LIQUID SURFACE AND INTERFACE WAVES

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LIQUID SURFACE AND INTERFACE WAVES

R. Loudon

To cite this version:

R. Loudon. LIQUID SURFACE AND INTERFACE WAVES. Journal de Physique Colloques, 1984,

45 (C5), pp.C5-93-C5-101. �10.1051/jphyscol:1984513�. �jpa-00224131�

JOURNAL DE PHYSIQUE

Colloque

C5,

suppl6rnent au

n04,

Tome

45,

avril

1984

page

C5-93

LIQUID S U R F A C E A N D INTERFACE W A V E S

R. Loudon

Physics Department, Essex University,

Co

Zchester

C04 3SQ, U . K .

Rgsumk

-

La f o n c t i o n de Green e t l e s p e c t r e de f l u c t u a t i o n s pour l e d6place- ment d ' i n t e r f a c e e n t r e deux d i f f g r e n t s f l u i d e s , visqueux e t compressibles, q u i s o n t s6par6s p a r une couche mince Q l a s t i q u e , o n t Q t B obtenus. Nous d i s c u t o n s l e s s p e c t r e s c a l c u l e s pour t r o i s c a s l i m i t e s de l a t h i o r i e ggndrale, avec a t t e n t i o n p a r t i c u l i 3 r e s u r l a s g p a r a t i o n d e s c o n t r i b u t i o n s des ondes c a p i l l a i r e s e t d e s ondes a c o u s t i q u e s .

A b s t r a c t - We d e r i v e t h e i n t e r f a c e - d i s p l a c e m e n t Green f u n c t i o n and f l u c t u a - t i o n spectrum f o r two d i f f e r e n t v i s c o u s and compressible f l u i d s s e p a r a t e d by a t h i n e l a s t i c f i l m . The c a l c u l a t e d s p e c t r a f o r t h r e e s p e c i a l c a s e s of t h e g e n e r a l t h e o r y a r e d i s c u s s e d , w i t h p a r t i c u l a r emphasis on t h e s e p a r a t i o n of t h e c a p i l l a r y and a c o u s t i c wave c o n t r i b u t i o n s .

I

-

INTRODUCTION

The well-known r e l a t i o n between t h e frequency w of a r i p p l e and i t s s u r f a c e wavevector Q f o r a f r e e l i q u i d s u r f a c e i s

where g i s t h e g r a v i t a t i o n a l a c c e l e r a t i o n , a i s t h e s u r f a c e t e n s i o n c o e f f i c i e n t and p i s t h e f l u i d d e n s i t y . The frequency i s mainly determined by g r a v i t a t i o n f o r s m a l l Q and by s u r f a c e t e n s i o n f o r l a r g e Q; t h e two f o r c e s have e q u a l i n f l u e n c e f o r water a t t h e wavevector 360 mel. Longitudinal bulk a c o u s t i c waves propagated p a r a l l e l t o t h e s u r f a c e with v e l o c i t y vL and wavevector Q have t h e d i s p e r s i o n r e l a t i o n

The main experimental t o o l i n t h e s t u d y o f t h e r m a l l y - e x c i t e d r i p p l e s is l i g h t - s c a t t e r i n g spectroscopy, where wavevectors i n t h e range

a r e most e a s i l y a c c e s s i b l e . For such wavevectors, wR i s t y p i c a l l y two o r t h r e e o r d e r s of magnitude s m a l l e r t h a n wL.

The i n t e r p r e t a t i o n of l i g h t - s c a t t e r i n g experiments r e q u i r e s t h e o r e t i c a l e x p r e s s i o n s f o r t h e power s p e c t r a of t h e t h e r m a l l y - e x c i t e d f l u c t u a t i o n s a t t h e l i q u i d s u r f a c e s o r i n t e r f a c e s . The l i g h t - s c a t t e r i n g c r o s s s e c t i o n i s i n most c a s e s p r o p o r t i o n a l t o t h e mean-square p e r p e n d i c u l a r displacement of t h e s u r f a c e from i t s f l a t , s t a t i c , e q u i l i b r i u m p o s i t i o n . The p l a n of t h e paper i s t o d e r i v e i n s e c t i o n I l t h e f l u c t u a t i o n spectrum f o r a r a t h e r g e n e r a l system of two f l u i d s s e p a r a t e d by a t h i n film. The subsequent s e c t i o n s a r e devoted t o s p e c i a l c a s e s of t h e g e n e r a l system. More complete d e t a i l s of t h e c a l c u l a t i o n and a review of experimental work a r e p u b l i s h e d elsewhere /I/.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984513

C5-94

JOURNAL DE PHYSIQUE

I1

-

INTERFACE FLUCTUATION SPECTRUM

F i g . 1 shows t h e arrangement of t h e l i q u i d s whose i n t e r f a c e r i p p l e spectrum i s t o b e e v a l u a t e d . The z-axis p o i n t s v e r t i c a l l y upwards, with t h e g r a v i t a t i o n a l f o r c e s d i r e c t e d i n t h e n e g a t i v e z d i r e c t i o n . The lower h a l f - s p a c e z < 0 i s occupied by a l i q u i d having d e n s i t y p and v i s c o s i t y 11. The upper h a l f - s p a c e z > 0 i s

occupied by a d i f f e r e n t l i q u i d whose p h y s i c a l parameters a r e denoted throughout by t h e same symbols a s f o r t h e lower l i q u i d b u t with primes a t t a c h e d t o them. Both l i q u i d s a r e assumed t o have i n d e f i n i t e e x t e n t p a r a l l e l t o t h e z - a x i s ; i n p r a c t i c e , t h e i r z dimensions s h o u l d be much l a r g e r t h a n t h e normal decay l e n g t h s of t h e i n t e r f a c e e x c i t a t i o n s .

The z = 0 s u r f a c e of s e p a r a t i o n o f t h e l i q u i d s i s a l s o assumed t o have i n d e f i n i t e e x t e n t , although it i s convenient t o a s s i g n t o i t a l a r g e a r e a A f o r n o r m a l i z a t i o n purposes. The two l i q u i d s a r e s e p a r a t e d by a t h i n f i l m of some t h i r d m a t e r i a l . The f i l m t h i c k n e s s i s taken t o b e very s m a l l compared t o t h e wavelengths of t h e i n t e r f a c e r i p p l e s and t o t h e c h a r a c t e r i s t i c wavelengths of bulk e x c i t a t i o n s i n t h e two l i q u i d s . Thus i n applying macroscopic f l u i d mechanics t o t h e system, t h e e f f e c t s of t h e f i l m a r e f e l t o n l y i n t h e boundary c o n d i t i o n s a t t h e f l u i d i n t e r f a c e .

A c a s e of p a r t i c u l a r i n t e r e s t i s t h a t i n which t h e f i l m i s formed from a monolayer of adsorbed molecules. An example of a system t h a t h a s been s t u d i e d by l i g h t s c a t t e r i n g i s a monolayer of f a t t y a c i d molecules on t h e s u r f a c e o f water with j u s t a i r above t h e monolayer. Forces of r e p u l s i o n between t h e molecules g e n e r a t e a s u r f a c e p r e s s u r e , which diminishes t h e s u r f a c e t e n s i o n a of t h e s u p p o r t i n g l i q u i d from i t s o r d i n a r y v a l u e . The s u r f a c e p r e s s u r e normally i n c r e a s e s w i t h t h e s u r f a c e number d e n s i t y y of t h e adsorbed molecules, and t h e magnitude of t h e e f f e c t i s d e s c r i b e d by a parameter / 2 , 3 /

The s u r f a c e t e n s i o n a t a molecular d e n s i t y y i s t h u s r e l a t e d t o i t s v a l u e f o r a f r e e s u r f a c e by

a = a

f r e e

- : 1

( ~ / y ) d y . ( 5 )

The parameter E p l a y s t h e r o l e of an e l a s t i c i t y s i n c e i t s p r o d u c t with t h e s t r a i n i n a f i l m deformed from e q u i l i b r i u m determines t h e r e s t o r i n g f o r c e . I t i s knownas t h e s u r f a c e d i l a t i o n a l modulus and i t a c t s only f o r d i s t o r t i o n s t h a t change t h e l o c a l molecular d e n s i t y . I t i s a complex q u a n t i t y f o r dynamic deformations, and

( 5 ) shows t h a t a i s a l s o complex i n t h e presence of a v i s c o e l a s t i c f i l m . Consider f i r s t t h e kinds of wave t h a t can b e propagated i n a bulk l i q u i d . The t r e a t m e n t below i s r e s t r i c t e d t o motions t h a t have a harmonic time dependence e x p ( - i w t ) s o t h a t t h e l i q u i d v e l o c i t y v and displacement u a r e r e l a t e d by

F i g . 1 - Geometrical arrangement of two l i q u i d s s e p a r a t e d by a t h i n f i l m .

The l i q u i d i s assumed t o be compressible with non-vanishing s t i f f n e s s components r e l a t e d t o t h e v e l o c i t y of l o n g i t u d i n a l a c o u s t i c waves by

The t h e r m a l l y e x c i t e d waves have small amplitudes and it is p e r m i s s i b l e t o u s e t h e l i n e a r i z e d e q u a t i o n s of f l u i d mechanics. Then E u l e r ' s e q u a t i o n expressed i n terms of t h e l i q u i d displacement i s /4,5/

where and q a r e r e s p e c t i v e l y t h e c o e f f i c i e n t s of bulk and s h e a r v i s c o s i t y , assumed t o b e cons-tants f o r a given substance.

Consider a p l a n e wave e x c i t a t i o n of t h e i n f i n i t e l i q u i d i n which t h e d i s p l a c e - ment has a s p a t i a l dependence e x p ( i q . r _ ) . The e q u a t i o n of motion ( 8 ) has two k i n d s of s o l u t i o n , corresponding t o t r a n s v e r s e and l o n g i t u d i n a l p o l a r i z a t i o n . The t r a n s v e r s e s o l u t i o n s have

The wavevector magnitude qT s a t i s f i e s ( 8 ) i f

and t h e square r o o t i s chosen t o have p o s i t i v e r e a l and imaginary p a r t s s o t h a t

These t r a n s v e r s e v i s c o u s waves a r e t h e r e f o r e c r i t i c a l l y damped i n space.

I n a p p l i c a t i o n s t o s p e c t r a measured by l i g h t - s c a t t e r i n g spectroscopy, t h e waves of i n t e r e s t have t h e i r wavevectors i n t h e zx-plane with an x-component Q t h a t i s r e q u i r e d t o be r e a l by t h e c o n d i t i o n s of t h e experiment. Let t h e z-component magnitude be denoted T, s o t h a t

and from (10)

The l o n g i t u d i n a l s o l u t i o n of ( 8 ) h a s

~ L x ~ L = o ,

where t h e wavevector magnitude qL i s given by q2 _L = pu2/[pv2 _L

-

^{i w ( 5}

^{+ 7}

4

^{n ) J} .

P o s i t i v e r e a l and imaginary p a r t s a r e a g a i n chosen i n t a k i n g t h e square r o o t . The imaginary p a r t i s normally much s m a l l e r than t h e r e a l p a r t and t h e propagating l o n g i t u d i n a l a c o u s t i c waves a r e o n l y l i g h t l y damped.

For wavevectors i n t h e zx-plane w i t h t h e x-component Q r e q u i r e d t o be r e a l , t h e z-component magnitude i s denoted L , s o t h a t

The l o n g i t u d i n a l damping h a s a s m a l l e f f e c t on t h e i n t e r f a c e f l u c t u a t i o n spectrum s o t h a t (15) g i v e s

JOURNAL DE PHYSIQUE

We now c o n s i d e r t h e boundary c o n d i t i o n s a t t h e i n t e r f a c e between t h e two l i q u i d s . Since t h e f l u c t u a t i o n s o f t h e i n t e r f a c e o u t of t h e z = 0 p l a n e a r e s m a l l , t h e boundary c o n d i t i o n s can b e e x p r e s s e d i n terms of t h e f l u i d displacements e v a l u a t e d a t z = 0. The c o n d i t i o n s t h a t t h e two f l u i d s remain i n c o n t a c t a r e

The remaining boundary c o n d i t i o n s e x p r e s s t h e matching of f o r c e s a t t h e i n t e r f a c e . They a r e d e r i v e d by Landau and L i f s h i t z /4/, by

arc fa-~oliner

/ 6 / , and i n g r e a t e r d e t a i l by Lucassen-Reynders and Lucassen /3/. The c o n d i t i o n s a r e

expressed i n terms of t h e s t r e s s t e n s o r s a and a ' . A component of t h e s t r e s s t e n s o r

f o r t h e lower f l u i d i s

-

The f o r c e s e x e r t e d on u n i t a r e a of i n t e r f a c e by t h e two f l u i d s must be e q u a l , except i n s o f a r a s t h e r e a r e a d d i t i o n a l f o r c e s a t t h e i n t e r f a c e i t s e l f . There a r e t h r e e k i n d s of i n t e r f a c e f o r c e . One of t h e s e i s t h e o r d i n a r y s u r f a c e t e n s i o n d i r e c t e d p a r a l l e l t o t h e s u r f a c e normal and with a magnitude p r o p o r t i o n a l t o t h e c u r v a t u r e o f t h e i n t e r f a c e . For r i p p l e s propagated p a r a l l e l t o t h e x - a x i s , whose amplitudes a r e much s m a l l e r t h a n t h e i r wavelengths, t h e s u r f a c e t e n s i o n f o r c e s a c t p a r a l l e l t o t h e z-axis. The g r a v i t a t i o n a l f o r c e s on t h e i n t e r f a c e a l s o a c t i n t h e z - d i r e c t i o n . The z-component boundary c o n d i t i o n i s t h e r e f o r e

The t h i r d k i n d of i n t e r f a c e f o r c e r e s u l t s from t h e v i s c o e l a s t i c p r o p e r t i e s of t h e t h i n f i l m ; a d i s t o r t i o n of t h e i n t e r f a c e l e a d s i n g e n e r a l t o a s p a t i a l l y - d e p e n d e n t molecular d e n s i t y y , and hence t o a s p a t i a l l y - d e p e n d e n t s u r f a c e t e n s i o n a. Thus f o r t h e r i p p l e geometry assumed h e r e t h e r e i s a f o r c e aa/ax p e r u n i t a r e a d i r e c t e d p a r a l l e l t o t h e x - a x i s , and t h e x-component boundary c o n d i t i o n i s

With t h e use of ( 4 ) we have

Suppose t h a t t h e number of adsorbed molecules remains unchanged when t h e f i l m s u f f e r s a displacement ux t h a t v a r i e s w i t h x. An i n i t i a l s u r f a c e d e n s i t y y i s changed t o

and hence

f o r s m a l l changes i n d e n s i t y . The right-hand s i d e of (21) i s transformed with t h e use of (22) and (24) t o read

Consider t h e e f f e c t of s u b j e c t i n g t h e lower l i q u i d t o an a p p l i e d f o r c e

d i r e c t e d p a r a l l e l t o t h e z - a x i s , where Q i s a r e a l and p o s i t i v e wavevector. The f o r c e couples t o a l i q u i d displacement u ( z ) e x p ( i Q x ) w i t h an i n t e r a c t i o n energy

H _{i n t} = -u (zo)*Fexp(-iwt)

.

(27)

The e f f e c t of t h e f o r c e on t h e motion of t h e l i q u i d s i s o b t a i n e d by s o l u t i o n of E u l e r ' s equation ( 8 ) w i t h (26) i n s e r t e d on t h e right-hand s i d e of t h e z-component of t h e equation.

The t o t a l displacement produced i n t h e lower l i q u i d by t h e a p p l i e d f o r c e has components

uX ( 2 ) = (iQ~/2Apw 2 ) { - e x p ( i ~ Iz-z0

1 ) +

exp ( i ~ Iz-z

1)

^{)sgn (z-z )}

where f a c t o r s exp(-iwt

+

iQx) a r e understood throughout and 1 f o r z > z

0

-1 f o r z < z

.

The y-component of t h e displacement i s z e r o . I n t h e above s o l u t i o n s f o r t h e x and z components, t h e f i r s t terms a r e t h e d r i v e n s o l u t i o n s and t h e y s a t i s f y t h e d r i v e n E u l e r ' s e q u a t i o n . The remaining two terms i n each component a r e t h e f r e e s o l u t i o n s ; t h e i r c o e f f i c i e n t s a r e a s y e t unknown q u a n t i t i e s t o be determined from t h e i n t e r f a c e boundary c o n d i t i o n s .

The displacement produced i n t h e upper l i q u i d c o n t a i n s o n l y f r e e s o l u t i o n s s i n c e t h e a p p l i e d f o r c e does n o t appear i n E u l e r ' s equation f o r z > 0 . The components a r e t h u s

with t h e s i g n s of theecponents chosen t o produce decay of t h e s o l u t i o n s a t l a r g e p o s i t i v e z . The wavevector z components a r e given by (13) and (17) w i t h t h e parameters f o r t h e upper l i q u i d s u b s t i t u t e d on t h e right-hand s i d e s .

I t i s convenient t o remove t h e x-components of t h e f o u r free-wave amplitudes i n t h e s o l u t i o n s f o r t h e displacements with t h e u s e of (9) and ( 1 4 ) . The boundary c o n d i t i o n s (181, (20) and (25) t h e n p r o v i d e f o u r simultaneous e q u a t i o n s f o r t h e f o u r unknown z-components. The s o l u t i o n s determine t h e displacement-displacement Green f u n c t i o n a t t h e i n t e r f a c e a s

I-

...

^{m 1}

where F i s taken a t z = 0 and

2 O 2

i p w L ⁺ i p ' w L ' 2

i p w T i p ' w T' EQ2] [ctQ2

+

( 0 - p l ) g -

- - ---

p2+TL Q 2 + - r ' ~ n

- n t

(TI 2-Q 2-2TtL')

12. ^' I

(34)

Q2

+

TL Q2

+

T'L'

C5-98 JOURNAL

DE

PHYSIQUE

An e q u i v a l e n t denominator h a s been d e r i v e d by ~ a r c i a - ~ o l i n e r /6/. The mean-square displacement of t h e i n t e r f a c e a t wavevector Q , frequency w , and temperature T i s o b t a i n e d from t h e f l u c t u a t i o n - d i s s i p a t i o n theorem

The e x p r e s s i o n f o r t h e i n t e r f a c e f l u c t u a t i o n spectrum provided by (35) can be compared r a t h e r d i r e c t l y with measured l i g h t - s c a t t e r i n g s p e c t r a . The g e n e r a l i t y of t h e e x p r e s s i o n however obscures t h e main f e a t u r e s of t h e t h e o r e t i c a l p r e d i c t i o n s . Subsequent s e c t i o n s c o n s i d e r t h r e e r e p r e s e n t a t i v e s p e c i a l c a s e s of t h e g e n e r a l theory.

I11

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SINGLE LIQUID WITH A FREE SURFACE

I t i s a simple m a t t e r t o remove t h e upper l i q u i d and t h e f i l m frorr, t h e t h e o r y of t h e p r e v i o u s s e c t i o n by s e t t i n g t h e i r parameters e q u a l t o zero. The s u r f a c e t e n s i o n a i s a r e a l q u a n t i t y i n t h e absence of t h e f i l m . The f l u c t u a t i o n spectrum

(35) can b e w r i t t e n with use of (10) and (13) i n t h e form i L

2 2

1-

⁽³⁶⁾

[ ( a ~ * / p ) +g] +w +4 ( n Q / p ) T (T-L)

This form i s p a r t i c u l a r l y convenient f o r c o n s t r u c t i n g g r a p h s , and F i g . 2 shows t h e s u r f a c e spectrum of l i q u i d mercury c a l c u l a t e d numerically. The q u a n t i t y p l o t t e d i s t h e spectrum (36) d i v i d e d by t h e f a c t o r k B ~ / * ~ p . The spectrum has two main f e a t u r e s , i n t h e r e g i o n s o f t h e c a p i l l a r y wave and a c o u s t i c wave f r e q u e n c i e s , and i t i s n e c e s s a r y t o show them i n s e p a r a t e p a r t s of t h e graph because of t h e l a r g e d i s - p a r i t i e s i n t h e i r s t r e n g t h s and f r e q u e n c i e s . The p a r t s form of course a s i n g l e continuous spectrum and t h e s h a r p l y r i s i n g f e a t u r e on t h e l e f t of F i g . 2 ( b ) i s t h e c o n t i n u a t i o n of t h e high-frequency wing of t h e peak shown i n Fig. 2 ( a ) .

I t i s n o t d i f f i c u l t t o e x t r a c t from t h e t h e o r e t i c a l spectrum (36) p a r t s t h a t correspond t o t h e f e a t u r e s shown i n t h e two p a r t s of F i g . 2. The component i n t h e v i c i n i t y of t h e r i p p l e frequency wR(2 x 108s-1) a g r e e s with an e x p r e s s i o n of Bouchiat and Meunier / 7 / ; t h i s p a r t of t h e spectrum h a s been measured by Bird and H i l l s / 8 / . The component i n t h e v i c i n i t y of t h e a c o u s t i c frequency w (1.5 x 10l0s-l)

L

F i g . 2

-

C a l c u l a t e d s u r f a c e f l u c t u a t i o n spectrum of l i q u i d mercury f o r a wavevector Q = 107m-'. (a) Region of capillary-wave frequency WR; (b) r e g i o n of a c o u s t i c wave frequency wL (from /5/).

a l s o a g r e e s w i t h e a r l i e r r e s u l t s / 9 , 1 0 / ; t h i s p a r t of t h e spectrum h a s been measured by D i l and Brody /11/.

IV - S I N G L E L I Q U I D COVERED BY A FILM

The f l u c t u a t i o n spectrum i n t h i s c a s e i s e a s i l y o b t a i n e d from (34) and ( 3 5 ) by removal of t h e p a r t s of t h e e q u a t i o n s t h a t i n v o l v e t h e upper l i q u i d . The

spectrum now depends a d d i t i o n a l l y on t h e f i l m d i l a t i o n a l modulus ^{E .} A s d i s c u s s e d i n s e c t i o n 11, both E and t h e s u r f a c e p r e s s u r e produced by t h e f i l m a r e g e n e r a l l y complex q u a n t i t i e s . We assume f o r s i m p l i c i t y however t h a t t h e imaginary p a r t s of t h e s e q u a n t i t i e s a r e n e g l i g i b l y small.

An i n t e r e s t i n g e f f e c t /12,13/ o c c u r s i n t h e component of t h e f l u c t u a t i o n spectrum c e n t r e d on t h e r i p p l e frequency

wR

i n t h e l i m i t of small v i s c o s i t y where

Fig. 3 shows t h e p r e d i c t e d v a r i a t i o n of t h e l i n e w i d t h Au of t h e r i p p l e spectrum with

Fig. 3

-

Dependence of t h e r i p p l e spectrum l i n e w i d t h on t h e r a t i o o f f i l m d i l a t i o n a l modulus t o l i q u i d s u r f a c e t e n s i o n f o r w a t e r a t Q = 5 104,-1.

t h e f i l m d i l a t i o n a l modulus E . The maximum i n t h e l i n e w i d t h v a r i a t i o n i s a s s o c i a t e d w i t h t h e propagation of l o n g i t u d i n a l waves i n t h e f i l m . I t can be shown /3,14/ t h a t t h e s e waves have t h e frequency

and F i g . 4 shows t h e v a r i a t i o n of uF with ^{E .} The f i l m waves a r e h e a v i l y damped, t h e i r frequency having an imaginary p a r t equal t o t h e r e a l p a r t w

,

which i s much l a r g e r t h a n t h e damping of t h e f r e e - s u r f a c e r i p p l e s shown by t h e gashed l i n e i n Fig. 4. The maximum i n t h e l i n e w i d t h i n F i g . 3 r e s u l t s from coupling of t h e f i l m and r i p p l e modes v i a t h e f l u i d v i s c o s i t y , t h e coupling e f f e c t s being p a r t i c u l a r l y s t r o n g a t t h e v a l u e of E f o r which

wF

=

uR. ^There i s t h e u s u a l mode s p l i t t i n g a t t h e crossover p o i n t b u t this i s n o t shown I n F i g . 4. A p p l i c a t i o n of t h e t h e o r y t o t h e i n t e r p r e t a t i o n of measured s p e c t r a r e q u i r e s allowance f o r t h e imaginary p a r t s of a and E /15-19/.

JOURNAL

DE

PHYSIQUE

Fig. 4

-

V a r i a t i o n of t h e l o n g i t u d i n a l frequency wF f o r water a t Q = 5 x

lo4,-l.

The h o r i z o n t a l l i n e s show t h e frequency w and damping 4 n Q 2 / p of t h e o r d i n a r y c a p i l l a r y wave on tl?e f r e e water s u r f a c e .

V

-

TWO LIQUIDS I N DIRECT CONTACT

The f l u c t u a t i o n spectrum a t t h e i n t e r f a c e of two l i q u i d s i n d i r e c t c o n t a c t , with no s e p a r a t i n g f i l m , i s o b t a i n e d by s e t t i n g ^& e q u a l t o z e r o i n ( 3 4 ) and ( 3 5 1 , and by t a k i n g ct t o be r e a l . T h i s c l e a r l y produces l i t t l e s i m p l i f i c a t i o n i n t h e g e n e r a l r e s u l t s . However, f u r t h e r s i m p l i f i c a t i o n s can b e made a s a r e s u l t of t h e s e p a r a t i o n of t h e spectrum i n t o two c o n t r i b u t i o n s i n q u i t e d i s t i n c t r e g i o n s of frequency. The s i t u a t i o n i s s i m i l a r t o t h a t a t t h e s u r f a c e of a s i n g l e f l u i d d i s c u s s e d i n s e c t i o n I11 and i l l u s t r a t e d i n Fig. 2 , w i t h t h e f i r s t c o n t r i b u t i o n i n t h e r e g i o n of t h e c a p i l l a r y wave frequency and a second c o n t r i b u t i o n i n t h e r e g i o n of t h e a c o u s t i c wave frequency of t h e bulk l i q u i d . We c o n s i d e r h e r e o n l y t h e second p a r t of t h e spectrum, i n t h e r e g i o n of t h e a c o u s t i c f r e q u e n c i e s wL and w i . These f r e q u e n c i e s a r e much l a r g e r t h a n t h e o t h e r c h a r a c t e r i s t i c f r e q u e n c i e s of t h e two l i q u i d s and t h e i r i n t e r f a c e . S u i t a b l e approximations i n ( 3 4 ) and ( 3 5 ) l e a d t o t h e i n t e r f a c e f l u c t u a t i o n spectrum

We suppose t h a t both t h e d e n s i t y and t h e sound v e l o c i t y f o r t h e upper l i q u i d a r e s m a l l e r t h a n f o r t h e lower l i q u i d and t h e form o f t h e p r e d i c t e d spectrum i s shown i n F i g . 5. With t h e h e l p of ( 2 ) and (17), t h e spectrum ( 3 9 ) can b e w r i t t e n

2 2 2

p ' v ' ( W -w ) ( W - w , 2 ) 4

= - L L L

3 2 2 2 2 2 , 2 2 2 w ' < W < w L L nAw p v ( W - w i )

+

P ' VL (WL-W )

L

The ~ 0 n t i n ~ 0 ~ s curve i n Fig. 5 shows t h i s spectrum i n u n i t s of kBTQ/aApwL; 3 t h e dashed c u r v e s show t h e f r e e s u r f a c e f l u c t u a t i o n s p e c t r a of t h e i n d i v i d u a l l i q u i d s i n t h e same u n i t s , s i m i l a r t o t h e spectrum i n F i g . 2 ( b ) .

Fig. 5 - Interface fluctuation spectrum in the acoustic frequency region for two liquids with v;/vL = 0.7 and p'v'/pv = 0.5.

The acoustic contribution to the interface spectrum is much weaker than the capillary wave contribution, similar to the single-liquid surface spectrum

discussed in section 111. There seem to be no measurements to date of the acoustic parts of interface fluctuation spectra with which the above theory can be compared.

References

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