Deformation- and rupture-controlled friction between PDMS and a nanometer-scale SiOx single-asperity

This work investigates the friction between polydimethylsiloxane (PDMS) and silicon oxide (SiOx) in single asperity sliding contact by atomic force microscopy (AFM). Two friction dependences on the normal force are identified: a tensile regime and a compressive regime of normal forces. In the compressive regime, friction is governed by the shear deformation and rupture of junctions between PDMS and SiOx. In this case, the shear strength τ ≈ 10 MPa is comparable with the cohesive strength of PDMS under compressive loading. In contrast, friction in the tensile regime is also affected by the elongation of the junctions. The single SiOx-asperity follows a stick-slip motion on PDMS in both normal force regimes. Statistical analysis of stick-slip as a function of the normal force allows determining the necessary amount of energy to break a SiOx/PDMS junction. Friction between a SiOx-asperity and a PDMS surface can be rationalized based on an energy criterion for the deformation and slippage of nanometer-scale junctions.


Introduction
Stick-slip friction is a phenomenon observed in various natural and technological situations, such as earthquakes [1], automobile brake systems [2], or articular joints [3]. Considering that dry contacts between natural and engineering surfaces initiate at their highest asperities, the following mechanisms are accountable for stick-slip friction: stick-slip motion may arise from the formation of cold-weld joints between asperities and their subsequent shearing up to rupture or from the brittle fracture of interlocked asperities for the cases of metals and rocks, respectively [4]. Stick-slip friction between stiff solids has been described successfully based on the thermally activated Prandtl-Tomlinson model [5]. In this model, a point mass asperity dragged on a corrugated surface potential jumps from one local minimum to the next. However, these models have limited validity when compliant surfaces, such as elastomers, are involved [6][7][8]. Despite their recent increased technological relevance, their tribology is far less understood than stiff solids, such as metals and rocks. A significant difference between metals and elastomers is that contacts with the later ones can persist in the range of tensile loading. In contrast, contacts with the former are often restricted to compressive loading [9]. Friction in the tensile load regime has important technological implications in robotics [10,11].
Stick-slip friction has also been observed in the sliding contact between soft adhesives and stiff solids (for example, Ref. [12]). For these partners, stick-slip has been attributed to the propagation of slow frictional waves, called Schallamach waves [13]. The propagation of Schallamach waves is affected by viscoelastic effects [14,15], the formation of surface wrinkles [16], and the sliding velocity [17,18]. In these studies, the stick-slip length was in the range of several 100 m.
With atomic force microscopy (AFM) and lateral or friction force microscopy (L/FFM) development, investigating the dynamics of single asperity contacts in reciprocal sliding has become possible [19][20][21][22]. First nanotribological studies by AFM have investigated ideal surfaces in ultra-high vacuum (UHV)-conditions, such as KF, KBr, mica, highly oriented pyrolitic graphite (HOPG), and single-crystalline metallic or semiconducting surfaces [23][24][25][26][27][28]. These pioneering investigations yielded unpreceded insights into the structure's role in the friction of stiff surfaces. More recently, dual-scale stick-slip friction on graphene/h-BN superlattices was observed and discussed based on the formation of Moire patterns and the accumulation and abrupt release of in-plane strain of the graphene layer [29].
Furthermore, nanoscale investigations of the tribological response of compliant polymer surfaces in single asperity sliding contact conditions have been performed by Gnecco and co-workers (for example, Refs. [30][31][32][33]). The authors observed the formation of surface ripples on polystyrene induced by plowing and locking effects. In Ref. [30], the formation of periodic ripples on polystyrene was explained by the competition between the plastic response of the polymer surface and its restoring elastic force of the contact. The first condition for rippling is the plastic indentation of an asperity into the surface. In analogy to the description of the atomic-scale stick-slip motion of a nanoscopic asperity on stiff surfaces with the Prandtl-Tomlinson model [5], the authors suggest that for the tip to jump from one locking position (indentation) to the next, elastic energy needs to build up in the contact and exceed the potential energy barrier arising from locking. In this scheme, the wavelength of the periodic ripples depends on the indentation rate or the applied normal force that determines the penetration depth and the sliding velocity that determines the dwelling time of the tip in an indentation. Furthermore, the formation of ripples is conditioned by a super-critical value of the indentation rate and sub-critical values for the sliding velocity and the lateral contact stiffness. For polystyrene, the authors experimentally determined the critical normal force and sliding velocity for rippling to be F n,c = 225 nN, and v s,c = 45 m/s. This work investigates the tribological response of polydimethylsiloxane (PDMS) in single nanoscale asperity sliding contact with SiO x by AFM. The presented results are discussed based on the shearing model of friction and the longitudinal deformation of PDMS/SiO x junctions. Furthermore, statistical analysis of the observed molecular scale stick-slip is performed to provide new insights into nanoscopic junctions' deformation and rupture mechanisms and their contributions to friction.

Materials and experimental method
A PDMS-sample was prepared from a two-part liquid component kit (Sylgard 184 silicone elastomer manufactured and distributed by Dow Corning, USA), consisting of a pre-polymer base and a cross-linking curing agent that were mixed to a ratio of 10:1. The mixture was poured on a flask dish and subsequently cured for 15 min at 90 °C. All subsequent experiments presented in this work were performed on the airside of the PDMS sample by AFM. The surface morphology of the as-prepared sample was measured by tapping mode (TM)-AFM using a stiff single-crystalline Si-cantilever (type NCHR, manufactured by NanoSensors, Switzerland). Figure 1 shows typical TM-AFM topography images of the PDMS sample recorded over scan areas ranging from 10 m × 10 m down to 1 m × 1 m. For all images, the roughness parameter R q was determined and found to be R q ≈ 1 nm.
In this work, the tribological response of PDMS was investigated in ambient conditions (T = 293 K, RH = 50%) by reciprocal sliding of a nanometer-scale single asperity of SiO x . To this end, lateral force maps were recorded with a CoreAFM (manufactured by Nanosurf, Switzerland) and a soft single-crystalline Si-cantilever (type CONTR, manufactured by NanoSensors, Switzerland). The sensitivity of the AFM-photodiode S was calibrated by recording five force-distance curves on a fused silica sample and calculating the average slope of the repulsive part of the curves. The bending stiffness of the cantilever C n was determined by thermal noise analysis [34], while its torsion stiffness was determined according to Gwt C h L  [35], where w is the width, t is the thickness and L is the length of The friction measurements presented in this work consisted of reciprocal sliding over an area A s = 10 m × 10 m successively scanned over a range of increasing normal force values F n = -50-25 nN. Four load-dependent friction measurements were performed with sliding velocities in the order of v s = 80, 40, 20, and 160 m/s. Thereby, topography, normal force, and lateral force maps were recorded in the forward and backward direction of the fast-scanning axis, perpendicular to the cantilever length axis. Friction force maps were subsequently computed from the lateral force images recorded in the forward and backward directions according to and their mean and standard deviation values were calculated. After each series of load-dependent friction measurements, a larger area including A s was scanned to determine topographical changes and wear during friction measurements. After completion of the friction measurements, the AFM-tip was imaged by scanning electron microscopy (SEM) to characterize its geometry and compared with a new tip of the same type. Figure 2 shows a typical lateral force image recorded in the forward direction with a normal force value F n = -15 nN and a sliding velocity v s = 40 m/s. Figure 2 also presents the lateral friction signals recorded in the forward and backward directions corresponding to the scan line indicated in white on the lateral force image. The experimental results in Fig. 2 indicate a stick-slip motion of the SiO x asperity. The stick-slip behavior was analyzed about the sticking events' slope and lateral force magnitude (see Fig. 3).   For each line scan in every recorded image, sequences of points with increasing lateral force values were determined with a dedicated MATLAB script. These sequences were then fitted with a linear function only if their length was larger than five points. Subsequently, the distribution of slopes, i.e., lateral contact stiffness * l -C values and the distribution of strength F l -values of the sticking events were calculated to determine their mean values and standard deviations by fitting with a gaussian function. In Ref. [36], the authors analyzed the effect of contact aging on the atomic stick-slip motion of a sharp AFM-tip dragged on a NaCl surface in an ultrahigh vacuum. There, the local lateral contact stiffness was estimated by calculating the derivative of the lateral force signal by the three-point finite difference method.

Results and discussion
The F f (F n )-plots shown in Fig. 4 for different sliding velocities v s = 20-160 m/s indicate two distinct friction regimes. In the range of normal force values F n = -50 --10 nN, one observes a linear increase of the friction force with the normal force, characterized by a slope  and intersection with the F n -axis * .
F For larger F n -values, the friction force can be fitted with the shearing model for adhesive friction. There, the friction force can be expressed as f where  is the shear strength, and is the contact area between the AFM tip and the sample and a c the corresponding contact radius. Furthermore, the load dependence of the contact radius between a smooth surface and a spherical asperity can be described with the Johnson-Kendall-Roberts (JKR)-model [36] There, F ad is the adhesion force, and E * is the reduced modulus of elasticity, which for isotropic solids is given , with E s,t , and  s,t being the Young's modulus and the Poisson's ratio of the sample and the tip, respectively, and R is the radius of the spherical asperity. In this work, the AFM tip was imaged after tribological measurements by SEM, and its radius was found to be R = 27 nm (see Fig. 5). Below, we evaluate the evolution of the tip radius based on the determined adhesion force values to correct the shear strength values obtained under the assumption of a constant tip radius. As summarized in Table 1 and plotted in Fig. 4, it is observed that  decreases logarithmically with increased sliding velocity. In contrast, no clear tendency is observed for the shear strength, F * , and the adhesion force F ad concerning the sliding velocity (see Fig. 4(d) and Table 1). Moreover, friction do not show any trend concerning the sliding velocity. A logarithmic decay of friction with increasing velocity, or equivalently with decreasing dwelling time, has been observed for various systems and associated with contact aging (see Refs. [36,38,39]). Mazo et al. [36] demonstrated how the contact area between a nano-asperity and a NaCl surface follows an exponentially saturating www.Springer.com/journal/40544 | Friction growth, whose kinetics is governed by the Arrhenius equation. Furthermore, the authors identified atomic attrition as the leading mechanism for contact aging, where, under the action of a normal force, a nanoasperity increases its contact area by a flow of atoms therein until the contact area is large enough to  Table 1).  support the applied load without further growth of the contact area. In the same line of thought, Petrova et al. [38] showed how the static friction force between polystyrene and glass and their associated shear strength depends on the contact time. The authors observed that this increase in static friction is related to a growth of the real contact area by plastic flow. Li et al. [39] found that the friction strength of a SiO x asperity in contact with a silica surface increases linearly with the logarithm of the contact time. However, the authors related frictional aging to changes in chemical bonding instead of the growth of the contact area by plastic creep. In this work, the maximum contact pressure can be evaluated according to n,max p  n ,max n,max n,max MPa. This value is by far lower than the yield strength of silica and PDMS. It is thus not expected that the contact area could grow by plastic creep mechanisms. Since our measurements were carried out in ambient conditions, a thin adsorbed water film at the PDMS/SiO x interface may have promoted their adhesive bonding. Villette et al. [40] investigated the effect of relative humidity (RH) on the spreading of PDMS on a Si wafer with a native oxide layer and found how water increases the spreading velocity of PDMS on hydrophilic surfaces. Thus, the author suggests that due to interfacial water, PDMS quickly wetted the AFM tip to form adhesive junctions with the SiO x tip faster than resolvable under the selected experimental conditions. According to Bowden and Tabor [41], shearing consists of the deformation and rupture of adhesive junctions. Most shearing experiments are conducted under compressive loading, in which case the evolution of the friction force as a function of the normal force matches well with the JKR model. In this work, the contact between a nanometer-scale SiO x -asperity and a smooth PDMS surface persisted beyond tensile forces F n < -50 nN. This finding suggests a much stronger adhesion between SiO x and PDMS than reported for glass and PDMS. Mergel et al. The results presented in this work reveal a transition from a linear relationship between F f and F n in the regime of tensile forces to a relationship that matches well with the JKR model in the regime of compressive forces. Given this work's tip radius R = 27 nm, an adhesion force F ad ≈ 9 nN was expected. The adhesion force values F ad ≈ 5 nN, determined by fitting the JKR model to our experimental values in Fig. 4, are thus slightly lower than expected from Ref. [9]. There, adhesion was attributed to van der Waals interactions and described by a Lenhard-Jones potential.
This view is contrasted by the difference in shear strength evaluated for PDMS/glass ( = 0.43 MPa) [9] and the value determined for PDMS/SiO . In analogy and with  = 0.5 for PDMS [44], one could expect which is very close to the value reported in this work.
In the compressive range of normal forces, friction is thus governed by the formation, shear deformation, and rupture of junctions between PDMS and SiO x . In where  is the applied longitudinal tensile stress and  corresponds to the strain rate sensitivity of PDMS. In unrelaxed conditions, i.e., at large strain rates or sliding velocities,  approaches to unity, while in relaxed conditions, i.e., at low sliding velocities, 

increases.
It is important to mention that in the presented experiments, the tensile force acting on the contact never exceeded F t . For that reason, the contact was  | https://mc03.manuscriptcentral.com/friction never fully ruptured. The author suggests that in the tensile regime the sliding contact was mediated by elongation and the storage of elastic energy in an adhesive junction, and its further drag or slippage. Both mechanisms are competing. PDMS appears to offer less resistance to elastic deformation of a junction than to its slippage. Once the elastic restoring force acting on the junction surpasses the friction force, slippage of the junction sets in until the elastic energy stored in the junction has been dissipated.
Furthermore, Figs. 5-10 indicate significant changes in the tip shape and PDMS surface properties during and after tribological tests. Figure 5 compares the shape of an unused SiO x -tip with one used for the friction measurements on PDMS shown in Fig. 4. After friction measurements, the tip became rounded. Figures 6 and 7 show topography images and lateral force maps recorded on PDMS corresponding to the results shown in Fig. 4 for a sliding velocity v s = 20 m/s. Moreover, Fig. 8 displays typical height profiles and friction loops recorded under the same conditions as the images shown in Figs. 6 and 7. The topography images in Fig. 6 are distinctly different from the topography images recorded in tapping mode shown in Fig. 1. In Fig. 6, the images are all affected by the stick-slip motion of the tip. Moreover, the topography images and their typical height profiles in Figs. 6 and 8 indicate a transition at a normal force value F n = -15 nN, that also corresponds to the transition from the observed linear dependence of the friction force on the normal force to its F n 2/3 -dependence. Below the normal force value F n = -15 nN, the topography images recorded on PDMS with a SiO x tip are affected by stick-slip through crosstalk. However, the topography images in this normal force regime, do not display clear sign of wear such as depression of the surface level or horizontal scars. Such features become first apparent for normal force values F n > -15 nN. Accordingly, it can be inferred that wear sets in at the transition from F n -dependence to F n 2/3 -dependence of friction. These observations are in good agreement with the description of friction based on adhesive junctions' formation, shearing and rupture for normal force values F n > -15 nN, as supported by the F n 2/3 -dependence of the friction force. In this view, beyond a threshold value of the normal force, an adhesive PDMS junction ruptures and may lead to material transfer to the SiO x tip, thus leading to the PDMS surface degradation observed in Figs. 6-10. This implies that in this regime, the adhesive strength between PDMS and SiO x is larger than the cohesive strength of PDMS. The fact, that post-mortem imaging of the tip did not indicate any sign material transfer, can be explained by the immediate degradation of PDMS under an electron beam. In contrast, for normal force values F n < -15 nN, the motion of the tip is understood to be governed by the elongation of an adhesive PDMS junction until it stores enough elastic energy to initiate its slippage; slippage then continues proceeding as long as the stored energy in the junction is dissipated. In Fig. 9, the surface of PDMS subjected to friction tests is lower than its surrounding. However, no pileup is observed. The measured friction maps show how friction increased for the tested areas (see Fig. 10). Figure 11 indicates that the height difference  w between the tested area and its surrounding decreases logarithmically with increasing sliding velocity. The  relative difference in friction force F f /F f inside and outside the tested areas also decays logarithmically with increasing v s . The abrasive wear of polystyrene has been investigated by Hennig et al. during nanoscale single asperity sliding contact by AFM [32]. There, the authors reported on the formation of spherical abrasion debris from polystyrene surfaces subjected to repeated scratching by AFM tips and explained it based on a three stages mechanism: formation of pileups upon indentation, particle nucleation upon the increase of the lateral force, and repeated grazing on the pileups, and detachment of a spherical particle. In this work, the tribological response of PDMS was dominated by the formation of adhesive junctions, their deformation, and their rupture. In the case of metals, the author showed how the transition from the shearing of adhesive junctions to plowing corresponds to a change in the slope of the F f (F n )-plot [46]. For pure shearing friction on metals, the friction force follows a 2/3 n F -dependence, while a linear dependence of the plowing force on the normal force has been observed on various metallic systems [46][47][48]. Furthermore, post-mortem imaging, as shown in Figs. 6 and 7, indicates neither ripples nor debris formation. Abrasion or plowing can thus be excluded as the leading wear of PDMS during the presented experiments. Instead, the author suggests that upon shearing of adhesive junctions between SiO x and PDMS, some material was detached from the surface and either dispersed into the environment or transferred to the tip. Postmortem SEM images of the SiO x tip, however, do not show any debris, but it is likely that such debris burned during the immediate irradiation by the electron beam.
Also, the dependence of the wear depth on the sliding velocity is associated with the strain-rate sensitivity of PDMS, also illustrated by the dependence on the velocity logarithm. For the time being, the origin of the decay of F f /F f with the logarithm of the velocity, indicated in Figs. 10 and 11, remains unclear.
Gotsmann and Lantz [49] investigated the abrasive wear of a silicon tip initially covered by a native oxide upon sliding on a smooth cross-linked polyaryletherketone (PEAK) surface. The authors applied normal force values ranging from 5 to 100 nN, sliding distances ranging from 100 to 750 m, and the sliding velocity was set to 1.55 mm/s. In this work, the total sliding distance is well below the distance investigated, e.g., ~0.5 m vs. up to 750 m. Also, the maximum sliding velocity in the present work was lower than in the reference (160 m/s vs. 1.55 mm/s). Also, in the present work, the load on a single tip varied from -50 nN to 25 nN, while in the reference, the load was kept constant for each tip. Furthermore, they reported on the absence of the PEAK sample wear, while, in this work, PDMS is found to wear out significantly. Despite these experimental differences, it is worthwhile to discuss and compare the results presented in this work with the findings made by them.
| https://mc03.manuscriptcentral.com/friction In Ref. [49], the authors monitored tip wear by measuring the adhesion force between the tip and sample after every 62 cm of sliding. In the same interval, they monitored the morphology of the PEAK surface by acquiring a topography image of the area subjected to sliding. Also, selected tips were imaged by SEM before and after sliding experiments. The authors reported that after a sliding distance of 750 m under a load force of 5 nN, the tip radius changed from 4 to 14 nm, while the corresponding volume loss was estimated as 1.5 × 10 4 nm 3 . It is important to note that the authors didn't observe tip fracture or wear of the polymeric surface at any moment during their sliding experiments. Instead, the evolution of the tip radius was observed to be a continuous process in which individual atoms from the tip gain sufficient energy to overcome their binding energy under the action of frictional shear stress. The authors formalized this view based on a thermally activated model, in which the action of frictional shear stress distorts the potential energy landscape and reduces the net energy barrier for an atom to leave the tip.
In this work, the tip radius was determined after tribological tests by SEM and compared to a used tip. The radius of the unused tip was determined as R = 3 nm, which agrees with the observation by Gotsmann and Lantz in Ref. [49]. After sliding on PDMS over ~0.5 m with velocities ranging from 20 to 160 m/s and under varying load values up to 25 nN, a tip radius of 27 nm was measured by SEM. The wear volume of the tip was evaluated as 1.8 × 10 5 nm 3 by comparing SEM images of a new tip with the tip used on PDMS after completion of the presented measurements. Although the sliding distance was much shorter in this work than in Ref. [49], the observed tip wear is much larger after measurements on PDMS than on PEAK.
As indicated above, the presented sliding experiments were recorded in the order of increasing normal load values. Totally, four series of measurements were performed, starting with v s = 80 m/s and continuing with v s = 40, 20, and 160 m/s. Assuming constant interfacial energy, this relative increase in adhesion force corresponds to a relative change of tip radius R = 7 nm, according to Accordingly, a final radius R = 10 nm would be expected. This value is much smaller than determined after measurements by SEM. Alternatively, using R = 27 nm as the final value of the tip radius would imply an initial value R 0 = 18 nm. In Fig. 4(d), the shear strength values plotted as blue symbols were calculated assuming a constant R = 27 nm. Table 1 presents corrected -values and considers tip radius values calculated from its final value (R = 27 nm) and the evolution of the adhesion force. The corrected values are also indicated in Fig. 4(d) as orange open symbols. After correction, the numeric-values changed only slightly.
The above results indicate a chemo-mechanical underlying mechanism for friction between SiO x and PDMS that favors the formation of junctions. Further, it appears that the magnitude of the friction forces depends on the cohesive strength of such junctions. A closer look at the lateral force images indicates that the sliding motion of a SiO x -asperity over PDMS is not continuous but consists of discrete stick and slip events (see Figs. 2 and 3). The occurrence of stick-slip motion can directly be related to the formation, deformation, and slippage or rupture of SiO x /PDMS junctions, depending on the normal force value. Figure 12 shows how the number of sticking events per 10 m × 10 m image depends on the normal force. This number almost linearly increases with the normal force in the tensile regime until it stabilizes to a plateau value in the compressive regime. Parallel to this observation, the lateral stiffness C l * and the force magnitude F f of the sticking events also increase linearly with the normal force in the tensile regime and then reach a plateau value in the compressive regime. These observations for C l * and F f appear reasonable since elongating the junctions in the tensile regime should result in their elastic destabilization in their transversal direction, i.e., in a shorter stick-slip length. This is confirmed by calculating the stick-slip length by , as shown in Fig. 12

Conclusions
This work investigated the friction between polydimethylsiloxane (PDMS) and silicon oxide (SiO x ) in single asperity sliding contact by atomic force microscopy (AFM). Two friction dependences on the normal force are identified: a tensile regime and a compressive regime of normal forces. The friction force increases linearly with the normal force in the tensile regime. In contrast, in the compressive regime, the friction force is proportional to the normal force to the power of two-thirds, in good agreement with the JKR model for shearing friction. Friction in the compressive is governed by the shear deformation and rupture of junctions between PDMS and SiO x . In this case, the shear strength  ≈ 10 MPa is comparable with the cohesive strength of PDMS under compressive loading. In contrast, friction in the tensile regime is affected by the elongation of the junctions. Specifically, it is suggested that elastic energy is stored within the junction until a threshold value to activate drag or slippage of the junction.
In both normal force regimes, the single SiO x -asperity followed a stick-slip motion on PDMS. Statistical analysis of stick-slip as a function of the normal force allows for determining the necessary amount of energy to drag or rupture a SiO x /PDMS junction.

Declaration of competing interest
The authors have no competing interests to declare that are relevant to the content of this article.
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