Capillary waves and ellipsometry experiments

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Capillary waves and ellipsometry experiments

D. Bonn, G. Wegdam

To cite this version:

D. Bonn, G. Wegdam. Capillary waves and ellipsometry experiments. Journal de Physique I, EDP

Sciences, 1992, 2 (9), pp.1755-1764. �10.1051/jp1:1992242�. �jpa-00246657�

J.

Phys.

I France 2 (1992) 1755-1764 SEPTEMBER 1992, PAGE 1755

Classification

Physics

Abstracts 68.10

Capillary

_waves

and ellipsometry experiments

D. Bonn and G. H.

Wegdam

Laboratory

for

Physical Chemistry, University

of Amsterdarn, Nieuwe

Achtergracht127,

1018 WS Amsterdam, The Netherlands

(Received 4 December 1991, revised 2

May

1992, accepted 2 June 1992)

Abstract. The inclusion of

higher-order

terms in the

capillary-wave

Harniltonian may reduce the contributions of these fluctuations to the

ellipsometric

coefficients. We show that the

renormalization of

capillary

waves at a fluid-fluid interface

by Sengers

and van Leeuwen

[Phys.

Rev. A 39 (1989) 6346]

using

the _wave vector-dependent ^surface tension that follows from the

coupled

mode

theory by

Meunier [J.

Phys.

France 48 (1987) 1819]

yields

_a

satisfactory

_agreement

with recent

ellipsometry

measurements

by

Schmidt

[Phys.

Rev. A 38 (1988) 567]. The interface is

viewed upon as an intrinsic interface broadened

by capillary

waves. We suppose ^{that the} cutoff

wave vector q~~~ that follows flom

mode-coupling theory

marks the dansition flom the short-

wavelength

bulk-like fluctuations that contribute _to the bare surface tension to the

long-

wavelength

capillary

wave-like fluctuations that contribute to the full surface tension. This enables _us to calculate, without any

adjustable

parameters, ^{both the ratio of the bare and} experimental surface tension and the universal _constant for the elliptical ^thickness of the interface. Both agree

remarkably

well with

experimental

values.

Introduction.

In the absence of _a suitable

first-principle theory

for the

description

of fluid-fluid

interfaces,

_a number of models have been introduced in order to calculate in

particular

the dielectric

profile. Having

found this

profile,

_{one can} in

principle

account for

ellipsometry

and

reflectivity

measurements of the fluid interface. The mean-field

theory

of _van der Waals

[Ii,

later refined and discussed

by

Fisk and Widom

[2], predicts

an

order-parameter profile

whose thickness iS _a function of the bulk correlation

length only. Also,

_aS has been Shown

by Buff,

Lovett and

Stillinger [3], capillary

_waves have to be

incorporated

in the

theory.

In tile

capillary-wave description

of the interface the

largest

and smallest scales of tllese _waves, q~i~ and q~~,

respectively,

have _to be defined to calculate the order

parameter profile.

The

unification of these two

points

of view _was

recently

reconsidered

by Sengers

and _van Leeuwen

[4], using

_a renormalization in q-space, where the difference between the bare and the

macroscopically

observed surface tension is attributed to the

capillary

waves.

Recently,

this

length-scale problem

_was also reconsiderd

by Meunier, using mode-coupling theory [5]. By

considering higher-order

_terms in the

capillary-wave Hamiltonian,

Meunier finds _a mechanism

for the

coupling

between the _waves. This

theory yields

_an

expression

for the _{wave vector-}

dependent

surface tension up to fourth order in q, which

implies

a natural cutoff _wave vector q~~ for _a

capillary-wave

dominated interface. The

q~-terra

thus _ensures convergence for the

integral

_over the

capillary

wave-fluctuations in d

=

3. In this paper we

show,

that when this surface tension is

renorrnalized, using

the q~~ obtained from

mode-coupling-theory,

both the surface tension and the

ellipticity

coefficients agree ^{with the}

experimental findings by

^Schmidt

[6]

and Moldover and co-workers

[7].

Renormalization of the surface tension.

In the

commonly

used

description

of _a fluid-fluid

interface,

_one

pictures

a thick

interface,

described

by

the Fisk-Widom

[2]

intrinsic

profile,

broadened

by thermally

excited

capillary

waves. The

roughness

^{of the interface due} to the

capillary

_waves is then

usually

described _{as a}

sum of Fourier

components ranging

from q =

0 _{to an}

empirical short-wavelength

cutoff q = q~~~. In _{a recent}

publication [4] Sengers

and _van Leeuwen discussed _a renormalization

procedure

to determine the

large

_wave vector cutoff q~~.

Starting point

in their paper is the column model for _an interface _as described

by

Weeks

[8] and, later, by

Evans

[9].

These

considerations

yield,

within the framework of mean-field

theory,

the

long-ranged

correlations in the interface _as

predicted by

Wertheim

[10],

and

thus, justify

the addition of

capillary

_wave

fluctuations _{to an} intrinsic interface.

Following

_a

suggestion by Kayser [I I]

the contribution of the

capillary

_waves to the total _excess free energy

density

is calculated

by

_a renormalization of the surface tension. This is achieved

by considering

the surface tension _{as a} function of

q~

and

proposing

that the shorter _waves renormalize the surface tension for the

longer

_waves.

Without

assuming

_an

explicit

functional form for the _wave vector

dependent

surface tension

«

(q~), Sengers

and van Leeuwen then solve the differential

equation

obtained for the surface

tension, using

the

boundary

condition that

«(q~)

_merges

_smoothly

_{with the bare surface}

tension «~ ^at q =q~~. In this way, on

integrating numerically

from different upper

boundaries

q~

=

q$~~

to

q~

=

0,

_one

gets

^the

dependence

of the

macroscopically

observed

surface tension

«(0)

_on the cutoff _wave vector q~~~.

Using experimental

values for

«(0),

the

empirical

cutoff _can then be evaluated. For further details _we refer to their article.

Mode

coupling.

Another

approach

to the

length-scale problem

_was followed

by

Meunier. In _a recent

publication [5]

he derived _a

coupled-mode expression

for the _{wave vector}

dependent

surface tension up ^to fourth order in q. The excess free energy associated with _an interface is then the

sum of _a

capillary

and _a

rigidity

contribution. The variation of the surface tension with the

wave vector can be obtained

through

_an

expansion

of the surface

fluctuations,

for small distortions of the surface. The

probability

that _a mode _q has _an

amplitude (~

^{is then} a function of all the modes of _{wave vector}

q'

_{~ q.} In this way, Meunier obtains the differential

equation

for the variation of the surface tension with the _wave vector :

d«

(q)

=

2 «q

dq ₍₁₎

where

« = 3/8 gTk~ ^{T is the}

rigidity

modulus of the interface

[5]. Integrating

this

function,

_we

can write for the average of the

squared amplitude

of the mode q :

(t()

=

~

~ ~.

(2)

6Pg +~"~

+ "q

This form

implies

a natural cutoff _wave vector q~~~ =

(«/«)~'~

_for the

capillary-wave

N° 9 CAPILLARY WAVES AND ELLIPSOMETRY EXPERIMENTS 1757

fluctuations in the interface. In this

view,

the

coupling

between the modes introduces _a natural cutoff for _a

capillary-wave

dominated interface.

Surface tension.

In this _paper the

expression

for the surface tension associated with the

capillary

_waves

obtained from

mode-coupling theory

is renorrnalized in the

Sengers-van

Leeuwen scheme. To calculate the contribution of the

capillary

waves to the

macroscopically

observed surface

tension

«(q

₌

0)

in the column

model,

_one divides the surface into unit cells of side

=

2w/(q~~~ f),

where q~~ is the cutoff _{wave vector} and

f

denotes the correlation

lengtll.

Following

the renorrnalization scheme of reference

[4],

_we use the

boundary

condition that the surface tension at the scale of the unit cell

equals

^the bare surface tension that follows

from the intrinsic

profile. Starting point

in _our discussion is the differential

equation

for the variation of the surface tension due _to the

capillary

waves :

d«cw(q)

=

2 «q

dq (3)

Using

the

boundary

condition that

«(q~~)

_{= «~.}

"cw(~)

" "b + "

(~~ ~lax) (4)

or, in terms of the reduced variables _:

iF~~(x)

= iF~ + R

(x

_x~~~

) (5)

where § _~

~2/~

~

R =

«/k~

_T~

(6)

and _x

=

(qi

)2

From references

[4]

and

Ill

the

change

in surface tension due to the

capillary

waves can be written _{as :}

kB

T

«~~~~

(q )

_{= «~ j} ^In

Z~~ (7)

L ^~

where «~~~~ is the

experimentally

measurable surface

tension,

d is the

dimensionality

and

Z~~

is the

partition

function associated with the

capillary

_waves

given by

_:

In

Z~~

=

~'~ ~~"

d(q')~

ln

~"~'~

_~

_~~~

_~~~'~~

(8)

2(2 w)~~

_q2 2

~rkB

^T

In first

approximation

the

capillary

_wave Hamiltonian in the

integrand

contains the bare surface tension «~,

independent

of the _wave number. As _was

pointed

out

by Stillinger [12], however,

the surface tension is in

reality expected

to be _a function of the _wave number q.

By considering

the

coupling

of the

modes,

Meunier finds _an

expression

for tile _wave vector-

dependent

surface tension up ^to fourth order in _q.

Contrary

_to the

procedure

followed

by Sengers

and _van Leeuwen _{we can use} this

expression

and

perform

the

integration

in

equation (8) directly.

In reduced units _:

i

~mm

_~,

(&cw(x')

x' +

6P#) (ii')2

"~~P~~~~ ~

~~

~

G

~~~

~

'~ 2 _w

Where

I

= 2 w/@~~~(@~~~ = q~~~

f), I'= ₍₂ if~)~"~

and

6pj

is the dimensionless

gravity

contribution. For further

details,

_see reference

[4]. Performing

the

integration using equation (5)

for

«~~(x') yields

_:

where _a

= R

(11')~/2

w

j3 ₌

(iF~ Rx~~) (11')~/2

ar

and _y

=

6pj (11')~/2

_w.

By setting

the lower

boundary

x ₌ 0 _{we can now} calculate the

integral

which

yields

the

dependence

of the

macroscopically

observed surface tension _on the choice of x~~. In

figure

I

we

plotted

tile surface tension _{as a} function of the cutoff. At tile natural cutoff

given by Meunier,

_q~~

=

1/(2.55 f),

we find

if(0)lif~

= 0.80. This is in

quantitative

agreement ^with the data used

by Sengers

and _van Leeuwen

[4, 7, 13],

who find

&(0~lif~

=

0.82±

0.15. In

figure

2 _we

plotted

the

dependence

of the renorrnalized surface tension

given by equation (10)

and the _«

mode-coupling

_» surface tension

(Eq. (5))

on the reduced _wave vector.

The

striking

result is

that,

after the

renormalization,

the variation of the surface tension with the _wave vector remains almost unaltered _: both calculations differ

by only

_a few

percent

in the

large-wavelength

^{limit. Note}

that, by choosing

the

boundary conditions,

_we made the two

curves coincide _at q = q~~~.

Thus, using equation (3)

^{for the} variation of the surface tension

due to the

capillary

_waves, and

choosing

the

appropriate boundary conditions,

_{we are} able _to renormalize the

capillary-wave

Hamiltonian. In this way, we obtain the total _excess free energy

density

_{as a} function of both the cutoff _wave vector and the _wave vector itself. This renormalization enables _us to

unify

the intrinsic

profile

and

capillary

_wave

description

of the interface. In _our view this _means that the

mode-coupling expression gives

_us the

capillary-

wave induced variation of the surface tension with the _wave vector, whereas the renormalized

fi n

16~~

_{o_}

o-1 02 0.3 0A 0.5 0.6

X~ax

Fig,

I.-The calculated

long-wavelength

surface tension relative to the bare surface tension _{as a} function of the reduced

squared

cutoff wave vector x~~ = (q~~~

f)~.

_{For q~~~}

= 1/(2.55 f_ ) _we find

«~~~~(0)/«~

= 0.80.

N° 9 CAPILLARY WAVES AND ELLIPSOMETRY EXPERIMENTS 1759

I

it

o-I o.15

Fig.

2. Solid line _: the reduced normalized surface tension &~~~~(x) = «~~~~(x)

f~

_{/kB T~}

as a function of the reduced _wave vector. Dashed line: the result of

mode-coupling theory:

&(x)=

J~

+ R(x

x~~).

Both calculations differ

only by

_a few percent ⁱⁿ the

large-wavelength

limit.

surface tension is the _one that would be measured

experimentally.

In the next two sections _we shall compare these

findings

with

experimental

measurements of the interface thickness.

Ellipsometry experiments.

The

ellipticity

p is the

imaginary _part

of the

reflectivity

ratio

r/r~

and is a measure of the

optical

thickness of _an interface. At the Brewster

angle

p ^is

given by [14]

:

(~2

_~

~2

_)l12 p ₌ Im

(r/r~)

=

'~ _~

~

(ll)

A

(n$ n~ )

where _~ is _a

system-dependent

characteristic

length

_:

~ =

dz ^~~~~~~ ^~~

~~~

^{~~ ~}

⁽

^l

2)

n

(z )

In the Fisk-Widom

theory

of the interface the

profile

of the refractive index is

given by

_:

n(z)=j(n~ +n_)+j(n~ -n_)X(fi). ₍₁₃₎

Where X

(y)

denotes the order

parameter profile,

which is

given by

:

xiii _-/tarn lil13-tanh~li)l~'~ ₍₁₄₎

and

f_

is the bulk correlation

length

the

two-phase region.

In order to compare results from different

systems

one has to remove the

dependence

of the

ellipticity

_on the correlation

length

and refractive indices of the

phases. Therefore,

_{one can} define the

quantity [6]

_:

XMF =

j _{II X~(y)]}

dy

_m 2.28

(15)

for the

system-independent

mean-field contribution to the

ellipticity,

where X

(y)

is

again

the

order-parameter profile

in the Fisk-Widom

theory

of the interface.

The

capillary-wave

contribution to the

ellipticity depends linearly

_{on q~~~,} and is

given by [15, 16]

:

n( n~

lkB ^T~

^q~~~

p ₌

3/4(

w/c

(l 6)

~~

(n(

₊

n~

_)~'~ "b 2 _~r

The total

ellipticity

_can then to a

good approximation

be written _{as :} p~~~ =

~'~

f

An

(XMF

+

x~~) (17)

A

where x~ has _a value of

m 2.28 and Xcw is

given by [17]

:

Xcw =

3 AM _w &

(18)

Where A

= q~~

f

and &

=

«f~/kB _T~. According

to

Sengers

and _van

Leeuwen,

A

=

0.748 and &

= 0.128. This

yields

_{Xioi = XMF} + Xcw m 3.67. However, the

experimental

data indicate that this constant should have _a value of about 3.0. Schmidt

[6]

shows that the

ellipticity

data from 3 very different

binary liquid

mixtures _can all be scaled to this universal constant.

Using

the natural cutoff q~~ =

1/(2.55 f)

from

mode-coupling theory,

_we find A

= 0.39. This

yields

_Xcw

=

0.74 and thus, _Xioi

=

3.02,

in excellent agreement with

experimental

data.

Reflectivity.

In recent papers on the connection between the

capillary-wave

Hamiltonian and the

optical properties

of fluid-fluid

interfaces,

much attention has been

paid

to the

reflectivity

measurements of the fluid interface

[4, 17-19].

The

reflectivity,

relative to the Fresnel

reflectivity,

is

given by

_:

R/R ~ ⁼ exp

(-

4

q~ (L _{~) )} (~~~

with q the _wave number and

(L _~)

^~'~ the interfacial

thickness, equal

to the _r.m,s.

displacement

of the interface in

capillary

_wave

theory,

and

proportional

to the bulk correlation

length

in mean-field

theory.

In

optical

^and

X-ray reflectivity experiments

_one

usually interprets

the measured thickness in _a combined mean-field and

capillary-wave

model _:

<Lit)

=

<Llw)

+

<Lsfi (2°)

where L~~~ is the measured total interfacial thickness. From

capillary-wave theory,

(Lj~)

is

given by

^the r-m-s-

displacement

:

(Lj~)

=

~~~

ln~(~~ ⁽²¹⁾

~ _~"

~in

Where q~~~ is a lower instrumental cutoff due _to the finite collection

angle

of the

experimental

setup

[20].

As _we have _seen in the

previous

section, the

integral

over the Fisk-Widom

profile yields L~~

= 2.28

f. Using

the relation if

=

«f~/kB _Tc,

where & is _a universal constant, we can write

equation (21)

close to the critical

point

_{as :}

N° 9 CAPILLARY WAVES AND ELLIPSOMETRY EXPERIMENTS 1761

which

yields, apart

^{from the}

logarithmic factor,

_{a r,m.s.}

displacement

that is also

proportional

to the bulk correlation

length.

Thus, the measured thickness _can be

expressed

_{as a} function of the bulk correlation

length only. Then,

if the

large-wavelength

cutoff q~~~ is known, the

comparison

with the

experiments

_can be made.

In their

optical reflectivity experiments

_on

cyclohexane/methanol

and

SF6, Beysens

and

Robert

[19]

estimate the

experimental

cutoff _q~i~ to be 1000cm~~

Using

_q~~~=

1/2.55f_

in

equation (22),

_we find

L~~m2.83f_,

and

thus,

from

equation (20)

_:

(L(~~)

^~'~ =

( (2.83 f_

)~ +

(2.28 f_

_)~)~'~

=

3.64

f_ Using

the known correlation

length

for the

binary

fluid system

cyclohexane-methanol f°

= 1.65

A,

we can compare this _to their

reflectivity

data

(note

^that their L~~~~ is twice _our L~~~). This

yields

2 L~~~ =

12.0 _t~ ~

(A _),

where t is the reduced temperature, t ₌

(T~ T)/T~

and _v

= 0.625. This is in excellent agreement with their data:

L~~~~

=12.4t~°.~~.

_{For the} measurements on

SF~ they

obtain

L~~~~ ⁼ 10.4

t~°.~~ (A).

_Our calculation

yields

2 _L~~~

= 8.3 t~

~(A),

in reasonable agreement.

Very recently,

_a number of

X-ray reflectivity experiments

_were

performed

_on

simple liquid/vapor

interfaces

[21, 22]. Among these,

Braslau _et al.

[21]

measured the

specular reflectivity

of

X-rays

from the surfaces of _water

(H20),

^carbon tetrachloride

(CC14)

and methanol

(CH~OH).

In

interpreting

their

data,

Braslau et al, calculated the contribution of the

capillary

_waves to the interfacial thickness

explicitly

^and subtracted this from the

measured value to obtain

L~~. However, fitting

the data for methanol in this way,

they

obtained _un

unphysically

small value for

L~~

of

roughly

one-third of the molecular diameter.

This Braslau _et al, attributed to either _an enhancement of the surface tension at these small

scales,

_or to the wrong choice of q~~, for which

they

chose q~~=

gT/r~,

where r~ is the molecular diameter. In _our

model, using equation (22)

_we can express the thickness

as a function of the bulk correlation

length.

A sensible estimate for the correlation

length

_can

be made from the

experimentally

established relation

[6] «o(f° _{)~/kB T~}

=

0.1047. For water

(«o

=

218 mNm~

[7])

this

yields f°

=

0.65

A, _{and, using f_}

=

f°

_{t~ ~, with}

v = 0.625 _we find

f_

₌ 0.95

A

_at

room

temperature. Thus, assuming

critical behavior up ^{to a} reduced temperature ^t m

0.5,

_{we can} estimate the bulk correlation

length

of the three

liquids. Using

the values for q~~~ from reference

[2 Ii,

_{we can compare}

equation (20)

with the measured values for the interface thickness. The results _are summarized in table

I,

from which it is clear that the

measured thicknesses _can

successfully

be accounted for

by

the combined mean-field and

capillary-wave

model without any

adjustable

parameters. ^Residual differences _can

possibly

be resolved

by

measurement of the bulk correlation

length.

Table I.

Assuming

critical behavior up ^to a reduced temperature t _m

0.5,

_an estimate

of

the correlation

length

_can be made and the theoretical calculation

ofequations (20)

and

(22)

can be

compared

with

experiment.

"0

(fllNlTl~~) #_ (A)

_qmin

(A~~) (L$t) ^(A) _(L)xpt) ^(A)

H20

218.0

[7]

0.95 4.5 _x

10-4

3.4 3.3

CC14

67.7

[23]

1.74 9.I x lo-S 6.5 6.0

CC14

67.7 1.74 4.5 _x 10-4 _6.0 5.1

CH30H

86.8

[24]

1.56 1.0 _x 10-3 5.3 4.8

Also,

from table I _we observe that the values calculated for the thickness _are somewhat

larger

than the measured values. The introduction of _a

rigidity

« leads _{to a} correction _on the

reflectivity.

The _r-m-s-

displacement

in _c-w-

theory

is then

given by [23]

_:

(Lj~)

=

~~

^~

ln

~)°~ /

(23)

"" qmin I + qmax "/"

Setting

_«

= 0

yields

the usual

expression

for the

reflectivity

and the introduction of _a

rigidity

thus reduces the calculated values of the _mean

squared displacement.

The introduction of the

rigidity

_{« =} 3/8 gTkB T shifts the calculated

reflectivity

in the

right

direction

by

_an amount of

approximately

O-I

A,

_{a very} small effect that _can

obviously

not be observed in this

experiment.

Measurements of the

optical properties

of _more

complicated

systems ^with a

vanishing

interfacial tension

(e,g.

_an

oil/water/surfactant/salt system)

where the surface tension _can be tuned

by varying

the salt concentration _so that the

rigidity

term dominates the

capillary-wave

Hamiltonian

might provide

_{an answer} to this

problem [25]. Finally,

it should be noted that the

reflectivity depends only logarithmically

_on both the cutoff wave vector and the

rigidity

modulus. Given

this,

it _seems difficult to draw any

rigorous

conclusions about the actual values of either the

large-wave

vector cutoff _or the

rigidity

constant

by fitting equations (20)

and

(25)

to the data.

Still,

the data _can be fitted

fairly

well _{to a} combined mean-field and

capillary-wave model,

without any

adjustable

parameters.

Discussion.

The Hamiltonian of

capillary

_waves at a fluid-fluid interface iS discussed. In first

approxi-

mation this Hamiltonian contains the bare Surface tension «~,

independent

of the _wave

number. AS _was

pointed

out

by Stillinger [12],

the Surface tension is in

reality expected

to be _a function of the _wave number q.

By considering

the

coupling

of the

modes,

Meunier

[5]

finds

an

expression

for the _wave

vector-dependent

Surface tension up ^to fourth order in q. This model holds for _a thin but

rough,

I-e-

capillary

_wave dominated

interface,

and

imposes

a

natural cutoff _wave vector q~~~. In order to

unify

the intrinsic

profile

and

capillary-wave

theories of the interface

Sengers

and _van Leeuwen

[4] give

_a formalism based _on the column model

by

Weeks

[8],

to renormalize the _wave vector

dependent

surface tension.

Having

found

equation (3)

^for the variation of the surface tension with the _wave vector _we can write

using

the

boundary

condition that _«

(q(~~)

= «~.

«

(q2)

= «~ ⁺ «

(q2 qj~~ ) (24)

where _we have _a cutoff q~~ =

1/(2.55 f)

from the

mode-coupling theory. Having

found _an

explicit expression

for the _wave

vector-dependent

surface tension, the renormalization in _q- space can be

readily implemented.

^{The full surface tension} can then be written _as the _sum of the bare surface tension and the

integral

_over the

capillary-wave

like fluctuations.

Integrating

this function from _zero to q~~~

yields

the

dependence

of the full surface tension _on the cutoff

wave vector q~~~.

A

viewpoint

_{one can now}

adopt

^is that at

large

scales the interface _can be viewed upon as

capillary-wave dominated,

whereas the intrinsic

profile

describes the

microscopic configur-

ations that contribute _to the bare surface tension. In this

view,

the natural cutoff that follows

directly

from

mode-coupling theory,

marks the transition from the short

wavelength

bulk-like

fluctuations to the

large wavelength capillary-wave

like fluctuations. We obtain the

macroscopic

surface tension _{as a} function of the cutoff _wave vector if these fluctuations _are renormalized in the

Sengers/van

Leeuwen

scheme, following

the

viewpoint

^of

Kayser [I I]

that the shorter _waves renormalize the

macroscopically

observed surface tension for the

longer

N° 9 CAPILLARY WAVES AND ELLIPSOMETRY EXPERIMENTS 1763

ones.

Intersecting

this _curve with the cutoff from

mode-coupling theory,

_we find

&~~~~lif~ ⁼

0.80,

in

good

_agreement with the

experimental

values

by

Moldover and co-workers

[7]

and Gielen et al.

[13]

who find if~~~~lif~ = 0.82 ± 0.15.

Also,

the

findings

_{agree very} well

with

ellipsometric

data obtained

by

Schmidt

[6]

on

binary liquid

mixtures. Schmidt shows that the

ellipticity

data from three very different

binary liquid

mixtures _can all be scaled to the

same universal constant. Since this constant

depends linearly

on the cutoff _wave vector, this is

a more severe test to determine what the value of q~~ should be than the

comparison

of the

theory

with

reflectivity data,

where the measured

reflectivity depends only logarithmically

_on

the cutoff _wave number. Another

approach

to this

problem

was followed

by

Blokhuis and

Bedeaux. The introduction of _a

rigidity

ⁱⁿ the

capillary

_wave Hamiltonian leads to a different

expression

for the universal

ellipticity

constant

[26]

:

3 _~ ^l12

~~~ ~ _" (Cr~ K

)~~~~~~~

~~~~

°~b

~~~~

Blokhuis and Bedeaux then

adjust

the values of the

rigidity and, consequently,

the cutoff

wave vector so that the

ellipticity

constant

equals

the _one found

experimentally,

X =

0.72,

and find @~~~ =

0.602,

R

=

1.2

[26].

However, if _we take the value for the

rigidity

constant _{« =} 3

kB

^T/8 _w that follows from

mode-coupling theory

_we find the

ellipsometric

constant Xcw = 0.81 which

yields

_Xtot

=

3. I.

Comparing

this with the data from reference

[6],

we still _{see a}

good

_agreement with the

experimental

data. This agrees with the

suggestion

from de Gennes and

Taupin [27]

that the total

rigidity

should be of the order

kB

T, but differs

about _one order of

magnitude

with the value found

by

Blokhuis and Bedeaux. We have _no

explanation

for this

discrepancy. Also,

the

comparison

with

reflectivity

data does _not

yield unambiguous

values for the cutoff _{wave vector} and

rigidity

constant.

The calculation

presented

here

yields

_a

satisfactory

_agreement between

theory

and

experiment,

without any

adjustable

parameters and shows that the choice of the cutoff _wave

vector q~~

=1/(2.55 f)

from

mode-coupling theory

is

supported by

the

finding

that the

ellipsometric

_constant, which

depends linearly

_{on q~~,}

equals

the _one found

experimentally

by

Schmidt _X

=

3.02,

whereas the

theory

of

Sengers

and _van Leeuwen

yields

_X

= 3.67.

Acknowledgements.

This work is part ^{of the research} program of the Foundation for Fundamental Research of

Matter and _was made

possible by

financial

support

from the Dutch

Organization

for Scientific

Research. We would like _to thank J.

Meunier,

E.

Blokhuis,

D. Bedeaux and J. V.

Sengers

for

helpful

discussions and

correspondence.

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