Adsorption–strain coupling at solid surfaces Jo¨rg Weissmu¨ller1,2
Thisbriefreviewinspectshowchemistryorelectrochemistryat asolid–vapororsolid–electrolyteinterfacecoupletothe mechanicsofthesolid.Emphasisisonthecomplementarityof twoatfirstsightunrelatedphenomena:ontheonehand, adsorptionorelectricchargingchangethelocaltangential stressesinthesolidsurface;ontheotherhand,atangential strainofthesurfacechangestheadsorptionenthalpyandthe chemicalorelectricpotential.Oneandthesamematerials parameterunderliesthesephenomena.Thephenomenology andtheMaxwellrelationsbehindthatobservationare discussedandtheunderlyingmicroscopicmechanisms addressed,withparticularattentiontosymmetryandsignof thecouplingcoefficients.
Addresses
1InstituteofMaterialsPhysicsandTechnology,HamburgUniversityof Technology,Hamburg,Germany
2InstituteofMaterialsResearch,MaterialsMechanics,Helmholtz- ZentrumGeesthacht,Geesthacht,Germany
Correspondingauthor:Weissmu¨ller,Jo¨rg(weissmueller@tuhh.de)
CurrentOpinioninChemicalEngineering2019,24:45–53 ThisreviewcomesfromathemedissueonSeparationsengineering:
advancesinadsorption
EditedbyGennadyGorandBenoitCoasne
https://doi.org/10.1016/j.coche.2018.12.012 2211-3398/ã2018PublishedbyElsevierLtd.
Introduction
Theinteractionbetweenchemistryandmechanicsiscru- cialformanymaterialsphenomena.Itsimpactonprocesses inthebulkofsolids isthoroughlystudiedandof eminent technologicalimportanceinmetallurgy.Asjustoneexam- ple, themartensitetransformation—whichreliesonthe elasticdistortionofacrystallatticetoenhanceitssolubility forinterstitials,suchascarboniniron—has beenexploited for millennia for hardening steel. The presentarticle is focusedonthecouplingbetweenchemistryandmechanics atsolidsurfaces,whichisamorerecenttopic.Sincemany adsorption phenomena involve ions, and since electro- chemistryprovidesaparticularlyconvenientandquantifi- able way of manipulating thechemistryat surfaces,the couplingphenomenaherealsoincludetheaspectoflocal electriccharging.Thus,anydiscussionofchemo-mechanical
couplingatsurfacesrequiresthatthemoregeneralissueof electro-chemo-mechanicalcoupling is included. It is evenmore important to realize that there are two complementary views on the science underlying the various coupling phenomena,whichwillnowbeaddressed.
The firstviewfocusesonthefact thatsolidsurfacesare afflictedwithlocalstresses—quantifiedbythephenom- enological parameter ‘surface stress’—and that these stresses change during adsorption or electric charging.
Thisphenomenoncanbeobservedinexperimentswhich monitor thechange in macroscopic dimensions of solid bodies as they areexposed to varyingenvironments— porous bodies swell or shrink, cantilevers bend. The surface-stress-induced deformation is of relevance for such diversetopics as actuation with carbon nanostruc- tures [1] and metal nanostructures [2] or sensing with microscale [3] and nanoscale structures [4] and the extraction of shale gas [5]. Its impact on the strength and even crystal lattice stability of nanoscale metals is under discussion[6,7].
Thesecondviewfocusesonchangesintheadsorption energy on solid surfaces when strain in the tangent plane is imposedon the surface.In the consequence, experiments observe the decoration of local stress concentrations at surfaces by adatoms [8] and the variation ofadsorptionenthalpies [9,10]orelectrode potentials [11,12] when surfaces are uniformly strained. Experiment also confirms this phenomenon asameansformodulatingthe reactivityofsurfacesin electrocatalysis [13].
Bothviewsareopento inspectionbyelectron-theoretic density functionaltheory(DFT) computation.Muchof the current interest in heterogeneous catalysis with strainedsurfacescanbetracedbacktoearlypredictions from that latter technique [14]. Furthermore, DFT confirms that the surface stress of metalsvaries during adsorption [15]and theworkfunctionof metalsin vac- uum[16]—closelyrelatedtotheirelectrodepotentialin electrolyte—as well as adsorption energies on metal surfaces[17,18]varyduring straining.
Here,webrieflysummarizethecurrentviewonelectro- chemo-mechanical coupling, with an emphasis on the complementarity of the above two points of view. In theinterestof simplicity,muchofthecurrentliterature isfocusedonsurfacesofhighsymmetry,whereattention canberestrictedtoisotropicstressandstrainandtoscalar materials parameters. Yet, interesting phenomena arise when lower symmetry is admitted. Our discussion
thereforestartsoutwithconsideringcouplingatgeneral, anisotropicsurfacesandintroducesisotropyintheplane of thesurface as aspecial case. The phenomenological descriptionofthesurfacebehaviorwillformthecenterof the discussion;it will be supplemented by remarkson atomic-scale processes and by selected examples for numerical values of materials parameters derived from experimentorfromabinitio computation.
This article touches on several points that have either been the subject of controversial discussion or that deserve more detailed discourse. Brevity requires that thereaderis referredto therelevant literature. Specifi- cally,thisrelatestothenotionthatstressatsolidsurfaces isquantifiedbythesurfacestressandexplicitlynotbythe surface tension [19], to the distinction between the Shuttleworth equation for surface stress in laboratory coordinates and the simpler defining equation in Legendre coordinates that is embraced here [20], and tothedescriptionofthemechanicalinteractionofafluid withasolidsurfaceintermsof,alternatively,the‘pressure tensor’inthefluidor,as here,thesurfacestress[21].
Fundamentals
Ouranalysisrestsonthediscussionofsurfacemechanicsin the seminal work by Gurtin [22] and its extension to electrode surfaces by Weissmu¨ller and Kramer [23].
Briefly,weconsiderasolidbodywhichmay benonuni- formly stressed in itsbulk and which is immersed in a uniformfluid.Aninterfacialfreeenergydensity,c,repre- sentsthelocalexcess(overthehomogeneousphases)in freeenergyperareaatthesolid–fluidinterface.Focusing firstonelectrolyte(superscriptE)asthefluid,wetakecEa functionof thestatevariables superficialcharge density (per area), q, andtangentialstrain, E. The fundamental equation,whichidentifiestheenergy-conjugatevariables electrodepotential,E,andsurfacestress,S,is
dcE¼EdqþS:dE: ð1Þ
Electrode surfaces can behave in an ideal capacitive manner,sothatqvarieswithEandthereisnoadsorption.
Here,asduringanidealelectrosorptionprocess,qandthe specific excess,G, (moleculesperarea)of adsorbateare linked since electric charge is here exclusively trans- portedbytheadsorptionofions.Thisalsoimpliesalink betweenEandthechemicalpotential,moftheadsorbate inthefluid:
dq¼zFdG ð2Þ
zFdE¼dm ð3Þ
withFFaraday’sconstantandzthesignedvalencyofthe adsorbedmoleculein solution.
Forsurfacesincontactwithgas(superscriptG),adsorp- tioniscontrolledbym,andthefundamentalequationis
dcG ¼mdGþS:dE: ð4Þ
Maxwellrelationsforelectrocapillarycoupling andsorption–strain coupling
Gokhshtein[11]firstpointedout thatthe samemate- rialsparameterofanelectrodesurfacedescribesonthe one hand the coupling between surface stress and charge density and on the other hand the coupling between electrode potential and strain. Haiss [24] demonstratedthatthemagnitude ofthecouplingpro- vides insights into electrode processes at a molecular level. The team of the present author has designed modern experimental approaches to verify the two sides of the underlying Maxwell relation and found excellent agreement [12,25]. Two variants of the Maxwellrelation emergenaturally from Eqn1 and4, namely
KE¼dS dq
E
¼dE dE
q
ð5Þforelectrolyteand
KG¼dS dG
E
¼dm dE
G
ð6Þforgas.
The tangential superficial tensors S and E describe a stress and strain, respectively, that act tangentially in theplane of thesurface and that can be anisotropic in the plane. The electrocapillary coupling parameters K haveanalogous characteristics. Assecond derivatives of thestatefunctionc,theKarematerialsparametersofthe surface.
Surfacesincontactwithgasorvacuumhavenodefined capacitanceandsocannot,asarule,becharged.Thereis herenoanalogontothecapacitivechargingofelectrode surfaces.However,theworkfunctionWprovidesamea- sure for the chemical potential of the electrons in the solid.Thecouplingoftheworkfunctiontostrainisinfact open to straightforward evaluation by DFT [16]. The relevantcouplingparameteris
KW¼1 F
dW
dE ð7Þ
forelectronexchange.
Strain-dependent adsorptionenthalpy
ThequantityKG ¼dm=dEjGinEqn6canbelinkedtoa strainderivativeoftheadsorptionenthalpy,Dhads.3This is readily illustrated for the example of the Langmuir isothermforadsorptionofnon-interactingmoleculesata surfacewithadiscretenumberdensity,Gmax,ofadsorp- tionsites perarea,
m¼m0þDhadsþRT ln½u=ð1uÞ; ð8Þ
withu=G/Gmaxthefractionalcoverageandm0thechem- ical potential in a reference state. Taking the strain derivative of m in Eqn 8 at constant G and, hence, at constant u,gives
Kads¼dm dE
G
¼dDhads
dE
u
ð9Þ
Thisresultcontinuestoholdwhen,inageneralizationof the Langmuirisotherm, interactions betweenadsorbate moleculesarepermittedand Dhadsisallowedto depend explicitly onu.
Isotropic surfaces
The surface stress must be isotropic in the plane of surfaceswith threefoldorhighersymmetry,suchas the (111) and (100) surfacesof face-centeredcubiccrystals.
ThenS¼fPwithfascalarsurfacestressparameterand Patangentialunittensorintheplane.Itfollowsthatthe electrocapillary or adsorption–strain coupling are also isotropic,andscalarcouplingparameters&(forelectrode processes and electron emission) and z (for adsorption fromgas)canbedefinedas
&¼traceKE¼df=dqje ¼dE=dejq ð10Þ
z¼traceKG¼df=dGje ¼dm=dejG ð11Þ zads ¼traceKads¼dDhads=de ð12Þ
&W¼traceKW¼F1dW=de ð13Þ
with e¼traceE the relative change in surface area (measured inlaboratorycoordinates) byelasticstrain.
Inelectrolyte,thetwosidesoftheMaxwellrelation,Eqn 10, havebeen exploredbycantileverbendingor porous metal expansion experiments that probe surface stress variationsandbydynamicelectro-chemo-mechanicalanal- ysis(DECMA),probingpotentialvariationinresponseto strain. Density functional theory provides data for the coupling at solid surfaces in contact with gas. For Au (111)invacuum,theDFTvalue[16]ofthework-function straincoupling,&=1.89V,isquiteprecisely(towithin 5%)equaltosurfacestress-chargeandelectrodepotential–
straincouplingparameters,whichare2.0V[26,27]and 1.9V[12],respectively.Theagreementprovidesstrong supportfortheMaxwellrelation,anditalsosuggeststhat the electrocapillary coupling parameters for charge- exchange at thesolid–vacuum interface, &W, and atthe solidelectrolyteinterface,&,connecttothesamephysics andagreeinmagnitude.
Figure1illustratesthecomplementarityofsurface-stress charge and electrode potential–strain coupling that is
Figure1
Complementarityofsurface-stresschargeandelectrodepotential straincouplingexploredbyactuationandsensingexperimentswitha porousmetalsample,Ref.[4].Topframe:scanningelectron micrographshowingmicrostructureofnanoporousgold,indicating idealizedconceptofa‘ligament’ofthemetalnetworkstructureasa cylindricalstrut.Lowerleftframeillustratesactuation,indicating contractionofligamentwhennegativecharge,dq,isdepositedonthe strutandcompensatedbypositivecounterionsinelectrolyteinthe porespace.Lowerrightframeillustratessensingwhereanexternally imposedlocalstraindeatconstantelectrodepotential,E,induces polarizationoftheinitiallychargeneutralelectrodesurface.CEand RE:counter-electrodeandreference-electrode,respectively.
3Energyorenthalpyofadsorption?Withattentiontofluids,phenom- enologicalthermodynamicsdistinguishesenergyandenthalpybytheir statevariableformechanics,namelyvolumeorpressure,respectively.
Adsorptionatasolid–fluidinterfacetypicallyinvolvesmixedboundary conditions:Theconstantpressureinthefluidfixesthenormalcompo- nentofthestressinthesolidwhereasthestiffnessoftheunderlying crystalfixesthetangentialcomponentofthestrain.Thus,thereisno obviousbasisforreferringtotheworkwhichisdoneduringadsorptionas either, an energy or an enthalpy of adsorption. Here weadopt— somewhatarbitrarily—theterm‘adsorptionenthalpy’.
embodiedintheMaxwellrelation,Eqn10,fortheexam- pleofanactuationandsensingexperimentwithaporous metalbody(nanoporousgold,Ref.[4]).Wettingthepore surfaceswithelectrolyteandtransferringchargebyapply- inganexternalvoltageleadstosurfacestressand,hence, macroscopicstrainoftheporousbody.Conversely,apply- ingamacroscopicloadleads tolocal strainofthemetal backbone; this polarizes the metal-electrolyte interface andleadsto chargetransfer [4].
The agreement between DFT for the work function strain responsein vacuum and experimental results for the electrocapillary coupling in electrolyte exemplifies that processes involving exchange of the same particle (electron, atom or molecule) have sensitively the same strengthofcoupling,irrespectiveofwhethertheprocess acts in gas or in electrolyte. The electrocapillary and sorption–strain coupling coefficients are then simply relatedby
&¼ 1
zFz ð14Þ
Specifically,for monovalentanions(z=1) theelectro- capillary couplingparameterin units ofV hasthesame numerical magnitude as the sorption–strain coupling parameterinunits ofeV.
Anisotropic coupling
The surface stress will generally be anisotropic in the planeofsurfacesoflowerthanthreefoldsymmetry.The (110) surfaces of face centered cubic crystals provide a simple example[28], whilemore complex structures of lowsymmetryarefoundinreconstructionsofmanyother crystalsurfaces.Thereconstructioncaninfactbedriven by the strife of the surface to reduce its energy by relievingthesurfacestress.Forinstance,terracesofclean Au(111) canundergo a ‘herringbone reconstruction’ by compressionalong theh110idirection[29,30],relieving thetensilesurfacestressofthebulkterminatedsurface.
As a consequence of the surface stress anisotropy, the orientationdistributionofdomainswithdifferentcrystal- lographicorientationalongthesurfacecouplestouniaxial strainsthat are externally imposed on the surface[31].
Lowsymmetryreconstructionsofinitiallyhighsymmetry surfaces are often the consequence of adsorption, as exemplifiedbythe‘stripedphase’of twofoldsymmetry onoxygencoveredAu(111)[32,33].
Vicinalsurfaces,whichareslightlymisalignedrelativetoa nearby low index orientation, provide another obvious exam- pleoflowsymmetry.Crystallographically,themisalignment isrealizedbystepedges.Theselineardefectsinteractby short-range,dipole-likestressandstrainfields[34,35,36].
Ontopofthat,stepsareassociatedwithlong-rangetangen- tialstressesthataddtothesurfacestressoftheterraces[37].
Inacoarserpictureofthesurfaceasanelasticcontinuum, corrugation or roughness may be anisotropic. Here, the amplitude and orientation of the corrugation affect the magnitudeandanisotropyoftheeffectivesurfacestressthat acts on the underlying crystal [38,39]. Some adsorbates attachpreferentiallyto stepedges, reducing theirenergy andtherebypromotingmorphologychangesofthesurface, between faceted(few stepedges) andrough (manystep edges).Cyclictransitionsbetweenroughandfacetedstates oftransition-metalornoble-metalsurfacesduringcatalytic reactionsprovidestrikingexamples[40,41].
Evenhighsymmetrysurfacescanofferadsorptionsitesof lowsymmetry. Forinstance, thebridgesite on(111) or (100) surfaces of fcc crystals connects two neighboring surfaceatoms.Thisisalocalconfigurationwithtwofold symmetry.Ithasrecentlypointedoutthatthecoupling betweenadsorption enthalpy and strain of such config- urationsdepend on thestrain direction[18]. In other words, even though the surface has three- or fourfold symmetry,coupling is notadequately described bythe scalar electrocapillary coupling parameter. Instead, the moregeneralcouplingofEqn9applies.Thisobservation isreconciledwiththeearlierconsiderationsonsymmetry when one considers that a uniaxial strain in the plane breaks the symmetry of the surface. It is tempting to dismissthis asasecond-order effect,so thatanisotropic couplingbecomesrelevantonlyatsufficientlylargestrain magnitude.Yet, surfacescienceprovidesobviousexam- ples for discontinuities of surface properties near high symmetrypoints. The pronounced cuspsin thesurface energy-versus-misorientationgraphsofvicinalsurfacesat their low index orientations [42] provide a prominent example. Furthermore, the electrocapillary coupling parametersofcleanandcapacitivelychargedsiliconsur- facesexhibitdiscontinuitiesarounde=0[43],whichcan berelatedto thesymmetry-breakingeffectof thestrain ontheelectronic bandstructure.Thisconfirmsthatthe symmetryofanimposedstrain,isotropicoruniaxial,may berelevantforthechangesinadsorption phenomena.
Signoftheelectrocapillary andsorption–
strain couplingparameters
Electrocapillary couplingparameters for capacitive pro- cessesattransitionmetalsurfacesareinvariablynegative valuedandintherange0.5to2V[44].Yet,thesignis notforceful,asexemplifiedbypositive-valued&inthesp- bonded metalsAl and Mg[45]. Furthermore, & for Si surfacescanexhibiteithersign,dependingonthenature (compressionor tension)of thestrain andontheorigin (conduction band or valence band) of the transferred electron[43].Relevantmicroscopicphenomena,assum- marizedinRef.[45],areadecreaseoftheFermiwave- number(&")and of thesurface dipolestrength (&#)by tensilestrain,arelaxationoftheoutermostlayerofatoms (whichcanbeinward[&#]oroutward[&"]),alongwith— typically—electron enrichmentin the bondingregions
in-betweenthesurfaceatoms(&#)andadepopulationof screening orbitals around the ion cores (of transition metals; &#) upon electron accumulation in the surface.
DFT studiesof sp-bonded and transitionmetalsreveal the relative contributionsof these partlyopposing phe- nomena.Yet,asimplepredictivetheoryforthenetvalue or ofsimply thesignof &isnotinview[45].
Inspiteofthecomplexityrevealedbydetailedstudiesof theelectrocapillarycoupling,itispopulartodiscussthe couplingbetweenstrainandadsorptionenthalpyinterms of asimpler approach termedthe d-bandmodel [14].
Withaneyeonlatetransitionmetalsurfaces,thatmodel restsonthenotionthattensilestrainnarrowsthed-band and,intheconsequence,increasestheenergyofitsstates, therebymaking thesurfacemore bindingforadsorbate.
Hence, z<0—a trend which is indeed qualitatively bornoutbymostof theknownsorption–straincoupling parametervalues.Forinstance,transitionmetalsurfaces typicallybecomemorebindingforhydrogen[10],oxygen species [46],and CO [9,14] whenstrainedin tension.
However, there are also important exceptions. Experi- ments on oxygen species electrosorption on Au(111) provide aprominent example:intheearlystagesof the electrosorptionprocess theadsorption becomesstronger upontensile straining(z<0),yetinthelaterstagesthe trend is inverted and tensile straining weakens the adsorption,hencez>0[47].Inthisinstance,thetrends canberelatedtosurfacereconstruction:Oxygenadsorbs initiallyonpre-existingsitesofthebulk-terminatedgold surface,andthisprocessisassociatedwiththeexpected, negative-valued z. Subsequently,as one possible mani- festation of the ‘replacement turnover’ process, gold atoms maymovefrom terrace-siteor step-siteintoada- toms sites and coordinate with oxygen. This recon- structed surfacehasapositive-valued z[47].
TheobservationsonOHonAuemphasizetherelevance ofstructuralchangesinthesurfacefortheelectrocapillary coupling.Asalreadymentionedabove,relaxation ofthe atomic positions of the substrate surface atoms in responseto either,tangentialstrainor electriccharging, is also an important contribution to the electrocapillary couplingduring capacitiveprocesses[45,48,49].
The discussion of chemo-mechanical coupling in this article is focused onstrong bonds. Yet, based on DFT datainRef.[50],Goretal.[21]haveinferredsubstantial
z-values also for van-der-Waals bonded adsorbates: For
CH4 and CO2 on graphene, they quote z=64 and 69mV,respectively.
Sorption–strain couplingduring underpotential deposition
Forcertaincombinationsoftwometals,theionsofonemetal insolutioncanbeelectrochemicallyreduced—atelectrode potentials positive of the bulk Nernst potential—to form up
tooneorsometimes twomonolayer thickadsorbatelayerson substratesfromtheothermetal.Thisprocessisreferredtoas underpotentialdeposition(UPD).Inviewofthewidespread notionthatthesorption–straincouplingisregularlynegative itisremarkablethatPdUPDonAumakesthesurfacestress more tensile (more positive f) [51]. Eqn 11hereimplies z>0.Thesignisimmediatelyrationalizedasanexpression of the coherencystressinapseudomorphiclayerwithalesser latticeparameterthanthesubstrate[51].Iftheatomsinthe layeraresmallerthanthoseofthesubstrate—asisthecase forPdonAu—thenthelayerisstrainedintension.This impliesatrendformorepositivefwhenthelayerisformed and,hence,atrendforz>0.Conversely,depositinglarger atomsresultsincompressioninthelayer,henceatrendfor morenegativefandforz<0.
Asawordofcaution,itisnotedthatthelinkbetweensize and adsorption-induced stress is by no means forceful.
Severaladditionaleffects maybesuperimposed:
ManyUPDlayersarereconstructed[52],asopposedto pseudomorphic.Reconstructionaffectsthestressesin thesurface,seeabove.
Even for pseudomorphic UPD layers, the effects of relaxation in the substrate and of charge-exchange betweensubstrateandlayercontributetothevariation insurfacestress,ontopofthemisfitstraineffect.The surface stressevolutionduring the UPDofBi onAu exemplifiestheseeffects[53,54].Indeed,thenumeri- calmagnitudeoftheelectrocapillarycouplingparame-
ter for Pd UPD on Au, &=0.14V, is substantially
smallerthanestimatedbasedonmisfitstrainalone[51].
UPD processes can involve the replacement of adsorbedanionsbythemetal[54],andthenetchange insurfacestresswillthereforedependonthedifference betweensurface stressesin therespective twoadsor- bate layers,metalversusanion.
Inthelimitofhighcoverage,someUPDlayersundergo a transitionfrom pseudomorphic (i.e. commensurate) to incommensurate.Again, BiUPD onAu fromelec- trolyte [53,55] provides an example. Gor et al. [21] emphasizethatthesurfacestressvariationinthelimit of a purelyincommensurate adsorbatelayer depends onlyontheadsorptionisothermforG(m)andisother- wise independent of theadsorbate–substrate interac- tions. Under these conditions, one finds simply df=Gdm.4Inotherwords,thetransitionfromcom- mensurate to incommensurate makes the size misfit irrelevant.
4ThisrelationbetweendfanddmisthedifferentialformofEqn25in Ref.[21]whenthereisnocommensurateadsorption.Thevariationin thesurfacestress,f,ishereidenticalto thevariationinthesurface tension,g,yetthisidentityholdsonlyundertherestrictiveconditionofa saturatedandincommensuratelayer.
Inspiteoftheimportanceofadditionaleffects,empir- icalstudies confirmthatthesizeoftheadsorbed atom or molecule can be decisive for the sorption–strain coupling. The role of size, which is illustrated schematically in Figure 2, is strikingly confirmed by studiesofthesurface-stresschargecouplingaswellas sorption–straincouplingfortheUPDofHonPd.Here again, several sets of experimental data can be com- pared: porous metal dilatometry suggests &=+1.17V [56] for the surface stress-charge coupling, while DECMA and electrosorption on strained pseudomor- phicPdmonolayerssuggests thepotential–straincou- pling&=+1.1V[57]and&=+1.40.2V[58],respec- tively. These values are in reasonable agreement.
Significantly, the sign and numerical magnitude of the coupling is correctly predicted by a continuum mechanicsmodelthatanalyzesmisfitstraininaH-rich
superficial layer, relying on the partial molar volume ofhydrogeninbulkPdandonPd’selasticparameters as only materials parameters: this model yields
&=+1.15V[56].Inotherwords, thesizemisfitishere
akeyparameter governing signandmagnitude ofthe sorptionstrain coupling.
Alink betweencontinuum approaches and atomistictheory
The success of continuum approaches in linking the enthalpyof adsorption tomechanicsisnotsurprising in viewofthewell-knownsuccessofsimilarapproachesfor theenthalpyofabsorption(orenthalpyofsolution)inthe bulk.BuildingontheorybyMottandNabarro[60]and byEshelby[61],Miedema[62]confirmedthattheelastic partoftheenthaplyofsolutioninbulkphasescanbewell estimatedfromtheenergyofthemisfitstrainfieldinthe
Figure2
Mechano-chemicalcoupling-relatingatomic-scaleinteractionstothesignofthephenomenologicalcouplingparametersandtoexperimental signaturesinthemacroscopicworld.Adsorbedmoleculescanrepresentcentersofdilatationorofcontraction,leadingtocompressiveortensile localstresses(‘eigenstress’)inthesurface,asdepictedinthecentralrow.Externallyimposedtensilestraininteractsconstructivelywiththe compressiveeigenstressofdilatationcenters(exemplifiedinthetoprow,leftframe),enhancingthebindingstrength.Theoppositeappliesto contractioncenters(rightframe).Dilatationcentersgiverisetocompressivesurfacestress(f<0,bottomrowleft),leadingtomeasurable deformationofmacroscalecrystals—forinstancethebendingofcantileversasillustratedschematicallyinthebottomrow.Contractioncenters inducef>0andoppositedeformation(bottomrowright).Thisconnectsmeasurementsofsurfacestressbycantileverbendingorbydilatometry tothecouplingbetweenadsorptionenergyandstrain.
ReproducedfromRef.[59].
matrixcrystal whenthesolute atomistreatedasamis- fittingsphere.Thestepfromstrainfieldtoenergysimply involvesintegratingthestrainenergydensitythatcomes fromthelocaldistortionofthecrystallatticearoundthe soluteatom.Thehydrostaticcomponentofthatstraincan be compressive or tensile, depending on whether the solute locally expands (interstitials or large substitu- tionals) or contracts (small substitutionals) the matrix (Figure2,centralrow).Atthesurface,adsorbingadatoms which act as centres of expansion (e.g. H onPd) or of contraction (e.g. Pd on Au) makes the surface stress, respectively, more compressive or more tensile [51,56].
Thisisillustratedin Figure2,bottomrow.
The Maxwell relation, Eqn 11, immediately links the above-mentioned trends for solute with differentmisfit strain to sign and magnitude of thecoupling, z, of the adsorption enthalpy to an externally imposed strain.
Continuum mechanics clarifies the origin of this link:
whereas the contribution ofmisfit strain to the energy of adsorption in theabsence of external forces may be represented by a volume integral over the local misfit strain energy density, externally imposed deformations generateanextraenergytermthatistheintegraloverthe product of the local misfit strain and the externally imposed stress [63]. This reasoning is confirmed by a recentatomisticevaluationoftheinteractionofthelocal stress fields aroundadsorbates (‘eigenstress’): Atom-by- atomsummationoftheinteractionenergyintegralrepro- ducesthelinkbetweenthesignofthemisfitstressaround adsorbates and the sorption strain coupling strength [18].Since the Maxwell relation includes the surface stress,andsincesurfacestressleadstomeasurabledefor- mationofmm-sizeorcm-sizecantileversorporousbod- ies, experimentsexploring surface-induceddeformation are immediatelylinkedtotheatomic scaleinteractions.
Figure 2illustratesthatlink.
Conclusion
The results compiled in this article emphasize that chemo-mechanical or electro-chemo-mechanical cou- pling at the interface of a solid with vacuum, gas or electrolytecanbedescribedbyasmallsetofparameters.
Essentially,foreachindividualprocess—namelyadsorp- tionofaspecificatomorionorexchangeofanelectronon aspecificsurface—thereexistsasorption–strainorelec- trocapillarycouplingparameterthatquantifiesboth,the impact of the surface process on the stress within the surface andtheimpactof asurfacestrainonadsorption enthalpyandelectricorchemicalpotential.Theseparam- eterscanoftenberepresentedasscalars,butlowsymme- tryprocessesthatrequiretherepresentationbysuperficial tensorsarenotuncommon.Atthestateoftheart,various experimentaltechniquesareavailableforquantifyingthe coupling. Electron-theoretic density functional theory can also readily quantify these phenomena. The desorption–strainorelectrocapillarycouplingparameters
allowexperimentalistsandtheoreticianstoquantitatively comparetheirresults.Majoradvancesinthefieldmaybe expectedifthisopportunityissystematicallyexploitedin futurework.
Acknowledgement
ThisworkwassupportedbytheGermanResearchFoundation(DFG), grantWe1424/16-1.
References and recommendedreading
Papersofparticularinterest,publishedwithintheperiodofreview, havebeenhighlightedas:
ofspecialinterest ofoutstandinginterest
1. BaughmanRH,CuiCX,ZakhidovAA,IqbalZ,BarisciJN, SpinksGM,WallaceGG,MazzoldiA,DeRossiD,RinzlerAG, JaschinskiO,RothS,KerteszM:Carbonnanotubeactuators.
Science1999,284:1340-1344.
2. Weissmu¨llerJ,ViswanathRN,KramerD,ZimmerP,Wu¨rschumR, GleiterH:Charge-inducedreversiblestraininametal.Science 2003,300:312-315.
3. BoisenA,ThundatT:Design&fabricationofcantileverarray biosensors.MaterToday2009,12:32-38.
4.
StennerC,ShaoL-H,MamekaN,Weissmu¨llerJ:Piezoelectric gold:strongcharge-loadresponseinametal-basedhybrid nanomaterial.AdvFunctMater2016,26:5174-5181.
Illustratesthequantitativeequivalencebetweensurface-stresscharge andpotential–straincouplingbycomparingactuationandactivestrain sensingbehaviorofmacroscopicbodiesofnanoporousgold.
5. KaracanCO:Swelling-inducedvolumetricstrainsinternaltoa stressedcoalassociatedwithCO2sorption.IntJCoalGeol 2007,72:209-220.
6. DiaoJK,GallK,DunnML:Surface-stress-inducedphase transformationinmetalnanowires.NatMater2003,2:656-660.
7. MamekaN,MarkmannJ,Weissmu¨llerJ:Ontheimpactof capillarityforstrengthatthenanoscale.NatCommun2017, 8:1976.
8. GsellM,JakobP,MenzelD:Effectofsubstratestrainon adsorption.Science1998,280:717-720.
9. SchlapkaA,LischkaM,GrossA,KasbergerU,JakobP:Surface strainversussubstrateinteractioninheteroepitaxialmetal layers:PtonRu(0001).PhysRevLett2003,91.
10.
KiblerLA,El-AzizAM,HoyerR,KolbDM:Tuningreactionrates bylateralstraininapalladiummonolayer.AngewChemIntEd 2005,44:2080-2084.
Astrikingdemonstrationofhowstrainofasolidsurfacecansubstantially changeadsorptionenthalpy.
11. GokhshteinA:Chargeacquirementbyanelasticallystrained electrode.DoklAkadNaukSssr1969,187:601.
Thispaperwasthefirsttohighlightthecomplementaritybetweensur- face-stresschargeandpotentialstrainresponse.
12. SmetaninM,KramerD,MohananS,HerrU,Weissmu¨llerJ:
Responseofthepotentialofagoldelectrodetoelasticstrain.
PhysChemChemPhys2009,11:9008-9012.
Presents a modern experimental approach for quantifying how the potentialofanelectrodevariesinresponsetostraining.
13.
DengQ,SmetaninM,Weissmu¨llerJ:Mechanicalmodulationof reactionratesinelectrocatalysis.JCatal2014,309:351-361.
The firstdirect and quantitative experimentaldemonstration that the activityofthesurfaceinheterogeneouscatalysiscanbemodulatedby straininthetangentplaneofthecatalystsurface.
14. MavrikakisM,HammerB,NorskovJK:Effectofstrainonthe reactivityofmetalsurfaces.PhysRevLett1998,81:2819-2822.
This paper is influential in thefield ofheterogeneous catalysis with strainedsurfaces.Itoutlinesthed-bandmodel,rationalizinghowstrain ofatransitionmetalsurfacesshiftstheadsorptionstrength.
15. FeibelmanPJ:First-principlescalculationsofstressinduced bygasadsorptiononPt(111).PhysRevB1997,56:2175-2182.
16. UmenoY,ElsasserC,MeyerB,GumbschP,NothackerM, Weissmu¨llerJ,EversF:Abinitiostudyofsurfacestress responsetocharging.EurophysLett2007,78:13001.
17. SethuramanVA,VairavapandianD,LafouresseMC,AditMaarkT, KaranN,SunS,BertocciU,PetersonAA,StaffordGR,GuduruPR:
Roleofelasticstrainonelectrocatalysisofoxygenreduction reactiononpt.JPhysChemC2015,119:19042-19052.
18. KhorshidiA,VioletJ,HashemiJ,PetersonAA:Howstraincan breakthescalingrelationsofcatalysis.NatCatal2018,1:263.
Discussesthecouplingbetweenadsorptionenthalpyandstraininato- mistictheory.Pointsoutthatsurfacesitesoflowsymmetrycanhavean anisotropic enthalpy–strain coupling; confirms that they sign of the enthalpystraincouplingdependsonthesignoftheeigenstress.
19. KramerD,Weissmu¨llerJ:Anoteonsurfacestressandsurface tensionandtheirinterrelationviaShuttleworth’sequationand theLippmannequation.SurfSci2007,601:3042-3051.
Adetailedsynopsisofthevariousobservationsthatrequireadistinction betweensurfacestressandsurfacetension,alongwithacriticalexam- inationofobjectionstotheconceptofsurfacestressthathavebeen raisedintheliterature.
20. CahnJW:Thermodynamicsofsolidandfluidsurfaces.In InterfacialSegregation.EditedbyJohnsonWC,BlakelyJM.Metals Park,OH:ASM;1978:3-23.
21.
GorGY,HuberP,Weissmu¨llerJ:Elastocapillarityinnanopores:
sorptionstrainfromtheactionsofsurfacetensionandsurface stress.PhysRevMater2018,2:086002.
Shows that adsorption-induced deformation phenomena which have beendiscussed interms of‘solvationpressure’ may equivalentlybe understoodastheconsequenceofsurfacestress.
22. GurtinM,MurdochAI:Acontinuumtheoryofelasticmaterial surfaces.ArchRationMechAnal1975,57:291-323.
Theclassicpaperonstressandstrainatthegeneral,nonuniformand curvedsurfaces,analyzedfromthepointofviewofsolidmechanics.
23. Weissmu¨llerJ,KramerD:Balanceofforceatcurvedsolid metal–liquidelectrolyteinterfaces.Langmuir2005,21:4592- 4603.
AppliesasimplifiedandlessmathematicalvariantofGurtin’sapproachto electrodesurfaceswhichcanbeelectricallychargedorwhichexperience adsorptionprocesses.Providesacomprehensiveaccountofthefounda- tionsofthetheoryinthepresentarticle.
24. HaissW:Surfacestressofcleanandadsorbate-covered solids.RepProgPhys2001,64:591-648.
Reviewsthemechanicsofsurfaceswithemphasisonelectrodesandon atomicscaleinteractionsbehindelectrocapillarycoupling.
25. Weissmu¨llerJ:Electrocapillarityofsolidsanditsimpacton heterogeneouscatalysis.In Electrocatalysis:Theoretical FoundationsandModelExperiments.Advancesin
ElectrochemicalScienceandEngineering.EditedbyAlkireRC, KiblerL,KolbDM,LipkowskiJ.Weinheim,Germany:Wiley-VCH;
2013:163-220.
26. SmetaninM,ViswanathRN,KramerD,BeckmannD,KochT, KiblerLA,KolbDM,Weissmu¨llerJ:Surfacestress-charge responseofa(111)-texturedgoldelectrodeunderconditions ofweakionadsorption.Langmuir2008,24:8561-8567.
27. LafouresseMC,BertocciU,BeauchampCR,StaffordGR:
Simultaneouselectrochemicalandmechanicalimpedance spectroscopyusingcantilevercurvature.JElectrochemSoc 2012,159:H816-H822.
Provides an in-depth description of surface stress measurement by cantileverbendingapproaches,includingoriginalaspectsofdynamics.
28. OlivierS,Tre´gliaG,Sau´lA,WillaimeF:Influenceofsurfacestress inthemissingrowreconstructionofFCCtransitionmetals.
SurfSci2006,600:5131-5135.
29. BarthJ,BruneH,ErtlG,BehmR:Scanningtunneling microscopyobservationsonthereconstructedAu(111) surface:atomicstructure,long-rangesuperstructure, rotationaldomains,andsurfacedefects.PhysRevB1990, 42:9307.
30. NarasimhanS,VanderbiltD:Elasticstressdomainsandthe herringbonereconstructiononAu(111).PhysRevLett1992, 69:1564-1567.
31. MenFK,PackardWE,WebbMB:Si(100)surfaceunderan externallyappliedstress.PhysRevLett1988,61:2469-2471.
32. MinB,DengX,PinnaduwageD,SchalekR,FriendC:Oxygen- inducedrestructuringwithreleaseofgoldatomsfromAu(111).
PhysRevB2005,72:121410.
33. MinBK,AlemozafarAR,BienerMM,BienerJ,FriendCM:
ReactionofAu(111)withsulfurandoxygen:scanning tunnelingmicroscopicstudy.TopCatal2005,36:77-90.
34. MarchenkoV,ParshinAY:Elasticpropertiesofcrystalsurfaces.
SovPhysJETP1980,52:129-131.
35. AlerhandOL,VanderbiltD,MeadeRD,JoannopoulosJD:
Spontaneousformationofstressdomainsoncrystalsurfaces.
PhysRevLett1988,61:1973-1976.
36. Mu¨llerP,SaulA:Elasticeffectsonsurfacephysics.SurfSciRep 2004,54:157-258.
Reviewsthemechanicsofsurfaces,includingadsorptionandthestress fieldsarounddefects.
37. LiWN,DuanHL,AlbeK,Weissmu¨llerJ:Linestressofstepedges atcrystalsurfaces.SurfSci2011,605:947-957.
38. WangY,Weissmu¨llerJ,DuanHL:Mechanicsofcorrugated surfaces.JMechPhysSolids2010,58:1552-1566.
39. DengQ,GosslarD-H,SmetaninM,Weissmu¨llerJ:
Electrocapillarycouplingatroughsurfaces.PhysChemChem Phys2015,17:11725-11731.
40. VendelboSB,ElkjærCF,FalsigH,PuspitasariI,DonaP,MeleL, MoranaB,NelissenBJ,vanRijnR,CreemerJF,KooymanPJ, HelvegS:Visualizationofoscillatorybehaviourofpt nanoparticlescatalysingcooxidation.NatMater2014,13:884.
Thein situ transmissionelectron microscopyin thiswork provide a strikingillustrationofcyclicchangesinthesurfaceatomicstructurein responsetoadsorption.
41. KamiuchiN,SunK,AsoR,TaneM,TamaokaT,YoshidaH, TakedaS:Self-activatedsurfacedynamicsingoldcatalysts underreactionenvironments.NatCommun2018,9:2060.
42. HerringC:Sometheoremsonthefreeenergiesofcrystal surfaces.PhysRev1951,82:87.
Aclassicpaperonhowthesurfacetensionofacrystalvariesasthe functionofthecrystallographicorientationofthesurfacenormal.
43. HoppeS,MichlA,Weissmu¨llerJ,Mu¨llerS:Ab-initiomodelingof electromechanicalcouplingatSisurfaces.JApplPhys2014, 116:073507.
44. AlbinaJM,ElsaesserC,WeissmuellerJ,GumbschP,UmenoY:Ab initioinvestigationofsurfacestressresponsetochargingof transitionandnoblemetals.PhysRevB2012,85:125118.
45. MichlA,Weissmu¨llerJ,Mu¨llerS:Electrocapillarycouplingat metalsurfacesfromfirstprinciples:ontheimpactofexcess chargeonsurfacestressandrelaxation.Langmuir2018, 34:4920-4928.
Summarizesthepresentunderstandingofhowstrainaffectstheelectro- nic and atomic structure of metal surfaces, and how these effects propagateintoelectrocapillarycoupling.
46. MavrikakisM,StoltzeP,NorskovJK:Makinggoldlessnoble.
CatalLett2000,64:101-106.
47. DengQ,GopalV,Weissmu¨llerJ:Lessnobleormorenoble:how strainaffectsthebindingofoxygenongold.AngewChemInt Ed2015,127:13173-13177.
48. WeigendF,Weissmu¨llerJ,EversF:Structuralrelaxationin chargedmetalsurfacesandclusterions.Small2006,2:1497- 1503.
Illustrateshowelectricchargingleadstorelaxationatacrystalsurface anddiscusseshowthisrelaxationfeedsintotheelectrocapillarycoupling.
49. UmenoY,ElsaesserC,MeyerB,GumbschP,WeimuellerJ:
Reversiblerelaxationatchargedmetalsurfaces:anabinitio study.EPL(EurophysLett)2008,84:13002.
