II. MODELING TECHNIQUES

Similar to previous work,^{Reference Crowson, Farkas and Corcoran33,Reference Crowson, Farkas and Corcoran34,Reference Farkas, Caro, Bringa and Crowson39} we constructed an atomistic model of np-Au using a template generated by a phase field simulation. In this technique, spinodal decomposition is modeled using the Cahn–Hilliard equation^{Reference Cahn and Hilliard40} under periodic boundaries in all three directions and the resulting order parameter distribution is used to guide the arrangement of atoms in the model. Although np-Au is usually synthesized through de-alloying^{Reference Erlebacher, Aziz, Karma, Dimitrov and Sieradzki41} and not through spinodal decomposition, the morphologies generated by these two processes have been shown to be comparable.^{Reference Crowson, Farkas and Corcoran33} The particular model used in this study is a single crystal with a relative density of 0.67 and ligament diameter just under 2 nm. The model is periodic in the [100] and [010] crystal directions and terminates with free surfaces normal to the [001] direction. Its dimensions are 107 × 107 nm in the former (periodic) directions and 21 nm in the latter direction. It contains approximately 10 million atoms.

Atomistic simulations were performed using the LAMMPS software.^{Reference Plimpton42} An embedded atom method^{Reference Daw and Baskes43} potential was used to describe interatomic bonding in Au.^{Reference Foiles, Baskes and Daw44} The structures were equilibrated at T = 1 K for 500 ps. During indentation, the temperature was maintained constant at 1 K to avoid any thermal effects. The pressure was kept at zero independently in the [100] and [010] directions using a Nosé–Hoover thermostat and barostat. After relaxation, a nondeformable spherical indenter of 30 nm radius was lowered onto the sample surface in the [001] direction. To prevent the model from simply translating downward under the applied load, the positions of atoms within an approximately 1-nm-thick layer on the surface opposite from the indenter were kept fixed. The technique for indentation is detailed in Ref. Reference Farkas45 and involves representing the indenter by the force it exerts on atoms in the np-Au model. The indentation process was carried out to a maximum penetration depth of 15 nm in 1.5 ns with an MD time step of 1 fs. Thus, our indentation speeds are much faster than those in experiments, as dictated by the available computing resources and the limitations of the MD technique.

Model structure visualization and crystal defect identification were performed using the OVITO software.^{Reference Stukowski46} Dislocations were analyzed using the Dislocation Extraction Aalgorithm (DXA).^{Reference Stukowski and Albe47} This technique involves constructing a closed path around a dislocation defect and mapping it to a perfect crystal lattice to compute the Burgers vector. The procedure is independent of the structural details of the dislocation core, it always yields the true Burgers vector of a dislocation, and no previous knowledge of possible dislocation types is required. Because surfaces can have great impact on, and, at the same time, are influenced by the deformation of such materials, monitoring the surface area and porosity are important analysis tools, and were also performed with the OVITO package.

Figure 1 shows the structure of the model in its as-constructed state. The image at left [Fig. 1(a)] was obtained using a surface mesh created around the ligaments, whereas the image at right [Fig. 1(b)] shows atoms colored by the centrosymmetry parameter.^{Reference Kelchner, Plimpton and Hamilton48} The relaxed structure contains stacking faults that formed during relaxation. The average ligament diameter was determined to be ∼1.76 nm using the AQUAMI software, which was specifically developed for analyzing porous microstructures.^{Reference Stuckner, Frei, McCue, Demkowicz and Murayama49} The complete distribution of ligament sizes in the model is shown in Fig. 1(c). This non-Gaussian distribution is similar to that reported recently for the distribution of ligament sizes in experimental studies.^{Reference Badwe, Chen and Sieradzki50} In particular, it is important to note that the ratio of the distribution width to the average ligament size that we find in our digital samples (approximately 1/3) is consistent with those found in the experiments.

FIG. 1.

(a) 5-nm-thick cross section parallel to the free surface showing the relaxed structure of the as-constructed np-Au model: (a) surface mesh illustrating the ligament morphology; (b) atoms colored by the centrosymmetry parameter, showing free surfaces in red and stacking faults in green. (c) Distribution of ligament diameters determined using AQUAMI.

To understand the specifics of the densification process, the evolution of pore shapes and dimensions was also characterized using AQUAMI.^{Reference Stuckner, Frei, McCue, Demkowicz and Murayama49} Sequences of cross-section images representing successive stages in the compression simulation were rendered using OVITO. Each image was created from the 3D sample using a 2-nm slice cut parallel to the indentation direction and centered under the indenter tip. A connected components algorithm was used to uniquely label each pore. Individual pores were tracked and consistently labeled in all cross-sections with care taken to identify the coalescence, splitting, and collapse of pores. The area of each pore was determined from the number of pixels representing the pore in each image. Pore height and width were determined by the number of pixels spanning the pore parallel and perpendicular to the compression direction, respectively. The orientation and aspect ratio of individual pores were also determined from the covariance matrix of an ellipse fitted to each pore.

IV. DISCUSSION AND CONCLUSIONS

We have investigated the response of np-Au to spherical indentation using atomistic modeling. Our simulations indicate a ductile, plastic behavior of nanoporous Au under compressive stress. Our results also show clear localized densification in the region under the indenter. We find that the densified region starts to form at indentation depths of about 10 nm for our model, representing an indentation depth of about five times the ligament diameter of the foam. The thickness of this densified region increases with indentation depth, and in our simulations, it is equal to about half the indentation depth. The indentation simulations show no strong long-range effects in the region below the densified layer. This suggests that the effects of the indentation are mostly collapse of the pores immediately below the indenter with no long stress fields. No crack formation was observed nor were any other phenomena that may suggest brittle behavior under compression, evidencing the intrinsic ductile behavior of nanoporous Au under compression. Our observations are in general agreement with experimental results.^{Reference Hakamada and Mabuchi13,Reference Biener, Hodge, Hayes, Volkert, Zepeda-Ruiz, Hamza and Abraham38}

Our work yields hardness–indentation curves exhibiting qualitatively similar features to those observed during uniaxial compression.^{Reference Ruestes, Farkas, Caro and Bringa11,Reference Rodriguez-Nieva, Ruestes, Tang and Bringa32} For both types of simulations, the observed Young’s modulus is of the same magnitude as the value predicted by scaling laws, and the experimentally determined values.^{Reference Biener, Hodge, Hamza, Hsiung and Satcher1} The hardness values that were observed in the present work are also in agreement with previous studies, suggesting that the yield strength of the individual ligaments in nanoporous Au approaches the intrinsic yield strength of Au. The onset of hardening is correlated with the emergence of a densified layer under the indenter with relative density approaching unity. This layer forms by the collapse of pores within the indented volume, first by uniaxial compression and later by triaxial shrinking.

Rodriguez-Nieva proposes that the qualitative features of mechanical response in porous materials may be distinguished based on relative density.^{Reference Rodriguez-Nieva, Ruestes, Tang and Bringa32} They state that “low density” materials, such as np-Au, behave like networks of nanowires. In their view, the deformation of such solids ultimately proceeds by similar mechanisms to those observed in isolated nanowires, including elastic straining to high limit stresses due to dislocation starvation,^{Reference Rodriguez-Nieva, Ruestes, Tang and Bringa32} deformation by shearing along individual slip planes, and lack of hardening. By contrast, they expect materials with relative density above ∼0.9 to behave like a composite of a fully dense matrix and individual voids. Since the spacing between the voids is small by comparison to their radii, the plastically deforming regions surrounding them are expected to overlap and interact, giving rise to hardening via the usual mechanisms of crystal plasticity.^{Reference Kocks and Mecking55}

Our observations are broadly in accordance with the view of Rodriguez-Nieva et al.^{Reference Rodriguez-Nieva, Ruestes, Tang and Bringa32} We see no hardening in the initial indentation response of np-Au with an initial relative density of 0.67. However, once the relative density in a layer of material below the indenter exceeds ∼0.9, hardening is observed. The densified layer contains numerous dislocations, consistent with forest hardening. Moreover, we see that the mechanism of pore collapse changes from uniaxial to triaxial with continued densification. Triaxial collapse is consistent with the behavior of voids embedded in otherwise fully dense crystals under uniaxial external loading.^{Reference Carroll and Holt56} This triaxial shrinkage is due to the emission of dislocations on multiple slip planes intersecting the void.^{Reference Davila, Erhart, Bringa, Meyers, Lubarda, Schneider, Becker and Kumar57–Reference Bulatov, Wolfer and Kumar59} The multiplicity of active slip planes is thought to be one of the factors contributing to hardening during collapse of isolated voids.^{Reference Ruestes, Farkas, Caro and Bringa11,Reference Davila, Erhart, Bringa, Meyers, Lubarda, Schneider, Becker and Kumar57,Reference Tang, Bringa, Remington and Meyers58}

A quantitative analysis of the connectivity of the void space within the nanoporous structure is beyond the scope of the current work. Nevertheless, we do expect that a change in the connectivity of void space accompanies the densification of the material under the indenter. As a first approximation, the threshold density at which this transition occurs may be estimated using percolation theory. The precise density at which percolation occurs depends on the details of the percolation model. In 3D, many models exhibit percolation thresholds at 10–20% density of the percolating entity.^{Reference Torquato60} Thus, we estimate that when the relative density of our nanoporous structure rises above 80–90% (i.e., when the void fraction drops below 10–20%), the interconnected void space breaks up into isolated, nonpercolating pores.

The foregoing discussion suggests that the onset of hardening in np-Au under compression is related to the commencement of forest hardening. To test this conclusion, it would be interesting to investigate the compression response of nanoporous solids composed of amorphous materials. Unlike crystalline solids, amorphous materials do not harden upon plastic deformation. Rather, they undergo shear banding.^{Reference Schuh, Hufnagel and Ramamurty61,Reference Maaß and Löffler62} Therefore, we expect that a nanoporous amorphous metal would also exhibit a propensity toward flow location as well as no hardening, even when its relative density exceeds ∼0.9.