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Acta Materialia 59 (2011) 7645–7653

Coarsening by network restructuring in model nanoporous gold

Kedarnath Kolluri ⇑, Michael J. Demkowicz

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received 11 March 2011; received in revised form 22 August 2011; accepted 23 August 2011Available online 15 October 2011

Abstract

Using atomistic modeling, we show that restructuring of the network of interconnected ligaments causes coarsening in a model ofnanoporous gold. The restructuring arises from the collapse of some ligaments onto neighboring ones and is enabled by localized plas-ticity at ligaments and nodes. This mechanism may explain the occurrence of enclosed voids and reduction in volume in nanoporousmetals during their synthesis. An expression is developed for the critical ligament radius below which coarsening by network restructur-ing may occur spontaneously, setting a lower limit to the ligament dimensions of nanofoams.� 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Coarsening; Foams; Nanoporous metals; Modeling

1. Introduction

Nanoporous metals are sponge-like metallic structureswith open-cell network structures. They are comprised ofinterconnected ligaments of nanometer-scale characteristicdimensions. Nanoporous metals have attracted attentionboth for their intrinsic scientific interest [1,2] and due totheir potential use as actuators [3], biosensors [4], fuel cellelectrodes [5] and in bone tissue engineering [6]. To realizethese promising uses, a fundamental understanding of thestability and morphological evolution of nanoporous met-als is necessary. Based on atomistic simulations of modelnanoporous gold, we suggest a coarsening mechanism thatmay explain certain puzzling experimental observations,namely the occurrence of voids enclosed in ligaments andthe reduction in volume during synthesis of nanoporousgold (np-Au).

Nanoporous metals are commonly synthesized by deal-loying, in which less noble components of an alloy are elec-trochemically dissolved. For example, np-Au is synthesizedby dissolving silver from a silver-rich Au–Ag alloy [7].Other examples include synthesis by dealloying of np-Pt

1359-6454/$36.00 � 2011 Acta Materialia Inc. Published by Elsevier Ltd. All

doi:10.1016/j.actamat.2011.08.037

⇑ Corresponding author. Tel.: +1 413 577 0136.E-mail address: kkolluri@mit.edu (K. Kolluri).

from Pt–Ag alloys [8] or Pt–Si alloys [9,10], and np-Cufrom Al–Cu alloys [11]. Initial understanding of the forma-tion and coarsening of np-Au was based on on-lattice sur-face diffusion of Au atoms [12]: as Ag atoms are removed,Au atoms diffuse on the surfaces thereby exposed andattach to terraces and hillocks, causing the np-Au to coar-sen. Such a model, however, is not sufficient to explain allexperimental observations on np-Au formation. An on-lattice model assumes that the number of atomic sitesremains constant, and therefore predicts that the volumeof the np-Au remains fixed during dealloying, whereasthe volume of samples reduces by as much as 30% in somesynthesis procedures [1,13]. A coarsening model based onlyon surface diffusion would predict that enclosed voidswould not form, whereas ligaments in some np-Au sampleswere found to contain voids [14]. Furthermore, surfacediffusion-dominated coarsening would predict that regionswith large positive or negative curvature such as ligamentpinch-off regions are quickly smoothed out. Remnants ofligament pinch-off, however, have been observed in exper-iments [34].

Because several np-Au samples were found to containlattice defects such as stacking faults, twins and disloca-tions [1,13,15,16], it has been suggested that localized plas-tic deformation may cause volume reduction during

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7646 K. Kolluri, M.J. Demkowicz / Acta Materialia 59 (2011) 7645–7653

dealloying. Nonetheless, a detailed picture of how plasticdeformation by dislocation glide—a volume conservingprocess—may lead to volume reduction and the formationof internal voids is lacking. Based on atomic-scale simula-tions of model np-Au, we suggest a mechanism by whichplastic deformation may lead to coarsening of nanoporousmetals. The mechanism we suggest could also explain otherphenomena observed during dealloying in experiments,namely densification of np-Au and formation of enclosedvoids.

The formation and coarsening mechanisms of np-Au aredifficult to determine by experiments alone since the spatio-temporal resolution required to directly observe the relevantprocesses often exceeds current capabilities. Unfortunately,a realistic representation of dealloying chemistry and timescales is also beyond the current capabilities of atomic-scalesimulations. We therefore investigate the morphologicalevolution of a model np-Au structure that, while it doesnot directly correspond to the experimental dealloying pro-cess, shares some important structural properties with realnp-Au. Using atomistic simulations, we find that this modelnp-Au coarsens by restructuring its open-cell network bycollapse of neighboring ligaments onto each other. The liga-ment collapse is made possible by concurrent localized plas-ticity at ligaments and nodes. The restructuring of the np-Aunetwork causes volume reduction and coarsening and onoccasion leads to formation of voids completely enclosedin ligaments. Such a coarsening mechanism may be opera-tive in the early stages of formation of real np-Au and mayexplain some experimental observations that a surface diffu-sion-dominated mechanism alone cannot. In addition, theproposed mechanism predicts a critical ligament radiusbelow which such plasticity-mediated coarsening wouldoccur spontaneously, setting a lower limit to the ligamentdimensions of nanofoams.

2. Model and methods

In our atomistic simulations, interatomic interactionsare modeled using an embedded atom method (EAM)potential for gold [17]. (A review of EAM potential meth-odology can be found in Ref. [18] and a recent review ofinteratomic potentials for metals can be found in Ref.[19].) This EAM potential was fit to the equilibrium latticeconstant, sublimation energy, bulk modulus, elastic con-stants and vacancy formation energy of face-centered cubic(fcc) Au [17]. It also predicts well other properties such asthe melting point and that Au is most stable in the fccstructure [17,20]. This EAM potential also agrees well withthe universal equation-of-state by Rose and co-workers[21] and the experimentally determined radial distributionfunction of liquid gold [20]. Additionally, the EAM poten-tial used in this study predicts correctly the trends in sur-face energy for different surface orientations [17,20],which is crucial in studies of nanoporous metals since sur-faces make up a large fraction of the material.

The stable stacking fault energy of 6 mJ m�2 predicted bythis EAM potential is much lower than the stacking faultenergy of 32.5 mJ m�2 for real gold [20]. Nonetheless, theunstable stacking fault energy of 102.9 mJ m�2, which gov-erns the barriers that must be overcome for glide dislocationnucleation, predicted by this EAM potential is nearlyidentical to 101.8 mJ m�2 predicted by a more-recentEAM potential [22]. Therefore, we expect that the lengthsof stacking-fault ribbons in simulated np-Au would belonger than in real np-Au, but the qualitative nature of dis-location nucleation with the potential used in this study[17] would be the same as that with the newer potential [22].

To create a model nanoporous structure, N = 500,000atoms were placed in a simulation cell under periodicboundary conditions. The size of the simulation cell waschosen to obtain np-Au of desired relative density,qrel ¼ q

qAu. For example, to create np-Au of relative density

qrel = 0.2, the simulation cell in each dimension was chosen

to be aAu �ffiffiffiffiffiffiffiffi

N4�qrel

3

q¼ 34:883 nm, where aAu = 0.408 nm is

the equilibrium lattice parameter of pure gold at 0 K.Initially, an expanded fcc structure with lattice parameterequal to aAuffiffiffiffiffi

qrel3p was filled with atoms after which each atom

was moved randomly with displacement in each of thethree directions chosen by uniform sampling in the range(�1 nm, 1 nm). The system was subsequently relaxed atconstant volume using conjugate gradient potential energyminimization (PEM) [23] for 150,000 iterations.

The initially random distribution of atoms spontaneouslyaggregates into a foam-like structure, similar to thoseobserved in previous simulations of processes ranging fromhomogeneous liquid–vapor nucleation in fluids [24–26] tomacroscale galaxy redistribution in the universe [27]. Inaddition to the aggregate foam-like structure, the simulationcell also contains free-standing clusters of atoms. These clus-ters form because the distance between their surfaces and theinterconnected foam exceeds the range of the EAM potentialused in this study and because there are no long-range (e.g.electrostatic) or body (e.g. gravitational) forces in ourmodel. Most free-standing clusters contain 1–200 atoms.For example, foam-like structure made with qrel = 0.2 con-tains a total of 1543 clusters with 2 clusters containing about1300 atoms, 12 clusters containing between 500 and 1000atoms, 30 clusters containing between 200 and 500 atoms,and each of the rest containing less than 200 atoms. A totalof 9% of the atoms are contained in these free-standing clus-ters for a foam-like structure for qrel = 0.2.

The resulting foam-like structure is a solid. The first,third and fourth peaks in the pair correlation function,while broader than that for a deformed single crystal, wereclearly discernible and centered around the values expectedfor fcc Au. The second neighbor peak was reduced due tothe presence of a large percentage of atoms at free surfaces.This structure was annealed at 300 K (maintained constantby velocity rescaling [23]) for �0.8 ns using moleculardynamics (MD) simulations [23] with a time step of

K. Kolluri, M.J. Demkowicz / Acta Materialia 59 (2011) 7645–7653 7647

0.8 fs, during which the np-Au coarsens. As a consequence ofcoarsening, most free-standing clusters of atoms becomeattached to the np-Au. The remaining few free-standing clus-ters are removed and the potential energy of the resultingstructure is minimized again. In order to confirm that the pres-ence of clusters has negligible effect on coarsening, a represen-tative nanofoam resulting after removing the few remainingclusters was annealed at 300 K for another�1 ns. The nano-foam continued to coarsen with no appreciable change incoarsening rate (Fig. 2a), confirming that the free-standingclusters have negligible effect on coarsening. Three differentfinal relative densities, qrel ¼ q

qAu2 f0:19; 0:3; 0:42g, were con-

sidered in this investigation. Annealing was performed at bothconstant volume and at constant zero pressure.

Diffusion, a thermally activated process, is not expectedto occur during the simulations described above. No ther-mally activated phenomena can occur during potentialenergy minimization. MD simulations performed at300 K would require a run of several nanoseconds for a sin-gle atom jump to occur, given migration barriers typicallyfound in metals [17]. Therefore, we expect negligible sur-face diffusion in the timescale of our study (about 1 ns).Consequently, the phenomena observed in our simulationsare caused by processes other than diffusion.

Simulations were performed using LAMMPS [28]. The liga-ment and pore radii of the nanoporous structures were deter-mined using pore-size distribution functions [29]. Fig. 1shows a visualization of a typical np-Au structure obtainedusing this procedure. Structural features such as atoms in fccenvironments, stacking faults, twins and dislocation cores werecharacterized using common neighbor analysis [30]. Prevailingcrystallographic orientations along free surfaces were deter-mined by analyzing surface radial distribution functions[23]. AtomEye [31] and VisIt [32] were used for visualization.

Fig. 1. A representative model np-Au with qrel = 0.19 formed by relaxationof an initially random distribution of atoms followed by annealing for�0.8 ns. Ligaments, nodes and pores are indicated. Two representativeligaments in which the curvature changes sign are also shown.

3. Structural features of model np-Au

The model np-Au, like that shown in Fig. 1, is always opencell with interconnected ligaments. As with most real np-Au,the model structures are crystalline and fcc with the latticeparameter corresponding to that of fcc Au [1,7,15,33] andthe ligament surface normals of model np-Au are predomi-nantly in the crystallographic h111i direction [34]. Like somereal np-Au [33], our model np-Au systems are polycrystallinewith grain sizes comparable to ligament dimensions. Themodel np-Au contains lattice defects such as dislocations,stacking faults and twin boundaries; such defects have beenobserved in real np-Au [1,13,15,16]. Like some real np-Au[14], three or four ligaments typically meet each other at nodesin the model np-Au we obtained (determined by visual inspec-tion; see Fig. 1). The local curvatures of each ligament in themodel np-Au vary, with curvatures changing from positive tonegative, sometimes more than once along a single ligament(determined by visual inspection; see Fig. 1). Similar varia-tion in mean curvature was observed in real np-Au as well[14,34,35].

The average ligament and pore sizes in model np-Austructures are much smaller than those found in experi-ments. The average ligament radii in model np-Au are inthe range 1.5–2.1 nm, while the average pore radii are inthe range 6.7–4.8 nm. By contrast, ligament radii derivedexperimentally ranged from 5 to 50 nm for the same rangeof relative densities [14–16,35–37]. A possible consequenceof this difference is that our model may be more represen-tative of np-Au in the initial stages of dealloying where lig-ament radii may be much smaller than those finallymeasured [14–16,35–37]. Despite this discrepancy, just asfound for some experimentally synthesized np-Au struc-tures [14], the ligament and pore radii of model np-Au scalewith relative densities in the same manner as in conven-tional open cell foams [38]. We found that the relative den-sities of our model np-Au are related to ligament and poreradii by the vertex-corrected scaling expression given byEq. (1) [38] where rl and rp are the ligament and pore radii,respectively, with C2 = 4.38 ± 0.23 and D2 = 4.93 ± 0.58:

qrel ¼ C2

rl

rp

� �2

1� D2

rl

rp

� �� �ð1Þ

Model np-Au increased in density when annealed at zeropressure. An increase of �33% in density was observed fornp-Au that was initially 19% dense and an increase of�25% in density was observed for np-Au that was initially30% dense. Furthermore, enclosed voids are observed inmodel np-Au with qrel 2 {0.3, 0.42} annealed at constantvolume. The average radius of the enclosed voids rangedfrom 0.65 to 1.35 nm. For comparison, voids with radiiranging from 0.5 to 2.5 nm have been observed in realnp-Au with 16 nm radius ligaments [14].

All of the structural and topological features present inmodel np-Au have been observed in differently synthesizedreal nanoporous materials, as described above. It appears

7648 K. Kolluri, M.J. Demkowicz / Acta Materialia 59 (2011) 7645–7653

that the main discrepancy between the model np-Au andreal nanoporous materials is that no one real nanoporousmaterial exhibits all of the structural features of the model.How this discrepancy may reflect differences in propertiesrequires further investigation, but we believe that it wouldnot alter qualitatively the results presented in thismanuscript.

4. Coarsening during annealing of model np-Au

Model np-Au samples coarsen during annealing at300 K. Fig. 2a shows the evolution of the average liga-ment size and number of surface atoms during annealingof a 19% dense np-Au at 300 K and constant volume.Atoms in free-standing clusters are counted neither whencomputing the number of surface atoms nor in determin-ing the ligament radii shown in Fig. 2a. The vertical linemarks the point where remaining free-standing clusterswere removed. Subsequent annealing of the nanofoamshows that the clusters have a negligible effect on nano-foam coarsening. As annealing proceeds, surface atomsreduce in number while simultaneously the average liga-ment radius increases. We found that the np-Au coarsensby collapse onto each other of neighboring ligamentsresulting in a net reduction of free surface area and for-mation of thicker ligaments. Two representative examplesof ligament collapse during annealing of a 19% dense np-Au at 300 K and constant volume are shown in Fig. 3a–c and d–f, where each panel is a sequence of snapshotsleading to one collapse event. In both cases, the collapseis aided by plastic deformation at nearby nodes. InFig. 3a–c, dislocation glide enables shearing initially atthe bases of adjacent ligaments connecting to the samenodes and then in the ligaments themselves, allowingtheir ”displacive” movement toward each other andeventual collapse into one ligament.

In Fig. 3d–f, the indicated ligament pinches off, followedby collapse of other nearby ligaments onto each other. The

Fig. 2. Evolution of average ligament radii and percentage of atoms at free surquenching randomly placed Au atoms and (b) volume-conserving uniaxial comthe left of the vertical line in (a) represents annealing of np-Au containing freenp-Au with all free-standing clusters removed.

ligament pinch-off occurs by dislocation glide across theligament cross-section. As in the example in Fig. 3a–c, herealso plastic deformation enables network reconstruction.Although the ligament pinch-off creates additional free sur-face area, the associated energy increase is more than com-pensated by the decrease in energy arising from theremoval of free surfaces due to accompanying collapse ofneighboring ligaments onto each other.

While there appear to be two regimes with differentcoarsening rates in Fig. 2a, they differ primarily in the num-ber of collapse events. Initially, the ligaments are suffi-ciently thin and close that the collapse rate, and thereforealso the coarsening rate, is high. As the average ligamentand pore sizes increase, the average distance between liga-ments also increases. The increased ligament size impliesthat a higher plastic work rate is required to maintain agiven coarsening rate. Increased pore size also implies thatgreater plastic work is necessary for equal number of col-lapse events. Therefore, the coarsening rate reduces as theligament radius increases. Since the coarsening rate is likelyto be dependent on both ligament and pore radius, andbecause the ligament and pore radius can be related to eachother by the relative density as in Eq. (1), the rate of coars-ening by network restructuring is likely dependent on therelative density.

No appreciable migration of surface atoms by thermallyactivated diffusion was observed during the annealing pro-cess. In model np-Au, atoms that are at the free surface atthe end of the annealing process moved on average byabout 0.45 nm from their initial neighbors. For coarseningto occur by surface diffusion, however, the average surfaceatom migration distance would have to have been of thesame order as the ligament length or the pore radius, whichis in the range 6.7–4.8 nm in our simulations. The observedatomic motion is likely due to surfaces relaxing to theirlowest-energy orientations, which may aid in the onset oflocal plastic deformation and thereby contribute indirectlyto coarsening [39].

faces during (a) annealing at 300 K of np-Au (19% final density) formed bypression of 19% dense np-Au annealed for �0.8 ns. The part of the plot to-standing clusters, while the part to the right is for continued annealing of

Fig. 3. Several examples of collapse of ligaments during (a–f) annealing at constant volume and (g–i) volume-conserving deformation of 19% denseannealed np-Au. In (a–c), the collapse of two neighboring ligaments onto each other is enabled by plastic deformation by dislocation glide, first at the baseof the ligaments and then within the ligaments. In (d–f), pinch-off of the marked ligament accompanied the collapse of adjacent ligaments onto each other.The horizontal lines in (e and f) are guides to the eye showing the direction of ligament collapse. In (g–i), two pinched-off ligaments collapse onto oneanother. In all cases, ligament collapse is made possible by concurrent local plastic deformation at both ligaments and nodes.

K. Kolluri, M.J. Demkowicz / Acta Materialia 59 (2011) 7645–7653 7649

5. Coarsening during deformation of model np-Au

Since localized plastic deformation is required for coars-ening by the mechanism described above, it may beexpected that application of an external load may also pro-mote coarsening. To test this hypothesis, volume-conserv-ing uniaxial compression (�zz < 0, �yy = �xx > 0) wasperformed at zero temperature on np-Au samples that wereannealed for �0.8 ns, with small remaining clustersremoved, and the resulting structure relaxed by potentialenergy minimization. Model np-Au was compressed alongthe z direction at strain increments of 0.99% while the x

and y cell dimensions were extended such that the total vol-ume of the simulation cell remained constant. The strainapplied to the model np-Au was therefore purely deviator-ic. After each strain increment the model was relaxed atzero temperature by PEM until the maximum force onany atom was less than 5 pN. All the samples were foundto coarsen during the deformation (Fig. 2b). At sufficientlylow applied tensile equivalent strains �dev, local plasticdeformation occurs but does not cause ligament pinch-offor collapse. Upon further deformation, however, ligamentpinch-off and associated network reconstruction do occurand cause coarsening. Just as during annealing, where

events like those shown in Fig. 3a–c and d–f take place,pinch-off and collapse of neighboring ligaments onto eachother are also observed upon deformation, as illustratedin Fig. 3g–i.

In addition, model np-Au at all densities investigatedexhibited an elastic–perfectly plastic stress–strain response(Fig. 4a) reminiscent of the behavior of conventional metal-lic foams [38] and qualitatively similar to that found in exper-iments on np-Au [36]. A critical tensile yield strength for goldof rs = 3.864 GPa may be backed out by fitting the averageflow stress (Fig. 4b), which is also the average yield stress inthe case of materials exhibiting elastic–perfectly plasticresponse, in the applied strain range 0.1 6 �dev 6 0.3 withthe Gibson–Ashby scaling equation for plastically collaps-ing foams. The Gibson–Ashby scaling equation, Eq. (2)[38], relates relative foam density (qrel), average plastic yieldstress of the foam (r) and the yield strength of the foam mate-rial (rs) where C = 0.3 [38].

rrs¼ Cq3=2

rel ð2Þ

Assuming that the ligament axes are on average orientedalong the h110i directions (the Schmid factor for slip along{111} h112i is then �0.47), a critical resolved shear stress

Fig. 4. (a) Elastic–perfect plastic response to volume-conserving uniaxial compression of model np-Au of different densities. (b) Gibson–Ashby scalingrelation [38] used to back out critical tensile yield strength of model gold (see text).

7650 K. Kolluri, M.J. Demkowicz / Acta Materialia 59 (2011) 7645–7653

for deformation of 1.8 GPa is obtained and is in goodagreement with the critical shear stress of model single-crystal Au used in our simulations (rideal

s ¼ 1:92 GPa forthe same slip system). The critical shear stress for modelsingle-crystal Au was calculated by a simple shear deforma-tion on a {11 1} plane along a h112i direction of a super-cell single-crystal Au using the same EAM potential [17]used to simulate model np-Au. Some experimentally mea-sured yield stresses in np-Au were in good agreement withthe Gibson–Ashby scaling expression when modified toaccount for ligament size dependence of yield stress [40].Modification of the Gibson–Ashby scaling relation toaccount for size effects is not required in our study as theligament radii in the model np-Au considered do not varyover a wide enough range.

6. Enclosed voids in model np-Au

A coarsening mechanism involving reconstruction of thenanoporous network may elucidate experimental observa-tions that cannot be explained by coarsening through sur-face diffusion alone. For example, internal voids may formwhen collapsing ligaments are not perfectly contiguous.Fig. 5 shows sections of a 30% dense np-Au as it coarsensduring annealing at constant volume. The np-Au in the toppanel is sectioned to make visible an internal void. InFig. 5a–d, which shows successive stages of coarsening,surface atoms on both ligaments and internal voids are col-ored black, while the lighter atoms are inside ligaments andnodes. The bottom panel, Fig. 5e–h, shows a three-dimen-sional view of the section marked in Fig. 5a. In the bottompanel free surfaces are colored in transparent red whileinteriors of ligaments and nodes are colored transparentgreen. As ligaments shown in Fig. 5e collapse, some regionsalong these ligaments are closer to each other than others,as indicated by the arrows in Fig. 5f. If the collapsing sur-faces completely enclose other surfaces that are furtherapart, voids may form. Such a scenario is played out inFig. 5f–h where the regions marked with arrows collapsesequentially to leave an internal void, marked in Fig. 5h.

The mechanism described above contrasts with arecently proposed surface-diffusion-based alternative

explanation for formation of enclosed voids [44]. In thisalternative explanation, based on kinetic Monte Carlo sim-ulations performed on simulated dealloyed nanoparticles,it has been suggested that surface diffusion in nanoporousstructures leads to pulling away of material from saddle-point curvature ligaments, with the geometric effect ofreducing the topological genus. Such pulling away of mate-rial leads eventually to ligament pinch-off associated withRayleigh instabilities, and this also leads to bubble forma-tion. In our view, the mechanism proposed in Ref. [44]offers a competing, though not contradictory, explanationto that presented here.

7. Critical ligament radius for spontaneous coarsening

Our simulations show that coarsening in some nanopor-ous metals may arise from network restructuring accompa-nied by localized plasticity. For nanoporous structures withsufficiently small ligament radii, the rate of surface energyreduction during coarsening may be greater than the rateof plastic work. If so, coarsening by this mechanism mayoccur spontaneously without thermal activation until liga-ment radii reach a critical value. For ligament radii beyondthis value, the rate of plastic work exceeds the rate of sur-face energy reduction. To develop an estimate of this crit-ical radius, we consider a nanoporous material consistingof ligaments with average ligament radius R and averageligament length L, which is proportional to the pore radius.Suppose that collapse of a fraction p1 of the ligaments leadsto an increase dR in the average ligament radius and corre-sponding decrease in surface area. As shown in Fig. 3,some of these events are accompanied by pinch-off ofnearby ligaments, which increases the free surface area.To account for this, we assume that a fraction p2 of the lig-aments that pinch off do not collapse. We also assume thatthe density remains constant during coarsening, thoughthis is not strictly necessary.

The incremental decrease in the per-ligament surfaceenergy due to the reduction in the surface area is then givenby

dEsurface ¼ ½C0p12pcL� p24pcR�dR ð3Þ

Fig. 5. Coarsening during constant volume annealing of a section of 30% dense np-Au. (a–d) Black atoms are at the surfaces of ligaments or internal voidsand other atoms are within the ligaments and nodes. (e–h) Three-dimensional view of the section marked in (a) where free surfaces are shown intransparent red and ligament interiors are shown in transparent green. The collapse of ligaments in (e) leads to the formation of an internal void marked in(h). The corresponding sectioned region of the void is marked in (d). (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

K. Kolluri, M.J. Demkowicz / Acta Materialia 59 (2011) 7645–7653 7651

where the first term corresponds to the surface area re-moved by the collapse of ligaments and the second is theadditional surface area created at pinched-off ligaments.C0 is a measure of the extent of collapse of ligaments ontoeach other and reaches its maximum value C0max

¼2�

ffiffiffi23p¼ 0:74 at constant np-Au density and when all

ligaments collapse onto their neighboring ones withoutforming any enclosed voids. At constant density, L andR are related by Eq. (1). For simplicity, we retain onlythe leading-order term R � LC02

ffiffiffiqp

rel, where C02 ¼ C�1

22 is

obtained by fitting data from our simulations. The incre-mental decrease in surface energy per ligament can thenbe rewritten as

dEsurface ¼ 2pp1

K1

C02ffiffiffiqp

rel

cRdR; where K1

� C0 � 2p2

p1

C02ffiffiffiqp

rel

� �ð4Þ

The plastic work required for ligament collapse may beestimated as follows. Since the localized plasticity is causedby dislocation glide, the average plastic work for each dis-location is the amount of work required to move a disloca-tion across a ligament. The glide of one dislocation causes adisplacement equal to the Burgers vector b of the disloca-tion. Since the average displacement required for ligamentcollapse is proportional to the pore radius, the number ofdislocations that must glide through a ligament on average

is 1K0

Lb, where L is the distance a ligament must be displaced

to collapse into one of its neighbors and K0 is a constantparameter of order 1. Therefore, the associated incrementof plastic work per ligament is given by

dW plastic ¼ p1½sbð2pRdRÞ� 1

K0

Lb

� �ð5Þ

where s is the flow stress.Coarsening by the postulated mechanism will occur spon-

taneously ifdW plastic

dR <dEsurface

dR . To compute the critical ligamentradius beyond which the postulated mechanism does notoccur spontaneously, we set the two terms equal and get

R� ¼ Kcs; where K ¼ K0 C0 � 2

p2

p1

C02ffiffiffiqp

rel

� �ð6Þ

The critical radius R* below which coarsening by net-work restructuring occurs spontaneously decreases withplastic flow resistance, increases with surface energy, anddepends on the nanoporous network topology and the sta-tistics of ligament collapse and pinch-off events through theconstant K. Furthermore, Eq. (6) predicts a network topol-ogy-dependent critical pinch-off fraction beyond which thesurface area increase is too large to be compensated by lig-ament collapse. Thus, it may be possible to design nano-porous networks that are less susceptible to coarseningby the proposed mechanism, even though ligament collapsemay still occur at isolated locations during annealing and

7652 K. Kolluri, M.J. Demkowicz / Acta Materialia 59 (2011) 7645–7653

ligament collapse may lead to coarsening when externalloading is applied.

For a numerical estimate of the critical radius, weassume that collapsed ligaments greatly exceed those thatpinch off (K = K0C0) and that all ligaments fully collapse(C0 = 0.74). We assume the flow resistance to bes = 800 MPa. This is the lowest flow stresses for Au nano-pillars observed in compression of 250 nm radius Au cylin-ders [41], although other studies suggest that the relevantflow resistance could be as high as 2.5 GPa [42]. We alsoassume that on average the ligaments must be displacedby half the pore radius to collapse onto an adjacent liga-ment, i.e. K0 = 2. For cf111gAu

¼ 1:25 J m�2 [43], we getR�Au � 3:2 nm. For the range of permissible values for sand K0, the critical radius R�Au ranges between 1 and10 nm. This range of predicted critical radii for np-Au liesbelow the average ligament radii reported in all experi-ments on np-Au of which we are aware [14–16,35–37,40],suggesting that coarsening by the mechanism describedhere may be operating very early in the dealloying andcoarsening process.

8. Conclusions

Atomistic simulations reveal that coarsening of modelnp-Au may occur by network restructuring caused byneighboring ligaments collapsing onto each other. Thisprocess is accompanied by localized plasticity at nodesand within the ligaments themselves. An expression isdeveloped for a critical radius below which this mechanismmay operate spontaneously, suggesting that synthesis ofnanoporous materials with ligament radii below of a fewnanometers might not be possible unless their surfaceenergy, flow resistance, and network topology are tailoredto increase the plastic work rate during coarsening beyondthe rate of associated surface energy reduction. Althoughdirect experimental verification of whether coarseningoccurs by collapse of adjacent ligaments is not currentlypossible, available experimental data provide indirect sup-port to the mechanism we suggest: ligaments with high cur-vature corresponding to pinch-off events have beenobserved in nanoporous samples, even when no externalload is applied [34]; enclosed voids form in nanoporousmetals [14]; and np-Au is known to densify during dealloy-ing, annealing [1] and deformation [33]. Continued devel-opment of multi length- and time-scale characterizationmethods may eventually enable validation of the coarsen-ing mechanism proposed here by direct observation [45].

Acknowledgments

This work was partially supported by a user grant fromthe Center for Integrated Nanotechnologies (CINT) at LosAlamos National Laboratory (LANL). We thank A. Misra,A. Antoniou, A.S. Argon, W.C. Carter and L.J. Gibsonfor many useful discussions and helpful insights as well asK.J. Van Vliet and M. Kabir for help with computing

resources during the initial stages of this work. K.K. thanksR.E. Baumer for help with visualization tools.

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