Effect of stress on vacancy formation and diffusion in fcc systems: Comparison between DFT calculations and elasticity theory

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Connétable, Damien

and Maugis, Philippe Effect of stress on vacancy

formation and diffusion in fcc systems: Comparison between DFT calculations

and elasticity theory. (2020) Acta Materialia, 200. 869-882. ISSN 1359-6454

Effect of stress on vacancy formation and diffusion in fcc systems:

Comparison between DFf calculations and elasticity theory

Damien Connétable

a

_,

_..

_{. Philippe Maugis}

b

•CJRJMAT UMR 5085, CNRS-INP-UPS, ENSIACET, 4 allée Emile Manso, BP 44362, Toulouse Cedex 4 F-31030. France

b Aix Marseille Univ, CNRS, IM2NP, Marseille, France

ARTICLE INFO

ABSTRACT

Keywords: Fcc systems Vacancy Stress effects DFT Elasticiry theory

This paper discusses the effect of stress on the solubility and diffusivity of vacancies using the elasticity theory of point defects. To support the discussion, results are compared with DIT calculations to ver­ ify model accuracy. The particular case of vacancies in aluminum is discussed in detail (DFf-elasticity), while three other metallic fcc systems -Ni, Cu and Pd - are discussed through the elasticity approach only. Different types of loading were considered: hydrostatic, multi-axial and shear stresses. ln the case of a uni-axial loading, two different directions were investigated: the first along a main crystallographic direction, i.e. (001], and the second perpendicular to the dense plane (111). ln order to quantify the effect of stress on diffusivity, the diffusion coefficient of each configuration was expressed for further calcula­ tions. By analyzing the symmetry break during the loading process, non-equivalent atomic jumps were identified and diffusion equations obtained. A multi-physic approach was carried out by combining first­ principles calculations, to study atomic-scale processes, and a multi-state formalism. to obtain exact dif­ fusion equations. Results show that elasticity accurately captures the effects of stress on vacancy diffusion and solubility and an application method is presented.

1. Introduction

The solubility and diffusion mechanisms of point defects are

key parameters in many fields, such as phase transformation

[ 1 ]

,

corrosion

[2]

. plasticity

[3]

, oxidation

(4]

. creep,

etc.

Vacancies are

particularly interesting because they are involved in many mecha­

nisms and determine physical properties. For instance, vacancies

participate in dislocation motions

[S]

. radiation damage healing

and in the diffusion of most atoms in solid solution in materials

[6]

. Numerical simulations, and specifically

Off

calculations, are

approaches that can be used to study them in detail.

ln most cases found in the literature, simulations are carried

out at ambient pressure, i.e. the lattice and atomic positions are

bath relaxed to obtain the atomistic parameters at zero pressure.

Formation and migration energies are thus computed stress free.

Still, other authors have carried out their simulations at a fixed

strain (with the same lattice parameters). This type of approach

necessarily leads to the use of simulations in which the stresses on

the system are inevitably different. The parameters thus obtained

are then used in equations to compute vacancy solubility or

dif-• Corresponding author.

E-mail addresses: damien.connetable@ensiacet.fr (D. Connétable),

philippe.maugis@im2np.fr (P. Maugis).

hnps://doi.org/10.1016/j.actamat.2020.09.053

fusivity, solute diffusion,

etc.

If thermal expansion is disregarded,

these data should be representative enough to describe atomistic

processes.

However, in real materials, internai or external stresses (or

strains) are not always equaJ to zero. During thermomechanical

processing or product use, internai or external stresses appear in

the grains and may reach a magnitude of a few GPa. Structural de­

fects (such as solute atoms, dislocations. precipitates, grain bound­

aries) also induce stresses inside the material. These stresses or

strains could therefore strongly impact aJI physical processes in

which the amount and mobility of vacancies are involved. lt is

therefore necessary to have a clear description of the effects of

stress and strain in order to accurately describe the behavior of

materials at the atomic scale. This

will

also help developing new

materials and the ever-finer interpretation of the phenomenologi­

cal behavior of matter.

This issue has been present in the literature for many years,

but it is only recently that approaches have been developed to ra­

tionalize the effect of stress on vacancies. These effects have been

studied for a long time only in the case of semiconductor systems

for which experimental data are available. The effect of stress has

been studied in the case of self-diffusion, vacancy diffusion, etc.

For example, Aziz

(7,8]

focuses on the effect of pressure on the

diffusivity of solutes and vacancies in semiconductors. He

formai-izedtheeffects ofhydrostaticpressureandbi-axialstress on dif-fusionusinga thermodynamicapproach includingatomic param-eters.Amore formal approachis tostudyvacancy orsolutes us-ingthe elasticitytheory.Clouet andVarvenne [9–11] rationalized theelasticitytheory byproposing a reliableapproach toevaluate accuratelyandeasily theelastictensor P whicharekey parame-terswithelasticconstantsofthesystem._P quantifiesrather pre-ciselytheelasticeffectsinduced bytheinsertion ofanatom ora defectina crystal. Moreover, such an approachcan also beused topredicttheeffectsofcomposition,temperatureandmechanical loadingonsolubilityanddiffusivity.Ithasbeencarriedoutinthe specific case of carbon insertion in iron [12,13]. This is interest-ingbecause,inabcc-Fesystem,carboninsertioninduces strongly anisotropicelastictensorwithhighvalues.

To includethe effectofstress in mattertransport, Trinkle de-veloped[14]amethodtotakestressintoaccountusingthe elastic-itytheory.Tchitchekovaetal.[12]alsodevelopedamethodto de-terminetheenergybarriersforinterstitialdiffusionunderastress field.

Themethodologypresentedhereinaftersucceedsintakinginto accounttheeffectofstressstatewithinfcc-metalsonvacancy sol-ubilityanddiffusivity. The impliciteffects oftemperature on lat-tice parameters are ignored for simplification purposes. To illus-trateour approach, thecase ofaluminum is studied indetail by comparingelasticitytheory andDFTsimulations. After validation, otherfccmetalswillbediscussed,i.e.Ni,CuandPdstructures,for whichonlytheelasticityapproachwillbegiven.

Inthisstudy,thedifferentloadsthatwereconsideredshouldbe sufficientlyrepresentativeofthemainstatesofstress.Thesimplest case,whichishydrostaticstress,butalsomulti-axialstresseswere examined.Twocasesofuni-axialloadingwereconsidered,thecase ofaloadingalongoneofthemainaxesofthefccphase,[001]for example,andinthedirectionofthedenseplane, i.e. [111].Thecase ofbi-axialloadingwasalsoconsidered.Ineachofthesecases,DFT calculationsarecompared tothe elasticitymodelin termsof va-cancyformationenergyandvacancymigrationenergy.Then, con-centrationanddiffusioncoefficientsarecalculatedtoquantifythe effectofstress.Thisisalsoanopportunitytodiscussvacancy coef-ficientsinsystemsundergoingadeformationbystudyingexplicitly the effect of deformation by characterizing the symmetry break. Thisworkshowsthattheelasticitytheoryisabletocapture qual-itativelyandquantitativelythephysicsoftheload.

The DFT procedure is described in Section 2 andthe elastic-itymodelinSection3.Preliminaryresultsarethereafterpresented anddiscussed in Section 4. To conclude, Section 5 examines the comparisonbetweenDFTcalculationsandelasticitymodelresults fordifferentloadings.

2. DFTApproach

2.1. Theoretical calculation method

Atomistic calculationswereperformedusingdensityfunctional theory (DFT) with VASP (Vienna ab initio simulation package) [15]. Self-consistentKohn-Sham equations [16]were solved using projector augmented wave (PAW) pseudo-potentials [17], in the framework of the Perdew-Burke-Ernzerhofexchange and correla-tion functional (PBE [18]). For Al, Cu and Pd, calculations were performedwithoutspin effects,contraryto nickelwhich exhibits a magnetism equal to about 0.72

μB

per atom. The plane-wave energy cut-off was set to 600 eV. Equivalent 12 _{× 12 × 12} Monkhorst-Pack meshes [19]were used to sample the first Bril-louinzoneofsuper-cells(2 × 2× 2).Additionalsimulationswere performedwithdenserk-meshgrids (upto 24 × 24 × 24)for theAlsystem. Thevalue ofthevacancyformationenergyisthen slightlymodified;thedifference,about10meVforthe2 × 2× 2

super-cell, remains negligible in view of the numerical cost. Re-gardingthevaluesoftheelastic tensorparameters, thedifference isevensmaller,lessthan0.1%.

Toperformacalculationforagivenloading(uni-axial,bi-axial, shear or hydrostaticstress), the components of the stress tensor

σ

ij were fixed based onthe loading duringrelaxation(super-cell shape, volumeandatomic positions). Forinstance,in thecaseof uni-axialloadingalongthe z direction,aseriesofcalculationswere conducted with different fixed values of

σ

zz stresses while the other stress components were set to zero. The internal energies (and vibrational energies for the hydrostaticloading) fora given stresswerethenobtainedthroughsimulations,forthesystemwith defect as well as for the reference state. For each configuration (with and without point defect), lattice parameters and energies werefittedasafunctionofstresswithasixth-orderpolynomial.

Consideringthetransformationsthatoccurunderconstant tem-perature T anduniformstresstensor

σ

,thethermodynamic equi-libriumisreachedwhentheGibbsenergyofthesystem G [

σ

, T ]is atitsminimum.Inthepresenceofvacancies, function G includes theGibbsenergyofformationofamono-vacancy, G f_[

_σ

_{,T ].} Forma-tionofavacancyinacrystalimpliestherelaxationofthe crystal latticearound thevacancy,thecreationofanad-atomatthe sur-face ofthe crystaland a changein thephonon spectrum.In the presentapproach,theGibbsenergyofformationthereforeconsists ofthreeterms:

Gf_[

σ

_,T]=Hrel_[

σ

_]+pV

at+Gfvib[

σ

,T]. (1) Thefirstterm H rel_[

_σ

_{]istheenthalpyofrelaxation:}

Hrel[

σ

]=Uf[

σ

]− V

σi j



rel

i j (2)

where U f_[

_σ

_{] isthechangeininternalenergyduetotheatom} re-laxation around the vacancy,



rel

i j is the uniform strain resulting fromtherelaxationand V isthevolumeofthesystem.Thesecond term pV at inEq.1isthework ofthesystemagainst theexternal pressure p whenthead-atomiscreated. p =−1

3

σ

ii isthepressure, and V atistheatomicvolumeofthelattice.Thelastterm G f_vib[

σ

, T ] isthechangeinvibrationalGibbsenergyduetothevacancy.Inthe linear elasticityapproach, the vibration effects willbe neglected, sothattheenthalpypartoftheGibbsenergyofformationwillbe expressedas:

Hf_[

σ

_]=Uf_[

σ

_{]− V}

σi j



rel

i j + pVat. (3) 2.2. Formation energies

FromDFTcalculations,therelaxationpartoftheinternalenergy ofvacancyformationinacrystalsubmitted toauniformstress

σ

canbedeterminedby:

Uf_[

_σ

_]=Ebulk+vac o [

σ

]− N− 1 N E bulk o [

σ

]. (4) E bulk+vac

o [

σ

]and E bulko [

σ

]are theDFT energiesofa super-cell com-posed of N atomic sites, with andwithout vacancy, respectively, computedforagivenstresstensor

σ

. G f

vib[

σ

,T ]correspondstothe phononcontributiontotheGibbsenergyofformationcalculatedat temperature T .Forsimplificationpurposes,itwasonlytakeninto accountforthehydrostaticloading.Itisgivenbythestandard re-lationship

Gf

vib[

σ

,T]=Gbulk+vacvib [

σ

,T]− N− 1

N G

bulk

vib [

σ

,T] (5)

wherethevibrationalGibbsenergyisexpressedby: Gvib[

σ

,T]=kBT  n,q ln



2sinh



h

ω

n,q[

σ

,T] 2kBT



. (6)

k B istheBoltzman’sconstant,

ω

n,qarethefrequenciesofallatoms inthewavevectorq andthemode n .Interatomicforceconstants

(IFC) were computed using the finite displacements method on 2 × 2× 2 super-cells,tolimit numericalcostsinduced by sim-ulations. G vib[

σ

,T ] wasthen computedusing 20 × 20 × 20q -meshes.

Inallloadingssubsequentlyconsidered,thereisonlyone non-equivalent vacancy. In the dilute approximation, the sitefraction of vacancies at equilibrium, C v, is therefore related to the Gibbs energyofformationthroughthewell-knownequation:

Cv[

σ

,T]=exp



−Gf[

σ

,T] kBT



. (7)

Quasiharmonicandanharmoniceffectsareneglectedherefor sim-plification purposes. However, it must be kept in mind that to haveanaccuratedescriptionofwhathappensathightemperature, theseeffectsmustimperativelybetakenintoaccountinthe calcu-lationoftheformationfreeenergyasshownbytherecent theoret-icalworks[20,21].Stresseffectsmustthereforebeaddedtothese effects.Butthisisoutofthescopeofthispaper.

2.3. Atomic jumps

Tostudymigrationprocesses,CLIMB-NEBsimulations[22]were conducted on2 _{× 2× 2} super-cells, containing32 atoms. The energy of the transition state was calculated using 600 eV and 12 × 12× 12k-meshes to samplethefirst-Brillouinzone. The migration enthalpywastakenasthe enthalpydifference between thetransition-stateandtheinitialconfiguration.Thephonon con-tributionstotheGibbsenergywerecomputedfortheinitialstates (is) and the transition states (ts) to compute jump rates. The phonopy package was then used [23] to compute inter-atomic forces.

Theelementarymechanismofan atomicjumpoccursata fre-quency



expressedby[24,25]:



[

σ

,T]= kBT h Zts Zis exp



−Hm[

σ

] kBT



. (8)

Inthisequation, H m_[

_σ

_{]isthemigrationenthalpyofthejump, i.e.} theenthalpydifferencebetweenthetransitionstateandtheinitial state.Itisexpressedas:

Hm_[

σ

_]=Um_[

σ

_{]− V}

σi j



m

i j (9)

where U m_=U f

₍

_ts

₎

_{− U}f

₍

_is

₎

_{is the migration energy and}

_

m₌



(

ts

)

−



(

is

)

isthestrainofmigration. Z isthe partitionfunction linkedtotheGibbsfreeenergyofvibrationthrough:

Gvib=−kBTlnZ. (10)

Therestofthestudyconsidersthattheratio_ZTS/ZISdependslittle ontheloading.

3. Linearelasticityapproach

Equations presented below are inspired from the work of Maugis etal. [13]and derive fromthe linear elasticitytheory of point defects[26,27].Theeffectofan appliedstrain orstresscan be achievedusingtwo quantities:the elasticdipoletensorofthe point defect(vacancy ortransitionstate),P, andtherank-4 stiff-ness tensorofthe lattice,C.Theywere bothcomputedusingDFT calculations,asdetailedinSection4.

Whenastress

σ

isappliedtoacrystalofvolume V containing a point defect,the resultinguniformstrain of therelaxed crystal is,withreferencetothestress-freedefect-freecrystal:

i j

=Si jkl



σkl

+Pkl V



(11) where S=C−1 is the compliance tensor. In thisequation, S i jkl

σ

kl istheelasticresponse ofthedefect-freecrystaland S i jklPVkl isthe

relaxationstrain



rel

i j ofthedefect.Underanuniformstrain



ij,the followingquantityneedstobeaddedtotheinternalenergy: Uel₌ 1

2V

i j

Ci jkl

kl

− Pkl

kl

. (12) Thesecondterminthisequationaccountsforthedefect-strain in-teraction. Bycombining Eqs 11 and12, it can be found that the elastic contribution to theformation energy of the defectequals −1

2VPi jS i jklPkl.Thisquantity can beviewedasacorrection ofthe formation energy when performing DFT calculationson a super-cellof finitevolume V .Its value doesnot depend onthe applied stress,thereforeaccordingtotheelasticitytheorythefollowing re-lationshipcanbeestablished

Uf_[

_σ

_]=Uf_{[0]. (13)}

Thiswillbeverifiedlaterinthisstudy.

Consider now the volume of relaxation tensor of the defect Vkl=Pi jS i jkl andthescalarvolumeofrelaxation V rel=tr

(

Vkl

)

. Us-ingtensorV,theeffectofstressontheenthalpy functioncanbe computed, provided the vibrationaleffects are neglected. For in-stance, fromEq. 2 the stress-dependentrelaxation enthalpy ofa vacancyis:

Hrel_[

_σ

_]=Uf_{− V}

kl

σkl

. (14)

Itcanbenoticedthat thestress-freeenthalpy H rel_{[0]corresponds} totheformationenergy U f_{.Eq.14definesthechangeinrelaxation} enthalpyresultingfromtheappliedstress:



Hrel=−Vkl

σkl

. (15) Similarlythechangeinformationenthalpyofavacancyis



Hf_=−V

kl

σkl

+pVat. (16)

UsingEq.7theeffectofstresson theequilibriumsitefractionof vacanciesisgivenby:

CV[

σ

,T]=CV[0,T]exp



−



Hf[

σ

] kBT



. (17)

Regardingatomicjumps,similarlytotheabovementioned rea-soning,themigrationenergyisstress-independent:

Um_[

_σ

_]=Um_{[0], (18)}

andthe migration enthalpy is affected by the applied stress ac-cordingto:



Hm₌₋

_

_V

kl

σkl

(19)

where



V kl=



P i jS i jkl isthemigration volumetensor. We intro-ducedthedifferenceinelasticdipolebetweenthetransitionstate, Pts_{,andthestablestate,P}vac_:

_

_P

i j=Pi jts− P vac

i j .Ifthe transition-statehadanisotropicsymmetry,themigrationvolumetensor



V kl wouldbereducedtothescalartensor 1₃V m_{I,where V} m_=tr

_(

_V

kl

)

is thevolume of migration. This isnot the casehere, as willbe showninwhatfollows.The effectofstress onjump frequencyis givenby



[

σ

,T]=



[0,T]exp



−



Hm[

σ

] kBT



. (20) 4. Preliminaryresults

4.1. Ground state properties of fcc systems

As seen above, different parameters are required to use the elasticitytheory,inparticulartheelasticconstantsandlattice pa-rameters of the perfect crystal. They were computed from the primitivecells(for elastic constantsthe linearresponse was em-ployedusingVASP). Resultsare summarizedinTable1and com-paredtotheliterature.

Table 1

Comparison between theoretical/experimental data of lattice parameters ao , in ˚A, and elastic constants Cij , in GPa, of Al, Cu, Ni and Pd

systems. The bulk modulus, B = (C11 + 2C 12)/ 3, and shear modulus, C = (C11 − C 12)/ 2, are also given.

ao C11 C44 C12 C B Al 4.04/4.05 [28] 105/116 [28] 33/31 [28] 65/65 [28] 22/26 78/82 [28] Cu 3.62/3.61 [28] 193/168-173 [29,30] 85/61-76 [29,30] 134/118-129 [29,30] 29/24 [29,30] 154/138 [28] Ni 3.52/3.52 [28] 272/253 [31] 125/122 [31] 158/158 [31] 57/47 [31] 196/189 [31] Pd 3.94/3.87 [32] 202/234 [32] 60/71 [32] 153/176 [32] 25/29 [32] 169/181 [28] Table 2

Stress-free crystal: vacancy formation energy ( U f ), migration energy in first- and second-

nearest neighboring position (labeled U m

1nnand U 2mnn ), in eV; formation and migration vi-

bration enthalpies ( H f

vib and H vmib ), in meV. For the 54-atoms case, the rhombohedric rep-

resentation was used.

nb super-cell k -mesh Uf _Um 1nn U2mnn Hvfib Hvmib atoms size 32 2 × 2 × 2 12 × 12 × 12 0.628 0.570 2.325 -8 -26 54 3 × 3 × 2 10 × 10 × 15 0.660 0.549 - - -108 3 × 3 × 3 10 × 10 × 10 0.639 0.595 2.389 -2 -216 6 × 6 × 6 8 × 8 × 8 0.636 0.607 - -

-DFT results(latticeparameters,elasticconstants)ofallfourfcc systemsstudiedarefoundinexcellentagreementwith experimen-talandtheoreticalliterature [28–30,32].Inthecaseofpalladium, Amin-Ahmadi et al. [33] alreadyshowedthat thegroundstate of fcc-Pdismagnetic,incontradictionwithexperimentalfindings.For thatreason,itwasdecidedtoconsiderPdasnon-magneticfor fur-thercalculations.FurtherdiscussionswillbebasedontheDFT val-uesfoundinthisstudy.

4.2. Vacancy characteristics

The Al system is considered representative enough to be fo-cusedon.Thefollowingsectionsummarizestheadditional param-etersusedintheelasticmodel, i.e. vacancyformationenergy, mi-grationenergy, etc .As theAl systemwillbe stressed,itis impor-tanttobesurethatthecommondescriptionofdiffusion parame-terswillbepreciseenoughtodescribethegroundstateproperties ofthevacancy.

Thesizeeffectofthesuper-cellwasfirstconsidered.Testswere therefore conducted. DFT results, formation and migration ener-gies, forthree sizesofsuper-cells are giveninTable 2. Itcan be notedthat,atthisstageofapproximation,theeffectofthe super-cell size on the relaxation energy is small,within a range of 10 meV.The use ofsmall super-cellswasconsidered to be accurate enough.The vacancyformationenergyis equaltoabout0.64eV, inagreementwithearlierworks[20,34,35].Thisvaluecorresponds tothelowtemperaturevalueofthevacancyformationenergy,as alreadyexplained,an-harmonic effectsare alwaysneglected here, seeGlensketal.[20].

Todescribediffusionattheatomicscale,itwasconsideredthat thevacancyjumpstoitsfirst-nearestneighboringAlatom:witha jumprate



12,asdepictedinFig.1.

Themacroscopicdiffusioncoefficient D ofthevacancyis there-foregiven in terms of elementary parameters: lattice parameter, a ,andfrequency,



12.Bothdependon stress(for instance p ) and temperature(T ). Asa functionof p and T (inthiscase, thestress doesnotchangethesymmetryofthesystem), D isthusgivenby:

D[p,T]=a2_[p,T]

_12

_{[p,T]. (21)}

In addition to the usual jump,



12, the jumps towards the second-nearest Al neighbor, labeled



2nn, were also considered. Equationofdiffusion21shouldthus bemodifiedinto(for simpli-ficationpurposes, p and T wereomitted):

D=a2

₍

_12

₊

_2

nn

)

. (22)

Theeffectofsuper-cellsizeonmigration energies(to 1nn and 2nn sites) wasstudied. Atambient pressure(i.e. p =0GPa), NEB calculations were done forboth trajectories.Results are summa-rized in Table 2. In both jumps, the trajectory is direct and the transitionstate islocated inthe middleofthe path(as inall fcc systems).Resultsalsoshowthatthemigrationenergytowardsthe 2nn positions, about 2.3 eV, is significantly higher than for the direct jump,0.6eV. From a probability standpoint,



2nn is much smallerthan



12andcanbeneglectedinfirstapproximation,even understress.Inwhatfollows,onlyfirst-nearestneighboringjumps werethereforeconsidered.

Resultsforthe1nn jumparefoundinexcellentagreementwith experimental and theoretical findings. Mantina et al. [34] (and references therein) for instance obtained a value of 0.6 eV. In Appendix A,thevacancy concentration anddiffusivitywere plot-tedusingDFTvalues.

Data (formation and migration energies) for Ni [36,37], Cu [20]andPd[38,39],Table3,arefoundinexcellentagreementwith theoretical literature. In the case of Ni and Cu [40], the agree-ment with theexperimental data is rathergood, but not forPd. Forpalladium, the experimental valuesof vacancy formation en-ergymeasured athightemperatureisatleast0.5eV higherthan ours(calculatedat0K)[41,42].Thisdifferencecanbeexplainedas inthecaseofaluminum.The anharmoniceffectsathigh temper-atureshould haveastrong impactandthusmodify thevaluesof thehightemperatureformationenergies.Consequently,therestof thestudyconsidersthatvacancycharacteristicsresultingfromDFT resultswillbeconsideredaccurateenoughtocapturetheeffectof stress.

4.3. Elastic dipoles

In order to complete the basic properties, elastic dipole ten-sors,_P,were computed. Todoso,firsttheperfectsystem (with-out defect)wasoptimized andthen one vacancywasintroduced. Atomicforceswerethusminimizedwithoutrelaxingtheshapeand volume of the cell. The residual stress tensor,

σ

res_{, was used to} computeP, asdescribedbyVarvenne[10].Largesuper-cellswere used, i.e. up to 216 atoms, see Table 3. As explained above, two tensorsareneeded:oneforthestablestateofthevacancy,and an-other forthetransitionstate.ThedipoletensoroftheAl vacancy intheconventionalcubicframe,iswrittenas(ineV):

Pvac₌

_{−2.62 0 0} 0 −2.62 0 0 0 −2.62

=−2.62I, (23)

Fig. 1. From left to right: detail of atomic jumps for hydrostatic ( p ), [001] ( σzz ) and [111] ( σ111 ) uni-axial stresses, and for shear stress in plane [010] ( σxy ). Letters on figures

correspond to length of boxes.

Table 3

DFT values according to the super-cell size: elastic dipole tensor P (in eV), image interaction energy E d (meV), volume of relaxation V rel (in ˚A 3 ),

migration energy, U m in eV, formation energy U f (in eV) and vibration energy U f

vib (in meV). Values from the elasticity theory: volume tensor of

relaxation V and volume of relaxation V rel (in ˚A 3 ). In the case of the Al system, a 6 × 6 × 6 super-cell generated from the primitive cell, labeled p ,

was also used. I is the identity matrix.

super-cell size P (DFT) V (elast.) Ed Vrel (DFT/elast.) Uf Uvibf Um

Al vac 2 × 2 × 2 -2.31 I -1.57 I 6 -5.01/-4.71 0.642 -8 3 × 3 × 3 -2.52 I -1.72 I 2 -5.17/-5.15 0.639 -2 6 × 6 × 6 p _{-2.62 I -1.79 I 1 -5.82/-5.36 0.636} -ts 2 × 2 × 2 ⎛ ⎝−1 . 72−0 . 38 −0 . 38−1 . 72 0 . 000 . 00 0 . 00 0 . 00 1 . 42 ⎞ ⎠ ⎛ ⎝−4 . 66−0 . 92 −0 . 92−4 . 66 0 . 000 . 00 0 . 00 0 . 00 7 . 94 ⎞ ⎠ 19 -1.78/-1.38 1.212 -26 0.570 3 × 3 × 3 ⎛ ⎝−2 . 12−0 . 19 −0 . 19−2 . 12 0 . 000 . 00 0 . 00 0 . 00 1 . 89 ⎞ ⎠ ⎛ ⎝−5 . 89−0 . 47 −0 . 47−5 . 89 0 . 000 . 00 0 . 00 0 . 00 10 . 18 ⎞ ⎠ 9 -1.56/-1.60 1.234 - 0.595 6 × 6 × 6 p ⎛ ⎝−2 . 29−0 . 21 −2 . 29−0 . 21 0 . 000 . 00 0 . 00 0 . 00 1 . 62 ⎞ ⎠ ⎛ ⎝−5 . 89−0 . 50 −0 . 50−5 . 89 0 . 000 . 00 0 . 00 0 . 00 9 . 77 ⎞ ⎠ 2 -2.62/-2.02 1.243 - 0.607 Cu vac 2 × 2 × 2 -3.59 I -1.25 I 12 -3.87/-3.74 1.065 -8 3 × 3 × 3 -3.68 I -1.28 I 4 -3.53/-3.84 1.039 -ts 2 × 2 × 2 ⎛ ⎝−3 . 69−0 . 80 −0 . 80−3 . 69 0 . 000 . 00 0 . 00 0 . 00 1 . 45 ⎞ ⎠ ⎛ ⎝−5 . 34−0 . 76 −0 . 76−5 . 34 0 . 000 . 00 0 . 00 0 . 00 8 . 62 ⎞ ⎠ 12 -2.62/-2.07 1.753 -19 0.688 3 × 3 × 3 ⎛ ⎝−4 . 09−0 . 44 −0 . 44−4 . 09 0 . 000 . 00 0 . 00 0 . 00 1 . 49 ⎞ ⎠ ⎛ ⎝−5 . 83−0 . 41 −0 . 41−5 . 83 0 . 000 . 00 0 . 00 0 . 00 9 . 33 ⎞ ⎠ 4 -2.54/-2.32 1.779 - 0.740 Ni vac 2 × 2 × 2 -4.41 I -1.20 I 12 -3.81/-3.60 1.445 -8 3 × 3 × 3 -4.67 I -1.27 I 4 -3.80/-3.88 1.430 -ts 2 × 2 × 2 ⎛ ⎝−4 . 52−0 . 37 −0 . 37−4 . 52 0 . 000 . 00 0 . 00 0 . 00 1 . 73 ⎞ ⎠ ⎛ ⎝−3 . 59−0 . 24 −0 . 24−3 . 59 0 . 000 . 00 0 . 00 0 . 00 5 . 19 ⎞ ⎠ 27 -2.35/-1.99 2.478 -19 1.033 3 × 3 × 3 ⎛ ⎝−5 . 09−0 . 13 −5 . 09−0 . 13 0 . 000 . 00 0 . 00 0 . 00 2 . 59 ⎞ ⎠ ⎛ ⎝−4 . 29−0 . 85 −0 . 85−4 . 29 0 . 000 . 00 0 . 00 0 . 00 6 . 52 ⎞ ⎠ 11 -2.39/2.06 2.487 - 1.057 Pd vac 2 × 2 × 2 -5.79 I -1.83 I 23 -5.82/-5.48 1.176 -3 3 × 3 × 3 -5.98 I -1.88 I 7 -6.35/-5.66 1.155 -ts 2 × 2 × 2 ⎛ ⎝−5 . 14−0 . 32 −0 . 32−5 . 14 0 . 000 . 00 0 . 00 0 . 00 1 . 02 ⎞ ⎠ ⎛ ⎝−7 . 70−2 . 37 −1 . 37−7 . 70 0 . 000 . 00 0 . 00 0 . 00 12 . 48 ⎞ ⎠ 10 -3.88/-2.91 2.031 -27 0.855 3 × 3 × 3 ⎛ ⎝−5 . 56−0 . 46 −0 . 46−5 . 56 0 . 000 . 00 0 . 00 0 . 00 0 . 55 ⎞ ⎠ ⎛ ⎝−7 . 77−0 . 62 −0 . 62−7 . 77 0 . 000 . 00 0 . 00 0 . 00 12 . 21 ⎞ ⎠ 8 -3.92/-3.33 2.027 - 0.872

whereIistheidentitymatrix.Forthetransitionstate,calculated forajumpinthe[110]direction,resultsgive:

Pts₌

_{−2.29 −0.21 0} −0.21 −2.29 0 0 0 1.62

. (24)

Asexpectedfromthecrystalsymmetry,thedipoletensorofthe vacancy,Eq.23,hasanisotropicsymmetry. Conversely,thedipole tensorofthetransitionstatealongdirection[110],Eq.24,exhibits anisotropy: the[001] axisis compressed during thejump, while

[010] and[100]axesareundertension.Furthermore,aslightshear stress of _−0.21 eV occursin the (001) plane. As a consequence, diffusionunderappliedstressisexpectedtobeanisotropic.

Thedipoletensorofmigrationalongdirection[110]isobtained fromEqs.23and24:



P=

_{0.33 −0.21 0} −0.21 0.33 0 0 0 4.24

, (25)

Table 4

Parameters for the vacancy diffusivity.

Al Cu Ni Pd

Uf [eV] _{0.636 1.039 1.430 1.155}

Um [eV] _{0.607 0.740 1.057 0.872}

Sf [J/K/mol] _{9.77 11.44 8.00 4.84}

D0 (10 −6 ) [m 2 / s ] 4.12 1.40 2.68 3.97

from which the volume tensor of migration can be derived (in eV/GPa):



V=

_{−0.026 −0.003 0} −0.003 −0.026 0 0 0 0.072

, (26)

or,asafunctionoftheatomicvolume V at:



V=

_{−0.25 −0.03 0} −0.03 −0.25 0 0 0 0.70

Vat, (27) with V at=16.5 ˚A3.

From thesymmetryoftensor



V,itcanbeseen thatthe mi-grationenergiesaresensitivenotonlytotheappliedpressureand normalstresses,butalsototheshearcomponentsinthereference frameofthecrystal.

As was done forAl, elastic tensors were calculated forCu, Ni andPdmetals,seeTable3.

This Tablealsoreportsthe volumeof relaxation(V rel_{) whena} vacancyis inserted in the system: the first quantity corresponds totheDFTvalue,whilethesecondwascomputedfromthe relax-ationvolume tensor. Resultsshow that both quantities are close: theelasticitytheory captureswitha goodaccuracytherelaxation volumeofthevacancyinthesystemsstudiedhere.Theinteraction energywithvacancyperiodicimages(calculationsdoneatconstant volume, E d) is always low,about10 meV,in agreementwith the factthatvolumeandpressureconstantcalculationsprovide equiv-alentresultsfrom108atoms.

The formation and migration energies computed from the largestsuper-cellsaregatheredinTable4(firstandsecondlines). DFT results obtained here are in agreement with the literature [40,43,44].Results alsoshow that P convergesince 3 × 3 × 3 super-cells.The restofthe studythereforeconsiders thesevalues tobeaccurateenough.

From Al to Pd,the elastic tensor ofthe vacancy andits tran-sitionstateincrease.The Nisystem, whichhasthehighestelastic constants,exhibits thelowest anisotropy among transitionstates. Therefore,theeffectofanisotropyondiffusionwasexpectedtobe small,especiallyforanon-hydrostaticloading.

From the additional data summarized in Table B.5, it can be concludedthat the four fcc metals of the studyexhibit a lattice contractionofabout1/3oftheatomicvolumewhenanatomis re-moved(V rel_−1

3V at).Hence, thegapformationvolumeisalways almostequalto V f_2

3V at.

The effectofdifferentstressesontheformationandmigration energiescannowbeinvestigatedindetail.

5. Hydrostaticstress

The effect ofhydrostatic stress onthe solubilityand diffusion of vacancies is the first point investigated. From an experimen-tal standpoint, the range of hydrostatic stress has no limit. The maximalpressureisonlylimitedbyphasetransitions.Under com-pression(resp.tension)adecrease(resp.increase)indiffusivityis expected. Forthis purpose,the differentquantities introduced in Section3, i.e. internalenergies,enthalpies ofvibrationand migra-tion,andjumprates,werecalculatedusingDFTonasetofstresses foraluminumonly.Forthesystemwithandwithoutvacancy,the volume and energy were calculated as a function of stress. DFT

Fig. 2. Formation and migration energies (top) and enthalpies (middle) in alu- minum as a function of pressure, under hydrostatic loading. Bottom: lattice param- eter of bulk aluminum as function of pressure. Continuous lines: DFT; dashed lines: elasticity theory.

results were computed to generate a polynomial curve fit. From thesefits,theevolutionoflattice parameter,energyandenthalpy werededucedasafunctionofpressureusingEq.7(neglectingthe effectoftemperatureoninternalenergy). G f_{[p ]wasthencomputed} usingthesepolynomials.ResultsaredepictedinFig.2.

The formation and migration energies computed by DFT are foundalmostpressure-independent,exceptathigherpressures,as can be seen in Fig. 2 top. Yet even athigh pressure, > 4 GPa, the elasticity theory deviates only a little from DFT results. One way to correct this wouldbe to evaluate the effect ofstress on elastic tensors, but this is beyond the scope of this study. This

stress-independence was expected fromthe linear elasticity the-ory Eqs.13 and(18).The slightlypositivecurvature ofthelattice parameterisexplainedby thedeviationfromlinearity,Fig.2 bot-tom.In thisgraph, a _modrefers tothe latticeparameter computed fromtheelasticitytheory(Eq.11):

a[p]=ao



1−₃p_B



(28)

where B isthebulkmodulus.Toevaluatetheeffectofvacancy con-centration on lattice parameters, ₊a oC vP11vac/3BV at must be added (this correction is always the same in the uni-axial and bi-axial cases)

When the applied stress is hydrostatic, the equationsof elas-ticity can be simplified as follows, to account for the effect of pressure ontheenthalpies.The formationofa vacancyis accom-panied by a volume increase of the crystal, ₊V at, and a volume decrease dueto theatomic relaxationaround thevacancy, +V rel_. Hence, the total volume change during the formation of a va-cancyisV f_=V

at+V rel.Fromtheelasticitytheory V rel=tr

(

V

)

. Cal-culations give V rel_{=−0. 35V}

at, resulting ina volumeofformation V f_=0.65V

at.Then, according to Eq.16, thevariation



H f in for-mationenthalpyduetotheappliedpressureisalinearfunctionof pressure p (p inGPa):



Hf[p]=pVf=0.067p. (29) Similarly, applying Eq.19 to the case of hydrostaticpressure, with V m_=0.20V

at,yieldsthechangeinmigrationenthalpydueto theappliedpressure:



Hm[p]=pVm=0.020p. (30) Ascan beseen,theelasticitytheorypredictsthat bothenthalpies shouldincreaseastheappliedpressureincreases.

Concerning DFT results found, the formation enthalpy evolu-tion isinagreement withtheresults ofIyer etal.[45]. The evo-lution of H f _{is similar. However, quantitatively, since they used} the Perdew-Zungerfunctional (PZ81-LDA[46]), their energies are higher than those of this study.The formation energy decreases rapidlytoreachavalueof0eVforapressureof7.5GPa.Todeepen totheworkofIyer,theeffectofloadingonmigrationenthalpywas investigated.Resultsshowthesametrendasfortheformation en-thalpy,ascanbeseeninFig.2.Theheightofthebarrierdecreases with pressure, but the slope is smaller than from the formation energy.

When comparing DFT calculations and elasticity theory, it is clearthat theelasticitytheoryaccurately reproducesDFT calcula-tions ofboth migration andformation enthalpies. This is dueto thefact that themaineffectonenthalpy isincluded inthe pres-sureterm, Eqs. 29and30,asillustrated inFig. 2top that shows theplotoftheDFTinternalenergy U f_only.

From this, aconclusion canbe drawnconcerning theeffectof hydrostaticpressureonvacancies.Allcontributionsyieldthesame effect: when the pressure applied to the system increases, the solubilitydecreases (H f _{increases)andthevacancy diffusivity} de-creases(H m_{increases).Thisbehaviorisconsistentwith} experimen-tal results on mostmetals [47]. In addition,the good agreement between the elasticity model and DFT calculations, even at high pressures,isnoteworthy.Theresultobtainedisconsistentwiththe idea thatasthe systemexpands(p < 0),vacanciesformand dif-fuseeasier.Theincreaseoflatticeparameterinducesanincreaseof atomicdistances which facilitatesthe formationof vacanciesand reducesenergybarriers.

The last effect studied is that of pressure on vibrational en-thalpy. Until now, all vibrational aspects were neglected for sim-plificationpurposes,nevertheless,thepresentDFTcalculationsare ableto capturetheir effect,whichisnotyetthecaseofthe elas-ticity theory.Aswasdone fortheinternal energy, G vib of the va-cancyandofthetransitionstatewerecomputedfordifferent

tem-Fig. 3. Vibrational formation and migration enthalpies in aluminum as a function of hydrostatic pressure for different temperature.

peratures.Results ontheformation andmigrationvibrational en-thalpiesaredepictedinFig.3.

Results show that the effect of an hydrostatic loading is still negligibleonthevacancyformationenthalpyandlowonthe tran-sitionstate(afewmeV).Themaineffectremainsthe pV contribu-tion.Since the pressurerange ofhydrostatic stress isthe largest, theeffectofloadingonthevibrationalenthalpywillbeneglected intherestofthestudy.

Equilibriumconcentrationsanddiffusioncoefficientsofthe va-cancycannowbeplottedforallfccsystemsconsidered,seeFigs.4 and5,respectively.Resultsareplottedusingtheparameters sum-marizedinTableB.6inAppendixB.

Regardlessofthemetal undertension(resp.compression), the concentrationanddiffusivityofvacanciesincrease(resp.decrease). Theeffectismorepronounced inAl thaninother metals.Thisis a consequence of Aluminum’s relatively low elastic stiffness. For allmetals,the highestmaineffectofpressure isobservedatlow andintermediatetemperatures.Theincrease(inconcentrationand diffusivity) can reach several orders of magnitude at room tem-perature, butonly when the pressure is high. For a stress equal to ± 1GPa,thechangeinvacancyconcentration isaboutone or-derofmagnitudeonly. Theenhancement ondiffusivityis onlyof oneorderofmagnitude, intherangewherethemodelis numer-icallyaccurate.Themainnotableeffectofpressureistheincrease in vacancy concentration, in the caseof metals with low elastic stiffness.

Toconclude, theelasticitytheory can capturethephysics, but thebehaviorof thedifferentmaterials presentedhereislittle in-fluencedbythehydrostaticloading.

6. Uni-axial/bi-axialstress 6.1. Formalism

Thecaseofuni-axialandbi-axialstressesalongthemain crys-tallographicdirectionscan be treatedusingthe sameframework. Inthecaseofauni-axial(along,forexample,the[001] direction) orabi-axialstress(applyinganequivalentloadingalong,for exam-plethe[100]andthe[010]directions),thecubicstructureevolves into a tetragonal structure, where two directions are equivalent. The initial spacegroup No225 (F d ¯3m ) changes into spacegroup No139 (I4/mmm)forthe perfectstressedstructure. The atomsin thelatticearestillequivalent,thusthereisonetypeofvacancy.Eq. 7,whichgives thevacancyconcentration C V,isstill applicable in thepresentcase.Insucha configurationhowever,thedegeneracy ofthejump ratesislifted. Twodistinct jump rates,



12 and



13,

Fig. 4. Vacancy concentration as function of T in Al, Cu, Ni and Pd systems under different hydrostatic loadings in the range of -8 to +8GPa.

describethewholemigrationprocess:thefirstoneisinthesquare plane(labeled12)andthesecondoneisalongthethirddirection (13along z fortheuni-axialstress),asdisplayedinFig.1,b).There isasymmetrybreakinthevacancydiffusioncoefficientalongthe crystallographic direction. The diffusion coefficients in the plane, D x,y,andalongthe z direction, D z,arethusgivenby[48]:



Dx,y[

σ

zz,T]=a 2 2

(12

+

13

)

Dz[

σ

zz,T]=c2

13

(31) Inthecaseofauni-axialstress, a =b and c are,theperpendicular latticeparametersandthelatticeparameterinthedirectionofthe uni-axialloading(

σ

zz),respectively.Allquantities, i.e. a, c , C V,



12 and



13,dependon

σ

zzand T .

Diffusion coefficientscanbe approximated,by aTaylor expan-sionofeq.31,usingthefollowingequations:



Dx,y[

σzz

,T]=a 2 2

12

[0]



exp



−H12 kBT



+exp



−H13 kBT



Dz[

σ

zz,T]=c2

12

[0]exp



−_kH12 BT



(32)

where



12[0]isthestress-freeatomicjump.

Equivalentequationswereobtainedforabi-axialstress. 6.2. Results

6.2.1. Uni-axial stress

In thecaseofstress along[001], DFTandelasticitytheory re-sultsare drawn inFig. 6.As wasdone forhydrostatic stress, the evolutionoftheformationenergy,migrationenergies (perpendic-ulartoandalong thestress)andlattice parameters(a =b and c )

are represented. Contrary to hydrostaticstress, the pressures ap-pliedinuni-axialandbi-axialloadingsarelimited;inthisstudy,it waschosento limitthe stressvalue to ± 1GPa. The first discus-sionfocusesontheeffectofauni-axialstress.

When the metal is pulled (contracted), the system undergoes acontraction (expansion)inthedirectiontransversetothe stress direction, see Fig. 6. Thisis well explainedby the elasticity the-oryandrationalizedbyPoisson’sratio,

ν

.Fromthepresentresults, the ratio

δ

c /

δ

a ࣃ0.37 is found in excellent agreement with the theoreticalpredictionofPoisson’sratioforaluminum,whichis ap-proximatelyequalto0.34inDFT(computedfromelasticconstants) andexperimentally.Theevolutionoflatticeparameterscanbealso rationalizedwiththefollowingrelations:



a,b=ao



1− C12 6BC

σ

zz



c=ao



1+C11+C12 6BC

σ

zz



(33)

ascanbeseeninFig.6(a oisthestress-freelatticeparameterand C  theshearmodulus).

Concerningthesolubilityenergy,iftheinternalenergypart, U , isdirectlycomputedbyusingDFTresults,thecorrectionrelatedto theelasticenergyleadsto:

H=U− V



zz

σ

zz =U[

σ

zz]− V



c[σzz]−ao ao



σ

zz (34)

where V isthevolumeofthesystemand



zz theelongation (con-traction)under

σ

zz.Eqs.3and4arethenusedtocompute H f.

Thebehaviorobservedisthesameasforhydrostaticstress,i.e. a decreaseinformationenthalpy whenthe materialis pulled.As

Fig. 5. Vacancy diffusion coefficients as function of T in Al, Cu, Ni and Pd systems under different hydrostatic loading.

expected, aliftingofdegeneracyofthemigrationenergies canbe noticedduringatensileorcompressivetest.Anincreasein migra-tionenergyalong(001)andan oppositemechanism(decrease)in theplane[110]isobserved.

The energybarriersdecrease inthedirectionperpendicular to thestress,whiletheyincreaseinallotherdirections.

From theelasticity theory(Eq.29) theapplied stress modifies thevacancyformationenergyviathepressure p =−

σ

zz/3,i.e.



Hf_=−0.022

_σ

zz. (35)

AccordingtoEq.30,thetwomigrationenthalpiesaremodifiedas follows:





Hm 12=−0.074

σ

zz



Hm 13=+0.026

σ

zz. (36)

There isan excellent agreementbetweenDFT simulations and theelasticmodel.Contrarytowhatwasfoundforhydrostatic load-ing,theevolutionofvacancyconcentrationunderauni-axialstress isrelativelysmall;theformationenergydifferenceat1GPais be-low < 0.1eV.The effectof

σ

zz on vacancyconcentration is thus low,ascanbeseeninFig.7top.However,thesymmetrybreakin thediffusioncoefficientis,onthecontrary,morepronounced.

To clarify, the D x/D z ratio for different loadings is plotted in Fig.7bottom.Forhighpositiveloadings,asmallsymmetrybreak andanincreaseinvacancydiffusivitycanbeobserved.Thisiswhy itwouldbedifficulttoidentifyexperimentally.

Theresultsobtainedonthefourmetalscannowbecompared. The slopesof migrationandformationenthalpies aregatheredin Tables B.5andB.6.Resultsshow thattheeffectofstress depends

littleonthemetalspecies,allslopesareequivalent.Basedon alu-minumresults,itcanbededucedthattheeffectofuni-axial load-ing isthus small.Vacancyconcentration changeslittle asa func-tionofstress.

6.2.2. Bi-axial stress

Inthe caseofa [100]+[010]stress,simulations results are de-picted in Fig. 8. Results are very similar to those of a uni-axial loading.

Qualitativelyandquantitatively,uni-axialandbi-axialloadings lead overall tothe same result. Onthe other hand,the elasticity theory adequately captures the physics, reproducing DFT results accurately.Theevolutionofthelatticeparameterscanbedescribed usingthefollowingequations:



a,b=ao



1+ C11 6BC’

σxx



c=ao



1− C12 3BC’

σ

xx



(37)

Resultsusingtheelasticitytheory,with

σ

xx=

σ

yy, give:



Hf=−0.045

σxx

. (38) and





Hm 12 =+0.0051

σ

xx



Hm 13 =−0.046

σxx

, (39) With theseparameters, DFT results can once more be repro-ducedwithahighdegreeofaccuracy.Asinpreviouscases,itcan bededucedfromtheseresultsthatthevacancy(concentrationand diffusivity)infccmetalsislittleaffectedbyabi-axialstress.Itcan alsobenotedthatthevaluesof



H dependslittleonthemetal.

Fig. 6. Evolution of formation and migration energies as a function of the stress, under a uni-axial stress along [001], σzz (in GPa).

7. Alongthe[111]direction

For the second uni-axial loading, another direction was con-sidered:along[111],perpendicular to thedense plane.The F d ¯3m symmetry thus changes into a rhombohedric structure: space group R ¯3m_,No166.Inthisrepresentation,alllatticeparametersare equalandall anglesarealsoequalbutnot orthogonal.Twojump rates(



12and



13)arenecessarytodescribetheatomisticprocess ofdiffusion.Inthecubicrepresentationofthecell,thetwo param-eters



12 and



13 aredisplayedinFig.1c).Inthe rhombohedral cell,seeFig.1d),theexpressionofthediffusioncoefficientisthus

Fig. 7. Vacancy concentration (top) and ratio of vacancy diffusion coefficients D x / D z

(bottom) as a function of T for Al for different values of σzz .

transformedinto: D[

σ

111,T]=

a2 111

2

(

12

+

13

)

(40)

a 111 corresponds to the lattice parameters of the primitive unit-cell,theloadingisalongthe[111]direction.However,for simplifi-cationpurposes,a[111]loadingcanalsoberepresentedusingthe hexagonalrepresentationoftheunitcell,asdisplayedFig.1e).The stressisthusalongthenew z axisanddiffusioncoefficientscanbe writtenas:



Dbasal[

σ

zz,T]=2 a 2 basal 3

(

3

13

+

12

)

Dz[

σ

zz,T]=c 2 h 3

12

(41) where



13is thejumprateinthe new xy (basal) plane,and



12 theonealongtheaxial direction. a basaland c h arethenewlattice parameters ofthe box.Thissecond representationwaschosen to performcalculations(3 × 3× 2super-cell,i.e.54atoms).

Asforpreviouscases,thereisonlyonenonequivalentvacancy, but there are two distinct jumps. Results are depicted in Fig. 9. Here again, the formation energy increases when the metal is pulled.

However, the effect on diffusivity, on anisotropy and the ex-plicit stress dependence, is almost non-existent. DFT results are confirmedby theelasticitytheory.Theformationenthalpy depen-denceis:



Hf_=−0.022

_σ

111. (42)

whilemigrationenthalpiescanbeexpressedas:





Hm

12 =−0.0048

σ

111,



Hm

Fig. 8. Evolution of the formation (top) and migration (middle energies as a func- tion of stress, under a bi-axial stress along [100]+[010]. The evolution of lattice pa- rameters is also plotted (bottom).

Ascanbeseen,theeffectofauni-axialloadinginthedenseplane isnegligible,evenat1.5GPa.Allfccsystemsconsideredhereshow thesametrend,seeTablesB.6andB.5.

8. Shearstress

Lastly,theeffectofpure shearstress

σ

xy inplane(001)is dis-cussedThecubicsymmetryistransformedintoamonoclinic struc-tureC/2m,No12.Thereisstill onlyonetype ofvacancy,butthree jumpratesareneededtodescribethemigrationprocess,asshown inFig.1.Inthe xz and yz directions,thelatticeremainssquare,and inthe xy itisdiamond-shape.Thelengthofthelatticeis

neverthe-Fig. 9. Evolution of the formation (top) and migration (middle) enthalpies as a function of pressure, under a uni-axial stress along [111], σ111 (in GPa). The effect

on lattice parameters is also represented, bottom.

lessunchanged,diffusioncoefficientsarethusexpressedas:



Dx,y[

σ

xy,T]=a 2 4

(

13

a+

13

b+2

12

)

Dz[

σ

xy,T]=c2

12

(44)

From the elasticity theory, a shear stress has no effect on the formationenthalpy ofvacancies, since thehydrostaticpressure is null:

However,asmalleffectonenergybarriersisfound:



_

_Hm 12=0.0,



Hm 13a=+0.0061

σ

xy,



Hm 13b=−0.0061

σxy

. (46)

Shearstress

σ

xy hasno effect on the migration along [011], and oppositeeffectsalong[110]and[1¯10].

9. Conclusion

Thisarticlepresentsafullmethodtostudytheeffectofstress onvacancyformationanddiffusivityenergiesinfccsystems. Elas-ticitytheory wasrationalizedby usingDFT parameters and com-paredto first-principlescalculations. Results show that both ap-proachesleadtoequivalentresultsquantitatively untilafewGPa, andqualitativelybeyond.Thebenefitoftheelasticapproachisthat itallows reproducingloadingeffectsusingafew parametersfora reducednumericalcost ascompared tofull DFT calculations. Re-sults also show that the vibrational enthalpy changes little with theloading.Additionally,avacancydiffusionequationisproposed inthecaseoffccdistortedsystems.

In terms ofresults, theeffect ofstress (aswell asstrain)can stronglymodifyvacancyconcentration anddiffusivity,butmainly inthe case of hydrostaticloading. For highloads, other physical processes such as dislocations orinterfaces, can also change the amount of vacancies and their diffusivity. It wasshown that all

fourmetalshaveaverysimilarstressbehavior.Theanisotropy in-ducedbythedisruptionofsymmetryisconsideredlow.Tofurther provethispoint, experimentson softmaterials shouldbe carried out.Thepresentresultscanprobablybeextendedtoother materi-alsandinparticulartobccandhcpsystems.

These results can significantly improve the understanding of physicalprocesseswithin materialsundergoinglocalorglobal de-formations.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Acknowledgments

This work was performed using HPC resources from

CALMIP (Grant 2018 and 2019-p0749) and GENCI-CINES (Grant A0040910368).DCwouldliketothankCALMIPstaff membersfor theirhelpinusingHPCresources.

AppendixA. Parametersat0GPa

Results on the vacancy andtracer diffusivity andthe vacancy concentrationatambientpressureareplottedinFig.A.10.

tion and migration energies are given in Table3, vibrational en-thalpyandentropyareincludedinthecalculations.

Based on these results andthe vibrational partition functions of the transition and stable states, the vacancy diffusion coeffi-cient D (T )wascomputedandfitusinganArrheniusfunctionofthe following form D 0exp

(

−Q/kBT

)

. The tracer diffusion coefficients,

f ∗C 1v∗D 1v,werealsodrawn.

The vacancyformationentropy (athigh T ), S f_{, wasalso} calcu-lated. S f_{isexpressedas:}

Sf_=S

vib[

(

N− 1

)

.Al]− N− 1

N Svib[N.Al] (A.1)

wherethephononentropy, S _vib,correspondstothehigh tempera-turevalue(at0K, S vib=0).

In fine ,diffusioncoefficients(D o=3.610−6m 2/s andQ=1.24eV) areinexcellent agreementwithexperimental values,seeMantina [34]forinstanceforAl. S f_{1.17 k}

B or9.8J/K/mol,tobecompared totheexperimentalvaluerangingfrom0.7-1.1 k Bunit[34]. AppendixB. Results

Results of the elastic modelcomputed using DFT values from Table3aresummarizedinTablesB.5andB.6.Dataare computed DFTcalculationswith3 × 3× 3super-cellsizes.

AppendixC. Tri-axialcase

In the case of a tri-axial stress, the fcc system evolves into a Fmmm (No69) orthorhombic structure. Three non-equivalent jumps are used to describe vacancy diffusion:



12,



13 and



14. TheconventionisthesameastheoneusedinFig.1.Thediffusion coefficientsbecome: Dx= a2 2

(

12

+

13

)

(C.1) Dy= b2 2

(

12

+

14

)

(C.2) Dz= c2 2

(

13

+

14

)

(C.3)

where a, b and c arethedeformedaxes.

Table B.5

Atomic volume ( V at ), relaxation volume ( V rel ) and formation volume ( V f ) of a vacancy, in ˚A 3 .

Changes in formation enthalpy ( Hf , in eV) due to an applied stress ( p or _{σ, in GPa). From}

DFT calculations with 3 × 3 × 3 super-cell size and elasticity theory of point defects.

stress state Al Cu Ni Pd Vat 16.5 11.9 10.9 15.3 Vrel _{-5.36 -3.84 -3.88 -5.66} Vf _{11.1 8.06 7.02 9.64} Vf / V at 0.67 0.68 0.64 0.63 hydrostatic Hf _{0.067 p 0.049 p 0.042 p 0.058 p} uni-axial [001] Hf _{−0 . 022 σ} zz −0 . 016 σzz −0 . 014 σzz −0 . 019 σzz bi-axial [100] + [010] Hf _{−0 . 045 σ} xx −0 . 032 σxx −0 . 028 σxx −0 . 039 σxx uni-axial [111] Hf _{−0 . 022 σ} 111 −0 . 016 σ111 −0 . 014 σ111 −0 . 019 σ111 shear (001) Hf _{0. 0. 0. 0.} Table B.6

Migration dipole tensor ( P, in eV) and migration volume tensor ( V, in ˚A 3 ) for an atom jump in direction [110]. Changes in migration enthalpy ( _Hm , in

eV) due to stress ( p or σ, in GPa) for various jumps. From DFT calculations (3 × 3 × 3 super-cell) and elasticity theory of point defects.

stress state Al Cu Ni Pd P ⎛ ⎝−0 . 210 . 33 −0 . 210 . 33 0 . 000 . 00 0 . 00 0 . 00 4 . 247 ⎞ ⎠ ⎛ ⎝−0 . 44−0 . 41 −0 . 44−0 . 41 0 . 000 . 00 0 . 00 0 . 00 5 . 17 ⎞ ⎠ ⎛ ⎝−0 . 42−0 . 13 −0 . 42−0 . 13 0 . 000 . 00 0 . 00 0 . 00 7 . 26 ⎞ ⎠ ⎛ ⎝−0 . 460 . 42 −0 . 460 . 42 0 . 000 . 00 0 . 00 0 . 00 6 . 53 ⎞ ⎠ V ⎛ ⎝−0 . 49−4 . 12 −0 . 49−4 . 12 0 . 000 . 00 0 . 00 0 . 00 11 . 5 ⎞ ⎠ ⎛ ⎝−4 . 55−0 . 41 −0 . 41−4 . 55 0 . 000 . 00 0 . 00 0 . 00 10 . 6 ⎞ ⎠ ⎛ ⎝−0 . 083−3 . 01 −0 . 083−3 . 01 0 . 000 . 00 0 . 00 0 . 00 7 . 78 ⎞ ⎠ ⎛ ⎝−5 . 88−0 . 61 −5 . 88−0 . 61 0 . 000 . 00 0 . 00 0 . 00 14 . 1 ⎞ ⎠ hydrostatic H12 0.020 p 0.0094 p 0.0101 p 0.0145 p uni-axial H12 −0 . 074 σzz −0 . 0662 σzz −0 . 0486 σzz −0 . 0880 σzz [001] H13 +0 . 026 σzz +0 . 0284 σzz +0 . 0188 σzz +0 . 0367 σzz bi-axial H12 +0 . 0051 σxx +0 . 0568 σxx +0 . 0376 σxx +0 . 0735 σxx [100] + [ 010 ] H13 −0 . 046 σxx −0 . 0378 σxx −0 . 0297 σxx −0 . 0512 σxx uni-axial H13 −0 . 0088 σ111 −0 . 0014 σ111 −0 . 0033 σ111 −0 . 0023 σ111 [111] H12 −0 . 0048 σ111 −0 . 0049 σ111 −0 . 0040 σ111 −0 . 0074 σ111 shear (001) H12 0. 0. 0. H13a +0 . 0061 σxy +0 . 0052 σxy +0 . 0010 σxy +0 . 0077 σxy H13b −0 . 0061 σxy −0 . 0052 σxy −0 . 0010 σxy −0 . 0077 σxy

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