Plastic Poisson's Ratio of Nanoporous Metals: A ... - ACS Publications

Subscriber access provided by University of Glasgow Library

Communication

Plastic Poisson's Ratio of Nanoporous Metals: A Macroscopic Signature of Tension-Compression Asymmetry at the Nanoscale Lukas Lührs, Birthe Müller, Norbert Huber, and Jörg Weissmüller Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b02950 • Publication Date (Web): 05 Sep 2017 Downloaded from http://pubs.acs.org on September 6, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Nano Letters is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Graphic for Table of Contents 34x13mm (300 x 300 DPI)

ACS Paragon Plus Environment

Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 2 of 11

Plastic Poisson’s Ratio of Nanoporous Metals: A Macroscopic Signature of Tension-Compression Asymmetry at the Nanoscale Lukas L¨ uhrsa , Birthe M¨ ullera , Norbert Huberb,a , J¨org Weissm¨ ullera,b,∗ a

Institute of Materials Physics and Technology, Hamburg University of Technology, Hamburg, Germany of Materials Research, Materials Mechanics, Helmholtz-Zentrum Geesthacht, Geesthacht, Germany

b Institute

Abstract The suggestion, based on atomistic simulation, of a surface-induced tension-compression asymmetry of the strength and flow stress of small metal bodies so far lacks experimental confirmation. Here, we present the missing experimental evidence. We study the transverse plastic flow of nanoporous gold under uniaxial compression. Performing mechanical tests in electrolyte affords control over the surface state. Specifically, the surface tension, γ, can be varied in situ during plastic flow. We find that decreasing γ leads to an increase of the effective macroscopic plastic Poisson ratio, νP . Finite element simulations of a network with surface tension confirm the notion that νP of nanoporous gold provides a signature for a local tension-compression asymmetry of the nanoscale struts that form the network. We show that γ promotes compression while impeding tensile elongation. Since the transverse strain is partly carried by the elongation of ligaments oriented normal to the load axis, the surface-induced tension-compression asymmetry acts to reduce νP . Our experiment confirms a decisive contribution of the surface tension to small-scale plasticity. Keywords: nanoporous metal, mechanical properties, tension-compression asymmetry, small-scale plasticity, Poisson’s ratio, surface tension

1. Introduction Nanoporous metals made by dealloying consist of a uniform network of struts or ligaments with sizes that can be tuned between few nanometers and several micron [1– 5]. Macroscopic samples can be made that are highly deformable in compression and that, thereby, enable model experiments exploring small-scale plasticity and elasticity by reliable macroscopic testing schemes [6–9]. One of the issues under investigation is in how far the surface of small bodies influences their strength. In situ mechanical tests on nanoporous bodies immersed in electrolyte allow for reversible variation of the surface state through the control of the electrode potential while plastic or elastic deformation proceeds. In this way, the role of the surface can be singled out [10–13]. When applied to tensile tests, the strategy highlights a decisive impact of adsorbed monolayers of hydroxide ions on crack propagation and toughness [13]. Furthermore, in situ compression tests show that adsorbed hydroxide will significantly enhance the strength [10] and stiffness [12] or diminish the creep rate [11], an effect that can be repeatedly switched on and off during the deformation. Here, we exploit electrical switching of the surface state for exploring, by means of experiment, an intriguing aspect of small-scale plasticity: Atomistic simulation studies suggest that capillary forces can cause a tension-compression ∗ Corresponding

author. E-mail: [email protected]

asymmetry of the strength of small metal bodies, with strengthening in tension and weakening in compression [14–19]. This suggestion, if it could be confirmed by experiment, would touch upon the fundamental mechanisms governing the mechanical behavior of nanoscale objects. Yet, the experimental verification of the numerical studies remains to be reported. The role of the surface is confirmed in a numerical study that finds the trend for ”smaller is stronger” in micro-pillar compression experiments [20, 21] to break down at very small size, where the action of the surface results in ”smaller is weaker” [22]. Nonuniform deformation in response to the capillary forces [23] is believed to contribute to this effect [22]. Yet, while wrinkling due to surface stress has indeed been resolved by electron microscopy [24], experiments demonstrating a weakening due to capillary forces in nanoscale plasticity remain to be reported. In the above context it is significant that electrochemical cycles afford control over and significant variation of the capillary forces at electrode surfaces and so may affect the proposed surface-induced tension-compression asymmetry. Furthermore, as we shall point out, the effective macroscopic transverse plastic response of network structures such as nanoporous gold (NPG) to uniaxial load can be linked to a local tension-compression asymmetry in the flow stress of the individual ligaments. Our study exploits this context by using digital image correlation to measure the effective macroscopic plastic Poisson’s ratio of NPG

Preprint submitted to Nano Letters

September 4, 2017


Page 3 of 11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

and to detect it’s variation in response to electrically induced changes in the capillary forces. We show that the effects are quite significant, and that they can be naturally reproduced in finite-element simulations of networks with surface tension. Thus, our study provides the missing experimental confirmation of a surface-induced tensioncompression asymmetry of the flow stress in small-scale plasticity. The relevant capillary parameter is the surface tension, that is, the excess energy per surface area.

240 MPa. Furthermore, changing the surface area requires the same work irrespective of whether the deformation is by creep or by plastic flow beyond the yield stress. One may therefore expect a tension-compression asymmetry of the flow stress where the tensile strength is increased by 240 MPa, whereas the compressive strength is decreased by the same amount. This suggests a difference between the strengths in tension versus compression of 480 MPa for a 10 nm gold wire. Comparing tension and compression tests on nanoporous gold might appear as an option for verifying the suggested asymmetry. Yet, whereas the material’s perfect deformability in compression allows the relevant mechanisms of plastic deformation to be explored, tension leads to brittle failure and to a behavior governed by the microstructural heterogeneity in the network rather than the local deformation mechanisms in the ligaments [7, 9]. Thus, the comparison of tension and compression tests on samples of nanoporous gold may not promote our understanding of a possible tensioncompression asymmetry of the local deformation behavior at the ligament level. On the other hand, one may exploit that network solids deforming in response to uniaxial – say, compressive – external load may exhibit distributions of local stress states that do include both, compression as well as tension. Simulations of plastic compression of random networks show that struts aligned with the loading axis tend to be compressed, whereas orthogonally oriented ones tend to be stretched [34]. The macroscopic signature of this anisotropic deformation is a transverse plastic flow that is parameterized by the plastic Poisson’s ratio, νp . Our study explores in how far experiments on the transverse plastic response of nanoporous gold can shed insights into a possible tension-compression asymmetry of the flow behavior of the individual nanoscale ligaments.

2. Surface-induced tension-compression asymmetry? The impact of surfaces on plasticity has been demonstrated in a number of studies exploring the creep of larger, micron-size structures during the second half of the last century. Zero creep measurements on metal wires, pioneered by H. Udin around 1950 [25, 26] and later extended to multilayers [27], measure the surface tension, γ, via the tensile load required to suppress the spontaneous contraction of macroscopic metal wires by creep at elevated temperature. By definition, the surface tension represents an excess in energy, per area of surface, over the bulk energy of a body. Plastic elongation of a wire increases the net area, A, of its surface; this involves an energy change γδA. This energy needs to be supplied by an extra tensile traction ∆T , which acts over and above the traction that is required to overcome the conventional dissipative forces that resist the deformation. Vice versa, shortening the wire reduces the surface energy. This leads to spontaneous contraction by creep unless ∆T is applied for compensation. The dissipative forces vanish when the creep rate is zero, and for circular wires with the radius r one then finds [25, 26] ∆T =

γ . r

(1)

Note that the zero creep experiments demonstrate a tension-compression asymmetry in the mechanical behavior which results from the action of capillarity: the energetics of the surface acts to impede tensile creep but accelerates creep in compression. The phenomenon is confirmed by studies of engineering materials wetted by electrolytes. These reveal similarities between the electrode-potential dependence of γ and creep rate [28–30] or fracture stress [31]. The recent approach using in situ experiments with nanoporous gold suggest opportunities for exploring the analogous behavior for plastic flow and for structures with much smaller characteristic size. Zero creep experiments typically use wires a few tens of µm in diameter and very low stresses, in the order of 10-100 kPa. For nanowires Eq. 1 predicts much larger surface-related stresses: For the example of a Au nanowire with 10 nm diameter, taking 1 γ ∼ 1.4 J/m2 gives ∆T = 1

3. Experiment 3.1. Procedures Cylindrical samples of NPG, 1.92 ± 0.04mm in length and with an aspect ratio of 1.8 ± 0.05, were prepared via electrochemical dealloying cylinders of Au25 Ag75 master alloys in 1 M HClO4 solution. As in Ref. [35], the dealloying potential was 1.25 V vs. the standard hydrogen electrode (SHE). In order to remove residual silver and adsorbed oxygen species [36], the samples were polarized at 1.35 V vs. SHE for 20 min and subjected to 20 potential cycles (0.1 to 1.6 V at 5 mVs−1 ) ending at 0.8 V. Energy dispersive X-ray spectroscopy suggested residual silver contents < 1 at. %. aqueous electrolytes is near 90◦ [32], suggesting that γ varies little between the dry and the wet surface. In this work, we therefore use an average of the collected values of dry gold surfaces from Ref [33] as the estimated surface tension value, γzc = 1.4 J/m2 , at the potential of zero charge.

The wetting angle of clean and charge-neutral gold surfaces in

2


Nano Letters

A fraction of the samples was annealed at 300 ◦ C for 5 min in air. Mean ligament diameters, L, were identified via scanning electron microscopy as 40 and 70 nm for as-prepared and annealed samples, respectively. By measuring the specimens mass and outer dimensions, the mass density was determined as 4.9±0.3 and 5.3±0.4 g/cm−3 for L = 40 and 70 nm, respectively; this corresponds to solid fractions of 0.25±0.02 and 0.28±0.02, respectively. 0.3

0.6

0.9

1.2

1.6 zc

1.2

2

[A/m ] j

2

[J/m ] 1.5 V

mean value

0.4

negative scan

-0.4

(a)

1

0.0

scan rate [mV/s]

10

10 mV/s

-0.1

5 mV/s

Figure 2: Estimated surface tension, γ1.5V , of nanoporous gold at electrode potential 1.5 V versus potential scan rate. Values from positive- and negative-going potential scans of Fig 1 c). Note nearly constant mean value.

1 mV/s

-0.2 4

2

[C/m ]

positve scan

0.8

0.0

1.5

0.1

(b)

q

2

separate load/unload compression tests we obtained the elastic Poisson’s ratio, νE . In situ experiments were performed in 1 M HClO4 electrolyte using a compression test setup in which the NPG samples were wired as the working electrode, carbon tissue as counter electrode and homemade Ag/AgCl as reference electrode. The potential was switched during the continuous loading tests, whereas the load/unload tests were performed at various, constant potential values. Reproducibility of νE and νP trends and values was ensured through the measurement of a set of 20 samples. Cyclic voltammetry provided a basis for estimating the variation of the surface tension, γ, with electrode potential, E. The charge density, q, was obtained by current integration and the variation in γ estimated by integrating Lippmann’s equation [37, 38], Z E ˜ E. ˜ γ(E) = γzc − q(E)d (3)

0 1.5 2

[J/m ]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 4 of 11

1.0

(c)

0.5 0.0 0.3

0.6

0.9

E

SHE

1.2

1.5

[V]

Figure 1: Electrochemical characterization of nanoporous gold in 1M HClO4 , here for a sample with ligament size 40 nm. (a) Cyclic voltammograms of electrode current density j versus electrode potential ESHE (referred to the standard hydrogen electrode, SHE) at scan rates of 1, 5 and 10 mVs−1 . Scan direction as indicated by arrows. (b) Charge density, q, obtained from cyclic voltammogram by integrating j(E) after subtraction of Faraday current. (c) Estimated variation of surface tension, γ, obtained by integrating Lippmanns equation separately for positive going and negative going branches of the CV, as indicated by arrows. Yellow bars highlight the potential values adopted for the in situ mechanical tests. Value of γ at the potential of zero charge was adopted from Ref. [33].

Ezc

Here γzc denotes the surface tension at the potential, Ezc , of zero charge. Voltammetry with a slow scan rate (50 µVs−1 ) was applied to evaluate the contribution of Faraday currents, which were subtracted before current integration. Voltammetry at lesser scan rate was dominated by Faraday signals and did not yield useful cyclic voltammograms. The net surface area, A, in a sample was determined as A = QO /qref where QO represents the experimental oxygen electrosorption charge and qref = 390 ± 10 µC/cm2 the charge density value from Ref. [39]. For the sample of Fig 1, the net charge was 0.379 ± 0.015 C , which suggests the total surface area A = 970 ± 40 cm2 .

Single loading and load/unload compression tests used engineering strain rates of 10−4 s−1 . As in Ref. [35], we used digital image correlation (DaVis 8.2.0, LaVision) with virtual strain gages on the sample surface for the strain in loading, εk , and transverse, ε⊥ , direction. Details are given in the Supporting Online Material (SOM). Poisson’s ratios, ν, were calculated from true strain increments δε as δε⊥ . (2) ν=− δεk

3.2. Electrochemical characterization Figure 1 shows the electrochemical characterization of NPG with L = 40 nm in 1M HClO4 . Part (a) displays cyclic voltammograms (CVs) at scan rates 1, 5 and

When applied during plastic flow in conventional continuous compression tests, this approach yielded the plastic Poisson’s ratio, νP . In regimes of elastic unloading during 3


Page 5 of 11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Figure 3: In-electrolyte compression tests with nanoporous gold. (a) Experimental setup (mounted in a conventional testing rig), showing electrolyte-filled glass cuvette with optical windows for digital-image correlation strain measurement, sample (working electrode, WE), counter- (CE) and reference (RE) electrode. (b) Compression stress-strain graph with load/unload segments, recorded at constant electrode potential, E. Engineering stress σeng versus engineering strain εeng for a sample with ligament size L = 40 nm at E = 0.8 V. (c) Elastic Poisson’s ratio, νE , versus εeng as determined from unload segments in experiments as in (b). Data for different L and at different E, see legend. Based on several successive strain measurements during each unload, error bars display confidence intervals with a confidence level of 95 %. (d-g) Electrically modulated compression tests of nanoporous gold with L=40 nm (left column) and 70 nm (right column). Top row, compressive stress-strain graphs. Bottom row, plastic Poisson’s ratio, νP versus εeng . E was switched between capacitive (0.8 V, light regions in graph) and oxygen adsorption regime (1.5 V vs. SHE, shaded regions).

10 mV/s−1 . The two potential values selected for the in situ mechanical tests are marked; E = 0.8 V is within a region of clean surface whereas 1.5 V corresponds to one monolayer of oxygen species. Part (b) of the figure shows the transfer of electric charge per surface area, q, obtained by current integration using Ezc = 0.40 V [40]. Charge transfer is seen dominantly in the oxygen electrosorption regime.2 Part (c) of Fig. 1 shows the estimated variation in surface tension, as obtained by integrating Eq. 3 separately for the positive- and negative-going branches of the CV. The display uses literature data for γzc (see Section 2). The surface tension has an approximately quadratic variation (”electrocapillary maximum”) around Ezc , which is not resolved at the scale of the figure. The significant variations are seen during oxygen electrosorption. Owing to the considerable hysteresis in the CV, positive- and negative-going scans provide largely different results for γ(E), with the former underestimating the variation at equilibrium whereas the latter overestimates it.

As the function of the scan rate, Fig. 2 shows estimates of γ at E = 1.5 V (which is the value selected in our mechanical tests, see below) as obtained from the positiveand negative-going scans. We take these values – which slowly converge when the scan rate decreases – as upper and lower limits for the true γ1.5V . As a best guess at the true γ1.5V , the mean value remains essentially constant at 0.4 J/m2 , irrespective of the scan rate. This indicates that oxygen adsorption reduces the surface tension to roughly a third of its clean-surface value. 3.3. Elastic response Figure 3 illustrates the in-situ compression test setup and the elastic-plastic behavior of NPG during deformation experiments in electrolyte at constant E. Part (a) shows the sample positioned in an electrolyte-filled glass cuvette; the cuvette is mounted in a conventional testing rig that applies the load via a fused silica pushrod. Part (b) shows a compression stress-strain graph including load/unload segments, at E = 0.8 V and for a sample with ligament size L = 40 nm. The data confirms the high deformability as well as substantial work hardening and progressive elastic stiffening during increasing plastic deformation of mm-size samples of NPG [6, 8, 10]. The elastic Poisson’s ratio, νE , was determined from the axial and transverse strains in the unload segments of strain-time protocols such as the one of Fig. 3 (b). Details

2 Regimes of electrosorption at the positive end of our CVs represent oxidation processes at the gold surface [41] and may involve restructuring of the surface [42]. Prerequisite for the Lippmann equation, Eq. 3, to apply is that the oxidation is surface-specific, in other words, the net amount of deposited oxygen needs to scale with the net area of surface. In our experiments, this was confirmed by the finding (Fig. 1(b)) of consistent charge density at all scan rates. Furthermore, we verified that cyclic voltammograms at all scan rates were reproducible over at least ten successive cycles.

4


Nano Letters

are reported in Ref [35]. The present results are summarized in part (c) of Fig. 3. The data is for two different ligament sizes and for stress-strain experiments at two different (constant) electrode potentials, 0.8 V and 1.5 V. Error bars indicate intervals with a confidence level of 95 %. The data in the figure reveals no significant dependence of νE on E. Furthermore, the initial value νE = 0.20 ± 0.02 agrees with Ref. [35], which found νE = 0.18 ± 0.03 at small strain. We also confirm the slight increase of νE during compression and the lack of significant variation with L.

70 nm

0.16

[ - ]

0.14

P

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 11

0.12

40 nm

0.10

0.08

0.06

0.04 0.8

3.4. Electrochemically modulated plastic flow Figure 3 (d)-(g) shows the results of compression tests in which the electrode potential was alternated between two values, E = 0.8 and 1.5 V vs. SHE. Recall that the lower potential establishes an adsorbate free, capacitively charged surface whereas the higher one corresponds to a single monolayer of adsorbed oxygen species. The graphs of axial stress versus axial strain in Fig. 3 (d) and (f) are for samples with ligament sizes L = 40 and 70 nm, respectively. The sample with smaller L is stronger, consistent with earlier reports [6–9, 43, 44]. Furthermore, the effect of the potential variation agrees with Ref. [10]: The transition from clean to oxygen-covered surface substantially strengthens the material; this effect can be reversed and repeated, and the relative strengthening is more pronounced at larger plastic deformation. Figures 3 (e) and (g) show Poisson’s ratio as obtained from the transverse and axial strains according to Eq. (2). Immediately after potential jumps the variation of the surface stress induces elastic strains [45] which lead to excursions of the derivative of Eq. (2) and which cannot be linked to the transverse plastic flow. However, in intervals of constant potential the data assumes roughly constant or slowly varying values that do represent the plastic transverse response of the material and, thereby, the plastic Poisson’s ratio νP . The striking observation is that these νP values systematically jump to more positive when oxygen adsorbs and to less positive when oxygen desorbs. In analogy to the strengthening during oxygen electrosorption, the modulation of νP is fully reversible. The variation in νP is around 0.05, which corresponds to more than 40 % of the absolute value at L = 40 nm. The νP values are summarized in the bar chart of Fig. 4. Successive bars refer to the plastic Poisson’s ratios during successive potential steps, alternating between 0.8 and 1.5 V. The significance and reproducibility of the above-mentioned jumps are apparent. We now turn to modeling results, demonstrating that this behavior is readily reproduced and can be linked to surface effects.

1.5

E

0.8

SHE

1.5

0.8

[V]

Figure 4: Experimental average values of the plastic Poisson’s ratios, νP , of nanoporous gold with ligament sizes of 40 (gray, striped bar) and 70 nm (blue bar). Data from single loading compression tests potential steps between 0.8 and 1.5 V (light and shaded regions, respectively). Advancing deformation, ε, is indicated by black arrow. Red arrows emphasize trends.

network in the finite element simulation code ABAQUS. With the exception of surface effects (see below), all details of the implementation are described in Ref. [46]. In brief, the network struts are represented by predefined ABAQUS Timoshenko beam elements of type B31 and the connecting nodes occupy randomly displaced sites of a diamond cubic lattice. The parameter A˜ quantifies the displacement and, thereby, the amount of disorder. Representative volume elements (RVEs) contain 4 × 4 × 4 diamond unit cells and satisfy symmetry boundary conditions in three dimensions. The loading was implemented as a homogeneous displacement of all nodes on the top side of the RVE. To capture the boundary conditions of a compression experiment at uniaxial load, all nodes on the lateral faces are free to move. We chose A˜ = 0.23 for achieving an elastic Poisson’s ratio of νE = 0.18 [46]. The ratio of strut radius, r, to length, l, was r/l = 0.303, which allows to reproduce the experimental solid fraction, ϕ = 0.26. This r/l ratio also includes a correction for the strengthening effect, which is caused by the nodal mass, as proposed in [47]. Each strut is represented by a chain of 20 beam elements, each of which permits axial, bending, torsion and shear strains. Their elastic/plastic response is embodied in a constitutive law with linear work hardening, with the parameters Young’s modulus (Y B = 80 GPa for polycrystalline Gold B [48]), yield stress σY , and work hardening rate (or ”tanB B gent modulus”) dσflow /dεplastic = ET , i.e. the slope in the true stress - true plastic strain diagram (see also Fig. 5 (b) below).

4. Modeling

4.2. Implementing surface-induced tension-compression asymmetry

4.1. Procedures Our modeling approach to the transverse elastic response of nanoporous gold uses an implementation of the

As detailed in Section 2, plastic tension or compression change the surface area and extra forces are required 5


Page 7 of 11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

Figure 5: Finite element simulation: plastic deformation of network of beam elements with pipe/core configuration. Left column, mechanics of individual beams. (a) Schematic illustration of beam element. Irrespective of sign and magnitude of bulk stress, σ B , a tensile stress, σ P , acts in the skin. During plastic flow, the work of elastic deformation against this stress emulates the work done against the surface tension, γ, in a ligament. (b) Deformation behavior of beam element in axial tension and compression. Data for γ swichted ”on” (red, solid) and ”off” (blue, dashed). Capillary forces cause an asymmetry by strengthening the beam in tension and weakening it in compression. Center column, anisotropic strain in the network. (c) Randomized network of beam elements underlying our numerical study. The network mimics nanoporous gold. (d) Local strain state in a network at 25 % effective macroscopic (plastic) compression. Mean value of strain along the axis of individual beams vs. angle included between beam axis and global loading direction. Data for γ switched ”on” (red squares) and ”off” (blue points). Bold line: projected strain in an isotropic effective medium with plastic Poisson’s ratio νP = 0.08. Right column, effective plastic response of the network. Effective engineering stress, σeng , (e) and νP (f ), plotted vs. effective engineering strain εeng . Capillary forces are switched ”on” (white regimes) and ”off” (yellow regimes) during compressive loading. Data for structures representing ligament sizes of 40 nm (blue line) and 70 nm (red line).

acting on a beam cross-section. With σ P the stress in the pipe, an axial plastic strain increment δε of a beam entails the net work B δW = πr2 lσflow + 2πrtlσ P δε , (4)

to supply the extra energy. These forces differ from the stresses in the elastic-plastic constitutive law of the bulk ligament in as much as the latter change sign when the flow direction is inverted from tension to compression, whereas the former always represent a trend for compression, irrespective of the flow direction. The extra forces can be implemented into our model by a nano-skin configuration, Fig. 5 (a): The cylindrical, elastic-plastic beam elements are covered with pre-stressed and purely elastic pipe elements of wall thickness t that have a very low Young’s modulus, Y P . The pipe elements deform along with an axial strain of the beam and so contribute an approximately constant force,3 independent of the plastic strain of the beam, to the net integrated stress

where r, land t denote the radius and length of the beam and the thickness of the pipe wall, respectively, with t r. In order to match this to the extra work against the surface tension that leads to Eq. 1, namely δW surface = πrlγδε, thickness and stress of the pipe were chosen so that σP =

γ . 2t

(5)

The parameters of our model where set as follows: The pipe wall thickness to inner (i.e., beam-) radius ratio was t/r = 0.1. The pipe Youngs modulus was Y P = 10−3 Y B . Technically, ABAQUS allowed us to apply the pipe’s prestress through a thermal shrinkage in combination with a negative thermal expansion coefficient assigned to the

3 In principle, the extra forces – which are associated with the surface and which are implemented via the stress in the pipe elements of the model – do vary as the ligaments are elastically strained. This variation can be neglected since it is much smaller than the variation that comes from the potential jumps, see the discussion in the SOM.

6


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

pipe material. Changes in σ P during the deformation (see below) were implemented by switching the temperature (which has no other consequence for the simulation). Setting σ P = 0 allows for studying the elastic-plastic behavior of beams without capillary forces by using the very same model. As in Section 2, we took γzc = 1.4 J/m2 , with Eq. 5 and with t = 0.05L, this implied σ P = 350 MPa and 200 MPa for ligament sizes L = 40 nm and 70 nm, respectively. Figure 5 (b) shows applied stress versus strain for composite (skin/core) beams with ligament size 40 nm, comparing σ P = 0 and σ P = 350 MPa. The two values of σ P represent the conditions without and with surface tenB sion, respectively. For simplicity, yield stress σY and work B hardening rate ET in this example take identical values for B B both surface states, σY = 200 MPa and ET = 4 GPa. The P graph for σ = 0 (no surface tension) illustrates the linear work hardening behavior of the core. When σ P > 0 (with surface tension) it is immediately apparent that the flow stress in compression is reduced whereas that in tension is enhanced. This confirms that the composite beams reproduce the surface-induced tension compression-asymmetry that ensues from Eq. 1.

Page 8 of 11

B model’s values of σY and ET were adjusted for qualitative agreement of the RVE’s ”effective” stress-strain response with the experiments of Fig 3 (d)-(g). For L = 40nm, B good agreement was obtained with the same value, σY = 200 MPa, for both surface states. The work hardening rate B was fitted to ET = 4 GPa for the clean surface while it had B to be increased to ET = 6 GPa for fitting the experiment with oxygen-covered surfaces. In the same way, the material parameters were determined for the ligament size B L = 70 nm to σY = 50 MPa, independent of the applied B surface tension, and ET = 1.4 GPa and 1.8 GPa for the clean and the oxygen-covered surface, respectively. The results are shown in Figs 5 (e) and (f). Part (e) shows the axial compression stress-strain graphs for the two ligament sizes. The agreement with the experimental observation is striking. It includes the switch to higher flow stress during oxygen adsorption, the undercut when switching back from oxygen-covered to clean surface, and the enhanced amplitude of the flow-stress variation at larger strain. The agreement suggests that our simulation approach embodies the central physics of the experiment. We now turn to the plastic Poisson’s ratio as revealed by the simulation, Fig. 5 (f). Here again, the simulation reproduces the findings from the experiment in a semi-quantitative way: When going from clean to oxygencovered surfaces, νP in the model jumps upwards and then slowly decreases. The opposite process, from oxygencovered to clean, results in a downward jump of νP and then a slow upward or downward variation. The agreement with the experiment suggests that the switch in the model parameters, which we implemented in order to make contact to the experiment, at the same time incorporates the central physics behind the observed variation in νP . We emphasize that upward jumps in the plastic Poissons ratio are only seen when the surface tension is switched. By contrast, jumps in the work hardening coefficient at constant γ do not noticeably affect νP . This is illustrated by Fig. 2S in the SOM. The observation provides a strong support for our hypothesis: the changes in plastic Poisson’s ratio which are observed in our experiment are a signature of the change in surface tension. At higher strains, νP of the model is smaller than in the experiment and eventually approaches zero. This is consistent with the maximum in lateral displacement which is predicted by an earlier analysis of the deformation of randomized networks in Ref. [47]. The stress-strain response of the simulation deviates from the experimental stress-strain curve for strains higher than 20 %. This can be traced back to the formation of new contacts between neighbouring ligaments when NPG is densified in the compression experiments [49]. This effect was not implemented in the present model.

4.3. Anisotropy of local axial strain Figure 5 (c) illustrates the randomized network structure of our model. We have subjected L = 40 nm networks with and without surface tension to 25 % strain under uniaxial compressive load and evaluated the resulting average axial strain of the beams, as the function of their misorientation angle φ with the load axis. The results are shown in Fig 5 (d). Also shown is the φ-dependence of the projected plastic strain in an isotropic homogeneous medium with νP = 0.08, which is the value reported for NPG with a similar ligament size [35]. The axial strains in the γ = 0 scenario (blue circles) are seen to agree closely with the homogeneous medium result, with a trend for compression in the beams aligned with the load axis and tension in those oriented orthogonally. This finding confirms similar observations in Ref [34]. The anisotropy of the strain provides the basis for linking the local axial tension-compression asymmetry of individual beams with surface tension to the effective macroscopic flow field of the network. In fact, networks consisting of beams with surface tension have a significantly different φ-dependent strain (red squares). Here, the trend is for more compression at all φ. This emphasizes the trend for γ to enhance compression and lower values of plastic Poisson’s ratio. 4.4. Electrochemically modulated plastic flow Our simulation set up allows mimicking the electrochemically modulated plastic flow experiments. To this end, networks matched to L = 40 or 70 nm where uniaxially compressed and the stress in the skin switched so as to emulate the potential jumps of the experiment. The

5. Discussion In the focus of our study are the elastic and the plastic transverse response to uniaxial load, as parameterised 7


Page 9 of 11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

by the elastic and the plastic Poisson’s ratio’s, νE and νP . We find νE independent of the ligament size (this agrees with Ref [35]) and independent of the electrode potential, whereas νP is systematically smaller for i ) samples with smaller ligament size and ii ) electrode potentials that give higher surface tension. We also find that these observations are readily reproduced in simulations with networks in which the switching-on of surface forces introduces a local tension-compression asymmetry of individual struts. Preliminary to the discussion of the above observations we emphasize four central notions that are naturally confirmed by the mechanical behavior of our model structures:

elastic parameters [12]. The surface excess elasticity has a negligible impact on the effective Youngs modulus of NPG with clean surfaces [50], yet the oxygen-covered surface appears much stiffer than the clean one [12]. Since the stiffening acts symmetrically, irrespective of the deformation direction, we expect no effect of the surface on νE . This notion is fully consistent with our observation that νE is independent of the ligament size and independent of the electrode potential. Contrary to the elastic deformation, experiment and model find the plastic lateral expansion – as quantified by νP – strongly sensitive to the electrode potential. In this context it is significant that our electrochemical characterization shows that stepping the electrode potential to positive leads to a substantial reduction of the surface tension. The experiment therefore confirms our proposition that surface tension leads to a tension-compression asymmetry of the flow stress, which in turn reduces νP . Further confirmation comes from the finding – in our experiment and in earlier studies [19, 35] – that νP is more reduced at the smallest ligament sizes, where surface effects are most pronounced. We have implemented tension-compression asymmetry of the flow stress at the level of individual ligaments in a numerical model of the network solid. The mechanical behavior of the model network reproduces all experimental observations. This finding provides strong support for our conclusions. Contrary to the transverse mechanical response, the axial one (that is, the flow stress) is not completely explained as a consequence of changes in the surface tension: While the reduction in surface tension during electrosorption does qualitatively explain an enhanced flow stress, it will not explain the increase of the flow stress jump during increasing strain. In fact, Jin and Weissm¨ uller [10] have linked the increased flow stress of oxygen-covered nanoporous gold to a pinning effect of adsorbates to dislocation endpoints that move along the surface. This effect differs from the surface tension effects discussed above in i.) being dissipative, not linked to stored energy of surfaces, and in ii.) acting symmetrically, in the same way for tension as for compression. Jin and Weissm¨ uller suggested that the increasing magnitude of the flow stress jumps at higher plastic strain are linked to dislocation accumulation and interaction, which may for instance reduce the effective length of free dislocation segments traveling near the surface. In our simulation this is represented by a larger work hardening for the oxygen-covered state. This feature of the simulation also achieves good agreement with the experiment.

• Firstly, plasticity and surface effects are linked because of the trend for the deformation of elongated nanoscale objects to create (for elongation) or remove (for compression) surface area, A, thereby changing the energy. The surface tension, γ, thus helps the compressive shortening but impedes the tensile elongation. In other words, the work of deformation which can be associated with γ results in a tension-compression asymmetry of the flow stress of nanoscale solids. Figure 5 b) documents this tensioncompression asymmetry and its implementation in our model. • Secondly, the deformation state – axial tension or compression – of individual ligaments or struts in our material depends on their orientation relative to the load axis: macroscopic compression entails local compression for ligaments aligned with the load axis but a trend for more tensile deformation for ligaments orthogonal to the load axis. This has been pointed out in Ref [34] and is confirmed by our numerical analysis, see Fig 5 d). • Thirdly, the surface-induced local tensioncompression asymmetry of individual ligaments acts to diminish the value of νP . This is an obvious consequence of the suppression of the elongation of the orthogonal ligaments, since that elongation feeds directly in to the transverse plastic strain. • Finally, because γ controls the tension compression asymmetry, and because that latter quantity feeds into νP , the transverse plastic response can be switched along with the surface tension. We emphasize that the above notions imply that changes in the mechanical behavior of the ligaments or struts do not affect the transverse response if they act in a symmetric fashion, that is, in the same way for tension as for compression. In the light of the above observations, let us now discuss the invariance of the elastic transverse strain of NPG under potential variation. Changes of up to 8 % in the Young’s modulus by potential variation have been reported for experimental conditions similar to this work, and have been attributed to changes of the surface excess

6. Conclusions Our experiments on the impact of the electrode potential on the transverse mechanical coupling of nanoporous gold in electrolyte have revealed a distinct difference between elastic and plastic deformation. We find the elastic 8


Nano Letters

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Poisson’s ratio independent of the applied electrical potential. We argue that the stiffening which results from the surface oxygen coverage is symmetric, identical for tension and compression, thus leaving the lateral mechanical coupling unchanged. This concept also accounts for the reported ligament-size independence of νE [35]. Contrary to νE , the plastic Poisson’s ratio νP responds strongly to electrical surface modulation. By combining experiment and simulation, we can confirm that this behavior is the signature of a surface-induced tensioncompression asymmetry of the flow stress at the level of individual ligaments. We also link this argument to the ligament-size dependence of νP which was described in the literature [35]. The fundamental microscopic interaction term is the energy balance between the mechanical work done by the external load and the local work in increasing or decreasing the surface area when a ligament or a strut of the network structure is deformed. The relevant capillary term is the surface tension, which measures the excess in energy per area of surface. Our findings provide a direct experimental confirmation of the surface-induced tension compression anisotropy at small scale, which has been suggested based on atomistic simulation. Acknowledgement: This work was funded by Deutsche Forschungsgemeinschaft (DFG) through SFB 986, subprojects B2 and B4. Supporting Information Available: Details of strain measurement by digital image correlation, variation of surface tension during potential jumps, separating contributions of surface tension and work hardening rate to the plastic Poissons ratio in finite element simulations.

Page 10 of 11

[10] Jin, H.J., Weissm¨ uller, J.. A Material with Electrically Tunable Strength and Flow Stress. Science 2011;332:1179–1182. [11] Ye, X.L.L., Jin, H.J.J.. Electrochemical control of creep in nanoporous gold. Appl Phys Lett 2013;103:201912. [12] Mameka, N., Markmann, J., Jin, H.J., Weissm¨ uller, J.. Electrical stiffness modulation - Confirming the impact of surface excess elasticity on the mechanics of nanomaterials. Acta Materialia 2014;76:272–280. [13] Sun, S., Chen, X., Badwe, N., Sieradzki, K.. Potentialdependent dynamic fracture of nanoporous gold. Nat Mater 2015;14:894–898. [14] Diao, J., Gall, K., Dunn, M.. Yield strength asymmetry in metal nanowires. Nano Lett 2004;4:1863–1867. [15] Diao, J., Gall, K., Dunn, M., Zimmerman, J.. Atomistic simulations of the yielding of gold nanowires. Acta Mater 2006;54:643–653. [16] Zhang, W., Wang, T., Chen, X.. Effect of surface stress on the asymmetric yield strength of nanowires. J Appl Phys 2008;103. [17] Weinberger, C., Cai, W.. Plasticity of metal nanowires. J Mater Chem 2012;22:3277–3292. [18] Farkas, D., Caro, A., Bringa, E., Crowson, D.. Mechanical response of nanoporous gold. Acta Mater 2013;61:3249–3256. [19] Ngˆ o, B.N.D., Stukowski, A., Mameka, N., Markmann, J., Albe, K., Weissm¨ uller, J.. Anomalous compliance and early yielding of nanoporous gold. Acta Mater 2015;93:144–155. [20] Greer, J.R., Nix, W.D.. Nanoscale gold pillars strengthened through dislocation starvation. Physical Review B 2006;73(24):245410. [21] Volkert, C.A., Lilleodden, E.T.. Size effects in the deformation of sub-micron au columns. Philosophical Magazine 2006;86(3335):5567–5579. [22] Marian, J., Knap, J.. Breakdown of self-similar hardening behavior in au nanopillar microplasticity. International Journal for Multiscale Computational Engineering 2007;5(3-4):287–294. [23] Danescu, A.. The asaro–tiller–grinfeld instability revisited. International Journal of Solids and Structures 2001;38(26):4671– 4684. [24] Roy, A., Kundu, S., M¨ uller, K., Rosenauer, A., Singh, S., Pant, P., et al. Wrinkling of atomic planes in ultrathin au nanowires. Nano Letters 2014;14(8):4859–4866. [25] Udin, H., Schaler, A., Wulff, J.. The surface tension of solid copper. Trans AIME 1949;185:186–190. [26] Buttner, F.H., Udin, H., Wulff, J.. The surface tension of solid gold. Trans AIME 1951;191:1209–1211. [27] Josell, D., Spaepen, F.. Determination of the interfacial tension by zero creep experiments on multilayers – ii. Experiment. Acta Metall Mater 1993;41:3017–3027. [28] Pf¨ utzenreuter, A., Masing, G.. Zunahme der Geschwindigkeit des Plastischen Fliessens von Metallen im Elektrolyten bei der Electrochemischen Polarisation. Z Metallkunde 1951;51:361– 370. [29] Revie, R.W., Uhlig, H.H.. Effect of applied potential and surface dissolution on the creep behavior of copper. Acta Metallurgica 1974;22(5):619–627. [30] van der Wekken, C.J.. The effect of cathodic polarization on the creep rate of gold. J Electrochem Soc 1982;129:706–711. [31] Latanision, R., Opperhauser Jr., H., Westwood, A.. The influence of surface charge density on the fracture of zinc single crystal electrodes. Scripta Metall 1978;12:475–479. [32] Yokomaku, H., Satoh, W., Fukuda, J., Suzuki, H.. Electrowetting on gold electrodes with microscopic threedimensional structures for microfluidic devices. Journal of Applied Physics 2008;104:064910. [33] Umeno, Y., Els¨ asser, C., Meyer, B., Gumbsch, P., Nothacker, M., Weissm¨ uller, J., et al. Ab initio study of surface stress response to charging. Europhys Lett 2007;78:13001. [34] Van Der Burg, M.W.D., Shulmeister, V., Van Der Geissen, E., Marissen, R.. On the Linear Elastic Properties of Regular and Random Open-Cell Foam Models. J Cell Plast 1997;33:31–54. [35] L¨ uhrs, L., Soyarslan, C., Markmann, J., Bargmann, S., Weissm¨ uller, J.. Elastic and plastic Poissons ratios of

References [1] Li, R., Sieradzki, K.. Ductile-brittle transition in random porous Au. Phys Rev Lett 1992;68:1168–1171. [2] Erlebacher, J., Aziz, M.J., Karma, A., Dimitrov, N., Sieradzki, K.. Evolution of nanoporosity in dealloying. Nature 2001;410:450–453. [3] Senior, N.A., Newman, R.C.. Synthesis of tough nanoporous metals by controlled electrolytic dealloying. Nanotechnology 2006;17:2311–2316. [4] Jin, H.J., Wang, X.L., Parida, S., Wang, K., Seo, M., Weissm¨ uller, J.. Nanoporous AuPt Alloys As Large Strain Electrochemical Actuators. Nano Lett 2010;10:187–194. [5] Detsi, E., Van De Schootbrugge, M., Punzhin, S., Onck, P.R., De Hosson, J.T.M.. On tuning the morphology of nanoporous gold. Scr Mater 2011;64:319–322. [6] Jin, H.J., Kurmanaeva, L., Schmauch, J., R¨ osner, H., Ivanisenko, Y., Weissm¨ uller, J.. Deforming nanoporous metal: Role of lattice coherency. Acta Mater 2009;57:2665–2672. [7] Briot, N.J., Kennerknecht, T., Eberl, C., Balk, T.J.. Mechanical properties of bulk single crystalline nanoporous gold investigated by millimetre-scale tension and compression testing. Philosophical Magazine 2014;94:847–866. [8] Mameka, N., Wang, K., Markmann, J., Lilleodden, E.T., Weissm¨ uller, J.. Nanoporous goldtesting macro-scale samples to probe small-scale mechanical behavior. Materials Research Letters 2016;4:27–36. [9] Badwe, N., Chen, X., Sieradzki, K.. Mechanical properties of nanoporous gold in tension. Acta Materialia 2017;129:251–258.

9


Page 11 of 11

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Nano Letters

nanoporous gold. Scripta Materialia 2016;110:65–69. [36] Jin, H.J., Parida, S., Kramer, D., Weissm¨ uller, J.. Signinverted surface stress-charge response in nanoporous gold. Surf Sci 2008;602:3588–3594. [37] Grahame, D.C.. The Electrical Double Layer and the Theory of Electrocapillarity. Chem Rev 1947;41:441–501. [38] Kramer, D.. Dependence of surface stress, surface energy and surface tension on potential and charge. Phys Chem Chem Phys 2008;10:168–177. [39] Trasatti, S., Petrii, O.. Real surface area measurements in electrochemistry. J Electroanal Chem 1992;327:353–376. [40] Kolb, D., Schneider, J.. Surface reconstruction in electrochemistry: Au(100-(5 20), Au(111)-(1 23) and Au(110)-(1 2). Electrochim Acta 1986;31:929–936. [41] Angerstein-Kozlowska, H., Conway, B., Hamelin, A., Stoicoviciu, L.. Elementary steps of electrochemical oxidation of single-crystal planes of AuI. Chemical basis of processes involving geometry of anions and the electrode surfaces. Electrochim Acta 1986;31:1051–1061. [42] Vitus, C.M., Davenport, A.J.. In Situ Scanning Tunneling Microscopy Studies of the Formation and Reduction of a Gold Oxide Monolayer on Au(111). J Electrochem Soc 1994;141:1291. [43] Biener, J., Hodge, A.M., Hamza, A.V., Hsiung, L.M., Satcher, J.H.. Nanoporous au: A high yield strength material. Journal of Applied Physics 2005;97. [44] Volkert, C.A., Lilleodden, E.T., Kramer, D., Weissm¨ uller, J.. Approaching the theoretical strength in nanoporous au. Applied Physics Letters 2006;89:061920. [45] Weissm¨ uller, J.. Charge-Induced Reversible Strain in a Metal. Science 2003;300:312–315. [46] Roschning, B., Huber, N.. Scaling laws of nanoporous gold under uniaxial compression: Effects of structural disorder on the solid fraction, elastic Poisson’s ratio, Young’s modulus and yield strength. J Mech Phys Solids 2016;. [47] Huber, N., Viswanath, R.N., Mameka, N., Markmann, J., Weißm¨ uller, J.. Scaling laws of nanoporous metals under uniaxial compression. Acta Mater 2014;67:252–265. [48] Davis, J.R.. Metals handbook. ASM international; 1998. [49] Ngˆ o, B.N.D., Stukowski, A., Mameka, N., Markmann, J., Albe, K., Weissm¨ uller, J.. Anomalous compliance and early yielding of nanoporous gold. Acta Mater 2015;93:144–155. [50] Elsner, B., M¨ uller, S., Bargmann, S., Weissm¨ uller, J.. Surface excess elasticity of gold: Ab initio coefficients and impact on the effective elastic response of nanowires. Acta Mater 2017;124:468–477.

10


Plastic Poisson's Ratio of Nanoporous Metals: A ... - ACS Publications

Recommend Documents