Volume 123, number2 PHYSICSLETTERSA 20 July 1987

MELTING CURVE OF IRON UNDER PRESSURE

GangCHENInstituteofMaterialsScience,Jilin University, Changchun,PRChina

Received6 March 1987; revisedmanuscriptreceived13 May 1987; acceptedfor publication 19 May 1987Communicatedby D. Bloch

The dislocationtheory of melting is generalizedby consideringtheeffect of pressure.TheKraut—Kennedylaw is derivedbyusingthisgeneralizedapproachandappliedto iron. Thetheoreticalresultis in goodagreementwith experiment.

In thispaperweproposethe dislocationtheoryof ~ a2 3melting [lJ as a new approachto melting underpressure.The dislocationtheoryof meltinggivenby whereP is theexternalpressure,anda2p theuniformEdwardsandWarneris basedon the ideathata liq- compressionperunit volume.Thus,the free energyuidcanbedescribedin termsof ahighlyfaultedsolid of the crystalperunit volume isandthatthe meltingpointis wherethesolidbecomes

F (y, fl =Ay[By+C+ 1 —D+ln(l +y)/2yunstablewith respectto the statein which thereis alargenumberof suchfaults in the crystal. But the

—~Jytan~(l/~’y)+EJ, (4)theoryof EdwardsandWarneronly considersmelt-ing at zero pressure.Theeffect of pressureon melt- whereE= 9~t2dP/a~uN.ing is what the presentpaper will take into full ThemeltingtemperatureTmmaybeobtainedfromaccount, thefollowing equations,

Accordingto EdwardsandWarnerthefree energy F’’ ~ —0 (5

of the crystalperunit volumeis p’~YP’ m — , a

F0(y, T)=Ay[By+ C+ 1 —D+ln(l +y)/2y F~(y~,Tm)0, (Sb)

— fy tan’ (1/\/~)1. (1) whereTm is the melting temperatureat pressureP,andy~,is the reduceddislocationdensityat Tm. A

A detailedexplanationof this formulacanbe found differencebetweeny~,andYo canbe seenby corn-in the original paper [1]. The melting temperature paring (2a) with (5a), to which the main contri-T°~,(at zero pressure)can be calculatedfrom the bution is due to the termE.equations Examinationof (2a) and (5a), the derivativeof

F~(Yo, T°m)= 0, (2a) the free energyF with respectto y, gives the posi-tions of y~andy~,respectively,

F0(y0, T°m)=O, (2b) A’F~(y0,T~)

whereYo is the reduceddensityof dislocationsatmelting.The freeenergy(1) is thatat zeropressure. = 2By0+ C—D+ ~ [1 — ,fy~tan’ (l/fy0)]

If the crystalis underpressure,theeffectof pressuremustbe considered.This effectcanbe seenby add- = 0, (6a)ing to the free energy F0(y, T) an energy due toexternalpressure[11

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Volume123, number2 PHYStCSLETTERSA 20July 1987

A — ‘F,(y~,Tm) Mitzushimaseemsto neglectthe differenceof the

=2By,,+C—D+ 4[l —~[y~tan ‘(l/fy0)] +E valueof (afl/47t)(L~Vm/Vo)for different substances,andusesa constant0.05 insteadforeverysubstance.

= 0. (6b) Therefore,therelativeerrorbetweentheexpenmen-

From (6b) weknow thatwith increasingpressure tal valueandthetheoreticalone from (10) is abouty~decreases.Whenthepressureapproachesinfinity, 60% for iron.thevalueofy~approacheszero.EdwardsandWarner Thelatentheatat pressurePcanalsobe obtained

from the Clausius—Clapeyronequationhavegiventhe valueYo 0.2. Accordingly,whenthepressureincreasesfrom zero to infinity, y,, only Lm=(P+aflG/4~)z~Vm(P). (11)decreasesfrom 0.2to zero.Obviously,thevaryingofy~withpressureis sosluggishandinsensitivethatwe From (11) onecanunderstandthe variationof thecanneglectthedifferencebetweenYo andy~anduse latent heatwith pressureby meansof that of i~Vmy0 insteadofy~in (5b) aslongasthe pressureis not with pressure,but not from (10). So eq. (11) istoohigh. seeminglymoreaccurateandusefulformeltingunder

Making an accuratecalculationfrom (2b) and pressure.(5b),one obtains Bridgman’swork aboutcompressionof crystalsis

convenientlysummarizedas analytical fits of theTm= 7

0m (1+ 4,rP/a/JG), (7) experimentaldata,at a definedpressurerange,usu-

whereTm andT°~,are the meltingpointsat pressure ally of the formPandzero, respectively,fi is a constantrelatedwith (V—v

0)/v0 = aP+bP2. (12)

thenon-lineareffectsof thedislocationcore,G is theshearmodulus,anda is a constantformedby Yoas The isothermalcompressibilityat zero pressureis

given by — a, andthe pressuredependenceof thea= 1 + [1 +ln(l +y

0)/2y0 compressibilityis given by b. Therefore,the corn-— \[Yo tan—’(1/fyo) 1/C. (8) pressibility at pressureP can be approximately

expressedasNow (7) describesthe relationbetweenthe meltingtemperatureandthepressure.Accordingto Edwards ~(P) =x(0) — bP. (13)andWarner[I], theparametersYo andfi canbeeither Thus,eq. (12) canbe expressedasdirectlyestimatedfrom anaccuratecalculationorbyfitting themto thethermodynamicobservables.They P=B(P)( V0 — V)/ V0, (14)determinedYo and/i by fitting theexperimentalvalue whereB(P) isthebulkmodulusat pressureF,whichofthelatent heatwith thetheoreticalvalue.Thisfit is the reciprocalof the compressibility~(P).ting result is alsousedin the following calculation. Combining(7) with (14), oneobtains

Neglectingthe pressuredependenceof the shearmodulusG, wegain the latent heatat zeropressure Tm = I~{ 1 + [4irB(P)/afiG]A V/V0}. (15)from the Clausius—Clapeyronequationas We separatethebulk modulusB(P) into two parts.

Lm = (aflG/4it )~Vm, (9) OneisB(0) thebulk modulusat zeropressure,whichis pressureindependent.The other is pressure

wherei~Vm isthe volumechangein melting.Eq. (9) dependent.Then, we canchange(15) intorevealsthat thelatentheatis proportionalto thevol-ume changein melting. While Mitzushima [2] has Tm_T~m[l+Ci~V/Vo+Cf(P)IXV/Vo1, (16)given a formula in which the molar latent heatof wheremeltingis proportionalto the molar volume V0 ofthe crystal, C4itB(0)/aflG, (17a)

Lm0.O5GV0. (10)f(P)= ~ (bP/a)~. (17b)

By comparing (9) and (10), one finds that

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Volume 123, number2 PHYSICSLETTERSA 20 July 1987

Generally,b/a 10—6 bar’. Providedthe pressure of V0, whichisclosetotheexperimentalvalueof 3.5%is in sucha rangethat bP/a~ 1, eq. (16) becomes [5]. B(0)= 168 GPa[6], weget that C=2.8475,that

Tm=7~i(1+CEiV/Vo). (18) is,Tm= 1808(1 +2.8475~V/V0) K. (19)

Eq. (18) is theKraut—Kennedylaw [31. Here, theconstantC in (18) wasdeterminedby the experi- The experimentalmeltingcurve is [3]mental data in the, original form •of the Tm1786(l+2.8l33L~V/Vo)K. (20)Kraut—Kennedylaw, which canbe determinedthe-oretically. Fora crystalits meltingcurvecanbe pre- Thetheoreticalresult is in good agreementwith thedicted from the Kraut—Kennedy law with the experimentalvalue.It seemsthat formula (18) mayknowledgeof the elasticpropertyof the crystal. be usedfor predictingthe melting curve of crystals

TheKraut—Kennedylawis onlyanapproximation underpressure.of (16). The lower the pressure,the more accurate As comparedwith the Lindemann approachtothe Kraut—Kennedylaw is. An accuraterelationship melting,thedislocationtheoryofmeltingisbetterinbetweenTm/7°~.and(~V/V0) shouldincludehigher- respectof the mechanismof melting. Thus, otherordertermsof (~V/V0),which are equivalentlycon- problemsin meltingunderpressure,suchas the tn-tamedin the termf(F). Rosshaspointedout that pie point, canalso be studiedwith the dislocationthe Kraut—Kennedylaw is an adequatedescription theoryof melting.of the relation betweenTm/7

0m and (i~VI V0) with

higher-orderterms neglected[4]. With increasing I would like to thankProfessorZou Guangtianforpressurethe termf(P)becomesmoreimportantand the discussionswhich startedthisproblem.the linearrelationship(18) breaksdown.Thecurveof Tm versus(i~VI V0) is concaveto thetemperatureaxis.Kraut andKennedyindeeddiscoveredsucha Referencesrelation in the meltingof helium andargon [3].

Furthermore,we use (18) to derive the melting [1] S.F.EdwardsandM. Warner,Philos.Mag. 40 (1979)257.

curveof iron. First,weshoulddeterminetheparam- [2] 5. Mitzushima,J.Phys.Soc.Japan15 (1960)70.

etersaand/ieitherby directcalculationor by fitting [3] E.A. Kraut and G.C. Kennedy,Phys.Rev. Lett. 16 (1966)

them to the experimental value of latent heat. 608; Phys.Rev. 161 (1966)668.

EdwardsandWarnerhavefitted Yo andfi with the [4] M. Ross,Phys.Rev. 184 (1969)233.[5] C.J.Sm,thells,ed., Metalsreferencebook, 3rd Ed. (Butter-

expenmentalvalue of latent heat, from which one worths,London,1962).

obtainsthata valueof a18G/4xis 59 GPa.Using(9), [6] K.A. GschneidnerJr., in: Solid statephysics,Vol. 16, eds.F.we canfind the volume change~Vm is about 3.6% SeitzandD. Turnbull (AcademicPress,New York, 1964).

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