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Journal of Applied Research and Technology
www.jart.ccadet.unam.mx JournalofAppliedResearchandTechnology15(2017)449–453
Original
Mechanical properties of Ni-based solid solution alloys:
A first principles study
Ashish Pathak, Ashok Kumar Singh
∗DefenceMetallurgicalResearchLaboratory,Kanchanbagh,Hyderabad,India Received7October2016;accepted4May2017
Availableonline18October2017
Abstract
PresentworkdescribesthestructuralstabilityandmechanicalpropertiesofNi-basedbinaryandternaryalloys.Thesealloyshavebeenevaluated usingfirstprinciplesdensityfunctionaltheory(DFT)withingeneralizedgradientapproximation(GGA).Theequilibriumlatticeconstantvalues ofthe␥phaseinthesealloysareingoodagreementwiththeexperimentaldataobtainedbyX-raydiffraction.Thevaluesofformationenergyper atomoftheternaryalloysareconsiderablylowerthantheNi–16Crbinaryalloy.Thesealloyssatisfythemechanicalstabilitycriteriaintermsof elasticconstantsandpossessductilebehaviourbasedonsheartobulkratios.
©2017UniversidadNacionalAutónomadeMéxico,CentrodeCienciasAplicadasyDesarrolloTecnológico.Thisisanopenaccessarticleunder theCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Keywords:Ni-basedalloys;Firstprinciplescalculation;Elasticconstants;Formationenergy;Latticeconstant
1. Introduction
The main alloying elements of the important commercial Ni-basedsolidsolutionstrengthenedalloysareCr,Mo,Wand Fe(Bradley,1988;Davis,2000;Donachie,1984;Sims,1987).
Amongthese,Ni–CrandNi–Mobinaryalloysappeartobeideal systemsforsolid-solutionstrengtheningintermsofatomicsize andvalencydifference(z)(Delehouzee&Deruyttere,1967).
TheNi–CrandNi–Moalloysdisplayhighandlowstackingfault energy(SFE)values,respectively. Inaddition,theNi–Crsys- temshowslowerdegreeofsolidsolutionstrengtheningthanthe Ni–Mo(Mehta,Mukhopadhyay,Mandal,&Singh,2015a).The largedifferenceinSFEhasmarkedinfluenceonstrengthening mechanismandflowbehaviourofthesealloys.
Acorrelationamongmicrostructure, texture andmechani- calbehaviourofaternaryNi–16Cr–16Mohasbeenestablished recently(Mehta,Mukhopadhyay,Mandal,&Singh,2015b).It istobenotedthatthealloyNi–16Cr–16Mopossessesanopti- mizedeffectof boththe factorsi.e. SFE anddegreeof solid solutionstrengthening.TwootheralloysnamelyNi–16Cr–8Fe
∗Correspondingauthor.
E-mailaddress:singh[email protected](A.K.Singh).
PeerReviewundertheresponsibilityofUniversidadNacionalAutónomade México.
andNi–16Cr–4W havealsobeen investigated recentlywhich have modest values of SFE lie between Ni–16Cr–16Mo and Ni–16Cralloys(Mehtaetal.,2015b).Interestingly,theatomic sizeandvalencydifferenceof Ni–MoandNi–Ware moreor lesssame(Delehouzee&Deruyttere,1967).Itisimportantto mentionheretheatomicsizeofFeissimilartoNianditsaddi- tiontoNi–16CralloyhasmarginaleffectonSFE.Incontrast, thevalencydifferencebetweenNi–CrandNi–Feisquitelarge.
ThefirstprinciplescalculationsofNi-basedalloyshavebeen carriedoutbyseveralinvestigators(Chandran&Sondhi,2011;
Pathak,Mehta,&Singh,2017;Tsi,Morozov,Yu,Merinov,&
Goddard, 2016). These studies have been focused on differ- entalloysaswellasondifferentaspects.Pathaketal.(2017) haverecentlystudiedfirstprinciplescalculationofNi–Crand Ni–Mobinaryalloys.They haveobservedthatthe␥phaseof boththealloysarestableandfollowBornstability criteriain termsofelasticconstants.Boththealloysareductileanddisplay anisotropicbehaviour.
Presentworkisthusconcernedwithafirstprinciplesstudyof theNi–16Cr,Ni–16Cr–16Mo,Ni–16Cr–8FeandNi–16Cr–4W alloys. It istobe notedthat thisstudy is anextension of the Ni–16Crbinaryalloy(Pathaketal.,2017).Themajoralloying elements of the present alloys are Cr, Mo, W andFe which impart different degree of solid-solution strengthening and SFE.Thetwocommercialalloys,HastelloyC-276andInconel
http://dx.doi.org/10.1016/j.jart.2017.05.006
1665-6423/©2017UniversidadNacionalAutónomadeMéxico,CentrodeCienciasAplicadasyDesarrolloTecnológico.Thisisanopenaccessarticleunderthe CCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Table1
Equilibrium lattice constants and formation energy/atom for Ni–16Cr, Ni–16Cr–16Mo,Ni–16Cr–8FeandNi–16Cr–4Walloys.
Alloys Latticeconstants(Å) Formationenergy/atom(eV) Experimental Presentstudy
Ni–16Cr 3.5366 3.5040 −0.0060
Ni–16Cr–16Mo 3.5905 3.5667 −0.0125
Ni–16Cr–8Fe 3.5488 3.5190 −0.0115
Ni–16Cr–4W 3.5487 3.5455 −0.0194
600comprisetheseelements.Theselectedalloysarethepartof aprogramtodevelopnewalloyswithsimilar/betterproperties ofthecommercialalloysasmentionedabove.Thecomparative stabilityofthe␥(fccmatrix)inthesealloyshasbeenexplained.
The latticeparameter valueshave been compared withthose observed by X-ray diffraction technique. The mechanical propertiesinterms ofelastic constantsof thesealloysexcept Ni–16Cr–4Warealsoevaluated.
2. Methodology
TheQuantumEspressosoftwarebasedonfirstprinciplesden- sityfunctionaltheory(DFT)utilizingpseudopotentialtechnique hasbeenemployed(Giannozzietal.,2009;Hohenberg&Kohn, 1964;Kohn&Sham,1965;Vanderbilt,1990).Thegeneralized gradientapproximation(GGA)withPerdew–Burke–Ernzerhof formulation has been used to include exchange-correlation effects (Perdew, Burke,& Ernzerhof, 1996).The plane wave cut-off energy has been verified for convergence for all the alloysandaccordinglyplanewavecut-offenergyof40Ryhas beenused.Afterwards,fastfouriertransformalgorithmhasbeen taken up for the conversions of wavefunctions between real andreciprocallattices(Goedecker,1997).Inviewofthat,con- jugate gradient algorithm (Gonze,1996; Payne, Teter, Allan, Arias,&Joannopoulos,1992)hasbeenusedwithintheframe workofself-consistency.TheintegrationovertheBrillouinzone (BZ) has been carried out with the Monkhorst-Pack scheme (Monkhorst&Pack, 1976).The convergence withrespect to k-points has been verified for different alloys. The optimum values used for the present calculations are 10×10×10 for Ni–16Cr,Ni–16Cr–8Fe,Ni–16Cr–8Moalloyswhile4×4×4 forNi–16Cr–3W.Thestructuraloptimizationsofallthealloys for obtaining the minimumenergieshavebeen performed by relaxingthelatticeconstantofcorrespondingfcclattice.Inorder tocapturethedisorderedstructureeffectively,alloyingelements (Cr, Mo, Fe and W) havebeen placed randomly atdifferent locations within the supercell and the system energies have been computed.All the alloys withminimum systemenergy have been considered for further calculation. Convergence in present calculation has been obtained when the differ- encesbetweenenergiesintwoconsecutivestepsare lessthan 1.36×10−4meV.
Boththeformationenergyperatomandthelatticeconstants ofallthealloyshavebeencomputedandgiveninTable1.The energyofformationperatomhasbeendefinedas
Efor= 1
(x+y+z)(ENi−yCr−zN−xENi−yECr−zEN) (1) where,ENi–yCr–zNisthetotalsystemenergyofNi–yCr–zNalloy, Ni–yCr–zNhaving‘x’Ni-atoms,‘y’Cr-atomsand‘z’N-atoms, ENi,ECr,ENarethetotalenergyperatomintheirgroundstates forNi,CrandN(Mo,FeorW)atoms,respectively(x+y+z) denotesthetotalnumberofatomsconsideredinthesupercell.
Thenumberofatomsconsideredinthepresentcalculationsfor Ni–16Cr, Ni–16Cr–8Fe,Ni–16Cr–8MoandNi–16Cr–3W are 12, 12, 12 and32 atoms, respectively. The ElaStic code has beenutilizedtocalculatetheelasticconstants(Golesorkhtabar, Pavone,Spitaler,Puschnig,&Draxl,2013).
3. Resultsanddiscussion
Thecommonalloyingelementswithlargewt.%inNi-based solid-solutionalloysaremainlyCrandMo.ThecontentsofCr andMolieintherangeof10–20%inmostofthecommercial Ni-based solid solution alloys apart from other alloying ele- ments. Itisknownthataround16%ofboththeCrandMoin Niimprovesthealloythermalstability,fatiguelife,hightem- peraturestrength,lowexpansioncharacteristics,oxidationand corrosion resistance (Tawancy&Al-Hadhrami, 2012). Based onthis,theNi–16CrandNi–16Cr–16Mohavebeenconsidered for the present study.It hasbeen mentionedinliterature that the optimum compositionsof the Ni-basedalloyshould con- sistofanequalamountsofCrandMo(Tawancy&Asphahani, 1984).Inaddition,thetwootherternaryalloysi.e.Ni–16Cr–4W and Ni–16Cr–8Fe have also been investigated in the present study which has moderate values of SFE that lie between Ni–16Cr–16MoandNi–16Cralloys.TheamountofWiskept 4wt.%duetoitslimitedsolidsolubilityinNi.Interestingly,these alloysalsoformthecompositionalsubsetofseveralcommercial alloyssuchasHastelloyC-276,Inconel600etc.(Bradley,1988;
Davis,2000;Donachie,1984;Sims,1987).
A firstprinciples totalenergy calculationhasbeen carried outforboththebinaryNi–16Crandternary(Ni–16Cr–16Mo, Ni–16Cr–8Fe andNi–16Cr–4W) alloys. The fcc structure of thesealloyshasbeeninitiallyoptimizedbyvaryingthelattice constantvalueasafunctionofenergy.Thecorrespondingequi- librium latticeconstantforeach alloyhasbeenobtainedwith minimumenergy.Thelatticeconstantsofalltheternaryalloys havebeenmeasuredusing XRDtechniqueandcorresponding XRDpatternsareexhibitedinFigure1.TheXRDpatternsreflect marginal shift in peak positions indicating change in lattice parameter with different alloys. The equilibrium lattice con- stantvalues(Table1)areingoodagreementwiththeavailable experimentaldata(Mehtaetal.,2015a,2015b).
Thelatticeconstantvaluesofthepresentsolidsolutionalloys display anincreasefromthe pureNi(3.5239 ˚A)although the extentofincreasevarieswiththeatomicradiiaswellasthecon- tentofalloyingelements.TheatomicradiioftheelementsNi,
30 40 50 60 70 80 90 100 2θ ( ° )
Ni-16cr-8Fe
Ni-16cr-8Fe
Ni-16cr-4w
Ni-16cr-4w Ni-16cr-16Mo
Ni-16cr-16Mo
(111) (200) (220)
(311)
15 10
0 20 25
Intensity (X 1000 counts)
5
Fig.1. XRDpatternsoftheternaryalloys:(a)Ni–16Cr–16Mo,(b)Ni–16Cr–4W and(c)Ni–16Cr–8Fe.
Table2
Elasticconstantsand anisotropyfactor(A) forternaryNi–16Cr–16Mo and Ni–16Mo–8Fealloys.
Alloys C11(GPa) C12(GPa) C44(GPa) A
Ni–16Cr 310 178 141 7.0
Ni–16Cr–16Mo 301 175 120 9.3
Ni–16Cr–8Fe 290 179 118 9.5
Cr,Mo,FeandWare1.62,1.85,2.01,1.72and2.02 ˚A,respec- tively.The marginalincreaseinlatticeparameterof Ni–16Cr alloy can therefore be ascribed to the size of Cr. The alloy Ni–16Cr–16Moexhibitslargestvalueoflatticeconstantamong the presentalloys. Thiscan also beattributed toacombined contributionofatomicsizesofboththealloyingelementsi.e.Cr andMoinsolidsolution.Asmentionedabove,theatomicradii oftheseelementsarelargerthantheNi.Itistobenotedthat theatomicradiiofWisslightlyhigherthantheMo.Thelattice parameterofthealloyNi–16Cr–4Wissmallerthanthatofthe Ni–16Cr–16Mo.ItisknownthatthesizedifferenceofNiand Wisquitelarge(10.4%)butitseffectonlatticeconstantofthe alloyNi–16Cr–4WissmallduetosmallcontentoftheW.
The values of formation energy per atom of the alloys obtainedbyfirstprinciplecalculationaregiveninTable1.These arenegativeforallthebinaryandternaryalloys.Thisindicates thatthe ␥phaseof thepresent alloysisstable.The valuesof formationenergy per atom of ternary alloysare significantly lowerthanthe Ni–16Cralloy.It isimportanttomentionhere thatthe ternaryphase diagramsofNi–Cr–Mo, Ni–Cr–Feand Ni–Cr–Wsystemsexhibitreasonablywidesolubilityrangeof␥ phase(Bloom&Grant,1953;Nash,1991;Turchi,Kaufman,&
Liu,2006).Theformationenergyperatomvaluesalsoreflect thatthealloyNi–16Mo–4Wismorestablethanallotheralloys havinglowestamongothers.
Theelasticpropertiesofasinglecrystalofallthealloyshav- ingfcccrystalstructuresaredescribedbyelasticconstantsC11, C12,C44 andgiveninTable2.Thevaluesofelasticconstants of Ni–16Cr–4W could not be calculatedduetolarge system sizeaswellaswithlimitedcomputationalresource.Theelastic constantsvaluesexcept C12 reflect a marginaldecrease from binary to ternary alloys. The values of C11 are significantly higherthanthoseofC12 andC44 inallthealloys.Thenature ofmetallicbondingofcubicphasescanbepredictedbasedon Cauchypressures(Olijnyk&Jephcoat,2000).Thesearedefined
asC12–C44<0forcubicmaterials.ThenegativeCauchypres- sureindicatesmoredirectionalbondingwhereaspositivevalue exhibitsmetallicbonding.ThecalculatedvaluesofCauchypres- suresfortheNi–16Cr,Ni–16Cr–16MoandNi–16Cr–8Fealloys are 37, 55 and 61, respectively. These values clearly points towardsthepresenceofpredominantmetallicbondsinpresent alloys.Theextentofmetallicbondingincreasesfrombinaryto ternaryalloysandisthehighestforNi–16Cr–8Fe.
Thevaluesofelasticconstantscanbeutilizedtoenvisagethe mechanicalstabilityofthe␥phase.TheBornstabilitycriteri- onsforcubicmaterialsaregiveninEqs.(2)–(4)(Born,1940;
Fedorov,1968).Thecorrespondingcalculatedvaluesinunitsof GPaaregiveninparenthesisforNi–16Cr,Ni–16Cr–16Moand Ni–16Cr–8Fealloys.
C44 >0(141,120,118) (2) C11−|C12|>0(132,126,111) (3) and C11+2C12>0(666,651,648) (4) Thevaluesofelasticconstantsofthesealloysgiveninparen- thesissuggestthatmechanicalstabilityofthe␥phaseinthese alloysis quite high. Thisisalso inagreement with available experimental phase diagram wherein thesephases have been observedeitherinsingleortwoand/orthreephasefieldregions (Bloom&Grant,1953;Nash,1991;Turchietal.,2006).
The anisotropy factors (A) for all the alloys are given in Table2.
Forcubicmaterials A= 2C44
(C11−C12) (5)
The valuesofanisotropic factorsshouldbe1 forisotropic materials and away from 1 point towards the presence of anisotropyinelasticconstants.Thevaluesofanisotropicfactors obtainedinpresentalloysarefarawayfrom1.Thisindicatesthat theelasticanisotropyispresentinthesealloys.Itistobenoted that the single crystalwith fcc structure hasdifferent atomic arrangements along different crystallographic directions. The atomicpackingismaximumalong111whichismostclose packeddirection.Theextentofanisotropyincreasesfrombinary toternaryalloys.Boththeternaryalloysexhibitcomparableval- uesofA.Interestingly,thesealloysalsoexhibitthepresenceof plasticanisotropyinpolycrystallineforminhotrolledcondition (Mehtaetal.,2015a,2015b).
The effective elastic modulus of isotropic polycrystalline cubicmaterialscanbepredictedfromtheelasticconstantsbyfol- lowingtwoapproximations.Thesearenamely,theVoigt(Voigt, 1928)andReuss(ReussandAngnew,1929)thatprovideinfor- mation about the upper and lower limits. These are defined as
B=BV=BR=C11+2C12
3 (6)
GV= C11−C12+3C44
5 (7)
Table3
Polycrystallineelasticconstantssuchasbulkmodulus(B),shearmodulus(G) andYoung’smodulus(E)andPoisson’sratio(υ)forNi–16CrandNi–16Mo alloys.
Alloys B(GPa) GV(GPa) GR(GPa) G(GPa) E(GPa) G/B υ
Ni–16Cr 223 111 96 104 270 0.47 0.30
Ni–16Cr–16Mo 213 100 83 92 240 0.43 0.31
Ni–16Cr–8Fe 209 99 80 89 235 0.42 0.31
GR= 5(C11−C12)C44
4C44+3(C11−C12) (8)
whereBisbulkmoduluswhileGVandGRareshearmodulus values obtained by Voigt and Reuss approximations,respec- tively.
Theaveragevalueofthesetwoestimatesasmentionedabove is givenbyHill (1952)approximation for all thealloys. 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)
where BH, GH, E and ν are bulk modulus, shear modulus, Young’smodulusandPoisson’sratio,respectively.
Thecalculatedvaluesofbulk, shearandYoung’smodulus ofallthealloysaregiveninTable3.ThebinaryNi–16Cralloy exhibitsmaximumvalues.Modulusvaluesdecreasefrombinary toternary alloys.The alloyNi–16Cr–8Fedisplays lowerval- ues of B, G and E than the Ni–16Cr–16Mo. It appears that the valuesof bulk modulus obtained inpresent alloysfollow Hill’sapproximationsincethesearesameinVoigtandReuss approximations.Ontheotherhand,shearmodulusvaluesofall thealloysobeyVRHapproximationwhereinVoigtandReuss describe upper andlower bounds of the shearmodulus. The Hill’sapproximationontheotherhandprovidestheaverageof thesetwo.
Thevaluesof shearandbulkmodulus canalso beutilized topredictthebrittleandductilebehaviourofmaterials(Pugh, 1954).Thiscanbevisualizedbytakingtheratiosofsheartobulk modulus.TheratioG/Band<0.57isassociatedwithbrittleand ductilebehaviour,respectively.TheG/Bratiosofpresentalloys aresmallerthanthatof0.57.Thisreflectsthatthesealloysare ductile. It istobe noted that allthe three alloyshaveshown reasonablygoodductility(largeuniformelongationvalues)in hotrolledandannealedconditions(Mehtaetal.,2015a,2015b).
ThelargerdecreaseinthevaluesofG/Bratiosinpresentstudy can be ascribedto the presence of Cr and Moelements that decreases SFE significantly inNi-based solid solution alloys (Table3).It isknown that the decrease inSFEincreasesthe ductilityofthealloys.
4. Conclusions
1. The structural stability and mechanical properties of Ni- based binary and ternary alloys have been evaluated for thefirsttimeusingfirstprinciplesdensityfunctionaltheory (DFT)withingeneralizedgradientapproximation(GGA).
2. Thevaluesofformationenergyperatomoftheternaryalloys aresignificantlylowerthantheNi–16Crbinaryalloy.
3. Theequilibriumlatticeconstantvaluesofthe␥phaseinthese alloysareingoodagreementwiththeexperimentaldata.
4. Thesealloyssatisfythemechanicalstabilitycriteriainterms ofelasticconstants.
5. TheG/Bratiosofthesealloysindicatethepresenceofrea- sonablygoodductility.
Conflictofinterest
Authors are grateful to Ministry of Defence, Government ofIndiaforfinancialsupport.TheauthorsareindebtedtoThe Director,DefenceMetallurgicalResearchLaboratory(DMRL), Hyderabadforhisencouragement.Theyintendtheirthanksto DirectorAnurag,Hyderabadfortheprovisionofcomputational facilities andDr.R.Sankarasubramanian andDr.K.K.Mehta fortechnicalsupport.
Theauthorshavenoconflictsofinterest.
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