Surface Tension
10
Surface Science 62 (1977) 267-276 © North-Holland Pubhshlng Company SURFACE FREE ENERGIES OF SOLID METALS" ESTIMATION FROM LIQUID SURFACE TENSION MEASUREMENTS W R TYSON Ferrous Metals Section, Phystcal Metallurgy Research Laboratories, Canada Centre for Minerals and Energy Technology, Department of Energy, Mines and Resources, Ottawa, Canada and W.A. MILLER Department o f Metallurgy and Matertals Sctence, Unwerslty of Toronto, Toronto, Canada Recewed 25 May 1976, manuscript received m final form 24 September 1976 An equation is derived on semi-theoretical grounds which expresses the sohd-vapour sur- lace free energy as a function of the hquld surface tension and the sohd-hquld lnterfaclal free energy A means of calculating rehable values for the sohd-hqmd energy ts presented, which then allows an accurate estimate of sohd surface energy at the meltmg temperature, Tin, to be made for the large number of elements for which dependable hqmd surface tension data exist A method of esttmatmg surface entropy IS presented, and has been used to calculate the ener- gies typical of "average ", high-index surfaces at temperatures ranging from 0 K to T m It is felt that thls paper describes the most accurate method presently available for the calculation of the surface energy of sohds In the absence of direct experimental measurement 1. Introduction The surface energy of solids IS an important physical parameter m controlling a w~de range of phenomena, such as the stress for brittle fracture, the rate of sInter- ing, and the growth rate during particle coarsening. For tlus reason, there have been many attempts, both experimental and theoretical, to derive values of the surface free energy, summaries have been given recently by Blakely [1], LInford [2], and Mykura [3] The available experimental techniques are difficult and usually Inac- curate, with results from different workers often being In very poor agreement [4]. Theoretical methods are strnflarly complicated, although there IS some hope that recent advances wall produce reliable calculations both for simple metals [5] and for transition metals [6]. The experimental determination of absolute values of surface energy for any type of interface is both difficult and subject to numerous errors, the most severe 267
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An equation is derived on semi-theoretical grounds which expresses the sohd-vapour sur-lace free energy as a function of the hquld surface tension and the sohd-hquld lnterfaclal freeenergy A means of calculating rehable values for the sohd-hqmd energy ts presented, whichthen allows an accurate estimate of sohd surface energy at the meltmg temperature, Tin, to bemade for the large number of elements for which dependable hqmd surface tension data existA method of esttmatmg surface entropy IS presented, and has been used to calculate the ener-gies typical of "average ", high-index surfaces at temperatures ranging from 0 K to T m It is feltthat thls paper describes the most accurate method presently available for the calculation ofthe surface energy of sohds In the absence of direct experimental measurement
Transcript of Surface Tension
Surface Science 62(1977)267- 276 North-HollandPubhshlngCompany S UR F AC E F R E E E NE RGI E S OF SOLI DMETALS" ES TI MATI ONFROM LI QUI DS URF ACE TENS I ONME AS URE ME NT SWRTYS ON Ferrous Metals Section,Phystcal Metallurgy Research Laboratories, Canada Centre for Minerals and Energy Technology, DepartmentofEnergy, Mines and Resources, Ottawa, Canada and W. A. MI L L E R DepartmentofMetallurgy and Matertals Sctence,Unwerslty ofToronto,Toronto,Canada Recewed25May1976,manuscriptreceived mfinalform24September1976 Anequationisderivedonsemi-theoreticalgroundswhichexpressesthesohd- vapour sur- lacefreeenergyasafunctionof thehquldsurfacetensionandthesohd- hqul dlnterfaclal free energyAmeansof calculatingrehablevaluesforthes ohd- hqmdenergytspresented,which t henallowsanaccurateestimateof sohdsurfaceenergyatthemeltmg temperature,Tin,tobe madeforthelargenumberofelementsforwhichdependablehqmdsurfacetensiondataexist AmethodofesttmatmgsurfaceentropyIS presented,andhasbeenusedtocalculate theener- giestypicalof "average ",high-indexsurfacesattemperaturesrangingfrom0KtoT mItis felt thatthlspaperdescribesthemostaccuratemethodpresentlyavailableforthecalculationof thesurfaceenergyofsohdsIn theabsenceof directexperimentalmeasurement 1.I n t r o d u c t i o nThe s ur f acee ne r gyo f sol i dsISani mp o r t a n t phys i cal p a r a me t e r mc ont r ol l i nga w~der angeo f p h e n o me n a , suchast he st ressf or br i t t l e f r a c t ur e , t her at eo f sInt er- i ng, andt h e g r o wt h r at edur i ngpar t i cl ecoar s eni ng. F o r t l usr eas on, t her ehaveb e e nma n y a t t e mpt s , bot he x p e r i me n t a l andt he or e t i c a l , t oder i veval ueso f t he sur f ace f r eee ne r gy, s umma r i e s havebe e ngi venr ecent l yb y Bl akel y[1], LI nf or d[ 2] , and Mykur a [3]The avai l abl ee x p e r i me n t a l t e c h n i q u e s aredi f f i c ul t andus ual l yInac- cur at e, wi t h r esul t sf r omdi f f e r e nt wor ke r s o f t e n bei ngInver yp o o r a gr e e me nt [ 4] .Th e o r e t i c a l me t h o d s arest rnfl arl yc o mp l i c a t e d , a l t hought her eISs omeh o p e t hatr ecent advances wallp r o d u c e r el i abl ec a l c ul a t i ons b o t h f or s i mpl eme t a l s [5]and f or t r a ns i t i onme t a l s [ 6] .The e x p e r i me n t a l d e t e r mi n a t i o n o f a bs ol ut e val ueso f s ur f aceener gyf or a ny t y p e o f i nt e r f a c e isb o t h di f f i cul t ands ubj ect t on u me r o u s er r or s , t he mos t sever e 267 268W RTyson,W AMdler/Surfacefreeenergteso fsohdmetals ofwhichISusuallyloweringofthetruelnterfaclalenergybysurface-activecon- tarmnants.ThesemaybeIntroducedeitherasdissolvedImpurityfromwithinthe materialorasanadsorbingspeciesfromthesurroundinggaseousatmosphere Nevertheless,surfacefreeenergy(surfacetension)canbemeasuredmuchmore accuratelyforhqut ds thanforsolids(althoughnotnecessarilymoreeasily),and comprehensivetabulationsof hquld-vapoursurface tension 7LVareavailablefor a largenumberof elements[7].It is thepurposeof thispapertodescribebriefly how theliquidsurfacetension datamay beusedtoderivereliableestimatesof "average" sohd-vapoursurfaceenergies7sv, thebasisofthemethodwillbeanalyzedIn greater detailelsewhere[8] 2.Estimation of 7SV/TLV at themelting point,Tm Threemterfaclalfreeenergiesarerevolvedmsolld-hquld-vapourequlhbrium 7sv, 3'LV,andthesolld-lNuldlnterfaclalfreeenergy7SLInthissection,we attempttodeducetherelationshipsbetweenthesequantitiesmsinglecomponent systems Wemaywriteoneconditionlmme&atelywhichfollowsfromthewell-known observationthatmetalscannot normally besuperheatedabove T mThis Impliesthe absence of a nucleation barrierfor surface melting, hence ")'SV ~"~LV +')'SL( 1 )Notethatanytypeofhighenergydefectinthesohdsuchasdislocations,grain boundanes,surfacesteps,etc.,maybeabletonucleatemeltingwithoutanenergy barrierIn thepresentanalysishowever, weassumethata local region of a perfectly flat,Infinitesurfacemay alsonucleatemelting withnosuperheating,so thateq.(I) may beapplied. AttemperaturesclosetoTin,itwouldbeexpectedthatthesohdsurfacehas a structurewhichIS atomicallyroughIf thisroughening IS achievedwithoutdestroy- ingthecrystallographicnatureof thesurface, thenthesurfacecan beconsidered to be"crystalline"Iftherougheningandatomicrearrangementismoresevere,then thestateofthesurfacemaybemoreakintoaliquid,andasurface"melting" transitionmayoccurbelowTmThenatureof theprocesswhichactuallyoccur, willbedeterminedbythemagnitudes of thevarious freeenergy changes. Supposethatthesurface wereto remaincrystallineuptoT mThen crystallogra- phicrougheningwouldoccur,withanassociatedsurfacefreeenergy"yR. Suppose, ontheotherhand,thatsurfacemelting were tooccur attemperaturesbelowTmto adepthsufficientforthesurfacelayertobeconsideredas a bulk liquidwithinde- pendentsohd-hquldandhquld-vapourInterfacesThesurfacefreeenergy of such a solidwould be 7M =TLv+TsL +(GLGs) t ,W RTyson,W AMdler/Surface freeenergiesof s ohdmetals269 whereGL, G Sarethefreeenergiesperumtvolume of the hqmdandsohd,respec- tively,andtisthedepthof thelnterfaclalhqmd layer.At temperaturesbelowTm, thelasttermincreasesthefreeenergy andin order tominimize its magnitudetwill clearlybesmall,oftheorderofatmostafewmonolayerthicknesses.Hence,the lasttermtends tozeroatTm,as (G L -Gs) tends tozero Unfortunately,therearenoexpenmentaldatawithwhichtoestabhshrehably thestructureofsolidsurfaces attemperaturesclosetoT m.It is knownfromLEED studiesthatmetalsurfacesarecrystallmeattemperatureswellbelowTm,butitIs notknownwhetherthesurfacemeltsbeforeT misreached.Thestateofthe surface,andhence7sv, willbedeterminedbytherelativemagnitudesof ?'Rand 7M.Asshownabove,thesurfaceenergyof asolidcoveredbyathinmolten layer is "i'M =")'SL + ")'LV atTm, andso surface meltingshould takeplaceattemperatures equaltoandslightlylessthanTmunless7R~
