Journal of Applied Research and Technology
www.jart.ccadet.unam.mx JournalofAppliedResearchandTechnology15(2017)78–82
Short communication
A first principles calculation of Ni-16Cr and Ni-16Mo alloys
Ashish Pathak
a, Kumud Kant Mehta
b, Ashok Kumar Singh
a,∗aDefenceMetallurgicalResearchLaboratory,Hyderabad500058,India
bRDAQA(M),c/oDRDO,Hyderabad500058,India Received10May2016;accepted13October2016
Availableonline11January2017
Abstract
Thepresentpaperdescribesthestructuralstabilityandmechanicalpropertiesofthe␥phaseforNi-16CrandNi-16Moalloysusingfirstprinciples densityfunctionaltheory(DFT)withingeneralizedgradientapproximation(GGA).Theequilibriumlatticeconstantvaluesofthesealloysarein goodagreementwiththeexperimentaldataobtainedbyX-raydiffractiontechnique.Theformationenergyperatomvaluessuggeststhatthe␥ phaseofbothalloysisstable.TheNi-16MoalloyisenergeticallymorestablethanNi-16Cr.BothalloyssatisfytheBornstabilitycriteriainterms ofelasticconstantsandareassociatedwithductilebehaviourbasedonsheartobulkmodulusratios.Bothalloysshowananisotropicbehaviour.
TheNi-16MoalloyexhibitshigheranisotropythantheNi-16Crone.
©2016UniversidadNacionalAutónomadeMéxico,CentrodeCienciasAplicadasyDesarrolloTecnológico.Thisisanopenaccessarticleunder theCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords: Alloys;First-principles;Crystalstructure;Physicalproperties
1. Introduction
Ni-basedsolidsolutionalloyssuchasInconel617,Inconel 625, Inconel 686, Hastelloy S, Hastelloy C, Hastelloy C-4, Hastelloy C-22,HastelloyC-276, Nicrofer5923,amongoth- ers,consistof CrandMoas majoralloyingelements(Davis, 2000). The industrial applications of these alloysare largely inwroughtformssuchasplates,sheets,strips,extrudedtubes, wiresandflangesduetotheirexcellenthotand/orcolddeform- ingcapabilitywithmoderatetohighwork-hardeningrateand weldability. The wrought forms of these alloysare therefore producedbythermo mechanical processingof thecast ingot.
ThemicrostructuresandmechanicalpropertiesofNi-16Crand Ni-16Mowt.%(Ni-18CrandNi-8Moatom%)alloyshavebeen investigatedrecently(Mehta,Mukhopadhyay,Mandal,&Singh, 2015).Thepropertiesofbothalloysare quitedifferentinhot rolledandannealedcondition.The contentsof CrandMolie intherangeof10–20(wt.%)inallthealloysmentionedabove apartfromotheralloyingelements.Thelimitsofsolubilityof
∗Correspondingauthor.
E-mailaddress:Singh[email protected](A.K.Singh).
PeerReviewundertheresponsibilityofUniversidadNacionalAutónomade México.
CrandMoinNiare35–40wt.%andabout20wt.%,respec- tively(Takeuchi&Inoue,2005).Inaddition,thesizedifference betweentheNi-Moisfairlylargeincomparisontothatofthe Ni-Cr inviewof thefact that the atomicradii ( ˚A) of Ni,Cr and Moare 1.25,1.28 and1.36,respectively(Delehouzee&
Deruyttere,1967).Alinearrelationbetweenflowstressandlat- ticeparameterchangehasbeenobservedforanysinglesolute element (Cr, Mo, W, Fe, Co andCu) in nickel (Tawancy &
Al-Hadhrami, 2012).The Ni-based solidsolution alloyshav- ing around 16wt. % of boththe Cr and Modisplay marked improvementsinthermalstability,fatiguelife,hightemperature strength, low expansion characteristics, oxidation and corro- sion resistance(Tawancy&Al-Hadhrami, 2012).Inaddition, most ofmodellingworkofsolid solutionNi-basedalloyshas been carried out around this composition (Mishima, Ochiai, Hamao,Yodogawa,&Suzuki,1986;Roth,Davis,&Thomson, 1997).
An understandingof these binary alloysin terms of their stability,latticeconstantsandmechanicalpropertiesusingfirst principlescalculationisthereforequiteimportant.Thiscanin factbeutilizedtotailorthecompositionsandpropertiesofmulti- componentsolidsolutionalloyswithoptimizedproperties.The presentpaperisthusconcernedwithafirstprinciplesstudyof theNi-16CrandNi-16Moalloys.
http://dx.doi.org/10.1016/j.jart.2016.10.001
1665-6423/©2016UniversidadNacionalAutónomadeMéxico,CentrodeCienciasAplicadasyDesarrolloTecnológico.Thisisanopenaccessarticleunder theCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Table1
Density,equilibriumlatticeconstantsandformationenergy/atomforNi-16CrandNi-16Moalloys.
Alloys Density(gm/cc) Latticeconstants( ˚A) Formationenergy/atom(eV)
Experimental Experimental Presentstudy
Ni-16Cr 8.620 3.5366 3.5040 −0.0060
Ni-16Mo 9.098 3.5690 3.5580 −0.0396
2. Experimentaldetails
The alloy pancakes of 600g with nominal compositions Ni-16Cr and Ni-16Mo (wt. %) were prepared using non- consumablevacuumarcmelting.Themeltingwasrepeatedfour timestoensurechemical homogeneity.Theaverageanalyzed chemicalcompositionsofbothalloysobtainedbywetchemical andelectronprobemicroanalyzer(EPMA-SX100,CAMECA, France)techniquesareNi-16.4CrandNi-16.3Mo(wt.%).The densityofbothalloyshasbeenobtainedbyArchimedesprinci- ple.Pancakeswereunidirectionallyhotrolled(HR)upto70%
reductionat1150◦Candthenaircooled(AC).The materials weredeformed10%ineachpassandreductiondirectionwas keptstrictlyunidirectional.Thesampleswereplacedbackinto furnacefor15minaftereachpasstore-attaintherollingtemper- ature.Hotrolledmaterialswerethenannealedat1000◦Cand 1100◦CforNi-18CrandNi-8Moalloys,respectivelyfor5min invacuumfurnaceandsubsequentlyfurnacecooledinvacuum.
Microstructuralcharacterizationofbothalloysinhotrolled (3mm sheet) and annealed condition was done using opti- calmicroscope(OM).Thespecimenswerepreparedfollowing standard metallographic technique used for Ni-based alloys andetched withKallings reagent (5gm CuCl2 in 40ml HCl and60mlmethanolsolution). X-raydiffraction(XRD) study of hot rolled and annealed materials was performed using a Philips3020diffractometerwithCuK␣radiationequippedwith graphitemonochrometer.
3. Computationaldetails
Afirstprinciplesultrasoft(US)pseudo-potentialmethodin the frame work of the density functional theory (DFT) has been utilized withQuantum Espresso code in present calcu- lation(Giannozzietal.,2009;Hohenberg&Kohn,1964;Kohn
& Sham, 1965; Vanderbilt, 1990). Perdew–Burke–Ernzerhof formulationwithingeneralizedgradientapproximation(GGA) hasbeenconsideredforexchange-correlationeffects(Perdew, Burke,&Ernzerhof,1996).Boththeplanewavecut-offenergy andcut-offdensityof40Ryand400Ry,respectivelyhavebeen chosen inallcalculations afterthe verificationof the conver- gence for plane wave. The wave functions between the real andreciprocallatticeshavebeenconvertedusingFastFourier transformalgorithm(Goedecker,1997).Consequently,conju- gategradient algorithm has been employedwithin the frame work of self-consistency (Gonze, 1996; Payne, Teter, Allan, Arias, &Joannopoulos, 1992). The Monkhorst–Packscheme has been utilized for the integration over the Brillouin zone (BZ)(Monkhorst&Pack,1976).Theoptimumvalueusedfor
thepresentcalculationsis10×10×10afterensuringthecon- vergence. The differences betweenenergies or forces in two consecutivestepsarelessthan1.36×10−4meVor0.36eV/ ˚A, respectivelyhavebeenpresumedforconvergence.
Boththeformationenergyperatomandthelatticeconstant
‘a’ofNi-16CrandNi-16Mohavebeencomputedusingequiv- alentatom%(Ni-18CrandNi-8Mo)andgiveninTable1.The equilibriumlatticeconstanthasbeenobtainedbyvarying lat- ticeconstantandcalculatingtheenergycorrespondingtoeach latticeconstanttogettheEvs.acurve.Thelatticeconstantcor- respondingtominimumenergyinthecurveisregardedas an equilibriumlatticeconstant.Theenergyofformationperatom hasbeendefinedas
Efor= 1 (x+y)
ENi−yM−xEgNi−yEgM
(1) whereENi−yM is the totalsystemenergy of the Ni-16Crand Ni-16Moalloyshaving‘x’Ni-atoms,‘y’M(Cr/Mo),EgNiand EgM arethetotalenergyperatomintheirgroundstatesforNi andM(Cr/Mo)atoms,respectively.The(x+y)denotesthetotal numberofatomsconsideredintheunitcellthatare16and12 fortheNimatrixofNi-18CrandNi-8Moalloys,respectively.
The ElaStic codehas been utilizedtocalculate the elastic constant(Golesorkhtabar,Pavone,Spitaler,Puschnig,&Draxl, 2013).Thiscodeisabletocalculatethefullsecond-orderelastic stiffness tensors forany crystalstructure fromabinitiototal- energyand/orstresscalculations.
4. Resultsanddiscussion
ThedensityofbothalloysisgiveninTable1.Thisreflectsthe densityvaluesofNi-16CrandNi-16Moaresmallerandlarger thanthatofpureNi(8.9g/cc),respectively.Itistobenotedthat thedensityvaluesofpureCrandMoare7.19and10.28g/cc, respectively. It reflects that the density of the present alloys dependsmainlyonthedensityofindividualalloyingelements andtheirrespectiveweightfractions.
The optical microstructures of Ni-16Cr and Ni-16Mo are shown in Figure 1. The microstructure of the Ni-16Cr alloy in hot rolled andannealed condition reveals equiaxedgrains withlargevariationingrainsize(Fig.1a).Ontheotherhand, theNi-16Moalloydisplaynearlyuniformmicrostructurewith annealingtwinswithinequiaxedgrains(Fig.1b).XRDpatterns showthepresenceof␥phase(Nimatrix,Fm ¯3m)onlyinboth alloys(Fig.1c).Thelatticeparametersofthe␥phaseofboth alloysobtained fromXRDpatternsare giveninTable1.The latticeparameteroftheNi-16Moalloyislargerthanthatofthe Ni-16Crone.Thisobservationisnotsurprisingsincetheatomic radiiofNi,CrandMoare1.25,1.28and1.36 ˚A,respectively.
0
30 40 50
c a
b
60 70
2θ (º) 100 μm
100 μm
Ni-16Mo
Ni-16Mo Ni-16Cr
Ni-16Cr (111)
(200) (220)
(311)
Intensity (x 1000 counts)
80 90 100
2 4 6 8 10 12 14 16
Fig.1.Opticalmicrostructuresofhotrolledandannealedalloys:(a)Ni-16Cr,(b)Ni-16Moand(c)correspondingXRDpatterns.
Table2
Elasticconstantsandanisotropyfactor(A)forNi-16CrandNi-16Moalloys.
Alloys C11(GPa) C12(GPa) C44(GPa) A
Ni-16Cr 310 178 141 6.994
Ni-16Mo 270 175 116 9.522
The latticeparameter valuesof the␥ phaseof bothalloys havealsobeencalculatedusingfirstprinciplescalculationand given in Table 1. These are in good agreement and follows similartrend withlatticeparameter valuesobtained by XRD technique.
Theformationenergy peratomvaluesofNi-16CrandNi- 16Moalloys obtained by first principle calculationare given inTable1.Theformation energyperatomof bothalloysare negativeindicatingthatthe␥phaseofthepresentalloysisstable.
ThisalsoreflectsthattheNi-16Moalloyismorestablethanthe Ni-16Cronesincetheformationenergyperatomofformeris lowerthanthelater.Incidentally,thestackingfaultenergy(SFE) ofNi-16MoislessthantwothirdoftheNi-16Cr(Mehtaetal., 2015).Thisfollowssimilartrendtothoseofformationenergy peratomofbothalloys.
Theelasticpropertiesofsinglecrystalofthe␥phaseofboth alloysaredescribedbyC11,C12andC44andgiveninTable2.
TheelasticconstantvaluesofNi-16Crarehigherthanthoseof theNi-16Moalloy.Acomparisonoftheelasticconstantvalues ofpureNi(C11=261,C12=151andC44=132GPa)andpresent alloysindicatesthat the equalamountof Craddition ismore effectivethanMotoincreasethecompliances.Italsoappears thatthiseffectisoppositetothestabilityofthe␥phase(interms offormationenergyperatom)inthesealloysandextentofsolid solutionofbothelementsinNi.
Thenatureofmetallicbondingofbothalloyscanbepredicted based onCauchypressures(Olijnyk &Jephcoat,2000).This is defined as C12–C44<0. The negativeand positiveCauchy pressures are sign of more directional andmetallic bonding, respectively.ThecalculatedvaluesofCauchypressuresforthe
␥ phaseof bothexperimentalalloysarepositive. Thisclearly pointstowardsthepresenceofpredominantmetallicbondsin thesealloys.
The values of elastic constants canbe used to predict the stability of phases.The Born stability criterions are givenin Eqs. (2)–(4)(Born,1940;Fedorov,1968).The corresponding calculatedvaluesaregiveninparenthesis.
C44> 0(Ni-16Cr:141;Ni-16Mo:116) (2) C11–|C12| > 0(Ni-16Cr:132;Ni-16Mo:95) and (3) C11+2C12> 0(Ni-16Cr:666;Ni-16Mo:620) (4) ThevaluesgiveninparenthesisofEqs.(2)–(4)suggestthat the␥phaseofthesealloysismechanicallystable.Thisisalsoin agreement withavailableexperimentalbinaryphasediagrams whereinthe␥phaseisstableatCrandMocontentsofpresent alloys(Massalski,Okamoto,Subramanian,&Kacprzak,2001).
Theanisotropyfactor(A)forthe␥phaseofthesealloysis definedbelowandcorrespondingcalculatedvaluesaregivenin Table2.
Forcubicmaterials A= 2C44
(C11−C12) (5)
ThevaluesofAshouldbe1forisotropicmaterialsandaway from1pointtowardstheextentofanisotropyinelasticconstants.
ThevaluesofAobtainedinpresentstudyclearlyindicatethat
Table3
Bulkmodulus(B),Shearmodulus(G)andYoung’smodulus(E)andPoisson’s ratio(υ)forNi-16CrandNi-16Moalloys.
Alloys BV BR B GV GR G E G/B υ
GPa
Ni-16Cr 223 223 223 111 96 104 270 0.466 0.30
Ni-16Mo 202 202 202 90 74 82 217 0.406 0.32
thesinglecrystalofbothalloysexhibitconsiderableamountof anisotropy.However,theextent ofanisotropypresentinalloy Ni-16MoishigherthanthatoftheNi-16Cr.
The effective elastic modulus of isotropic polycrystalline cubic materials can be evaluated from the elastic constants byfollowingtwoapproximationsnamely,theVoigtandReuss (Reuss&Angnew,1929;Voigt,1928).Theseapproximations provideinformationabouttheupperandlowerlimitsthatare B=BV =BR= C11+2C12
3 (6)
GV = C11−C12+3C44
5 (7)
GR= 5(C11−C12)C44
4C44+3(C11−C12) (8)
whereBisthebulkmoduluswhileGVandGRareshearmodulus values obtained by Voigt andReuss approximations, respec- tively.
The average value of these two estimates, as mentioned above, is given by Hill approximation (Hill, 1952). The Voigt–Reuss–Hill(VRH)averagevaluesaregivenby
BH =BV +BR
2 (9)
GH =GV +GR
2 (10)
E= 9BG
3B+G (11)
υ= 3B−2G
2(3B+G) (12)
whereBH,GH,Eandνarethebulkmodulus,shearmodulus, Young’smodulusandPoisson’sratio,respectively.
ThecalculatedvaluesofB,GandEforbothalloysaregiven inTable3.TheNi-16Cralloyexhibitshighervaluesofmodulus thantheNi-16Moone.Acomparisonofmodulusvaluesofpure Ni(B=188,G=101andE=257GPa)withthepresentalloys indicatesthattheadditionofCrincreasesthesevalues.Onthe otherhand,anadditionofMoincreasesBanddecreasesGand Evalues.Mehta etal.haverecentlyinvestigated themechan- icalproperties of thesealloys along three sample directions, i.e., longitudinal (L or 0◦),45◦ (specimen axis at45◦ tothe rollingdirection)andtransverse(T or90◦)directions(Mehta etal.,2015).TheYoung’smodulusvaluesofthesealloyshave beencalculatedfromengineeringstress-straincurvesalongall thethreedirections.The averagevaluesof theYoung’s mod- ulusoftheNi-16CrandNi-16Moalloysare242and216GPa,
respectively.TheseareingoodagreementwithYoung’smodulus valuesobtainedinthepresentstudybyfirstprinciplescalcula- tion.ThePoisson’sratioofalloyNi-16Moislargerthanthatof theNi-16Cr.ThesevaluesarealsolargerthanthePoisson’sratio ofpureNi(0.27).
Thevaluesof shearandbulkmodulus canalsobeutilized topredictthebrittleandductilebehaviourofmaterials(Pugh, 1954).Thiscanbeenvisagedbytaking theratio ofGandB.
Theratio(G/B)>0.57isassociatedwithbrittleotherwiserelated withductilebehaviour.TheG/Bratiosofbothalloysarereason- ably lowerthanthe 0.57. Thisindicates that thesealloys are ductile.Theseareinagreementwiththeexperimentalresultsof Mehtaetal.(2015).Theyhaverecentlyinvestigatedtheorien- tationdependentflowbehaviourofthesetwobinaryalloysand reportedhighductilityintensiletestedsamples.
5. Conclusions
Themainconclusionsare:
1. BasedonformationenergyperatomvaluesandBornstability criteriaintermsofelasticconstants,the␥phaseofbothalloys isstable.
2. Theequilibriumlatticeconstantvaluesofthesealloysarein goodagreementwiththeexperimentaldata.
3. TheNi-16MoalloyisenergeticallymorestablethantheNi- 16Cronewhile thelatteroneismechanicallymorestable thantheformerone.
4. Bothalloysshowanisotropicandductilebehaviour.TheNi- 16MoalloyexhibitshigheranisotropythantheNi-16Crone.
Conflictofinterest
Theauthorshavenoconflictsofinteresttodeclare.
Acknowledgements
TheauthorsaregratefultotheMinistryofDefenceofGov- ernmentofIndiaforfinancialsupport.Theauthorsareindebted toDirectorDMRLHyderabadforhisencouragement.Theyalso thankDirectorANURAG,Hyderabadfortheprovisionofcom- putationalfacilitiesandDr.R.SankarasubramanianandShriA.
Mondalfortheirkindsupport.
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