197 3ApJ. . .186.1107S 4 The AstrophysicalJournal,186:1107-1125,1973December15 exist atsupercriticaltemperaturesandhighpressures.Boththetrendofexperimentalresults up to9.3kilobars,attemperaturesfrom26°100°K.Twodistinctfluidphasesareshown even tothelimitsofstabilitymolecularphases.Theresultsconfirmearlierpredictions separation persistsattemperaturesandpressuresbeyondtherangeoftheseexperiments,perhaps and ananalysisbasedonthevanderWaalstheoryofmixturessuggestthatthisfluid-fluidphase © 1973.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. Subject headings:gasdynamics—interiors,planetarymolecules solidification ofthemolecularphasesatsupercriticaltemperatures.Theimplicationsthisphase concerning theformofhydrogen-heliumphasediagraminregionpressure-induced principles. However,itisfairtosaythatasatisfactorytheoryofdensemixtures two elementswiththesimplestatomicandmolecularstructures,wouldseemto diagram forplanetaryinteriorsarediscussed. be amongthemixturesmostamenabletoatheoreticaltreatmentbasedonfirst From atheoreticalstandpoint,theyareofinterestbecausecomposedthe unlike molecules.Ontheotherhand,significantprogresshasbeenmadeindeveloping quantum nature,andinparttothelackofunderstandinginteractionsbetween molecular hydrogenandheliumhasyettobedeveloped.Thisisdueinparttheir pressure. Currentestimatesofthecentraltemperature andpressureofJupiter,for importance. Amongthecoldcosmicbodiescomposedlargelyofhydrogenandhelium understanding ofmixturescanbeexpectedtofollow. satisfactory theoriesofthetwopuresubstances—especiallyhydrogen—andabetter because oftherequirementforknowledge equilibrium andtransportproperties are thegiantplanetsJupiterandSaturn(DeMarcus1958;Wildt1961;Peebles1964; example, are~10°Kand~35megabars(Smoluchowski andHubbard1973). structures ofthesebodiesposesformidabletheoretical andexperimentalproblems, Hubbard 1968,1969,1970).Thedevelopmentofaccuratemodelsfortheinterior pressures betweenafewkilobarsand1or2megabars. of densemixturesseverallightsubstancesover widerangesoftemperatureand knowledge ofthepropertiesmolecularhydrogen andhelium,theirmixtures,at studies ofdensefluidsandsolids.Muchremains tobedone.Smoluchowskiand of meagerexperimentaldata,supplementedbythe gradualaccumulationoftheoretical Models developedduringthelastfewdecadeshave reliedontenuousextrapolation Hubbard haveemphasizedthatoneoftheprincipal unsolvedproblemsisthelackof by theU.S.ArmyResearch Office(Intra-ArmyOrderforReimbursableServices#AROD-3-73). Experiments onphasebehaviorinhydrogen-heliummixtureshavebeencarriedoutatpressures © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The propertiesofhydrogen-heliummixturesareinterestforseveralreasons. It goeswithoutsayingthathydrogen-heliummixturesareofenormouscosmological * SupportedinpartbyNASA (NASA-DefensePurchaseRequestNo.W13,142), and inpart PHASE EQUILIBRIAINMOLECULARHYDROGEN-HELIUM Science ResearchLaboratory,U.S.MilitaryAcademy,WestPoint,NewYork MIXTURES ATHIGHPRESSURES* Received 1973June11 I. INTRODUCTION W. B.Streett ABSTRACT 1107 197 3ApJ. . .186.1107S helium—the verypropertieswhichtosomeextentfacilitatetheoreticalstudies—cause practical problemswhichhaveseverelylimitedhisaccesstostatesofmatterpresumed point theexperimentalististemptedtoquoteSamuelJohnson:“Andbidhimgo with relativeeasetowhatheperceivesbetheverycentersofthesebodies.Onthis theoretical sidehas,forobviousreasons,farsurpassedthatontheexperimentalside. equation-of-state experimentsatpressuresinthemegabarrange.(Grigorievandhis In addition,therearetheproblemsofhydrogenembrittlement—thedeleteriouseffects Part oftheproblemisresulttheirhighvolatilityandsmallmolecularsize,which them torankamongtheleasttractableofallsubstancesforhigh-pressureexperiments. to existinplanetaryinteriors.Theatomicandmolecularsimplicityofhydrogen The theoretician,usinghisequationsasavehicleandthelawsofphysicsthermo- 4-75%, andtheexplosivelimits,limitswithinwhichdetonationwavescanbepropa- hydrogen-air mixtures.(Theflammablelimitsofhydrogeninair,byvolume,are enable themtoescapepasthigh-pressuresealsthateifectivelycontainotherfluids. Hell, toHellhegoes.”However,thisonlyrevealsthelatter’sfrustrationwith dynamics ashisguide(andwithanassistinrecentyears,fromcomputer),descends interest intheouterplanets,andpartlyasaresultofsuggestion(Ashcroft1968) surprising thatexperimentalmeasurementsofthehigh-pressurepropertieshydrogen gated, areapproximately20-50%.)Inviewoftheseproblemsitisperhapsnot that metallichydrogenmightbeahigh-temperaturesuperconductor.Moderndynamic hydrogen athighpressureshasdevelopedinrecentyears,partlyasaresultofgrowing helium underpressurehavebeenreported.Anintenseinterestinthepropertiesof (1924) at30°and65°Cpressuresto~13kilobars,thoseofStewart(1956) and heliumhave,untilrecently,beenlimitedtothedensitymeasurementsofBridgman of hydrogenonthemechanicalpropertiesmetals—andexplosivenature methods forresearchatveryhighpressures,includingshock-waveandexplosive as afunctionoftemperature.Nosignificantexperimentsonmixtureshydrogenand at 4.2°Kandpressuresto20kilobars,afewmeasurementsofmeltingpressure there remainsanimportantroleforlaboratoryexperimentsonhydrogenand.helium unambiguous, however,especiallywherepressure-inducedphasetransitionsarecon- hydrogen.) Theinterpretationoftheresultstheseexperimentsisnotentirely colleagues havereportedthefirstexperimentalevidenceofmetallictransitionin compression techniques(Grigorievetal.1972;Hawke1971),havemadepossible pressure of20kilobarsbarely“scratchesthesurface” ofthegiantplanets,static perhaps quiterapidlyifthepresentlevelofinterestinhigh-pressurehydrogenis cerned. Notwithstandingthesuccessesofthesedynamichigh-pressureexperiments, 1108 W.B.STREETTVol.186 by statichigh-pressuremethods.Withpresenttechniques,avarietyofexperi- helium-methane (Streett and Hill1970,1971
P F1 (kbars) (Mol % He)
T = 34.95
1.110 3.0 1.151 5.8 1.172 7.5
T = 38.88
1.379 2.4 1.462 7.3
T = 34.95 F2 1.806 99.1 2.634 99.5 4.240* 100.0 * Indicates a point on the melting curve of pure helium. portion of the system, separated from other homogeneous portions by boundary surfaces. 2. The number of independent intensive properties, p, of a two-component system at equilibrium is given by v = 4 — p, where p is the number of phases present. 3. If one considers pressure, temperature, and phase compositions (P, P, X\ X", X'\ etc., where the primes denote different phases) as the properties of interest, the phase behavior of a two-component system can be completely described by a three- dimensional P-T-X diagram. When the system consists of a single phase, the variables are P, P, and X\ all of which are independent. In other words, the pressure, tempera- ture, and composition of such a system can be varied independently and continuously. When the system consists of two phases in equilibrium, there are four variables, P, P, X\ and X\ only two of which are independent. If P and P are chosen as the independent variables, then there exist two functions, X' = X'(P,T) and X" = X"(P, P), which describe the P-P dependence of X' and X". These functions define two surfaces in P-T-X space. It follows from similar considerations that three phases are represented by three lines and four phases by four points. At fixed pressure and temperature it is the difference in composition which distinguishes between coexisting phases in P-T-X space. 4. The key to understanding the three-dimensional phase diagram lies in under- standing the locations, in P-T-X space, of the boundary lines at which the system has a single degree of freedom. These are {a) the two-phase boundary lines (melting, curves, vapor pressure curves, etc.) of the pure components, in the P-P faces of the diagram; {b) critical lines, which are lines in P-T-X space that identify states at which two phases become identical; and (c) three-phase lines, which are lines in P-T-X space which define the coexistence of three distinct phases. IV. EXPERIMENTAL RESULTS There are four distinct phases in the hydrogen-helium system in the region of interest here: two fluid phases, Pi and P2? and two solid phases, Sx and S2. The subscripts indicate whether hydrogen (subscript 1) or helium (subscript 2) is the major
© American Astronomical Society • Provided by the NASA Astrophysics Data System 197 3ApJ. . .186.1107S o n No. 3,1973PHASEEQUILIBRIAINH-HeMIXTURES1113 2 component ineachphase.Theplussymbolindicatesequilibriumbetweenphases; therms intherange26-100°K,andatamaximumpressureof9.3kilobars(kbar). hence F+referstoequilibriumbetweenthetwofluidphases,etc. region. Temperaturesoftheisothermsareindicatedbynumbersatrightdiagram.Legend: critical line. pressure portionofthehydrogen-heliumphasediagram.Circlesindicatedatafromthisexperiment. These resultsarerecordedintable1andplottedfigures2.Thereisgenerally equilibrium phasesintheregionF+^.Measurementshavebeenmadefor13iso- are shownonaP-Tdiagram.Table2isdiscussed inthefollowingsection.Theportion recorded intable3.Infigure3theboundarylines ofthevarioustwo-phaseregions are recordedintable2,andtheP-T-Xcoordinates ofpointsthecriticallineare below 100bars.AfewmeasurementsofFand intheregionsSx+FxandF satisfactory agreementwiththeearlierresultsof Streett,Sonntag,andVanWylen Data below~0.06kbarfromStreettetal.(1964)andSneed(1968).Dottedlineisthemixture (1964) andSneed,Sonntag,VanWylen(1968), whicharelimitedtopressures —O—, experimentalresults;—□,mixturecriticalline.(Seetextforfurtherdiscussion.) remained atlowtemperatures, theortho-paracompositionofhydrogen isassumed of thediagrambelowcriticaltemperature of hydrogen,showninfigure1,is to haveremainedthatof “normal”hydrogen(757ortho,257para). cylinders atroomtemperature.Duetotheshortlength oftimeoverwhichanysample discussed furtherhere.Thehydrogenusedinthese experimentswastakenfromgas (Streett andHill1970)helium-argon andHill19716).Itwill°tbe similar informtothecorrespondingportions of thediagramshelium-nitrogen 12 x 1 2 09 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The principalresultsofthisworkareexperimentallymeasuredcompositionsthe Fig. 1{left).—Pressure-compositiondiagramshowingisothermsforthelow-temperature,low- Fig. 2{right).—Pressure-compositiondiagramshowingexperimentalisothermsfortheFi+F The isothermsinfigure2 canbeviewedascontourlinescutintheP-T-X surfaces 2 197 3ApJ. . .186.1107S 7 in thehydrogen-vaporpressurecurve{leftedgeoffig.2)andmixturecriticalline which definetheequilibriumF+jP.Thesetwosurfaceshavecommonboundaries the three-phaseequilibriumS+FjFoccursatthattemperature.Thedashed {dotted line).Theportionofeachisothermtotheleftcriticallineliesini! lines GHandG'H'describe,respectively,theFphasesinthree-phase The horizontaldashedlineatthetopofeachisothermmarkspressurewhich presence oftheSxphase,butnottowithdrawsamplesforanalysis.Hencelocation region. Itwaspointedoutin§IIthatthisexperimentitpossibletodetectthe at thecriticalline,whichislocusofminimainisothermsaboveabout28°K. surface, andtheright-handportionliesinFsurface.Thesesurfacesjoinsmoothly 1114 W.B.STREETTVol.186 way ofsayingthatfluidheliumisalmostcertainlylesssolubleinsolidhydrogenthan in fluidhydrogenthisregion. known; however,italmostcertainlyliestotheleftofG'H'infigure2,whichisanother little doubtthatitoccursatstillhighertemperatures atpressuresbeyondtherange critical temperaturesofthetwocomponents(5.2° Kforheliumand33.2° the compositionspreadofP\+Fphaseseparation attemperaturesabovethe of thelinewhichdescribes*Siphase,inregionSx+F,isnotprecisely pressure differencebetweenthecriticalpointand thethree-phasepressure—increases hydrogen). TheresultsconfirmthattheFx+F separationoccursattemperatures up to3timesthecriticaltemperatureofhydrogen, andthetrendofdataleaves region isboundedathigher pressuresandtemperatures,itislikelytoend inaninter- section ofthecriticalline EFandthethree-phaselinesGHG'H\ commonly divergence ofthecritical lineandtheS+F(islandGH).If theF+ with increasingtemperature,atleastupto100°K. Thisisillustratedinfigure3bythe 12 of theseexperiments. 12 ±2 2 2 ±2 2 x2 ±2 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Among themoreremarkablefeaturesofthissystem arethecontinuedexistenceand It willbeseenfromfigure2thatthepressuredepth oftheFx+Fseparation—the 2 P-T-X PointsontheMixtureCriticalLinein (1968). (1964); pointsatP=0.052and0.082fromSneedetal. 100.00 7.3758 29.00 0.1747 28.20 0.08238 29.00 0.05232 28.47 0.1446 93.00 6.3358 77.61 4.3558 70.30 3.5258 61.50 2.5658 38.88 0.6857 34.95 0.4854 31.00 0.2749 31.00 0.02821 32.50 0.0178 84.82 5.2358 33.19* 0.0130 Note.—Points atP=0.017and0.028fromStreettetal. * Hydrogencriticalpoint. (°K) (kbars)(Mol%He) T PX Hydrogen-Helium PhaseDiagram TABLE 3 197 3ApJ. . .186.1107S No. 3,1973PHASEEQUILIBRIAINH-HeMIXTURES1115 A recentsurveybySchneider (1970),oftheexperimentaldataforphase behaviorin high pressures,including atleastadozeninwhichheliumisoneofthe components. is perhapsmoreappropriate,sincethephasesare moreaccuratelycharacterizedas pure componentsarenormallyconsideredtobe gases, isoftencalledgas-gasequilib- temperature limitsofstabilitythemolecularphase. latter twolinessuggeststhattheymaydivergewith increasingpressureaboveabout form adomeatthepointofintersectioninfigure2.Antheselines dense fluids,ratherthanasgases,undertheconditions atwhichtheseseparations rium, atermoriginallysuggestedbyvanderWaals. Theterm“fluid-fluidequilibrium” three-phase lines,EFandGHinfigure3,diverge; however,infigure2GHandG'H' nitrogen orhelium-methane,whichhavebeenstudiedto10kbar(Streetteial.1972; known asanuppercriticalendpoint.Atsuchintersection,GPTandG'H'would remaining boundarylines.) F andlines{GHG'H'infig.2).Inthehydrogen-helium systemthecriticaland experimentally, andhasbeenaddedhereonlytoshowitsapproximatelocationrelativethe represents anisothermsuchasthatshowninfig.4.(Thethree-phaselinehasnotbeenstudied regions. AB,hydrogenmeltingcurve;CD,heliumEF,mixturecriticalline;GH, occur. Fluid-fluidphase separations havebeenobservedinmanybinary mixturesat show aslightconvergenceovertherangeof experiment.Thecurvatureofthe of thecriticalandthree-phaselinesinP-Tdiagrambutalsobyconvergence Streett andErickson1972).Theintersectionisprecedednotonlybytheconvergence occurs inthehelium-argonsystemat~llkbar,butnonewasobservedhelium- three-phase lineSi+FiF;JE,S.Theverticaldotted 2 10 or12kbar.Ifso,theF+phaseseparation maywellpersisttothepressureand 12 2 12 The existenceoftwodistinctfluidphasesatsupercritical temperatures,atwhichthe © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig. 3.—Pressure-temperaturediagramshowingtheprincipalP-Tboundariesoftwo-phase 197 3ApJ. . .186.1107S treated inananalogousmanner,iftheirequilibriawiththesephaseswereknown. three separatetwo-phaseregions:Sx+FS,and.Thepresumed Their inclusionwouldaddtothecomplexityofdiagram.Thismatterwillbetaken have eachbeenassumedtoexistinasinglefluidphaseandsolidphase. emphasized that,'forthepurposeofthisdiscussion,molecularhydrogenandhelium It isbasedlargelyontheauthor’sexperimentsavarietyofgasmixtures,andalso fluid mixturesathighpressures,suggeststhatfluid-fluidphaseseparationsmaybe the threepossibleadditionalthree-phasecombinations,onlyonewhichoccursis reached inthisexperiment,impliestheexistenceofothertwo-andthree-phaseregions. existence oftheSphase(helium-richmolecularsolid),atpressuresabovethose up inasubsequentpaper.) (Additional phases—e.g.,metallicphasesordifferentcrystallinesolids—couldbe Krichevskii andfsiklis,whowerepioneersinhigh-pressureresearchongasmixtures F +S.TheexcludedequilibriaarethoseforwhichthecombinationFxoccurs (Krichevskii 1940;KrichevskiiandTsiklis1943;1946,1952).Itshouldbe section isdevotedtoan“educatedguess”abouttheformofremainingportions experimental dataforalimitedportionofthehydrogen-heliumphasediagram.This the ruleratherthanexceptioninmixturesofunlikemoleculesathighpressures. helium atanytemperature.Ifahydrogen-heliummixtureiscompressed,theFxphase The resultsreportedhere,andsimilarforothergasmixtures,suggestthatof on theelegantworkofSchneider(see,e.g.,1972)andexperiments of thediagraminregionpressure-inducedsolidificationmolecularphases. first formed,sothatequilibriumbetweenthetwophasesisunlikelytooccur. will disappear(i.e.,solidify)atapressurewellbelowthatwhichtheSphaseis simply becausethemeltingpressureofmolecularhydrogenislowerthanthat (helium-rich solid+hydrogen-richfluid).Thiscombinationprobablydoesnotoccur Sx +FS.Itfollowsthattheadditionaltwo-phaseregionsareand the fluid-solidtransitionregionofmolecularhydrogenandhelium.Thisfigureis points ofpurehydrogenandheliumatthe temperatureinquestion.Thethree- represented byanarea,equilibriumbetweentwophaseslines,and has beenreducedbyone;hencewithinthisfigureahomogeneoussinglephaseis key tounderstandingtheimportantfeaturesofP-T-Xphasediagramforthis in theseregions.ThustheFx+Fregioncerepresents thesaturatedFxphase,and connecting thesepointsarenot,strictlyspeaking, partofthephasediagram;theyhave phase equilibriumisrepresentedby thethreepointsè,c,and between threephasesbypoints.Pointsaandjare,respectively,themelting system. Withthetemperaturefixed,numberofdegreesfreedomsystem lines boundingthevarioustwo-phaseregionsdescribe themutuallysaturatedphases ed thesaturatedFphase;inS+Fxregion abrepresentsthesaturatedSxphase been includedonlytoshowtherelationbetween threecoexistingphases.Thesolid l9x2± 1116 W.B.STREET!Vol.186 2 would lieclosertothevertical boundariesofthefigure. three-phase equilibriumS+Fbythethree points/,g,andh.Thedashedlines 2 which meansthatifthe diagram weredrawntoscale,thesaturatedS and Slines solubilities areverylow,infactalmostnegligible, intherangeofthisexperiment, appreciable mutualsolubilitiesinthesolidphases. Thereisevidencethatthese and octheFxphase;etc.Forpurposesofclarity, the diagramhasbeendrawntoshow 2 2± 2 2± ±2 ± 2 V. PROPOSEDPHASEDIAGRAMFORTHEREGIONOFPRESSURE-INDUCEDSOLIDIFICATION The coexistenceofthreeseparatephases,SFand,impliestheexistence The discussionintheprevioussectiondealtmainlywithinterpretationof These considerationssuggesttheformshowninfigure4foracompleteisotherm © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem l92 A pointinfigure4represents ahydrogen-heliummixtureatfixedtemperature, i IN MOLECULARHYDROGEN-HELIUMMIXTURES 197 3ApJ. . .186.1107S 4 No. 3,1973PHASEEQUILIBRIAINH-HeMIXTURES1117 fixed pressure,andtotalcomposition.Theverticaldottedline,forexample,isthe locus ofpointsrepresentingmixtureswhosetotalmolefractionheliumis0.35. region (lineac)at34.95° and38.88°KontheFphaseinS + Fregion the diagram,furtherexperimentswerecarried out ontheFxphaseinSx+ this discussionthatareasinsidetheinteriorlinesof thediagramareexcludedregions,” to thelengthsoflinesegments5-9and5-8, respectively.Amixturerepresented horizontal linethroughthepointinquestionintersectssaturationboundarylines. helium. Thecompositionsofthetwocoexistingphasesarefoundatpointswherea Point 1representsahomogeneoussinglephase{F^)ofthiscomposition,whilepoints in thisexperiment.Toconfirmatleastpartofthe assumedformoftheremainder in thesensethat,atthistemperature,ahomogeneous singlephasecannotexist,at different compositions,butforwhichthetotalcompositionis0.35molefraction (line dg)at34.95°K.The resultsarerecordedintable2.TheS+Fdata showthat these areas. equilibrium, underconditionsofpressureandcomposition definedbyapointwithin by 4or6separatesintothreephases:b,c,andd; or/, g,andh.Itshouldbeclearfrom and anFphase(point9).Therelativeamounts ofthetwophasesareinproportion A mixturerepresentedbypoint5,forexample, separates intoanSxphase(point8) 3, 5,and7representtwo-phasemixturesinwhichthemutuallysaturatedphaseshave at 34.95°dgliesveryclose tothepureheliumboundaryandthat,consequently, points solidification insupercriticalhydrogen-heliummixtures.(Seetextfordiscussion.) 2 2x 2 x2 2 Only theFx+Fregion{cedinfig.4)hasbeen studiedformostoftheisotherms © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig. 4.—Suggestedformforanisothermextendingthroughtheregionofpressure-induced 2 ^He 197 3ApJ. . .186.1107S P-T diagraminfigure.3.Theseboundaries,atwhichthesystemhasasingledegreeof freedom, canberepresentedinthisfigurebyfunctionsoftheformP=P(T).The g, h,andylieveryclosetogether.Athighertemperaturesthismaynotbethecase. to theotherboundarylines.Theisothermoffigure4mightcorrespondatempera- rium S-L+FS,whichhasbeendrawnheretoshowitslikelylocationrelative determined experimentally,withtheexceptionofJK,linerepresentingequilib- each ofwhichisactuallythreelinesinP-T-Xspace.Allfigure3havebeen helium phasediagraminathree-dimensionalsketch.Themeaningofthisrather which thislineintersectsthetwo-phaseboundarylineshavebeenletteredinsame ture of,say,60°K,indicatedbytheverticaldottedlineinfigure3.Thepointsat boundary linesofinteresthereare:themeltinglinehydrogenAB; in P-T-Xspace.Theselines,shownasheavysolid,dashed,anddotteddescribe manner asthecorrespondingpointsinfigure4. to veryhighpressures. phases. BoundarylinesfortheS-^+Sregion,//:andhm,areshownasdotted complex diagramcanbegraspedifonevisualizetheimportantboundarylines of heliumCD;themixturecriticallineEF;andthree-phaselinesGHJK, at thetopoffigure4,toindicatelikelihoodthatthisphaseseparationpersists ture) mayoccurathightemperaturesandverypressures,evenforthefluid that justtheoppositebehavior(decreasingmutualsolubilitywithincreasingtempera- temperature; however,itwasnotedintheprevioussectionthatthereissomeevidence In general,onemightexpecttofindincreasingmutualsolubilitywith the conditionsforwhichsystemhasasingledegreeoffreedom.Theyare: melting curveofpurehydrogen,AB,intheleftP-Tfacediagram; the three-phaseregionSx+Fforexample,threephasesarerepresented two setsofthree-phaselinesGHandJK,eachconsistingthreeseparatelines.In curve ofpurehelium,CD,intherightP-Tface;mixturecriticallineEF;and is perpendiculartotheP-Tcoordinateplanes,henceeachsetprojectsasasingleline three-phase region^+£2•Eachsetofthreelinesliesinaruledsurfacewhich by thelinesG'77",G'H',andGH.Similarly,J"K",J'K',JKdefine in theP-Tdiagramoffigure3.Theletteringisconsistentthroughoutfigures3,4,and5. the sameformasisotherminfigure4,andanisobar,indicatedbydash-dot 2 line atthetopoffigure.Theisobarisaninvertedformisotherms,which 1118 W.B.STREETTVol.186 therms, Tx,hasbeenletteredtocorrespondtheisotherminfigure4.Theisotherms phase behaviorasincreasingpressure,andviceversa,expected.Oneoftheiso- 2 in thethree-dimensionalsurfaceswhichdefine mutuallysaturatedphasesinthe simply meansthat,ingeneral,decreasingtemperatureproducesthesameeffecton is definedbytwosurfaces:oneboundedEF andG'H'(theFxphase),one these surfaceslieinfigure5.TheequilibriumFx+ F,describedbythedataintable1, two phaseregions.Withtheaidoffigures3and 4itshouldbepossibletoseewhere and theisobararecontourlines,cutbyplanesof constanttemperatureandpressure, 1 bounded byEFandGH(theFphase).Theportion ceofisothermTxliesinthefirst within oneoftheenclosed regionsrepresentsamixturewhich,atequilibrium, consists described inananalogousmanner. The linesZ/anddglieinthesetwosurfaces. remaining two-phasesurfacescanbe those boundedby<7'//"andJ"K"(theSphase), andbyGHJ'K’(theFphase). by thetwo-phasesurfaces identifiesasinglehomogeneousphase.Ingeneral, apoint composition, atfixedtemperature andpressure.Apointoutsidetheregions enclosed of thesesurfaces,andedinthesecond.InSx +Fregionthetwosurfacesare 2 2 x 2 2 In figure5anattempthasbeenmadetoshowtheimportantfeaturesofhydrogen- The principalboundariesofthetwo-phaseregionsinthissystemareshownona © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The remaininglinesinfigure5aretwoisotherms{TandT^),bothofwhichhave x A pointwithintheP-T-Xdiagramoffigure5 identifies amixtureoffixedtotal 197 3ApJ. . .186.1107S P-T planes. No. 3,1973PHASEEQUILIBRIAINH-HeMIXTURES1119 identified bythepointsatwhichalinethroughoriginalpoint,perpendicularto lines andalinewhichpassesthroughtheoriginal pointandisperpendiculartothe following sequenceofeventswouldoccur.Atpoint 1themixturewouldconsistofa 0.35 molefractionofheliumcanbedescribedby theverticaldottedlineinfigure4. total composition.Forexample,theisothermalcompression ofamixturecontaining to examinetheeffectsofchangesintemperature andpressureonamixtureoffixed consists ofthreedistinctphases,foundattheintersectionsbetweenthree-phase the P-Tcoordinateplanes,intersectsnearbysurfaces.Ifaninteriorpointlieson phase disappeared,being transformedpartlyintothesolidphaseSand intoF. until point4isreached,atwhichthepressure would remainconstantwhiletheF into twofluidphases,andthequantityofsecond phase,F,wouldincreaseslowly If thecompressionwerecarriedoutslowlyenough tomaintainequilibrium,the at point6thepressurewould againremainconstantwhiletheFphaseis transformed one oftheruledsurfacesthatcontainthree-phaselines,itidentifiesamixture of atleasttwoseparatephasesdifferentcompositions.Theseare Between points4and6the relativeamountsofSxandFwouldchangevery little,and single homogeneousfluidphase,F.Atpoint2 the mixturewouldbegintoseparate hand P-Tcoordinateplane. are describedinthetext.)Fig.3isasideviewofthisdiagram,seenbyanobserverfacingleft- molecular hydrogenandhelium.NotethattheisothermsTiTinthisfigurehavesame form astheisotherminfig.4,andthatpointsonTiareletteredsameway.(Theremaininglines 2 ± 2 ± 2 2 2 1 2 Another methodofexplainingthephasediagram istoillustratehowitcanbeused © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Fig. 5.—Three-dimensionalsketchoftheP-T-Xphasediagramforsupercriticalmixtures 197 3ApJ. . .186.1107S figure 5,itcanreadilybeshownthatcoolingatconstantpressureproducesessentially into amixtureofandS-Thepreciseeffectsfurtherlargeincreasesinpressure for mixturesofverysimplemolecules.Thisdiscussionwillbeconfinedtoabrief component mixturesfromthepropertiesofpuresubstances,particularlyinter- composition. Itfollowsthatthechangesinphaseequilibriaproducedbyanysequence the samesequenceofstatesasisothermalcompression,formixturesabout hydrogen) wouldappearinthediagram. molecular potentialfunctions.Progresstowardthisobjectivehasbeenlimited,even determined ifthecompletephasediagramandsequenceofP-T-Xstatesareknown. are unknownatthistime.Itislikelythatotherhigh-pressurephases(e.g.,metallic of matteristhepredictionphysicalandthermodynamicpropertiesmulti- of P-T-Xstates—suchasthattheequationpathaplanetarybody—canbe features ofcriticalphenomenaandfluid-fluidphaseseparationsinmixturessimple survey ofanalyticalandsemitheoreticalmethodswhichhavebeenappliedtomixtures fluid-fluid). Hisexperimentshavealsoshownthatanenormousvarietyofcritical molecules athighpressures.TheremarkableexperimentalworkofSchneiderandhis of simplemoleculesathighpressures. between thethreetypesofequilibriainfluidsystems(liquid-gas,liquid-liquid,and colleagues (see,e.g.,Schneider1972)hasmadeitclearthatthereisalogicalcontinuity nature, includingthatofhydrogen-helium,canbepredictedqualitatively—oftensemi- behavior occursinrealsystems.Ataboutthesametime,VanKonynenburg(1968)and van derWaalsequationofstate.TheworkVanKonynenburg,whichreliedheavily 1120 W.B.STREETTVol.186 quantitatively—by meansofcomparativelysimpleanalyticequationsbasedonthe Scott (1972)havedemonstratedthatvirtuallyeverytypeofcriticallinefoundin around thebeginningofthiscentury,butwhichhadbeenhamperedbytediousness 2 van derWaalsform,byusingtheresultsofcomputerstudiessystemsmodel on computercalculations,broughttofruitionastudybegunbyvanderWaalshimself features offluidbinarysystemsformixturesmolecules ofequalcoresizes.Inother of handcalculations.UsingthevanderWaalsmodelasastartingpoint,Rigby,Alder, molecules whichhavehardcoresandbycalculatinghigher-orderperturbationterms. and others(Rigbyetal.1970)havederivedatheoreticallyexactexpressionofthe tive effect,however;andiftheyaretoolarge,the fluid-fluidimmiscibilitydisappears.) words, differencesinattractiveforcesbetweenmolecules ofthesamecoresizecan and theaugmentedformofRigby,isthatitpredicts virtuallyallofthequalitative predicts arapidriseinthecriticaltemperaturewith increasingpressure,andacritical tively, bytheaugmentedvanderWaalstheoryof Rigbyetal.forthecaseofamixture account foralloftheobservedbehavior.(Differences incoresizehavealargequantita- One oftheremarkablefeaturesvanderWaalsmodel,inbothoriginalform pressure rangeof~8kilobars, andthecriticalmolefractionreachesalimiting value value 3timesthecritical temperature ofthelessvolatilecomponent(hydrogen) overa the behaviorshowninfigures2and3.Themixture criticaltemperaturerisestoa mole fractionwhichapproaches~0.5inthehigh-temperature limit.Thisisprecisely of ~0.58onlyafewdegrees abovethecriticaltemperatureofhydrogen. (Asmall of hard-sphereandsquare-wellmoleculeswithequal coresizes.Inthiscasethetheory One oftheprincipalobjectivesacomprehensivetheorymolecularstructure By drawingalineofconstantcompositionintheplaneisobarattop Some successhasbeenachievedinrecentyearspredictingatleastthequalitative © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The presentresultsforhydrogen-heliummixtures appeartobedescribed,qualita- a) ThevanderWaalsTheoryofMixtures VI. THEORETICAL 197 3ApJ. . .186.1107S mole fractionof0.5.)Moreover,thecalculationssuggestthatcriticallinenever difference incoresizespresumablyaccountsforthedeparturefromtheoretical the molecularphases,thatis,tolimitsatwhichandatomicstructure phases. begin tobealteredbyeffectssuchasdissociation,ionization,andtransitionmetallic separation maypersistatleasttothetemperatureandpressurelimitsofstability intersects thethree-phaseregion(Alder1973).Inotherwords,fluid-fluidphase principle ofcorrespondingstates(BreedveldandPrausnitz1973),equations Ten Seldam1970),conformalsolutiontheory(Tan,Luks,andKozak1971),the No. 3,1973PHASEEQUILIBRIAINH-HeMIXTURES1121 Phenomenologically, thesemethodsarerelatedtothoseusedstudyandpredictthe on avarietyofmethods,including:thelatticegasmodel(Trappeniers,Schouten,and reported here. yet beenappliedspecificallytohydrogen-heliummixtures,butthereisnoreason dicting thequalitativefeaturesoffluidphasebehaviorinbinarymixtures.Nonehas state otherthanthatofvanderWaals(PeterandWenzel1972;Yorizaneetal.1971). A comprehensivetheoryofmeltinghasyettobedeveloped,evenforpuresubstances. believe thattheywouldnotpredictthequalitativefeaturesofexperimentaldata Myers, andGiddings1970).Thesemethodshaveallbeensuccessfulatleastinpre- solubility ofcondensedsubstancesindensegases(HaarandSengers1970;Czubryt, behavior inbinarysystems,carriedoutthislaboratory,constituteoneofthefirst data onmeltingbehaviorinmixturesunderpressure.Recentexperimentsphase systematic studiesofmeltingbehaviorunderpressureinmixturesgases. Moreover, thislackofprogressinthetheoryismatchedbypaucityexperimental dynamics stumblesagainstourignoranceoftheactualequationsstate.”Now, predicting thebulkpropertiesofpuresubstances whentheintermolecularpotential tions ofdensematter,basedontheMonteCarloandmolecular-dynamicsmethods. however, itmaybepossibletogetaroundthisproblembymeansofcomputersimula- densities. InthewordsofRakovsky(1927),“thepowerfulapparatusthermo- has beenthelackofknowledgeequationstatematterathighpressuresand 2 progress inthisareathefuture.Hubbard(1972) hasrecentlypublishedtheresults function isreasonablywellknown.Theapplication ofthesemethodstomixtureshas (For areviewofprogressinthisfieldsee,e.g.,McDonaldandSinger1970.)These from theoreticalormachine-calculated propertiesofwell-definedreference systems, time. Evenwiththefastestmoderncomputers,calculations forasinglepointonthe force computermethodsatpresentistherequirement forlargeamountsofcomputing and Watts1972)havedemonstratedthatthese methodsarecapableofaccurately substances whosemoleculesinteractaccordingtowell-definedpotentialfunctions. equation ofstateapuresubstancecanoccupy anhourormoreofprocessingtime. mixtures atpressuresinthemegabarrange.One ofthedisadvantagesthesebrute- only justbegun(SingerandSinger1972),it isreasonabletoexpectsubstantial Barker andothers(Bobetic1970;Barker, Fisher,andWatts1971;Fisher methods makeitpossibletocarryoutcomputer“experiments”onhypothetical what, bymakingitpossible toestimatethethermodynamicpropertiesof realfluids of MonteCarlocalculationsdesignedtosimulate thebehaviorofhydrogen-helium Recent developmentsin perturbation theoryhavealleviatedtheserequirements some- Other attemptstopredictfluid-fluidequilibriaandcriticalbehaviorhavebeenbased In thecaseoffluid-solidequilibriumingasmixturessituationislesssatisfactory. © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem One oftheprincipalstumblingblocksindevelopmentatheorymixtures b) OtherAnalyticalandSemitheoreticalMethods c) MethodsBasedonComputerSimulation 197 3ApJ. . .186.1107S phase diagramhavingessentiallythesameformasthatshowninfigures2-5was gas mixtures,andwehavediscussedcertainaspectsinearlierpapers(Streett1969, ponents. Itfollowsthatthephysicalstructureandhydrodynamicbehaviorofplanets tion; (2)meltingunderpressure;and(3)phasedensityinversions. giant planets.Thefeaturesofparticularinterestare:(1)thefluid-fluidphasesepara- tions ofthispointhavegraduallybecomeapparentinthecourseourexperimentson composed ofmixturesthesegasesmightbemorecomplexthanhasbeensuggested hydrogen-helium mixturesisconsiderablymorecomplexthanthatofthepurecom- planetary structures.First,ifconditionsfavoringtheseseparationsexistdeepwithin persist toveryhighpressuresandtemperatures,perhapseventhelimitsofstability several importantfeaturesofthisdiagramhavebeenconfirmedbyexperiment,it by modelsdevelopedaroundtheframeworkofabodypurehydrogen.Theimplica- such asthehard-spheresystem(see,e.g.,BarkerandHenderson1967; Barker 1968). seems worthwhiletoreviewsomeofitsimplicationsfortheinteriorstructures suggested asalikelyformforthehydrogen-heliumsystemathighpressures.Nowthat dynamic behaviorofthedensefluidregions.Second,suchseparationswouldcon- the atmospheresofJupiterorSaturn,gravitationalseparationphasesdifferent augmented vanderWaalstheorysuggestthatthefluid-fluidphaseseparationmay density mustbeanimportantfactorindeterminingthephysicalstructureandhydro- fundamental generalizationofthephaserulethatcoexistingphasesinmulticomponent upper atmospheres,derivedfromobservationaldata,maynotberepresentativeofthe stitute amechanismforthegravitationalseparationofhydrogenandheliumin of themolecularphases.Thispossibilitycarriesseveralimportantimplicationsfor where phasetransitionsoccurwithintheplanets.Indeed,itfollowsfrommost that significantdiscontinuitiesinheliumconcentrationcanexistattheboundaries hydrogen andhelium,thelowsolubilityoffluidheliuminsolidhydrogen,suggest dense fluidregionsbelow.Finally,theobservedpartialmiscibilityof outer fluidlayersoftheplanets.Thismeansthathydrogen/heliumratiostheir 1971; StreettandHill1970,1971Z?;etal.1971).Inseveralofthesepapersa different phasesisfixedbythechemicalpotentialsoftwosubstances.Thisatleast limit ofthermodynamicequilibrium,thedistributionheliumandhydrogenbetween 1122 W.B.STREETTVol.186 component systems,however, eachadditionalcomponentaddsanother degreeof is assumedtooccur,butinwhichhydrogenand helium areassumedtobeuniformly helium canbetreatedasfreevariablesinformulating planetarymodels.Amodel suggests thattherearelimitstotheextentwhich thequantityanddistributionof systems haveunequalcompositions.Thispointcanbestatedanotherway:inthe hence asingle-component planet,inwhichtemperatureisafunctiononlyof theradius, mixed throughout,thuscontainsabuilt-incontradiction. Amongthecurrentmodels such asthatofPeebles(1964),inwhichthemolecular-metallic transitioninhydrogen of largeconcentrationdifferencesatthemolecular-metallic interface. of Jupiter,onlythatSmoluchowski(1967)has takenintoaccountthepossibility solidifies atadepthwhich theequationofpathcrossesmeltingcurve. Inmulti- It isclearfromtheforegoingdiscussionsthatphasebehaviorofmolecular Both thetrendofexperimentaldataandtheoreticalanalysisbasedon © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem In asingle-componentsystem,meltingpressureis auniquefunctionoftemperature; b) MeltingBehaviorinTwo-ComponentSystems and Density InversionsbetweenFluidandSolidPhases VII. IMPLICATIONSFORPLANETARYSTRUCTURES a) Fluid-FluidPhaseSeparations 197 3ApJ. . .186.1107S freedom totheconditionsofequilibrium,andtwophasescancoexistoverfiniteareas in pressure-temperaturespaceor,otherwords,overextendedradialdistances within theplanet.Inahydrogen-heliumplanet,forexample,phasetransitionsmight phase stableathigherpressuresisthemoredensephase.Incaseofmixtures, be expectedtooccurapproximatelyatpressure-temperaturepointswheretheequation No. 3,1973PHASEEQUILIBRIAINH-HeMIXTURES1123 curves ofsingle-componentsystems. phase richestinthecomponentwithhighestmolecularweightisalmostcertainto At highpressures,wherethespacesbetweenmoleculeshavebeensqueezedout, however, thecompositionbecomesanimportantfactorindeterminingphasedensities. across atwo-phaseboundary.ItfollowsfromtheprincipleofLeChatelierthat of pathcrossesthree-phaselines,whicharetwo-componentanaloguesthemelting have thehighestdensity.Amonglightgasesthereis,ingeneralaninverserelation greater thevolatility.Henceformostbinarymixturesoflightgases“normal” between volatilityandmolecularweight,i.e.,thelower density relationbetweenphasesapplies:theorderofincreasingisgas,liquid, consequence, pureheliumismoredensethanhydrogenbyafactorof2or twice thatofthesecondcomponent,butmoleculesarenearlyequalsize.Asa general rule.(OtherswhichhavebeenstudiedincludeAr-NH,CO2-H2O,and solid. Thehydrogen-heliumsystemisoneofanumberknownexceptionstothis hydrogen-vapor pressureandsublimationcurves,wherehydrogenisinthecondensed Ne-CH.) Herethemorevolatilecomponent,helium,hasamolecularweightabout phases inthehydrogen-heliumsystemwasfirstobservedbyKamerlingh-Onnes—the with liquidhydrogenatabout20°K.Hefoundthatonraisingthepressureto over mostregionsofP-Tspace.(Theexceptionisthelow-pressureregionnear 2 phase, whereuponthetwoexchangedpositionsinsidecontainer.Sneedeta/.(1968) man whofirstliquefiedhelium—in1906,whenhebroughtheliumhasintocontact 40 atm,thehelium-richgasphasebecamemoredensethanhydrogen-richliquid state andheliumisatenuousgas.)Areversalintherelativedensitiesoftwocoexisting determined thelocusofpointsforwhichtwophaseshaveequaldensities.(Equality the experimentaldataintable1arefromregionaboveP-Tinversionline;that line inP-Tspace.)Exceptforthedataat26°Kandfirstfourpoints29°,allof is, thehelium-richfluidphaseFismoredensethanhydrogen-rich of densitybetweentwocoexistingphasesremovesonethedegreesfreedom of thetwo-phase,two-componentsystem,hencelocusequaldensitypointsisa to beconfirmedbyexperiment.Itisinstructiveconsidertheprobablerelative 3 that theheliumcontentofsaturatedFphaseincreases morerapidlywithincreasing its orderedmolecularstructure.However,formixtures ofSandFthediagramshows the meltingcurveofpurehydrogen,solidismore densethanthefluidbyvirtueof existing solidandfluidphasesinthehydrogen-heliumsystem,althoughtheyhaveyet 4 pressure thanthatofthesaturatedSphase.Itcan beshown(Streettetal.1971)that densities ofthetwosaturatedphasesin^+Fregionfigure4.Atpointa,on a heliumcontentforthefluidabout15molepercent greaterthanthatofthesolidis it asthebasisofanewtheory—calledCartesian divertheory—forJupiter’sGreat by morethan~15molepercent,theFphaseis more dense;andwhereverthediffer- implications oftheS+Fdensityinversionelsewhere (Streett1969)andhaveused ence islessthanthatamount,theSphase more dense.Wehaveexaminedthe ordered molecularstructure.Asaresult,wherever thecompositionsofandFdiffer sufficient toovercometheincreaseddensitywhich thesolidgainsbyvirtueofits features ofthehydrogen-helium phasediagramonwhichtheCartesiandiver theoryis 2 based, andtendtokeep the theory“afloat.” Red Spot(Streettetal. 1971). Theexperimentsreportedhereconfirmthe qualitative ± ±u 1 ± ± x± ± ± © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem For amodelplanetofpurehydrogen,thereisnoambiguityaboutrelativedensities Similar densityinversionsalmostcertainlyoccurathighpressuresbetweenco- 197 3ApJ. . .186.1107S isotherm.) Thesequenceofstatesencounteredinascendingalongtheverticaldotted dPjdT >0.(Suchafunctionismorerealisticequationofpathforplanetthanan taken throughthephasediagramforanarbitraryfunctionP=P(T),which intimately mixedinthisregion,sothatgravitationalseparationwouldnotoccur.A point 6,Sismoredensethan\however,thetwosolidphasesmightwellbe fluid, atalevelwhichthetwophaseshaveequaldensities.Itwouldprobablyalso probably resultinalayerofhydrogen-richsolidfloatingseahydrogen-helium layering wouldclearlytendtooccurbetweenpoints2and6,however, At point4thedensityrelationis(verylikely)F>andbetweenpoints estimate forSaturn.)Betweenpoints2and4,theFphaseismoredensethan 0.35 molefractionhelium,whichobeysthechosenequationofpath.(Accordingto line inthephasediagramwouldthencorrespond,ageneralway,tosequenceof tion whichischaracteristicoftheplanetasawhole.Thisbriefdiscussiongreatly result inanupper-atmospherecompositiondifferentfromthe0.35heliummolefrac- from themanyotherfactorswhichdetermineplanet’sstructure.Nevertheless,it states encounteredindescendingintotheinteriorofahydrogen-heliumplanet, an isotherm,itisreadilyseenfromfigure5thatcanapplyequallywelltoaslice other regionsofthediagraminfigure4.Althoughthiswasdrawntorepresent 6 therelationisF>AtpointdensitySAbove Smoluchowski andHubbard1973aheliummolefractionof0.35isreasonable transitions mayexertaprofoundinfluenceonthehydrodynamicbehaviorof hydrogen-helium mixtures. does pointoutsomeoftheimportantimplicationsphasetransitionsinmolecular oversimplified, sincetheeffectsofphasetransitionsobviouslycannotbeseparated fluid layersoftheouterplanets.Phasechangesaretemperaturedependent,and usually accompaniedbytheabsorptionorreleaseofsignificantquantitiesheat. have theiroriginindeep-seatedphasechanges,andthatconvection,drivenbythe x2 effects ofthesechangesreachtheuppercloudlayers,wheretheyproduceregions Hence itisconceivablethatsomeofthepuzzlingfeaturesJupiter’svisiblesurface 2x 2 outward flowofheat,isthemechanismbywhichhydrodynamicandthermal contrast visibleasspotsandstreaks. bringing tohisattentiontherelevanceofhydrogen-heliumphaseequilibriaplane- 2 tary physics,andforhiscontinuingsupportencouragementwhichhelpedmake these experimentspossible. 1124 W.B.STREETTVol.186 Babb, S.E.1963,Rev.Mod.Phys.,35,400. 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© American Astronomical Society • Provided by the NASA Astrophysics Data System 197 3ApJ. . .186.1107S © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem