Elastic properties of hard cobalt boride composite nanoparticles

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Acta Materialia 58 (2010) 6474–6486 www.elsevier.com/locate/actamat

Elastic properties of hard cobalt boride composite nanoparticles A. Rinaldi a,b,⇑, M.A. Correa-Duarte c,⇑⇑, V. Salgueirino-Maceira c,⇑⇑⇑, S. Licoccia a, E. Traversa a,d, A.B. Da´vila-Iba´n˜ez c, P. Peralta b, K. Sieradzki b a

NAST Center & Department of Chemical Science and Technology, Universita’ di Roma “Tor Vergata”, Via della Ricerca Scientifica, Roma 00133, Italy b Mechanical and Aerospace Engineering Department, Fulton School of Engineering, Arizona State University, Tempe, AZ 85287, USA c Departamentos de Fı´sica Aplicada e Quı´mica-Fı´sica, Universidade de Vigo, 36310 Vigo, Spain d International Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, 305-0044 Ibaraki, Japan Received 24 August 2009; received in revised form 7 March 2010; accepted 11 August 2010 Available online 6 September 2010

Abstract This paper reports on the determination of elastic and hardness properties of Co–B composite nanoparticles (CNP). Co boride materials are usually known for their functional properties (hydrogen catalysis, magnetism, corrosion, biomedics), but nanoscale dimensions also bring significant mechanical properties. In situ compression tests of 70–150 nm core–shell silica-coated Co2B CNP were performed for the first time with a nanoindenter in the load range 30–300 lN. The CNP modulus is comparable with the bulk material (ECNP = 159–166 GPa), but the hardness is as much as five times higher (4.5 ± 1.0 GPa). Both modulus and hardness (to a lesser extent) are found to increase with the applied pressure. The paper first addresses the limitations of ordinary contact analysis intended for single-phase NP, and then presents a hybrid Oliver–Pharr strategy suitable for CNP, where numerical modeling overcomes issues related to anisotropy and heterogeneity of the composite nanostructure that hinder the direct application of basic contact models. An alternative regression-based approach for estimating modulus and hardness is also considered for comparison. The importance of the model selection for the contact area A for accurate modulus and hardness results is emphasized. Besides typical Hertzian, geometrical and cylindrical area models, a new one is formulated from a “rigid-sphere” approximation, which turned out to perform best and consistently in this study, on a par with the cylindrical model. Finally, evidence of the magnetic nature of CNP and, unexpectedly, reverse plasticity is provided. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Core-shell nanoparticles; Transition metal; Intermetallics; Nanomechanics; Mechanical properties

1. Introduction Transition metal borides, as a class, are usually recognized as very hard materials with a high bulk modulus [1]. As new methods have become available to produce ⇑ Corresponding author at: NAST Center & Department of Chemical Science and Technology, Universita’ di Roma “Tor Vergata”, Via della Ricerca Scientifica, Roma 00133, Italy. ⇑⇑ Corresponding author. ⇑⇑⇑ Corresponding author. E-mail addresses: [email protected] (A. Rinaldi), maco [email protected] (M.A. Correa-Duarte), [email protected] (V. SalgueirinoMaceira).

these materials inexpensively at ambient pressure and low temperatures, as opposed to traditional high pressure and high temperature synthesis approaches, their mechanical properties make them attractive for deployment in innovative applications [2]. This paper examines the elasticity and hardness properties of cobalt–boride-based composite nanoparticles (CNP). In the past, Co–B materials drew considerable attention, mainly because of a number of functional properties relevant to a number of critical applications. Both in bulk form and (especially) as powder or nanoparticulate, Co borides have been actively researched as catalysts for hydrogen storage and fuel cell applications [3–10] because of the fine

1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.08.009

A. Rinaldi et al. / Acta Materialia 58 (2010) 6474–6486

electrochemical properties (e.g., electrochemical reversibility, high charge–discharge capacity) inherited from Co [11] and leveraged by the antioxidant effect of boron. The oxidation resistance makes Co–B also an interesting option for corrosion- and wear-resistant surface coatings [12,13]. Other applications seek to exploit these materials for their marked magnetic and (anisotropic) magnetostrictive properties [14,15] or for biomedics and drug delivery. From a mechanical viewpoint, the presence of boron is associated with the introduction of covalent bonds [1,2] and with a progressive amorphization of the solid structure of pure crystalline cobalt [2,16–19], which are both potentially strengthening factors. However, in comparison with other transition metal borides with far superior performance, Co–B alloy do not stand out for their mechanical properties, as demonstrated by the relatively limited and fragmented related literature, dealing exclusively with the mechanical properties of bulk Co–B samples (e.g., Refs. [20,21]). This paper presents a nanomechanics study suggesting that nanoscale Co–B samples may be substantially stronger, while retaining the aforementioned array of functional properties, making this type of material even more attractive for nanotechnology [7,10]. It reports on the improved mechanical properties of a novel nanosized Co–B-based material measured from in situ micro-compression tests performed with a nanoindenter. This provides a viable methodology for determining the elastic constants of other transition metal borides at the nanoscale. Broadly, nanomechanics is that part of nanoscience dealing with the study of mechanical properties on the nanoscale. Knowledge of the mechanical properties of volume-confined materials (e.g., thin films, nanoparticles (NP) and nanostructures) is an essential prerequisite for nanotechnology design of, say, micro- and nano-electromechanical systems (MEMS and NEMS) and for deploying nanotechnology safely and reliably. Much research effort is being directed not just towards the elastic behavior, but also towards the inelastic behavior (e.g., plasticity, damage, fracture) and the failure modes. Other aspects, such as hardness, friction, wear and adhesion, also fall within the scope of nanomechanics. Within this context, nanoindentation is currently the principal technique for investigating mechanical properties at the nanoscale. In spite of having long been used to study the surfaces properties of bulk samples as well as of thin films (e.g., Refs. [22–24]), only in the last decade has it become a unique tool permitting the in situ mechanical probing necessary to assess the reliability and durability of structural nanosized materials such as NP (e.g., Refs. [25–30]), nanowires (e.g., Refs. [31–33]) and nanopillars (e.g., Refs. [34–39]). Confining the present scope to NP, earlier compression tests on Si nanospheres were carried out first by Gerberich and co-workers [25], using a flat punch (derived by a conical diamond tip micro-machined by focused ion beam). The estimated hardness (i.e., 50 GPa) in dislocation-free

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NP with diameter 40–100 nm was found to range up to five times more than in bulk Si (i.e., 12 GPa), pointing out a marked nanoscale effect on the mechanical properties. In addition, the observations were qualitatively confirmed by other authors in subsequent work on other materials, e.g., silica and Ag [29,30]. While the experimental methodology has since improved and gained popularity over time, several difficulties and concerns still remain about both the fundamental deformation mechanisms and the data analysis techniques. The data analysis necessary to extract mechanical properties from indentation and/or compression tests of particles is, in general, neither a trivial nor a straightforward task, and usually requires a sophisticated inverse analysis based on concepts from contact mechanics. Young’s modulus and hardness are among the most important material properties for MEMS and NEMS design, e.g., Ref. [40]. Several simplified models are commonly used, none of them being the absolute best. Each model yields different predictions and may be better suited than others in a certain range of mechanical response and depending on the situation. This insufficiency arises partially from the presence of several “confounding” factors at the nanoscale, such as uncertainties about contact geometry (i.e., non planarity of plates, finite curvatures, uncertain shapes, roughness, friction, centering, etc.), sizedependent deformation, adhesion, plastic deformation and high NP hardness compared with the (elastic rather than rigid) substrate, which all pose significant challenges to modeling, and elude simplistic assumptions. Hence, for one given experimental data set, it is customary to evaluate and compare several estimations from different models (e.g., Hertzian or geometric contact [26]). It is not coincidental that only single-phase NP (e.g., Ag, Si, SiO2) appear to have been considered in the literature so far. In fact, despite the enormous variety of multiphase NP that can be synthesized nowadays (e.g., core–shell, onionlike, dimmers and aggregates described in Refs. [41,42]), such nanocomposite materials add significantly to the degree of complexity of mechanical behavior in terms of deformation and failure mechanisms associated with compression. Therefore, the same toolbox used for elementary NP cannot be applied to CNP directly. Instead, if possible, new methods need to be devised on a case by case basis. This scenario hindered the mechanical characterization of CNP. This paper undertakes a nanoindentation study on the mechanical properties of core–shell CNP and present a new strategy to obtain the elastic modulus and hardness of a single CNP from a modified Oliver–Pharr method. Several modeling options and challenges are discussed, duly linked to the corresponding situations reported for (crystalline and amorphous) single-phase NP. This proofof-concept study used 70–150-nm-diameter core–shell CNP made of a Co2B cobalt boride core coated with a thin 10-nm outer layer of SiO2, as described in Refs. [43,44] (the technological relevance of which was instead highlighted in Refs. [6,7,10]).

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2. Materials and methods 2.1. Synthesis and characterization of CNP The synthesis of silica-coated cobalt-based NP was performed as follows. A solution of cobalt chloride hexahydrate (0.4 M) (Fluka) was injected (0.1 mL), while being vigorously and mechanically stirred to an aqueous solution (100 mL) of NaBH4 (4.4 mM) (Riedel de Haen) and citric acid monohydrate (Riedel de Haen) (5  105 M). Immediately after the cobalt reduction, 400 mL of an ethanolic solution containing 20 lL of TEOS (tetraethoxysilane) (Aldrich) was added. The solution was centrifuged 15 min later, and the precipitate was redispersed in ethanol (20 mL). Transmission electron microscopy (TEM) measurements were performed on a Philips CM20 microscope operating at 100 kV and on a JEOL 1010 operating at 200 kV. Samples for TEM were prepared by depositing them on a carbon-coated copper grid. A group of CNP are shown in Fig. 1A, along with a 150 nm CNP and the diameter distribution (Fig. 1B and C). A large number of radial defects are visible in the Co2B core (90 wt.% Co and 10 wt.% B), the nature and role of which will be discussed in a forthcoming paper dedicated to the plastic behavior of CNP. Worth mentioning, though, the presence of such defects is an element of novelty compared with the defect-free NP previously subjected to mechanical testing. 2.2. Nanoindentation 2.2.1. Equipment and tests The nanoindenter suitable for this type of study typically works as a scanning probe microscope, similarly to an atomic force microscope (AFM) in contact mode, where the probe is replaced by a sharp and hard diamond tip which can be “actuated” to apply measurable forces to the material. A standard Triboindenter transducer (HysitronÒ) driven by a Nanoscope IIIa SPM controller (VeecoÒ) was used to perform tests at constant stress rate.

The transducer basically consists of a three-plate capacitor, with the two external plates being stationary and the mid one being attached to the diamond probe. Even though the maximum applicable load is 12 mN, this sensor is suitable for a much narrower load range, owing to a nominal force sensitivity of 1 nN and a background noise of 100 nN. The selected load range was between 0 lN and 300 lN, which allowed a single CNP to be compressed against a flat Si substrate without crashing it. The Triboindenter was outfitted with a relatively blunt cube corner diamond tip with a finite curvature radius of 200 nm. While early compression experiments [25] adopted flat tips to ensure planarity of the platens during compression tests, many researchers shifted to such finite tips radius owing to several advantages, e.g., Refs. [30,45]. First, a finite tip radius allows for scanning the zone underneath, which is crucial for fine positioning of the diamond tip on a CNP and for in situ testing. High-resolution scans, acquired using the indentation tool itself, provide images of the CNP before and after the indentation, where the imprints of the tip are clearly visible. For that, an area of 2  2 lm was scanned by the indenter tip in order to identify the nanostructures of interest, so that progressively smaller scans were then taken to “zoom into” a chosen CNP. A nanoindentation experiment was then performed on top of a NP and once indented, the CNP was immediately imaged by the same indenter tip. Fig. 2a and b shows these AFM images provided by a 60° conical diamond tip with 1 lm nominal tip radius of curvature before and after a nanoindentation using 30 lN force. Yang and Vehoff [46] pointed out that the indent position greatly influences the nanoscale hardness, so that the nanoindents carried out were always performed in the center of individual CNP. Apparently, the indentation created a plastic deformation on top of the NP, showing the preciseness of the positioning of the indentation. It also shows that the tip, despite having a large nominal radius of curvature, was able to indent the NP in this case and cause an indentation mark. Secondly, smaller tip radii enable more closely spaced CNP to be resolved and tested. To this end, an increase

Fig. 1. TEM analysis of CNP samples and size (outer diameter) sample distribution: (A) 150 nm CNP with 10 nm silica coating on a Co2; (B) core of 140 nm.

A. Rinaldi et al. / Acta Materialia 58 (2010) 6474–6486

Fig. 2. AFM scanning map of an indented CNP before and after testing, showing the sharp mark left on its top by the precise positioning of the tip (scale bar 200 nm).

in CNP density on a scanning area drastically reduces experimental operational and set-up time. The AFM map shown in Fig. 3 refers to a CNP sample with inter-particle spacing of the order of 100 nm, i.e., significantly smaller than that reported so far for similar tests. As evident from a comparison between Figs. 1 and 3, the AFM map needs to be de-convoluted (based on the tip radius) to render the correct in-plane geometry of the CNP [26,30]. However, when finite size tip radii are being used, compression data may be influenced by local deformation at large loads if local indentation becomes a sensible quota of the total strain. Typically, the tip curvature determines the fracture onset and the failure mode of the CNP. Therefore, limiting the maximum applied load and performing an AFM check of the CNP after each test are critical to assess such effects. Also for these reasons, the standard contact models developed for flat tips may be usable for tips with a large finite radius, but their accuracy progressively decreases as the tip radius reduces or the strain increases. Starting from a 15 lm scan window, a few target CNP were selected and then centered under the tip one by one by progressively reducing the scan size to 2 lm. Repeated scanning of the same CNP at the smallest speed (0.1 Hz) ensured no damage on the target during scanning, but adequate adhesion between the CNP and the Si substrate.

Fig. 3. AFM scanning map of a 2  2 lm2 with a relatively dense spread of CNP as tested here.

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After centering, the same tip was used to compress the CNP by ramping the loading up to a selected force, while collecting both the load P and the tip displacement “d” throughout the test. Fig. 4 shows a sketch of the contact geometry for a 100 nm CNP. The P–d data reflect the total deformation resulting from the contributions of all the contact elements, i.e., the CNP, the tip and the substrate (the machine compliance is filtered out from the P–d data due to standard pre-calibration [45,48]. Nonetheless, material properties of the CNP can be “extracted” via suitable contact models following different strategies, two of which are outlined next. 2.2.2. Approach 1: extended Oliver–Pharr method This well-known procedure is used to analyze compression tests of both simpler NP and indentation experiments on flat surfaces. It provides the (equivalent) elastic modulus and the hardness of the material from the analysis of the unloading slope of the P–d curve. The method is based on contact mechanics concepts (e.g., Hertz’s elastic theory [48] and Sneddon’s solution [49] for axisymmetric contact problems), which have been revisited and recast by Oliver and Pharr [50] for micro- and nanomechanics. For a single contact pair consisting of two deformable elastic solids, shaped as an axisymmetric solid (i.e., the CNP) and a (quasi-)flat half-space (e.g., the tip), the reduced Young’s modulus E can be estimated according to the elastic theory as rffiffiffiffiffiffiffiffiffiffi 1 p dP ð1Þ E ¼ 2 AðdÞ dd where A(d) is the contact area at maximum penetration and dP/dd is the derivative of the experimental curve taken at

Fig. 4. Sketch of contact geometry for a 100 nm CNP; only the section of half sphere is drawn.

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the load peak along the unloading path. Recall that E is the key parameter governing the elastic contact between two isotropic homogeneous elastic solids and is customarily defined for the contact pair under consideration as 1 1  v2TIP 1  v2CNP þ  ¼ E ETIP ECNP

ð2Þ

where {ETIP, mTIP} and {ECNP, mCNP} are the elastic properties of each contact element. Evidently, the unknown modulus ECNP can be determined from Eqs. (1) and (2) if the rest of the parameters are known. Note that the latter expression is equivalent to the typical Sneddon elastic correction done in pillars experiments to estimate the actual pillar response by removing the effect of the elastic foundation [34,39,49]. In fact, from  the CNP modulus is computed 1 Eq. (2) as ECNP ¼ 1=E  ð1  v2TIP Þ=ETIP  ð1  v2CNP Þ only after subtracting the contribution of the diamond deformation ð1  v2TIP Þ=ETIP from the gross 1=E . However, in the present case, a few approximations are enforced to adapt this method for CNP. First, discarding the finite curvature of the tip, the CNP is assumed to be squeezed in a symmetrical fashion between two flat platens, determining two contact pairs. Secondly, endorsing the rationale of Gerberich’s group [25] and Zou and Yang [30], a material symmetry for the platens is also assumed, and only the properties of the hardest materials are used in Eq. (2), yielding upper estimates for ECNP. Typically, ETIP = 1100 GPa and mTIP = 0.07 are taken for diamond, while ESi = 170 GPa and mSi = 0.218 are assumed for Si. The consequence of the “material and geometric symmetry” assumption (i.e., a double symmetry assumption) implies that half of the raw deformation d is assigned to each pair, and Eq. (1) needs to be applied not to the raw P–d experimental data, but rather onto P–d, with d=d/2. In other words, A(d) and dP/dd are measured on the P–d curve for one contact pair. The derivative dP/dd is taken as equal to dp ðm1Þ ¼ m  k  dMAX dd

ð3Þ

where {m, k} are statistical parameters from fitting a power law P ¼ k  dm to the unloading data near PMAX. One additional issue to overcome for CNP is the determination of vCNP in Eq. (2). Unlike single-phase homogeneous isotropic NP, the Poisson ratio of a composite nanosphere is not defined. However, an effective value can be assigned to the CNP by studying its elastic response. The finite element method (FEM) was used to establish a correspondence between the elastic deformation of a CNP and an “equivalent” homogeneous elastic isotropic, yielding the approximate value vCNP = 0.23 adopted in the calculation. This result was achieved using ESiO2 = 94 GPa and mSiO2 = 0.2 for silica, and the approximate bulk Co values ECo = 211 GPa and mCo = 0.32 for Co2B [11,51,52]. This was necessary because, as mentioned above, no systematic characterization of bulk Co2B boride seems to have been done, but seems reasonable based on the findings reported by Barandiar and co-workers [14] and on 90 wt.% content

of Co in the core. It is noteworthy, in view of the nanoscale dimension and of the defective nature of the amorphous Co2B core of the CNP, that the value of 211 GPa from crystalline Co should be expected to represent an upper bound, because smaller modulus values were reports for Co nanowires (191.4 GPa) [33] and for non-fully dense ultrafinegrained fcc/hcp cobalt (170 GPa) [53]. Besides the modulus ECNP, the Oliver–Pharr method also provides a measure of the CNP hardness HCNP (i.e., the mean withstood contact pressure) from the P–d defined as H CNP ¼

P MAX AðdÞ

ð4Þ

The accuracy of the estimated ECNP and HCNP strongly depends on the value of contact area A(d) plugged into Eqs. (1) and (4). This choice is complicated as much as crucial. Using standard nominal first-order area models (e.g., A(d) = 2.598d2 for a cube corner tip [47]) does not yield good results in the present test. To this end, some authors have pursued an accurate area calibration [30]. Instead, in accordance with Gerberich’s approach [25], A(d) = p  a2 is assumed here, where the contact radius a can be estimated according to selected contact hypotheses. Typical estimates [25,26,30] take into account the Hertzian, the geometric and the cylindrical contact hypotheses, which amount respectively to aH ðd Þ ¼ ðd  RCNP Þ 



1=2 

 1=2

ageo ðd Þ ¼ ð2d  RCNP  d  d Þ  1=2 4  R3CNP acyl ðd Þ ¼ 6ðRCNP  d Þ

ðlabel HERTZÞ

ð5aÞ

ðlabel GEOÞ

ð5bÞ

ðlabel CYLÞ

ð5cÞ

where RCNP is the CNP radius. While the first is suitable for small strains, the second and third perform progressively better for higher contact pressures, i.e., at the large d where local plastic deformations occur in the contact region. The cylindrical approximation presumes a CNP deformed in barrel-shaped fashion beyond certain loading.1 Besides these standard options (which all refer to a quasiflat tip), a fourth possibility is examined related to a flat deformable substrate conforming to a rigid spherical (RS) tip (Fig. 5) and approximately equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 aRS ðd Þ ¼ R2TIP  ðRTIP  d Þ ¼ ð2d  RTIP  d  d Þ1=2

ðlabel RSÞ

ð5dÞ

It is worth noting that Eq. (5d) is a duplicate of Eq. (5b), since the derivation is analogous. While the latter assumes the CNP to be curved and the platen to be flat and rigid, it is vice versa in the former. This does not seem absurd when considering that: (i) tip curvature might become influential

1

The estimate is derived from the condition 4p=3R3CNP ¼ 2pa2cyl  ðRCNP  d Þ, which imposes the volume equivalence between the initial CNP volume and a cylinder of height 2 ðRCNP  d Þ.

A. Rinaldi et al. / Acta Materialia 58 (2010) 6474–6486

Fig. 5. Hypothesis of contact between a conforming flat surface and a rigid-sphere (RS) as in Eq. (5d).

at very high pressure, and (ii) the hardness and Young’s modulus of the diamond tip are much higher than the CNP. Finally, it is recalled that this analysis is designed to hold even if the particle is plastically deformed, since the calculation are done on the unloading portion of the experimental curve. Consequently, trying to apply the method on the loading portion does not usually yield good results, owing to the bias of inelastic deformation and steady contact [47,50]. 2.2.3. Approach 2: bounding method for the elastic modulus Mook and co-workers [28] argued that the reduced modulus in Eqs. (1) and (2) is not constant, but rather increases with pressure. They found that a linear model, such as E ðpÞ ¼ E0 þ k  p

ð6Þ

fit their data for Si and Ti particles reasonably well. Parameters E0 and k are material parameters, with the former being the reduced modulus in the limit of zero-pressure. As a consequence, Young’s modulus of the NP would display a similar dependence. Furthermore, in their formulation, they speculate that Young’s modulus of the NP may be included between a lower bound (LB) and an upper bound (UB) limit, which are written as follows in the present notation: 2P ðRCNP  dMAX Þ pa2cyl dE 16P ¼ 2 3p ageo dE

ELB ¼

ð7Þ

EUB

ð8Þ

Both equations depend on dE ¼ ðdMAX  dR Þ, i.e., the elastic quota of maximum displacement dMAX , with dR the residual depression. Both equations express a ratio between some sort of mean pressure and the elastic strain. Referring to the cited work for the details, It is just mentioned here that ELB is obtained from an average contact pressure taken over an area of radius acyl , whereas EUB assumes a geometric contact and adopts ageo . In the present analysis, the concept of a pressure-dependent ECNP ðpÞ will be revisited and the suggested bounds from Eqs. (7) and (8) will be examined. 2.3. FEM simulations: algorithm for estimating vCNP The FEM is primarily (and necessarily) used here to get the value vCNP for Eq. (2). To this purpose, a 100-nm CNP (i.e., a Co2B core of 80 nm coated with a 10 nm outer layer

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of SiO2) was modeled in compression between two rigid platens up to 30% deformation (reflecting the actual strain range reached in the experiments) using ABAQUSÓ commercial software. Only the elastic properties were assigned to the CNP components. The geometric model was meshed with axisymmetric four-node quadrilateral elements. Some sample simulation results are reported in Fig. 6, showing the output fields for all the non-zero stress components {Sxx, Syy, Szz, Sxy} of the stress tensor S at the maximum load. Remarkably, in spite of the purely elastic setting, there is an evident tendency to barreling, which would probably accentuate if plasticity were also accounted for. The extreme values of the stress components were estimated to be of the order of tens of gigapascals. To determine vCNP , an equivalent homogeneous isotropic sphere of unknown properties fECNP ; vCNP g was sought that would match the behavior of the CNP within 5% accuracy in terms of three control parameters: (i) the total force response FY; (ii) the transversal displacement uX on the equatorial plane [0, 0, 1]; and (iii) the contact area AC. The simulation outputs of the actual CNP at the intermediate 20% strain were taken as reference values REF REF fF REF Y ; uX ; AC g A search algorithm was devised to perform the task systematically based on the following facts:  ECNP is linearly proportional to FY (i.e., DECNP / DEY ), but does not affect the transversal strain at fixed vCNP .  vCNP controls the transversal kinematics with minor influence on the force response FY. ðiÞ

ðiÞ

As a first step, after starting trial values fETRIAL ; vTRIAL g had been assigned to theequivalent sphere, the modulus ðiþ1Þ

ðiÞ

ðiÞ

was changed to ETRIAL ¼ F REF ETRIAL based on outY =F Y ðiÞ

put F Y . As a second step, the vTRIAL changed in a “trialand-error” fashion to minimize the %|DAC|. A simulation ðiÞ ðiÞ is run for any new pair fETRIAL ; vTRIAL g . Such a two-step process is iterated to convergence. The search took six iterations (Table 1) starting from metallic Co properties and and vCNP = 0.23 (Ref. yielded ECNP = 169 GPa ð6Þ ð6Þ fETRIAL ; vTRIAL g). This value of modulus agrees well with the above-mentioned values for nanoscale and not-fully dense Co samples [33,53] and is used as a reference value in the data analysis presented in the discussion. The effectiveness of this search approach is demonstrated in the results shown in Fig. 7, displaying the good agreement between the CNP model () and the final equivalent sphere (D) up to 30% strains in terms of force response and transversal displacement. The agreement is maximum at 20%, it being the optimization point. It is worth noting that a progressive divergence in AC (not plotted) was observed, starting at 30% strain, where kinematics becomes highly influenced by the composite structure of the CNP and AC becomes much larger than for the equivalent homogeneous material. The behavior of pure Co (h) or silica (s) is evidently much different.

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Fig. 6. FEM results of a purely elastic 100 nm CNP compressed between rigid platens at 30% strain. The four non-zero components of the stress tensor are shown. The model is axisymmetric around y.

Table 1 Result of the two-step search algorithm to estimate fECNP ; vCNP g which converged at iteration 6 (5% tolerance required for all three control parameters). ðiÞ

ðiÞ

Iteration

ETRIAL (GPa)

vTRIAL

Control 1 %DFY

Control 2 %DuX

Control 3 %DAC

1 2 3 4 5 6

211 161 161 161 161 169

0.32 0.32 0.29 0.26 0.23 0.23

34.0 1.9 0.7 2.9 5.0 0.3

24.5 24.6 16.6 9.5 2.4 2.4

0.5 21.6 2.1 3.9 5.0 5.0

While this purely elastic model is deemed appropriate to address the calibration issue, the results, such as the force response in Fig. 7, are not intended for direct comparison with actual test responses. To this end, detailed simulations accounting for plasticity, exact contact geometry and platen deformability are necessary and will be discussed in a forthcoming paper. 3. Results and discussion Tests were performed on 20 randomly selected CNP at a variety of loads between 30 lN and 300 lN. The tests are subdivided into seven groups as reported in Table 2, based

on the applied maximum load. Sample P–d curves, shown in Fig. 8 from groups at 30 lN, 40 lN and 60 lN, demonstrate the good reproducibility of the CNP response, especially at low strains. Overall, tests in all groups confirm the good repeatability of the response up to 60 lN (Fig. 8). Tests at higher loads point out that the CNP response during loading exhibits a characteristic shape divided into three regimes. The meaning of such a profile is related to the occurrence of plastic deformation, which falls beyond the scope of the current analysis. Instead, for the analysis of the elastic properties, one needs to focus on the unloading (elastic) portion of each response. 3.1. Elastic modulus and hardness: approach #1 For each test, Table 2 summarizes the main information and results from the analysis of the unloading data, such as particle diameter, test data (i.e., dMAX, dR, PMAX, dP/dd) and estimates of CNP modulus and hardness from Eqs. (5)–(9) (approach #1). The percentage R2 related to the polynomial fit in Eq. (3) is also reported as an indication of the quality of dP/dd. Table 2 highlights that ECNP and HCNP from Eqs. (1) and (2) tend to increase with the load levels, thus appearing to be force (i.e., pressure) dependent. But values obtained at high load are unrealistic.

A. Rinaldi et al. / Acta Materialia 58 (2010) 6474–6486 250 CNP

Co

Si

Eq. Sphere

FORCE (uN)

200

150

100

50

0 0%

5%

10%

15%

20%

25%

30%

35%

STRAIN CNP

Co

Si

Eq. Sphere

ux (nm)

3

2

1

0 0%

5%

10%

15%

20%

25%

30%

35%

STRAIN Fig. 7. Comparison among FEM results of CNP (), equivalent sphere (D), pure Co (h) or silica (s). The force response (TOP) and the transversal displacement (BOTTOM) are shown over a large deformation range at 10%, 20% and 30% strains.

Fig. 9 displays the average estimates of hECNPi and hHCNPi just from five data groups between 30 lN and 90 lN. Significant discrepancies exist between the four contact theories. 3.1.1. Low force estimate of modulus According to Fig. 9A, the Hertzian (s) and geometric (h) models yield values >300 GPa, which greatly overestimate the correct modulus based on the comparison with ECNP = 169 GPa from FEM and ECo = 211 GPa from the literature. The other two models (RS e and CYL d) perform best, especially up to 60 lN, and agree reasonably well with each other. Since the FEM simulations show evident barreling in the CNP at 20–30% strain (see Fig. 6), the adequacy of the cylindrical model is not surprising. Furthermore, the minimum load of 30 lN corresponds to strains of 10% and relatively high contact pressures. It is emblematic that other researchers, such as Gerberich et al. [25] or Zou and Yang [30], have used lower loads (<20 lN) to estimate the elastic properties of NP and, even then, the Hertzian model was overshooting. Therefore, one direct way to obtain good estimates for ECNP is to consider only tests of Group 1 in Table 2. Considering the cylindrical case at first, the average

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modulus value from the seven data points is hECNP–CYLi = 180 GPa, which agrees reasonably well with 169 GPa from FEM. Remarkably, the corresponding mean squared error (MSE) would be as large as MSE(ECNP) = 64 GPa, since it is magnified by the higher ECNP value corresponding to particle #7 (75 nm diameter). Apparently, the high ECNP observed could be due to the much larger pressures associated with the smaller CNP radius at the given load, which is consistent with the p-dependence argument from Mook et al. [28]. This is evident from the ECNP values of Group 1 in Table 2, which increase in stiffness as the CNP diameter reduces. Related to this behavior, Varghese and co-workers reported Co3O4 nanowires Young’s modulus to be sizedependent, exhibiting lower elastic modulus with increasing diameter (for diameter >100 nm) [54]. Ideally, to avoid this size-dependence issue, repeated measures aimed at estimating the modulus should be collected at constant maximum pressure, e.g., by keeping both CNP diameter and PMAX constant, as done in Ref. [27]. Hence, by discarding the outlier observation (CNP #7), the remaining set is more homogeneous and finally yields hECNP–CYLi = 159 GPa and smaller MSE(ECNP) = 31 GPa. Interestingly, the RS (e) model would render an estimate of hECNP–RSi = 166 ± 30 GPa, which substantially confirms the CYL model. Notably, to increase the robustness of this analysis, measures at lower loads should also be taken. Indeed, Zou and Yang [30] reported a 36% increase in the modulus of silica NP measured in the range 10–20 lN. If the lower limit were not appropriate because it was too large, it would make this method not accurate or even unfeasible. Checking both the extent of the deformation range (i.e., dMAX) and the AFM image after testing is crucial to assess this aspect (which, however, was not an issue here). 3.1.2. Extended load range estimate of modulus An alternative way to determine ECNP consists of extrapolating it indirectly from regressing the data points over a sufficiently extended load range. After observing the influence of pressure on the effective modulus E(p) at relatively high loads (up to and above 80 lN), Mook and co-workers [28] extrapolated E as the zero-pressure limit E0 from the linear model Eq.(6) fitted to their E–p data. In the same fashion, by acknowledging hECNPi(P), fitting a linear model to the CYL data series (d) in Fig. 9A yields ECNP–CYL (P = 0 lN) = 56 GPa at zero force, which is not accurate compared with previous estimates. Nonetheless, GEO data (h) in the same load range yield ECNP–RS (P = 0 lN) = 210 GPa. Analogous with Mook’s group, GEO and CYL estimates could be interpreted as upper and lower bounds, respectively, to ECNP, but the limited data set may undermine the efficacy of the method in this case. 3.1.3. Estimates of hardness A similar scenario is portrayed in Fig. 9B for the hardness. Considering CYL and RS estimates for consistency

6482 Table 2 Summary of main information, experimental data and results of tests on 20 randomly selected CNP, i.e., the particle diameter (D), the residual (dC) and the maximum displacements (dMAX), the derivative (dP/dd), the R2 of the power fit to determine m, the contact radius, the CNP Young’s modulus (E) and hardness (H) based on the four contact model. CNP#7 is an outlier and is discarded from mean and MSE of Group #1. Experimental

Group 1: PMAX = 30 lN

Group 2: PMAX = 40 lN

Group 3: PMAX = 50 lN

Group 4: PMAX = 60 lN

Group 5: PMAX = 90 lN

Group 6: PMAX = 100 lN

Group 7: PMAX = 300 lN

D (nm)

1

2

Geo

Cyl

RS

C

MAX

(nm)

dP/dd (lN nm1)

%R

(nm)

a (nm)

ECNP (GPa)

H (GPa)

a (nm)

ECNP (GPa)

H (GPa)

a (nm)

ECNP (GPa)

H (GPa)

a (nm)

ECNP (GPa)

H (GPa)

119

6

12

13.3

94

19

517

28.2

26

327

14.8

51

141

3.7

44

167

5.0

2 3 4 5 6 (7)

114 111 117 91 123 (75)

8 6 9 5 9 (4)

13 11 16 10 19 (9)

15.2 14.1 15.1 14.2 11.4 (16.2)

88 86 94 91 82 (90)

19 17 21 15 24 (13) Mean

595 626 504 776 293 (1544) 552

25.8 32.2 20.7 41.1 16.4 (61.0) 27.4

27 24 29 21 33 (18) Mean

372 387 325 466 200 (789) 346

13.7 17.0 11.1 22.8 8.9 (32.4) 14.5

49 48 51 39 54 (33) Mean

170 163 162 205 110 (310) 159

4.0 4.3 3.6 6.3 3.2 (9.4) 4.2

47 43 51 42 55 (38) Mean

182 185 163 191 108 (253) 166

4.4 5.3 3.7 5.6 3.1 (6.8) 4.5

8

81

6

13

14.9

97

16

780

49.5

22

476

27

36

243

9.9

46

179

6.0

9

94

6

13

17.6

91

17 Mean

939 860

42.8 46.2

24 Mean

548 512

23 24.9

41 Mean

254 249

7.6 8.8

46 Mean

221 200

6.0 6.0

10

122

7

28

16.1

77

29

354

17.8

39

244

10.0

57

156

4.7

68

127

3.3

11

115

21

28

21.6

93

28 Mean

574 464

19.3 18.5

37 Mean

379 311

10.9 10.5

54 Mean

235 195

5.3 5.0

67 Mean

180 154

3.4 3.4

12

111

13

23

24.1

91

25

843

29.9

34

517

16.7

51

290

7.3

61

230

5.1

13

111

14

21

21.9

95

24 Mean

744 793

31.7 30.8

33 Mean

464 490

17.5 17.1

50 Mean

259 275

7.4 7.4

59 Mean

211 221

5.3 5.2

14

115

22

32

34.7

95

30

1202

30.5

40

710

17.7

55

425

9.2

72

297

5.4

15 16

115 112

20 15

30 25

30.8 31.3

96 96

29 26 Mean

1001 1290 1164

32.6 40.5 34.5

39 35 Mean

609 730 683

18.8 22.8 19.8

55 52 Mean

366 402 398

9.4 10.6 9.7

70 64 Mean

265 302 288

5.8 6.9 6.0

17

80

22

32

26.9

95

25

1016

48

32

662

30

42

433

17

72

214

6

18

70

13

22

34.3

96

20 Mean

4878 2947

81 65

26 Mean

1767 1215

48 39

35 Mean

893 663

26 22

61 Mean

367 291

9 7

19

74

38

59

46.9

99

33

2102

86

36

1619

71

68

493

21

96

303

10

20

93

33

56

41.1

97

36 Mean

1186 1644

72 79

43 Mean

847 1233

52 62

60 Mean

479 486

26 23

94 Mean

263 283

11 10

A. Rinaldi et al. / Acta Materialia 58 (2010) 6474–6486

Outlier

CNP #

Hertz 

A. Rinaldi et al. / Acta Materialia 58 (2010) 6474–6486

6483

Fig. 8. Sample P–d curves from tests at maximum loads of 30 lN, 40 lN and 60 lN.

HERTZ

GEOM

RS

CYL

hhHCNP–CYLii = 7 GPa and hhHCNP–RSii = 5 GPa. Beyond such a load threshold, hHCNPi increases rapidly. Interestingly, unlike for the modulus, no pressure-enhanced theory, such as in Eq. (6), has ever been set forward in the literature, and there seems to be no evident physical rationale to do so, even though regressing the hHCNPi data linearly over the whole data range would give realistic zero-load estimates, i.e., HCNP–CYL (P = 0 lN) = 3.2 GPa for CYL and HCNP–RS (P = 0 lN) = 4.2 GPa. The validity of the regression approach applied to hardness remains controversial and can only be speculated upon at this stage.

a

1200

ECNP (GPa)

1000 800

y = 4.9 x + 210

600

y = 3.7x + 56 400 200 50

b

HCNP (GPa)

40

30

20

10

0 0

20

40

60

P (µN)

80

100

120

Fig. 9. Plot of (A) average hECNPi and (B) average HCNP from the four area models restricted to groups between 30 lN and 120 lN. The trend line is fitted to geometrical and cylindrical cases rendering a projected value of hECNPi at zero-load of E0-GEO = 210 GPa and E0-CYL = 56 GPa.

with the modulus analysis, the “lowest load approach” renders a mean estimate hHCNP–CYLi = 4.2 ± 1.1 GPa for CYL data (d) and hHCNP–RSi = 4.5 ± 1.0 GPa for RS data (e), which are about five times larger than the bulk value for cobalt (HVICKERS = 1 GPa; HBRINELL = 0.7 GPa [11,52]). A similar increase was observed also in Co nanowires (5.2 GPa) [33], which supports the presence of a nanoscale effect. When the load increases, larger values tend to be found, suggesting a pressure effect. Up to 90 lN though, the mean estimates hHCNPi for both CYL and RS models are of the same order (<10 GPa) and would yield the grand means

3.1.4. Remarks on modulus estimates from approach #1 In conclusion, from a methodological standpoint, the selected load range is a fundamental parameter of the experimental design and may direct the experimenter to choose between the two methods to get the modulus, namely: (i) the first based on the direct consideration of data at small loads; and (ii) the second based on a wider loading range, but relying on regression models to account for load or pressure dependence. Regardless of the selected method, it is important to underline the role of FEM simulations for the validation of the test results. As no reference value exists for the CNP modulus, the FEM estimate constitutes a useful first approximation reference value. Furthermore, the macroscopic properties of pure cobalt and (amorphous) silica are known to vary independently considerably. Metallic crystal cobalt (considered here in place of Co2B) usually exists as a mixture of two allotropes at ambient temperature, and the subtle transformation from one form to the other causes a scatter in the physical properties [55,56]. The silica, in contrast, is affected by surface absorption and/or reaction with water molecules at room temperature [30,57], which may be responsible for the variation in modulus and hardness of the external layer of the CNP. Such a key role of simulations as companions to experimental tests should not be overlooked. As a concluding remark, the good performance of the RS model proposed here for the first time is pointed out. In particular, according to Fig. 9, both modulus and hardness from the RS assumption do not appear to be as sensitive to load as in the other three models.

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3.2. Elastic modulus: approach #2 In order to validate the above results further, based on the Oliver–Pharr analysis, for the modulus one can also apply the bounding method (approach #2). The upper and lower bounds for ECNP(p) computed from the E estimates from Eqs. (7) and (8), respectively, are plotted in Fig. 10, confirming the marked pressure-dependence highlighted in Fig. 9A. Fitting the linear models in Eq. (6) on the two data sets renders the zero-pressure higher and lower bounding estimates of ECNP–UB(0) = 134 GPa (based on geometrical contact) and ECNP–LB(0) = 72 GPa (based on cylindrical contact), which appear to be effectively lower (although the upper bound is at least of the same order) than both their Oliver–Pharr’s counterparts in Fig. 9 and reference values (i.e., FEM and ECo). To check against regression artifacts, the regression analysis was repeated on a more limited pressure range (e.g., <60 GPa as [28]), which on one side substantially confirmed such discrepancy, but on the other revealed that these bounding values can vary greatly, depending on the selected pressure range. In conclusion, this method in the present study was less accurate in comparison with approach #1 and should be used with care. Like any regression model, it is inherently sensitive to the pressure range, and its applicability is evidently related to the adequacy of the area model it relies upon and to the linear dependence on pressure stated in Eq. (6). Validity checks about the regression range and the area models are a necessary part of approach #2. But only a thorough statistical analysis of sufficiently large data sets, purposely collected over an extended pressure range, would allow it to be ascertained in the future whether Eq. (6) is appropriate or whether another model for the pressure dependence (e.g., a power law) should be sought at the nanoscale. 3.3. Other factors contributing to modulus determination

study that should be undertaken to identify the exact deformation mechanisms. It is unlikely that the mechanical behavior of the CNP can be explained in the conventional terms of plasticity and material strength, owing to the complex nature of the material. This can be clearly understood by considering two possible contributing factors such as, for example, the magnetic or magnetostrictive nature of Co2B and the evidence of “so-called” reverse plasticity. 3.3.1. Magnetic and magnetostrictive properties The magnetic properties of Co and Co–B alloys is well documented [14,15,58–60]. If such materials are magnetically excited, a fraction of the energy is stored as elastic energy, of which the magnetostrictive effect is the visible effect. (Magnetostriction is the term applied to the dimensional change observed in a body being magnetized.) In turn, applying a mechanical load causes rotations and displacements of the magnetic dipoles which result in magnetic interactions. This mechano-magnetic coupling has an effect on the elastic modulus (reflected in macroscale samples also as a change in the ultrasound velocities). The actual effect of course depends on several parameters, such as internal friction, degree of amorphization and magnetic state, which indirectly depend on sample size. The CNP have a complex nanostructure, with a magnetic core made up of aggregates of superparamagnetic nanostructures which determine a non-zero resultant dipolar magnetization, as confirmed by the magnetization measurements in Fig. 11. This confirms that the magnetostrictive coupling is possible and that a quantitative study may be pursued in the future. 3.3.2. Reverse plasticity Reverse plasticity consists of a partial recovery of the plastic deformation after a certain elapsed time after testing [26]. A total recovery of the plastic deformation in silica NP after a 30-lN indentation was reported previously [30]. Discovering it in the CNP was a surprise. Its occurrence was

The mechanical properties measured in the present experiments are an important first step towards a deeper 800 LB UB

700

ECNP (GPa)

600

y = 8.8757x + 134.31 500 400 300 200 100

y = 0.4817x + 72.214

0 0

20

40

60

80

p (GPa) Fig. 10. Plot of average upper and lower bound estimates of ECNP from Eqs. (7) and (8). The regression lines from Eq. (6) are plotted, rendering the statistical zero-pressure estimates ECNP–UB = 134 GPa and ECNP–LB = 72 GPa.

Fig. 11. Magnetization measurements of silica-coated cobalt boride CNP showing an open hysteresis loop at room temperature (T = 300 K).

A. Rinaldi et al. / Acta Materialia 58 (2010) 6474–6486

6485

Fig. 12. AFM profiles (left) showing absence of global permanent deformation after two repeated tests (right). The top row shows AFM images of the CNP before test (A), after hit#1 (B), and after hit#2 (C) – scale bar is 250 nm. On the bottom, pair-wise comparison of cross sections (cross section B is the reference profile), showing a total strain recovery after hit#1 and some flattening observed locally on the CNP at the contact zone after hit#2. The unsymmetrical profile after the last hit may result from a small eccentricity in tip positioning or from the failure of a weak spot in the composite nanostructure.

proved by compressing one CNP twice at the same load level of 40 lN and imaging with the nanoindenter before and after each test. The two loading curves plotted in Fig. 12 (right) are almost identical, indicating that the (small) plastic deformation measured in each curve was reversibly reabsorbed after some time. The shape recovery is confirmed by the AFM micrographs in Fig. 12 (left). This phenomenon is relatively new and not completely understood. In dislocation-free metal nanostructures, such a plastic deformation is most likely caused by the nucleation or “insertion” of a number of dislocations at the contact surface. In the CNP, however, the occurrence of reverse plasticity came as a surprise owing to the presence of an amorphous core filled with radial defects. In fact, this phenomenon has been reported only in dislocation-free crystalline NP so far. This reversible phenomenon appears as a viscoelastic effect which may also play a role in the determination of the elastic modulus of the CNP, possibly giving rise to a strain-rate dependence, especially in dynamic testing. TEM-monitored in situ tests may be conducted in the future to improve understanding of this pseudo-plastic phenomenon and relate it to the complex nanostructure of the CNP and its fundamental deformation mechanisms (i.e., inter-aggregate plasticity, densification, magnetic coupling, etc.). 4. Conclusions The measurement of the mechanical properties of Co– B-based NP were reported here for the first time. In situ compression tests of 70–150 nm core–shell NP were performed and analyzed to obtain Young’s modulus and hardness according to two approaches borrowed from the literature. One approach (approach #1), which combines the Oliver–Pharr method and standard contact theory, is normally intended for single-phase NP, but was extended here to composite CNP as well. The FEM was deployed to provide an equivalent Poisson ratio vCNP for Eq. (1),

and a numerical algorithm was purposely formulated to this end. In general, FEM or other numerical techniques appear to be instrumental in overcoming issues related to anisotropy and heterogeneity of composites by providing those quantities required in the models, but none the less unknown (or ill-defined) in complex nanostructures. From a methodological standpoint, such a hybrid philosophy, combining analytical, experimental and numerical elements, bears broader significance in the pursuit of the mechanical characterization of other existing and/or forthcoming complex nanocomposite materials (core–shell, onion-like, dimmers, aggregates, etc.). The study confirmed that the accuracy of modulus and hardness estimates from approach #1 may heavily depend on the selected model for contact area A(d). As it turns out from the comparison of four alternative options (see Eqs. (5a)–(5d)), the cylindrical and the rigid-sphere approximations, the latter of which is novel and as yet unreported, seemed to perform best and consistently here. Accordingly, calculations indicated that at the lowest load (30 lN) the silica-coated Co2B have a modulus comparable with bulk material (ECNP = 159–166 GPa), but much higher hardness, in agreement with trends in simpler NP. This paves the way to highly functional Co–B nanomaterials endowed with enhanced mechanical properties in addition to the usual array of functional (e.g., electrochemical, anticorrosion, biocompatibility and magnetic) properties. It was finally shown that the present results need to be interpreted in connection with other advanced phenomena, such magnetostriction and reverse plasticity, the evidence of which was also demonstrated here. That possibly represents a useful starting point for the next experimental endeavor in the study of this innovative and promising material. Acknowledgements The authors gratefully acknowledge that support of this work was provided by the Ira A. Fulton Engineering School of Arizona State University. M.A.C.-D. and V.S.-M. also

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