ELLIPSOMETRY OF LIQUID SURFACES

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ELLIPSOMETRY OF LIQUID SURFACES

D. Beaglehole

To cite this version:

D. Beaglehole. ELLIPSOMETRY OF LIQUID SURFACES. Journal de Physique Colloques, 1983, 44

(C10), pp.C10-147-C10-154. �10.1051/jphyscol:19831030�. �jpa-00223487�

JOURNAL DE PHYSIQUE

Colloque CIO, supplément au n°12, Tome W, décembre 1983 page C10-147

ELLIPSOMETRY OF L I Q U I D SURFACES D. Beaglehole

Physios Department, Victoria University of Wellington, Wellington, mew-Zealand

Résumé - Les propriétés telles que la densité, la composition et l'orien- tation moléculaire sont altérées aux abords de la surface que forme un liquide avec sa vapeur, avec des parois et entre deux phases liquides, par rapport à leurs valeurs au sein du liquide. L'ellipsométrie peut être utilisée afin d'étudier ces modifications , et dans cet article nous

passons en revue la mesure et l'interprétation du coefficient d'ellipticité.

Abstract - Properties such as density, composition and molecular orientat- ion are altered near the surface which a liquid forms with its vapour, with walls and between liquid phases, from values well inside the liquid. Ellips- ometry can be used to study these changes and the paper reviews measurement and interpretation of the coefficient of ellipticity.

Studies of liquid surfaces by ellipsometry have a long history. Jamin /l/

made the first measurements in 1851. At that time interest lay in whether the reflectivity for the p polarised wave was indeed exactly zero at the Brewster angle as predicted by Fresnel's Equations. Jamin found rather erratic values, but by 1891 Rayleigh / 2 / was able to state definitively that the surface of water showed a reproducible small amount of elliptical polarisation. Defining ~p the co-

efficient of ellipticity as "p = Im(r_/rs) at the Brewster angle where Re(rp/rs)=0, his final value for clean water, measured with sunlight, was + .00042. Since his time water has been studied many times, and a variety of other liquids as well:

Author "pxlO11 water A(A°) other liquids Reference

* 1 2

3

* 4 5

* 6 7

* 8 Part of the variation between workers is due to difficulties in cleaning the sur-

face, and water is particularly difficult since organic impurities rapidly pre- cipitate, but part is also due to instrumental problems, since such small values of ~p corresponding to intensity reflectivities ~ 10"⁷ require high quality optical components. The work of Bouhet matches that of Rayleigh in thoroughness, while that of Kizel covers a wide variety of organic liquids. Drude /8/ explained the residual elliptical polarisation in terms of departures of the dielectric constant profile from the step profile assumed in the derivation of Fresnel's Equations (Rayleigh /9/ also cites the earlier work of Lorenz, and Van Ryn), obtaining

Jamin (1851) Rayleigh (1892)

Raman and Ramdas (1927) Bouhet (1931)

McBain Bacon and Bruce (1939) Kizel (1956)

Kinosita and Yokota (1965) present author (1980-)

-58 4.2 7.2 4.0 3.3,4.2 10.4

3.0

sunlight sunlight

5460 Hg arc

6328

Light here is incident from medium 1, e(z) is assumed isotropic and a function ofz

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19831030

C10-148 JOURNAL DE PHYSIQUE

perpendicular to the surface only, and the discussion is only for transparent liquids ( c real). The above is the first order expansion of p i n terms of </A where 5 is the characteristic distance over which departures from the step profile occur. Ellipsometry studies are thus sensitive to dielectric constant profiles, which are themselves dependent upon density, composition and molecular orientation in the surface region. We discuss first free surfaces, then rigid and layer surf aces ; the distinction between these liquid surfaces will become clear in the discussion.

Free surfaces

Here we are interested in the surfaces between two equilibrium phases, for instance the liquid and vapour phases of a pure liquid, or the a and 6 phases of a partially miscible binary liquid mixture. These surfaces are inherently "free"

since they can carry transverse propagating waves, the capillary waves, which will be excited thermally. This complicates the interpretation of 5. If these waves are neglected, then ~ ( z ) will follow the density variation between liquid and vapour, or the concentration variation between a and f3 phases. Note that the sign of

p

and hence 0 indicates whether a monotonic variation in E

is appropriate, since for c15 E 5 ^{EZ T)} is negative.

To estimate a thickness of the interface Drude /8/ suggested using a layer model with E =

G2

^{SO that n/tD =} - Writing E~ = ( E ~ + E ~ ) / ~ , AE = E ~ - E ~ , if << 1 then q/tD =

-

ae2/4 E ~ . Other "realistic" continuous profiles

give thicknesses t between 10% and 90% of AE which are fairly close to t ~ . For instance the Fermi (or tanh) profile has

e ^{z / g}

E = --- z,6

,

q = ( ~ ~ - ~ ~ ) 6 l o g - ^€2

,

t = 4 . 3 9 4 6 ;

l + e €1

for << 1, q/6 = - A E ~ / E ~ . Lekner /lo/ and Beaglehole /11/ give examples of other profiles.

Figure 1 summarises values of t and t/d for the liquid-vapour surfaces of Kizel's /6/ homologous series of organic alcohols and acids (at room temperature) and includes values for other liquids measured by the present author. Here t has been derived from

p

data assuming the Fermi profile, Kizel's X (unspecified) has been taken as 6000 A', and d the molecular diameter has been estimated assuming that the molecules are spherical and fill 45% of the volume.

Fig.1 - t/d and t versus d from Kizel's data (Reference 6) and measurements by the present author a

.

The anomalous value t/d = 1, d = 3 AO is for water.

d 2

2

4 6

(%I

'4

0 .

.

_A

0 .

C . & o r

.b

I -

. .

o alcohol

n acid B.

1 I I 1

While there is a systematic trend in t/d with d, the thicknesses are

apparently independent of molecular type and chain-length, suggesting that the surface thickness is determined by the smallest molecular dimension rather than an average size.

As a liquid is warmed the vapour density approaches the liquid density, with equality at the critical temperature Tc where there is no distinction between the two phases. The interface is therefore expected to increase in thickness as AT = Tc-T tends to zero. The temperature dependence of p for liquid argon is shown in Figure 2. /11,12/ t has grown from 7 Ao at 8S°K to 15 A' by 120°K.

Fig.2 - The temperature variation of for liquid argon from Refs. 11,12.

The full and dashed lines refer to capillary models of the interface, see above reference.

5 -

1 I I I

8 0 90 100 110 120

T'K

At high temperatures the pressure becomes large making pmeasurements difficult.

An equivalent almost constant pressure system is found in binary liquid mixtures where the compositions of the a and B phases become equal at the critical temperature, called here the consolute point. Figure 3 shows a measurement of B for the clB interface of the cyclohexane-aniline system /13/, where for AT = 0.3O the corresponding t is about 700 A " .

Fig.3 - The temperature variation of 7 ^{for the} aB interface of the cyclo- hexane aniline binary liquid mixture, Reference 13.

10

5

rr)

0

- -

L

-

I

- -

- - ap

cyclohexone aniline

- ' 2

- .,P'*/

-

^{/ w}

-e--"o O*/

-

" ' "

^{I 1 1 1 1 I I I I I}

10 5 I 0.5 0.1

AT0C

CIO-150 JOURNAL DE PHYSIQUE

This interpretation of p, as we have mentioned, neglects the effects of capillary waves, which will roughen the surface and modify the amplitude reflect- ivities. Indeed, it is not yet clear how best to picture the interface when these waves are taken into account - whether these waves when summed over all wave- vectors to a cut-off hax of the order of the inverse thickness form themselves the total interface assumed in the continuous profile /14/,or whether the surface should be pictured as a continuous intrinsic profile superimposed upon the

waves /IS/. When capillary waves are present it was argued by Beaglehole /11/ that waves with wave vectors between qo

-

^{g 6 2} and qmax produce first order imaginary contributions to the reflection amplitudes leading to an

~ B T ( E ~ - E ~ ) ~ - -

T O

EI+E1

^{%ax '}

where a is the surface tension. This result was subsequently also given by Zielinska Bedeaux and Vlieger /16/; it is somewhat unsatisfactory since it depends sensitively upon qmaX which is not well-defined in the capillary model. When the surface is represented by both intrinsic and capillary terms, these are additive to first order. The estimates for liquid argon suggest that once the capillary waves are taken into account there is little left over for an intrinsic profile, while for the a6 interface the two contributions appear to have about equal magnitudes.

The interpretation of p is subject to a second caution. If the molecules are preferentially ordered near the surface and have an anisotropic polarisability

EX will differ from E, in the surface region. Then /17,18/

and

7

becomes sensitive to surface molecular orientation. Such surface ordering would appear to be the explanation for the anomalous value of t/d for water shown in Figure 1 which was estimated neglecting such a possibility. If the second term in nhas a value corresponding to t/d

-

2, the first term must

be contributing a positive value to q of about half the isotropic magnitude.This leads to an estimate of complete ordering over 10 AD near the surface if the anisotropy in polarisability is taken as 2%, with orientation such that EX>EZ.

Castle and Lekner /19/, and Lekner /lo/ have considered the possibility that local field effects may cause surface anisotropy in E even for spherical molecules such as argon. There is considerable subtlety in cancellations in the theory, but finally little effect is to be expected.

Rigid Surfaces

Here we are interested in surfaces formed by a liquid with an essentially inert boundary, for instance a liquid or liquid mixture in contact with a wall, or the liquid-vapour surface of a binary liquid mixture near the clB consolute point.

The characteristic dielectric constant profile will show a step at the boundary (taking place over a molecular dimension) and then an extended region as E tends to the equilibrium value well inside the liquid. Thus one can associate an qo with the initial step (and this can usually be estimated with fair confidence from pure liquid values or values far from the consolute point) while the major part of r] is contributed by the extended region. If in this extended region E-E;, then /20/

where T, is the integrated deviation of E from E?. For mixtures E depends upon mixture concentration, say as E = ax+b, so that

rE

= ar, where I' = dz(x-x,) is

J ^-

the Gibbls Adsorption, the total excess concentration in the surface region, which is a quantity otherwise difficult to determine. Typical composition profiles and

values for

r

have been given by Beaglehole /21/.

In the case of molecular ordering near the surface we can define an order parameter s(z) _{= k c 3} cos28-1, where 8 is the orientation of a molecule with respect to the z direction. Then E ~ - E ~ = €2 Aa/ai s(z) so that /22/

where

rs

is the total excess surface order, and Aa/ai the fractional polarisability anisotropy.

Early studies of the liquid vapour surface of binary liquid mixtures were made by Rusanov and coworkers /23/, who measured the concentration dependence of ji at a few temperatures for a variety of mixtures. Beaglehole studied the temperature dependence for the cyclohexane-aniline /20,24/ and cyclohexane-methanol /25/

systems in more detail. As the system approaches Tc it becomes easier for one component to be depleted from the surface, and p show a characteristic peak at Tc, Figure 4. 'I was found to be proportional to log AT, which differs from the theoretical prediction of I' = AT-'3/15/.

51°3~

cyclohexane methonol

A divergence of with temperature was also observed with the liquid-vapour surface of the liquid crystal 5CB as the liquid was cooled in the isotropic phase to the isotropic-nematic phase tsansition temperature /22/, Figure 5. This was clearly a log AT divergence in

rs

as the liquid developed surface-ordering above the bulk transition temperature.

2 I

1

^{~ 5 ~ ~ l o g A TFig.5 -} The variation of with temperature for the liquid vapour surface of 5CB. The peak occurs at the isotropic-nematic phase transition temperature

.0155 (Reference 22).

0 A T

4 -

3:

:

^{Fig.4 -} A typical vari-

ation of with temper- ature for the liquid vapour surface of a binary liquid mixture, here

I I I cyclohexane-methanol.

Layer Surfaces

35 40 45 5 0

A5

^{T oC} (Reference 25).

In some binary liquid mixtures it is favourable for a thin equilibrium layer of the heavier 6 phase to cover the liquid vapour surface formed by the a phase.

This is the "wetting" situation. A good approximation for TI then is to use a constant E throughout the layer. Detailed profiles and i? have been calculated by Beaglehole/26/. Recent studies have measured the temperature dependence of the thickness of the wetting layer /27,25,28/,and Beaglehole /25/ and Schmidt and

C10-152 JOURNAL DE PHYSIQUE

Moldover /28/ have confirmed Cahnls p r e d i c t i o n /29/ t h a t t h e w e t t i n g l a y e r should d i s a p p e a r a b r u p t l y a t a temperature f a r from t h e c o n s o l u t e p o i n t .

Another example o f l a y e r s a r e s u r f a c e a c t i v e l a y e r s which occur f o r i n s t a n c e when monolayers o f i n s o l u b l e f a t t y a c i d s form on t h e s u r f a c e of w a t e r . Bouhet /4/

a s p a r t o f h i s e x t e n s i v e s t u d y o f l i q u i d s u r f a c e s h a s observed t h e s e l a y e r s by e l l i p s o m e t r y . Here one can a s c r i b e an a n i s o t r o p i c d i e l e c t r i c constant t o t h e s u r f a c e a c t i v e l a y e r , but a b e t t e r approach i s t o a n a l y s e t h e changes i n r e f l e c t - i v i t y d i r e c t l y i n terms of molecular p o l a r i s a b i l i t i e s /30,31/.

Thick I n t e r f a c e s

For t h i c k i n t e r f a c e s t h e l i n e a r r e l a t i o n between p and is i n a d e q u a t e . The next term i n t h e expansion i s 0 (c/X) 3 , b u t t h e r e i s no general t h e o r e t i c a l express- i o n f o r t h e form o f t h i s term, f o r e i t h e r t h e Drude o r t h e rough s u r f a c e model.

The Brewster a n g l e a l s o d e p a r t s from t h e s t e p p r o f i l e v a l u e , t h e change i n Bgbeing o(€,/A)', and t h i s v a r i a t i o n h a s been given by Lekner /32/. For t h e Drude model it it e a s i e s t t o determine t h e g dependence on t h i c k n e s s by numerical c a l c u l a t i o n u s i n g t h e m a t r i x f o r m u l a t i o n f o r t h e r e f l e c t i o n p r o p e r t i e s o f an a r b i t r a r y , o n l y z dependent p r o f i l e , a s d e s c r i b e d by Abelbs /33/, which breaks t h e p r o f i l e i n t o a s e r i e s o f t h i n s u b - l a y e r s . For t h e f r e e continuous Fermi p r o f i l e t h e s e c a l c u l a t i o n s show t h a t i s l i n e a r i n S t o v a l u e s a t l e a s t o f 6/X

-

^{0 . 1 .} For t h e r i g i d s u r f a c e an example of t h e p v a r i a t i o n w i t h t h i c k n e s s i s shown i n F i g u r e 6 , f o r t h e s t e p e x p o n e n t i a l p r o f i l e where i n t h e extended r e g i o n E = E , + ( E , - E , ) ~ - ~ / Z ~ , and

rE

⁼ ( E ~ - E ~ ) z ~ . For z=0, E = E~ t h e s u r f a c e v a l u e . For zo small t h i s s u r f a c e responds a s an i n t e r f a c e between E, and E,; f o r zo l a r g e a s an i n t e r f a c e between

E,. Thus p r e t u r n s t o z e r o a s zo -t w w h i l e 8g and RS(BB) v a r y between v a l u e s c h a r a c t e r i s t i c o f t h e s e l i m i t i n g p a i r s o f E.

step exponential

Fig.6 - The v a r i a t i o n o f 7 and €IB with

rE

f o r t h e s t e p e x p o n e n t i a l p r o f i l e a denotes t h e l i n e a r approximation between p and

n ,

C.G. t h e Charmet-deGennes approximation. R and

BB

a r e t h e f r a c t i o n a l changes i n Rs(€IB) and Bg between t h e v a l u e s correzponding t o t h e l i m i t i n g p a i r s o f E , s e e t e x t .

Charmet and deGennes /34/ have r e c e n t l y developed a p e r t u r b a t i o n t h e o r y f o r such a r i g i d s u r f a c e p r o f i l e which i s s u i t a b l e when ( E - E , ) / ( E ~ - E ~ ) < < 1.

T h e i r r e s u l t f o r i s given by

q = - i k z w

'E2-E1' r E ( - 2 q ) where r E ( k ) = Re ~ Z ( E - E , ) ~ , and q =

; 6,

cosB,

E 2

i s t h e normal component o f t h e l i g h t wave v e c t o r i n t h e extended r e g i o n . This simple e x p r e s s i o n i s remarkably good i n p r e d i c t i n g t h e v a r i a t i o n o f

5

w i t h zo, Figure 6.

Weakly absorbing and scattering media

Weak absorption causes rs and r to have small imaginary components and thus induces elliptical polarisation evgn for a step profile. Thus the interpretation of

7

in terms of dielectric constant profiles requires the absorption to be small.

Scattering probably acts in a similar manner, and since liquids become opalescent in the region of critical points this effect must be watched with caution. Writing the refractive index of medium 2 as n,+ik, with k, << n,, for the step profile

which sets limits on the amount of absorption or scattering which can be tolerated in comparison with the profile contribution to p.

Experimental Methods

The modulation ellipsometer /35,11/ has proven to be a very convenient instrument to determine

D.

Since it gives the real and imaginary components of r /rs directly, it can be set easily at Bg. Systematic errors in

p

have been

&und in practice to be

-

5x10-', while the noise with a 3 second time constant is less than ~xlo-'. It is extremely difficult to remove residual phase shifts from the container walls, etc. but working at the Brewster angle makes their effects negligible. High vapour pressures can induce inhomogeneous strains in cell walls, which are reduced with spherical cells. The cell must be supported so that mechanical vibrations do not disturb the surface, and a cell shape which is always unstable to liquid convective flows is convenient for variabletemper- ature work.

Conclusions

For transparent liquids the

p

characterisation of the surface is a null measurement, in the sense that a finite value o f 7 indicates deviations from the step profile. This contrasts with other optical characterisations of surfaces such as the reflectivity where it is deviations from finite values, here the Fresnel reflectivity, which depend upon the profile, and in this case the bulk dielectric constants must be accurately known. For the reflectivities, the lowest order deviations are O ( ~ / X ) ~ and are thus inherently less sensitive to the profile, and depend not on the single rl characterisation but on two others in addition /32/.

Again p measurements determine the sign of

n ,

which provides variable information about the surface, which is lacking in intensity measurements.

Acknowledgements

Dr John Lekner stimulated the author's interest in liquid surfaces, and has been a constant source of encouragement and assistance which it is a pleasure to

acknowledge. The computer programme for the numerical calculations was developed by B.M. Law (Reference 36).

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