On factors defining the mechanical behavior of nanoporous gold

ContentslistsavailableatScienceDirect

Acta Materialia

journalhomepage:www.elsevier.com/locate/actamat

Full length article

On factors defining the mechanical behavior of nanoporous gold

Birthe Zandersons

^a

, Lukas Lührs

^a

, Yong Li

^a,b

, Jörg Weissmüller

^a,b,∗

aInstitute of Materials Physics and Technology, Hamburg University of Technology, Hamburg, Germany

bInstitute of Materials Research, Materials Mechanics, Helmholtz-Zentrum Hereon, Geesthacht, Germany

a rt i c l e i nf o

Article history:

Received 30 September 2020 Revised 3 May 2021 Accepted 5 May 2021 Available online 18 May 2021 Keywords:

Nanoporous gold Dealloying

Mechanical properties Scaling laws Young’s modulus Strength

a b s t r a c t

Nanoporousgold (NPG)madebydealloying takesthe formofanetworkofnanoscalestruts or“liga- ments”.Itiswellestablishedthatthematerial’smechanicalbehaviorisstronglyaffectedbyitsligament size,Landbyitssolidvolumefraction,ϕ^{.WeexplorethemechanicalbehaviorofNPG,withanempha-} sisonestablishingaconsistentdatasetwithcomparableLbutcoveringasignificantrangeofinitialϕ^. SpecimensarepreparedfromAg-Aumasteralloys withtheirAuatomfraction,x⁰_Au intherange0.20–

0.35.Sincedealloying replacesAgwithvoids,ϕ ^{may beexpectedtoagreewithx0}Au.Yet,spontaneous plasticdeformationeventsduringdealloyingcanresultinmacroscopicshrinkage,decouplingϕ^fromx0Au. Thisraisesthequestion,howdoϕ^{and x0}Auseparatelyaffectthemechanicalbehavior?Weconfirmtwo recentsuggestions,namelyi)amodifiedRoberts-Garboczi-typescalinglawforYoung’smodulusversusϕ ofthematerialinitsas-preparedstateandii)therelevanceofanapparent“load-bearingsolidfraction”

forYoung’smodulusaswellasstrength.Yet,remarkably,wefindthatstiffnessandstrengthoftheas- preparedmaterialshowamuchbettercorrelationtox⁰_Auascomparedtoϕ^{.Thiscanbeunderstoodasa} consequenceofthemicrostructuralchangesinducedbyshrinkage.Studyingthemicrostructureevolution duringannealing,wealsoconfirmthesuggestionthatcoarseningentailsanenhancedlossinstiffness for sampleswith lessersolidfraction. Thisfindingconfirms concernsabout the notion ofself-similar coarseningasageneralbehaviorofdealloying-madenetworkmaterials.

© 2021TheAuthor(s).PublishedbyElsevierLtdonbehalfofActaMaterialiaInc.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

1. Introduction

Nanoporousmetalsmadebydealloyingaremacroscopicmono- lithic bodies that consist, at the nanoscale, of a homogeneous network of struts or “ligaments”. The materials, and specifically nanoporous gold (NPG), are under study as model systems for small-scale mechanicalbehavior [1]. As-preparedNPG takes liga- mentsizes,L,ofafewtensofnanometers[2–5].Annealinginduces coarseningthatlets Lincreaseuptovaluesofseveral100nm[6–

9].

Much of the interest in using the material as a model sys- tem stemsfromthe notionofself-similar coarsening.Size-effects on the mechanical behavior could then be studied in isolation, while the architecture of the nanoscale network remains invari- ant.Thefindingofpower-lawrelationsbetweentheyieldstrength,

σ

^{y,andLanditsdiscussionintermsof}size-inducedstrengthening exemplifies this approach [10–12]. Experimental tomographic re-

∗Corresponding author at: Institute of Materials Physics and Technology, Ham- burg University of Technology, Hamburg, Germany.

E-mail address: weissmueller@tuhh.de (J. Weissmüller).

construction [7,13–15]andseveralnumerical studies[16–19]sup- port the notion of self-similar coarsening. Yet, evidence against that notion has recently been accumulating. Studies of the evo- lutionof thestiffness ofNPGcan only be understoodif thenet- workisallowedtoprogressivelydisconnectduringcoarsening[20]. Furthermore, large-scale atomistic simulations of coarsening ex- ploringan intervalofparameter spacerelevanttoexperimentdi- rectly evidence the disconnection [21]. The disagreement about self-similarity appears to originate in a pronounced dependency ofthetopologyevolutionduringcoarseningonthesolid(volume) fraction,

ϕ

^{. While networkswith}

ϕ

0.3 tend to maintaintheir connectivityduringcoarsening,inkeepingwiththenotionofself- similarity,networkswith

ϕ

0.3tendtodisconnect[21].Asstud- iesofNPGhavebeenconsideringnetworkswith

ϕ

^oneachsideof

thethresholdvalue0.3,thedifferentfindingsmaysimplyoriginate in slightlydifferent solid fractions in the individual experiments.

Thisadvertisesaninterestinexperimentsprobingtheevolutionof connectivityduringcoarseningasthefunctionof

ϕ

^.

Liuetal.[22]varied

ϕ

^byinterruptingthedealloyingprocessof AgAuPtatvariousstatesofAgdissolution.Theyfoundthedecrease ofthestiffnessuponcoarseningsubstantiallymorepronouncedat

https://doi.org/10.1016/j.actamat.2021.116979

1359-6454/© 2021 The Author(s). Published by Elsevier Ltd on behalf of Acta Materialia Inc. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

lesser

ϕ

^{,andtheyattributethelossinstiffnesstovariationsinthe}

network connectivity. Thisiswell compatiblewiththe aboveno- tion,yetthesystematicvariationofthecompositionwith

ϕ

^inthe

experimentofRef[22]complicatesthediscussionoftheirobserva- tionsintermsofacorrelationbetweenthelossofconnectivityand

ϕ

^{alone.Asystematicstudyofthelinkbetween}

ϕ

^{andthestiffness}

variationduringcoarseninginelementalnanoporous goldhasnot beenreported.

Irrespectiveoftheissueofself-similar ornon-self-similarevo- lution duringcoarsening,network solidsmadeby nanoscaleself- organization– as,forinstance,bydealloying– exhibitasystematic correlationbetweentheir

ϕ

^andtheconnectivityalreadyintheas- prepared state.Thisis reflectedby compilations, inRefs[22] and [23],ofliterature data[5,12,20,22,24–36]forthe stiffnessofNPG.

Specifically,thecompilationssuggestthatthevariationofeffective Young’s modulus,Y^eff,with

ϕ

ⁱⁿas-preparedsamplesisnotcom- patible withtheexpectation forstructures withconstantconnec- tivity, asembodied inthe Gibson-Ashbyscaling lawfor thestiff- ness.Astudysystematicallyvarying

ϕ

^{indealloyedFe-Cr[37]con-}

firmsthisobservation.

Soyarslanet al.[23] proposea physicallymotivatedmodel for the microstructure ofNPG, whichlinks therelation betweenY^eff and

ϕ

^toquantifiablemicrostructurecharacteristicsandspecifically to a scaledtopological genus.Thatstudyproposesa scaling rela- tion that accountsfora lossofpercolation– and,hence,forme- chanical disintegration– at a finite solid fraction. While thelink betweentheconnectivityofas-dealloyedNPGanditssolidfraction iswellconsistentwiththeexistingbodyofresearch[1],asystem- atic study,based on aset ofelemental nanoporous gold samples andwithconsistentpreparationandcharacterizationconditions,of thislinkandofthescalingrelationbySoyarslanetal.hasnotbeen reported.

Here, we study the mechanical behavior of as-dealloyed and coarsenedNPGspecimenspreparedfromAg-Aumasteralloyswith their goldatomfraction,x⁰_Au,intherange0.20–0.35.Sincean ide- alized dealloying process replaces all less-noble atoms(here Ag) with voids, values of

ϕ

^{may be expected in that same range.}

Ourstudydoes,however,accountforshrinkageduringdealloying, which canmodify

ϕ

^,independently ofx⁰_Au.Carefulcontrol ofthe preparation conditionsinour studyaffords a consistentligament size, L,independent of

ϕ

^{or x0}Au. We confirm the findings by Liu et al.andbySoyarslan etal.andspecificallythe notionofvaria- tions in connectivitybetween sampleswithsame

ϕ

^. Remarkably, our results point towards a systematicinfluence ofthe composi- tionofthemasteralloyonthemechanicalbehavioroftheporous material.Thus,studiesofthemechanicalbehaviorofNPGneedto accountfortheimpactofx⁰_Au,ontopofthatof

ϕ

^.

2. Scalinglaws

2.1. Constantconnectivity

As a basis for discussing the relation between stiffness and strength of nanoporous solids and their solid fraction,it is con- venient to compilethe mostimportant scalinglaws.Afrequently usedbenchmarkforY^eff(

ϕ

)^{instructureswithfixed}connectivityis theGibson-Ashbyscalinglaw[38],

Y^eff=C1Y⁰

ϕ

^{2 (1)}

withY⁰ Young’s modulus ofthe massivematerial, andC₁ a con- stantthattakesthevalue1foropen-cellfoamsingeneral[38]and specifically for NPG [39]. Similarly, the Gibson-Ashby scaling law forthemacroscopicyieldstrength,

σ

^y,links

ϕ

^{forNPGtothelocal}

ligamentstrength,

σ

y⁰,as

σ

^y=C2

σ

y⁰

ϕ

^{3/2 (2)}

withC₂=0.3forNPG[39].

2.2. Accountingforconnectivityvariationindealloyednanoporous metals

Thescalingbetweenmechanicalcharacteristicssuchasstiffness orstrengthandthesolidfractionofporousmaterialsisstronglyaf- fectediftheconnectivityvariessystematicallywith

ϕ

^andspecifi-

callyifapercolationthresholdismetatfinite

ϕ

^{[23,40–43].Foras-}

preparedNPGwith

ϕ

<0.5,ithasbeendemonstratedthatamod- ifiedversion [23] oftheRoberts-Garboczi scaling lawforrandom porousmicrostructures [42]isin excellentagreementwithlitera- turedatafortheYoung’smodulus.ThemodifiedRoberts-Garboczi lawtakestheform[23]

Y^eff=C3Y⁰

ϕ

⁻

ϕ

P

1−

ϕ

^P

m

( ϕ

^<0.5

)

⁽³⁾

with

ϕ

P thesolidfractionatthepercolationthreshold.Theunder- lyingmodelisaleveledGaussianrandomfield,originallyproposed forearly-stage spinodallydecomposedstructures [44].Here,

ϕ

P= 1−erf(^2−1/2)≈0.159 [23].Fitting the parameters to as-prepared NPGresultsinC₃=2.03±0.16andm=2.56±0.04[23].

NotingthatNPG’sexperimentalvaluesofY^effareregularlysub- stantiallylowerthanthepredictionsoftheGibson-Ashbylaw,Refs [20,22]proposetheintroductionofanapparentload-bearingsolid fraction,

ϕ

^{lb,whichisalwayslessthantheactualsolidfraction,}

ϕ

^.

Inthatapproach,

ϕ

^{lbisdeterminedfromthe}experimentalstiffness ontheassumptionthatEq.(1)holdsfortheload-bearingstructure.

Inotherwords,

σ

^y=C2

σ

y⁰

( βϕ )

^{3/2 (4)}

where

β

^{denotesanapparent}load-bearingfractionfactor,whichis definedsothat

β

⁼

ϕ

^lb

ϕ

^{, (5)}

andwheretheapparentload-bearing solid fractionis determined fromtheexperimentalY^effvia

ϕ

^lb=

Y^eff Y⁰

1/2

. (6)

Experimentalvaluesof

σ

^{yagreewellwithEq.(4).}

It may be emphasized that the approaches underlying Eqs.

(3)and (6)are not mutuallyexclusive, they justobtain informa- tiononthenetworkconnectivityfromdifferentsources.Thelatter equationisbasedonanempiricalcharacterizationofthenetwork basedonitselasticbehavior,whereastheformerassumesaphysi- callymotivatedsystematiccorrelationbetweenthenetwork’scon- nectivityanditssolidfraction.

2.3. Accountingforshrinkageduringdealloying

Motivatedbytheexperimentalresultsofthepresentstudy(see below),weconsideryetanothermodificationofthescalingbehav- ior.Thismodificationisdesignedtoaccountfortheshrinkagethat isobservedtoaccompanydealloyingunderawiderangeofexper- imentalconditions [45–49].Whileourargumentmayapply more generally,wedesignatethelessnobleandmorenobleelementsas AgandAu,respectivelyforsimplicity.

Asthefirststepofdealloying,weconsidertheformationofan initialstructure thatisfreeofshrinkage.Forthisinitialstructure, weproposethat1.)allAgisremovedsothattheinitialsolidfrac- tion,

ϕ

ini,agreeswiththeAuatomfractioninthemasteralloy,x⁰_Au; 2.)themodelofRef [23]applies,includingspecificallyEq.(3).As the next step,we allow forshrinkage.Thereby, thesolid fraction increases from

ϕ

ini to the final solid fraction of the as-prepared

sample,

ϕ

^{.Here,weproposethat 3.)the}connectivityremains in- variantduringtheshrinkage.

The above propositions lead to the following suggestions:

Firstly,Y^effof theinitial structure (no shrinkage) obeysEq.(3),yet withx⁰_Au substitutedfor

ϕ

^{.Inotherwords,}

Y^eff=C₃Y⁰

x⁰_Au−

ϕ

P

1−

ϕ

P

m

. (7)

Secondly, since the shrinkageis atfixed connectivity, Eq. (1)ap- pliestomaterialsthathaveundergoneshrinkageinthesensethat Y^eff scales withthe shrinkage-induced increase in

ϕ

^{2. Shrinkage}

willthenintroduceacorrectiontoEq.(7)intheform Y^eff=

v

²C3Y⁰

x⁰_Au−

ϕ

^P

1−

ϕ

^P

m

. (8)

The parametersC₃ andm inEqs.(7) and(8)agreewiththose in Eq.(3).Thedensificationparameter

v

isgivenby

v

=

ϕ

x⁰_Au; (9)

itrepresentsavolumeshrinkageratio,inotherwords,theratioof as-preparedNPGvolumeovermasteralloyvolume.

Equation(8)isequivalenttousingtheGibson-Ashbyscalinglaw Eq.(1)with

ϕ

^{correctedbytheapparent}load-bearingfractionfac- tor

β

=

C3

1 x⁰_Au

x⁰_Au−

ϕ

^P

1−

ϕ

P

m/2

. (10)

Note, that

β

^is independent of whether or not there has been shrinkage. Note also, that the relations discussed in the present subsectionbreakwiththenotionthatY^effisuniquelydetermined by

ϕ

^{.Instead,theyallowforx0}Au asanadditionalparameter.

3. Procedures 3.1. Samplepreparation

MasteralloysAg-Auwithx⁰_Au= 0.20,0.25,0.30,and0.35were preparedasinRef [35].Inbrief, arcmeltingwasfollowedby ho- mogenization via annealing for five days at850 ^◦C in an evacu- atedandthensealedglasstube.Wires0.95mmindiameterwere drawnandcutintocylinders1.9mminlength.Vacuumannealing for 3hat600 ^◦C followed forrecovery.Todealloy theprecursors produced in this way,we used the two mostcommon methods:

electochemicalcorrosionandfreecorrosion.

Electrochemicaldealloyingusedathree-electrodesetupin1M HClO₄ with a Ag wire counter electrode and a Ag/AgCl pseudo- reference electrode. We found that producing crack-free, mono- lithicsamplesrequiredaninitialcurrentdensityof5mA/cm²(re- ferredtotheareaofthemacroscopicsamplesurface).Ineachcase, the dealloying potential, E_D wasadjusted tomeet that condition.

E_D wasdetermined froma linearpotential sweepat ascan rateof 5 mV/son aseparate precursorsamplewithknownsurfacearea.

Afterdealloying,residualsilverwasremovedbymaintainingapo- tential 0.1 V above E_D until the currentfell below 15 μA. After- wards,40potentialcyclesfrom0.1to1.6Vvs.SHEat5mV/swere applied,endingat0.8Vsoastoensureanadsorbate-freesurface.

Thesampleswerethenrinsedinwateranddriedinair.Theresid- ual silver content of all electrochemically annealed sample types was∼1at%.

We found the required E_D toincrease from1.13 to 1.28V vs.

SHE for master alloys as x⁰_Au increasedfrom 0.20 to 0.30. While samples with x⁰_Au = 0.20 and 0.25 could be dealloyed at ambi- enttemperature,avoidingcrackformationinprecursorswithx⁰_Au= 0.30requiredthedealloyingtemperaturetoberaisedto50^◦C.For

x⁰_Au=0.35wewereunabletoestablishanelectrochemicaldealloy- ing protocol that wouldyield fully dealloyed and crack-free NPG samples.

Forfreecorrosion,themasteralloysweredealloyedin65wt.%

HNO₃ at ambient temperatureand the resulting NPGthen rinsed inwateranddriedinair.Forthisprocess,after25daysthex⁰_Au= 0.35masteralloysevolvedtoNPGwitharesidualsilvercontentof

∼3at.%.Prolonged exposurein acidof up to sixweeksgave no furtherreductioninresidualAg.Masteralloyswithx⁰_Au=0.30and 0.25yieldNPGwith∼1at%silverafter4and2daysoffreecorro- sion,respectively.Formasteralloyswithx⁰_Au=0.20,freecorrosion wasnotapplicable,sincethespecimensdisintegratedintopowder upondealloying.

Toevaluate the possibleeffect ofadsorbates on free corroded NPG,we subjected some ofthe free-corrosionsamples to 20po- tentialcyclesbetween0.1Vand1.6Vvs.SHEatscanrate5mV/s, stopping at 0.8 V. This mimics the protocol at the end of elec- trochemical dealloying, andleaves the samplesurface clean,free ofadsorbate. We confirmedthat this conditioningtreatment had noimpactonthemechanicalresponse.Themechanicalcharacteri- zation showninthe figures ofthiswork were obtainedwiththe straightforward free corrosion protocol without the conditioning treatment.

3.2. Structuralcharacterization

On average, the nanoporous samples had a length of 1.77± 0.12 mm anda diameter of 0.94±0.06 mm. By measuring geo- metricaldimensionsand mass,

ϕ

^wasdetermined. Forthe deter- minationof

ϕ

^duringdeformation,seeSection3.3below.

The ligament size, L was determined as the diameter of the struts in the nanoporous samples, as obtained by analysis of scanning-electron-microscope(SEM)images.Foreverysample,di- ameters ofmore than50 ligamentswere measured fromeach of atleasttwoimagesoffracturesurfaces.Lisspecifiedasthemean, and the standard deviation is specified as the uncertainty (error bar).Theresidualsilvercontentwasdeterminedbyenergydisper- siveX-Rayanalysis,basedonaveragesfromatleasttwosamplesof everyindividualdealloyingtechniqueandelementalcomposition.

3.3. Mechanicaltesting

Single loading and load/unload compression tests with engi- neering strain rates of

ε

˙=10⁻⁴s⁻¹ where evaluated from a to- tal of 60 samples, using the uniaxial compression test setup in- troducedinRef.[50].Digital ImageCorrelationwiththesoftware DaVis8.2.0 by LaVisionwasused to determinethe strain by vir- tualstraingagesonthesamplesurface,asdescribedinRef.[50,51]. Sincethedeformationinandtransversetotheloaddirectionwas known,themacroscopicdimensionsofthesamplescouldbedeter- minedduringtheentiremeasurement.Withthisinformation,the evolutionof

ϕ

^duringcompressionwasevaluated.

We determined Y^eff in various states of deformation as the slopeofstraightlinesofbestlinearfittotheentireunloadregime oftherespectiveload/unloadsegment.Inotherwords,Y^effisspec- ifiedasthesecantmodulus.

When subjected to compression, as-prepared NPG exhibits an early yield onset and a continuous elastic-plastic transition [1,33,34].In theinterestofameaningfulseparationofelasticand plastic deformation, the yield strength,

σ

y was measured as the stressat∼1%plasticstrain.Thisisconsistentwithearlierstudies [5,52].TheinitialvalueofY^effwasdeterminedfromasubsequent unload segment, at ∼ 1.5% plastic strain. We found (see Section 4)that thevariationofflow stressaswell asY^eff issmallinthat strainrange.

Table 1

Properties of as-dealloyed nanoporous gold samples that were used for load/unload compression tests. Au atom fraction, x ⁰_Au, densification parameter, v, solid fraction of the uncompressed samples, ϕ, ligament size, L , Young’s modulus at a plastic strain around εpl= 0 . 015 , Y ^eff, and apparent load-bearing fraction factor, β. At least five samples of each type were investigated as the basis for the data of this table.

x ⁰_Au v [-] ϕ[-] L [nm] Y ^eff[GPa] β[-]

electrochemical corrosion 0.20 1.43 ±0.04 0 . 29 ±0 . 01 33 ±6 0 . 35 ±0 . 07 0.23 ±0.03 0.25 1.06 ±0.06 0 . 26 ±0 . 02 32 ±8 0 . 38 ±0 . 07 0.26 ±0.02 0.30 1.02 ±0.01 0 . 31 ±0 . 01 29 ±8 2 . 25 ±0 . 42 0.55 ±0.05 free corrosion 0.25 1.22 ±0.07 0 . 31 ±0 . 02 49 ±13 0 . 50 ±0 . 11 0.26 ±0.04 0.30 1.02 ±0.02 0 . 31 ±0 . 01 31 ±7 1 . 73 ±0 . 33 0.48 ±0.06 0.35 1.02 ±0.01 0 . 36 ±0 . 01 46 ±9 6 . 53 ±0 . 42 0.80 ±0.03

3.4. Atomisticsimulationofrelaxation

The atomistic simulationof relaxationused procedures ofour previousworkforgeneratingrealisticnanoporousinitialstructures [21,53], for implementing the relaxation by molecular dynamics (MD)[34]andforanalyzingtheconnectivity[21].Wehereprovide a briefsummary;details areprovided intheSupporting Informa- tion.

The initial structures–with solid fractions 0.20, 0.25, 0.30 and 0.35 andwithidentical ligamentsize,3.3 nm–weregeneratedby a variantofJohn Cahn’sleveledwave model[44],modifiedtoaf- fordperiodicboundaryconditionsin3D [23].Aface-centered cu- bic (fcc) crystal lattice wasinscribed into the solid phase, using the stress-freebulk latticeparameter(a=408nm)oftheMDin- teratomic potential. The simulation boxes hadan edge length of 200aandcontainedbetween6.4and11.2×10⁶ atoms,depending on

ϕ

^.

The relaxation by MDused the open-source softwareLammps [54] with an embedded-atom method (EAM) potential for Au [55]andperiodicboundaryconditionsin3D.UsingaNosé-Hoover thermostat andbarostat [56,57],the structures were relaxedfirst athermallyandthenthermallyat300Kfor2ns.Theopen-source softwareOvito[58]wasappliedforvisualization.

The net topological genus, G, of structures with volume V was determined by the open-source code CHomP [59]. G repre- sents the numberof connections in the sample. The topology of a microstructure may be described by a scaled value, g, of the genus,amaterial-specificandsize-independentquantitythatspec- ifiesthenumberofconnectionsinarepresentativestructuralunit [13,14,23,60–62]. The scaled genus of as-prepared NPG has been experimentally quantified [13,14,61] and its variation with

ϕ

^as

well as its impact forY^eff(

ϕ

) ^emphasized [1,5,20,22,23,61,63,64]. Furthermore, experimental and modeling data for the variation of gduringannealingare available[13–17,19,21,65].We identified the sizeofthe representativestructuralunitswiththemeandis- tance,˜L,betweenthecentersofneighboringligaments[23],hence g=GL˜³/V. The value of L˜ is implied by the wavelength used in constructingtheleveled-wavestructureandbytheoverallvolume changeduringtherelaxation.

4. Results 4.1. Microstructure

Figure 1 shows representative scanning electron micrographs (SEM) forall sampletypes.Theresultingligamentsizesarelisted inTable1.Electrochemicalcorrosionprovidesforparticularlycon- sistent L. Yet, in spite of the larger scatter in the free corrosion samples,alldataisconsistentwithourdealloyingprotocolsresult- inginsensiblysimilarLforeachsampletype,around30–40nm.

Figure2plotsthesolidfraction

ϕ

^versusx0Au.Thefigureisbased on sets of 9 samples of x⁰_Au= 0.20, 22 electrochemical corroded

Table 2

Properties of annealed nanoporous gold samples. Listed are the Au atom fraction, x ⁰_Au, the densification parameter, v, the solid fraction of the uncompressed samples, ϕ, the ligament size, L , Young’s modulus at a plastic strain around εpl= 0 . 015 , Y ^eff, the apparent load-bearing fraction factor, β, and the number of tested samples. At least four samples of each type were investigated as the basis for the data of this table.

x ⁰_Au v[-] ϕ[-] L [nm] Y ^eff[GPa] β[-]

0.25 1.15 ±0.10 0.29 ±0.03 116 ±28 0.19 ±0.05 0.17 ±0.02 0.30 1.03 ±0.02 0 . 31 ±0 . 01 115 ±37 1.36 ±0.29 0.43 ±0.04 0.35 1.03 ±0.01 0 . 36 ±0 . 01 106 ±28 5.75 ±0.19 0.75 ±0.01

and14freecorrodedsamplesofx⁰_Au=0.25,28electrochemicaland 10free corrodedsamplesofx⁰_Au = 0.30and29samplesof x⁰_Au = 0.35,showingmeanvalueandstandarddeviationineachset.The twoparameters agreeforthehighersolidfractions,indicatinglit- tleshrinkage.Yet,thereisatrendfor

ϕ

^{ofsampleswithlesserx0}Au

tobeincreased.Inother words,sampleswithlesserx⁰_Au aremore pronetoshrinkage.Atx⁰_Au=0.25,theshrinkageismoreseverefor thefreecorrosionsamples.Masteralloyswithx⁰_Au=0.20exhibited particularly severe densification (

ϕ

substantially larger than x⁰_Au) when dealloyed electrochemically, while they disintegratedupon freecorrosion.Thebehaviorisconsistentwiththenotionthatthe percolationlimit ofthenetworksolidisreachedat

ϕ

≈0.16[23], whichisonlyslightlybelowthevalueof0.20.

Annealingin airat300 ^◦C suppliedsampleswith coarsermi- crostructures. Annealing times where 15 min for samples with x⁰_Au=0.25and30minforthosewithx⁰_Au=0.30and0.35.There- sultingligamentsizesarelisted inTable2. Thesizesherescatter around110nm.

4.2. Stiffnessofas-dealloyednanoporousgold

Figure 3 plots the stress-strain graph of an exemplary load/unloadcompression test onan as-dealloyedx⁰_Au=0.25sam- ple. The figure illustrates that values of Y^eff in various stages ofdeformation were determined assecant moduli by analysisof unload/reload segments. Figure 4(a) illustrates how these values evolve during the compression ofrepresentative samples.As de- scribedinSection3.3,wetakethevalueofY^effataplasticstrainof around 0.015asrepresentative ofthestiffnessinthe as-prepared state.

Figure5 presentsanexemplary map ofthedisplacementfield onthesamplesurface, hereforasamplewithx⁰_Au=0.20atcom- pressiveengineeringstrainof0.30.Itisseenthatthedisplacement gradient isuniform overthe entiresample. The observationsup- portsthe homogeneityof the samplesand theuniformity of the deformation.

Figure4 (b)plotsY^effforas-prepared samples,asobtainedby averaging the experimental results forseveral specimens ofeach sample type. These values are also listed in Table 1, where the properties ofall samples used forload/unload compression tests

Fig. 1. Scanning-electron-microscope images of fracture surfaces with identical scale. Upper row: nanoporous gold made by electrochemical corrosion of Ag 1−xAu xwith (a) x = 0 . 25 , (b) x = 0 . 30 , (c) x = 0 . 20 . Lower row, samples made by free corrosion of Ag 1−xAu xwith (d) x = 0 . 25 , (e) x = 0 . 30 , (f) x = 0 . 35 .

Fig. 2. Solid fraction, ϕ, of as-prepared nanoporous gold versus Au fraction, x ⁰_Au, in master alloy. Mean value and standard deviation for sets of at least 9 samples of each type. Dealloying method is indicated in legend. Dashed line: identity, applica- ble when there is no densification during dealloying.

are summarized. Within the range of solid fractions covered by our sampletypes,Y^eff can be seento varyby morethan one or- derofmagnitude.Yet,thegraphofY^effversus

ϕ

^{doesnotreveala}

strongcorrelation,nordoesitsuggestagreementwithanyone of thescalinglawsEq.(1)or(3),whicharerepresentedbysolidlines inthefigure.

Remarkably, the experimental Y^eff exhibit a much better cor- relation withthe Aufraction,x⁰_Au in themaster alloy.This isap- parent inFigure 4(c), whichreveals aconsistent trendforY^eff to increase with x⁰_Au. The solid linein that figure isthe scaling law ofEq.(7),inotherwords,Eq.(3)withx⁰_Au substitutedfor

ϕ

^.This

graphagreeswiththetrendinthedata,suggestingthatthemas- teralloycompositionisrelevantforthemechanicalbehavior,even whenthesolid fractiondeviatesfromx⁰_Au asaresultofshrinkage duringthedealloying.

As an unbiased measure for correlation, we have computed Spearman’s rankcorrelation coefficient for theexperimental data inFig.4(b)and(c).Wecomparedthecorrelationoflog(^Yeff)^with

Fig. 3. Engineering stress, σ, vs. engineering strain, εeng, of an load/ unload com- pression test of dealloyed Ag 75Au 25. Inset: details of a load/unload segment; red line indicates linear fit. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

ϕ

^tothatwithx0Au.Thecorrelationcoefficientevaluatesto0.88for

ϕ

^{andto0.97forx0}Au.Thisconfirmsthesignificantlybetter corre- lationbetweenthemechanicalbehaviorandthemasteralloycom- position,asopposedtothesolidfraction.

A notable exception fromthe agreement betweenEq. (3)and experimentisthematerialwithx⁰_Au=0.20,whichissubstantially stifferthan predicted.Thisdeviationcoincideswiththeverysub- stantial shrinkage of that sample. This might be a consequence of the gold content in the precursor being close to the percola- tion limit, which might lead to extended restructuring processes alreadyduringdealloying.

4.3. Strengthofas-dealloyednanoporousgold

As abackground forourdiscussion ofthe plasticdeformation behavior,Fig. 6(a)showsthestress-strainresponseofrepresenta- tive specimensin single-loadingcompression tests. Weobserve a

Fig. 4. Effective Young’s modulus, Y ^eff, determined by unload regimes of load/ un- load compression tests of nanoporous gold made by dealloying Ag-Au master alloys with a molar Au fraction of x ⁰_Au= 0.20, 0.25, 0.30 and 0.35. Color code and legend in (b) and (c) identify x ⁰_Auand dealloying method, respectively. (a) Y ^effvs. engineer- ing stain, εengof representative samples. (b) Y ^effat a plastic strain around 0.015 vs.

solid fraction, ϕ; averages over several samples of same type with error bars indi- cating the standard deviation. Solid lines: gray, Gibson-Ashby scaling law, Eq. (1) , and red, modified Roberts-Garboczi scaling law, proposed by Soyarslan et al., Eq.

(3) . (c) plots the data of (b) vs. x ⁰_Au. Red line: Eq. (3) with x ⁰_Ausubstituted for ϕ^{. (For} interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

trend for the strength andflow stress to increase withx⁰_Au. This agreeswithearlierresultsfromnanoindentationtests[11,67].

Figure 6(b)exploresthevariationof

σ

^{y with}

ϕ

^.Thedatarep-

resentsaverages andstandard deviationsofseveralspecimensfor each sampletype;x⁰_Au isindicated bylabels.AsforYoung’s mod- ulus,

σ

^{yincreaseswith}

ϕ

^,yetthecorrelationisagainpoor.Inpar- ticular,sampleswithlowerx⁰_Au areseentoexhibitlowstrengthin view oftheircomparativelyhigh

ϕ

^{.Thegraylineshowsthemod-}

ifiedGibson-Asby lawproposed byRef. [37].This modelisbased onEq.(2)forwhichthesolidfractionisreplacedbytheapparent

Fig. 5. Homogeneity of deformation. Photograph of NPG, electrochemically deal- loyed from Ag 80Au 20, in situ at 30% engineering strain. Axial (projected on the load axis) displacement component from digital image correlation is indicated by color code. Load surfaces can be seen at top and bottom of image. Bright spots: speckle pattern, applied for enhanced contrast, as in [66] .

Fig. 6. Strength and flow behavior. Color code designates Au atom fraction in mas- ter alloy, see labels in (b). Legend in (c) specifies corrosion method. (a): Engineering stress, σ, vs. engineering strain, εeng, for representative samples. (b): Strength, σy, obtained as σat 1% plastic strain, vs. solid fraction, ϕ^{. Average} values are given for sets of several samples of the same type, with standard deviations indicated by er- ror bars. Gray line: scaling model proposed by Ref. [37] . (c): σyas in (b), plotted versus Au atom fraction in master alloy, x ⁰_Au. Gray line: Eq. (4) combined with Eq.

(10) .

Fig. 7. (a) Yield stress at ligament level, σy⁰, vs. plastic strain, εpl, of nanoporous gold made by dealloying Ag 1−xAu xwith x = 0.25, 0.30 and 0.35 (see color code and legend in (b)). σy⁰was obtained by combining the Gibson-Ashby scaling law, Eq.

(2) , with the reloading yield stress, σ, of load/unload compression tests,. Triangular symbols: Eq. (2) with the true solid fraction, ϕ; pentagonal symbols: Eq. (2) with the apparent load-bearing solid fraction, ϕ^lbof Eq. (6) . Note the much better agree- ment in the latter case. (b) Symbols: experimental σ vs. ϕ^lb. Line: Gibson-Ashby law Eq. (2) , using σy⁰= 1 . 7 GPa from (a).

load bearingsolid fraction

ϕ

^{lb (Eq. (6)). Forthe requiredYeff we}

usedthemodifiedRoberts-Garboczilaw,Eq.(3).

Here,asforthestiffness,itisagaininstructivetoplottheme- chanical characteristics against x⁰_Au as opposed to

ϕ

^{. That plot,}

Fig. 6(c), shows a substantially improved correlation. Spearman’s rankcorrelationcoefficientishere0.97forlog(

σ

^y)^withx0_Au^,acon- siderablystrongercorrelationthanwhatissuggestedbythevalue, 0.70,forthecorrelationoflog(

σ

^y)^with

ϕ

^.

Section2.3introducesahypothesisabouttheimpactofshrink- ageduringdealloyingontheconnectivity,leadingintoEq.(10)as a prediction forthe apparent load-bearing fraction factor,

β

^.The

solid line in Fig. 6(c) shows the prediction of Eq. (4), using Eq.

(10) for

β

^{. This} representation is seen to match the experimen- tal trendwell. The observation confirmsthat x⁰_Au, rather than

ϕ

^,

is the moreappropriate materials characteristics determiningthe mechanicalpropertiesofas-preparedNPG.

4.4. Apparentload-bearingsolidfraction

Experimental dataforthe yieldstrength ofNPGmaybe com- bined with scaling equations, such as Eq. (2), to evaluate the strength,

σ

y⁰, orflow stress atthescale ofthe ligamentsof NPG.

Thatstrategyisofinteresttoourworkbecausetheligamentsizes aresensiblyidenticalforallsamples,soweexpectthe

σ

y⁰toagree forallsamples,independentoftheirsolidfraction,connectivityor deformationstate.Whetherornotthisexpectationisbornoutpro- videsaconsistencycheckforthescalinglawinquestion.

Figure 7(a)showsthe evolutionof

σ

y⁰ withplasticstrain,

ε

pl, forselectedsamplescoveringthefullrangeofx⁰_Auofourstudy.The

Fig. 8. Apparent load-bearing fraction factor, β, vs. Au atom fraction in master alloy, x ⁰_Au. Data points represent averages over several samples of the same type, with error bars indicating standard deviation. Note overlap of two data points at x ⁰_Au= 0 . 25 . Red line: βcalculated by Eq. (10) .

opensymbolsarebasedonexperimentaldatafor

σ

^andonEq.(2),

with

ϕ

^{determinedfromtheaspreparedsolidfractionalongwith}

thelongitudinalandtransverseplasticstrainsduringcompression.

Here,thepredictionfor

σ

^{0variesstronglywitheachofthesetwo}

parameters,x⁰_Au and

ε

pl.Inviewoftheexpectationthat

σ

y⁰ should takeonauniversalvalue,thedataconfirmsthatEq.(2)initsorig- inalformdoesnotappropriatelydescribethestrengthofNPG.The situation is quite different when the apparent load-bearing solid fraction

ϕ

^{lb is}substitutedfor

ϕ

^{in Eq.(2).The closedsymbolsin}

Fig. 7(a)represent the

σ

^{0 obtainedin that way. Here,}

ϕ

^{lb is de-}

terminedfromtheexperimentaldataforY^eff alongwithEq.(6).It isseen that the data isnow inmuch better agreement withthe expectationofconstant

σ

y⁰.Thisconfirms

ϕ

^{lbasaparametergov-}

erningthestrengthofNPG,inagreementwiththesuggestionsin Refs[20,22].

Averaging – at1% plastic strain – over the ligament strength represented by the closed symbols in Fig. 7(a) suggests

σ

y⁰= 1.7 GPa for our material, which is in accordance with Ref. [20]. Thisvalue underlies thesolid linewhich representsthemodified Gibson-AshbybehaviorinFig.6.

Thesignificanceof

ϕ

^{lbisalsoapparent inthegraphoftheef-}

fective,macroscopicflow stress versus

ϕ

^lb,Fig.7(b).Representa- tivespecimensofeachsampletypeareincludedinthisplot.Ade- tailedviewoftheindividualsampleswithx⁰_Au=0.30and0.35re- vealsadeviationfromthequadratic,Gibson-Ashby-type variation thatemergeswhenEq.(2)isevaluatedwith

ϕ

^replacedby

ϕ

^lb.Yet,

sampleswithx⁰_Au =0.25andsmaller,especiallytheelectrochemi- callycorrodedones,dofollowthatvariationindividually.Moresig- nificantly,theoveralltrendfordataspanningmorethanoneorder in

σ

^{is consistent with the} Gibson-Ashby-type law. Thereby, our observationconfirmsthesuggestedscalingof

σ

^with

ϕ

^lb.Weem-

phasize,however,that anyapparent agreementbetweendataand scalinglawmaydeceive:substantialdifferencesinthemicrostruc- tureandinthe mechanicalbehavior aresimplyparameterized by

ϕ

^{lb.Atthepresentpointinourdiscussionofthestrength,thatpa-}

rameterisempirical,governedbytheratioofY^effandY⁰,tobede- terminedinaseparateexperiment.Inotherwords,thegoodagree- mentdoesnotimplyanaprioripredictivepoweroftheapproach.

Anapproachtowardspredicting

ϕ

^{lbhasbeendiscussedinSec-}

tion 2.3, whereit isembodied in Eq.(10)for theapparent load- bearingfractionfactor,

β

^{.Fig.8verifiesthatequation,bycompar-}

Fig. 9. Molecular dynamics simulation of relaxation. Scaled topological genus, g, versus solid fraction, ϕ. Solid line: theory [23] for the leveled-wave initial struc- tures. Symbols: evolution of gand ϕ during relaxation of structures with differ- ent initial solid fractions, as indicated in legend. Each structure starts sensibly on the theory line and then evolves towards increasing ϕas the relaxation time pro- gresses. All structures were relaxed under identical conditions and for identical time periods, see Supporting Information for details. Some data points overlap. Dashed horizontal lines represent guides to the eye for constant values of inital g. Insert:

exemplary rendering of simulation volume, here for initial ϕof 0.35, after relax- ation.

ing itspredictions to

β

^{valuesdetermined fromthe}experimental Y^eff.TheunderlyingexperimentaldatabaseisshowninFig.4and listedinTable1.Figure8confirmsthepredictionthat

β

^increases

systematicallywithincreasingx⁰_Au,andtheexperimentalvaluesare indeedseentoscatteraroundthetheoryline.

Itisofinteresttofocusonthesampleswithx⁰_Au=0.25,which exhibit substantial differences in their initial

ϕ

^{, as apparent in}

Fig.4(b).Inspiteofthosedifferences,the

β

^{valuesarenearlyiden-}

tical.Inotherwords,weobservepracticallynoimpactofshrinkage on

β

^{. That isconsistent with the notion that the} connectivityis established early-on duringthe dealloying, andremains invariant duringlatershrinkageevents.

4.5. Relaxation,shrinkage,andconnectivityevolution

Molecular dynamics simulations ofthe relaxation of NPG ex- plored the interrelation between volume shrinkage and the con- nectivity, asparameterized bythe scaled genus g. Thesimulation studied the same four

ϕ

^{as the} experiment, namely 0.20, 0.25, 0.30, and0.35. The initial structures were constructed by means of theleveled-wavemodel. Thatmodel,whichalso underliesour analysisofthestiffnessandspecificallyEq.(3),hasbeenfoundto quantitatively agree with experiment on as-prepared NPG in re- spect to connectivity, Young’s modulus, and strength [23,68]. In agreement withthe experiment, the results(Fig. S2a inSupport- ing Information) show progressively largervolume contraction as

ϕ

^{is reduced.We found that the relaxation brings} practically no relative increase inthe netgenus Gforlarger

ϕ

^{.Yet,G increases}

substantiallyatlesser

ϕ

^(Fig.S2b).

Figure 9 summarizes the findings for the relaxationin a plot of gversus

ϕ

^{, following theevolution of each ofthe four model}

structures duringtherelaxation.Thefirstdatapoint(atlowest

ϕ

⁾

foreachstructureisseentoberightonthetheorylineforthelev- eledwave model,validatingourimplementationofthatmodel.At the largest

ϕ

^{value,0.35, thetracesofg}(

ϕ

)^{are sensiblyhorizon-} tal, confirmingthat thedensification hereleavestheconnectivity unchanged. Astheinitial valueof

ϕ

^{isreduced, theincrease in}

ϕ

is enhanced and at the same time the traces of g(

ϕ

) ^{slope up-} wards,indicatingthattherelaxationwillenhancetheconnectivity

Fig. 10. Exemplary depiction of a reconnection event in nanoporous gold with an Au content of 0.20 as a result of relaxation. Images show molecular dynamic simu- lations of a nanoporous structure before (upper image) and after relaxation (lower).

Numbers refer to initially individual ligaments that connect during the relaxation event.

ofstructures withlesser initial solid fraction.Wefound that this trendisparticularlystrongforthestructurewith

ϕ

=0.20.

The observationsfromMD alsopoint towardsan explanation:

many “dangling” ends of disconnected ligaments were observed in the low-density structure. The shrinkage can bring the dan- glingends incontactwithneighboringregions ofthesolidstruc- ture, thereby establishing new connections. Several examples of this process are marked in Fig. 10. That process is less likely in thedenserstructures,becausetheystartout wellconnected,with muchfewerdanglingends.

4.6. As-dealloyedvs.coarsenednanoporousgold

We now discuss the evolution of the mechanical behavior in response to coarsening, focusing on sample types that have un- dergonenegligible shrinkageduringpreparation,so thattheir as- preparedstatehas

ϕ

≈x⁰_Au.Figure11(a)plotsthemeanvalueand standarddeviationofY^effofthesampletypesfromthissetversus

ϕ

^{.As introducedin Tables1 and2,the initialandfinal ligament}

sizesareconsistentlyL≈40and110nm,respectively,forallsam- pletypes.Thedataconveysthat annealingcanintroduceadensi- fication,particularlyforsampleswithsmallx⁰_Au.InrespecttoY^eff, thefigureillustratesagaintheenhanced complianceatreduced

ϕ

inthe initial state.Moreover, the arrowsillustrate that annealing reducesY^eff.The remarkableobservationis that thisreduction is smallforsampleswithx⁰_Au=0.35,whereasthereisapronounced decreaseinY^effforsampleswithlowerx⁰_Au.ThereductioninY^effis relativelylarger forsamples withsmaller

ϕ

^{.Thatisobviousfrom}

thelarger downwardshiftofthedatapointsforsmaller

ϕ

^,along

withthelogarithmicordinatescaling.

The last-mentioned observation is further illustrated by the graphinFig.11(b),whichshowstheapparent load-bearingfrac- tionfactor

β

^{,Eq.(5),inadisplayanalogoustoFig.11(a).Thedata}

Fig. 11. Comparison of as-dealloyed and annealed nanoporous gold with varying Au amount in Ag-Au master alloy, x ⁰_Au. (a) Effective Young’s modulus, Y ^eff, at low strains, normalized with Young’s modulus of massive gold, Y ⁰= 79 GPa [69] , vs.

solid fraction, ϕ. The stated percentage quote the drop in stiffness due to coars- ening. (b) figures β= (Y ^eff/Y ⁰)^1/2/ϕ vs. x ⁰_Au. The data points are averages over all samples of the same type, with error bars indicating the standard deviation.

is here plotted versus x⁰_Au in order to emphasize its assignment tothesampleidentity.Thisfigureunderlinesthat thefindings for the variationofY^effinresponse toannealingimplyavariation of theapparentload-bearingfraction,inotherwords,achangeinthe networkconnectivity.Thatvariationisrelativelylargerforsmaller solidfractions.

5. Discussion

The present study investigates the mechanical behavior of nanoporous gold (NPG) made by dealloying Ag-Au, withan em- phasisoncoveringawiderangeofinitialAuatomfractions,x⁰_Auin themasteralloyand,thereby,awiderangeofsolidfractions,

ϕ

ⁱⁿ

theporousmaterial.We exploittheestablishednotionthatvaria- tions intheeffectiveYoung’smodulus,Y^eff,provideasignatureof variationsinthemicrostructure ofthematerial’snetwork ofliga- ments, andspecificallyforvariationsintheconnectivity. Thereby, ourexperimentsonthemechanicalbehavioraffordconclusionson howthe microstructuredependsonx⁰_Au and

ϕ

^intheas-prepared state, andon how themicrostructure evolveswhen the ligament size,Listunedbyannealing-inducedcoarsening.

Inthe contextofourwork,variationsofthemechanicalprop- erties areonlyindirectlyrelatedtovariationsinthescaled genus g, namelythroughthe apparent load-bearingfraction factor,

β

^of

thenetwork.Thatparameterisempiricallyaccessiblefromtheex- perimentalY^eff alongwithEq.(6).Qualitatively,anydecreasein

β

canonlybeunderstoodwithaconcomitantdecreaseing.Thislink underliesourdiscussion.

5.1. As-dealloyedstate:variationofconnectivitywithsolidfraction

The data in our work confirms the established notion [1,5,20,22,61,63,64,68] that scaling lawsfor network structures of constantconnectivity,suchastheGibson-Ashbylaws,donotpro- videanadequatedescriptionofthevariationofstrengthandstiff- ness of NPG with its solid fraction.Our results also support the approach ofRefs [20,22],in which theGibson-Ashby law forthe strength,Eq.(1),isappliedtoanequivalentnetworkthataccounts only forthe load-bearing struts. Tothis end, the true solid frac- tionisreplacedwithanapparent,load-bearingone,whichcanbe empiricallydeterminedbasedonexperimentaldataforY^effandon Eq.(6).The agreement withthe experimental strength datasup- portsthenotionofaneffective,load-bearingnetwork.

Thehighsignificanceoftheparameter

ϕ

^{lbsuggestsan inspec-}

tionofitsrelationtothenetworkmicrostructure.Oneobviouscon- cept relatesdifferencesbetween

ϕ

^{lb andthe truesolidfraction}

ϕ

todisconnectednetworkstruts,whichcarrynoload.Thatnotionis consistentwiththeexpectationthat coarsening– whichisalways a part of the microstructural evolution during dealloying – re- quiresligamentstopinchoff.Pinch-off eventscreatedisconnected struts, whichwill persistuntil theirmaterial isredistributed into theload-bearingparts ofthe networkbydiffusion.Yet,theintro- duction of disconnectedstructures is not the only wayin which thestiffnessofanetworkcanbereduced.Relevantchangesinthe topologycanincludedifferencesinthemeannumberofstrutsper node[63],orinthegeometryofload-bearingrings[70].The sig- nificance of those microstructural descriptors for the mechanical behaviorofthe randomnetwork ofNPGremains tobe examined indetail.

As another word of caution, we have pointed out that stud- ies predicting

σ

^{y based on}

ϕ

^{lb so farrequired empirical datafor}

Y^eff. That approach leaves the issue of a systematic dependence of thescaled genus – and, hence,of the load-bearing solid frac- tion– onthetruesolidfractionand/orthemasteralloycomposi- tionasanopenissue.Astronghypothesisonasystematicdepen- dencebetween g and

ϕ

^{underlies the modified}Roberts-Garboczi scaling law of Ref [23], Eq. (3). The law accounts for a system- aticlossinconnectivitywithdecreasingsolidfractioninNPG.Our considerationsontheimplicationofthislast-mentionedapproach for the apparent load-bearing fraction factor,

β

^{are well consis-}

tent with the data. This suggests that the leveled wave model ofRef. [23] catchesessential aspects of thephysics that leadsto the systematic decrease in stiffness with solid fraction in deal- loyed NPG.That agreement betweenscaling law andexperiment has already been pointed out, with previously publishedexperi- mentaldatafromvariousteamsasthebasis[23].Thepresentwork strengthenstheagreementbyadding,forthefirsttime,adataset withaconsistentsynthesisandcharacterizationprotocol.

5.2. As-dealloyedstate:impactofshrinkageonconnectivity

Shrinkage duringdealloying lets the solid fraction,

ϕ

^{, endup}

larger thanthe non-dissolvedatom fraction,which isideallyx⁰_Au. Asexemplifiedinourstudy,thepresenceornotofshrinkage,and itsnumericalamount,dependonthedealloyingconditions.Specif- icallyourexperimentspointtowardsmoreshrinkageatlesserx⁰_Au, andthisisconfirmedbyourMDstudyofrelaxation.

Remarkably, we find samples that have undergone shrinkage consistentlymorecompliantthansamplesofsamedensitythatdid notexperience shrinkage.Furthermore,wefindthat stiffnessval- uesdofollowsystematictrendswhenplottednot versusthetrue solid fraction,

ϕ

^{,butversus x0}Au.We hypothesizethat thecharac- teristicnetworkconnectivity– asitunderliesspecificallythemod- ifiedRoberts-Garboczi scaling law Eq.(3) – is established in the early stagesofdealloying, beforethe onsetof shrinkage.Further-