Chatterjee, S

Gopal, E S R

Vani, V C

English

IACS Institutional Repository

IndlonJ. fhyt. «SA (4) 276-285 (1991)Scattering of light by capillary waves in critical wetting interfacesS ChatterjeeIndian Institu te o f Astrophysics. B an0alore-56O 0 34 . Ind iaandV C Vani and E S R G opal-instrum entation and Services U n it, Indian In s titu te o f Science,B an g a lo re -560 0 1 2 , IndiaReceived 1 4 M arch 1 9 9 1 , accepted 21 M arch 1991A b s t r a c t : The intensity d istribu tion in scattered lig h t, due to scattering by capillary w aves in critical w e ttin g interfaces, is derived . It is seen th at the in tensity is confined w ith in an angle < « * > around the speculard irectio n , o being the surface tension of the interface. The q u a n tity < e * > is seen to be independent o f th e thickness o f th e layer. A tem poral m easurem ent, m easuring the tem poral rate of change o f scattered intensity in any d irection is seen to probe the dispersion re la tion o f th e cap illary w aves and is hence very sensitive to the layer thickness, for very th in layers.K e y w o rd s ; W ettin g , critical po in t, in terface, lig h t scattering.P A C S N o s ; 6 4 .7 0 J a . 6 8 .1 0 . - m, 7 8 .3 5 .+ cI. IntroductionW etting phenomenon, near the c ritic a l p o in t o f a b inary liq u id m ixture is an in te resting phenomenon where a heavier liq u id resides on to p o f a lig h te r liq u id . The observation o f w e ttin g has been reported in many system s (Cahn 1977, M oidover and Cahn 1980, Vani et of 1983). O bservations on the e ffect o f hydrodynam ic in s ta b ilitie s in the w e ttin g layers has been reported by us (C hatterjee et a l 1985). W henever the ce ll dim ensions are sm all enough such th a t hydrodynam ic in s ta b ilitie s are absent, an im portant ro le is played by the surface ca p illa ry waves in determ ining the th ickness o f the w e ttin g layer. C hatterjee and Gopal (1988) have discussed the e ffect o f the cap illa ry waves on the th ickness o f the w e ttin g layer and in pa rticu la r, have shown that because o f the influence o f the ca p illa ry waves, the th ickness o f the w e ttin g layer goes to zero as one approaches tha c ritic a l p o in t, provided the forces are o f long range type. For^Present A d d re ss : Physics D epartm ent, Indian Institu te o f Science, B a n g a lo re -5 6 0 0 1 2 . In d ia .276C a h n -H lllla rd type short range forces, the thickness has a com plicated dependence o n t= f ( 7 - T .) /T , ) I.A s has been show n (C hatterjee et a l 1985) s ta b ility o f the w e ttin g layer is very c ritic a lly decided by the d ispersion re la tion o f the cap illa ry waves. In the above paper, the authors considered the w e ll know n d ispersion re la tion« ® (lc )= (1//))[(g d p + o k *)k ta n h (k /)]I being the th ickness o f the w e ttin g layer. T h ii show s tha t co*(k) ~ (< r/p )k*lfo r very th in layers and leads to a free energy con trib u tio n F,Scatttrittg o f light by capillary waves etc 277capillary I-where X changes from I in extrem e lo w t lim it to 1/2 fo r higher t values. This pa rticu la r behaviour o f Fo,p,u,ry w ith respect to I is an im portant determ ining facto r as fa r as the dependence o f I on t is concerned. Hence, w e consider that the s ing le m ost im portan t ju s tific a tio n in favour o f our 'ca p illa ry w ave ' model can be obta ined, if a d ire c t evidence o f the above d ispersion re la tion can be found fo r the surface ripp les.W ith th is m o tiva tion in m ind, w e discuss the p o s s ib ility o f experimental ve rifica tio n o f the presence o f ca p illa ry waves. We take lig h t as the probe and consider the scatte ring o f lig h t by ca p illa ry waves, w h ich render the surface irregu lar (Beckmann and S pizzichino 1963). We derive in the fo llo w in g , an expression fo r the in te ns ity o f the reflected lig h t from a w e ttin g interface and show th a t it fo llo w s a d is tin c t sca ling behaviour w h ich may be ve rified experim entally.We propose tw o experim ents, in one o f w h ich a broad beam o f lig h t is inc iden t on the w e ttin g interface and the reflected beam is converged using a lens and then projected on a screen. The w e ttin g surface being irregu la r, because o f displacem ents caused by cap illa ry waves, the reflected beam w ill proceed in random d irections and hence the p o in t o f focus on the screen w ill broaden ou t in to a patch i f the exposure tim e be large. In th is case, the w id th o f the patch fo llo w s a fixed scaling behaviour. In the second experim ent, w e have a narrow beam o f lig h t incident on a given spot in the w e ttin g interface. The movement o f the reflected lig h t on the screen is determ ined by the t i l t o f the surface. This gives the behaviour o f grad The tem poral changes o f Grad ( depend on m (K) d ispersion and hence e x p lic itly on th e th ickness o f the w e ttin g layer and should fo llo w the behaviour derived in the fo llo w in g section. The tem poral changes in the in tens ity are related to the tim e evo lu tion o f the displacem ents o f the surface ripples. W e show th a t th is in both cases one stud ies the change in coherence properties o f the reflected lig h t. In the firs t experim ent, one studies the d is to rtio n o f the w avefront on re fle c tio n . In the second one, w e take the tem poral change in coherence fo r a very narrow reflected beam o f lig h t.278 S Chotterjee V C Voni and E 5 R Gopal2 . T h e o ry2 a . L ig h t s c a tte rin g b y c a p i l la r y w aves : s ta tic e xp e rim en tF igure 1 shows the w e ttin g s itu a tio n in a b inary liq u id m ixture . L^ and L , are the tw o liq u id s and L ^ L , is the interface show ing the b u lk phase separation. A th in w e ttin g layer o f form s on to p o f L and L ,L^ is the w e ttin g interface.L, l iF ig u re I . Schem atic diagram of the w ettin g phenom ena.W e consider the d is to rtio n o f the Interface, due to cap illa ry waves and the scatte ring o f a plane wave fro n t o f lig h t by these d is to rtio n s on the w e ttin g interface.F ig u re 2 . Sketch o f th e scattering geom etry.The c ^ illa r y waves at the surface are considered to be a random rough surface in tw o dim ensions. A s shown in Figure 2, le t a plane fro n t be incident on th e interface.Let $ko be the w ave ve cto r o f the scattered lig h t, w ith 5(s«, s„, s,) being the d ire c tio n o f sca tte ring . Let s°, s°, s° be the d ire c tio n cosines fo r the specular d ire c tio n and f(x , y) be the e leva tion o f the surface at any p o in t x, y.Then the in te n s ity in any d ire c tio n (s«, Sy, s i^) isScattering o f light by cap illa ry waves etc 279w herel(v )= (E r/A « ) e ‘ « ( x , - x J e J * ( y , - y . ) e ‘ » K (x „ y , ) - f ( x „ y ,)]dxdyX = X i — Xay = y i - y *(X i, y j , (Xg, ya) being p o in ts in the x, y plane.£ = T o ta l energy inc iden t on the surface.A =2w /ko = w ave length o f lig h tv = k o ( s - S o )r= R e fle c tio n co e ffic ie n t o f the surfaceW e know (^ 1 - la ) to be a zero mean random variable, whose d is trib u tio n is Gaussian. Then it can be show n.w ithand< e ; ’’( f i - l a ) > = e ; ”*<lx -|a)»>/2 (6)where ^ ( l i — la ) " ^ is the average over a ll possib le rea lisa tions o f the interface. The in tegra l in (1 ) is therefore,l (v » , v„) = (Er/A») \ ^ e * ' > ( x , - x , ) . e ^ ” ( y ^ - y , ) . e ; ^ \ ( ( ^ - i , ) » > / 2 dxdy(7)W e now Fourier analyse,l ( x . y ) - 2 l ( p . (8 )N ow , the d isplacem ent f(x , y) being real, w e m ust havee ( p .q ) = i ( -p , -q ) (9)A lso , w e consider the Fourier com ponents to be correlated as,< i(p'. q 'm p r q) >= s(p-p' )Hq-q' ) (iQ)U sing (8 ) to (10) one o b ta ins,< I i x - i .pf a280 S Chdtterjee V C Voni and E S R CopalTo evaluate / ( v „ Vy) (Landau and L ifsh itz 1959) w e m ust now evaluateK. I I*From (1 1 ), w e firs t ca lculate the therm al average fo r < | f(p , q) |* >, fo r the c a p illa ry w aves w h ich w e now proceed to do.2b. C a lcu la tion o f the rm a l averages ( | ^ (p , q) > •*To ca lcu la te the therm al averages < |^ (p , < l) i*> w e firs t ob ta in the expression fo r the energy o f the ca p illa ry waves.Let us assume th a t a displacem ent ^ ( r , t ) = ( ( k , t ) cos ( k . r ) be created a t any p o in t r = ( x , y) on the in terface, located at z = 1 . We know th a t the ve lo c ity po ten tia l fo r the flo w o f the flu id , a t any he ight z, is g iven by (Landau and L ifs h itz 1959),^ = ^ ( k , T) cosh kz (12)= V, = k ^ (k , t ) sinh kz (13)N ow the Z'Component o f the velocity at the surface is given byT)k sinh k l= a f(k , t ) /d t (14)^ (k , t ) = ( ( k . t) /(k sinh k l) (15)V ,= k (1/k sinh k l)) sinh ( k z ) i ( k , t ) (16 )The k in e tic energy is then g iven by.^ (p /2) .v j d xd yd z= (1 /4 )A p f*(k , t)/(k /a n h k l) (17 )A being th e area o f the surface.P otentia l energy has a co n trib u tio n both from the buoyancy term and the surface tension term .The co n trib u tio n to the po ten tia l energy from the buoyancy term is the w e ll know n g rav ita tiona l energy.(P .£ .)fl= (1 /2 )g d /> 5 ^ I « k , t) |«dxdy (18 )The co n trib u tio n from the surface tension term is ca lcu la ted from the product o f th e increase in surface area and surface tens ion.The surface area a t any instant is given by,^ ^ d x d y ( 1 + ^ * + f ; ) » ' « ( 1 9 )•VThe change in area,(SO)Sca tte ring o f lig h t by c a p illa ry waves etc 281on su b s titu tin g and s im p lify in g y ie ld s ,( 1/2 ) ( f*+ fJ )d x < /y = A /2 ^ k k „ - k , ) (2 i )Thus the surface energy is ,I ' ' • ) I? ( 2 2 )The to ta l energy o f the system is g iven by ^ d in g the term s in (17), (18) and (22), w h ich g ives us.{ 1/2 (gdp+< rk» ) 1 f(k ) j* + ( ^ /( k tanh k l)) i f(k )(* (23)Com paring (23) w ith th e expression fo r thefenergy o f a sim ple harm onic o sc illa to r and from the theorem o f e q u ip a rtitio n o f onergy, fo r a sim ple harm onic o sc illa to r, w e fin d the d ispersion re la tio n to be<»*(k)= (1 /P )(gd P+ ok •)k th (k /) (23a)< fC .E .> = = K ,T /2 (24)w e have.A = (K ,T /2 ) ( g J p + o k * ) (25)2c, l(Vg, Vy) fo r c a p illa ry waves :The q u a n titie s < 1 f(k ) |* > be ing know n, w e now sub stitu te (25) in ( 11) to get < 1 > = (k .T /8 r (2 ) ) (1 /T ) [K * „ /2 - (g J p /2 a ln (g J /.+ a K iJ ) ]R »(26)W ith R* = ( X i- X a ) * + ( T i- Y , ) * and K a ,x being an upper cu t o ff wave vector fo r the ca p illa ry w aves. As can be seen from (26 ), the q u an tity < ( f a - ^ a ) > 'is not determ ined by the h e ig h t o f the interface.Thus w e have from (26 ),exp ( - v j < ( f i - f . ) ’ > /2)= e x p ( - v ;a * R * /2)whereo « = (K ,T /8 r(2 ))(1 /ff) [K * „-(g d /> /2 a ) In ((g d p + a K “ „ ) / ) ] (26.1)Thus, the in te g ra l in (7 ) can be eas ily w ritte n as,/(V ., V v)= (A 2»r/o*) exp ( - ( v J + v ; ) / 2v ;o « ) (27)The Gaussian nature o f (27 ) show s th a t the in te ns ity is maximum along the specular d ire c tio n and th e in te n s ity is broadened to a patch o f size< v ? + v ! > = 2 r V (28)or to an angle e around the specular d irection , such tha t< s !n *a > = 2 lc * fl* /k *= 2 < i* (29)It is seen th a t a * is independent o f the w avelength o f lig h t. Th is conclusion is in fa c t a resu lt o f the approxim ation — w h ich alw ays arises inthe lo w R lim it. How ever, fo r R -> «>, coi^stant and th is shouldg ive rise to a k dependence in a ‘ .Thus i f F is the d istance from the source to the screen, the size o f the patch SX is g iven by, = f« < s in * (30)from the expression fo r a ‘ ,a® 1 /« « (31)the size o f the patch should scale as w h ich should be the resu lt expected from th is s ta tic experim ent, where a broad beam o f lig h t is focussed on the w e ttin g interface. A s m entioned in the in troduction , the average in tens ity p ro file ob ta ined above contains average over a ll rea lisations o f the interface and hence has no in form ation about the dynam ics.2d. Light sca tte ring by cap illa ry waves : dynam ic e x p e rim e n t:To ob ta in in fo rm ation about the dispersion re la tion o f the ca p illa ry waves, a dynam ic experim ent is required. I f w e note the tem poral rate o f change f'(t) wesh a ll be studying the behaviour o f ^(x, y). This e x p lic itly involves the dispersion re la tion as w e show be low .D iffe ren tia ting eq. (2 ) w ith respect to tim e,l(*. K)=(E/A*) e‘ « (x , -x . ) .e J ”( y , - y , ) V ( f , - ^ . ) e ‘ « ( i , - ( , ) d x d y(32)W e consider the case along the specular d irection i.e , v . = v „ = 0 . This gives,1(0, 0 ) = (E r/A » )l v , ( f , - f * ) e J ’- ( ^ ,- f ,)d x d y (33)The integrand in (33) is a zero mean random variab le , so th a t < 1(0,0) > - 0 . H ow ever, the quan tity < f(0 , 0 )* > can be easily estim ated as fo llo w s .We have seen from (26 and 26.1) the quan tity in the exponent correlated w ith in R - ^ r o such th a t(1 /2 )v ;a * r; = 1 o r28 2 S Chatterjee V C VanI and E S R Copalf 2 = 2 / ( v ? o « ) (34)283(35)(36)(37)Scattering o f lig h t by ca p illa ry waves etc Thus, over a b lo b o f rad ius to , w e have[ 6 ( x i - x , ) + f v ( y i - y , ) ] r ;where and are also random variables, correlated w ith in the b lob.Thus fo r every b lob ,< [ \ \ W x - i . ) ] « i x d y ] * > ~ > + < / ; > ] r » = < «TOThe interface has N ~ A /r* b lobs. Hence from the' law o f large numbers, < [l(0 ,0 ) ]* > = (£ r /A * )« y ,* N < *= (E r/A ‘ )» v ;(A /r * ) t< i‘ > + < f ;> ]r « := (£ r /A « )« v J A < i;> r*= 4 ( £ r / A « ) M < > / { v ; o * )Now< > = < f ‘ > + = 2 ' * ^kw hich from (25) readskC max= (kaT /p ) ^ k * th k ld k (38)OFor ko ,„l«^1 one fin d s> ~ ( IcbT/p) ( C .J ) /6 (39)i.e, it fo llo w s I, increasing as t increases,w h ile fo r kni*xl > 1 w e have t < (d o /k)^ '*CP = (k B T /p )(1 /5 )[k ;;,-(1 /l' ^ x« sech»xdx)] (40)Oi.e. there is a s lo w rise as I increases o r as t increases.Equation (37) show s that «c A. A lso < / > t A . Hence for effectivedetection o f < /*■} in the background o f < / > one m ust have the ra tio < !■ >/< / >* large i.e . the size o f the lig h t beam m ust be small*3. Effect due to finite thickness of the interfaceThe to ta l dependence o f < ( • > also contains the* re fle c tiv ity o f the interface. In th is respect, the m odel o f the in te rfac ia l thickness is necessary, to ascertain there fle c tiv ity o f th e interface. We consider here the d ie le c tric constant to vary accor­d in g to the Epstein m odel (Epstein 1930)« (z )= (‘ i+ « .) /2 + C (« x -« .) /2 ] th (z /d ) (41)where 2d is the in te facia l th ickness, i t is im m ediately seen th a t c(z) -> as z oo and e(z) -> e , as z oo. The re fle c tiv ity in such a case is g iven by (8 ),[$h(7rkd(ni - n ,) /2 ) /s K n k d (n ^ + n ,) /2 ) ]* (42)I t is seen th a t fo r n k d /2 I, r is approxim ated as,r = ( n i - n . / i i i + n , ) *r g to o (43)be ing decided by the d ifference o f the re fractive indices and being insensitive tothe in te rfa c ia l th ickness.For n k d /2 > 1, on the other hand, d ie re fle c tiv ity fo llo w s ,r= e x p [ - n k d ( U i+ n ,) ] (44)being insensitive to the re fractive index d ifferences, bu t fo llo w in g the in te rfac ia lth ickness.N ow w e know , d ~ d o t" * '* w h ich means th a t w ith decreasing t, the re fle c ti­v ity fo llo w s a decay,r ~ e x p ( - d o t - * '* /A ) . (45)The m agnitude o f < i* > thus depends c ru c ia lly upon the q u a n tity k „ l . For a ll p ra c tica l purposes one m ust have km < l/d o r kml < 1/d <1, in order th a t an interface can be defined. Hence equation (40) is the va lid equation fo r a ll ranges. A lso if one perform s experim ents a t t> 10~°, in the v is ib le range, one fin d s cond itions proper fo r the app lica tion o f eq. (43).Thus, in these cases, re ta in ing o n ly the t-dependent term s, one has.28 4 S Chotterjee V C VanI and E S R Gopal00< [? (0 ,0 )]*> /A < t* * 't* ‘‘ [k ;^ - (1 / l» )^ x» sech*xdx] (46)w here /5 ~ 0 .33 , /» ~ 1.33, k^ t * '* and / ~ 0 .66 ,Thus the firs t tw o factors in (44) g ive ^ on inserting the fa c to r[ — ] In (44) one fin d s < 1 (0 ,0 )]•> to fa ll faster than t®’ * * as T« is approached, o w in g to the faste r va ria tio n o f /~® term ( ~ t " “ ’ ») th a t k ^ ( ~ t " ) as t 0 .Thus as the tw o term s become com parable < /*(0,0)® > w o u ld a b ru p tly vanish.4. ConclusionThe m o tiva tio n o f the present paper is to explore the p o s s ib ility o f d ire c t observation o f ca p illa ry waves in the interface o f w e ttin g layers. The ir existence can be easily 'seen ' in th e scatte ring o f lig h t by the interface o f th e w e ttin g layers. In absenceScattering o f light by capillary waves etc 285o f the d is to rtio n o f the in terface, one w o u ld o b ta in a perfectly specular re flection , m a in ta in ing the angle o f incidence equal to the angle o f re flection . The cap illa ry wave d is to rtio n s , w o u ld cause departures from th is specular cond ition . In contrast, the scattered beam w o u ld be seen to arrive a t angle o, w ith respect to the specular d ire c tio n w here < « > = 0 and This sca ling is seen to be determinedby the sca ling law s o f th e surface tension and beingindependent o f th e layer th ickness. The dynam ic experim ent invo lving the measurement o f < I* > is seen to be strong ly t dependent as g iven by equations (38) and (40 ). A com bination o f these tw c | sim ple experim ents w o u ld unm istakably estab lish the im portance o f ca p illa ry vjfaves in determ ining the s ta b ility o f w e ttin g layers. »Recently, D ie te rich and Schack (1 9 j|7 ) have considered a case where the interface is o f m esoscopic nature i.e ., the in terface cannot be considered by a w e ll defined th ickness and stud ied the ease l o f specular re flection . The in terp lay o f cap illa ry and the m esoscopic in terfaces in ie fle c tiv ity o f surfaces prom ises to be one o f rich prospects.AcknowledgmentThe authors are g ra te fu l to IFCPAR fo r fund ing a pro ject and supporting th is w ork. ReforencetBeckmann P and Splzzlchlno A 1963 Scattering o f Electromagnetic from Rough Surfaces (London: Pergamon)Cahn J W 1977J. Chem. Phys. M 3667Chatteriee S, Van! V, Guha 8 and Gopal E S R 1985 J. PhysUiue 46 1533 Chatterjee S and Gopal E S R 1988 J. Physique 49 675 Dieterich S and Schack R 1987 Phys. Rev. Lett. 58 140 Epstein Paul S 1930 Proc. N a tio n a l. Acad. Scl, 16 627Landau L D and Lifshitz E M 1959 F lu id Mechanics (Oxford : Pergamon) Chap 1 ,7 Moldover M R and Cahn J W 1980 Science 2071073VanI V, Guha 8, Gopal E 8 R and Madhusudana Rao 8 1983 Phys. Lett. 99A 441

Scattering of Light by Capillary Waves in Critical Wetting Interfaces

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Scattering of Light by Capillary Waves in Critical Wetting Interfaces

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