Royal Society Publishing

Micromechanics of friction: effects of nanometre-scale roughness

Qunyang Li, Kyung-Suk Kim


Nanometre-scale roughness on a solid surface has significant effects on friction, since intersurface forces operate predominantly within a nanometre-scale gap distance in frictional contact. To study the effects of nanometre-scale roughness, two novel atomic force microscope friction experiments were conducted, each using a gold surface sliding against a flat mica surface as the representative friction system. In one of the experiments, a pillar-shaped single nano-asperity of gold was used to measure the molecular-level frictional behaviour. The adhesive friction stress was measured to be 264 MPa and the molecular friction factor 0.0108 for a direct gold–mica contact. The nano-asperity was flattened in contact, although its hardness at this length scale is estimated to be 3.68 GPa. It was found that such a high pressure could be reached with the help of condensed water capillary forces. In the second experiment, a micrometre-scale asperity with nanometre-scale roughness exhibited a single-asperity-like response of friction. However, the apparent frictional stress, 40.5 MPa, fell well below the Hurtado–Kim model prediction of 208–245 MPa. In addition, the multiple nano-asperities were flattened during the frictional process, exhibiting load- and slip-history-dependent frictional behaviour.


1. Introduction

From the first paper published by Amontons in 1699 (see Dowson 1998) to recent observations on molecular-level tribology (Homola et al. 1990; Krim et al. 1991; Carpick et al. 1996a; Lantz et al. 1997), many aspects of friction still remain a mystery. As the mystery stems mostly from the complexity of multi-scale frictional processes, research activities have become vivacious to uncover the molecular origin of the processes and bridge the understandings of the phenomena at different length scales. With the advent of modern interrogating techniques, e.g. surface force apparatus (SFA; Tabor & Winterton 1969; Israelachvili & Tabor 1972), atomic force microscope (AFM; Binnig et al. 1986; Mate et al. 1987) and quartz-crystal microbalance (Krim & Widom 1988), coupled with the fast advancement of molecular-level simulations, more and more details about the molecular basis of friction have been revealed. These developments have been well reviewed by Carpick & Salmeron (1997), Dedkov (2000), Gnecco et al. (2001) and Gao et al. (2004).

From the molecular to the macroscopic level, the frictional responses exhibit distinct characteristics at different length scales. Three typical configurational characteristics of frictional sliding at different length scales are schematically depicted as types I, II and III in figure 1a, representing the frictional sliding of a molecular contact, a single-asperity contact and a multi-asperity contact, respectively. Corresponding characteristic responses of the frictional forces as functions of the normal loads are sketched in figure 1b as curves I, II and III on the friction versus load plane. In §2, details of each response together with four multi-scale mechanics issues in friction will be reviewed. In addition, bridging between the responses at different scales will also be highlighted.

Figure 1

(a) Three configurational characteristics of frictional sliding at different length scales. (b) Corresponding response characteristics of the frictional force F as functions of the normal load P: (I) a molecular contact, (II) a single-asperity contact, (III) a multi-asperity contact; α is the molecular friction factor, Embedded Image where τ0 and A are the adhesive friction stress and the contact area, respectively, Pa the pull-off force and μ the Amontons' friction coefficient.

2. Overview on multi-scale mechanics issues in friction

(a) Friction at the molecular level (type I)

For molecularly intimate contact, Bowden & Tabor (1954) suggested that friction was due to the shear strength of cold-welded junctions and proportional to the true contact area F=τfA, where τf and A were the intrinsic friction stress and the true contact area. They also suggested that τf would be constant. Adopting the notion of pressure-dependent shear strength of plastic flow (Bridgman 1937), the intrinsic friction stress was later considered to be pressure sensitive by Briscoe & Tabor (1978) for friction with a third-material interlayer. Subsequently, Sorensen et al. (1996) observed the similar pressure sensitivity in their molecular dynamics (MD) simulations for direct solid–solid contact. In these works, the molecular friction law was described by τf=τ0+αp, where τ0 is the adhesive friction stress of the interface, α the molecular friction factor and p the normal pressure applied at the contact interface. Pietrement & Troyon (2001) called α a friction factor, and we will call it a molecular friction factor in this paper. According to the molecular friction law, the frictional force F increases linearly with the normal load P for a fixed true contact area A, which is shown as solid curve I in figure 1b where F0=τ0A.

In general, the adhesive friction stress τ0 depends on the adhesive interaction as well as the surface registry of the contacting surfaces. MD simulations with ideal molecular interaction potentials (e.g. harmonic potential for wall atoms in the solids and Lennard-Jones potential across the interface used by Müser & Robbins 2000) suggest that the friction stress vanishes for incommensurate large-area contact. However, abundant experimental observations (e.g. see collection of data in the reviews by Carpick & Salmeron (1997), Dedkov (2000) and Gnecco et al. (2001)) indicate that the friction stress does not always vanish for incommensurate adhesive large-area contacts. Possible atomic-level mechanisms responsible for the discrepancy are under extensive investigation (e.g. Ashkenazy et al. 2001; Friedel & Gennes 2006; Merkle & Marks 2007a,b).

Similar to the adhesive friction stress, the molecular friction factor α is also sensitive to lattice configurations. For example, in an MD simulation (Harrison et al. 1992) of two diamond (111) surfaces sliding over each other, the resultant friction coefficient was approximately 0.52 for a slip along the well-aligned direction Embedded Image while the value reduced to only 0.02 for a slip along the poorly aligned direction Embedded Image. In contrast to a number of MD simulations (e.g. Harrison et al. 1992; Müser & Robbins 2000), only a limited number of experimental estimates of α have been reported for direct solid–solid contacts (Pietrement & Troyon (2001); also see the review by Carpick & Salmeron 1997).

So far, the molecular friction factor α could have been measured only with great uncertainties owing to two major difficulties. First, the contact area of a commonly used paraboloidal probe is always enlarged as the normal load increases. The large uncertainty in the contact area estimation makes it quite difficult to determine an accurate value of α. Second, the value of α is usually very small and the accuracy of the frictional force measurement has not been high enough to unveil the real value of α.

In this work, for the first time, the adhesive friction stress τ0 as well as the molecular friction factor α between gold and mica was measured directly and accurately. The results provided direct evidence of a strong frictional resistance with small pressure sensitivity between two incommensurate surfaces.

(b) Friction of a single-asperity contact (type II)

At a larger scale, the frictional response of a single-asperity contact depends on the size and the shear strength of the adhesive contact junction. The contact size is mostly determined by the elastic response of adhesive contact, while the shear strength of the junction is primarily sensitive to the molecular friction law that we discussed above and slip instability mechanisms. The elastic response of an adhesive contact has been well characterized by Johnson, Kendall & Roberts (JKR; Johnson et al. 1971), Derjaguin, Muller & Toporov (DMT; Derjaguin et al. 1975) and Maugis (1992) models, depending on the Maugis parameter (λ in Maugis 1992). A typical frictional response of a single-asperity contact is shown as the dot-dashed curve II in figure 1b, where Pa is the critical pull-off load. With the wide applications of AFM and SFA, a considerable amount of experimental efforts have been carried out to investigate the single-asperity friction (see Carpick et al. 1996a; Lantz et al. 1997; and also the reviews by Carpick & Salmeron 1997, Dedkov 2000; Gnecco et al. 2001). In one of the pioneering works, Carpick et al. (1996a) measured the friction of a platinum-coated AFM tip on a mica surface in an ultra-high vacuum (UHV). The friction was then used to estimate an effective friction stress, assuming that the friction stress is independent of normal contact pressure, and the adhesive contact area follows Maugis model. The effective friction stress, in the range of 270–910 MPa, was believed to be attributed to the adhesive friction stress τ0.

The interpretation of the data by Carpick et al. (1996a) plausibly implied that the adhesive friction stress was insensitive to the contact-size variation. The contact area in their experiment was smaller than a critical value, approximately 14 nm in radius, under which elastic deformation of the contact shearing could not accommodate dislocation-assisted slip along the interface. In other words, below the critical contact size, all the atoms in the contact area slipped concurrently with a constant friction stress. For a finite contact size larger than the critical value, Hurtado & Kim (1999a) predicted that the stress concentration at the contact edge would induce instability of non-uniform frictional slip. At the small length scale, the instability implies interfacial dislocation emission. Then, the instability would lead to the dependence of the nominal friction stress (frictional force divided by the contact area) on the contact size. However, Homola et al. (1990) observed that the friction stress (25 MPa) between mica and mica measured by an SFA was independent of the contact size at a large scale, approximately 30–80 μm in contact radii (Hurtado & Kim (1999a) cited this range mistakenly by 40–250 μm). This observation prompted Hurtado & Kim (1999b) to make a model of multiple-dislocation-cooperated (MDC) slip for frictional processes with large contact size. Indeed, the MDC slip model provided a constant friction stress at the large scale. However, the constant friction stress of 25 MPa observed in the SFA experiment might be mainly caused by the water-molecular interlayer, not by the dislocation mobility. Although the absolute value of the dry friction stress might not be 25 MPa, the HK model (Hurtado & Kim 1999b) still holds in its formalism for a properly chosen level of effective Peierls stress. In spite of the model predictions (Hurtado & Kim 1999a,b) for the size-scale dependence in single-asperity friction, there have been no experimental data available in the intermediate contact-size scale.

In the present study, the scale dependence of the friction stress was measured experimentally in the intermediate contact-size range, and the results showed that the friction stress was indeed size dependent; however, the dependence was quantitatively different from the prediction of Hurtado & Kim (1999a) when nanometre-scale roughness and/or a third-material interlayer were involved.

(c) Friction of a multi-asperity contact (type III)

As length scale gets even larger, surface roughness becomes inevitable and the intimate contacts are established only between individual small asperities (Bowden & Tabor 1954). Regarding the roughness effect, new approaches are needed to bridge the frictional response from single-asperity contact to multi-asperity contact. For non-adhesive elastic contacts, Archard (1957) concluded in his paper that the average micro-contact state hardly changes whatever the load, only the number of micro-contacts varies, and the multiple micro-contacts would alter the nonlinear frictional response towards a linear response, in general. The linear frictional response is shown as the dotted curve III in figure 1b, where the slope μ is the macroscopic friction coefficient. The later studies (Greenwood & Williamson 1966; Greenwood & Tripp 1967, 1971) attributed the linear response of rough surface friction, as an integrated behaviour of nonlinear single-asperity responses, primarily to the statistical nature of the level-set area variation under the contact loading. Based on the Greenwood & Williamson (GW) type of statistical asperity description, Fuller & Tabor (1975) studied elastic rough surface adhesion using the JKR model. The GW approach was further applied to study rough surface adhesion and friction for elastic–plastic contact by Chang et al. (1988) using the DMT model and by Chowdhury & Ghosh (1994) using the JKR model. Recently, Adams et al. (2003) and Adams & Müftü (2005) incorporated size-dependent single-asperity friction (Hurtado & Kim 1999a,b) to develop a scale-dependent model for multi-asperity elastic contact and friction.

Although the adhesion can be considerably reduced depending on the surface roughness, the intersurface interactions can still be quite adhesive if the roughness is on the nanometre scale. It is not clear how the nanometre-scale roughness will affect the frictional response especially for curved-surface adhesive contact. In most of the theoretical models (Chang et al. 1988; Chowdhury & Ghosh 1994), macroscopic plasticity was adopted for the analysis of asperity deformation. However, it has been recognized that the mechanical properties of materials are strongly size-scale dependent (Wang et al. 2006; Ward 2007) and the classical continuum plasticity may not be applicable for nano-asperity contacts.

Macroscopically, it has long been noted that friction displayed a complicated dependence on sliding and load histories. These characteristics have been described by the empirical rate and state-variable-dependent constitutive formulations (Dieterich 1978; Rice & Ruina 1983; Ruina 1983; Prakash & Clifton 1992). However, the operative underlying micro-mechanisms are little understood and the physical interpretations of the state variables are still vague.

In the present study, we reported on the effects of nanometre-scale roughness on the frictional responses viewed from two different perspectives. One was the response of a micrometre-scale asperity with nanometre-scale roughness, which was called micro pseudo single asperity (M-PSA). The other was the response of an isolated nano-asperity surrounded by low-height neighbouring bump asperities, which was named nano pseudo single asperity (N-PSA). We carried out a meticulous N-PSA friction experiment to measure the intrinsic frictional properties and to investigate the detailed mechanisms of the frictional evolution in the sliding processes. In addition, an experimental study was performed to investigate the roughness effects of M-PSA in assessing the size dependence of the nominal friction stress. A detailed model study of the roughness effects on the M-PSA frictional response is to be presented elsewhere.

(d) Effects of a water layer on friction

In addition to the three size-scale characteristics discussed previously, it has been observed that a water layer between the contacting surfaces can play an important role in determining the frictional response at multiple length scales. For example, Dieterich & Conrad (1984) reported that a small amount of humidity could alter the macroscopic frictional responses appreciably. In general, the water layer can affect the frictional behaviour in two aspects. When the contact pressure is small, the interlayer is kept between the contacting surfaces and the water behaves like a lubricant (Homola et al. 1989; Israelachvili 1992; Pertsin & Grunze 2006). This lubrication effect alters the frictional response at the molecular level significantly. However, if the contact pressure is high enough, the interlayer is squeezed out and the two solid surfaces come into direct contact, losing the lubrication effect. If a water layer is condensed from the air, a meniscus is formed surrounding the contact junction (see Xu et al. 1998) and exerts a Laplace pressure on the contacting pair. This capillary effect changes the adhesion profoundly (e.g. McFarlane & Tabor 1950; Christenson 1988; Israelachvili 1991). The capillary effect depends on the humidity and wettability of the contacting surfaces; it can contribute considerably to the frictional response at the single-asperity level. We will show in this paper that with the assistance of the capillary adhesion between gold and mica surfaces, the true contact pressure of the N-PSA is significantly amplified to a few gigapascals to plastically deform the nanometre-scale gold asperities. As a consequence, the flattening of nano-asperities leads to roughness evolution, causing the load/history-dependent responses of both N-PSA and M-PSA frictions.

In this work, the effects of the nanometre-scale roughness on the frictional behaviours of the M-PSA and N-PSA are investigated through an experimental study of friction between gold-coated microspheres and a mica surface. The materials and methods of the experiments are described in §3 and the experimental results are summarized in §4. Sections 5 and 6 are devoted to the analysis of the experimental results and the discussion of new understandings with regard to the aforementioned four multi-scale mechanics issues. Finally, a conclusion is presented in §7.

3. Experimental procedures

For both M-PSA and N-PSA friction experiments, the lateral force microscopy (LFM) mode of an AFM (AutoCP, Park Scientific Instrument) was used to slide a gold-coated probe on a mica substrate in 30% humidity air as shown in figure 2a. The probe, a borosilicate glass microsphere, was attached to an AFM cantilever by an epoxy resin (Devcon 2-ton Epoxy). The microspheres purchased from Duke Scientific Corporation had a 14.5 μm nominal diameter, and the AFM cantilever from Veeco Corporation had a nominal spring constant of 17 N m−1. After the microsphere was attached to the cantilever, a 5 nm titanium adhesion layer followed by a 50 nm gold layer was deposited by electron-beam evaporation on the surfaces of the probe assembly in vacuum at room temperature.

Figure 2

(a) Schematics of M-PSA and N-PSA friction experiments. (b) Bottom view of the AFM scanning configuration. (c) Typical friction loop from an LFM measurement: calibrated lateral force versus scan distance; the half width of the loop w is the friction. (d) Reverse imaging schematic.

The friction experiments were carried out by scanning a 30×30 nm2 area on a freshly cleaved mica surface with the probe. The scan area is schematically depicted as a dashed-line square with four corners at A11, A1n, An1 and Ann in figure 2b. The LFM data were taken within 512 round-trip raster lines (Ak1 to Akn, k=1, 2, …, 512). On each raster line, the lateral force signal was recorded digitally as a function of scan distance also at 512 points, i.e. n=512. During each raster scan, the normal load was held fixed. One round-trip raster provided a friction loop on the plane of lateral force versus scan distance. A typical friction loop of an M-PSA friction experiment is shown in figure 2c. The half width w of the friction loop is taken to be the frictional force under the applied normal load. The normal load was varied every 20 friction loops, keeping the normal load constant in each set of the 20 loops. Average values of each set were used to obtain the relationship between the frictional force and the normal load. The scan rate of all the friction tests was 90 nm s−1.

For the tests, the normal force constant of the AFM cantilever was calibrated with the Sader's method (Sader et al. 1999), by deflecting the cantilever against another rectangular cantilever of known stiffness and against a hard surface. The lateral force constant was calibrated with the newly developed diamagnetic lateral force calibrator (Li et al. 2006). The experiments were carried out in air where the local temperature and relative humidity near the specimen were measured to be 28±2°C and 30%, respectively.

Surface coatings of the probes with two different topographies were prepared to study the effects of nanometre-scale roughness on the frictional behaviours. One coating was the M-PSA and the other was the N-PSA, as previously mentioned. For the M-PSA tests, the gold surface coating on the microsphere was used as deposited. The topography of the as-deposited gold layer will be shown in §4 (figure 3b). For the N-PSA test, the surface of the microsphere was modified by indenting the surface with a sharp silicon reverse tip (figure 2d) with a radius of curvature less than 10 nm. The local indentation generated a crater that pushed out a high central-spike asperity surrounded by several slightly lower bump asperities, which will be shown in detail in §4 (figure 4b). The topographic images of the M-PSA and N-PSA surfaces were collected by a reverse AFM imaging technique, both before and after the tests. A schematic of the reverse AFM imaging is shown in figure 2d. For the imaging, the gold-coated probe was scanned over a grating of sharp tip arrays (TGT01 silicon grating, MircoMasch). Owing to the sharpness of the tips (typical radii less than 10 nm), a three-dimensional profile of the microsphere itself could be traced. Non-contact mode AFM was employed for the reverse imaging in order to minimize possible damage and contamination of the microsphere.

Figure 3

(a) Frictional force versus normal load in the M-PSA experiment: increasing (squares, experiment; solid line, Maugis-BT model) and decreasing (triangles, experiment; dashed line, Maugis-BT model) normal load processes, respectively, with best fits of the experimental data using the Maugis-BT model. (b) Three-dimensional AFM topographies of the gold-coated microsphere (i) before and (ii) after the M-PSA friction, line profiles across the top of the microsphere (iii) before and (iv) after the test.

Figure 4

(a) Frictional force versus normal load for the N-PSA test using a modified probe; test sequence: A0→Ap→As⇒B0→Bp→Bs⇒C0→Cp→Cs⇒ scan for 1024 raster lines under a low load ⇒D0→Dp→Df⇒ high compressive load approximately 10 μN⇒E0→Ep→Ef. (b(i)(ii)) Three-dimensional topographic images of the modified gold-coated microsphere around the contact region and (b(iii)(iv)) the corresponding line profiles across the middle section of the N-PSA before and after the friction test.

4. Experimental results

(a) M-PSA frictional response

Figure 3a shows a load-history-dependent response in the frictional force versus normal load relationship observed in the M-PSA friction experiment. Each data point in figure 3a represents an average value of 20 measurements, as mentioned previously. The square data points show the variation of the frictional force corresponding to the increasing normal load from approximately −2000 up to 2700 nN. The triangular points display the variation of the frictional force in line with the subsequent decreasing normal load. The M-PSA friction cycle clearly exhibits history-dependent apparent friction strengthening, i.e. the frictional resistance at the same normal load becomes higher once the probe experienced high normal loads and/or excessive apparent frictional slips. The trend shown in figure 3a is opposite to that shown for the nanometre-scale single-asperity friction weakening between a platinum tip and mica in UHV. Carpick et al. (1996a) observed that the friction stress drifted from 910 down to 270 MPa continuously when the platinum tip experienced successive frictional sliding under various normal loads.

In addition to the friction strengthening, the M-PSA friction shows a significant adhesion effect, exhibiting frictional resistance under negative external normal loads. As the normal load P varies from −2 to +3 μN, the friction increases from 0.3 to 1.1 μN, approximately. The strong adhesion is considered to be mainly caused by the capillary effect of the water meniscuses condensed from the 30% humidity air. Unlike Archard's (1957) prediction for non-adhesive multi-asperity friction, the M-PSA with nanometre-scale roughness still exhibits a nonlinear response similar to that of single-asperity friction. The response can be fitted with Maugis' model (1992) coupled with the constant friction stress conjecture of Bowden & Tabor's (1954), shown as solid and dashed lines in figure 3a for increasing and decreasing normal loads, respectively.

Changes of the nanometre-scale topography of the gold surface were measured by reverse AFM imaging before and after the friction test, as described in §3. Figure 3b displays the three-dimensional AFM topographic images of the gold surface in figure 3b(i)(ii) and the corresponding line profiles in figure 3b(iii)(iv) before and after the friction test. As shown in the upper-left frame of figure 3b, the gold film fabricated by electron-beam vapour deposition at room temperature exhibits a columnar morphology with a root-mean-square (r.m.s.) roughness of 2.00 nm in an area of 1×1 μm2 around the top of the microsphere. The radius of the microsphere is best fitted to be 7.95 μm from the line profile in figure 3b(iii). Clearly, the surface topography after the friction tests, shown in the figure 3b(ii), indicates flattening of the nano-asperities within a diameter of approximately 300 nm around the contact region of the microsphere.

(b) N-PSA frictional responses

A series of the N-PSA frictional responses are presented in figure 4a. The loading procedures are summarized as follows. As the normal load increased from a slightly positive value right after the probe engaged at A0, to 2100 nN at Ap in figure 4a, the friction varied gradually from 50 to 300 nN, similar to the frictional response of a single-asperity contact with a nanometre-scale sharp tip (Carpick et al. 1996a). However, the friction did not change much, while the normal load decreased subsequently from 2100 nN at Ap down to the pull-off (P=−400 nN) point at As, except for the small range where the normal load is very close to the pull-off value. A similar series of tests were carried out again, scanning a new area of the mica surface with the same probe. In these tests, the friction varied from B0 up to Bp, when the normal load increased. The variation of friction was smaller than that of the A-series test. Upon decreasing the normal load, the variation of friction became more negligible than that observed in the A series. However, the pull-off force changed notably from −400 nN down to −1050 nN at Bs. The trend continued to the third C-series test, with a larger pull-off force (−1600 nN) at Cs. After the C-series test, the probe was allowed to rub the mica surface continuously for 1024 raster scans at approximately 100 nN normal load before the D-series test was conducted. In the D series Embedded Image, the friction and its variation increased, when compared with those of C-series test, and the history dependence of the frictional response became pronounced again. Right after the D-series test, approximately 10 μN normal contact load was intentionally applied between the gold and the mica surfaces. Then, the probe was subsequently used in the E-series friction test Embedded Image, which showed a similar adhesive frictional response of a single-asperity contact, close to the previous M-PSA response.

To sum up, we observed the following distinct N-PSA frictional characteristics of gold nano-asperities sliding on a mica surface. (i) The frictional response depended highly on the history of the loading involved in the friction tests, i.e. the history of the external normal pressure and the net slip distance. (ii) Variation of friction was hardly sensitive to the normal load for A, B and C series during the unloading process. But, at the same time, the pull-off forces changed drastically for these tests, which indicated that the variations of apparent adhesion and friction were decoupled. (iii) A number of frictional scans under a low external normal load made the frictional response depart from the flat C-series mode to the D-series mode, which was sensitive to the normal load again. (iv) An application of a large contact pressure altered the frictional response dramatically to the E-series mode of an M-PSA type of frictional response.

The three-dimensional AFM topographic image of the modified surface of the N-PSA, taken before the tests, is shown in the figure 4b(i) along with a line scan in figure 4b(iii). The topographic image of the final flattened state of the N-PSA and a line profile, taken after the friction test, are shown in figure 4b(ii)(iv), respectively. The sharp contrast between the three-dimensional topographic images in figure 4b indicates that the asperities were flattened substantially during the friction tests. The flattening is believed to be responsible for the evolutions of adhesion and friction.

5. Analysis of the frictional responses of M-PSA and N-PSA

(a) Fitting of the M-PSA response with the Maugis-BT model

The friction data shown in figure 3a for both increasing (squares) and decreasing (triangles) normal loads are best fitted by the solid and dashed lines, respectively, with the Maugis-BT model. The fitting was made within the framework of Maugis' (1992) single-asperity adhesive contact model assuming that the friction stress was constant for each dataset. Once the friction stress τf is assumed to be constant, τf=τ0, the contact radius a can be expressed in terms of the frictional force F as Embedded Image. Maugis' model provides a set of implicit but closed-form expressions (Maugis 1992), for the relationship between the normal load P and the frictional force F in terms of three parameters, i.e. the cohesive-zone strength σ0, the critical interaction distance q and the friction stress τ0. The expressions can be reduced to a single functional form as Embedded Image, where E* and R represent the effective elastic constant and the asperity radius, respectively. Using the functional form, we define an error function asEmbedded Image(5.1)and minimize the error function to get the best-fit parameter values of σ0, q and τ0. The superscript ‘exp’ in equation (5.1) stands for the experimentally measured values, and m the number of data points, i.e. 20 and 19 for the present two sets, respectively. The best-fit values of σ0, q, τ0 and λ are obtained to be 2.3 MPa, 22.3 nm, 40.5 MPa and 0.013 for the solid curve of increasing normal load and 2.0 MPa, 33.1 nm, 40.8 MPa and 0.010 for the dashed curve of decreasing normal load. This is the Maugis-BT fit mentioned previously.

(b) Morphological characteristics of N-PSA

The three-dimensional topographic image of the N-PSA surface is examined by a sequential virtual sectioning starting from its peak. Variations of the area and the second polar moment, I2, of the cross-section are plotted in figure 5a,b, respectively, as we increase the sectioning depth. For simplicity, the N-PSA contact will be idealized as an axisymmetric problem in the following analysis to understand the underlying mechanisms. A schematic of the effective radial profile of the N-PSA is shown in figure 5c, where the asperity is depicted as a central spike sticking out of axisymmetric wavy undulations. From figure 5b, the initial height of the central spike, h0, measured from its immediate base is determined to be 1.0 nm. From the cross-section data shown in figure 5a, one can see that the central spike has a pillar-like shape with a slightly rounded end. The root of the central spike on the base is identified as the first transition point of the area and the second polar moment variations in figure 5a,b. The second polar moment of the cross-section increases abruptly, exhibiting the second transition in figure 5b, as soon as the sectioning plane begins to pass the peak of the first neighbouring asperity ring bump. Therefore, the second transition point, h1=2.9 nm, is identified as the height difference between the peak of the central spike and the peak of the first neighbouring asperity ring bump. After the loading cycle A in the N-PSA experiment, the central spike is believed to be plastically flattened by approximately 1 atomic layer (0.2 nm) to form a flat-end pillar, whose radius a is determined to be 19.4 nm from the area profile in figure 5a. For quantitative modelling, the effective radial profile of the N-PSA around the first ring bump, g(r) for r>a, is assumed to be approximately a parabola superposed by a decaying sinusoidal undulation which is described asEmbedded Image(5.2)It is noted that g(r) is measured relative to the top of the central spike in the undeformed configuration as depicted in figure 5c. The amplitude, Δ, and sinusoidal wavelength, λ, of the undulation are chosen to be 4 nm from the r.m.s. value of the surface roughness and 60 nm from the average radial location of the five neighbouring bumps. The other profile parameters, c1, c2 and β, are estimated to be 0.00029, 0.98 and 1.4 to best fit the area and the second polar moment profiles in figure 5a,b.

Figure 5

(a) Cross-sectional area as a function of virtual sectioning depth for the N-PSA; initial height of the central spike, h0=1.00 nm, and height difference between the peaks of the central spike and the first ring-bump asperity, h1=2.9 nm. (b) Cross-sectional second polar moment as a function of virtual sectioning depth for the N-PSA. (c) Schematic of the effective radial profile for the N-PSA; a, radius of the central spike; g(r), profile of the wavy undulations.

(c) Formulation for capillary adhesion in the N-PSA contact

As previously mentioned, when exposed to ambient air a mica surface is always covered with a thin layer of water due to the high level of surface charge (Xu et al. 1998). In dilute solution, the water molecular layer can be squeezed out of the contact region if the contact pressure exceeds the critical electric double-layer repulsive pressure of the monolayer of water. This pressure is approximately 2 MPa (Pashley 1981). If we consider the hydration effect of the possible potassium ion adsorption, the repulsive pressure required to squeeze out the last monolayer of water does not exceed 20 MPa (Pashley & Israelachvili 1984; Homola et al. 1990). In contrast, the local contact pressures of both N-PSA and M-PSA are believed to reach more than 2.5 GPa during the friction test to cause plastic deformation of the asperities (Wang et al. 2006). Therefore, the condensed water layer is expected to be squeezed out to form water meniscuses around the contact junction as the gold asperities contact the mica surface. The capillary pulling force due to the meniscuses will impose an extra pressure on the contact junction in addition to the notional contact pressure exerted by the external load. The capillary effect on the N-PSA frictional responses is modelled quantitatively as follows.

Figure 6a shows a schematic of the N-PSA contact model considering the water capillary effect. In this figure, P represents the external load, a the contact radius of the central-spike asperity, q the meniscus height, rm the radius of the outermost meniscus and Embedded Image the gap in the deformed configuration. As previously discussed, the N-PSA contact is idealized to be axisymmetric. After flattening of the central spike, the interface gap g0(r) in the undeformed configuration is represented byEmbedded Image(5.3)where r denotes the radial distance from the contact centre, Δh the reduction of the central-spike-asperity height caused by either dislocation plasticity or tribo-diffusion. Then, the gap in a deformed configuration can be expressed asEmbedded Image(5.4)where uP(r) and uC(r) are the gap-closing displacements caused by the external normal load P and the capillary force, respectively. Assuming that the contact area remained constant due to the pillar-shaped configuration of the central spike, the first-order solutions of the displacements near the contact area are derived from Tada's handbook (Tada et al. 1985) and Maugis' (1992) asEmbedded Image(5.5)where Embedded Image is the effective plane strain Young's modulus, with E and ν representing Young's modulus and Poisson's ratio, respectively, and the subscripts 1 and 2 indicating the probe and the substrate, respectively. The generic function Embedded Image in equation (5.5) is given asEmbedded Image(5.6)where Embedded Image and Embedded Image are, respectively, the inner and outer radii of the meniscus filling the ring area of Embedded Image and σL denotes the Laplace pressure inside the meniscus.

Figure 6

(a) Schematic of the N-PSA contact model considering capillary effect. (b) External normal load Embedded Image versus the outmost meniscus radius Embedded Image; Embedded Image, Embedded Image and Embedded Image, points of maximum normal load; A′, B′ and C′, points of pull-off instability; Embedded Image, Embedded Image and Embedded Image, calculated pull-off forces. Pillar heights for continuous (thin lines) and separate (thick lines) capillary fillings are as follows: solid lines, 0.8 nm; dashed lines, 0.5 nm; dot-dashed lines, 0.2 nm.

(d) Capillary meniscus evolution in the N-PSA friction

When the capillary condensation around the contact junction is in thermodynamic equilibrium with the vapour environment, the height of the meniscus can be estimated as Embedded Image, where ρK is the radius of curvature of the meniscus and θ1 and θ2 are the wetting angles of water on the probe and substrate surfaces, respectively. The radius of meniscus, ρK, is given by the Kelvin equation Embedded Image, where kB, T, v0, γ, pp/ps are the Boltzmann constant, absolute temperature, molecular volume, surface tension and relative vapour pressure of water, respectively. For a relative humidity of 30%, the Kelvin radius under the experimental condition is estimated to be ρK≈0.44 nm (or qK≈0.66 nm with θ1≈60°, θ2≈0°), for γ=72.8 mJ m−2 and Embedded Image. However, it has been noted that the macroscopic assessments systematically underestimate the water condensation on a freshly cleaved, charged mica surface (Xu et al. 1998). Therefore, the Kelvin radius is considered only as a guideline for observing qualitative trends of water condensation under various humidities at such a fine scale.

In our N-PSA modelling, the height of the capillary meniscus, q, is evaluated to be 2.75 nm (or equivalently ρ=1.83 nm) by the meniscus adhesion analysis described in the following paragraphs, to fit the three experimental pull-off loads at As, Bs and Cs shown in figure 4a. The Laplace pressure inside the meniscus follows the Young–Laplace equation σL=γ/ρ to be 39.7 MPa. As far as the water condensation is concerned, three types of interaction tractions other than the capillary Laplace pressure can arise across an intersurface gap in the range of 0.8 to 2.75 nm. These tractions are produced from the attractive van der Waals force, the repulsive electrical double-layer force and the repulsive hydration force caused by the potassium ions on the mica surface. However, the experiment by Pashley & Israelachvili (1984) and the theory of DLVO (Derjaguin & Landau 1941; Verwey & Overbeek 1948) indicated that the pressures due to these interactions would be within a few megapascals and would decay very fast as the intersurface gap increased (Israelachvili 1991). These tractions would be localized only in a narrow annular region around the central pillar. Therefore, we neglect the DLVO and hydration interactions and consider only the capillary effect in the analysis presented in §5c.

Figure 6b shows the theoretical size evolution of the capillary meniscuses during the friction test. The three solid, dashed and dot-dashed curves represent the relationships between the non-dimensional external load Embedded Image and the non-dimensional outermost meniscus radius Embedded Image for three different heights of the central pillar, i.e. 0.8, 0.5 and 0.2 nm, correspondingly. The relationships are provided by equation (5.4)–(5.6) for a fixed value of q. In this figure, the thin curves represent the case for which the meniscus is assumed to fill continuously between the central contact edge, r=a, and the outermost edge of the meniscus, r=rm. The thick curves correspond to the case that the meniscuses can exist around the central pillar and the first ring bump separately as depicted in figure 6a.

The thick curves indicate that as the external load P (or Embedded Image) decreases, the meniscus size rm (or Embedded Image) shrinks until the meniscus configuration becomes unstable at A′, B′ and C′. Then, the subsequent unstable shrinkage of the meniscus triggers the full detachment of the contact, exhibiting the macroscopic pull-off phenomena at Embedded Image, Embedded Image and Embedded Image. The critical pull-off load depends on the value of the meniscus height q. To best match the theoretical instability loads at Embedded Image, Embedded Image and Embedded Image with the experimental values of As, Bs and Cs in figure 4a, q is evaluated to be 2.75 nm. The resultant instability loads at Embedded Image, Embedded Image and Embedded Image are calculated to be 590, 1090 and 1600 nN.

As the external loads increase, the thick curves merge with the thin curves, indicating that the two meniscuses around the central pillar and the first ring bump are joined together and the meniscus fills the gap continuously. At the maximum external load of 2100 nN, the non-dimensional radii Embedded Image's of the outmost meniscus, at Embedded Image, Embedded Image and Embedded Image in figure 6b, are determined to be 5.65, 6.48 and 7.00 for the three central-pillar heights. These values of Embedded Image's will be used to calculate the true normal contact pressure on the central pillar in §6.

6. Discussions

(a) Overview of the frictional responses of N-PSA

The four distinctive characteristics observed in the N-PSA experiment are considered to be caused by two major underlying mechanisms—flattening of nano-asperities and capillary effect of nanometre-scale water condensation. More specifically, the flattening of the asperities mainly accounts for history-dependent characteristics of (i), (iii) and (iv), and the capillary effect is mostly responsible for the adhesion-strengthening characteristics of (ii). The general picture of the N-PSA friction is described as follows. The central-spike asperity was plastically deformed to form a flat-end pillar during the normal load increasing process of the A series. The central pillar was gradually flattened during the B and C series. For these periods, the capillary adhesion increased, but the friction was maintained at the same level because the contact area was hardly changed. As the flattening progressed further in the D series, the height reduction of the central pillar was large enough so that the flattening was spread from the central pillar to its immediate base and the first neighbouring bumps. It is believed that the spreading of the flattening caused the apparent friction strengthening in the D-series test. Finally, the extreme normal loading applied just before the E-series test could flatten the asperities so much that the frictional response converged to that of the M-PSA friction.

(b) Capillary effects and true contact pressure in the N-PSA friction

In our capillary adhesion analysis, it is conceived that once the water layer is squeezed out of the N-PSA contact region under the high pressure, the capillary condensation contributes only to the normal adhesion but hardly to the lateral frictional resistance. According to the analysis, the wetting area of the capillary condensation, or Embedded Image, increases for a given external load when the height of the central pillar is reduced, as shown in figure 6b. The height reduction also increases the capillary adhesion, i.e. the pull-off force. However, the intrinsic friction stress of the central-pillar contact hardly changes when the height of the pillar is reduced, since the contact area of the central pillar is assumed to barely vary during the height reduction process. In this model analysis, the evolving pull-off forces and the constant frictional force are consistent with those observed in the A, B and C series of the N-PSA friction experiment.

The contact area of the central pillar would experience a maximum external contact pressure of pext=1.78 GPa from the maximum external load of 2100 nN during the tests of A, B and C series. Although this pressure is extremely high when compared with the macroscopic yield strength of bulk gold, it is known that metals can withstand much higher stress as their sizes are reduced to the nanometre scale. It was reported that (Carpick et al. 1996b) a platinum tip could only be blunted to form a circular plateau up to 9–10 nm in radius under a high load of 1000 nN, which corresponded to an average pressure of 3.1–3.9 GPa. Wang et al. (2006) measured the flattening pressure of a truncated (114) facet gold pyramid asperity experimentally as 2.5 GPa for a contact area of 54 nm equivalent radius. The MD simulation by Ward (2007) also predicted the flattening pressure of a truncated gold pyramid asperity as approximately 3.5 GPa for an equivalent contact radius of 19 nm. Then, how was the asperity flattened during the tests of the A, B and C series? The theoretical modelling presented in figure 6b shows that the meniscus capillary force can exert an extra pressure, e.g. Embedded Image, on the contact area in addition to the external contact pressure. Taking into account the capillary pressure, the true maximum contact pressures with Embedded Image in the tests of A, B and C series, corresponding to Embedded Image, Embedded Image and Embedded Image in figure 6b, could reach up to 3.68 GPa. This pressure is exactly in the range of generating surface dislocation plasticity.

In the above discussion, the estimated values are approximations, since the geometries of the asperities are idealized as axisymmetric ripples and the kinetic effect of the water evaporation is neglected in the analysis of the capillary adhesion. However, an important outcome of this analysis is that the water meniscus can generate much of the required contact pressure to induce plastic deformation in gold on the nanometre scale. This capillary effect is believed to play a similar role in increasing the contact areas in the macroscopic friction tests, where the humidity was found to be one of the important factors responsible for the time-dependent frictional behaviours (Scholz & Engelder 1976; Dieterich & Conrad 1984; Titone et al. 2001). However, as mentioned previously, water can also act as a lubricant depending on the normal pressure and the surface properties. For example, in Dieterich's experiment (1984), the frictional force reduced approximately 30% when a small amount of humidity was introduced for two rocks sliding on each other under an external pressure of 1.7 MPa.

(c) Molecular-level response in the N-PSA friction

As capillary forces are involved in the N-PSA friction test, the molecular friction law must include the capillary-induced contact pressure pcap in addition to the external contact pressure pext, asEmbedded Image(6.1)The adhesive friction stress τ0 between gold and mica in air is measured to be 264 MPa from the unloading response of the C series of the N-PSA test shown in figure 4a. As discussed in §5, it is probable that the water molecules would be completely squeezed out under the extremely high pressure (up to 3.68 GPa) for the N-PSA test. Therefore, the friction stress τ0 measured in this experiment is the value of a direct contact between gold and mica.

By measuring the variation of the friction with respect to the change of the true contact pressure of the central pillar, we can accurately calculate the molecular friction factor α. If we consider only the external contact pressure pext for the C-series test, the apparent molecular friction factor αapparent can be fitted to be 0.0166 from the unloading response shown in figure 4a for the range of the external normal pressure between 0 and 1.78 GPa. However, if we take into account pcap, the true contact pressure varied from 0.95 to 3.68 GPa and the friction stress from 274 to 304 MPa, which corresponded to the true molecular friction factor α of 0.0108.

The molecular friction factors for various molecular interlayers between mica flats were measured by SFA at a relatively larger scale (see table 1 in Carpick & Salmeron 1997). However, it has been difficult to measure α experimentally at the nanometre scale (e.g. Pietrement & Troyon 2001). The special configuration of N-PSA minimizes uncertainties in evaluating the contact area and enables a direct measurement of α. However, it is worth noted that the value of α, 0.0108, is an upper bound of the molecular friction factor, since the N-PSA pillar contact area may increase slightly as the normal load increased.

(d) Frictional response of M-PSA

The frictional response of the M-PSA contact is quite different from the theoretical predictions of the non-adhesive contact models. Because the roughness is on a nanometre scale, the M-PSA friction exhibits a strongly nonlinear feature similar to that of a smooth single-asperity friction with adhesion. It is for this reason that the probe is called a micro pseudo single asperity (M-PSA). Many researchers (e.g. Cain et al. 2000; Meurk 2000; Bogdanovic et al. 2001) have used microspheres for frictional studies regarding them as smooth single asperities. However, without knowing the details of the surface morphology and the underlying physical interactions between the surfaces, it is almost impossible to distinguish the response of pseudo single-asperity friction from that of smooth single-asperity friction. The M-PSA friction data could be fitted well with the Maugis-BT model of single-asperity friction, as shown in figure 3b, in particular for the data of increasing normal load. The Maugis-BT fit yields a cohesive-zone strength of 2.3 MPa and a critical interaction distance of 22.3 nm for the cohesive zone employed in the Maugis' model (1992).

In spite of the apparent matching, the fitted critical interaction distance 22.3 nm is too large when compared with the van der Waals interaction range, typically 2 nm; therefore, the cohesive zone cannot come from van der Waals interaction. If we regarded the cohesive-zone strength of 2.3 MPa as the Laplace pressure of a water capillary, the height of the meniscus would be 47.5 nm from the Young–Laplace equation. The critical interaction distance of 22.3 nm is also inconsistent with the estimated meniscus height. Either value of 22.3 or 47.5 nm is too large when compared with the proper value of the meniscus height of 3 nm measured in the N-PSA friction experiment. Furthermore, the Maugis-BT fit value of the friction stress, 40.5 MPa, is substantially smaller than the N-PSA friction stress, 264 MPa. Even considering the size effect, the effective M-PSA friction stress of the Maugis-BT fit is still much less than the theoretical prediction of approximately 208–245 MPa made by Hurtado & Kim (1999a) for smooth single-asperity friction. The improper fitting of the M-PSA friction data, with a smooth single-asperity model, produces the friction and adhesion parameters inconsistent with the underlying physical micro-mechanisms. The fitted parameter values are not molecular-level intrinsic quantities; instead, they are effective values distorted, or modulated, by the nanometre-scale roughness. If the molecular-level intrinsic values are to be extracted from the M-PSA friction data, the nanometre-scale roughness has to be incorporated properly in the modelling.

In addition to the nonlinear feature of single-asperity-like response, the M-PSA data also exhibit a load-history-dependent frictional response, i.e. friction strengthening. This friction strengthening effect is believed to be induced by the morphology evolution of the mating surfaces, caused by the flattening of nano-asperities under the excessive normal and lateral loading, as discussed previously for the N-PSA friction.

(e) Size-scale effects in pseudo single-asperity friction

As discussed above, the friction stress was measured to be 264 MPa for a small contact area in the N-PSA friction test, while the nominal friction stress of M-PSA was measured to be 40.5 MPa for large apparent contact sizes. Since the measured friction stresses exhibit apparent size-scale dependence, it is informative to compare the two values with the predictions of Hurtado & Kim's scale-dependent friction model. The N-PSA and the M-PSA experimental data are plotted in figure 7, as a circle and a thick dotted line, respectively, in Hurtado & Kim's (1999a,b) map of the scale-dependent friction stress. In this map, a denotes the contact radius, b the Burgers vector of the interface dislocation, G the effective bi-material shear modulus, η the Rice–Thomson parameter and τp the effective Peierls stress of the interface dislocation.

Figure 7

Friction stress versus contact size for a variety of experimental data and the model prediction of Hurtado & Kim (1999a): a, contact radius; b, Burgers vector of the interface dislocation; G, effective bi-material shear modulus; η, Rice–Thomson parameter; τp, effective Peierls stress of the interface dislocation; the N-PSA data are highlighted by symbol N as a solid circle and the M-PSA data are highlighted by symbol M as a thick dotted line. Dashed line, SDA slip; up triangle, AFM (Carpick et al. 1996a); down triangle, SFA (Homola et al. 1990).

According to the HK model of dislocation-assisted frictional slip processes, the sliding mechanisms of single-asperity friction can be classified into three categories, i.e. concurrent slip, single-dislocation-assisted (SDA) slip and MDC slip, depending on the contact size. These three regions are shown in figure 7. The boundary between the concurrent- and SDA-slip regions is depicted as line l1. The other boundary between SDA- and MDC-slip regions is represented by line i1. In figure 7, the scale dependence of the friction stress in SDA-slip region can be described by a series of dashed lines depending on the Rice–Thomson parameter η (see Hurtado & Kim 1999a,b; for details). The AFM friction data measured by Carpick et al. (1996a) lie inside the concurrent-slip region. In this range, the friction stress is independent of the contact size, as shown in figure 7 by two sets of up triangles connected by a line. However, as mentioned earlier, the friction stress drifted from 910 down to 270 MPa, as the platinum tip was contaminated by the potassium from the mica surface during the test. The range of drifting is indicated by a vertical double-sided arrow. The SFA friction data measured by Homola et al. (1990) locate inside the MDC-slip region and are plotted as down triangles connected by a line.

In figure 7, the status of the N-PSA friction is within the concurrent-slip region. If the slip process is mainly controlled by interfacial dislocation motion, the scale dependence would follow the thick dashed line that corresponds to the Rice–Thomson parameter η approximately 3.8 in the HK model. In other words, if the friction test were carried out with an atomically smooth direct contact, the data of the M-PSA friction would have lied in the intermediate SDA-slip region, following the line of η=3.8. However, the nominal friction stresses of the M-PSA friction exhibit a premature drop from 264 to 40.5 MPa as the contact size increases from that of the N-PSA to the M-PSA.

The experimental data in figure 7 suggest that there could be some other possible mechanisms of frictional processes than interfacial dislocation motion, which would be responsible for the apparent scale dependence of the friction stress. Those include scale-dependent mechanisms of water (or third-material)-layer-assisted slip, nanometre-scale-roughness-assisted slip, tribo-diffusion-assisted slip and bulk dislocation-assisted slip. However, the friction stress governed by the water (or third-material)-layer-assisted slip is primarily sensitive to the normal pressure required to squeeze out the water layer from the contact zone, not to the contact size. If the experimental apparatus or procedure makes the normal pressure depend on the contact size, the water (or third-material)-layer-assisted slip can exhibit apparent size-scale effect. On the other hand, the nanometre-scale roughness can cause strong size-scale effect directly. As previously discussed, the nominal friction stress of the M-PSA friction is an effective one modulated by the nanometre-scale roughness. Observation of the topographic evolution of the nano-asperities on the surface of the M-PSA indicates that direct contacts were made at the nano-asperity tips during the M-PSA friction test. Thus, the apparent scale effect observed between the N-PSA and M-PSA tests are considered to be caused by the nanometre-scale roughness. The experimental data suggest that the scale dependence of the nominal friction stress of the M-PSA friction must be expressed as a roughness-parameter family of curves, instead of Rice–Thomson parameter family of curves. More experiments and modelling analyses are needed to explore this scale-bridging issue that depends on the amplitude and wavelength of the nanometre-scale undulations on the M-PSA surface.

7. Conclusions

The following major conclusions have been determined for the multi-scale mechanical characteristics of the frictional processes involving the nanometre-scale roughness.

  1. At the molecular level, the adhesive friction stress τ0 has been measured in the N-PSA friction for a direct contact between gold and mica in air. The measured value of τ0, 264 MPa, is a significant frictional resistance for an incommensurate contact, and it is consistent with those of similar but UHV experiments (see Carpick et al. 1996a; Lantz et al. 1997; also the reviews Carpick & Salmeron 1997; Gnecco et al. 2001).

  2. The value of α, 0.0108, is the first direct and relatively accurate experimental measurement of molecular friction factor for a direct incommensurate solid–solid contact. The value is close to the range predicted by MD (Harrison et al. 1992), but below the lower end of the range.

  3. In an ambient environment, water condensation and capillary effect play important roles for interfacial contacts involving nanometre-scale roughness. In the N-PSA friction experiment, the condensed water layer was considered to be squeezed out of the contact area due to the extremely high contact pressure, 3.68 GPa, well above the critical pressure (approx. 20 MPa) to squeeze out the last single layer of water. However, the capillary effect of the water meniscuses dominated the apparent adhesion, i.e. the pull-off force. It is believed that the water layer did not cause the molecular-level lubrication. The measured adhesive friction stress value, 264 MPa, strongly suggested that a direct contact between gold and mica was achieved in the 30% humidity air.

  4. Even though metals (e.g. gold) could have a much higher yield stress (e.g. 3.68 GPa) on the nanometre scale (e.g. for a contact radius of 19.4 nm), the resultant Laplace pressure of the capillary was capable of adding an extra high pressure (e.g. 1.9 GPa) to the external pressure (e.g. 1.78 GPa), to deform the nano-asperity plastically.

  5. The frictional response of the M-PSA experiment exhibited strongly nonlinear and load-history-dependent characteristics. Flattening of the nanometre-scale asperities was clearly observed; solid–solid direct contacts were made at the tips of the asperities with high local contact pressure. Evolution of the asperity flattening was responsible for the load-history-dependent characteristic.

  6. The nonlinear characteristic looked like the one for a single-asperity friction with smooth-surface contact. However, the 2 nm r.m.s. roughness of the M-PSA reduced the nominal friction stress to 40.5 MPa from the predicted values (Hurtado & Kim 1999a), 208–245 MPa, of smooth-surface single-asperity friction. The predictions of Hurtado & Kim are based on interfacial dislocation motions for the frictional slip of single-asperity contact. However, our new results indicate that the nanometre-scale roughness might significantly alter the scale dependence of the nominal friction stress.


The authors are grateful to reviewers for their thorough interrogation of the manuscript and constructive suggestions. K.-S.K. thanks Prof. K. L. Johnson for his mentorship to motivate the current research. This work is supported in part by the Nano and Bio Mechanics Program, under award CMS-0511961, and in part by the MRSEC Program, under award DMR-0520651, of the National Science Foundation.


    • Received December 12, 2007.
    • Accepted January 28, 2008.


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