HIGHER ORDER CORRELATION FUNCTIONS IN SUPER-CRITICAL XENON : EXPERIMENTAL EVIDENCE FROM LIGHT SCATTERING AND NON-LINEAR DIELECTRIC MEASUREMENTS

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HIGHER ORDER CORRELATION FUNCTIONS IN

SUPER-CRITICAL XENON : EXPERIMENTAL

EVIDENCE FROM LIGHT SCATTERING AND

NON-LINEAR DIELECTRIC MEASUREMENTS

R. Huijts, J. Michielsen, J. van der Elsken

To cite this version:

HIGHER ORDER CORRELATION FUNCTIONS I N SUPER-CRITICAL XENON

:

EXPERIMENTAL EVIDENCE FROM L I G H T SCATTERING AND NON-LINEAR D I E L E C T R I C MEASUREMENTS

R.A. Huijts, J.C.F. Michielsen and J. van der Elsken

bboratory for PhysicaZ Chernistry, University of Amsterdam, Nieuwe Achtergracht 127, 1018 WS Amsterdam, The NetherZands

Abstract

-

Higher order d e n s i t y c o r r e l a t i o n f u n c t i o n s are measured i n super- critical xenon by d e p o l a r i z e d l i g h t s c a t t e r i n g (DIS) and by measurement of t h e non-linear d i e l e c t r i c effect (I(IDE). The r e l a t i o n between the c o r r e l a t i o n functiom meaeured by both techniques is d i s c u s s e d .

An a b r u p t &ange i n the .econd moment o f t h e DLS spectnmi is report&. Rridence of t h i n effect i n the NLDE experiments is aleo found. The e i g n i f i c a n c e o f t h i s experimental o b s e r v a t i o n is discussed.

There is a gmwing i n t e r e s t i n h i g h e r order c o r r e l a t i o n f u n c t i o n s , both £rom the t h e o r e t i c a l and from t h e experimental p o i n t o f view. A l a r g e amount o f theoretical work has been done a l r e a d y i n c o n t r a s t w i t h experimental work.

MDst o f t h e e x p r i m e n t s done on the h i g h e r o r d e r c o r r e l a t i o n f u n c t i o n s are oomputer experiments, and r e l a t i v e l y few o t h e r experimental techniques are a v a i l a b l e .

To e x p l o r e t h i s f i e l d we performed two t y p e s o f experimente: a ) d e p o l a r i z e d l i g h t s c a t t e r i n g (DLS)

b ) measurements o f the non-linear d i e l e c t r i c e f f e c t (NLDE).

Both techniques are q u i t e complicated and g i v e r e a u l t s which may be d i f f i c u l t to i n t e r p r e t e e p e c i a l l y f o r molecular f l u i d e . To avoid complications we have chosen to s t u d y

a simple

atomic f l u i d , xenon, i n t h e s u p e r c r i t i c a l region. Xenon has a h i g h atomic p o l a r i z a b i l i t y and such a critical p o i n t that it i a p o s s i b l e t o m a s u r e a l a r g e d e n s i t y range a t e a a i l y a t t a i n a b l e c o n d i t i o n s . We r e p o r t measurements along t h e

293 K isothenn, i.e. 1.01 times the critical temperature, at d e n s i t i e s o f 0.1 t o 1.8

timea t h e critical d e n s i t y .

Only a v e r y s h o r t d e s c r i p t i o n of the experimental DLS set-up is given, f o r a f u l l d e s c r i p t i o n see elsewhere /1/. The d e p o l a r i z e d l i g h t s c a t t e r i n g e p e c t r a are recorded in the 90° e c a t t e r i n g geometry, ueing the 514.5 nm l i n e o f an argon-ion laser. The i n c i d e n t light is p o l a r i z e d p a r a l l e l w i t h respect t o t h e s c a t t e r i n g p l a n e . Both p o l a r i z a t i o n d i r e c t i o n s o f the s c a t t e r e d l i g h t are d e t e c t e d . The Stokes-aide of the e p e c t r a is mewured from the e x c i t i n g l i n e up t o 150 wavenumbers. A c o r r e c t i o n f o r d e t a i l e d balance is applied.

me

=non ie contained i n a thennoatated ce11 i n which the p r e s s u r e c a n b e v a r i e d o v e r a l a r g e range up t o e e v e r a l hundreds o f bars. Due t o s p u r i o u s s c a t t e r i n g o f

C9-130 JOURNAL DE PHYSIQUE

d u s t , w a l l a , etc. it is impossible t o m a s u r e p r o p e r l y d e p o l a r i z e d s p e c t r a c l o s e t o the e x c i t i n g laser frequency. So the spectra are e x t r a p o l a t e d £rom f o u r t o z e r o wave- numbers by f i t t i n g w i t h a Lorentzian p r o f i l e /2/. As can b e s h m t h i s procedure o n l y has a minor i n f l u e n c e o n the i n t e g r a t e d i n t e n s i t i e s , and s p e c t r a l moments. I n Pig. 1 an example o f the measured s p e c t r a is given. Measuring isochromes made it p o s s i b l e t o compare the i n t e n s i t i e s o f the v a r i o u s spectra recorded a t d i f f e r e n t c o n d i t i o n s .

Pig. 1

-

DepoZartzed Ltght scattertng spectrum of xenon at 293K f o r denstttes of 1.22

(Zower curve) and 1.44 (upper curve) ttmes the crtttcaZ denstty. The tntenstty ts on a retattve scaZe tn arbttrary untts.

For the non-linear dielectric experimenta xenon is k e p t at v a r i a b l e p r e s s u r e s between the e l e c t r o d e s o f a thermoetated c a p a c i t o r . T h i s c a p a c i t o r fornia part o f a n oscilla- t i n g c i r c u i t . By applying a h i g h v o l t a g e electric f i e l d a c r o s s the c a p a c i t o r the per- m i t t i v i t y of xenon i n c r e a s e s , which r e s u l t s i n a frequency change o f the o s c i l l a t i n g c i r c u i t . This frequency change can be a t t r i b u t e d t o the non l i n e a r dielectric effect. By the v e r y mail s e p a r a t i o n o f the e l e c t r o d e s o f the measuring c a p a c i t o r , and the a p p l i c a t i o n of the pulaed W-field,electrostriction is avoided. Thus, almoat no f i r a t order e f f e c t s are measured. A d e s c r i p t i o n of the a p p a r a t u s is given elsewhere /3/.

III

-

RESULTS

Depolarized l i g h t s c a t t e r i n g from simple f l u i d e is a second o r d e r e f f e c t . This means t h a t a straight-forward connection e x i s t a with h i g h e r o r d e r c o r r e l a t i o n f u n c t i o n s . The main d i f f i c u l t y i n e x t r a d i n g t h i s connection i a the lack o f a f u l l y a a t i s f a c t o r y theory b o t h f o r the i n d u c t i o n mechanism, as f o r the local f i e l d .

U s e o f the dipole-induced d i p o l e approximation f o r t h e i n d u c t i o n mechanism and

assuming p o i n t p o l a r i z a b i l i t i e s , makes it p o s s i b l e t o d e r i v e t h e following r e l a t i o n f o r the scattered i n t e n s i t y /4/

I( a ) ( , , , ) a [dt

e-i&[bko

[bk"

azP(k' )aZ@*(k" ).

Where a is the a t & C p o l a r i z a b i l i t y and P ( k ) and Pz stem £rom the F o u r i e r t r a n s f o m o f the dipole-induced d i p o l e t e n e o r . The Becond Legendre polynomial Pz is due t o the SyImnetry of l i g h t e c a t t e r i n g . So the power spectrum i e determined by time c o r r e l a t i o n s between p a i r s o f d e n s i t y f l u c t u a t i o n s with o p p o s i t e wavevectors.

f l u i d .

Fig. 2

-

The second s p e c t r a t moment of depotartzed Ltght s c a t t e r t n g from senon a t 293K t n a r b t t r a r y u n t t s , as a f u n c t t o n of t h e d e n s t t y d t v t d e d by t h e c r t t t c a L d e n s t t y

(Uots). The t r t a n g t e s r e p r e s e n t t h e second s p e c t r a t moment c o r r e c t e d f o r t h e t n t e r n a t f t e t d , a t s o t n a r b t t r a r ~ u n t t s .

I n F i g . 2 the t u z > is given f o r the T/Te1.01 isotherm as a f u n c t i o n o f reduced d e n a i t y . The d e n e i t i e s are o b t a i n e d £rom m a t a a v a i l a b l e i n the l i t e r a t u r e / 7 / .

Besides a n i n c r e a s i n g v a l u e o f the three p o i n t w r r e l a t i o n f u n c t i o n w i t h d e n s i t y there t o be a d i s c o n t i n u i t y i n t u 2 > f o r p/pcg1.34.

It may be argued t h a t the d i s c o n t i n u i t y arises £rom the fact that no c o r r e c t i o n f o r

the d e n s i t y dependent i n t e r n a l f i e l d i e used. lb make s u r e that t h i s argument is not v a l i d we a p p l i e d a c o r r e c t i o n given by (t+2)./81. The p e m i t t i v i t y a is measured a l o n g the aame isotherm as the D L S - e v r i m e n t s /3/. The i n t e r n a l f i e l d c o r r e c t i o n v e r s u s the d e n s i t y is a smooth f u n c t i o n i n d i c a t i n g t h a t no critical phenomena are

involveâ

.

The t h i r d o r ü e r d e n s i t y c o r r e l a t i o n f u n c t i o n can be m i t t e n as a sum o f two tenus:

where g, is the n - p a r t i c l e d i s t r i b u t i o n f u n c t i o n and N t h e number o f p a r t i c l e a . The first term on the r.h.s. is r e l a t e d t o t h e static s t r u c t u r e f a c t o r , which is known f o r a simple f l u i d / 8 / and does n o t show any d i s c o n t i n u i t y above the c r i t i c a l p o i n t . It must be concluded that the sudden i n c r e a s e i n t h e cmz? must f i n d i t e o r i g i n i n the last term.

The r e l a t i o n for t h e second m e n t , ( 2 ) can be w r i t t e n under s i m p l i f y i n g c o n d i t i o n s

as

The combined geometical f a c t o r s r e s u l t i n favourably weighted c o n t r i b u t i o n s o f w a v w e c t o r s k' and kW that make a n a n g l e around 2a0 o r

l l o O .

Equal magnitude ie favoured becauee of the o s c i l l a t o r y behaviour o f the spherical B e s s e l f u n c t i o n j,(ku)

i n which u is a cut-off parameter, u s u a l l y taken a6 the a t d c diameter /4/.

C9-132 JOURNAL

DE

PHYSIQUE

the i n c r e a s e of the

l l o O

c o n f i g u r a t i o n s i n t h e t r i p l e t c o r r e l a t i o n function.

The t h e o r y of t h e NLOE runs along the same l i n e s as i n t h e case o f DLS. The -rimental r e s u l t s c o n t a i n i n p r i n c i p l e c o n t r i b u t i o n s fram d e n s i t y - c o r r e l a t i o n s of v a r i o w o r d e r s /9/. Expansion o f t h e p o l a r i z a t i o n and of the i n d u c t i o n mechanism i n povers o f the d e n s i t y makes it p o s s i b l e t o s e p a r a t e these orders.

The first term i n the expansion l e a d s t o

where ae is the incremant o f t h e p e r m i t t i v i t y due to electric f i e l d E.

~ ( 2 )

i e the static s t r u c t u r e f a c t o r and w,(q

a

weight-factor.

I n the l o n g wavelength l i m i t o,(k) is a d e l t a f u n c t i o n at %PO l e a d i n g to

where XT is t h e i s o t h e r a i a l c o m p r e s a i b i l i t y and a

is

the relative d i e l e c t r i c c o n s t a n t . T h i s t e n u is the classical e l e c t r o s t r i c t i o n e f f e c t .

The second term i n the expansion ist

The c o r r e l a t i o n f u n c t i o n involved i n these tenu ifs the connected part o f a f o u r p o i n t d e n s i t y c o r r e l a t i o n function. w , ( ~ * , ~ ) a g a i n is a weight-factor depending on the a c t u a l i n d u c t i o n mechanism, comparable w i t h DLS.

A f i t o f the experimental NLDE-reeults i n first and second o r d e r effects, f a c t o r i n g o u t the t r i v i a l terme such ae the p e n u i t t i v i t y leads t o an e q u a t i o n

w i t h a-3.5 ~ O - ~ O

m2/VZ

and bP1.7 10-20 mz/V2.

The fact that a and b are o f e q u a l magnitude is due t o the experimental set-up. Only the 'li+o c o n t r i b u t i o n of the f i r s t order t e n u is aeen /a/.

If the c o n t r i b u t i o n ( e l e c t r o s t r i c t i o n ) would be preaent. t h i s would l e a d to v a l u e s as i n d i c a t e d i n Fig. 3 by the daahed l i n e .

Fig. 3

-

nie non-Ltnear dte2ectrtc effect of renon at 293 K tn absotute untts

(#/te)

p o i n t d e n s i t y c o r r e l a t i o n function hae a d i s c o n t i n u i t y at p = 1.34 pc. So t h e 1iJIllE-function muet have a maximum a t t h e same d e n s i t y . The remalte as given i n Fig. 3 c l e a r l y indicates t h a t t h e NLDE-result at t h i s d e n s i t y is s i g n i f i c a n t l y h i g h e r t h a n t h e f i t t e d curw.

Pig. 4 show8 the l o c a l i s a t i o n o f the sudden i n c r e a m o f t h e second spectral manent aa determùied along the 293 and 298 K i e o t h e r m i s by DLS. The p o i n t a f a l l on a epiooth extension o f t h e vapourization curw i n t h e PT diagram /IO/.

Along the sanie l i n e anomalies i n t h e a d i a b a t i c 8ounü v e l o c i t y /Il/ and i n t h e heat c a p a c i t y at c o n s t a n t p r e s s u r e /12/ haw been found. The evidence f o r an a b ~ p t change i n t h e h i g h e r o r d e r d e n s i t y c o r r e l a t i o n functiona abow t h e critical p o i n t may add to t h e understanding of t h e s e phenanena.

7

apour

0.9

T/T,

1.1

~ i g . 4

-

nie tocus of- the sudden tncrease" tn the second moment of the depobartzed Ltght scattertng spectra tn the PT-dtagram of senon tn reuuced untts.

T h i s wrk is part o f the research programa of t h e Poundation f o r Fundamental Research of Matter (POM), and o f t h e Netherlands Poundation f o r Chemical Reaearch (SON) and

wae made p o s s i b l e by f i n a n c i a l support froei t h e m t h e r l a n d s Organisation f o r Pure Reaearch (zwo).

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J.,

Van Zoonen, P. and Hichieleen, J.C.P., Chem. Phys. ïett.

106

(1984) 252, 115 (1985) 230.

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ml.

Phys. (1978) 365.

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be p i b l i s h e d .

/7/ S t r e e t t , W.B. ,Sagan, L.S. and Staveley, L.A.K., J. Chem. Therarod. 5 (1973) 533 Michels, A., Wassenaar, T. and Iauwerse, P., Physica- (1954) 99.

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a

(1942) 261. nikoiaj, P.G. and Pings, C.J., Chem. Ph-.

g

(1967) 1401. Haeman, A., Phyaica

a

(1973) 499.

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