Hydrogen Induced Vacancy Clustering and Void Formation Mechanisms at Grain Boundaries in Palladium

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Acta Materialia

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Full length article

Hydrogen induced vacancy clustering and void formation mechanisms at grain boundaries in palladium

Jonathan M. Polfus

^∗

, Ole Martin Løvvik , Rune Bredesen , Thijs Peters

SINTEF Industry, Sustainable Energy Technology, PO Box 124 Blindern, NO-0314 Oslo, Norway

a rt i c l e i nf o

Article history:

Received 8 May 2020 Revised 5 June 2020 Accepted 8 June 2020 Available online 12 June 2020 Keywords:

Density functional theory Vacancies

Hydrogen

Grain boundary segregation

a b s t ra c t

Hydrogenhasasignificantimpactontheformation ofvacancies,clustersand voids inpalladiumand othermetals.Theformationofvacancy-hydrogencomplexesinbulkPdandat³ând⁵^grain^bound- arieswasinvestigatedusingfirst-principlescalculationsandthermodynamicmodels.Equilibriumvacancy and clusterconcentrationswereevaluatedas afunctionoftemperatureandhydrogenpartial pressure basedonthe Gibbsenergyofformation includingvibrationalandconfigurational entropies.Vacancies werefoundtobesignificantlystabilizedbyassociationwithinterstitialhydrogen,leadingtoenhanced concentrationsbyseveralordersofmagnitude.Vacancyclusterswerefurtherstabilizedatgrainbound- aries, withequilibrium concentrationsreachingsitesaturationfor clusters comprisinguptothree vacancies.NanovoidswereinvestigatedbasedonWulff constructions fromcalculatedsurfaceenergiesof the(001)and(111)terminationsasafunctionoftemperatureandcoverageofhydrogenadsorbates.

Themoststableterminationchangedfrom(111)invacuumto(001)inH₂,andthesurfaceenergies wereloweredduetohydrogenadsorbates.Consequently,voidswerealsostabilizedinthepresence of hydrogen.Coalescingofvacanciesintonanovoidswasfoundtobethermodynamicallyunfavorabledueto thelossofconﬁgurationalentropy.Itwasthereforeconcludedthatenhancedconcentrationsofvacancies and clustersdoesnotdirectlycausetheformationofvoids.TheformationofvoidsinPd-basedmem- braneswasdiscussedintermsofmicrostructuralcharacteristics,and strainduetochemical expansion andplasticdeformation.

ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

The role of hydrogen onthe structuraland functionalproper- tiesof metallic materials has long been a subject of interest for a broad range of applications.Metals are in many cases directly exposedto hydrogen-containing environments, forinstancewhen used as sensors [1], hydrogen-selective membranes [2], catalysts [3]and hydrogenplasmafacingmaterialsinnuclearfusionreactors [4].Moreover,hydrogenisatypicalproductofcorrosionandoth- erwiseomnipresentasthemostabundantelement intheknown universe.

Exposuretohydrogencan leadtostructuraldamageanddete- riorationofthemechanicalintegrityofthematerial.Dependingon thespeciﬁc metal and environmentalconditions, thedegradation can manifestitself macroscopically in the formof embrittlement [5,6],cracking[7,8],blisteringandporeformation[9,10].Combined witharangeofmicrostructuralchanges,oneoftheprimarymech-

∗ Corresponding author.

E-mail address: [email protected] (J.M. Polfus).

anismsbehind thesedegradation processes isthe enhanced con- centrationofmetalvacanciesintheformofcomplexeswithinter- stitialatomichydrogen. Thisphenomenon– referredto assuper- abundantvacancies– wasﬁrstreportedbyFukai etal.forPdand Ni[11,12],andlaterforarangeofothermetalsandrelatedalloys [13].

The vacancy complexes can cluster into nanovoids or cavities, eventually becoming large enough to contain molecular and gaseous H₂ [14,15].Formationoflarger clustersby coalescenceof mmonovacancies andn interstitialhydrogencanbe describedby thereaction

mv₁+nH_i= vmHn (1)

In some cases, the H₂ pressure can build-up inside the voids andact asa driving force forfurthergrowthby plastic deforma- tionofthesurroundinglattice[16].Inrelationtothepossibilityof formingsuchpressurizedbubbles,thereisacrucialdifferencebe- tweenthebehaviorofdifferentmetals,whichcanbedistinguished bytheGibbsenergyfordissolutionofinterstitialhydrogenrelative to H₂ gas. Fora systeminequilibrium, thechemical potential of hydrogenshouldbeconstantthroughout,andthepressureofcon- https://doi.org/10.1016/j.actamat.2020.06.007

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roundingatmosphereexceptwhenthesystemispushedfaroutof equilibrium atvery shorttimescales,e.g., nanosecond laser pulse heatingofNbD_0.7andVHx[20,21].Otherwise,thechemicalpoten- tial ofhydrogen is allowed to equilibrate and thisprocess is not kinetically hindered in palladium dueto the fast kinetics of adsorption/desorption. In this context, the formation of bubbles in palladium containing the tritium isotope of hydrogen may serve as an instructive example.Tritium is highlysoluble in palladium duetoitschemicalsimilaritywithhydrogen,butdecaysnaturally over timeto³He, whichexhibitsverylow solubilityanddiffusiv- ity.Associationbetween³Heandvacanciescanbeanticipatedand furthernucleationtobubbleswithincreasing Hepressureeventu- allyresultsincracks throughwhichthegascan escape[22].This processthereby representsa similarbubbleformationmechanism as insupersaturated metals that exhibit relatively low diffusivity of hydrogen. Anotherrelated bubble formation process has been reported for N-doped ZnO andZnO–GaN semiconductors due to evolutionofN₂ bubblesafterionimplantation[23] andannealing [24],respectively.Inthesecases,thedifferenceinsolubilityarises fromthedistinct oxidationstateofnitride anionsinthehostlat- ticeandmolecularnitrogen.

InparticularforPd-basedH₂separationmembranes,theforma- tionofvoidsandpinholesduringoperationoverlongtimeperiods canresultinreducedselectivityandlifetime[25–28].Forinstance, cavities andpinholes ofup to tens ofnanometershavebeen ob- served inconjunctionwithincreasedtrans-membraneN₂ leakage under1–5barhydrogenat300–450°C[29,30].Notably,thevoids mainly formed along grain boundariesand at triple junctions in the ﬁlms. Inunderstanding the mechanismsofvoid formation,it is importantto alsoconsider the role ofchemical expansion due to dissolution of hydrogenand transitionto hydride phases; The evolution ofinternalstress inPdﬁlms subjectedtohydriding cy- cles can resultin increased dislocation densities andirreversible microstructural changes [31,32].Higher dimensional defects such asgrainboundariesthereforeappeartoplayan importantrolein void formation, possiblyas nucleation sites forvacancies to coa- lesce.

Ab initio calculations have been utilized to gain atomistic in- sightintotheformationofvacancy-hydrogen complexesofdiffer- entsizeinseveralsystems.Nazarovetal.performedacomprehen- sive studyofhydrogeninteractingwithmono-anddivacanciesin twelve different fcc metals including Pd; the results consistently showed energeticallyfavorable trappingof multiple hydrogenin- terstitialsinthevacancies[33].Vekilovaetal.[34]reportedsimilar ﬁndings forvacancy-hydrogencomplexes inPd.You etal.considered sixfcc and sixbcc metals andconcluded that vacancyfor- mation wasparticularly favorablein V,Nb,Ta, Pd andNi dueto low formationenergiesofthecomplexeswhichfurtherdecreased with the number of hydrogen [35]. Several similar studies have been undertaken onthe formation ofmono-and divacancyclus- tersandthe interactionwithhydrogen inW [36,37],Mo[38],Ni [39,40],andCu[41].Gengetal.investigatednanovoidsofupto27 vacanciesin

α

^-Fe^and^modelled^the^pressure ^of^molecular^hydro-

genaftertheinnersurfaceswerecoveredwithhydrogenadatoms [42].Hou etal. recentlyinvestigated bubbleformation inW due

presenceofedgedislocationsin

α

^-Fe,annihilationofvacanciesand complexeswasfoundtobefavoredoverclusteringthatcouldlead tonanovoids[47].

Inthepresentcontribution,we furtherinvestigatevacancyand void formation mechanisms in the bulk and along grain bound- ariesofpalladiuminthepresenceofhydrogenbymeansofdensity functional theory (DFT) calculations. The Gibbs energy of forma- tionofvacancy-hydrogen complexes andclusterswithupto four vacancies(m = 1–4), were considered in the bulk as well as in the proximity of the ³ ⁽¹ ¹ ¹⁾ ând ⁵ ⁽² ¹ ⁰⁾ ^grain ^boundaries,whichareamongthemostabundantsmall-angleboundaries inPd[48].Equilibriumconcentrationswere thereby calculatedas afunctionoftemperatureandhydrogenpartialpressure.However, avacancy-hydrogencomplexandananovoidcanbeconsideredto bequitedifferentfromathermodynamicperspective;whileforma- tionofacertainamountofvacanciesatfinitetemperaturesresults fromthegaininconfigurationalentropyofthecrystal,ananovoid represents an increase in surfacearea and is thermodynamically unstablerelativeto thesinglecrystal.Therefore,theenergetics of voidformationwasalsoevaluatedbasedonthesurfaceenergiesof theinnersurfacesofnanovoidinthepresenceofhydrogen.

2. Methodology

2.1. Densityfunctionaltheorycalculations

DFT calculations were performed using the projector aug- mented wave method implemented in the Viennaab initio sim- ulation package (VASP) and the generalized gradient approxima- tiondueto Perdew, BurkeandErnzerhof(GGA-PBE) [49–51]. The calculationswere carriedoutwithaplanewave cut-off energyof 500 eV andan electronic convergence criterion of10^–5 eV. A^- centered8×8×8Monkhorst-Packk-pointgridwasusedforthe cubicFm¯3munitcell[52].Theatomicpositionswererelaxeduntil theresidualforceswere within 0.01eV ˚A^-1 (0.02eV ˚A^-1 forcells containingdefectclustersatgrainboundaries).Theoptimizedbulk latticeparametera₀ was3.944 ˚A.

Defect clustersinthe bulk were studiedusing 4× 4 × 4su- percells(256 atoms)anda corresponding2×2× 2k-pointgrid.

Inaddition to isolated metal monovacancies, vacancyclusters vm

withm =2–6nearest neighborvacancieswereintroducedinlin- ear,trigonalplanar, tetrahedralandoctahedralconﬁgurations.Hy- drogen wassuccessively introduced on tetrahedral or octahedral sites asfar apart aspossible in the clusters, forming v_mH_n clusters(Fig.1).Adjacenttometalvacancies,thesesitesbecomesym- metricallyequivalentto3-foldhcpsitesonthe(111)surfaceand 4-foldsitesonthe(001) surface,respectively.Theconcentration dependentdissolutionenthalpy ofhydrogenintobulk Pdandthe associatedchemicalexpansionwascalculatedforﬁllingof1–16of the32 octahedralsites ina 2× 2 ×2 supercellincludingrelax- ationofthelatticeparameter.

Thevibrational frequencies ofPd andhydrogeninterstitials in bulk andassociatedwith avacancy were calculatedusing theﬁ- nitedisplacementmethod.Beforethesecalculations,thevibrating specieswerefurtherrelaxedtoanaccuracyof10^–4 eV ˚A^-1.

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Fig. 1. Bulk Pd structure with hydrogen in tetrahedral and octahedral interstitial positions adjacent to a metal vacancy (v). The hydrogen sites are structurally equivalent to 3-fold (1 1 1) and 4-fold (0 0 1) surface sites, respectively. Only the Pd atoms directly adjacent to vacancies are shown in the ﬁgures throughout this pa- per.

The ³⁽¹¹¹⁾ând⁵⁽² ¹⁰⁾ coincidencesitelatticebound- arieswereconstructedwiththeboundaryplanesseparatedby5(1 11)layers(13.7 ˚A)and14(210)layers(13.2 ˚A),respectively.Due totheperiodicboundaryconditions,thecellscontainedtwoequiv- alent grain boundaries. The super cell size was relaxed in steps of0.1 ˚A inthe directionperpendicular to thegrain boundary (z- axis),whichresultedinanoptimizedgrainseparationthatwasun- changedfor³ândêxtended^by^0.4 ^˚A^per^boundary^for^5.^De- fectclusterswere considered insupercells withan interface area of16.7˚A×19.3 ˚Afor³⁽⁵⁷⁶âtoms)ând^15.8Â^˚×17.6 ˚Afor⁵ (480atoms)andthecorrespondingk-pointgridswere2×2×1.

Surface energies were calculated for the (1 1 1) and(0 0 1) terminations and the contribution from atomic hydrogen adsor- bateson 3-fold and 4-fold sites,respectively, were included.The calculationswere performedusing slabs with a surface unit cell of√

2a₀ ×√

2a₀ andthicknessof7atomiclayers(28 atoms)and 6×6×1k-pointsampling.

2.2.Thermodynamicmodels

2.2.1. Formationofvacancyclusters

TheformationenergyofavmHnclustercanbeexpressedas

^Ev^fmHn=E^tot_v_m_H_n−E_bulk^tot +mE_Pd^tot−n 2

E_H^tot

2+kTln

(

^p^H2

)

(2) whereE_v^tot

mH_n andE_bulk^tot arethe totalelectronic energies ofthede- fectiveandbulksupercellsascalculatedbyVASP,respectively,kis Boltzmann’sconstant, T isthe temperature and pH₂isthe partial pressureofhydrogen.The totalenergies ofpalladium andhydro- genwere deﬁned fromthe bulk structure (per Pd atom)and H₂ molecule,respectively.

The vibrationalentropy ofthev_mH_n clustersandbulk Pdwas calculatedfrom the vibrationalfrequencies ofthe atomic species accordingto[53]

S^vib=−k

j

β

j

exp

β

j

−1−ln

1−exp

−

β

^j

(3)

where

β

j=h

v

j/kT,hisPlanck’sconstantandv_jarethevibrational frequencies.TheentropyofformationofthevmHnclusterswascal- culatedaccordingto

^Sv^fmHn=l

^S^fv+n

^S^fH (4)

where^S^fvwascalculatedasthedifferenceinvibrationalentropy betweenthatofaPdatomadjacenttoavacancyandthatofabulk Pd atom,and lwas thenumber ofPd atomsadjacentto the va- cancycluster(TableS1).SincebulkPdwasusedasreferencestate inEq. (1), therewasno changein vibrationalentropy forthePd atomsthatwere removeduponclusterformation.^S^fHwascalcu- latedasthedifferencebetweenthevibrationalentropyofa3-fold or4-fold Hatom ina v1H1 cluster andtheentropy ofmolecular H₂ which wastakenfromthermochemical tables[54]. Additional defectinteractionsandchangesinthevibrationalfrequencieswith increasedclustersizeandhydrogencontentwerenotconsidered.

The vibrational frequencies were also used to calculate the zero-pointenergy(ZPE)ofvmHn,bulkPdandmolecularH₂accordingto

ZPE=

j

h

v

j

2 (5)

Theformationenthalpyofv_mH_nclusterswascalculatedaccord- ingto

^H^fvmHn=

^Ev^fmHn+l

^ZP^E^v+n

^ZPÊH (6) where^ZPE^was^calculatedⁱⁿ^the^same^wayâs^for^theêntropies.

The change in ZPE of Pd upon vacancy formation in the grain boundarieswasassumedto bethesameasinthebulk, although onecouldexpectsomedifferencesforthe⁵^boundary^due^to^the more open structure. The binding energyof the vm clusterswas calculated relative tothe isolated vacanciesincludingZPE contri- butionsaccordingto

^E^bvm =E_v^tot_m −mE_v^tot+l

^ZP^Ev (7) The Gibbs free energyof formation of the v_mH_n clusters was calculatedas

^G^fvmHn=

^Hv^fmHn−T

^S^fvmHn−TS^conf_v_m_H_n (8) whereS^conf_v

mHn=kln(ⁿ) ^is^theconﬁgurationalentropy ofnhydro- genatomsdistributedoverthetotalnumberofsitesinthevacancy cluster,N,accordingto

ⁿ= N!

n!

(

^N⁻ⁿ

)

^! ⁽⁹⁾

Forsimplicity,andduetothe closeproximitybetweenhydro- gen on adjacent 3-fold and 4-fold sites, the clusters were only ﬁlledwithhydrogenoneither3-foldor4-fold.Thenumberofhy- drogensitesNarelistedinTableS1.Theequilibriumconcentration ofvmHn clusterswascalculatedaccordingto[55]

cvmHn=

z mexp

−

^G^fvmHn/kT

1+_m^zexp

−

^G^fvmHn/kT

⁽¹⁰⁾

wherez is the numberofconﬁgurations of thevm cluster (Table S1).Theconcentration(sitefraction)ofvmHnclusterswasthereby limitedtothenumberofPdsitesinthebulkorgrainboundaries.

2.2.2. Surfaceenergies

Foradsorptionofhydrogenontoasurfacesite^∗,i.e., ¹₂H₂+ ∗ =

∗H,theequilibriumhydrogencoverage,H,canbeexpressedfrom theequilibriumconstant,K_H,accordingto

H= KH√pH2

1+KH√pH2

(11)

KH=exp

−

^HH^ads

kT

exp

^S^adsH

k

(12) where ^HH^ads is the coverage dependent adsorption enthalpy in- cludingZPEcontributions,calculatedbyVASP.Thevibrationalpart

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Fig. 2. Relaxed structures and binding energies of vacancy clusters (dark spheres): a) linear v 2, b) triangular v 3, c) tetrahedral v 4, and d) octahedral v 6.

oftheadsorptionentropyandZPEwereobtainedinthesameman- nerasforthevmHnclusters,whilethechangeinvibrationalprop- erties of Pd upon adsorption could be neglected due to the low mass of hydrogen. The conﬁgurational entropy of the adsorbates wascalculatedas

S_H^conf=−k

(

Hln

(

H

)

⁻

(

¹⁻

H

)

^ln

(

¹⁻

H

)

⁽¹³⁾

Equilibrium hydrogen coverages were obtained by numerical solution of Eqs. (11–13) due to the interdependence of H and ^Hâds_H (H) ând ^Sâds_H (H)^. ^Surface ênergies ^with contributions fromhydrogenadsorbateswerecalculatedaccordingto

γ

hkl

(

H

)

⁼

1 2

E_slab^tot −rE_bulk^tot

+

H

^HH^ads−T

^SH^ads

A (14)

where E^tot_slab is the total energyof the slab unit cell, r is the ratio between the numberof Pd atomsin the slab and bulk cells, A isthe areaof thesurface, andhklare theMiller indicesofthe speciﬁcsurface.Thecalculatedsurfaceenergiescanbeusedtoes- timatetheenergyrequiredtogenerateavoidofacertainsizeand shape. Since the inner walls ofa voidare equivalent to the sur- facesofaparticle,theequilibriumshapeofthevoidcanbefound bygeneratingparticlesbasedonWulff constructions[56].Amini- mumenergypolyhedronsatisﬁesthefollowingcriterion

d_hkl

γ

hkl

=cW (15)

where d_hkl is the distance to a speciﬁc surface fromthe polyhe- droncenter,andthec_Wparameterdeterminesthesizeofthepoly- hedron. Thepressureofmolecular hydrogeninthevoids wasthe sameasinthesurroundingatmosphereasdeﬁnedfromthe con- stantchemicalpotentialofhydrogenthroughoutthesystem.

3. Results

3.1. Vacancyclustersinbulk

The different conﬁgurations of vacancy clusters are shown in Fig. 2. The corresponding binding energies showed a favorable

−0.04eVforv₂andupto−0.14eVpervacancyforthev₆cluster.

Thev₁H_n,v₂H_n,andv₄H_n clusterswereconsideredfurtherdueto theirrelativelyhighnumberof4-foldsitespervacancy(TableS1).

These clustersare shown inFig. 3 withthe 3-fold and4-fold sitessaturatedwithhydrogen.The3-foldhydrogensitesaremore numerous and more closely spaced within the cluster than the 4-fold ones. The 4-fold hydrogen sites reside within the atomic planes of Pd surrounding the vacancy cluster and the extent of unfavorableH–Hinteractionsathigherconcentrationscanthusbe expected to be lower.These considerations are supported by the

Table 1

Change in zero-point energy for H (3-fold and 4-fold) and Pd in v mH nclusters relative to bulk Pd and molecular H 2.

Species Cluster ZPE / Ev

3-fold 4-fold

H v 1H 1 0.029 −0.056

v 1H 8/v 1H 6 0.054 −0.047

v 2H 1 0.027 −0.055

Pd v 1 −6.48 ×10 ^–4

calculatedformationenthalpiesoftheclusters.ForthevmHn clusters with 3-fold hydrogen, there was a notable shift in the en- thalpycurves’slopearoundn=4–6,whichiswheremoreclosely spacedsitesareﬁlled(Fig.4a).Incontrast,theformationenthalpy ofthe clusterswith 4-fold hydrogenexhibited an essentially linear behavior as hydrogen wasadded. Overall, the formation en- thalpiesincreasedsubstantiallywithclustersizeduetothecostof forming Pd-vacancies,while this waspartlyalleviated by the in- creasednumberofhydrogensitesduetotheexothermiccontribu- tionofaddinghydrogen.Theformationenthalpieswere generally more exothermic for the 4-fold conﬁgurations, especially for the higherhydrogen contents. The formation enthalpies of the vmHn

clustersnotexplicitlycalculatedweretakenbylinearinterpolation betweenadjacentpoints,i.e.,linesinFig.4.

The ZPE contributions to the formation enthalpies of selected v_mH_nclustersarelistedin

Table1.Notably,^ZPE^wasêxothermic^for^4-fold^hydrogenând endothermicfor3-foldhydrogen, whichdemonstratesthe impor- tance of including ZPE contributions in determining dissolution andadsorptionenthalpiesforhydrogen.Aswiththeformationen- thalpies,thechangein^ZPE^with^hydrogen^content^was^quite^sig- nificantforthemonovacancyclusterswith3-foldhydrogen.Inthe caseofclusterswithmorevacancies,we assume that ^ZPE^does notchangenoticeably,sinceitisessentiallythesameforv₂H₁and v₁H₁.Furthermore,a lineardependenceoncoverage accordingto thev₁H_nvalueshasbeenusedin^Hv^f_mH_n forallclusters.Thecal- culatedvibrationalfrequencies andZPE are provided inTable S3.

Fig.4cshowsthecontributionstotheformationentropyofthein- dividualspeciesin a v₁H₁ cluster: hydrogen in3-fold and 4-fold conﬁgurations, and a Pd atom adjacent to the vacancy. The for- mationentropyoftheclusterswaspredominatedbythecontribu- tionfromhydrogen,primarily duetotheloss ofthetranslational androtationaldegreesof freedomofthe H₂ molecule.The vibrational entropyof hydrogen in the cluster was highestfor the 4- foldsite,andtheentropicpenaltywasthereforelowerbyasmuch as0.2eVper hydrogenat800°C.Incomparison,thecontribution fromtheincreasedvibrationalentropyofthePdatomssurround-

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Fig. 3. Relaxed structures of vacancy clusters (as shown in Fig. 2 ) saturated with hydrogen on 3-fold sites: a) v 1H 8, b) v 2H 12, c) v 4H 12; and 4-fold sites: d) v 1H 6, e) v 2H 8, f) v 4H 12.

Fig. 4. Formation enthalpy of v mH nclusters as function of the number of hydrogen atoms n in a) 3-fold and b) 4-fold conﬁgurations, and c) contributions to the formation entropy of a v 1H 1cluster from hydrogen and the vacancy as a function of temperature.

ingthevacancyremainedafractionofthecontributionofasingle hydrogen.

The calculated concentration of v_mH_n clusters is shown as a functionofinversetemperatureinFig.5.Forthev₁Hn cluster,the conﬁguration fully saturated with 4-fold hydrogen predominated uptoabout15°C(Fig.5a).The hydrogencontentdecreasedwith increasing temperature as the entropic contribution, nT^S^fv_mH_n, became increasingly important. Eventually, v₁H₁, v₁H₂ and the puremonovacancyexhibitedsimilarconcentrationsat800°C.The concentration ofthe v₁Hn with3-fold hydrogenwassigniﬁcantly lower,inparticularatlowertemperatures.Comparedtothesingle vacancyclusters,theconcentrationsofthev₂Hn andv₄Hn clusters remainedseveralordersofmagnitudeloweratalltemperatures.

As shownfor v₁Hn in Fig. 6a, the concentration of individual vmHn clusters followed a linear relationship with the hydrogen partialpressureonalogarithmicscale,i.e.,log(^cv_mH_n)∝^H^fv_mH_n∝

nlog(^pH₂)^. Accordingly, the total concentration of v₁Hn clusters increasedsubstantially with p_H₂ asthe clusterswithsuccessively higherhydrogencontentbecamepredominating.Nevertheless,the concentration ofthev₂Hn andv₄Hn clustersremainedseveralor- ders of magnitude lower (Fig. 6b). Notably, the concentration of v1Hn with 3-fold hydrogen showed a signiﬁcantly lower dependenceon p_H

2 comparedto the4-foldclusters. Thisdifference re- ﬂectsthenon-linearregimein^Hv^f_mH_n asthe3-foldsitesatclosest proximityareﬁlled(n=5–8inFig.4a).

Overall,theconcentration ofvacanciesinthebulk wassigniﬁ- cantlyenhancedinthepresenceofhydrogen,particularlyattem- peraturesbelow800°CandatH₂ partialpressures aboveapprox.

0.1 bar H₂. Nevertheless, monovacancy clusters, v₁Hn, predomi- natedlargerclustersunderalltheconsideredconditions.Thus,the results showed no strong tendency forvacancies to agglomerate intolargerclustersornanovoidsinthebulk.

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Fig. 5. Equilibrium concentration of clusters as a function of inverse temperature for an atmospheric hydrogen partial pressure of 1 bar H 2: a) sum of v 1H n and the contribution from each n , and b) similar sums for all the considered v mH nclusters. Solid lines are v mH 0in the order m = 1,2,3,4 from the top, and the black solid line is the sum of all vacancies. Square and triangular symbols represent 4-fold and 3-fold hydrogen, respectively.

Fig. 6. Concentration of vacancies and clusters as a function of p H2a) v 1H nat 673 K and 873 K with contribution from each n at 673 K, and b) all the considered v mH n

clusters at 673 K. The thick solid line is the sum of all vacancies. Square and triangular symbols represent 4-fold and 3-fold hydrogen, respectively.

3.2. Vacancyclustersatgrainboundaries

The optimized structures of the ³ând ⁵ ^grain ^boundaries are shown inFig. 7. The³^twin ^boundaryîs ^simply â ^stacking fault in the fcc lattice andthe Pd site atthe interface layer dif- fers frombulk only inthe symmetryof the12-foldcoordination.

Ontheother hand,the⁵^boundaryîs^moreôpenând^distorted, and four differentPd-sites were considered for vacancy segrega- tionandclusterformation.Fig.8showsthelocalstructurearound monovacanciesatthegrainboundarysitesandthecorresponding

segregationenergy,i.e.,totalenergydifferencebetweenacellwith avacancyresidingatthegrainboundaryandinthebulk.Theseg- regationenergieswereinsigniﬁcantorpositiveforthe³^bound- aryandsites1,3and4inthe⁵^boundary,^and^canⁱⁿ^general^be understoodby considering thecoordination environment andex- cessvolumeofthe sites[48].Asubstantial segregationenergyof

−0.67 eVwasobtainedforsite2nearthe⁵^boundary, ^and^this sitewasthereforeconsideredfurtherwithadditionofhydrogen.A v₂ clusterwasconstructedbyintroducinganadditionalequivalent vacancyon the other side of the boundary plane and a v3 clus-

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Fig. 7. Optimized structures of the a) 3 (1 1 1) and b) 5 (2 1 0) grain boundary unit cells with eye guides that highlight the orientation of the grains.

Fig. 8. Relaxed structure and segregation energy of Pd vacancies (dark spheres) in the a) 3 grain boundary and b–e) sites near the 5 grain boundary according to the numbering in Fig. 7 .

Fig. 9. Relaxed structures of saturated v mH nclusters associated with higher dimensional defects: a) 3 v 1H 6(4-fold), b) 5 v 1H 6(4/5-fold), c) 5 v 2H 11(4/5-fold), and d) 5 v 2H 11(3/4/5-fold).

terwasmadewithathirdvacancy onsite1inordertoobtaina triangulargeometry.

The consideredvmHn clustersatthe³^and⁵^grain^bound- ariesare depictedin Fig.9,andthe corresponding formationen- thalpies are presented in Fig. 10. Hydrogen was added to 4-fold sitesforallclusters,andfouradditional3-foldsiteswereﬁlledfor v₃H_ninordertocompletelysaturatetheinnersurfaceofthisclus-

ter.Comparingtheformationenthalpyofv₁Hninthe³^boundary withthatofthebulk(dashedlineinFig.10),therewasaclearsta- bilizingeffectatthe³^boundary^for^largerⁿ^.^The^main^difference betweentheseclustersistheconnectivitybetweenthe4-foldsites, whichinthe³^boundary ^share^edges ⁱⁿ^a^pairwise ^manner,^re- sultinginshorterH–Hdistancesof2.5 ˚Aalongonedirectioncom- paredto2.9 ˚A forv₁H_n inbulk (Fig.9a). The stabilizingeffectof

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Fig. 10. Formation enthalpy of vmHn clusters at 3 and 5 grain boundaries as function of the number of hydrogen atoms. Hydrogen were in 4/5-fold coordination except the last four in 5 v 3H n.

theclusterwassignificantlylargeratthe⁵^boundary,^partly^due tothe segregationenergyofthe vacancy.Ultimately,the segrega- tionenthalpiesamounted to−0.4eVand−1.4eVforv₁H₆tothe ³ ând ⁵ ^boundary, respectively, and−1.9eV for v₂H₈ to the ⁵^boundary^,^thusîndicatingâ^strong^tendency^forsegregationof theclusterstowardsthegrainboundaries.

The concentration of v₁H_n clusters at the ³ ⁽¹ ¹ ¹⁾ ^grain boundaryisshownasafunctionofinversetemperatureandcom- paredtothe bulkvaluesinFig. 11.Despitethesimilarityinlocal structure and the negligible segregationenergy of pure monovacancies,theconcentrationofv₁Hnclusterswassubstantiallyhigher at the ³ ^boundary ^compared ^to ^bulk, ^especially ^towards ^lower temperatures.Consideringthe v₁H₆ cluster inparticular,theneg-

Fig. 11. Concentration of v1Hn clusters (4-fold) at the 3 (1 1 1) grain boundary, with the bulk values shown for comparison, under 1 bar H 2atmosphere.

ativeformationenthalpyisclearlyevidentfromtheactivationen- ergyatlowertemperatures.

Theconcentration of vmHn clustersis furtherincreasedat the ⁵⁽² ¹⁰⁾ ^grain ^boundary,ⁱⁿ ^part^due^to^the ^addedsegregation energyof the isolated vacancy. As shown in Fig. 12, the consid- eredsite was saturated withv₁Hn clusters over the whole tem- peraturerangeat1barH₂.Inotherwords,thevacantsitecanbe consideredasastructuralpartofthegrainboundaryunderthese conditions.Forthev₂Hn andv₃Hnclusters,theconcentrationsbe- came saturated belowabout 175°C and20 °C, respectively. This isinremarkablecontrasttothecorrespondingbulkconcentrations whichwasattheir highestvaluesof10^–8 and10^–16, respectively, at800°C(Fig.5b).Clustersofallthreesizesthuscompeteforthe same sites at the ⁵ ^boundary ^at ^the ^lowest temperatures.The

Fig. 12. Concentration of v mH nclusters (4/5-fold) at the 5 (2 1 0) grain boundary under 1 bar H 2: a) v 1H n, b) v 2H n, and c) v 3H n. The dashed lines indicate ﬁlling of the 3-fold sites in v 3H n.

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Fig. 13. Equilibrium concentration of v mH n clusters as a function of the atmospheric partial pressure of hydrogen p H2at the 3 (1 1 1) and 5 (2 1 0) boundaries at 673 K.

predominatingclusteratagiventemperaturemaybeevaluatedby considering the Gibbs free energy of the clusters. From this, the v₃Hn clusters becamefavored over v₂Hn clustersbelow−233°C, asapparent alsofromtheminorgainin^H^f ^for^v3H₁₅compared to v₂H₁₁ (Fig. 10). Similarly, the v₂Hn clusterswere favored over v₁Hnbelow37°C.

Fig.13showstheequilibriumclusterconcentrationsasafunc- tionof p_H

2 at673 Katthe⁵^boundary. ^The^saturation ^concen- trationwasreachedatapressure aslow as0.2barforv₁Hn,and atabout5barand30barforv₂Hn andv₃Hn,respectively.Forthe ³^boundary, ^the ^saturation ^limit ^of ^v1H_n wasreached atabout 530bar.

3.3.Innersurfacesofnanovoids

The thermodynamicparametersofhydrogenadsorptiononthe (001)and(111)surfacesarepresentedinFig.14.Forthe(001) termination,theadsorptionenthalpywasessentiallyunaffectedby neighboringadsorbates. On the other hand,interactions between

adsorbates were prominent forthe (1 1 1) termination, and the surfacewasconsideredtobesaturatedwithhalfofthesitesoccu- piedduetothepositiveadsorptionenthalpy.Theresultingamount ofhydrogenadsorbatesperunitareawasfourhydrogenatomsper 31.1 ˚A²and 26.9 ˚A²for the (00 1) and (11 1) surface,respectively.

IntermsofH–Hinteractions,theoverallbehavioroftheadsorption enthalpiesinFig.14awasfoundtobesimilartothatobservedfor ﬁlling ofthe respective 4-fold and3-fold sites invmHn (Fig. 4b), aswell asprevioussurfacestudies[57].^ZPE^was−0.075eVand 0.030eVforthe(001)and(111)terminations,respectively.The adsorptionentropyforthe(111)terminationwasessentiallythe same asfor the 3-fold sites in v₁Hn and slightly more favorable for(0 01) terminationincomparison tothe 4-foldsitesin v1Hn

(Fig.4c).

Fig. 14c shows the calculated equilibrium hydrogen coverage andsurface energyas a function oftemperature in1 bar H₂ for both terminations. The (0 0 1) surfaceremained essentially saturated with hydrogen over the whole temperature range up to 1100 K. On the other hand, the coverage of the (1 1 1) surface decreased signiﬁcantly withincreasing temperature above 500 K since the adsorption enthalpy and entropy were both less favorable.Whilethe(111)terminationwasthemoststableinvacuum, the(001)terminationwassigniﬁcantlystabilizedinthepresence ofhydrogentotheextentofbecomingthemostfavorabletermina- tion.Thus,restructuringofthepalladiumsurfacescanbeexpected inhydrogencontainingatmospheres.

Fig. 15 shows the shape of larger nanovoids based on Wulff constructionsusingthecalculatedsurfaceenergies invacuumand 1 barH₂ at673 K,aswell ascorresponding atomisticmodels of nanovoidsofspeciﬁcsizes.Adsorptionofhydrogenclearlyleadsto a higher fraction of (0 0 1) facets, which further increaseswith increasingtemperatureduetotherelativedecreaseinsurfaceen- ergycomparedtothe(111)termination(Fig.14c).Theatomistic models are discrete in size and in the ratio betweenfacets. The theoretical facet ratioaccordingto theWulff shapes could there- forenot bereplicatedexactlyintheatomisticmodels,particularly forthesmallestsizes.

The calculated surface energies of voids of discrete size are shownincomparisontotheGibbsformationenergyofvmHnclus- ters in Fig. 16. The energy of the voids increases proportionally with the surface area (∝r²) and becomes orders of magnitude larger than the moststable vacancy clusters. There is signiﬁcant stabilizationofthevoidsinthepresenceofhydrogenduethere- ducedsurfaceenergywithhydrogenadsorbates.Basedonthelarge

Fig. 14. Thermodynamic parameters for adsorption of hydrogen on (1 1 1) and (0 0 1) surfaces: a) Coverage dependent adsorption enthalpy with insets of the fully covered (1 1 1) surface (top) and (0 0 1) surface (bottom), b) adsorption entropy as a function of temperature, and c) surface energy and equilibrium coverage as a function of temperature. The horizontal lines designate the surface energy of the pristine surfaces.

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Fig. 15. Wulff constructions based on the calculated surface energies in a) vacuum and b) 1 bar H 2at 673 K (c), and the corresponding atomistic models of b) 55 vacancies and d) 63 vacancies.

Fig. 16. Gibbs energy of formation of vacancy clusters and nanovoid energies as a function of effective radius in vacuum and in 1 bar H 2at 673 K. The effective radius of the clusters was deﬁned from the unit cell volume per atom/vacancy, and from the equivalent radius of a sphere with the same surface area as the Wulff surface for the voids. The dashed lines are square ﬁts of the void energies, and the inset shows that the formation energies of the clusters lie below these curves.

energeticaldifferences,itisreasonabletoexpectseparateunderly- ingmechanismsfortheformationofclustersandnanovoids.

4. Discussion

The presenceof pointdefects, suchasvacanciesin palladium, atfinitetemperaturesis athermodynamic necessityduethegain inconfigurationalentropyofthesystem.Thedefectconcentrations aredetermined bytheir Gibbsenergyofformation(Eq.(10)),and the monovacanciesare significantlystabilizedby association with interstitial hydrogenontheinternal 3-foldand4-foldsitesofthe v₁H_n complexes.Thus, theconcentration ofmonovacanciesis enhanced byseveralorders ofmagnitudeinthe presenceofhydrogen. However, coalescing of vacancies into larger clusters repre- sentsareductionintheconfigurationalentropyofthesystem,i.e., the principal cause for the presence of point defects in the first place. Thisaspect can be appreciated by considering theequilib- riumconstantofcoalescingmmonovacanciesintoalargercluster, i.e.,thereactioninEq.(1):

K= cvmHn

c^m_v₁ pⁿ_H^/²

2

=exp

−

^E^bvmHn

kT

(16)

where^E^bv_mH_n isthebindingenergybetweenmonovacancies and interstitialhydrogenatagiventemperature.FromEq.(16)itisap-

parent that the concentration of vmHn clusters follows a power- law dependence of the monovacancy concentration, c_v^m

1, which predominates the exponential dependenceon binding energy for larger clusters. Accordingly, the concentration of clusters of increasing size is quickly suppressed, as evidenced by the present results(Fig. 5b).It can thereforebe concludedthat theincreased concentration of (mono)vacancies in the bulk is a not a direct causeoftheformationofnanovoidsorporesinhydrogencontain- ingatmospheres.

Inthe case ofgrain boundaries, vacanciescan be further sta- bilizedonspeciﬁcsitesthatexhibitnegativesegregationenergies.

Particularlyinthe presenceofhydrogen,theconcentration ofva- canciesmayreachfulloccupancy forsuchsites,aswascalculated forthev₁Hn,v₂Hn andv₃Hn clustersassociatedwiththePd2site atthe⁵⁽²¹⁰⁾^boundary⁽^Fig.¹²^).^Due^to^theinterconnectivity ofthePd2sites,thesevacanciesmayformchannelsalongthegrain boundarywithdiametersofaround2.5–4.1 ˚A(assuminganatomic Pdradius of ^√₂²a₀ fromtheclose-packedplane).Channels ofthis sizemayconstitutefastdiffusionpathsforatomichydrogen,while possiblyalso allowing Knudsendiffusion ofsmallmoleculessuch as He used for probing trans-membrane leakage [30]. The max- imum size of voids formed by equilibrium concentrationsof va- cancyclusterswill,nevertheless,belimitedtothesiteswithinab- solutenearestvicinityofthegrainboundarywherestructuraldis- tortionsandsegregationenergiesmaybesubstantial.Accordingly, thelargestdiameterofsuchvoidsorchannelscanbeexpectedto beabout0.5nminmostcases.

Withrespect tothe formation oflarger voids,the surfaceen- ergyofthepredominating(001)and(1 11)facetswasfoundto besigniﬁcantlyreducedinthepresenceofhydrogenduetostabi- lization by adsorbates (Fig. 14c).Nevertheless, surfacesand voids remain thermodynamically unfavorable relative to bulk, meaning that voids will not spontaneously form inpalladium singlecrys- tals.Voidscantake theplaceofother higherdimensional defects inpolycrystallinematerialandtherebyeffectivelyreducetheirrel- ative energy. Moreover, formation of voids in the

μ

^m-range ^has

beenlinkedto plasticdeformationinPdmembranessubjected to pressuregradients [28].The originofthe formationofnanovoids andporesthereforeappearstobecloselylinkedtomicrostructural features and the presence of strain in the as-deposited ﬁlms, as well asunderoperation. Severalreportshavedetailedtheforma- tionofvoidsduetoplasticdeformation,i.e.,intheabsenceofhy- drogen[58–61].It isimportanttoconsiderthat palladiumunder- goessigniﬁcantchemicalexpansionduetodissolutionofhydrogen, withadditionalcontributions fromvacancy clusters. Forinstance, the 1D chemical expansion reaches 3% for PdH_0.5 (FigureS1), in goodagreementwithexperimentaldatafromFeenstra etal.[62]. Theresultingstrain inpolycrystallinematerialscanberelievedby accommodatingmicrostructuralchanges,andvoidsmaybecomea

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relativelylowenergyalternativeinthepresenceofhydrogen.Thus, small-grained polycrystalline palladiumand materials that other- wiseexhibitmicrostrainandlowcrystallinityshowsparticularpo- tential for forming and retaining stable hydrogen-covered voids.

The Pd-based membranes subject to void formation in previous studiesexhibitedthesemicrostructuralcharacteristics andsignifi- cantgraingrowthoccurredduringoperation[26,28–30].According totheseconsiderations, itisproposed that voidformationin Pd- basedmembranesmaybealleviatedbyannealinginhydrogen-free atmosphereso thatcrystallization,grain growthanddensification mayproceedin absenceofchemical expansion andundercondi- tions where voids are significantly less stable. While it is chal- lengingtodirectly characterizevmHn complexes andclusters,the roleofgrainboundariesandmicrostrainontheformationofvoids maybesystematically studied insingle- andpolycrystallinesam- plesannealedinhydrogen-containingatmospheresusingtransmis- sionelectronmicroscopy.

In summary,theformationofvacancy clustersandvoids have differentorigins. The role of hydrogenon their stabilityis, however, similar in that hydrogen terminates the under-coordinated metal atoms of the inner surface of the cluster or void. While the increased concentration of vacancies in the presence of hydrogen mayimprove the kineticsof microstructural changes,the presentworkindicates thatthey donotdirectlycausetheforma- tionofvoids.Hydrogenthereforehasaconfounding effectonthe observedvacancyandvoidformationphenomenainpalladiumand similarmetals.Furtherstudiesonhydrogeninducedvacancyclus- teringshould consider therole of chemical expansion andstruc- tureswithorderedmetalvacancies[12].

5. Conclusions

Palladium vacanciesare signiﬁcantlystabilizedinthepresence of hydrogen due to favorable association between the vacancies andinterstitial atomic hydrogen into vmHn clusters. Accordingly, the concentration of vacancy clusters is enhanced by several or- dersofmagnitude.Atgrainboundaries,theconcentrationofclus- ters can reach site saturation and form structural channels with diameters ofup to around 0.5 nm. Even at highhydrogen pressures,largerclustersandnanovoidsremainthermodynamicallyun- stableduetothelossofconﬁgurationalentropyuponcoalescingof monovacancycomplexes.The formationofnanovoids inPd-based membraneswasascribedtomicrostructuralcharacteristicsofpoly- crystalline materials, combined with strain induced by chemical expansiondueto dissolutionofhydrogenandplasticdeformation duetopressuregradients.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingﬁnan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto inﬂuencetheworkreportedinthispaper.

Acknowledgments

FinancialsupportfromtheResearchCouncilofNorway,through theCLIMITprogram(ContractNo.281824),andHYDROGENMem- TechASisgreatlyappreciated.Computationalresources werepro- videdby Uninett Sigma2undertheprojectnn9259k.Theauthors wishtothankDr.PatriciaA.Carvalhoforfruitfuldiscussions.

Supplementarymaterials

Supplementary material associated with this article can be found,intheonlineversion,atdoi:10.1016/j.actamat.2020.06.007.

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