On the impact of capillarity for strength at the nanoscale

The interior of nanoscale crystals experiences stress that compensates for the capillary forces and that can be large, in the order of 1 GPa. Various studies have speculated on whether and how this surface-induced stress affects the stability and plasticity of small crystals. Yet, experiments have so far failed to discriminate between the surface contribution and other, bulk-related size effects. To clarify the issue, here we study the variation of the flow stress of a nanomaterial while distinctly different variations of the two capillary parameters, surface tension, and surface stress, are imposed under control of an applied electric potential. Our theory qualifies the suggested impact of surface stress as not forceful and instead predicts a significant contribution of the surface energy, as measured by the surface tension. The predictions for the combined potential-dependence and size-dependence of the flow stress are quantitatively supported by the experiment. Previous suggestions, favoring the surface stress as the relevant capillary parameter, are not consistent with our experiment.


Supplementary Note 3: Work of deformation
When a dislocation line segment dl is displaced in its glide-plane by the vector δs, then the work done against the local acting stress S net is δW = (S net ·nδA)·b, where b denotes the Burgers vector and nδA = dl × δs with δA the area covered [4]. Therefore, when a glide-plane of finite area is swept by the dislocation, the net work scales with the area-integral of S · n, the traction.

Supplementary Note 4: Cyclic voltammetry of nanoporous gold in different electrolytes
Supplementary Figure 1 shows cyclic voltammograms of nanoporous gold (NPG) in the electrolytes of our study. This data was used for defining the interval of nominally capacitive charging, for use in the in situ compression tests.

Supplementary Note 5: In situ compression tests during capacitive processes
On top of the in situ compression tests shown in the main text, additional tests on NPG samples were performed in smaller potential intervals, excluding the region of oxygen electrosorption. The purpose of these experiments was to verify that i), the sign inversion of the flow-stress potential response is recovered when the potential step sequence is inverted and that ii), consistent results for the response are also obtained in the initial stages of plastic deformation, at small strain. Supplementary Figure 2 illustrates the results of an experiment with mean ligament size L = 40 nm in 1 M HClO 4 and of another experiment with L = 30 nm in 0.5 M H 2 SO 4 . Parts a and b of the figure show the stress-strain curves (red graphs) along with the potential step protocols (blue graphs). In both compression tests the potential scan started with a hold at the potential of zero charge, E zc , for NPG in the respective electrolyte. The potential, E, was then decreased to a more negative value and subsequently stepped upward (anodic) to an upper vertex of 1.0 V. In Supplementary Fig. 2a the scan direction was then inverted, so that the potential step series continued negative-going (cathodic). In Supplementary Fig. 2b the potential cycles were started at lesser strain, at about 4% of engineering strain.
As can be seen from Supplementary Fig. 2c, the trends for the flow-stress potential coupling parameter δσ/δE versus E are indeed reproducible on the anodic and cathodic step series: Consistent with the graph for 1 M HClO 4 in Fig 6a of the main text, the coupling approximates zero when E zc is approached from below, and the behavior is consistent during the backwards scan. The numerical values of the coupling during anodic and cathodic scan differ. This is consistent with the flow stress increase during ongoing compression and with our observation, see the main text, that the coupling scales with the net value of the flow stress. In fact, the two branches coincide in the plot of the normalized response parameter, σ −1 δσ/δE, in Supplementary  Fig. 2d.
Supplementary Figure 2b shows that the findings from the main text apply also when the potential jumps As in the main text, we also compared the experimental coupling parameters to the theory of Equation (12), which is represented by the solid lines in Supplementary Fig. 2c. For the experiment of Supplementary  Fig. 2b the response near the pzc is well compatible with the theory. The data derived from the experiment of Supplementary Fig. 2a shows a smaller-than-predicted δσ/δE during the anodic potential scans, whereas the later, cathodic series gives a stronger response than predicted. In view of the good agreement of the normalized parameters with those in Fig 6a we conclude that the behavior of these two samples is consistent with the reports in the main text, except for a somewhat lesser or greater flow stress.

Supplementary Note 6: Surface area variation during compression
Supplementary Figure 3a displays the surface area, A, that was evaluated in situ during compression in 1 M HClO 4 . The experimental procedure was as follows: We used the capacitance ratio method [5], which determines A from A = C/c DL with C the net capacity and c DL the specific double-layer capacitance, c DL = 40 µF/cm 2 [6]. Electrochemical impedance spectrometry (EIS) in the frequency range 0.1 − 1 Hz and at the DC potential 0.8 V provided C via the procedures from Ref [7]. The impedance spectra were run continuously during the whole deformation experiment and the DC potential of 0.8 V was shortly (60 − 80 s) imposed on the sample during recording of each single EIS data point. The values of A were then normalized by the solid volume (sample mass divided by mass density of gold) of the NPG samples.
The graphs of A versus the compressive strain in Supplementary Fig. 3a show a consistent reduction of area during plastic deformation, consistent with a key assumption in our theory. The graph of relative change in surface area during compression, Supplementary Fig. 3b, affords a verification of the consistency. Superimposed to the data is here a graph represent- ing the molecular dynamics simulation (MD) of plastic deformation of NPG during compression in Supplementary ref. [8] (Fig. 5b there). It is seen that the graphs from the experiment and simulation are in good agreement, specifically in the early stages of compression. Note that the area may be affected by the formation of new contacts between ligaments through cold welding. Yet, this process is expected to prevail only at the later stages of compression. The variation in surface area is therefore significant as an indication of thickening of the ligaments during compression, see our theory.
It is known that exposure to electrolyte and potential can leads to coarsening of the ligaments of NPG [9,10]. Changes in microstructure before and after applying the potential steps of the experiment in Supplementary Fig. 2 have been explored in the following way: An as-prepared sample was cleaved and one of the two halves was exposed to potential cycles as in an in situ compression test, whereas the other saw no contact with electrolyte. Both halves were then investigated side-byside in a scanning electron microscope and the identical image analysis was used to quantify a distribution of sizes. Supplementary Figures 4a and b shows the corresponding scanning electron micrographs (SEMs). No change in ligament diameter is apparent to the naked eye.
For a precise assessment of possible coarsening, we performed a stereologic analysis of the images by the image processing software ImageJ [11]. SEMs of larger areas, containing roughly 1000 ligaments, were binarized using identical threshold values and the resulting images were then analyzed by ImageJ's BoneJ Thickness plugin [12,13,14]. In its implementation for 2D images, the algorithm uses the method of Supplementary ref. [15], computing a local size measure as the diameter of the largest inscribed circle at any point and specifying the mean size as the average of the diameter distribution.
Analysis of the mean ligament size in the images of Fig 4a and b supplied the values 41.7 and 43.1 nm, respectively. In view of the microstructural heterogeneity that is present even in the quite well-defined structure of NPG, the two numbers may be considered as consistent with little, if any, coarsening. The relative change in ligament size is certainly smaller than the relative change in specific surface area of Supplementary Fig. 3. This implies minimal or no effect of the applied potentials on the porous network in our study. The observed decreasing of the surface area may thus be dominantly attributed to the compressive strain.

Supplementary Note 7:
Link between flow stress and yield strength in nanoporous gold A connection between flow stress and yield strength of NPG is very well illustrated by a compressive stressstrain diagram with unload-reload cycles (Supplementary Fig. 5). In the compression test, a dry sample with ligament diameter of about 40 nm was continuously loaded with engineering strain rate of 10 −4 s −1 and subjected to intermediate load/unload segments. During reloading, a strain level was incrementally increased followed by unloading up to a minimum stress value of ∼ 2 MPa. The further details of the experiment can be found in Supplementary ref. [16]. Upon reloading the NPG specimen suffered plastic yielding at the flow stress value that was reached just before the unload. This implies that the flow stress at a certain state of plastic strain represents the yield strength of the material in that strain state. Thereby, it emphasis a strong coupling between strength and plastic flow in NPG.