SUPPLEMENTARY INFORMATION

Transcription

1 Supplementary Information for Manuscript: Nanoscale wear as a stress-assisted chemical reaction Supplementary Methods For each wear increment, the diamond indenter was slid laterally relative to the silicon asperity. The sliding distance interval between high-resolution out-of-contact images (from which volume loss was calculated) was most commonly 100 nm, but in some tests was shortened initially to be more sensitive to the early stages of wear, and lengthened later in the tests to achieve longer total sliding distances. No effect of the variation of the sliding interval on the rate of volume lost was observed. The adhesive force between the tip and sample was measured upon separation of the contact. Upon retraction of the flat diamond punch, the cantilever deflected (as shown in Supp. Fig. 1) until the restoring force of the bending cantilever beam exceeded the adhesive force in the contact. Since the spring constant k cantilever was known from calibration 23, then the adhesive force F adhesive was calculated using: F adhesive = k cantilever Δx pull off (Supp. Eq. 1) where Δx pull-off is the bending of the cantilever at the point of pull-off. Supplementary Figure 1: The adhesive force in the contact was ascertained by measuring the deflection of the calibrated cantilever. a, The indenter and AFM tip are shown in contact at the position of sliding. b, As the indenter was retracted, the tip/sample adhesion caused the calibrated cantilever to deflect by Δx. The real-time video (captured at 30 frames per second) allowed the identification of the maximum deflection Δx pull-off at the specific moment of contact separation (during post-processing). To quantify the worn volume, each high-resolution out-of-contact still image was traced to measure the profile of the probe tip after each interval of sliding (Supp. Fig. 2a). For all of the still images, the profiles upper (unworn) portions were then aligned spatially using an optimization algorithm that minimizes the mismatch parameter, defined as the average lateral displacement of the unworn portions of the profiles (Supp. Fig. 2b). Once aligned, the profiles were integrated using a method of disks (Supp. Fig. 2c) to calculate a three-dimensional volume (Supp. Fig. 2d). The volume lost was then NATURE NANOTECHNOLOGY 1

2 calculated by comparing the volumes of the asperity at various points in the wear test. The largest source of uncertainty in the measurement of volume lost arose from improper alignment of the profiles. Changes in beam conditions and other TEM artifacts can cause slight tracing differences even in unchanged portions of the asperity, which in turn leads to imperfect alignment. Thus, the uncertainty was quantified by recomputing volume loss at deviations +/-5% away from the optimum value of the mismatch parameter. An additional source of uncertainty arose from the integration using the method of disks, which assumes local (not global) axi-symmetry (i.e., the asperity has a circular crosssection at every height). Tilting in the TEM about the axis of the asperity prior to testing was used to confirm that this source of error was small compared to the alignment uncertainty discussed above. Supplementary Figure 2: Lattice-resolved out-of-contact images of the tip were used to calculate the instantaneous volume lost at various points throughout the wear test. a, The high-resolution TEM images were traced to extract the profile of the asperity at intervals throughout the wear test. b, Profiles of the same asperity after different amounts of wear were aligned using an optimization routine. Here a reference profile (solid, red) is compared to a worn profile, which is shown both in good alignment (dotted, black) and in poor alignment (dashed, blue). Horizontal green lines schematically indicate the measured displacement between the two profiles, which is used to calculate the mismatch parameter used for optimization. c, Once aligned, the two-dimensional profiles were integrated using a method of disks to create a threedimensional shape (d), from which the volume was measured. Supplementary Movie Legends In the online component of the Supplemental Information, a sample video of the in situ wear testing can be viewed. A screen shot and the caption for this video are included below: 2 NATURE NANOTECHNOLOGY

3 Supplementary Video 1: A sample video demonstrates an individual increment of the wear test. A polished, single-crystal diamond substrate (bottom, non-electron transparent) was displaced along the vertical actuation axis (designated on screen shot) to bring it into contact with a sharp silicon AFM probe (top). The AFM tip was observed to snap into contact with the diamond (a phenomenon well known in atomic force microscopy 31 ), then the diamond substrate was slid laterally with respect to the silicon tip. Finally, upon separation of the contact, the adhesive force caused the AFM cantilever to bend, and the bodies separated at a different vertical position from where snap-in occurred. By measuring the maximum deflection of the calibrated 23 cantilever, the magnitude of the tip-sample adhesive force was determined (as explicitly shown in Supp. Fig. 1). (Quicktime, 37 seconds) Supplementary Data Figure 2 in the print version of the paper shows an example of aligned traces, which demonstrates the gradual evolution of the wearing asperity. Supplementary Figure 3 demonstrates this for all four of the AFM probes studied over their respective wear tests. The sliding interval between adjacent profiles varies (as discussed above) and thus the amount of profile recession varies. In all but one case, the change from one profile to the next is gradual (in many cases just one nm of recession or less) and relatively uniform, without evidence of fracture. Even if there are undetected fracture events, the total distance of recession puts an upper bound on the size of fractured pieces of typically 1 nm or less. The one case where fracture clearly does occur is marked with an arrow in Supp. Fig. 3b, and demonstrates removal of a large amount of the tip, with a preferred crystallographic orientation of the fracture plane that is distinct from the sliding plane. This event occurred while a significant amount of oxide was still present on the tip and does not influence the analysis of worn volume of crystalline silicon described in this Letter. More generally, this demonstrates that fracture can be detected when it occurs, but also that the majority of wear occurred in the absence of fracture. NATURE NANOTECHNOLOGY 3

4 Supplementary Figure 3: Overlaid traces show that silicon wear is gradual. Initial (red) and subsequent (black) profiles have been overlain to show the evolution of the tip over the course of testing. Solid lines indicate that the wearing material is crystalline silicon (as determined through TEM imaging), dotted lines indicate wear of the initially-present oxide. For tips shown in a and c the oxide was removed in the TEM during tip alignment, prior to wear testing. Sliding intervals for the silicon correspond to the spacing of points on the x-axis in Fig. 3a and Supp. Fig. 5a. In only one case (during wear of oxide) is there clear evidence of fracture (in b, orange arrow), demonstrated by sudden removal of a larger amount of material with evidence of a preferred crystallographic plane different from the sliding plane. As discussed in the print version, DMT contact mechanics was used to calculate contact parameters. This method requires knowledge of the geometry of the tip (available from TEM images) and fitting this geometry to a parabola of the form 32 : z(r) = r2 (Supp. Eq. 2) 2R tip where z is the height of the tip profile at any radial position r, and the tip radius R tip is extracted from the axisymmetric parabola that best fits the geometry. In order to perform this fitting, the spatial positions of points on the tip profile were extracted using a MATLAB routine, then the best-fit parabola was found using a least squares fit to the measured profile. The profiles were traced up to a vertical height of roughly 5 nm, since tip/sample interaction forces are predicted to die off by this separation 33. Additionally, every profile was traced at least five separate times to ensure reproducibility of R tip. The average value of R tip from these multiple traces was used in the analysis, with the standard deviation as a measure of the uncertainty. Naturally, there were in some cases small deviations away from a perfect parabola, which would lead to errors in the stress calculation. However, in general, the probes were very well approximated by parabolas at all intervals throughout the wear tests, as shown in Supp. Fig. 4. Asperity shapes that remain parabolic despite large volumes of wear have also been observed in ref NATURE NANOTECHNOLOGY

5 Figure 4: Parabolic profiles were fit to the near-tip geometry of the probe at every stage in the wear test. Tip profiles were extracted from high-resolution still images taken periodically throughout the sliding, and each profile was fit with a parabola. Two probes are shown here in their initial, pre-test states (a, c), and then the same probes are shown at the end of testing (b, d, respectively). It is clear from these representative images that the probes are well approximated by a parabola throughout the wear test. Figure 3 in the print version shows the silicon wear data for the four different AFM probes. For ease of comparison between datasets, only the first 1 µm of sliding is shown. Two of the probes were slid for more than one µm; the entirety of the data is shown in Supp. Fig. 5. As with the zoomed-in presentation of the data, variations in slope are seen within a single dataset and also among the different probes. Supplementary Figure 5: Wear measurements for silicon shown to 4.5 µm of sliding distance. a, Only the data from the first 1 µm of sliding are plotted in Fig. 3a of the print version; here the results are plotted out to the end of testing (>4.5 µm, in one case). (Colors and shapes of data points correspond to those of Fig. 3.) Similar trends are still visible, including widely varying local slopes. b, The Archard-like presentation of the data is shown to longer sliding distances than in Fig. 3b. The inadequacy of a linear fit in describing the wear data is confirmed. Note that in accordance with Figure 3, only the wear of crystalline silicon is shown; sliding tests involving the removal of the oxide have been excluded. NATURE NANOTECHNOLOGY 5

6 Supplementary Equations To compare the kinetics of wear to Eq. 1 of the print version, one must calculate the reaction rate Γ atom-loss [s -1 ] and mean normal stress in the contact σ normal [Pa]. The stress is calculated using Eq. 2b. Since the reaction rate (Γ atom-loss, Eq. 2a) is the time-rate of reaction for a single atom, it is determined by dividing the number of atoms removed by wear in a given measurement interval (N lost ) by the product of the number of atoms that were in contact (N contact ) and the sliding time of that interval (t slide ) as follows: N Γ atom loss = lost V = lost ρ Si V = c lost (Supp. Eq. 3) N contact t slide ( A contact ρ surf,si(100) )t slide A contact t slide where N lost is expanded as the product of the volume lost (V lost ) and the atomic density of silicon (ρ Si ), and likewise N contact is expanded as the product of the contact area (A contact ) and the areal atomic density of the silicon (100) surface (ρ surf,si(100) ) (the tip s orientation is verified by diffraction measurements in the TEM). For clarity, the density terms are collected as a pre-factor and replaced by the constant c. This approach (calculating a rate of tip change that depends on time) thus predicts that for a fixed sliding distance, the volume removed will be inversely proportional to the sliding speed. This is equivalent to the approach used by Gotsmann and Lantz (Eq. 2 of ref. 10) in every respect but one: Gotsmann and Lantz identified the frictional shear stress as the activating stress for wear, whereas in the present work, compressive stress is used for the reasons discussed in the print version. The shear stress can exhibit a dependence on speed, and if so, that will add an additional dependence of wear on sliding speed. However, Gotsmann and Lantz s experiments were at constant speed and thus the predicted speed dependence was not verified. There have not been many investigations into the effect of sliding speed on nanoscale wear. A speed dependence was seen experimentally for wear of a platinum tip sliding against diamond-like carbon by Bhushan and Kwok 34, whereby the worn volume increased roughly logarithmically with speed at low sliding speeds, then leveled off. This is in contrast with the present model and with our experimental results obtained at two speeds, as discussed in the print version. However, in that work, the authors mention that wear consists of adhesive, abrasive, and tribochemical wear, and they explicitly state that asperity fracture and particle generation occurs in their experiments, which are carried out at higher loads ( nn) than the present work. Therefore, it is not surprising that a different speed dependence of wear is observed. The quantity A contact in Eq. 2a,b and Supp. Eq. 3 is calculated using DMT adhesive contact mechanics, as follows 27,28 : 2 A contact = πa contact = π 3R F tip adhesive 4E c where a contact is the contact radius, R tip is the radius of the asperity, and E C is the composite modulus (defined as E C = 2 3 (Supp. Eq. 4) [( 1 ν 2 tip ) E tip + ( 1 ν 2 surface ) E surface ] 1, where ν and E designate the Poisson s ratio and elastic modulus of the tip and sample materials 6 NATURE NANOTECHNOLOGY

7 load, so only F adhesive appears in Supp. Eq. 4. The use of DMT contact mechanics 28 is justified by a calculated Maugis parameter 27 λ of 0.17 (calculated in a manner similar to 35 using literature values for ν and E, a tip radius of 22 nm (the mean value throughout all testing), a work of adhesion of J/m 2 (the average of the measured values) and an equilibrium separation of 0.3 nm (ref. 36). Note that while 0.17 falls outside of the strict limit for DMT behavior of less than 0.1, the deviation in calculated parameters between λ=0.10 and λ=0.17 is just 3% (ref. 37). The uncertainty values of σ normal and Γ atom-loss are calculated by the method of propagation of uncertainty 38. The activation parameters ΔU act and ΔV act are extracted from a best fit of Eq. 1 to the measured data as shown in Fig. 3. As discussed, the pre-factor Γ 0 was set to s -1, corresponding to atomic vibration frequencies. ΔV act is independent of the choice of prefactor, and ΔU act has only a weak dependence. Pre-factors of and s -1 produce activation energies of 0.85 and 0.97 ev, respectively, these represent the largest contribution to the uncertainty in the calculated value and are therefore used as the bounds of error for the value (i.e., ΔU act = 0.91±0.6). Supplemental Information References 31. Seo, Y. & Jhe, W. Atomic force microscopy and spectroscopy. Rep. Prog. Phys. 71, (2007). 32. Johnson, K. L. Contact Mechanics. (Cambridge University Press: Cambridge, UK, 1985). 33. Israelachvili, J. Intermolecular and Surface Forces. (Elsevier, Inc.: Oxford, UK 2011). 34. Bhushan, B. & Kwak, K. J. Velocity dependence of nanoscale wear in atomic force microscopy. Appl. Phys. Lett. 91, (2007). 35. Grierson, D., Flater, E. & Carpick, R. Accounting for the JKR-DMT transition in adhesion and friction measurements with atomic force microscopy. J. Adhes. Sci. Tech. 19, (2005). 36. Rabinovich, Y. Adhesion between nanoscale rough surfaces I. Role of asperity geometry. J. Coll. Int. Sci. 232, (2000). 37. Carpick, R. W., Ogletree, D. F. & Salmeron, M. A general equation for fitting contact area and friction vs load measurements. J. Coll. Int. Sci. 211, (1999). 38. Taylor, B. N. & Kuyatt, C. E. Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. (National Institute of Standards and Technology). NATURE NANOTECHNOLOGY 7