of partial

LIGHT SCATrERINGIN BI NARYGASHIXnJRE S:

EVIDENCE OF FAST AND SLOW' SOUNDHODES

By

LuoHongda, B,Sc

Athesis submitted in partial fulfillment of therequirements for thedegreeof

HasterofScience

Departmentof Physics HellorialUniversity of Newfoundland

August1990

St.John's Newfoundland

1+1

The authorhasgranted en Irrevocablenon- exctusiveIioenceallowingtheNaliooalLJbrary ofCanadato reproduce,loan.cfcstributeOfSEll copies of hislherthesisby any meansandin any formor format.makingthistfcstsavailable

to Interested persons,' .

The al.tthor retains ownership ofthecopyright in hisfher thesis.Neither the thesis00(

substantial extracts fromit maybeprinted(l(

otherwise reproduced withouthislherper.

mission.

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a

Iadisposition des personoes interessees.

L'auteurCOOSM'eIa proprietedudroitd'auteur quiprotegesathese.NiIathesenides extralts substanliels de

ceue-cr

ne delvenl 6tre imprim6sau autrernentreproduits sans son autortsatoo.

In this theah Brillouinscatte r ingdata are presented for th reedifferentbinary gas mixturescomposedof specieswith wide l ydi f fe r en tMasses,i.e.AI'+H:dmu / mllz...20),SF,+eH.

<tnsFs/lllcll. "9),andSFe+HZ(tlls F 6 /~z..73).Underthe expe rime nta l co ndi ti o ns predictedby the theoryafast sound modecontr ibut io n to theli gh t scatteringspectra of AI'+H2. mixtures has been detected. Unlikethe ordinary soundmode the fast soundmode pr ops g a t e s onlyinthe lightcomponent with a velocity higherthan tha tobtained usinghy dr odyna mi e the o ry. However , theatte mpts to observe thesame eff e c t in IdxtureaofSFe ... eH. undersimila r condi tions vere unsuccess ful.An an a logousslowsound mode con tr i b ution tothe spectr aofSF6+H1mixtures ha s been clear l y identi f i ed andit ....as found thbmodewas, ingeneral,much easi e r tode te c t .

Anev parameter calledtheeffect ivemeanfreepa t hva s also in t r od u c edinorder to obtain II more consistentcharacte riza t ion ofthedynanlicalbeha vi or in binar ynlix.tu r e swithdis pa rate

i i

ACKN01lLEDGE/'lENTS

I wouldliketo expressIllysinceregutitudes to my supervisor Dr . H. J. cteucer for guidan ceat everystep of this expe x faenc,His experience has been of great help to me throughout th isproject and inthe prepa rationof thisthe sis.

I wishto thankmy supervisor Dr. H,Kieftefor his careful reading of this thesis,his encouragement andvalu ab l ehe l pin the finalstage of thethesis,

Inpar t icula r, I am gratefu lto Dr.A.Campaof th e Rockefel le rUnivereity, and Dr,J. A.Zollwagof the~choolof Chemi calEngi ne e ring , Cornell UniVersi ty for send i ng lIIe the theoret icalcalcul a tions conce r ni ng ourexper imen t,

Thanks goto Dr,J.K.C, Lewis and Dr . N,Richfor thei r helpful di sc ussion snd adviceand toIllyfriends V.Askerpour and Dr. Ri ch ardIt,J. Gouldingforwh a t thevhave donefo r me during Illy stay at Memori al University of Newfoundland.

Ialsowishto gratefully acknowledge the financi al support of my work from Memorial Uni ve r si t y ofNewfoundland.

iii

2.1 Dens it)' Corre l ati onFunctionl&

BrillouinScattering Chap t er1.

1.1 1.2 1.3 1.4

Chapte r2.

2.2

In troduc t ion

Fluc t ua t i on.&Sril louinScat te r i ng ThaK. anFreePaths:inSinnyHixtura . Hydr odyn a,lIic&Kinetic Regllllu llyna mic s of Dlsparate-Hass Gas Hhltures

TheFast&510..,Sound Hoda .

Tha or y

Hydrodyn a lllicTheo ry

12

12 22 2.3 KineticThe ory&Predictionof a FaItSound

Hodein Dispara te-Han Mixture s 25

Chaptor 3. Apparatua and£xperil'llen t a l Procedur e

OpticalSy stem

iv

33

33

J.2 Data Acquis i tionand Stab il izatio n

syseee(MS-I) 19

J.J Gas HandlingSystem 4l

Chapter 4. Experimenu.l Res ul tsandDiscussio n

"

4.1 GeneralRe mar ks

"

4.2 TheFu t Sound HodeInHz+Ar Mixtures 47

4.3 CH4+SFa^Mixtures 54

4.4 TheEff ect iv e Mean Free Path

"

4. ' The Slow SoundMode in SF, +Hz Hhture!l

4.' Conclu5ions 66

Referenc ee 72

Appendix Pub licat i onRepr i nt 76

LISTOFFIGURES

Fig.1.1 Classical pic ture ofI1ragg reflection .The incidentlightwaveis scattered by the densit.ywave1nt.he medium.

Fig. 1.2 Schematicrepresen tationof the lOcatt e r e d light epeecrua fraila fluid.

Fig.2.1 The scatteringangles definedin relation 18 (2.10)

Fig.3.1 Schematicd1s.gralllof theexperimenta lsetup . 35

Fig. 3.2 Typical sp e c trumre cor de d in I1rillouln 42 sc atteringexpe rimen t.

Fig .3.3 Schematic d1s.gr.ll1ll of the gashan d li ng syseee, 44

Fig. 4.1 Spect raforAr +Ha lIIixtures with a H2ba s e SO pr ess u reof 9.0ba rs,

Fi g.4.2 Spectra for Ar+H2mixtureswith a "2ba se 51

prassure of "1.7 bars.

Fig.4.3 Spectra for Ar+H1 lIIixtures wi t h at!1base sa pressure of6.3 bars.

Fig.4.4 Spectrafo r SF6+CH,mixtureswithII CH,

"

bas e pressureof 6.Jbars.

Fig .4.5 SpeetraforSF6+C;H~ mixtures'11t h IICH~ 57 base pressureof 3.7bars.

Fig.4.6 SpRct ra for SF6+H1 mixtureswithIl SF6 ⁶⁸ basepressure of 1.7bars.

vii

Chapter 1. In troduction

i.1.1. FLtJCTlJATIONS "IIRlu.ooI NSCAlTEll ING

The scatteringofligh t generallyarheaalitberesult or opt ' c al inhomogeneitie sin the scatteringmedium.The physical reasonsfor thelen er8tlonof opticalinhomog eneitiesin the scatte r i n g lIIediUlll are vari o u s.ForeX/I.IIIpl.,contalilna t i o nby for eign sub s t a nce. 11 one possibility.However,even insub stance s whi charecOCIpla t l l yfr••fr~anykind of fo uigncontamin . nt a. su t t ered Hshtcan baob••eveealltheresultof theata t btlcal cha ra c t e r of theraalmot i on .In thi sca . ..til,opt i cal inho.llKlgen e i t i e.l arecaus edbythe local fluctuation.of the optical dhlactricconstant,which are directlybrough tabout, in tum,bythefluctuatloruIn loc aldensityofthesubsta nc e.In contrastwith theimpurity sc atter ing and R...nIca t te ri n g ,this kin dof lightBcatU ri ng,arising from randollthermalfluctuat ions is called Rrillouinscatteri ng !l].

According to Brillouin,thespontaneOUlildensity£luctuationlll in themed i um can bedeco .posedintodifferentplane -waveFourier

.1.

ceeponents,These plane-wave apropag atinginthe med i Wl caus",a periodicdenslty changealong thepropagat iondirec tions. Scatte r ed lightisbr ought abou t bythe diffractionof the inciden t plane monochro mat i c llghtwavewhichis cOJ1si dl!lredto be diffractedby thedens ity maxi lllaofthe thermal501.1(, .1lO'ave in the same wayas X- r sys ar e sca ttered bya crys tal. Ifleoisthewavevec t or ofthe inci dentlight,kris the wavevecto r of the scatte red liGht andK i.thewave vectorof theKth Fourier ecepe neneof thesound wav e caus ed by the spontaneousloc aldend ty fluc tuatiofUI of the med i um in direc ti on

i ,

thenthecond it io nfor sc a tte ri ng is givenby the Rraggrelati on [Fi g.I.l )

K-±<0, •

k,> .

^{I.1)

where the ehoice of slgna referstoeit he rof two oppositely directedsound wa ves .Becausethedensl ty max i llla ar e travelling withtheveloci t y ofsound inthefluid (fl ui d refersto both gas andliquid her e),the frequencyof the scattere d lightwUI eonsequently baDoppler sh i f ted_If1010 istheanguhrfrequen cy of theincidentlight wave,IoIf isthatofthesca tte red wave,v. is thesound vel oci t y,,isth e sc a t te ringangle. n is th e refractive index ofthe lIedi um,anl1c is theveloc i t y of ligh tin vacuum, thentoahigh degreeof approximation,thefreq uencyshift is gi venbythe woll-knolmBrillouin re lation

~^{_ "If• "1}₀^_±2wo:.n sln(~).

.2.

(1.2)

'Jr.J.J Claninl picture of Ira" nn.ctton. Ttle inddant.

Ulht. waveIelcect..n'

bJ'

t.h.'.ndtJ weve in t.he . .di_•. 1r.e1athewev.nctor of che incident. U&ht.It,h t.hewe.,.

dendtJ vlve. eneS ,1a t.heleatt.erll\l ancle.

3.

Although sound waves of many differentwavelen gthsand any propagatio ndirection ar e present in the medium,inan actual llghtsca t te r i ng exp erteencfor eachsc at te rin g angle(Jother than zero only ehesoun d wav eswithwavevectorgivenby(1. :~can be observed,Accordingto (1.2),the scatteringspectrumconsists of on l ytwo sharp l1ne.~with frequency sh Hlillof ±.:1<.l,respectively , frolll the centralfrequen c ywu'In reality, the spect rumofthe sc a t t er e d light alsoincludesa central linewith no frequency .IIh i l t. Lan dec. and Placzek(2) gavethe followingexpl a na t i on for the origin ofthe centr alpeak .They pointedou t thatt.:"t:h erm al fluc tuati ons inafluidwerecOPlposedof two par t s,namely, isoba ric fluctuationscause dbyent r opy fluctuat i onsatcons t a n t press ur e,and adiabaticf1uctuationa cau s e dby pressure fluctuatiQnsat cons ta nt en t r opy. Fluetuationsofthe fintty pe are unorganizedin ti me,90they are not propagal;edintheform of wavesand eons equentlythe centralline eanbee)'~lain..ttas elastic Rayl eighscattering . On the otherhand, thepre s surefluctuations repres entrandollloeal eompre s sionsor rarefa ctionswhich, as a cons equen ce of theelas t i c properties of the mediUDl, do not remaIn n"edbut travel through the fluid and give rise to'"shifred doublet. Because ofrhe ex1steneeof dissipativeproces seswhich da mp out the ela s t icwavesin the med ium,the doubl etccep cnenes haveneaaere Une widths.The nonzerowidth of the centr.alline ca n be explainedin termllof hlUllt conduction and diffusion.A typiea.l Brillouinspec t rum issh own in Fig.1.2 .

•4.

o 100 Frequency shift

800 I---~

.~ .. c .5 !

Fig.1.2 Sehemat i c rep re s enta t Ion of theBcet te re dligh t apec t r\Jlll frop;,•fluid: long- da sh edlIn erepresentsthe und isplacedRlI.yle1ghCOlllpone n t iahor t -da shed lineeep r e s enes th e St oke sandan t I - Sto k es &rilloulncomponents ;dotte d 11ne indicates.thenon-p r opllgHt l ne; fourth compone nt whichIs much leu intense thanoth en[I];solid line in dic ate sthe aggregat e intenaitydistributioninthe sca tter e d,pect rUlll.

•5.

§.1.2.TIlE MEANFREE PATIIS IN BINARYfUXTUltES

Theoret.icallythe spontaneousde nd t y fluctuat.ionsin t.he mediU/ll canbe decomposedinto densit.y plane-wavesof different wave lengths.In reality, however,onlyt.hose density waves with wave lengths larger thanor eorepareble to the eeen fre e patht existin the mediUll.Usuall y che product. of K and mean free patht is used to characterizethe conditions of thes8lllples(see§.1.3). The meanfreepath in a low-densityone-componentfluid is given by

(1.3)

where(I isthe har d sphere diamete r of the part lc leandnisthe nUlDber density.

For the binarymixtures t.he sit.uat.ion ia somewhat.eoepHeaced because thereare several possibledefinitions for themean free path. For example ,i tispossiblet.o define alISaa fre e path with respect to collisionsbetween the specific types ofparticles. Here we int roducetwo",eanfreepathsl1 and(1definedbyP]

(I -(t;~+t;~)-I ,

t

1-

(t;t ...

t;~>"l,

(1.4a) (1,4b )

where llJ(l,J-1, 2 ),theaveragedis t a nce travelled by a

.6.

part i c le of.pe c ie . i betweenit.two.ucce s. iv e co1lllions with par tic le.~f.pe ci. .j. i. . .Iv e nby thefollow ingrela..:on

where nl,171 and^_.Ire p r es en t thenW'llberden!li ty,herl!.aphere

If aparti cla of spac ie' 1travelsa dh ta nc eL,acc ordi ng to thedefi n itio nofll J ' th o averagenUlDbera ofcol lision.it expe ri en c es with parti c les of species1 and2aregi venbyLi lli andLillare . p ec tiv el y .The.v erage totalnumber-ofcollision.ia given by Lllll+LIlli' HenceL1 ,whichequalstoL!(L/ll1+L/ll~)' canbe understoodasth e average dis t a ncebet we entwosuccessive col11.10 n .exp e rie nced bypa rt.ic l e s of.pee h. 1.However , as we viII seein Chapter 4, forthebinary .. btur. .wi t h large lila••

difference_L1andL,are in.uf f1 c h nt to characteriz.ethe conditions of binary fluid •.A possible .ol ut i on is viathe -.ffectiveICe anfre e path-whi chw11 1 be definedanddiscussed later.

§. 1.3. HYDRODYNAKIC~KINET IC REGlKES

In asimp lefluid, ..cihlen.i onlen p.u8meterKL(Lis th e llIeanfre epathofparti cle.in thefluid) iaus edto char8 ccerl ze

.7.

thenatur e of thedensityfluctu ationsto beprobe d{4].The in d i v i dual particles in the flu i d travela finite distance.(

be t we entneer-par ticl e co l lis i ons ...bic h ca use loc a ldensi t y fluctuat i o ns .SinceKisthewav e vect o r of the fluctua tions to be prob ed, a smallvaluefor Klmearvi!thatwi thinthe leng t h scale IjK(thewave-le ngth oftherma lsound wave) many c.Jllisionsoccur.

Sointhe limi tofK.t«l(hyd rod ynam ic reg ille ) . che deviationfrOlll local thermodynamic equll i briUIIIis verysmalland hyd ro dynamic theorycanbe used.On theotherhand, for Kt -l(k inetic regi me ), particles under gofewcol l h io ns over th e11Kle n gt h scale and tihe loc al equilibriumassumption cea s e s tobevalid.One mustuse kinet icchearyto obta i n aDor e ge nerelBnd more lDicroscop i c des c ript i on.Theregi on Kt»l is thefree -s t r eatllingre gionan d yieldsthe well-known Dop p ler profile, which 18not of inte rest here.

Ina bins ry flu i d, the situationIs 1II0recomplica te dsinc e th e r e ar eseveralpossible defini t i on softhelIIE1a n fr e epath.Mor e de taileddis cus sions wil l be give nin Cha pt er4.

§. 1.4 . DYNAMICSOF DI SPARATE-MASSGAS KIXTURES TheFastandSlow SoundKol'.es

Theinvestigat io nof binarydisp arat e ' lIIa ss mixtur e s,wh ere thetwocOlllponen ts hav e verydiffe rentma SS6S ,be gan In1960.At

.8.

thattime CUd ( 5)potnted outthatin a dilute dlsparllte·.ass .ixture theslow.."ch angeofki ne t i cene r&), be t we en two eeep en enee shoul dleadtotva te.peratu res:one is a..ocla te d with thelIgh t componentandano t he rh a. .ocle te d withthe he. vy coaponent.In th eae ee 1970a,wh.n Huck and Joh ns on (6 ) studied thebeha v i or of forc edsoundIIlOdu indilut ebinary dispa rat e-INII ga. lIi xtu re.of

their stud1e. ,theypredictedtheexla te nc:.of two diffe re n t forcedsound lIIode a, n. .el y a fastsound IDOde and a slovsound ec d e.the fas t sound mode b associatedonlywiththe light component and propag a t e svlth ave l ocit y grea te rthan that expectedfor hydrody nami chehaviorof the _ix t ura, whilethe sl ow .oun dlIode Sa Ju. ttheoppoaite: it1&aa.oc iated withthe heavy co mpo nen t andpropagat. awit h a lower va loc:i ty.In the expe r i_nta con du c t e d by !owlarand Johns on (7) in dilute bina ry. b tur e. of H.

and X" a ahArpincr•••e in aou ndvel oc itywa .fou ndund eran ex te rna l dSatu rb a nc ewithfr equenci u w>lO' H:z endHe .ale fractions hlghe rthan0.45.Hore rec ently ,in1986. Boase.J.cucc1 and Ronc:hen t t 1(8)found a fast propagating ao und1II0deineomputer aimula t i onsof LiPb liqu i dal loys withla rgeatOld c. . . . differenc e(.'1l/~1.l .30) .Theypredi c ted. forhigh fre quenc::ie sand la r g e waveItWllbenbeyond the hydrodynaJllicregi. a ,andunderthe co nd iti on ofhI ghconee n t rat i on of theligh tcomponent , thatthe fast sound couldbeobs ervedby tbe usinginela sticne ut r on sc a tte r i n gtech ni que.ThSa was eonfitllled inexpe r i llen u by Hontifrcoi.1,\lesterhulj andHa. n ( 9 ).In theirlataa t paper,Call1pa

. . .

andCohenllOJconsidered light scatteri ngin dilutedisp arate -mas s bi na r y gaslIIbttures with hi gh concentrationsof the light componentand predictedthata fastsound mode shouldalso be obs e rv e d as the pres enceofs1depea ksor extendedsh ou l de rswh ich ar e onlyass ociated withthe lighter compone n tin the scattering crossse c ti on for visible light.For a binary mixture, the scetteredlight intensitycanbe expressedasa weightedave rage of partialdynamicstructurefactors, (see detaileddiscussion in Chap.2)

where subscripts1and 2 refer to thelighterandtheheavier spe c ie srup. c t i v.l )' .'"I and Xl deno te th.lII018cularpolarizability and thelIIole fractionof component1. K and ",arewave numberan d angu l ar frequenc),to be probed,SlJ(K,w)are part i al dynamic struc t u r e factors.The first and thethird termsin(1 .6)are associatedwiththe lighterend heavierspeciesalone and the secondis amixed term.As we cansee from expression (1.6),the polari z1bilitiesoftwo componentspl ay tepereene rolesin ligh t sca t t e r i ng. In orderto observe the fast 1D0dewhi chis only associated withthe lighter component ,01/01should be aalar geas pOll8ib le(atleas tcomparable ).Conside ringbothmassand polar izabi1ityratios, the mbtturecombinationwhichCampa and Coh e n suggestedto studywasargonan dhydrogenwitha lIIasaratio of 1lI...~/ma 1"20and polarizability rat i o ofa-8/QA~"'O .5.

.10.

Althoughseveral llght-scattering expariments had previously beenconduetedon binarydisparate-masagas mixtures by Cornall fUJ, ClarkI12,13)and Le.tamendia(l4 ) ,the experimentalconditions and the mixtures theychose we re unfavorable[01"thaobservation o[ thefastsoun d mode. For eXlIlllple.inClar kand Letamend ia's exper i ments, th eyus ed xenon and heliwa("'x./~ ."'33)llIixturl'!s With a pola rh:at.lonratio of(1 . .1°110"19.6,muchhigherthan thatof Ar and H2•Besidesthis,accordingto Campa'scaleulations , in order to ebs e'rve the fast sound 1Il0de the gas density sh ou l d be IllUch lower and ala r ger scattering angle isalsopreferred. In this thesis,wa willpresent recent;light scatteringresultsfor dilute Hz and Ar mixtures,which provideevidence fo rth e existenceof a fast Plods.Our sttelllptsto observeth e sallie effectinmix turesof CH, andSFI (tIIg'I/IllcIl,",9, Qsrll/ocll,-1.8)wereunsuccessful, Howeve r,for th e gas lIixtu resSFI and Hz (tllar/lIla

z"73), an analog ousslO'IlIode was observed.Compared wi t h the fast mode, i t seellls thst:the dow IlIOde iamucheasierto detect.

Before presenting theexperimenta lresults,itill necessaryto giv e a mor e detaileddescrip tion ofthetheo ry1nvolved in Br il l oui n scattering.Inth efollowi n g chap ter,we will out li ne thetheoreticalbackgroundpertaining to the therlilal ligh t scattering.The experillle\lta laapects willbe contained1n Chapter3 (Apparatus&Expe r 1mante l Proccdu t"e)and Chapt e r4 (Experiment alResults&Discussion).

.11.

Chapt er 2. Th eory

J.2.1. DENSITY CORRELATIONI'UNCTIOIS 6r,lULLOUlNSCo\TTERING

Thedensi t yfluc tu ationof fluids 15 usua l lydllScl:'l be d In thelanguageof spa ce·t l l1le correlation functlon aI15 ] .A space- tll11ecorrela ti on funet i on11 defineda. the thermodynamicave r age of theproductoftwo dyn.lllli calvariable .,eachofwhi ch -xp resstl.

theinsta n taneousc!ev..",t !onofIIfluidpropa r tyfrollIt a equ i l l · brllJJ11 value.ta pa r ti c u larpoin tinspac.andtilDe.Ceneral ly

C...(r,t; r',to)_C,<'A (r',t')"6(r,t»

- cu c l r · r· l.

t·to), (2.1)

where

U(r,t) ..A(r,t)•<A(r ,t» . 6B( r , t )..lI(r,t)•<8(r , t» .

are thefluct ua t i on.in the d)'TI&lllcalvariable .A{r,t)and8(r,t ).

•12.

Thefa ctor Coisa con.\ltantdefined fo r theconv en hnc eof the spe cific cor r elat io nfunct ions. Theangle bracke ts represent av erage s over the phas e coor d i na tesof allthe lllolecules Inthe fluidwithsn equi l i ht:i ume naeebkeas the wei ght ingfunc ti on,i.•.

<A( r, t)~

^-

Z(JJ~V,N) -I

er ACr,t) · e· jlH( r) , (2.2)

wherercollec tivelyindicates all thephase spacevariab les , H(r) i_theHami l t onianoft~ee.ye t efl withN par t i cle llin volUllleV, ,B"C k!T)"1 ,andZ(P,N, V) is thepar t ition func t ion of theSY.\Itern.A space-ti mecorre latio n func ti onisth ere f ore a func t i on ofspace andtime , and it describesth ethe rma l flu c t ua t i onswhich edst spo nt ane ous l y In the equ ll I b r IUIIIsyst em.

Using cla u icalel ec t ro dynam i c theo ry the relatIonbetwe en the approp r ia teco r r elation function s an d the int ensity of Brillouin sCBtte r i n g can be obt ained.In order tosi lllp li f ythe dh;cuu i on we assumethBtth efluidsarecompos e d of optI ca lly ls otropic1l101ecule s , e.g.thein ertga se s.In this case the fluctuations of thedielectricconstan t causedbyorientationsof tho 1lO1e cui e s in the medIum do not need to be considered .

ConsIderinga fluid withan av erag edielectric cons ta n t co' thedIele ctric const ant w111flu ctua t efro lll place topla c e and fromti lleto timegIvIngrhe toinholJlogeneities.The instantaneousdi e l ec tric cons t an t t(r ,t)can beas sWIledtobe

.13.

equal!:> theavenge(0 pl usafluctuationpart Ac(r,t ), I.e.

I(r ,t) ..It+AI(r, t) , (2.3)

when rh the podtlon In the Ica t te ri ns .e41 \111.'nilvol Ullleof the lledlUIIIh ...-d tohe largesothat su r fat eeff. c tslIlaybe neg l ected.Be c au 5Ithe -.ollcu le. areis otropic ,( andA(are dmpl i f1 ed toheIcalars inlteadof tensora (16 ).

UsLng thlalocaldl.lectriccons tantin theHa x\lellequ a t i on s for anonconducting,noruaagn etlcmedlUJI,wehave

{ V_Vx_•V_D)(_{.. 0}E....(1/c^Z)(a 'D/8 t ' ) ^(2.4)

",h. n Eh the elect ric fh14 vec t o r Inth e _dl UIII,.ndD11the Hnvell4i . placeDen tveeeer "hlchs.tisfies thefo110\11nl nlat10n

D...(r,t)•E.

'nI.equati ons tha tneedtoIHtlolve dare, there f o re (2.5)

In cla1ll 1c alelectrodyn4lD1e pertu r ba t i on theory , the sca t tered eteecrtefield E c.nbe expande d 1n termsofthe Inci den tf1eldKo plush11he rorderte nuwhlchonlyIncl udethelighttca t te re d In

.14.

oth erthan thedi rect ionoftheincidant l1r,ht,l.e .

E-!o + E1+ t; · ·· · ·· · · · ·

He rewe assUllHl that the inte ns ityof. ut t e re d Ur,htvit h (2 . 7)

Beatt eringans la,..011 alllallc:oraparedto thein tena i t y of the incide nt llsht;thh conditionunbegen eraUy real b ed inguea

Sublltitut ingequ a ti ons(2.3)and(2.7) Intoequa tion s (2,6) and setting aWIIIof th e aame or de rof~gnil:udeequa l tozero,we ob ta i naseriesofequa tions:

VZ",-

3- . ~ -

0

c "

...• • .etc.,

(2.8a)

(2.8b)

I!o,whichrep re aentatheincidentli ghtwavewill (acco rd in gto theapprox!Dlation)betrans lIit t .d throughthafluid wit hou t attenuation.£1 can b. consld.:!red asthere .ultofaalngle acattering proce..:I.e.In th a quantUII Dlechanl cald. a crl ptlon . photons represen t ed by £1 are sca t t eredon lyonce byphonon sll7].

Th.temof Dlost intereat In (2.7)h t;" th eling le scattering eera .The contributions ofEaand ot her hi gh er order sClltt.ring

.1 5.

terms areverysmall exc ept inthe neigbljourhoodofa critical point,

The incident fieldis assUlDedto be a planemonochromatic wave of the fQrm

Eo-Aoexp(iko'I'-iWot), (2.9)

when koisthe prQpaga tionvactor ofthe incident light,and^Wo isits angularfrequenc)' ,SinceEosatisfies (2,8a),we abtdn the relationleo-(I:O) I/2·("'o/ c )

E" in equation (2.8b)eanbe solv edfor by doubleFourier

transformationin spaee andti. e,lJhenR,the distan<:efro.tho origin toth e pointofobservation,is muchlar gerthan the la r ges t dimens ionofthescatter i ngmediUIII. the space-ti_

Fourie r tl'llnsforlllofE1(K,ltlt)is givenby[18]

xJ.,dr3 re XP(u ::'r) ]

~2

^[bC(r,t) a XP(- iwot )].(2.10s) 21(It,t)...-

~

_R ^,

~

_4cI(^.exp(ikcR)

,16.

wh e n theSpliceIn t e gr ation1. overthe veh.eofthe sutte rlng

Incidentandscattere dli ght,• 11 theangl ebe tween the deetrlc ve ctorofthe IncidentwaveEoend theprope g3 t lonve ctorof the scattered"Ive~, 1M (,Ir11 the angu lar frequencyofthe lJcet t ered "ave(F1S.2.1).Inprsctleethe vari ationofl&(r,t). which 11 causedby fluctuAt i ons In the_dh~,11 Mleh 110\18r compare d to the fre que n c yof theInciden t l1&ht(Ibout IOUHz) . t t lder ivatives with re.peet ....tl.e can eon. equ e ntlybe Ignor ed compare d totheti lDe.der lva t lvelof theIncidentfield,Ind (2.10b) bec ome '

(2.11)

The Intensityof th ese e t t ere dlightcanbedefinedby the re latlon(19),

I(K, wpt)-

~ I <~(It, t) ' ~(It,t'+

t»exp(lw,t ') dt',(2.12 )

'Where th eangularbracketa denotetheaverar;"with re sp ectto a .ta tlona ryequil i br ium.ense mb le.E~^isthe con j uge te of£1 'lJe ah o not efrotl theprop erties of thecanonicalens emb le,th a tth e ensembleave ra s eof • dynamicalvar i ableact ually do e . notdepe nd

.17.

i- _k.

Incident.

light

SC.1ttered light

·f

FJg.2.1 The angle. usedIn (2.10) are defIned her..tbI electricfield in the IncIdent be. 18 along ther.

direction,andtheInc I de nt be_ propagat e .in thex direction.Iand' are the angl. . betweenthe prop_guion direction of the scatteredbeall. aOO the zandJI:axe . respectively .

18.

one, that la,

(2.1 )

Ualngthe !;"eeult(2.ll). (2.U)and(2.13 ),we thueobta in the frequencydistribu t ion of thescatter e d llr,ht al

Conal deringabi nary ..btu!;"e whichIs composedof Nt 1I01eculeaof.pee les 1 andNzlIol ecul e s of.pe c le e 2 Invol ume V (1fNzooO,'Weean eaaUyobtainthe resul tfora daplefluid) , the av era ge partillnWllb er densities.lire~-Nt/V,n:-N1/Vand the tota lrlI.lIlIberofmo l.tul ea 11N-N1+Nz •'Letr~and r:denot e the posi tions ofaole eu le numberi ofepec l e s1and 2,re s pe cttvely.

Then the die le c tr i c con s t.n tc.n be expanded Ina tayl orser IesIn the loea lden.tty aboutthe.verag'densi tyn&(i ..l, 2 ) so th a tlit In (2. 3) beeo....

.19.

vherei•1.2,... . ,NI;j-1,2, ·· · ·,Np

Bysubs t i t uti on of (2.15)into(2.14),'lieobtain

··•• ..(2.1 6 ) Afc er fureherdlllj)l1 !lu tion (2.1 6) c:anb.vrltcen . .

1(1.,"')orNI"~Sll(I., w)+ N!"~Szz(I.,w)+^(N1M! )1 / 1"t " ! SU( I.,,,,,) +(N1N!)I /!(lI"!Su( J::,w) (2.17)

wherew..Iolt·ltIt ,andell(wh ichb propor t io nalto (8c/ant )n l_nA (19]) b thepol a rl u bil1tyofspe des1.TheSIJ(i,j -1,2),which are ueua11ycalledthe·pa rti al dynamicst ru c t u refactou·,are definedby:

.20.

where

(2.19 )

Sincefor clas :Ji calfluidsinequ llib riUlllallthe Slj (K,w)are real , and are evenfunctionsofw, we have

SlJ (R, w) - SjI (K,w),(l,j - 1, 2) (2.20 )

and ass\llIIing thedllu t e fluidis isotropic ,SlJ(K ,w)dependa only on

K-IKI .

SO all partIaldyn ami cstr uc t ure fa c tor s also sllt i s f y

SiJ(R,w)-SlJ(K,w). (i,j- 1,2) (2.21)

Using (2. 20 ) and(2. 21 ) we canell511 y see (2. 17) is jus t (1.3) us edin Chapt er1.

Itisclearlysh olom in(2,17) that the intensity of scatteredlight for binaryfluidsisproportional to theweight ed SW'II of the partial dynamicst ru c t u r e factors givenby the space-tilleFourier transformof the densi tycorrel ,ationfunctions. From the defLnltionof dynamic structure factor(2. 18 ) we can also see that 511 (or^Sa) isonly relatedto themicroscopicpr ope r t i es

.21.

ofthe l1gh ter(or theh'avh r)IIp e des Ina binaryIIb t urtlvh ile thecro. .te rti 5u 11 nlatedto thebo t h.Ho,tof thetheoretical atudiesconce rning th.rmAl ligh tsca tter i ngconc entrateon using di f fe re nt .... thods to calc u latethe partialdyulllic st ru c t u re factors .thereafterthetheoreticalre s ul t.c.n becompa re dvl th experbent.A.vu Dentlon edin Chapter 1. In theliliit vhe re Kt <<1thedendty fluctuations Ine flui dere gov.rnedby hydrodynamictheory.vtllle for thecuevh e re Kt_l.ontheoth e r ha nd, kineti ctheoryh nec e ss ary .Inthefollowing twosectio ns vewill IIho w how thepart i a ldyn amic struc tu re raetor s are obt a i nedinthes,two di ffe r en t re g illes.

I.2.2.HYDRODYNAM IC'nfEORY

In the bydrod)'f'Qlcde .cr l p t ion. the fluctua t i onsofthe loeeldie lectricCONtenter e reletedtothefl llCtv.atlon s ofa compl e t eseeof loedthenaodynamlcqua l'ltl th satsuch as pre ssure,tellperat u re and conce:ttre t l on,I.e.

(2.22)

.22.

Any th r eein d epende n tvar i3 b lea willauf f i c e for binary flu i ds.

butcertal nchoices ofvar i a b l es may provide.lIloreconv enie nce for thecalculationthan others [ 20 ].

Thespac eendtilllores pons e ofthe systemto a deviation from equlli b rlUllliscalculated by usingthelinear h e d hydr odynll.mi c equationsand initi alvalue sfor corr elationfuncti onsprovlde d by thermodynamicfluctuation theory.If (T,p,c)ar e chose n,then thes e equat i onsar e thecont i nui t yequat i on,

[~]+PoV·v-O.

thelongitudinalpart ofthe Navi er-S tokes equa tion, (2 . 2 3)

the diffusion equa tion,

andthe energytran sport equat i on,

(2.25)

Inthese equations,all the equilibrium values are denotedbya subscrip tzero .pis the den3ity.visthe ma ss vel oci t y ,jJisthe

.23,

che.icalpotentiA lof the. h tu rel2 1l,'I.and'I...arethe shea rand volu.e vIsc ositie.respec tively, Dillthe dlffu.!il1oncoefficient,S istheentr opy,Il'11thethenalconductivity,kt iatheth e r_ l diff usionratio ,andIt,11 • the t".-odyna-i eqU8ntitydefi ne d by

Hencei fwerewriteeqU8tions (2.23)- (2.26)intermsof the apetta! Fouri ertran.f orm.,we ca n obtai np(K, t),T(K,t) and e(K,t) inter lllsof init iAl fluc t uat io nsp(K),T(K)ande( I:). Meanwhile thespa tia lFourie rcr ans f o rlllof thelocaldielectr ic co ns ta n t canb• • •pr. . . .da.

c(K,t)-

[ ~]. ,tP(K, t)+( ~ ]p,..

T(K,t) +[

~

^{]",C(E.,t )(2.27 )}

Usin g(1.17)and(1.14) , th efinal expruaioncanb.obta in ed in thefo n.

where 1.01'

Ao" ". ,

101'lU'1. andWwiare coe f ficients dep end i n g onK.TheRaylei gh peokcanno tsimply be consider. ds.the supe rp os i tion of th.flu ttwoLorentziansshown 1n (2.28) .In fac t , as the resu ltof thecouplingeffec tsbetw sendiffusionlind hutflow1n abinaryfluid, theact ua lcentra l puk isthe re sultantof• .uchIl101'.compl ica te dsuperposi t ion whichconsLsts

.14.

ofsixLorent ziaoa ( a u re f.20 for more deta il) .

In thehydr odynamic des c ri p t i on thelocalequilibrium assumption isus e d, which.ea ns tha tateach point in aflu i d the samethermodynlllDic rela t i ons can beus e d torelat edifferen t themodynamicvariab les,andthefluidisconsideredtobehave li ke a continuUlll.So th ehydrodynamic theory 18prope r only for sllIaUKandIIIunderth iscond it io n ,and themi c ro s c op i c deteilof local structur ecanbe igno red .

s,

2.3. KINETICTllEORY&PREDICT I ONOF A FAST·SOutmKODE INDIS PARATE· HASS KIX11JRES

In adilutefluidwhereKt",I,hyd r odynami cthe ory CellS1l.1Ito be anappl icablethe ory . Kine ti c theory,which invo lvesamor e de ta ile d microscop ic de s cription, isnee ded. Aa will beBeen later,kinet ictheoryis 1II0re genera land includes hydrodynamic theoryas a spec ificcase . Inthis sect ion,usingkinetic theor y, the dynamic st ruc t u refactor for simpl eflu i disded uc ed .

Ki net i ctheor y is bas...~ontheBol tz ma nn equat i?n whi ch involves tha diet r ib ut io nfunc t ion f( r,v,t) .Inaflu id this distributi on func ti on is define d insuch way . th atf(r,v ,t) t1r tr.v isthe averageflUIIIberof parti c lesatth et inside thevolumeAr aroundposition r,and with velocities wit hin thecange!J.V around

.25.

v.Because the interllolec ularinte r a ctionrange iswchslIIa lle r than theave r ag eintermol e cu lardis u. nc ein a di l ute flui d ,itis unli ke ly th a t1I0rethsn two particl e awillcollideat the. .me time. Thereforeunder the molecu l a r - cha o s assUlllptionthe Bol t zm&nn equation(2 .29)givesus the ti meevolution off(r, v , t ) :

~f(r,v,t)+V' Vf(r,v , t)-Jd31'J_{dn •},,(I, lv ' lI'j) .Iv . v l x(f (r,v',t)f(r,'"tt )- f(r,v,t ) f ( r , v , t )} (2.29)

wherev' and v' (whic hare theveloci ti e s of two partic les,after abin a r y col11sion, within i ti a l velocitiesv andv,respectively) depend onth ei r initialve l oci t i esand thesc a t t e ring angle (I,f1) in polar coordina tes(po la raxis para lle,i. tov -").0 in(2 . 29) isthesolidangle and I1(I, lv - ..

I)

isthe differential collision cross section.

ThelIlost importancresul t of theBol cz mann equat i onh that for anygiven initial dis t r i butionfunct ion,f(r , v,to).the snlu t i onof(2.29)willapp r oac hthe absoluteequllibriUlll Maxwell dbtr ibution functi on:

(2.30) loIh e re IIisthe av era genUlllber densi tyof thefluid. Thl.res ul t is also known as ".Bol tzma nn 'sH· t heo re m".The asswoptionwhichfol l ows isChatequa t io n(2.29)can bel inea riz ed because the dis t rib u ti on fun:: t i onf(r,v,t)15 close toitsequllibr1 U/1lfo rlll, fM(v).Ifwe

.26.

defin e a small perturba ti on of the distributi onfunc tion as bf(r , v , t). thenf(v , r, t) -bf( v. r.t)+fll(v).Substitutionof the latter fortllint o (2.29).whilecon side ri ngonlythe first order of bf ,yields thelinea rized equatio n for bf(r,v ,t )i

t-lbf( r,v, t)]+v·Vl bf(r.v,t»)- ....B[bf(r .v.t)l. (2,31)

where~isaline ar integ ra lope rator whichactaonly onv.

Nowweint r oduc ethe dynll.llicvar i ablei(r, v, t)whiehisgiven by

(2, 32)

It ca n b. see n tha t i(r.v,t) de sc r ibestheparticlea' position and veLoc Ley dls tributi onat ti me t,and that itaensemb le aver ageis justequ altotheequ ilibri UIIIdis tributi onfunctionf ll(v ),while itspe rtur bationbi (r ,v,t)- i(r,v.t)• fH(v )sa tis fi esthe equation(2 . 31).Accordi n gtotheOn.sager re gr es sion hypot hesisin ste t i sticalmech anicsIl 9 ],thedecay of mic r osc opic spontaneous fluctu at i on s of a dynamic variable,andth edecay of its correlationfunction ,on theaverage, followthe 5aflle li nea rized equation . Therefor ewe can replac ebf (r,v,t ) in(2.31) withthe correlationfunction

C(lr·r'r.v.v '.t) -v<t.i( r ,v , t) t.t( r ' ,v',0». (2.33) end obta i n

.27.

Ftc(lr.r'I,v,v',t)+v'VC( lr · r' l ,v, v',t)

-ABC(lr. r' l,v, v', t ). (2,34)

Because of the 1sotropiccond itionin dilute fluids,the correlationfunctionin(2.3 3 ) dependsonly on the modulusof the relative position ,and the factor Vintroduc e dthe r e is for calculationconvenience, The n thespatialFouriertrsnsform of (2.33) ca nbe wri t t en88

C(K,v,v',t)- Ji lr.r' l exp{-iK·(r.r ')}C(lr-r'I,v,v' ,t)

where

ll.tu~.,v, t)

-l' t 6 (V-vp ( t ») ex PI . i K'rp (t) ] . (2.36)

and becauseof hotropyat equilibriUbl ,C(K,v,v ',t )depends on K - ltel.The equationfor C(K,v,v ',t) 18 ob tained froll theFourier transfom of equation(2.34)

~C(K,v,v ',t)+ite'vC(K,v,v' ,t)- B( K) C(K,v, V',t), (2.37) wher e IHK)is aUnearinte gral operato r.Us i ng (2. 18),(2.19) (consid eri ngcee pen en es l,j arethe slllIle), (2.32),and(2.33)the Btucturefactorcan iWDediatelybe obtainedas

.28.

S(K,w)..~^{Id t}eXP ( iwt)[Id' Vd3v'C(K,V,V',t )] , (2. 38 ) soth e keyproblemrellla1ningis tofind4'Way of calculating C( K,v , v ',t).

Since the BoltzlIlannequa tionisva lidonl yin the limit of dilute gases, based onahard- sphereecd eICohe nand Campaused a mod i f iedEnsKogthe ory in order toobt a i n the moregen eral resul t s whicharevalid for both dil u t e and dense fluids.And ins t e a d oftryingto obtainC(K,v ,v ' ,t ) directly frollltheequa t i on :r1l11ih .r to(2.37)which provedto be quit e difficult,they attemptedan indirect waybycal c ulat in gan innnita set of correla tionfunc tionsI22,23).ThecOlllpu tat i onsof partial struc ture factonSlj (K,w) areperfo1"llledby aspect r a l decompositionof a time-evolutionoper atorl.z(It ) 1n temsof db creteeigenmodell.The finalresultfor the partialstructure fac t orcanbeexpr esse d asaSUIIIof Lorentzian.:

(2.39)

where the sum run. over theei ge nvalues z"(K)of

Lz

(K) ,and i,j..1,2 denot etwodifferent eompor'lOm tGof the mixture,AI.l ,n (K) ar ethe amplit ude s correspon d i ng tothe eigenvaluez..(l),Nisthe numberofei genva luesus ed,whichischos enon the basis ofthe Bhatnagar-Cro.s-Krooklnethod[23].These dtffeI'ilntelgeRlllodes ca n be interpretedasth e differ entehannels by which the fluctuations deea yinctee .

.29.

InClolllpa' .calc ulat io n theeigenvalue s of theki n e ti c operator4(K)hll in totwotypes.Onetyp e 15 realandanother is complex.Bec ause ofthe exis tence ofdampin g processes ,th e real parts of both types arene ga tiv•.Thecomplex eigenvalue s alwayscomein conjugatepairs and the i rre al parts re pr e s e nt dampingpr oce s s e s whUe imagi nary par t srep r e s e ntpr opaga ti on pr oc ess e aintwoopposite direc tIonI'.

In tho hydrodynamic limit . (La .for K of suffieientsmall value)all th e 8IIIplitudea Ail ,e(K) are real and contt:ibut lo nsto

the SUlll(2.39 )come onlyfromfour eigenmode swI t h followIng

eIgenvalues

and

~)(K)- _OIKz,

~.(K)__ OzKz,

(2.404)

(2.40b) (2.40c)

wher ec. isthe adiabat icvel ocity ofsoundwhich is independent ofK, and

°

1,D2, rare positivecoefficientsassoclatedwI ththe damping pr oc e ss. Thereforefor K ..0,substit u t io nof(2.40)into (2.39)shows tha tthe kinetic resultisinagreementwith hydrodynlllllic result:(2.28).The descrlp t i on Ineeresof discr e t e lIIodes isthus a generalizationofthehydrodyn8lllic theory.

.30.

When the conditions arebeyond the hydrodynalllicregi me, the :;;itustionbecomes Illore complicated.Contribut ionsfrom higher propagating modes must be takenint o account, andsome of the amplitudes Al.j,n(K) canbe complex.The eigenvalues fordiffusive modes are stUI real, but theeige nva luesof the propagatingaode s come in complex conjugatepnira and eigenvalues Ilhould be written in the generalfOB

(2.41)

wherethene gs t i v e real part'Tn(K) represents the damping, and the imaginarypartlola (K), whichisthe dispersion relation for the n-thmode,determines thepropagation velocityof the mode assoicated with:tn(K) (thegr oup veIocd ey of the n-thmode c.Ia givenbyclwa(K)/cIK ).Contrary tothe hydrodyn8lllic lildt,w(K) can no longerbe expressedIn simple formas thepr oduc t of K andt. whi chillindependentof K.

Inth e calculationfordUute ,dispa rate-mass ,binary mlxtu>:>es ofAr+HiI,Campafoundtha t underthecondit ionsofhi gh concentrationof theligh t e rcomponen t and K larger than a certain value, one of the eigenval ueshe obtained had an imaginary part wn (K)for wh ic h the groupvelo.;:1tydld(K)/dKwa sconsiderab l y large rthanthatof other1I0des,Calculation ofthe part ial dynamic scrceeure fa c t ors showedthat thisfas t 1I10de was only associated withtheUght ceeponentand wouldgivean observable

.31.

shoul d eror pe a konlyinthepa r t i a l dynallllic .t ruc t u refacto r Sll(K.w).Nosh oul dero~peakwould appear inSu( K.w)and Su(K....).Thla.houlde~or peakin Sll (K.w)l. .d•• fora proper choiceof thetwoco-POnenU(m.cOllbl na t lon ofAr +H2 va•

• 'Ju e s t ed by CaIIpa).to a.hou l der in the diff are ntialcross sect i on for !r Ulou lnsca tt ad ng. Thisll1lplle.th.t thefastIIOde isassocia tedonlywi t h thelight ceep onent;•

•32.

Chapt er 3 . App ar at us and Exper i ment al Procedure

The overallarran gementof apper a tulusedtost udy Bri lloui n scat t e rin g frolll thegas mixture sIs schemat i callydepIcted11'1.

Fig.3.1.An Ar+laserwa s used8S the incident ligh t sewree.The spectrometerconsi:>tedof a piezoebctrieal ly.scannedFabry -Pero t inter f"tolleter, ,pllo t olllui tipiter tubeandaDat a Acquh i tionsnd Stablizatlon System(DAS-l).Thespectral datawe refirst ac cumulated for severs1hour s In theDASlZIemoryand then trans mitted toapersonalco.-purerfor furth erproteuing.Mo r e detailisgiven Infollowingsections.

I.3.1. OPTICALSYSTEII

Rec au s e ehs fr e que n c y shi ftsobs e rv edin Br i llou i n sca tt ering are verysmall(le ss thanlOGHzinour experiment)Ithe incid e nt rad i ation mus t:behighlymonochromati c.Theexcitingrad i a t i o n use dinour experimentwa sprov i dedbyan Ar+ laser (Coher en t

.3 3,

FIg.J.l Sehelllat1cdla gr AII of theexperimentalsetup.

SllIIIple cell:

B5 belllll-s pll t ter:

AT par tially pen e t ratingglass attenuater : electromecha nical shut te r: IU' mic roprocessori GF glassfiber :

nar row-bandgrat i ngfilt er : F·P Fabry·PerotInt erferollleter; PH pinhole :

PM thenaoele etr ice lly - c oo h d photollUltlplhr;

DAS dat aac qu b l t l on and atabUzatlon .yate.; H1,H2,H3.85,an d 86 lIlirrou;

H4 unco atedrefle ct or; APIandAP2 ape r t uru ; L1,1.2,13. and 1A len ses.

34.

Radiation,HodelInnova90-5)whieh WIlS operatedin a single cavitylllode witha nominalwavelengthof514. 5nmand output power of 30.°.500111"". Selectionof the outputlight frequency was made possible by using an intracavityprism.Anintrac avi t y etalonwas installedin side the resonatorfor obtaining8.narrow laser linewidth. The tUt of the etalon could be adjustedto obtaina singlemode.After severalhourswa rm-u p,boththe frequencyand power outputof thela lla rwe s general ly quite steble.No furthe r attemptwas lIIade for la s e r stabilization in thisexperiment.

Sinceli ght scatteringexperimentsat low or moderate pressure(less than 10 atm.inou r exper:iment)involved measu rementsofli gh tat verylow intendty levels,long data aqu is ition ti mes (from 5to24hou r s )we r erequire d.In orderto minbdzlItheeff ectsoflongte rm drif t.in the interf eromet.e r alignment,it wasconseque nt lynecessarytoutilize thefe e db a c k con t ro l capabilityof the DAS-lwh erea refere ncela s er algn a lwas provided as follows:The laller be am wasfirs t. dhidadinto two par ta by usi ng a beamap litt.e r (BS),ofuncoa t edglass. The nthe reflectedlightwas attenuate dandtransmi tte dthroughan opt.ica l glaaa -fiber (GF)whenth e shutter (8)W,U Iopen.The outp utof glass-fiberwa, accurata lydi rectedalong theop ticalaxisof spectrometerbya second uncoa te d re flec t or (K4),and fInally collec tedinthe OM-lfor feedbackcontrol(seesection3.2).

The ma i nlaserbeam was focusedbylensLI, and then

.36.

reflec ted bymir r or H2 andaprism into thecenterofscatter i ng cellC.Thescat tering angl e formed bytheinciden t beam andthe scattered lightwa s about 157.5°.This eonfigurat1.onalso permittedusto observe the spectrWII cor respondingto a smaller scatter ingangle of 22.5°byaddingtwomirrors H5,H6 as shown inFi g . 3.1,whilekee ping the distancefromLlto thecentre of th ecellalmos t unchan ged .The sca tte r ed ligh t passing throu gh apertu re APIwa.collec tedby thelen s L2and focu se d atapoint wh i ch was justcoincidentwith the virtua l image of thebri ght head oftheglass - fiberproduced byH4.Mi rror KJ wasus ed to directthesca tteringlighttoaFab r y- P e r o t (FP)interferometer (Burle ig h,Hodel RC-llO).r wa s anadju stablenarrow-ba nd monoch romat icgrati ngfilter(KRATOS,Mo del GM100-2 ),whic h wa s adjusted only to transmi t the light aigna lnear th e wavelength 5l4.5l'\lllandtorejectunwanted Raman ra to....ti on frOID theeaepj ee. Be caus a of th elowint en si ty of thesc a t tere d ligh t,it wa s neces sa rytolI.ininliz ethe intens i ty of the strayligh t.The aperturesAPI andAP2wereus e d to preventunwant ed stray light fromgetting in to the spectrome te r.This factwasalso considered inthedes i gn of the scatte ringcell .

Ape r ture AP2,pinho lePH,le ns e s L3and lA,theFabry-Pero t interfe rODle t e r FP,the ph?tOlllUltipliertubePKanda

·thermos taticallycontrolled heaterwereallcontained in II styr ofoam box coveredwith blac kpolyethyl ene. Inadd ition ,e large pieceofhe avy black clo t h wasdra ped over thebox .The

.37.

purpos e of thiswasfirsttoexc l ud est ra y l1gh':andsecond lyto provideprotec t ion fr omtemperatu r e flu c t uations ofthe surrounding s110as to mainta inthetempera tureinsidethe box ar ound20°C. A 4cIn hole was cut in thestyrofoambox$0that the len s L3 coul ddi rect bo th thescat tere d lightfrolll cel l andthe refere nc e las e r signa l toth eFab r y-Pero t int e r f e r olilete r.

In our exper i ment l , the Fabr y-Perotinterfe r omet erconsi s ted of twonatmirr on (with flat ne s s es of A/2 00and reflection ra tios of 98, )mounte d para lle lto each ot he r inanad j ustab le su pe r inv ar asselllb ly.If thecav ity betwe entheIlli r r orsis illUII.ina t edwitha be amof mono ch roma ticUghc,it willtra ns mit thebe/UII onlywhenthe relation (3.1) issat isfied, I.e.

ex- 2ndcos' (3.1)

where IIIis the orderof interference ,A.iswavelengthof the incidentlight,n is the ref ract ive indexof mediUIIIbetwe en the two mirrora,d is the mirrorspa c i n g , and'istheangle fOr1lled by theinciden tbeamand thenormal Hneof thepletes (in the present expe rimen ts, n...1,,..0).Thefront mirro rvasmounte d onanadj us tablemount usingthre eextremelyfine differen t i al micrometer adjustmentassemblie s.There a r mirrorwas supported by threeItacks of pie zo ele ctric tr ansduce r s placed ar oundits cirCUIIlferenc e .Theape c t r um of the scatteredlightwas obtained by cha ngi ng the voltageappli edto thepiezoelectric tra ns ducers ,

.38.

wbich consequentlychangedth eIpaci ng bet "'eent"'o plat e...rrrees• ThefringesproducedbytheInter f eren cebe cveentheparal lel .i r r orswerefocundonapi nh o le(PH )in. t a nedInfr ontof the photorault ip Ue r.Thedi _ terof pinh oleva. illpOrta nttothe over a llfine n eof the 'pect ro_ ter . If too lergeitbroa dened the Uneandi ftoo IlUl l it cutdownth etnnslllitte d intendty(24). Theapprop riat e sin ofthepinho l e.",bleh"'. .generallyeeteeeee by trielander r or , ra ngadfrolllOOJlUlto20 0,..11end the ob served finea sewas abo u t4).ThephotollUltipl hr tube(ITTHodel,FII130 ) was mou n te d ineth e r moele t ric ally-cooled,RF·lb ie l de d chlllllber, wh i chmaint ained the cath odetellperatureat ·20^oCand reducedthe clarkco un tof thephotomul t i plierto ebout 1eoun tperaeecnd,

I.~.2.DAn AQUISl"l'IOMANDSTABILIZATIONSYSUX (DAS-l)

The longexperl llle nu lruNneededfor lowintel\ll1tyspectra required hi&h ata bUityof boththeInterfer_ t er and las er.

Howe v er,change s of inter f erOllete r alignmen tendfre que ncy dr ifting ofthe lalerlIbich could signi fi cantlybroadenthe instrw:llental l inewid t h arelargelyuna voidab le, it vas nec es sary to compensat e forsu c h drlf t:lngaf fe cts.This .... .madepossib l eby usingthe DIlS-I1nconj u nc t i o n ...ith a referenc e.i gna l whichwas lIIent i on el!pr e v i ously.

TheMsterclockof theDAS·lprtJue e.pulae....hieh areused

.3 9.

to stepthe data accumula t i onthr oughsuccess ivechannel s ofa 1024 channe lscale r.Thesepulsesarethe ms elve s accumula te dand subsequent lyconverted throug ha DAC to producethescanning ramp voltage whichisappliedsimultaneou slytoall threeofthe piezoelectrictransducers onwhiClh one of theFab r y ·P e rot mirrors ismounted.Asi n gle cycle in the repe t i tiv escanningprocessvas usuallycomple t edin lessth an5secon ds,andthe ampli tud eof therampwascho s e n to coversl ight ly lQorethan two fullorde rs of inter f eren c e (Le.so that th reesuccessive Ra yl ei gh peaks coul d beobs e rv e d ). The high lylinea rrelationbetwe e n appliedvoltsge and piczoelec t ..t e extensionresultsin acor re !lpond i ng relation betwean channel number andfrequ e ncy (seeEq.3.1for 8_0).

COlllF"nsat io nfor ins t rumen t aldri ft s Isnedepo ssibleby the pr ov isi on forextern a ladjustmentandcontrol of the zerole vel of the voltage ramp.Itisthuspo s d bleto selecta re f erenc epeak in theobserved spe ctrum and , thr ough feedbac k con t ro l,force the peakto maintai nttlpos i t i onatapre s e ltc t ed channel nUllbe r . In thopresent.eas e,however,the scatte redlightintensity wa s not sufficient for thepurpose,andit wasnec e ssa ry to pr ov ide a referencesignalof high e r int ensi ty thr ou&h dir e c t samplingof thela s etbeae,

The elec t r omechan i c a l sh u t te r S wa scontrolledbyaseparate etercpeecessoe,MP,which uti lizedthe di gi t al clockof the DAS·l 8Sti meba s e.It va sprogrammedtoopen the shutte rat a po i nt (usual lynearth e endof eachsweep)wherere sonant eraneet.ss ten

at thelaser freque neyv••• bou t tooc cu r,and tocl oa. It i_ d hte ly af terscann ingthelaserprofUe10 that noneo[ th b hi ghInt e ns it y lignal coul d con t...,l nateth e(10\1Int en sl ty)

!iri llou i n spectI1Ja. Any ten4e ncyfor thl. re f er encepeak todrl£t avay[ro mit.pre. allcted locetionve. dete ctedbytheDM·lwhen oper.lt l ng in its fe edback con t r ollIIO<le,andenepproprhte adj ustmen t to theze rolevelof the Icann i n,r&llpva.

autolll!lt i c al l y ..adetocompansa te for thedrift.Aal.nar but somewha t morecomp l icated. .th odisal so provid edfor opt lllllzation ofthe instrumentalf!nesu.Detailsof how the feedba ck controls areach i ev edcanbe found elsewh ere [ 25 , 2 6 ].

Ano tl.er import antfeat ureof »AS-lh sepantedscsnni ng Whichpertl,i U dif fenn tac annin g speed. fOTseeumula ting eounCsin theselecce d.nSi o ns.In our exper i_ntetwose gJHntsweresetfor theDAS·l.The _in..pent for the lpect na wever ein t er es tedin va•.at betweencha nnelnwlbera310-730 ,andthela lllraepan t , which va. used topr ovideenough countafor thereference leser lignal,va aset betweenchann elnumbers925-955.The spee d retio betw eenthefaitan4sl owpor t i on of therPlpva .99,.0 that for eachsca nalllOat90'of tiaevas usedintheiliaIn sepent.The spectr UIII v.aaccumul.ce d in a totaltime of betwe en5 to 24 hou r s. A typical spectrum obtianedfrom the DAS·l11eh ollO in FlS'3.2.

.41.

1200

,-- ---T...,

~

c

"

o o

O+---=::...--r--....:::::'---...L~

o 500

Channel number

1000

Fl.••3.2 Typical Ipect:rwrol'8co rded Inarl11 0ul n IC.l:tur ln gexperi llent (CH. +SF. lIixt urawtch CIt,ba..

pl:elS8ur e of 6.)bar and x".-0.08).

'2.

s.

3. 3.GAS HANDLING SYSTDI

Ablock diagr Ul ofthegas handli n gsystelllisshownin Fig.3. 3 .The highpuri ty(>99.99,>gasesus ed in ourexpe rime nts we r e prov i d e d by Ha the son GasProduc t s.ABour don -tube gauge wi th an estimatedaccur a cyof to.OSbar wa sus ed to measurethegl9 pressures.

The sarapl egas e s, foreXlllDpleSFeandC~,wer emixed usi n g th e followingproc e dur e s . Referi ng toFig .3.3 ,fint weclose d valves V3 ,V4,V5 ,V6,andV7, open ed valvesVi,V2 ,and va , turned on thero t ary vecuuapump.The scatteringcellandthe connected tube scoul dbe evacuatedtoapre s sureofabout10.²torr.

Thenw.clo ••d valveva , ope ned V4,an d.IIlowly filledthe scat te rin g cellwithCH_ inorde r to reduc ethe gasturbulenc e wh i ch coulddisturbany dust insi deth e cell.Whenthe gauge reach edthebase pre s surene eded,we cl osedva lves VIandV4, opened WJ , and turned on the rota ryvacuumpueptoevacuatethe conne ctingtubes,The procedurefollowedWilltocloseva , open V3 and fillth e connectingtubes wit h SFauntllthereadingonth e gaug ewa. alittlebithi gher thanth epre s sureinddethe cel l, the n closeV3 , ope ne dVIandadded Sfsinto the ctll.~yrepe ating thisproc e ss we ob taine dthe gealIbt.t ure a ofdi f fe r en t parti al pr essure ratio s.Weneeded tobecar efulduring thewholepr oc e ss to keep th e pressure insi de the connectingtube sa littlehigh er thanthe presureinsidethecell.

,43.

f

f , - i

,

I

. ^!

I ^..

" Il

.; ^~

I f ^:i

I ^{a !}

!

~

~ ~

i

!

J

i

I !

~

"

~

The enti re experilllent we re perforlHdinan air-conditioned rooll.No ••traprecautions wer e tllken to keeptheseatteri n& eell atcon. tent temper atu r e_

.45.

Ch apter Ii. Experimental R esult s and Discussion

§,4.1. GENERAL REMARKS

The methodused.todetect the fn tOJ:slow modeco n t ribut i o n toa give nspe ctr um wa sto compareth eobserved so un dvel oc i t y

v :

and the sou ndspeed".Cla lculatedusingthehyd J:odyn aml ctheor y (K ... 0). ,,;wasdeterminedbythe BrU lou inequation,

v ; ..

..,,>./[251n(I/2) } , (4. 1)

where"Bistheobserved freq uencyIIMft, ),isth ewav e l engt h of the inci den tligh t in the mediUlll,and 'isthe sca tteringangle.

In i t ia lly,the lawof partLa l pr e ss ur eswa s essueee to beva lid for all mix tures st udied,and valuesof".were ca l culated viathe ideal~asr..1IIt10n11'. . .(..,p/p)I/Z.wbere,. ..tp/c., . pis the pressure,andpIsthe massdensl t:y(forII;givenmixture) . Su bs e quent l yIDOr eaccuratevaluesfor ". were obtainedthr ou gh col labor ationwithDr.J.A.ZollwegoftheSch oolof Chemical

.46.

Engineering ,CornellUniver.!l1ty(2 7 ].Zol l weg's tec hniqu e utlliz;ed the bastc relation[28] ,

(4.2)

whereV.istheIlola r volumeof chemixture,H isth e average molecular weight ,and/c.is the adiabatic compr e a s i b ll i t y.Because the actual average molecularwe ightfo r adh pa r a t e - mas s mixture couldbe s1gnificantlydi fferent fro m that cal culatedonthe basis of partialpressures,the measured prusuraswere first used to eei.eutetethe concentrat ionof eac hspecies viathe virial sea ee equation. Then the sound speedisobtainedusi n gthe virial

(4.3)

wherep is aga i n th eto t al pressu re,C:is theideal - g a shea t capacity, and S, thesecondvirialcoeffic ien tforth e mixture, is obtainfld byusing Hayden-O 'Conne llmethod(29J. Asit turned out, th e valuesof"'.obta ine d by thesecondmetho d wer enot:

si g nificantlydifferentfrolllthe in itia lcalculations,wi t h one or two except ions.

§. 4.2.TIlE FASTSOUNDKeDE IN Hz+ArMIXTURES

A dilute gas mixture which is compoaedof two species with a

.47 .

larg e_!ISdi ffere nc ere sp ond s toather _ lfluctuationwit h propaga t i ng dendtr",.ves.Un der cond i ti on s of h1ghconcent r. tio n ofth elight erCOllpOnent, •densityvaveofhighfraquency .nd ahor t",.ve leng th_i shtonly be supported bythe lightar pa r ti c les

"'ithou c thep. rt i cipationofthe heav ierp.r tidell1nthe eerteec t ve~t ion.In thb easethe group veloc i t yof the lIixture isve ry do.etotha soundvelodtyin thesinSh-co_pone nt flui d whi ch is obtainedby relaOvd of allhea vierpsrticlea .nae dynamicsofthe tvoCOlPponentsu'e thu llpartid lyaepa r a te d. The eontr ib u t i on oftheheavier componentto thepr opagationmode is negligiblebecausethe he av i e rpar ti c les areunable tofolloW' the fast sound IlIOd.whi eh11ce ueedby the rapidosc illatioM ...ociatedwith theLiShterceepenene,

Thew. vevect orKofthelocaldensityper turbationto be detec t e d,correllpondinseoascat ter i ngangle" sa thlie s the relati on(1. 1).Sotheleng t hsu leof theeheraalfluct....ationto beprob edb give nby "..2..jX-.1/ (2.1n (l/2 ») ,whe r eAis the wsvelensthof inc id entlishtinthelIedi ...he... . ther. stsCll:n d mode, wh i ch correspond.to a short wavelengthperturbationin the fluid ,cu se.to prop. Sate ifIth slianerchanacertai n vtl ue ( 21), a largeaca t te ri .,g angle(157.5°)vas select e din our exper i ments. Foreachaarie s ofexperimentstha118lJIpiecel l ",a . firs tchargfldwith enoughoftheligh terIIpe ele. 10 that well- defLnedBrilloui npe a k. eou l d beob s e rv e d inthespeet rUlll.This spec t rull wa ssubaequentlymon i t or e d al luc ee. dve inerementsof

.48.

the he avier spe c ieswere adde d into the call.Be cau s eofthe fas t soundacdeeffect,it wa sexpec t ed th a tthepodtion ofth e BrU lo uin peakwouldremain substant ially unthsnged, or shi f tIlluch le s s thanexpe ctedon the ba sh of the hyd r ody nBlIli ctheory.

Threegroups of spe ctra were collected fordiff e r e nt Hzbase pr e s sur e s.Themixtur e s,cond itionsarelis t e dasfollows:

O:sxz:SO.23, O. 2lgtl:SO.27,Kt z..O.0 7,PI-9.0bars, lnsxz:SO.18 ,O.27sKt₁:sO.3 2 , Ktl-0 .09,Pl-7.7bars, O.sXz.so.17,O.33 sKt1:s0.39. Ktz-O .lO,Pl-6.3bara,

wh ereXlistheconc e nt r at i on ofspe cieai (1 and 2 refer tothe ligh tand heavy ape c ies , re sp ectively) and t_l isthe mean frea path of species i obt ainedusingtheconve n t io nal waywh ich has beendiscusse d in Chapter1.The f1 value sthat wereusedinth e calcul a t i ons were(71-0.297nm andf1z- 0. 340nm for Hzar." Ar respectiv ely. The cont ribut ion of the fa st soun dmod e to the spectrUllwasobserveliin all threecasesfor lowcon c en tra t i ons of Ar(Fig. 4.1-4.3). As the concentrationof th e heavyspe c ies inc reased, th eBrillouin peak.sweredecidedlybroadened due to the severeat tenua t ionofsound. Forxz~O.lthe Brillouinpeak s became unr ecog niz able.However,forxz:sO.l.it is clearthatthepeak pollition s on thescaleofnor mali zed frequencyshi f ts(thesolid trtangl e s)differincraa singlyfrom positions (t he opentriangles) obtain edus in ghydrodynamic the ory(K"O)astheconcen tra t i onof

.49 .

150 , - - - -- - - . . . ,

"j ..

t Normalized shift

2

nor-Uudto uniqoat thepolltcloQ oft:2wBr ill ou i npea ktOI' purelIa<uch horb ont a l dlvlllon cou••pond.tIto9.2CHz).Only the (Stok. . ) rallon

of

down-.hUtdfreque-Myh .hawn.TM opentrtenll. . inclte.t.theftOmdbed 'hif ta obtainedba..d

Thearrov indlcat•• thenomaUzedpodtlon of the Brillouin plllal:;for punlor.Th.ac a n adasansI. ••,157. 5·.

50.

I"

III

:l _c

" ^o

o

150

r---~

2 Normalized shift

'1••

".J

Spect ra lorAr+Ha _latun_'11th. fbadHI pn..ura

of7.7ban,S. .Fla.4.1for additIonal let and.

51.

150

i .. _... ..

C :J

o

o

1 Normalized shift

2

FII_ 4.J Spectra forAr+~ab.tur. .with.fbed Ha preuur.

of6.3ban,SeeFlS'4.1foraddltlonalle&end..

52,

the heavy compone nt is increasod.

Based on ourexperimental re s u l tsPO),Campa us e d thehard sphere lllodel and parforlDedcalculations [31 ]topr odu c e theoretical Brillouinspec t r afor Ar+Hz mix t u r e swi t h aHzba se pressur eof 6.3bar s. He fou nd.thattherelative podtio nllof theobserved peaks, withre sp ec tto the peakpositionscalculatedon the basis of hydr odynamics(the rat i o ofthesol i dtr ianglepositi onstothe open tr iangle posi t i on s in Fig.4.3),agreedwith his calculated spectra (Table4.2).Healsofound forx2..0. l4,as observed experimentally,that there isnovisi b lecontributionfromthe fast soundI10de inthe Brillou i n spectrum.

Table 4.2 Compa risonofthePre.en t ExperbnentalRe eul u lind ClIIJlpa's Calcuilltionllforan Ar+HzKixtur ewithaH2.

Ba sePre s sure of6. 3 bars

x,

0.05 0.10

Observed Reaul t

1.30 1.51

Campa's Cal cu latio n

1.20 1,46

As showninFi g . 4.1-4.3,considerabladifficulty wa s experi encedindet ecti ng the fastmode contr ibution as a we ll·

definedspectra l fe a t u r e . Thismightbe due to(1)a relatively highpolarh:ab ili tyof Ar(IIA ~"'1.64xIO ·2.'/cm3 ,O'&z..0.8 2)(10·24

/ c1ll3 O'A~/O'iZ"'2)wh i ch fa vors the second andthi r d terlllS in (1.3),

.53.

(i i) broad e ningof thespe ctral featurebeca us eofthe$t r ong attenuation ofthefa s taoundmodecause dby the heavierpattic l e s , and (iii)the loW' abso lu tepolariza b il i t yof H2.whi chle ads to a loW'signa l lev e lintheden sityrangeof int ere st . Thereh.ti v e l y high int e nsi ty of the central pe a kalsoindicat edtha t the data

"lightbe contami na t edby $t rey light.

s.

4.3. CH,+SF, KIXTURES

Bea rin gthe fore goi n gpointsinmi nd,mixtures ofCH,+SF, were selectedforfur th erinvestigation s.The po la rb.ab i l1tyof CH,isabo ut threetimesgreate r thantha t ofH2. (acll,/a8

z" 3.2 ). In addi t ion,the ,",ola r izab il i ty rati o of SF,andCH~iaab ou t the samef,Sthat ofA r: andHz(Qsr,/Oc. ~ "'1.8 ).However,the lUISra ti o fo r the new combination(IIIsr/ IIlcI,"9. 1 )is lowere droughlybya factorof 2. Twogroupsofspect r a vet o col lectedfor eH,bas e prassu tesof 6,3ba ts and3. 7bar s ,with conditions asfollows:

Theuvaluesusedinthecslc ul a tions oft 1 we reul...O.3B9rua,

"2",O.46i1nm.Compa r i ng th e condi t i ons forCH ~+SFswith thosefo r

"2+Ar, itshouldbenoted that theywerequit e simila r. Beca u s e of thelargeincrea s einpolarizab ili ty.asexp e c t ed , the

.54 .

flrillou i npeaksof therecorded sp e c t ra werewell·defined. Neve r t he less,nosigni fi c an t fastmode con tr ibution.. asdet ec t able in eithe rcas e .Assholrit\ inFig.4. 4 and Fig. 4.S, forthespe ctr a correspond ingto th e CH, ba s epre s sure of 6.3ba rs ,the identified fr e qu ency sh iftsofth e spe c t r al peaksagr e edwiththe calcul ated re s ult sbasedon hy dr odyn a mi c the o r y.For spectr awith theCM,ba s e pres sureof 3.7bars,the obs erv e d freq uen cy sht f t sdidnotdiffe r sign i fica n t ly fr om thehy dr odyn8.llli c prediction.Usi ngthe hard- sphe relllode l , ClUIlpa agai ncon firme d that, under theexp e rillent a l cond itions cho s en above, no fastmode shou ldbe observedIn these cas e s [3 1].Furt h ermor e,itwas conc l udedth atthecond1t ions requ i r ed forde te c t io n ofth efas t modecontr i butio n forSFe+CH, mi x tu r eswere un l i kelyto beachie v ab leinpractice.

A pos siblepoin tof impor t ancehar aisthatthest~cture.of the SFe and eH,mol ec ules are lIIor e complica t ed thanthoseofAr andH1, and thevalidityof thehard·sphe re1Il0de l lIlaybe que s tionable.Howev e r ,becau seIIIsre /fllA ~"3.7andlIlee./mp, 1"7.9,the ave r a geve l oc i t y of boththeSFI andCH. molecules are conside r ab l y lower(VA ~/vS'I"2, vFJ/ vn; -2.8) .At the !lame etee, thet.va l ues for SFe+CH••which dep endonthe den si ti e s andth e diBllletersof theparticle s, remain almos tunchanged.Itwas consequen tlyexpected th a t devia tion sfrofllthe hard-spheremodel sho u l dno tbesign i fic an t (Le,because ofthe low densityand slowerspee d of the part i cle sin themed i um, thecollisiontimeis much lowar tha n theaeanfree time).Theobs erved resul ts also

.55.

~C :::I

o

o

1000,---~

Normalized sh ift

'1.8... Sto\•• spec t n forSFI~.btun awit h . flxadCIt.

p.utlalpr . ..ureof 6.3bar senc! fractionalconee ntrationsof SF, ••Ulte d.tachhoriz ontal diYb lonrepru .nUfre quency .hlf t of1.l CMz.Thautanc l a rI'Il rk,rahave the1_'. . .ni n l I' inF1S.4.1.

56.

1000 , - - - -- - -- - -- - - - ,

~C

-= _o o

No rmalized shift

FJ,.4.' Stoke.spect ralor S',.aI.wi th.fixed!~partial pn uuraof3. 1bu•. S.. Fil.4.4for additional h ,and.

57.

provedthis.Hencethe negativeresultfor theSFs +CH~mi x t u r e rDightbecause d bya smalle rIDa^{/ 1IIl}1rae te.

In th eabov,two cases( Hz+Ar&SF,+<:H. )themethodusedto charact e rizetheeltpe rim e ntalcondit i on s for the gas milttureswith dispa r at e ma sses (I. e .in temsof thet.valu esobtainedusing the convention alway ) providesno insightinto the distinctly differentruults.It wasconsequentlydecided to investigate alter n a t echar a c te r i z a t io n criteria.The following discussion intro duc e sanef f ec t i vemes n freepathwhich provide s alDor e cons is te ntdes c ription .

I.4.4 TIUtEFFECTIVE KEAN FREEPADIS DISCUSSIOIf

Firs titisnece ssarytoin t rodu c e the lIleanpersistenceratio 0iJ.which is: definedas the ae en ratio of theve l oc i t y component ofpart i clei along itsincidentdirectionto its originalvel oc i ty aftera colUsion wit hSoparticlej.01Jisgiven by(3]

whereHi -IId( lIIi+IlIJ)'(i,j -1,2). 1111 ,lII_Jare the massesof two differentparticles,e.g.the H: molecules and Ar atoms in a binarygaseous sys t em.Using(4.4), 0iJvaluesweracalculatedfor thest ud i edmix tures. The re sul tsarelistedIn Table4.3whare it

.58.

shou ld be nohd that ,duetothelargll lila••differe nces .Ozl.. 1 aoo012 - O.1. ••th eve1 oe it,.vec torof aheav lerparti cleal_st does not cha nge after it collld••wi th aUlh t erpa~·t1cl e,and the vel odt,. vactorof a l1pter~r tlcleIe totall,. ch.nsadafta rit collideswithaheavi er partic le .

Tab le 4.3 The HeanPers is tenc eIIatio fo r theStudh d Mixtures

0"

", "

^{tH, SF,}

" ,

^{0.406 2. 64xlO·a 7,01xI 0'3}

0.937 0. 406

CR, 0.406 5.89xlO· a

SF, 0.982 0.871 0.406

Given apart i c le of .peeles i !DOvin ginthe ga.ofspecies j, af te ra tI\IlIberofcollisions wit h particles of.pe ele sJ.the ve l ocit y victorofpartic le iis rando. iz ad.The averegeflUllber of eolUslons required forrandomization,i.e.nlJ'canbe obtained usingthe Poh El on dl.atri butio n:

(i,J-1,2) (4.5)

The calculatediil._J for thestudied mixture . angivenin Table 4.4.

As shown in Table 4.4,i fthe coll isions occurbetwee n

,59.

identical par U cles , thl!velocityvector is randomize dafter les s than twocollisions(nll-1,68).I fthe collisionscecur between unlike partic leswithlsrgemass diffe rence ,theveloe i t yof the lighter pertic lesisgreatlychangedillll.llediatelyafter a co11idon with a heavier one, while tho:! velocityvecto rof a heavier particlecanberandOllli z e d ;)illyaftera long series of collisions

Table4.4 TheAverageRandomi za t i on Co llis io n for theStudied Mixtures

nlj H, A, CH, SF,

M, 1,68 1,03 1,01

15.9 1.68

CH. 1,68 1,.:16

SF, 54 . 6 7.73 1.68

with light e rpa rtic les . It sh ou lda1.5obe notedthatunlik enu, nZi isve r y senllitive to the change oflIIa s s ratio m2/ml'Compared to the Ar +H2mix t ur e , thelIlass rati oof theSFe.CH~mix t ure decr e a s e s byroughl y afact or of2;n21 also dec r e a s e s bya factor

For the pictu retobe comple te,it mustbare co gn ized thatIn abinarymixturethecollt~ionscanocc u rnotonlybetve enth e differentspecLea bu talsobe tween the same specIes . SoIn s t.ea dof using 0ijin fon.ula(4.8) it is moreappr op r i a t e touse a

.60.