Quantitative interpretation of anomalous adsorption effects in a critical binary mixture

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Quantitative interpretation of anomalous adsorption effects in a critical binary mixture

G. Zalczer

To cite this version:

G. Zalczer. Quantitative interpretation of anomalous adsorption effects in a critical binary mixture.

Journal de Physique, 1986, 47 (3), pp.379-382. �10.1051/jphys:01986004703037900�. �jpa-00210216�

G. ZALCZER

Service de Physique des Atomes et des Surfaces, ^Centre d’Etudes Nucléaires de Saclay, ⁹¹¹⁹¹ Gif-sur-Yvette Cedex, France

(Reçu ^{le 23} juillet 1985, révisé le 20 dgcembre 1985, accepté ^{le 20} dgcembre 1985)

Résumé.

Par simulation numérique ^d’une pré-

cédente expérience montrant l’existence d’une anomalie critique de la couche de

mouillage ^dans

^un

mélange binaire,

^nous

avons

montré _{que les} données sont compa- tibles

avec

le profil prédit par Fisher ^et de Gennes

avec un

exposant compris ^entre 0.3 et 0.6, ^{et aussi}

^{avec un}

profil

expo-

nentiel dont l’amplitude ^varie

^comme

t03B2.

Abstract.

By performing

^a

^numerical simula- tion of

a

previous experiment showing ^the

existence of

a

critical anomaly ^of wetting layers ⁱⁿ

^a

binary mixture,

^we

^have shown that the data

are

consistent with the pro- file predicted by ^Fisher and de Gennes with

an

exponent in the range 0.3-0.6,

^as

well

as an

exponential profile ^with

^an

amplitude varying

^as

t03B2.

Interaction of

a

binary mixture with

a

surface.

Binary liquid ^mixtures exhibiting

^a

critical phase separation point (at tempe-

rature Tc and concentration c) ^{have been} widely ^studied

^as

^model systems ^of phase

transitions since they ^allow precise expe- riments and belong ^{to the}

^same

^universa-

lity ^class

^as

^{the 3} dimensional Ising

^mo-

del / ¹ /. ^{The order} parameter ^{is the}

^con-

centration difference 0 - Oc. ^{Close to the}

critical point the coexistence

curve

is

expressed as 1$ - wc) m IT - Tcla ^with

a 0.325 and the correlation length ^at oc

as E0 IT/Tc - 1 I-v with v = 0.63.

M.E. Fisher and P.G. de Gennes

re-

cently ^studied / ² / the interaction of such

a

system ^with

^a

^solid wall, ^where

^one

of the two components (N) ^is preferentially

adsorbed. They

^were

^{able to show} that,

^as

a

function of the distance

z

from the wall, the concentration profile ON ^{of N behaves}

as

follows :

i) for

z ~

ao,

^a

typical molecular dis- tance,

^a

^normal adsorption phenomenon

("proximal" regime) occurs, which depends

on

the chemical nature of the system.

ii) for

^z >

ao,

^a

universal behaviour is

observed : 0 - $b =-(o - b) S (z, E) , ^where

at T = Tc, ^the profile decrease

^as

^{z-b with}

b = S/v (wb ^{is the bulk} concentration). ^At T # _Tc this perturbation ^is destroyed by

the fluctuations beyond ^{the length E, and} replaced ^then by

^an

exponential profile

exp (- z/E) (Fig. ^{1), with}

^an

amplitude proportional ^to tvb

⁼

t8.

The model proposed by ^D. Beaglehole / ³ / leads to

a

rather similar profile

but with

an

exponent of 1 instead of 8/v 0.516.

Several experiments have been per- formed / ^4-6 /. ^The results of ref. / ⁵ /

are

sensitive only ^to the first moments of the

excess

concentration. A direct evalua- tion of the first moment exponent p =v(2-b) is possible ^but

^no

further information

on

the profile

^can

^be obtained. The value p = 0.88 + ^{0.10 obtained there} corresponds

to b - 0.60 + 0.15. The results of ref.

/ ⁶ /

^are

shown ^to be consistent with the Fisher-de Gennes model but the value b = 1 is not strictly ^excluded.

The results of ref. / ⁴ / ^{have not}

been quantitatively interpreted yet. ^Indeed

the data cannot be simply ^{related to the} parameters ^of

^a

^{model. On} the other hand

they might contain additional information,

even

though features much smaller than the

wavelength ^of light cannot be directly put

in evidence. In this letter, ^{I shall} pre- sent

a

numerical simulation of this expe- riment, and show that is results

are con-

sistent with

an

exponent b in the range

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703037900

380

Fig. 1 - Sketch of the concentration pro- file showing ^the three behaviours.

I) w = 1 close to the wall.

ii) -b power law decrease ^at intermediate distances.

iii) exponential decay far from the wall.

For the computation, the power law has been extended to lo, where it reaches the values (() = ¹ (dotted line).

0.3-0.6 and check whether

more

detailed information

can

be extracted.

Recall of the experiment.

The principle ^{of the} experiment (Fig. ^{2) is}

^as

following,. ^A S-polarized

laser beam is shone onto

a

cell containing

the critical mixture

so

that total reflec- tion

occurs.

The fluorescence of the

excess

component ^excited by ^the field is measured.

However, ^{due to the} concentration profile,

the condition for total reflection is not met at the wall-liquid interface but in the liquid ^at

^a

^location depending

^on

^the profile, ^which greatly increases the

sen-

sitivity ^{to the} long range part ^{of the} profile.

On the other hand the molecular size first layer will neither affect signi- ficantly ^the propagation ^{of the} light

^wave

nor

contribute appreciably ^{to the total}

fluorescence intensity.

The profile is therefore mainly ^de-

fined _{(Wo -} by _Wb) ^the _ao exponent ^{b and the product} which

^we

^{choose to express}

as

(1 - wb) 10 b ^where 10 will be the adjus-

table parameter. The actual profile ^used

is wN = ^{1 for} lo ^and N = ^{(1 -} b) (z/l )-b

for

z >

lo ^(the

^{same as}

^{used in / 6 /).}

Fig. 2 - ^a) Refractive index profile ^and

value k /ko ^{crossing at the point Ztr where}

the condition for total reflection is met.

b) Path of

a

light ray, in ^terms of geome- trical optics. ^c) Electric field profile

deduced from a).

The mixture is the well studied ni- trobenzene and n-hexane one, with nitro- benzene attracted by the silica wall / ⁷ /

(wN ^will

^now

denote the nitrobenzene volume fraction). The values of the parameters

were

taken from / ⁸ /. ^The length Eo ^is 2.65 A. The refractive index had to be

cor-

rected for the different wavelength ^and

^was

taken

as

Computation.

If

we

choose the y axis in ^the plane

of incidence, the electric field obeys ^the propagation equation :

and it

can

be considered

as

the product ^of

a

y-dependent ^factor F(y) obeying

and

a

z-dependent

^one

^G(z) obeying

where

n

stands for therefractive index and k the wave-vector in

vacuum.

0

This last equation

^can

^{be solved} only numerically. When its second member is negative the solution is

a

propagative

wave, otherwise

an

evanescent

wave.

The refractive index profile and the electric field

are

sketched in Fig. ^{2. At the wall} discontinuity, ^the boundary conditions for

s-polarization require the electric field

and lo, ^this computation ^is performed ^for

all relevant experimental points, ^{the "ar-} bitrary unit" factor optimized ^{and the}

^mean

quadratic ^deviation printed.

Results.

The calculation has been performed

for

a

set of exponents ⁱⁿ the range 0.2-1 and for lengths lo in the range 0.05 A - 20 A (down to 3.10-3 Â for b = 0.2). Table I reports ^the

^mean

quadratic deviation of the fits to the experimental ^{data. The}

framed

area

encloses the region ^{where the}

b and lo ^values

^are

consistent with the

experiment. ^This region includes the Fisher- de Gennes theoretical value (0.516) together

with reasonable values of the lengthlo(1-5A)

but clearly excludes the value of 1 for the exponent. The deviations of the fits

are

reported ⁱⁿ Fig. 3 for different expo- nents and the best lo value for each expo- nent, illustrating the results

are

consis- tent with 0.3 b ^0.6.

Computations ^for sightly ^different

incidence angles have also been performed,

but allowing ^for

^an

uncertainty

^on

^this parameter ^{does not} appreciably ^{widen this}

range.

In order to check the sensitivity ^to

small size features,

^we

have tried simple exponential profiles, ^with

^an

amplitude proportional ^to tB, ^{similar to} the Fisher- de Gennes profile ^{at distances} greater ^than E and with the

same

moment variation. The

mean

square deviation between the data and the model

can

be made

as

small

as

the best values of Table I by

^a

suitable choice of the amplitude prefactor. ^This experiment

does not prove directly the existence of the power law part ^{of the} profile. ^{It is}

worth noticing however that the field pro-

pagation equation ^{for the} p-polarization

involves also the dielectric constant gra- dient. Calculations and experiments ^{in the}

purpose of investigating ^this possibility

are

presently under way.

Conclusion.

By simulating numerically the expe- riment of Beysens ^{and Leibler} using

^a

para-

metrically defined concentration profile,

we

have been able to show that the profile exponent b lies in the range 0. 3 b _ ^0.6,

in good agreement ^{with the} Fisher-de Gennes prediction ^{and is} disagreement with that of

Fig. 3 - Plots of the calculated and expe- rimental values (l.h.s.) and their diffe-

rences

(r.h.s.) for different values of the exponent b and the best value of lo ^for

each exponent. ^The dotted line through ^the

calculated points ^is only

^a

visual aid.

TABLE I - Mean square difference between

experimental and calculated data. The

framed

area

encloses the acceptable

region ^{for b and} 10.

382

Beaglehole. ^An empirical exponential pro- file, ^with

an

amplitude varying ^{tS is also}

consistent with the data. This experimental technique ^is capable ^of quantitative pre- dictions and computations

^are

^{under in}

order to apply ^{it to other} systems.

Acknowledgements.

I gratefully acknowledge ^fruitful

discussion with D. Beysens.

Bibliography.

1. For review

see

Phase Transition : Status of Experimental and Theoretical

Situation, Cargèse (1980), ^ed. by ^L.

Levy, ^{J.C. Le} Guillou and J. Zinn Justin (Plenum Press), ^1981.

2. Fisher M.E. and de Gennes P.G., C.R.

Acad. Sci. Paris (287B) (1978), 207.

3. Beaglehole D., ^{J. Chem.} Phys. 75, ¹⁵⁴⁴

(1981).

4. Beysens ^D. and Leibler S., ^J. Phys.

Lett. 43, ^{L133 (1982).}

5. Schlossman M., ^Wu X-L. and Franck C., Phys. ^Rev. B31, ¹⁴⁷⁸ (1985).

6. Heidel B. and Findenegg G.H., J. Phys.

Chem. 88, ^{6575 (1984).}

7. Copper D.E., ^{Ph. D. Thesis M.T.} (1980) unpublished.

8. Zalczer G., Bourgou ^{A. and} Beysens D.,

Phys. ^Rev. A 28, ^{440 (1983).}