3. AFM indentation for quantitative hardness measurements

The nanometer-scaled plastic deformation of Au(111), Pt(111), and Pt57.5Cu14.7Ni5.3P22.5 metallic glass was investigated by AFM indentation and subsequent nc AFM imaging. For indentation and imaging, diamond-coated silicon single-crystalline cantilevers were used (type CDT-NCLR, manufactured by NanoSensors, Switzerland). For the cantilever used on Au(111), the bending stiffness was found to be Cn = 55 N/m. For AFM indentation of Pt(111) and Pt57.5Cu14.7Ni5.3P22.5 metallic glass, a single cantilever of the same type as on Au(111) was used, whose normal stiffness was found to be Cn = 46 N/m.

Prior to the measurements on Au(111), the sensitivity S of the photodiode was calibrated by recording a force-distance curve on nanocrystalline diamond, consisting in an initial retraction of the z-scanner by 50 nm away from the sample surface and a subsequent series of approach and retraction by the same distance at a velocity of 0.3 μm/s. These parameters were set to avoid tip damages during contact between the diamond-coated tip and the nanocrystalline diamond sample. The sensitivity of the photodiode was then determined by fitting the repulsive part of the force-distance curve with a linear function. In contrast, before AFM indentation on Pt(111) and Pt57.5Cu14.7Ni5.3P22.5 metallic glass, the sensitivity of the photodiode is calibrated in the noncontact mode of AFM, according to Ref. [22]. Thereby, we considered a conversion factor for the vibration energy of the cantilever determined from the optically measured deflection [17].

nanometers. During scanning over a surface, changes in tip-sample distance due to sample topography result in changes in the amplitude and in a frequency shift of the cantilever resonance. To measure topography amplitude and/or frequency shift can be tracked by a feedback loop to keep the cantilever oscillation in resonance. Contact mode imaging relies on short-range interaction forces between the tip of a cantilever and the sample surface, the nature of which can be adhesive (attractive forces) or elastic (repulsive forces). During scanning, local changes in topography yield changes in the contact force between sample and surface. In this case, topography can be measured by tracking the normal contact force with a feedback loop to keep the

Figure 4. Topography images recorded by nc AFM on (a) Au(111), (b) Pt(111), and (c) Pt57.5Cu14.7Ni5.3P22.5 metallic glass

In this chapter, we present results obtained on single-crystalline metal and on metallic glass surfaces. An Au(111) polycrystalline thin film deposited on mica by physical vapor deposition was purchased by Phasis GmbH, Switzerland, and measured in ambient conditions (see Chapters III–V). Also, a Pt(111) surface and the surface of a Pt57.5Cu14.7Ni5.3P22.5 metallic glass were prepared for measurements in ultrahigh vacuum. The (111) surface of a platinum single crystal, purchased by MaTeck, Germany, was prepared by several cycles of Ar sputtering and annealing at 1000C. This resulted in the formation of 50–100 nm wide atomically flat terraces. A Pt57.5Cu14.7Ni5.3P22.5 metallic glass master alloy was prepared according to [21] and subsequently melt-spun. The amorphousness of the as-prepared metallic glass ribbons was confirmed by X-ray diffraction (XRD) with Cu Kα radiation and differential scanning calorimetry (DSC). To remove its native oxide layer, the surface of an as-prepared metallic glass ribbon was

All three sample surfaces were imaged by noncontact (nc) AFM to determine their respective RMS roughness Rq (see Figure 4). For atomically flat Au(111) and Pt(111), we found Rq = 0.407 nm and 0.372 nm, respectively, caused by atomic steps between terraces and adsorbates in the case of Au(111). For the Ar-sputtered Pt57.5Cu14.7Ni5.3P22.5 metallic glass, we found Rq = 0.375 nm.

The nanometer-scaled plastic deformation of Au(111), Pt(111), and Pt57.5Cu14.7Ni5.3P22.5 metallic glass was investigated by AFM indentation and subsequent nc AFM imaging. For indentation and

prepared by gentle Ar sputtering for 5 min with an energy of 1 keV.

3. AFM indentation for quantitative hardness measurements

contact force constant.

32 Contact and Fracture Mechanics

surfaces.

AFM indentation measurements consisted in recording the cantilever deflection upon extension of the z-scanner of the AFM. Owing to the tilt angle of the cantilever about the sample surface, a tilt correction was applied by moving the lateral scanner by Z tan w during a vertical scanner extension Z, where w = 13 is the tilt angle [23]. In this work the extension length Z of the zscanner was varied from 10 to 160 nm in the case of Pt(111) and Pt57.5Cu14.7Ni5.3P22.5 metallic glass and was set to Z = 150 nm for Au(111).

The plastic deformation of the three samples was analyzed based on nc AFM topographical images of the remaining indents and on the force-penetration curves. Typical topographical images of indented surfaces are shown for Au(111), Pt(111), and Pt57.5Cu14.7Ni5.3P22.5 metallic glass in Figure 5. For each indent, the projected area was determined by masking the area with threshold height values. This analysis was performed with the indentation analysis function of the software package Gwyddion [24]. It is, however, important to note that due to convolution effects with the shape of the tip, the size of indents imaged by nc AFM is underestimated (this effect is more pronounced for smaller indents). Also, in the case of Pt57.5Cu14.7Ni5.3P22.5 metallic glass, the prominence of the pileups makes an accurate determination of the projected area more difficult and less accurate.

The force-penetration (Fn – δ) curves were calculated from the recorded force-distance (Fn – Z) curves (see Figure 6). The principle of AFM indentation relies on the fact that the surface to be indented is softer than the AFM tip. In this case, an extension of the z-scanner leads, besides a deflection D of the cantilever, to a penetration of the AFM tip into the sample surface by the penetration depth δ = Z D.

Figure 6 shows a series of nc AFM images of Au(111), Pt(111), and Pt57.5Cu14.7Ni5.3P22.5 metallic glass surfaces after AFM indentation. In the case of Au(111), all indentations were performed with the same maximal load Fn = 7 μN and a same loading rate dFn/dt = 16 μN/s. For Pt(111) and Pt57.5Cu14.7Ni5.3P22.5 metallic glass surfaces, indentation is shown that was performed with varying maximum normal force values between Fn = 0.8 μN and Fn = 6 μN.

Figure 5. Nc AFM topography images of (a and d) Au(111), (b and e) Pt(111), and (c and f) Pt57.5Cu14.7Ni5.3P22.5 metallic glass surfaces after AFM indentation; in (d–f) the projected area was masked and calculated to determine the hardness values of each material.

For Au(111) two series of indentation measurements with the same maximum load values Fn = 7.2 μN but with different tips are shown. Within both series, the shape and size of the remaining indents are very similar. For Pt(111) and Pt57.5Cu14.7Ni5.3P22.5 metallic glass, remaining indents were only observed for maximum load values Fn > 0.8 μN. For these two materials, the projected area of the indents is observed to increase with the maximal load.

Figure 8 shows the load dependence of the projected area Ap for Pt(111) and Pt57.5Cu14.7 Ni5.3P22.5 metallic glass. The projected area Ap of indents is found to be much smaller for Pt57.5Cu14.7Ni5.3P22.5 metallic glass than for Pt(111). Further, we used the load dependence of Ap to calculate the hardness of Pt(111) and Pt57.5Cu14.7Ni5.3P22.5 metallic glass, according to dAp/dFn = 1/H. For Pt(111), we obtained H = 1.14 � 0.09 GPa. For Pt57.5Cu14.7Ni5.3P22.5 metallic glass, we obtained H = 7.3 � 2.4 GPa. These values are larger than the measured ones by nanoindentation with a Berkovich diamond tip (see Ref. [15] for more details). This can be explained by tip convolution during nc AFM imaging that results in an underestimation of the

Figure 6. Nc AFM topography images after AFM indentation measurements with the indicated normal force on Au(111),

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In the case of the indentation on Au(111), shown in Figure 5, we found Ap = 4703.52 nm2

corresponding to Fn = 7.2 μN. Using the classical expression for the determination of hardness,

was estimated from the noncontact AFM images shown in Figure 6 with the free SPM data analysis software Gwyddion (Figure 9). The half-opening angle of the as-reconstructed indenter was determined to be α = 67.21�. The hardness was then calculated according to

calculations deliver virtually the same value: HAu(111) = 1.5 GPa.

<sup>p</sup> tan2α δð Þ max�δel <sup>2</sup> <sup>¼</sup> <sup>1</sup>:46 GPa [25], where <sup>δ</sup>max is the maximal penetration depth in Figure 7(a), and δel was taken as the penetration depth at the first pop-in event in Figure 7(c). Both hardness

Ap ¼ 1:53 GPa. Alternatively, the shape of the AFM tip used to indent Au(111)

,

projected area.

Pt(111), and Pt57.5Cu14.7Ni5.3P22.5 metallic glass.

we obtained <sup>H</sup> <sup>¼</sup> Fn

<sup>H</sup> <sup>¼</sup> Fn <sup>3</sup> ffiffi 3

In the case of Au(111), almost no pileup can be observed. In this case, clear dislocation can be identified around indents. In the case of Pt(111), small pileups can be observed. More importantly, above an indentation load Fn = 3 μN, the indent exhibits a chevron-like shape that was never observed on the two other samples and which attribute to anisotropic elastic relaxation of Pt(111). The pileups around indents on Pt57.5Cu14.7Ni5.3P22.5 metallic glass are much more prominent than on Au(111) or Pt(111). This indicates that the plastic deformation of Au(111) and Pt(111) was accommodated over much longer distances than in the case of the metallic glass. This view is also supported by the observation of dislocation lines on Au(111) that extends hundreds of nanometers away from the indents. In the case of the Pt57.5Cu14.7Ni5.3P22.5 metallic glass, plastic flow appears to be closely confined around the indenting tip.

Figure 7 shows indentation curves recorded on Au(111), Pt(111), and Pt57.5Cu14.7Ni5.3P22.5 metallic glass. In the case of Au(111) and Pt(111), the force-penetration curves overlap with each other, demonstrating the good reproducibility of the method. For those two materials, also the indentation curves show clear pop-ins that are attributed to the activation of dislocations. For Pt57.5Cu14.7Ni5.3P22.5 metallic glass, the force-penetration curves do not show any of pop-in. In this case, the deformation appears to be continuous.

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Figure 6. Nc AFM topography images after AFM indentation measurements with the indicated normal force on Au(111), Pt(111), and Pt57.5Cu14.7Ni5.3P22.5 metallic glass.

For Au(111) two series of indentation measurements with the same maximum load values Fn = 7.2 μN but with different tips are shown. Within both series, the shape and size of the remaining indents are very similar. For Pt(111) and Pt57.5Cu14.7Ni5.3P22.5 metallic glass, remaining indents were only observed for maximum load values Fn > 0.8 μN. For these two materials, the projected area of the indents is observed to increase with the maximal load.

Figure 5. Nc AFM topography images of (a and d) Au(111), (b and e) Pt(111), and (c and f) Pt57.5Cu14.7Ni5.3P22.5 metallic glass surfaces after AFM indentation; in (d–f) the projected area was masked and calculated to determine the hardness

In the case of Au(111), almost no pileup can be observed. In this case, clear dislocation can be identified around indents. In the case of Pt(111), small pileups can be observed. More importantly, above an indentation load Fn = 3 μN, the indent exhibits a chevron-like shape that was never observed on the two other samples and which attribute to anisotropic elastic relaxation of Pt(111). The pileups around indents on Pt57.5Cu14.7Ni5.3P22.5 metallic glass are much more prominent than on Au(111) or Pt(111). This indicates that the plastic deformation of Au(111) and Pt(111) was accommodated over much longer distances than in the case of the metallic glass. This view is also supported by the observation of dislocation lines on Au(111) that extends hundreds of nanometers away from the indents. In the case of the Pt57.5Cu14.7Ni5.3P22.5

Figure 7 shows indentation curves recorded on Au(111), Pt(111), and Pt57.5Cu14.7Ni5.3P22.5 metallic glass. In the case of Au(111) and Pt(111), the force-penetration curves overlap with each other, demonstrating the good reproducibility of the method. For those two materials, also the indentation curves show clear pop-ins that are attributed to the activation of dislocations. For Pt57.5Cu14.7Ni5.3P22.5 metallic glass, the force-penetration curves do not show any of

metallic glass, plastic flow appears to be closely confined around the indenting tip.

pop-in. In this case, the deformation appears to be continuous.

values of each material.

34 Contact and Fracture Mechanics

Figure 8 shows the load dependence of the projected area Ap for Pt(111) and Pt57.5Cu14.7 Ni5.3P22.5 metallic glass. The projected area Ap of indents is found to be much smaller for Pt57.5Cu14.7Ni5.3P22.5 metallic glass than for Pt(111). Further, we used the load dependence of Ap to calculate the hardness of Pt(111) and Pt57.5Cu14.7Ni5.3P22.5 metallic glass, according to dAp/dFn = 1/H. For Pt(111), we obtained H = 1.14 � 0.09 GPa. For Pt57.5Cu14.7Ni5.3P22.5 metallic glass, we obtained H = 7.3 � 2.4 GPa. These values are larger than the measured ones by nanoindentation with a Berkovich diamond tip (see Ref. [15] for more details). This can be explained by tip convolution during nc AFM imaging that results in an underestimation of the projected area.

In the case of the indentation on Au(111), shown in Figure 5, we found Ap = 4703.52 nm2 , corresponding to Fn = 7.2 μN. Using the classical expression for the determination of hardness, we obtained <sup>H</sup> <sup>¼</sup> Fn Ap ¼ 1:53 GPa. Alternatively, the shape of the AFM tip used to indent Au(111) was estimated from the noncontact AFM images shown in Figure 6 with the free SPM data analysis software Gwyddion (Figure 9). The half-opening angle of the as-reconstructed indenter was determined to be α = 67.21�. The hardness was then calculated according to <sup>H</sup> <sup>¼</sup> Fn <sup>3</sup> ffiffi 3 <sup>p</sup> tan2α δð Þ max�δel <sup>2</sup> <sup>¼</sup> <sup>1</sup>:46 GPa [25], where <sup>δ</sup>max is the maximal penetration depth in Figure 7(a), and δel was taken as the penetration depth at the first pop-in event in Figure 7(c). Both hardness calculations deliver virtually the same value: HAu(111) = 1.5 GPa.

Figure 7. (a–c) Indentation curves and (d–e) magnification in the low load regime recorded on (a and d) Au(111), (b and e) Pt(111), and (c and f) Pt57.5Cu14.7Ni5.3P22.5 metallic glass.

4. AFM scratch test for friction and wear measurements

Figure 9. Estimated tip shape of the indenter used on Au(111).

<sup>2</sup> Cl h <sup>L</sup> SVl.

according to Fn <sup>¼</sup> CnSVn and Fl <sup>¼</sup> <sup>3</sup>

Wear and friction experiments were performed on Au(111) at room temperature and in ambient conditions (T = 293 K, RH = 40%) by friction force microscopy (FFM) [26] with diamondcoated silicon cantilever (CDT-NCLR, manufactured by NanoSensors, Switzerland). The normal and lateral stiffnesses of the cantilevers, Cn and Cl, were determined from the geometrical beam theory; for the cantilever used on Au(111), we found Cn = 50 N/m and Cl = 6954 N/m. The sensitivity of the photodiode S was obtained by recording a force-distance curve on a noncompliant surface and fitting its repulsive part with a linear function. The normal and lateral forces were calculated from the vertical and lateral voltages of the photodiode, Vn and Vl,

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Wear and friction measurements consisted in reciprocal sliding over the same area As = 2.5 � 2.5 μm<sup>2</sup> successively scanned over a load range Fn = 20–4600 nN. The topography and the lateral force were recorded during the forward and backward cantilever motion along the fast-scan direction (v = 10 μm/s). Amplitude-modulated noncontact AFM topography images of the area subjected to tribological testing were recorded before and after measurements and compared to extract the wear volume by integration. Topographical changes during tribological testing were analyzed by correlating successively recorded topography images with the initial topography image recorded at the lowest load (Fn = 20 nN). Thereby, we used the corrcoeff function of the MATLAB software package to extract a correlation factor R. The slopes of the R(Fn)-plot were further used to identify the transitions between wear mechanisms. Friction force images were calculated from the lateral force signals recorded in the forward and backward direction

Figure 8. Indentation load dependence of the projected area Ap for (left) Pt(111) and (right) Pt57.5Cu14.7Ni5.3P22.5 metallic glass.

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Figure 9. Estimated tip shape of the indenter used on Au(111).

Figure 8. Indentation load dependence of the projected area Ap for (left) Pt(111) and (right) Pt57.5Cu14.7Ni5.3P22.5 metallic

Figure 7. (a–c) Indentation curves and (d–e) magnification in the low load regime recorded on (a and d) Au(111), (b and e)

Pt(111), and (c and f) Pt57.5Cu14.7Ni5.3P22.5 metallic glass.

36 Contact and Fracture Mechanics

glass.

### 4. AFM scratch test for friction and wear measurements

Wear and friction experiments were performed on Au(111) at room temperature and in ambient conditions (T = 293 K, RH = 40%) by friction force microscopy (FFM) [26] with diamondcoated silicon cantilever (CDT-NCLR, manufactured by NanoSensors, Switzerland). The normal and lateral stiffnesses of the cantilevers, Cn and Cl, were determined from the geometrical beam theory; for the cantilever used on Au(111), we found Cn = 50 N/m and Cl = 6954 N/m. The sensitivity of the photodiode S was obtained by recording a force-distance curve on a noncompliant surface and fitting its repulsive part with a linear function. The normal and lateral forces were calculated from the vertical and lateral voltages of the photodiode, Vn and Vl, according to Fn <sup>¼</sup> CnSVn and Fl <sup>¼</sup> <sup>3</sup> <sup>2</sup> Cl h <sup>L</sup> SVl.

Wear and friction measurements consisted in reciprocal sliding over the same area As = 2.5 � 2.5 μm<sup>2</sup> successively scanned over a load range Fn = 20–4600 nN. The topography and the lateral force were recorded during the forward and backward cantilever motion along the fast-scan direction (v = 10 μm/s). Amplitude-modulated noncontact AFM topography images of the area subjected to tribological testing were recorded before and after measurements and compared to extract the wear volume by integration. Topographical changes during tribological testing were analyzed by correlating successively recorded topography images with the initial topography image recorded at the lowest load (Fn = 20 nN). Thereby, we used the corrcoeff function of the MATLAB software package to extract a correlation factor R. The slopes of the R(Fn)-plot were further used to identify the transitions between wear mechanisms. Friction force images were calculated from the lateral force signals recorded in the forward and backward direction according to Ff <sup>¼</sup> Fl,fwd�Fl, bwd <sup>2</sup> . In the case of Au(111), the probability distributions were calculated fitted with a Gaussian curve to provide the mean value and the standard deviation (see Figure 10). For the same ranges of normal force values as identified from the R(Fn)-plots, coefficients of friction (COF) were determined from the linear slopes COF <sup>¼</sup> dFf dFn .

Figure 11 shows topography and friction force images simultaneously recorded on Au(111). Plastic deformation was observed to start at a load value Fn = 129 nN as indicated by the occurrence of dislocation lines in the corresponding topography image. Increasing the load to Fn = 259 nN resulted in an increased number of dislocation. In this load range, surface

> topography features such as atomic steps remained clearly visible. This indicates in this load range that the sliding contact was rather governed by shearing and not plowing. In the load range Fn = 517–1295 nN, atomic steps were no longer observable, and a ripple structure was developed. In this load range, the mechanisms governing the sliding contact are considered to have a transition from shearing to plowing. In the range of the highest load values, Fn = 1942– 4531 nN, pileups at the left and right side of the topography images became clearly observable. In this case, the governing mechanism was plowing. The three load ranges indicated above are illustrated in the R(Fn)-plot, each of them being characterized by a different slope of decrease

> Figure 12. (a) Cross correlation factor R between the initial topography image in Figure 11(a) and the successive topography images recorded at the indicated load, (b) load dependence of friction, and (c) topography images of the area

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dFn (see Figure 12(a)).

Figure 11 also shows the friction force images corresponding to the topography measurements shown in the same figure. These images were further analyzed to determine the average friction force and its standard deviation (see above). Figure 12(b) shows the friction force Ff plotted against the normal force Fn. In the same figure, the error bars correspond to the standard deviation of the measurements. In agreement with the different load regimes determined in Figure 12(a), the Ff(Fn)-plot can be divided into different load ranges which corresponds a

the area tested by contact AFM shown in Figure 11. The scratched surface exhibits pileups at the edges of the area scanned in contact. The corresponding wear volume was determined by integration of the height signal using the MATLAB software package. We calculated a wear

Sliding friction experiments on Au(111) were performed in ambient conditions (T = 293 K, RH = 40%) by FFM with a soft gold-coated AFM cantilevers of the type CONTSC-Au

Prior to friction experiments, the sensitivity of the AFM photodiode S was determined following the same methods as above. The bending and torsion stiffnesses Cn and Cl of the cantilever were determined by thermal noise analysis. The cantilever stiffnesses are listed in Table 1.

volume Vw = 0.0811 μm3 corresponding to an average wear depth of δ<sup>w</sup> = 13 nm.

. Figure 12(c) shows a noncontact AFM topography image of

with increasing normal load <sup>Ρ</sup> <sup>¼</sup> dR

subjected to tribological tests (see Figure 2(a)).

coefficient of friction COF <sup>¼</sup> dFf

dFn

5. Atomic-scale sliding friction measurements

(manufactured by NanoSensors, Switzerland).

Figure 10. Probability distributions of friction force values measured at different normal force values. Each probability density distribution was fitted with a Gaussian function (red lines) to extract the mean friction force values and the corresponding standard deviation values.

Figure 11. (a) Topography and (b) friction force images successively recorded on the same area of an Au(111) surface at the indicated loads.

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according to Ff <sup>¼</sup> Fl,fwd�Fl, bwd

38 Contact and Fracture Mechanics

corresponding standard deviation values.

the indicated loads.

<sup>2</sup> . In the case of Au(111), the probability distributions were calcu-

dFn .

lated fitted with a Gaussian curve to provide the mean value and the standard deviation (see Figure 10). For the same ranges of normal force values as identified from the R(Fn)-plots,

Figure 11 shows topography and friction force images simultaneously recorded on Au(111). Plastic deformation was observed to start at a load value Fn = 129 nN as indicated by the occurrence of dislocation lines in the corresponding topography image. Increasing the load to Fn = 259 nN resulted in an increased number of dislocation. In this load range, surface

Figure 10. Probability distributions of friction force values measured at different normal force values. Each probability density distribution was fitted with a Gaussian function (red lines) to extract the mean friction force values and the

Figure 11. (a) Topography and (b) friction force images successively recorded on the same area of an Au(111) surface at

coefficients of friction (COF) were determined from the linear slopes COF <sup>¼</sup> dFf

Figure 12. (a) Cross correlation factor R between the initial topography image in Figure 11(a) and the successive topography images recorded at the indicated load, (b) load dependence of friction, and (c) topography images of the area subjected to tribological tests (see Figure 2(a)).

topography features such as atomic steps remained clearly visible. This indicates in this load range that the sliding contact was rather governed by shearing and not plowing. In the load range Fn = 517–1295 nN, atomic steps were no longer observable, and a ripple structure was developed. In this load range, the mechanisms governing the sliding contact are considered to have a transition from shearing to plowing. In the range of the highest load values, Fn = 1942– 4531 nN, pileups at the left and right side of the topography images became clearly observable. In this case, the governing mechanism was plowing. The three load ranges indicated above are illustrated in the R(Fn)-plot, each of them being characterized by a different slope of decrease with increasing normal load <sup>Ρ</sup> <sup>¼</sup> dR dFn (see Figure 12(a)).

Figure 11 also shows the friction force images corresponding to the topography measurements shown in the same figure. These images were further analyzed to determine the average friction force and its standard deviation (see above). Figure 12(b) shows the friction force Ff plotted against the normal force Fn. In the same figure, the error bars correspond to the standard deviation of the measurements. In agreement with the different load regimes determined in Figure 12(a), the Ff(Fn)-plot can be divided into different load ranges which corresponds a coefficient of friction COF <sup>¼</sup> dFf dFn . Figure 12(c) shows a noncontact AFM topography image of the area tested by contact AFM shown in Figure 11. The scratched surface exhibits pileups at the edges of the area scanned in contact. The corresponding wear volume was determined by integration of the height signal using the MATLAB software package. We calculated a wear volume Vw = 0.0811 μm3 corresponding to an average wear depth of δ<sup>w</sup> = 13 nm.

#### 5. Atomic-scale sliding friction measurements

Sliding friction experiments on Au(111) were performed in ambient conditions (T = 293 K, RH = 40%) by FFM with a soft gold-coated AFM cantilevers of the type CONTSC-Au (manufactured by NanoSensors, Switzerland).

Prior to friction experiments, the sensitivity of the AFM photodiode S was determined following the same methods as above. The bending and torsion stiffnesses Cn and Cl of the cantilever were determined by thermal noise analysis. The cantilever stiffnesses are listed in Table 1.


In this work, the radius of curvature R of the AFM tip was determined by scanning electron microscopy (SEM) after the friction experiments (see Figure 2 and Table 1) using a Helios 600i DualBeam FIB-SEM manufactured by FEI, Netherlands. A value R ≈ 25 nm was found and used to fit the experimental Ff(Fn)-plots. In Figure 2, a circle with a radius of 25 nm is

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Atomic-scale stick-slip was observed and statistically analyzed. The analysis consisted in lineby-line calculation of the power spectral density (PSD) function of each recorded Fl,fwd images using the pburg function of the MATLAB software package. The calculated PSD functions corresponding to each line were averaged to provide a single PSD function out of one Fl,fwd image. This statistical analysis transforms a signal in real space into a one-dimensional reciprocal space (k-space) signal, from which characteristic wavelengths λ ¼ 2π=k can be

Figure 14 shows the load dependence of friction on Au(111) with an Au-coated tip. For this tribological couple, a shear strength value τ = 24.21 MPa and an adhesion force value Fad = 25.8 nN were calculated. Also, Figure 3 shows a typical FFM image and corresponding forward and backward traces that exhibit periodic atomic scale stick-slip. In the following, the averaged power spectrum density (PSD) functions of the friction signals recorded at different loads were evaluated (see Figure 15). The PSD function corresponding to a typical friction measurement on Au(111) with an Au-coated tip shows a peak at a wavenumber k = 21.36 rad/nm. Neither the position nor the amplitude of this peak was found to change upon increasing load, except for Fn = 10 nN, in which case two slightly less prominent peaks were observed at k = 20.11 rad/nm

Correspondingly, a characteristic wavelength λ<sup>2</sup> = 0.294 nm was calculated that well matches with the interatomic distance of Au in the [110] direction (a[110] = 288 pm). The small discrepancy arises from the numerical approximation of the PSD function. The peak in the PSD functions was also found to split into two equidistant peaks at Fn = 10 nN, with corresponding wavelength values λ<sup>3</sup> = 0.277 nm and λ<sup>1</sup> = 0.312 nm, respectively. These peaks may correspond to the herringbone reconstruction of the Au(111) surface and the resulting different tilt angles

Figure 14. (a) Load dependence of friction and corresponding fit with a function of the type Ff = τAc(Fn), where τ is the shear strength and the real contact area Ac is expressed according to the JKR model [29]; (b) typical FFM image and

of the fcc and hcp domains with respect to the unreconstructed surface [30].

(c) corresponding forward and backward traces exhibiting atomic scale stick-slip.

overlaid to demonstrate the validity of this value.

and k = 22.62 rad/nm (see Figure 15).

identified.

\*\*Estimated data from SEM measurements.

Table 1. Cantilever properties.

The friction experiments consisted in recording the lateral deflection signal of the AFM cantilever in both forward and backward directions of the x-scanner. The experiments consisted in scanning an area of 10 � 10 nm<sup>2</sup> with a normal load in the range Fn = 0–10 nN (Figure 13).

For each measurement, the friction force was calculated according to Ff <sup>¼</sup> Fl,fwd�Fl, bwd <sup>2</sup> , where Fl,fwd and Fl,bwd are the forward and backward images of the lateral force, respectively. Subsequently, the calculated friction force image was averaged line by line, and a corresponding error was calculated as the standard deviation from the mean value using the MATLAB software package. Moreover, the shear strength τ and the adhesion force Fad were calculated by fitting the Ffð Þ Fn -plot with the function Ff ¼ τAcð Þ Fn , where we consider τ to be constant and Ac(Fn) is the normal force-dependent real area of contact between surface and tip (see Ref. [27]). Based on the Johnson-Kendall-Roberts (JKR) theory, the real area of an adhesive contact between a spherical elastic body and the flat surface of an elastic body can

be expressed as Ac <sup>¼</sup> <sup>π</sup> <sup>R</sup> E∗ � �<sup>2</sup>=<sup>3</sup> ð Þþ Fn � Fad 2Fad þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Fadð Þþ Fn � Fad ð Þ 2Fad 2 � � q <sup>2</sup>=<sup>3</sup> , where Fad is the adhesion force between the two elastic bodies [28], R is the radius of the spherical body, <sup>E</sup><sup>∗</sup> <sup>¼</sup> <sup>1</sup>�ν<sup>2</sup> 1 <sup>E</sup><sup>1</sup> <sup>þ</sup> <sup>1</sup>�ν<sup>2</sup> 2 E2 h i�<sup>1</sup> is the reduced modulus of elasticity, and Ei and ν<sup>i</sup> are Young's moduli and Poisson's ratios of the two elastic bodies involved in the contact [29]. The resulting F<sup>2</sup>=<sup>3</sup> <sup>n</sup> dependence of the friction force has been experimentally verified in Refs. [6, 27]. The following values were used for Young's modulus and Poisson's ratio: EAu = 75 GPa and νAu = 0.44.

Figure 13. SEM image of the gold-coated AFM tip used friction measurements on Au(111).

In this work, the radius of curvature R of the AFM tip was determined by scanning electron microscopy (SEM) after the friction experiments (see Figure 2 and Table 1) using a Helios 600i DualBeam FIB-SEM manufactured by FEI, Netherlands. A value R ≈ 25 nm was found and used to fit the experimental Ff(Fn)-plots. In Figure 2, a circle with a radius of 25 nm is overlaid to demonstrate the validity of this value.

Atomic-scale stick-slip was observed and statistically analyzed. The analysis consisted in lineby-line calculation of the power spectral density (PSD) function of each recorded Fl,fwd images using the pburg function of the MATLAB software package. The calculated PSD functions corresponding to each line were averaged to provide a single PSD function out of one Fl,fwd image. This statistical analysis transforms a signal in real space into a one-dimensional reciprocal space (k-space) signal, from which characteristic wavelengths λ ¼ 2π=k can be identified.

The friction experiments consisted in recording the lateral deflection signal of the AFM cantilever in both forward and backward directions of the x-scanner. The experiments consisted in scanning an area of 10 � 10 nm<sup>2</sup> with a normal load in the range Fn = 0–10 nN (Figure 13).

Cantilever type Cn [N/m] Cl [N/m] L\* [mm] R\*\* [nm]

CONTSC-Au 0.685 136.24 225 25

Fl,fwd and Fl,bwd are the forward and backward images of the lateral force, respectively. Subsequently, the calculated friction force image was averaged line by line, and a corresponding error was calculated as the standard deviation from the mean value using the MATLAB software package. Moreover, the shear strength τ and the adhesion force Fad were calculated by fitting the Ffð Þ Fn -plot with the function Ff ¼ τAcð Þ Fn , where we consider τ to be constant and Ac(Fn) is the normal force-dependent real area of contact between surface and tip (see Ref. [27]). Based on the Johnson-Kendall-Roberts (JKR) theory, the real area of an adhesive contact between a spherical elastic body and the flat surface of an elastic body can

the adhesion force between the two elastic bodies [28], R is the radius of the spherical body,

and Poisson's ratios of the two elastic bodies involved in the contact [29]. The resulting F<sup>2</sup>=<sup>3</sup> <sup>n</sup> dependence of the friction force has been experimentally verified in Refs. [6, 27]. The following values were used for Young's modulus and Poisson's ratio: EAu = 75 GPa and νAu = 0.44.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4Fadð Þþ Fn � Fad ð Þ 2Fad

� � q <sup>2</sup>=<sup>3</sup>

is the reduced modulus of elasticity, and Ei and ν<sup>i</sup> are Young's moduli

2

<sup>2</sup> , where

, where Fad is

For each measurement, the friction force was calculated according to Ff <sup>¼</sup> Fl,fwd�Fl, bwd

ð Þþ Fn � Fad 2Fad þ

Figure 13. SEM image of the gold-coated AFM tip used friction measurements on Au(111).

be expressed as Ac <sup>¼</sup> <sup>π</sup> <sup>R</sup>

<sup>E</sup><sup>∗</sup> <sup>¼</sup> <sup>1</sup>�ν<sup>2</sup> 1 <sup>E</sup><sup>1</sup> <sup>þ</sup> <sup>1</sup>�ν<sup>2</sup> 2 E2 h i�<sup>1</sup>

\*

Manufacturer's data.

40 Contact and Fracture Mechanics

Table 1. Cantilever properties.

\*\*Estimated data from SEM measurements.

E∗ � �<sup>2</sup>=<sup>3</sup> Figure 14 shows the load dependence of friction on Au(111) with an Au-coated tip. For this tribological couple, a shear strength value τ = 24.21 MPa and an adhesion force value Fad = 25.8 nN were calculated. Also, Figure 3 shows a typical FFM image and corresponding forward and backward traces that exhibit periodic atomic scale stick-slip. In the following, the averaged power spectrum density (PSD) functions of the friction signals recorded at different loads were evaluated (see Figure 15). The PSD function corresponding to a typical friction measurement on Au(111) with an Au-coated tip shows a peak at a wavenumber k = 21.36 rad/nm. Neither the position nor the amplitude of this peak was found to change upon increasing load, except for Fn = 10 nN, in which case two slightly less prominent peaks were observed at k = 20.11 rad/nm and k = 22.62 rad/nm (see Figure 15).

Correspondingly, a characteristic wavelength λ<sup>2</sup> = 0.294 nm was calculated that well matches with the interatomic distance of Au in the [110] direction (a[110] = 288 pm). The small discrepancy arises from the numerical approximation of the PSD function. The peak in the PSD functions was also found to split into two equidistant peaks at Fn = 10 nN, with corresponding wavelength values λ<sup>3</sup> = 0.277 nm and λ<sup>1</sup> = 0.312 nm, respectively. These peaks may correspond to the herringbone reconstruction of the Au(111) surface and the resulting different tilt angles of the fcc and hcp domains with respect to the unreconstructed surface [30].

Figure 14. (a) Load dependence of friction and corresponding fit with a function of the type Ff = τAc(Fn), where τ is the shear strength and the real contact area Ac is expressed according to the JKR model [29]; (b) typical FFM image and (c) corresponding forward and backward traces exhibiting atomic scale stick-slip.

Author details

Address all correspondence to: arnaud.caron@koreatech.ac.kr

surfaces. Tribology Letters. 2010;39:19-24

metals. (Submitted to Acta Materialia)

22(42):425703-1-425703-9

044004-1-044004-4

135506

faces. Journal of Materials Research. 2013;28:1279-1288

KoreaTech—Korea University of Technology and Education, Republic of Korea

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Figure 15. (Left) Typical power spectral density function calculated from FFM measurements on Au(111) with an Au-coated tip and (right) load dependence of the power density function as a function of the normal force Fn.
