S (oo ). ^The functional form of S (z ) ^{in the} interfacial

region ^has yet ^{to be} determined, although ^several investigations [7-11] ^{have been made with the nema-}

tic material above its nematic/isotropic ^transition temperature. Measurements in this case are facili- tated by the fact that birefringence ^now only ^exists

within the interfacial region itself ; ^{this can be}

measured by obtaining ^the phase ^{shift between}

transmitted ordinary ^and extraordinary ways. How- ever, ^more recently total internal reflection ellip- sometry [11] ^{has been used to} investigate ^{the iso-} tropic - ^substrate interface, ^{from which some} knowledge ^{of the} shape ^of S (z ) ^{can be inferred.}

Measurements on the interface with the bulk L. C. in the nematic phase ^{are less} copious ^{as in this}

case the dominant birefringence ^arises from the bulk nematic phase. Semi-qualitative measurements at the nematic-substrate interface have been obtained

by ^{Mada and} Kobayashi [12] ^and by Salamon and Skibinski [14] ^who both show an enhanced order parameter ^{near to the} substrate, ^{but obtain no}

information on the skindepth ^of the interfacial

region. ^The only other work by ^an optical ^{method is} by ^{this author} [13] who has measured the optical reflectivity ^{of the} nematic-substrate interface. Con- siderable deviation from Fresnel reflectivity ^{is ob-}

served in the case of 7CB which leads to estimates of enhancement of the order parameter by - ^{15 % near}

the wall and of density by -11/2 ^{%. If a} simple

linear refractive index profile ^is assumed, ^Refer-

ence 13 indicates a full skindepth ^{of - 800} A. These results compare favourably ^{with methods} [25, 26]

which obtain S (z) ^{from the force} separating ^the

walls of an L. C. layer ^{as a function of thickness.}

The method used in reference 13, ^{however is} relatively insensitive to the assumed shape ^of S (z ). ^In this paper, which is essentially ^{a continua-}

tion of that work, the classical technique ^of ellip-

sometry [17] ^{is used,} which under certain circum- stances (e.g. varying ^incident angle ^or varying wavelength) ^can produce information on the index

profile of the film under investigation.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490120211100

2112

2. Experimental ^method.

The liquid crystal cell is shown in (Fig. l(a)). ^The

cell is filled with 7CB whilst in its isotropic phase (to

prevent ^flow alignment) and consists of two prisms

with their enclosing ^{faces at a} slight angle. ^This wedge shape enables the reflection from _{the upper}

prism ^{face to be} selected ; the other reflection is

ignored. ^The upper prism ^{is made of flint} glass (Schott type F2) ^{and has a} hypotenuse microgrooved by briefly polishing unidirectionally ^{with diamond} polishing paste. ^{The lower} prism ^{has an} obliquely

coated layer of SiO such as to reinforce the planar alignment (perpendicular ^{to the} plane ^of Fig. 1)

caused by ^the microgrooving.

Prior to assembly, ^both prisms ^are subject ^to

intensive cleaning (the ^upper prism after microgroov- ing ; ^{the lower} prism prior ^to coating) ^{as follows :} (i) preliminary ^short ultrasonic cleaning (- ¹⁰ min) (ii) overnight ^{soak in} detergent (iii) several ul- trasonic rinses in distilled water (iv) several ultrason- ic rinses in 18 Mil cm deionized water. To be

pronounced satisfactory ^the optical surface should then be observed to drain-dry perfectly uniformly

i.e. without breaking ^into droplets.

Under high power microscopy ^the microgrooving

is seen to occupy ^a negligible fraction of the area of the unidirectionally polished surface. It therefore still behaves as a specular reflecting surface which is verified by the very low fraction ( 10- 7) ^of light

scattered off the surface. The microgrooving ^how-

ever, does assist in obtaining ^nematic alignment right through ^{the cell} (average thickness 100 J.Lm).

Uniform alignment ^after assembly ^was easily

checked by examination through ^crossed polarizers.

Before filling the cell with 7CB, however, ^the quality

of the F2 glass surface is itself investigated ^as

described below.

The L. C. cell is investigated ^{for a} variety ^of

incident angles by ^the apparatus ^{shown in} (Fig.1(b))

which is a conventional manual ellipsometer [17,.

The whole apparatus ^was enclosed in a temperature stabilised enclosure (not shown) enabling readings

to be taken throughout ^{the nematic} range. Ellip- sometry, ^unlike reflectometry, is sensitive to the

phase ^{as well as the} intensity ^{of the reflected} light. ^If rp and rS ^{are the} (complex) amplitude reflectivities

Fig. ^1.

(a) Liquid crystal ^cell investigated ; n is director orientation. (b) Ellipsometer used in the measurements.

L - polarized ^{laser ; C - 1/4 wave} compensator plate ;

L.C.

L.C. cell shown in (a) ; ^{A -} Analyzer ; ^{D -} photodiode ^detector.

for the TM and TE polarizations, ^the respective phase ^shifts 8 p, ⁸

^{are given by :}

An ellipsometer ^{measures two} quantities ^{L1 and Q} given by :

For a given ^incidence angle, ^A and Q are ^obtained by adjusting the incident polarization angle, ^{P and} analyzer angle, ^{A to obtain a} null response in the detector. This occurs for 4 independent values of P and A for a given compensator angle C, given by [17] :

These four null conditions were obtained for each

setting ; ^{this not} only provides ^{a check on the} reproductibility ^{of A} and Q but also leads to elimi- nation of certain systematic ^{errors in the} ellipsome-

ter, by ^a process known ^as 4-zone averaging [17] :

We also need to correct for the effect of the side faces of the prism. Assuming each side face to be a

simple dielectric interface [20] ^the amplitude ^trans-

mittance will have no phase retardation and there- fore d is unaffected. It is easily shown that the measured Q ’ ^{obtained from} equation (4) ^{is related}

to the actual Q for the nematic substrate interface

by :

tan Q

cos2 (i 1- i2) ^tan .p’ (5)

where i1 and i2 ^{are the} angles ^{shown in} (Fig. 1(a)).

Values of A and qi ^{were obtained as described} above, firstly for the unfilled cell to obtain these parameters ^{for the clean} reflecting ^{surface and} then,

with the cell filled as described earlier with 7CB.

Quite small values of A were observed for the empty cell (~ ^{1° at} 40° incident angle) compared ^{with the}

values for the filled cell (differing ^from the Fresnel value by 20-40°). Nevertheless these empty ^cell

measurements were unexpectedly important ^{as we}

shall see below.

3. Analysis of results.

Typical ^{results for two} temperatures ^{are shown in}

(Fig. 2) showing ^{the variation of A} and Q ^with

incident angle, 60. ^{In order to obtain} information on

the variation of S and density ^{in the} interfacial

region ^{we need to} compute ^the reflectivities rs ^and rp ^for given refractive index profiles for both E and 0 rays. This then leads ^to the expected values of d

and 0 through equations (1) ^and (2).

The computation ^{of the} optical reflection at an

interface having ^a varying refractive index is a well- researched problem [15, 16, 21] having ^a variety ^of

methods some of which were summarised in our

previous paper [13]. ^The optical geometry ^{chosen in} this investigation (optic ^axis perpendicular ^{to the} plane ^of incidence) ^means that the E and 0 rays remain essentially independent of each other. In

more complicated situations the 4 x 4 matrix method of Berreman [21] would be necessary ^to solve the problem ; ^{in this} case we can use the 2 x 2 matrix, method described in the reviews by

Abeles [16] and Jacobsen [15]. ^Then optical proper- ties of the layer ^{are described} by ^{a matrix} [Mij], ^and

the reflectivity ^{r is} given by :

with

Fig. ^2.

Ellipsometric ^data (points) and best fit curves

(for ^{best fit case in Table} I), ^for temperatures ^{of 32.65 °C} and 40.0 °C. (a) ^A

5 p - 5,

^{as a} function of incident

angle, 00 ^{within the} upper glass prism (b) 41

tan-1 rplrs I ^{as a} function of 00.

Here no is the refractive index of the first medium

(F2 glass) and n2o, n2 E the ordinary ^{and extraordi-}

nary refractive indices of the bulk L. C., 00 ^and 02 ^are the ray directions in the two media respect- ively.

The matrix elements are also different in the two cases being ^{related to} the assumed refractive index

profile ^{in the} interfacial region, n10(z) in the TM

case and nl E (z ) ^{in the TE case. Profiles} (shown ⁱⁿ Fig. 3) ^{tried were :}

Linear profile

Inverse parabolic profile

2114

Parabolic profile

Exponential profile

Step profile

Modified Landau-de-Gennes See reference [2].

(P

^{0 or E} polarization).

Here No, NE ^{and d are} adjustable parameters ^whilst n2o and nz E ^are the bulk refractive indices of 7CB [23].

Fig. ^3.

^Sketch graphs showing the functional forms tried for nlo (z ) ^and nl E (z), ^the refractive index variation in the interfacial region ; ^{Inv. Par.}

^Inverse Parabolic ; Exp.

Exponential ; ^Lin.

Linear ; ^Par.

Parabolic.

In the case of the linear profile, ^the Mij ^{could be}

computed by ^{the series} expansion ^method (for d ..: A) outlined in section 3.1 of reference [13],

which was fast in computer time. The other profiles

were not analytically ^tractable by ^this method, ^and

were analysed by ^the step-matrix ^method in appen- dix 1, ^{where the} profile ^is subdivised into a large

number of small elements, ^{each element of which}

can be regarded ^as having ^{a linear} profile. ^The

accuracy of the computations ^{were checked} by comparing results with those obtained by ^direct integration of the EM wave equation ^{described in}

the appendix ^{of reference} [13].

4. Analysis of results.

Comparison ^between experimental ^and theoretically

calculated values of d and Q ^was achieved by minimizing the function M :

where yi ^and Ii ^{are the} experimental and theoretical values respectively (of ^{either A and} qi) ^{for a} given

00, and Ui the estimated errors.

It was found quite easy ^to obtain good ^{fits to the A}

data or the 1/1 ^data separately, especially ^{with three}

parameters (No, NE, d) ^{available to} adjust. ^How-

ever much more difficult is to obtain a simultaneous fit to both curves with the same parameters. ^To obtain such a simultaneous fit it was found _necessary

to correct for the effect of the surface finish of the F2

glass substrate. Fortunately ellipsometric ^{data had}

been obtained for this surface prior ^to filling ^the cell ; ^{this was found to be} compatible ^{with a} glass

surface profile sloping linearly ^{from the bulk value}

of 1.61656 to surface value of 1.449 over a distance of 140 A, which is very similar ^to the results of Yokota et al. [24] ^{on the effect on} glass ^{surfaces due}

to surface leaching during polishing. The character- istic matrix [Mij] ^{for the} nematic-substrate interface is then obtained by simple multiplication [22].

where [Mglass ] ^{is the} characteristic matrix of the leached glass ^surface layer, ^and [MLC] is the matrix for the nematic interfacial region computed ^as

described in section 3.

With this procedure ^{it was now found} possible ^to

find good simultaneous fits to the Q ^{and A} data, ^for unique ^{values of} No, NE ^{and d for a} given profile.

Moreover the quality ^{of the} simultaneous fits shows small albeit definite preferences ^{for some} profiles

over others. Table I shows the best fit values of

No, NE, d ^and Mo, ^the (lowest) value of M for the refractive index profiles ^{shown in} (Fig. 3) ^{and de-}

fined in section 3. Apart ^{from the results for 38.5 °C}

which curiously ^{show little} preference, ^{the trend at}

all four other temperatures ^{is a} preference ^{for a .} long-tailed shape ^and against sharp ^cut-off profiles.

Thus best fits were obtain for the exponential shape, parabolic ^a poor second-best, ^linear profile ^an

intermediate fit, ^{and inverse} parabolic ^and step

profiles progressively ^{worse fits. A} profile ^derived

from the Landau-de-Gennes type theory ^{of Allender}

et al. [2] ^{was also tried but the results were identical}

to those for the exponential profile ^{and are not} separately listed. The theory ^{had to} be modified by allowing ^a density variation of the same shape ^{as the}

order parameter ^variation.

The results for the exponential (or ^modified Landau-de-Gennes) profile ^{are shown in} table II, together with those for the linear profile ^for compari-

son with our previous measurements [13]. ^{This in-}

cludes the surface/bulk values both for density ^and

order parameter ^{which can} be calculated from

No, NE, n2o and n2 E [13, 29].

Apart from the data at 32.6 °C (which may be

affected by proximity of the nematic-solid transition)

Table I.

Best fits ^to experimental ^data for ^various re fractive ^index profiles ^and temperatures

Observed tc

^{42.55 *C}

the trend seems to be for d to increase steadily ^with temperature, and surface density and order par- ameter ratios to decrease.

Measurements were also taken above the nematic-

isotropic transition temperature ^but uncertainty ⁱⁿ

the glass ^surface profile prevented meaningful ^con-

clusions being obtained. This is because in this case

the skindepth ^{of the} glass ^surface profile (70 A ^half-

width) ^is comparable with that of the L. C. surface

layer.

5. Discussion.

The results shown here are in qualitative agreement with our previous ^work [13] ^{which indicated an} unexpected large interfacial region, ^{viz., d-}

700-1 000 A assuming ^{a linear} profile. In this work for a linear profile, d ranges from 400 A _up ^to 650 A

near the transition temperature. Thus results here,

would seem to give values of d lower by ^{about a}

factor of 0.6. However in adition to any sample ^to sample variations, ^{one would} expect ^a lower d for the present ^{results as the} previous ^{work did not}

include the effect of the initial state of the glass

substrate which, ^{as we have seen} (Sect. 4) ^{can be} represented by ^{a surface} layer having ^{a linear} profile

of thickness 140 A. This might ^also explain ^the higher surface order parameters ^{observed in this} work (11-25 % enhancement for a linear profile compared ^{with the} previous ^{values of 9-17 % en-} hancement).

Perhaps, ^more important, ^{is to} compare results with theory. Best fits for exponential ^{or modified}

Landau-de-Gennes profile ^are given in table II and range from d = 135 A ^at 34.2 °C to 205 A at 40.0 °C.

The theory by Allender et al. [2] predicts :

2116

Table II.

Calculated values of density ^and order parameter enhancement from ^{the best} fit ^data for ^an exponential and linear profiles respectively. Statistical errors are random from (temperature) ^{run to} run, whilst the systematic ^{error is due to} uncertainty ^{in the} substrate surface profile ^{and is} therefore ^{common to all runs.}

and

Taking T, - ^{T* = 1.4 °C and )}

⁵ A [7]

equation (9) yields ^{values of d} ranging ^from

d

6.2 A at 32.6 °C increasing monotonically ^to

d

20.9 A at 40.0 °C. These are well over an order of magnitude ^{less than our} experimental values. The

more sophisticated ^{mean field} theory of Telo da Gama [3] unfortunately only ^{treats the} homeotropic

case. The large skindepths reported here have also been demonstrated in the case of 5CB by measuring

the force between two substrates forming ^{a sandwich}

of the liquid crystal [25, 26] ^reference [26] ^describes

both a medium-range and short range regime. ^{In the}

short range, the force oscillates ^{over a} period ^of

~

25 A indicating local smectic ordering ^{which de-}

cays away completely after - 100 A. The medium range force exhibits a decay ^of near-exponential shape having ^a 1/e decay ^{distance of = 200} A.

Whilst our measurements clearly ^{cannot detect the}

short range fluctuations, they would appear ^to agree very closely with the medium range force distribution which is related to an order parameter variation of similar shape.

Other work which support ^these conclusions have been detailed in the discussion of our previous

paper [13] but it is worth mentioning again ^{the work} by Gannon and Faber [27] who infer smectic layering

at the nematic-vapour interface from surface tension measurements on 8CB and 5CB, and the X-ray

diffraction results of Leadbetter et al. [28] ^{who infer}

local smectic ordering extending ^over

-

150 molecules in the case of 7CB. There seems to be ample ^evidence therefore, ^that simple ^models

based on the Landau-de-Gennes free energy expan- sion are totally inadequate ^{in the case of} 7CB, ^which

has a large interfacial region

sometimes described

as quasi-smectic.

Our measurements in this paper ^{are not} sensitive to any smectic layering, ^{but we can} confidently point

to a strong tendency ^to aggregation in 7CB which leads to a higher ^ordered layer ^{of - 200} A thick at the nematic-substrate interface, and which may be condensed solid phase ^or locally smectic in form.

The results also indicate a possible elevation of the

density ^{near to the} interface. The result at 32.6 *C for the exponential profile ^{indicates an increase of}

1.5 % - nearly twice the standard deviation of the

systematic ^error (due ^to uncertainty ^{in the} glass

surface profile). ^{However as the} temperature ^rises the density rise decreases and becomes barely signifi-

cant compared ^{with the} systematic ^{error at 40 °C.}

Conclusion.

The nematic substrate interface has been measured

by ellipsometry ^{which has} proved ^a superior ^tech- nique ^to simple reflectometry in that the method is

moderately ^{sensitive to} the refractive index profile

of the interface. Assuming ^an exponential profile yields ^a 1/e decay distance, varying ^{from 135} A to

250 Å. Values of order parameter ^and density ^{at the}

surface _appear ^to be enhanced by up ^to 40 % and

1.5 % respectively. ^A theory which includes a ten-

dency for 7CB molecules towards medium range

aggregation is necessary ^to explain ^{the results.}

Acknowledgments.

Thanks are due to Mr Paul Winning ^for making ^a

substantial number of preliminary measurements, and to good ^technical back-up by ^{Mr. Bob Dawson.}

I am also grateful ^for helpful correspondence ^from

Tom Faber (Cavendish Laboratory, University ^of Cambridge) ^{and John} Bunning (Trent Polytechnic), regarding the refractive indices of 7CB. Free samples

of liquid crystals ^are gratefully acknowledged ^from BDH, Poole, Dorset. As always, ^I acknowledge support ^{from the} Dept. ^of Physics ^and Applied Physics, University ^of Strathclyde.

Appendix ^1.

COMPUTATION OF THE CHARACTERISTIC MATRIX BY THE

STEP-MATRIX METHOD

The index-var-

ying layer ^is imagined ^{to be} subdivided into slices of width 6z each slice perpendicular ^to the normal (z direction) ^{to the} interface. E.M. wave propagation ^is

first through ^{medium of} refractive index, no, then

through ^the interfacial region ^of depth ^{d and refrac-}

tive index n (z), ^then finally emerging into the bulk

liquid crystal. ^For convenience the subscripts ^are omitted, ^{but in} practice the index profile n(z) ^is

different in the TE [nl E (z ) ] ^{and TM} [n 10 (z ) ] ^cases.

Consider the slice between z and z + 5z having

refractive index changing ^from n1 = n (z ) ^to

n2

n (z + 6z). ^{For thin} layers (Sz ’" 10 Å A ) we

can take the n variation within the slice to be linear,

furthermore we can evaluate the transfer matrix

[m ] for the slice using the series expansion ^method

to 1st order only [15] :

where, in the TE case :

and in the TM ^case

It can be shown [22] that the characteristic matrix for several layers is obtained simply by multiplying ^each

matrix together. ^{Thus :}

Starting ^with the unit matrix at z

0, ^{we can now}

obtain the matrix for the whole layer by successively applying equation ^{A.4 until we reach z}

^{d, the end}

of the interfacial region.

For profiles ^{with an infinite} tail such as the

exponential profile ^{or the} (similar) Landau-de-Gen-

nes [2] profile, ^the step-evaluation ^{needs to be taken}

well beyond ^the skin-depth parameter, ^{d. For the} profiles examined in this paper, sufficient _accuracy

was obtained by taking the evaluation out to values of z/d" ^10. Unfortunately this slows up the compu- tation by ^{the same factor. For this reason, it is} advantageous ^to having ^a varying step width,

6z which becomes longer ^{as the} gradient ^{for the} profile decreases. In the exponential ^{case this is}

obtained by ^the simple instruction :

where 6z’ is the ^next step ^{and 5z the} previous. ^This procedure ^{was found to} speed up the computation considerably ^without significant loss of accuracy.

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