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Contact dependence and velocity crossover in friction between microscopic solid/solid contacts Joshua McGraw, Antoine Niguès, Alexis Chennevière, and alessandro siria Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.7b03076 • Publication Date (Web): 20 Sep 2017 Downloaded from http://pubs.acs.org on September 23, 2017

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Contact dependence and velocity crossover in friction between microscopic solid/solid contacts Joshua D. McGraw,† Antoine Niguès,∗,‡ Alexis Chennevière,¶ and Alessandro Siria‡ †Département de Physique, Ecole Normale Supérieure/Paris Sciences et Lettres (PSL) Research University, CNRS, 75005 Paris, France ‡Laboratoire de Physique Statistique de l’Ecole Normale Superiéure, UMR CNRS 8550, PSL Research University, 24 Rue Lhomond 75005 Paris, France ¶Laboratoire Léon Brillouin CEA, CNRS, CEA Saclay, 91191 Gif-sur-Yvette, France E-mail: [email protected]

Abstract Friction at the nanoscale differs markedly from that between surfaces of macroscopic extent. Characteristically, the velocity dependence of friction between apparent solid/solid contacts can strongly deviate from the classically assumed velocity independence. Here, we show that a non-destructive friction between solid tips with radius on the scale of hundreds of nanometers, and solid, hydrophobic self-assembled monolayers has a strong velocity dependence. Specifically, using laterally oscillating quartz tuning forks, we observe a linear scaling in the velocity at the lowest accessed velocities, typically hundreds of micrometers per second, crossing over into a logarithmic velocity dependence. This crossover is consistent with a general multi-contact friction model

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that includes thermally activated breaking of the contacts at sub-nanometric elongation. We find as well a strong dependence of the friction on the dimensions of the frictional probe.

Keywords Nanoscale friction, surfaces, interfaces, atomic force microscopy

Introduction Over the last decade, thanks to the development of high performance experiments, it has been possible to disentangle different contributions to nanoscopic friction arising from van der Walls 1 , electronic 2 , phononic 3 , crystallographic 4 and instrumental 5 factors. In contrast to observations at the macroscale, where the friction is weakly dependent on the material properties, the collected results above demonstrate that dissipation at the nanoscale is strongly impacted by physical and chemical characteristics of the surfaces 6–9 . As we show here, the sliding velocity also plays a dominant role on the measured friction 10–12 , and a crossover between two asymptotic regimes, linear to logarithmic, can be observed on increasing the velocity. Modelling friction at the nanoscale commonly follows the ideas of Prandl and Tomlinson 12–15 . In such models, a single particle is subject to a periodic force field representing the solid surface. A spring force is superimposed onto the periodic one, representing an atomic force microscope (AFM) cantilever 16 to which the particle (e.g. the terminal atom) is attached. Upon translation of the spring at a velocity, v0 , the particle successively hops the periodic barriers into adjacent minima of the periodic field. Importantly, the kinetic hops may be thermally activated and this single-particle model leads to a marked velocity dependence on the friction 13 . In particular, a logarithmic velocity dependence was observed in ultra-high vacuum experiments of single-contact asperity friction by Gnecco et al. 10 . More 2


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generally, multi-contact models of rough surfaces were developed by Persson 17 —and generalized in particular by Braun and Peyrard (BP) 18–20 — retaining the main ingredients of the Prandl-Tomlinson approaches 10–15 with thermal activation. To measure friction at the nanoscale, high performance tools such as the friction force microscope (FFM) 16 and variants 4 have been developed. This instrument allows for measurement of the load dependence of the friction for nanoscale and atomic contacts, as well as its velocity dependence 10 . The weak forces expected in these nanoscale friction scenarios require the use of a flexible cantilever in the FFM making it susceptible to mechanical instabilities (i.e. snap in). Alternatively, quartz tuning forks 21–25 employ i) stiffness typically three orders of magnitude larger than a FFM cantilever, making them robust against snap in and ii) a high quality factor thus reaching force precision at the piconewton level. By oscillating the tuning fork parallel to a substrate, the friction between microscopic contacts becomes accessible 23–25 . Further to the need for tools reaching nanometric precision, model systems for the investigation of friction at the nanoscale are necessary. In this context, self-assembled monolayers (SAMs) are of particular interest 26,27 , as they are robust and can be prepared with the atomic scale roughness inherited from the underlying substrate to which they are attached 28 . SAMs have been used for controlling wetting properties 29 and influencing interfacial fluid dynamics 30,31 , as well as for their tribological performance 1 . In this work, we focus on the friction between the surface of alkylsilane SAMs prepared from dodecyltrichlorosilane (DTS) and electrochemically etched 32 tungsten tips with radius 700 nm. A schematic of the experimental setup along with representations of the sample surfaces are shown in Figure 1a), with details in the Supporting Information. The principle result of our study is that we have measured a strong velocity dependence of the friction force between these contacts. All of the data reproduce quantitatively the velocity dependence predicted by BP 20 for a system which contains a large number of contacts susceptible to thermal activation. Namely, a linear dependence on the velocity is observed at low ve-

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idea that a stronger conservative (e.g. normal) force gives rise to a higher dissipation. We constantly return to a reference frequency shift, δfref = 0.5 Hz, following each measurement of a {δf, V } pair to obtain a corresponding reference dissipation force FD,ref . Representative measurement sequences are shown in Figures 1b) and c) for a = 0.15 nm and 4 nm. There it can be seen that the excitations, Vref , required for the maintenance of δfref do not change with time. We therefore determine that our measurement is non-destructive. The quartz tuning fork can be modelled as a mass-spring resonator, whose resonance profile will be modified by the interacting forces. As the interaction of the oscillator with its environment is modified, one observes a change in both the frequency and the amplitude at resonance for a given excitation 33 . The resonance shift δf is related to the conservative force response, FC , whereas the excitation required to maintain a constant amplitude is related to the total dissipation force, FD :

∂r FC = 2kTF

kTF a FD = √ 3Q0

δf , f0

V −1 V0

(1)

.

(2)

Here ∂r denotes differentiation with respect to the direction of oscillation, and kTF is the tuning fork spring constant. While Equation 2 was derived for purely viscous damping by Karrai and Grober in ref. 33 , the relation also holds on the condition that all dissipation forces are proportional to the applied voltage and that the reference dissipation associated with V0 is purely viscous; see also Comtet et al. 34 . The quartz tuning forks used here couple ultrahigh stiffness of kTF = 40 kN m−1 , preventing snap in, and a low intrinsic dissipation as characterized by high quality factors, here Q0 ≈ 7 500 in air, making them ideal tools for the study of nanoscale friction. In the Supporting Information, we demonstrate that in the frictional mode used here, the force gradient in Equation 1 is the elastic tip/sample stiffness parallel to the substrate. 5


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In Figure 2a), we show the dissipation obtained at δfref as a function of the oscillation amplitude for the same experiment represented in Figure 1. At each value of a, there are fourteen superimposed instances of FD,ref . At small oscillation amplitudes the data show a strong increase in the frictional dissipation with amplitude, followed by a levelling off of the friction approaching a = 10 nm. In this work, we assume that the typical velocity of the oscillator, v0 = 2πf a with 20 ≤ v0 ≤ 2 000 µm s−1 , which is simply linear in the amplitude, is proportional to the constant velocity used by BP 18–20 in their development of the velocity dependence of friction. In Figures 2b) and c), we have reproduced the data shown in part a) of the same figure to emphasize the scalings. At low velocity FD increases linearly, while at higher velocities the dependence on the amplitude is logarithmic. In the Supporting Information we demonstrate that these scalings are observed over the whole range of accessed frequency shifts. In Figure 3a), we show the dependence of friction on both the oscillation amplitude and the imposed frequency shift. There it is seen that at a given oscillation amplitude, higher imposed frequency shifts result in higher friction, as also seen in Figure 1. Independent tips and samples give qualitatively similar results, with linear to logarithmic scaling crossovers, see the Supporting Information. The data in Figures 2 and 3a) are furthermore well represented by the modelling (orange lines), which we shall now describe. Theoretical description The BP model 18–20 is based on an elastically deforming ensemble of Nc contacts. Each contact has a specified spring constant, ki , characterized by an average kk ≡ hki i, and breaks at a critical elongation distributed around an average x∗ . The model assumes an energy of elongation at break that is not very different from the typical thermal energy: kk x2∗ ∼ kb T ; here we use the nomenclature b =

kk x2∗ . 2kb T

A typical attempt frequency

for thermally activated contact breaking, ω, is assumed. The ensemble sum of such contact forces thus determines the overall friction. With an ensemble of contacts containing a distribution of stiffnesses, the BP model leads naturally to a linear scaling of the friction with velocity, FD = ζv, when the velocity

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Figure 2: a) Total dissipation force as a function of the oscillation amplitude at the reference frequency shift, δfref = 0.5 Hz for the DTS-coated Si wafer represented in Figure 1. In b) and c), the same data as in a) are shown with the with double-logarithmic axes and semilogarithmic axes, respectively. is much smaller than a critical value, v ≪ v∗ =

ωx∗ 1+b/3 , C0 (1+b)2

√ where C0 = 4(3 π)−1 is a

normalisation constant related to the contact breaking distribution. A logarithmic scaling of the friction with the velocity follows the linear one. These are exactly the scalings we observe experimentally shown in Figures 2b) and c). In accordance with the BP model for a singular distribution of breaking elongations 20 , single-atom contact experiments have shown purely logarithmic dependencies —as detailed by Gnecco et al. 10 and Sang et al. 11 — over several orders of magnitude in the velocity. In Figure 2a), we show the velocity dependence of the friction predicted by BP imposing kb T = 4.1 × 10−21 J (i.e. ambient temperature), ω = 6.3 × 106 Hz and with Nc kk = 18 ± 4 N/m, and x∗ = 0.58 ± 0.03 nm (see Supporting Information for fitting details). The model provides an excellent fit to the data over the whole range of imposed oscillation amplitudes 35 . Figure 3a) shows similar fitting quality for several imposed δf , and Figures 3b)–d) show the resulting fitting parameters for all the accessed δf . Analysis and interpretation The dynamic friction data in Figure 3a) show a systematic change in the velocity dependent friction resulting from a change in the frequency shift. The data are also in excellent quantitative agreement with fits to the BP model for each particular

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Figure 3: a) Dissipation force as a function of the oscillation amplitude for independent values of the frequency shift indicated in the legend (values in Hz). Lines represent predictions of the BP model; all but the data set in squares (δf = 0.21 Hz) are fit using ω = 6.3 × 106 Hz. b) – d) Extracted fitting parameters for the full measurement series represented in a) using kb T = 4.1 × 10−21 J. Grey points indicate data sets for which ω(δf ) 6= 6.3 MHz. value of δf . As a first check on the parameters characterizing the velocity dependent friction models, we note that by changing the frequency shift set point, we impose on the experiment the lateral stiffness (Supporting Information); the feedback control adjusts the z-piezo to maintain an appropriate contact in order to ensure the desired set point. Thus if the model is correct we expect to obtain δf 2kTF ∼ Nc kk . f0

(3)

In Figure 3b), we show that for all of the experimentally accessed frequency shifts, a linear scaling of Nc kk with δf is observed, the former obtained from the model fitting. A numerical prefactor of 9.8 ± 0.3 is found for Equation 3, with the origin on the fit forced to the origin. 8


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10 5 0 0

2000

4000 6000 Nc

8000

Figure 4: Friction force as a function of the number of contacts, the latter determined by comparison between the BP model and our experimental data at various δf , see Figure 3a). The legend shows a, the oscillation amplitude. In addition to the expected linearity between δf and Nc kk , we have found that to a good approximation: the microscopic contact stiffness is constant for all of the imposed frequency shifts, kk = 0.0056 ± 0.0003 N/m; and the values of ω and x∗ reported in Figures 3b) and c) are constant except for the weakest interactions. Thus, combined with the constant kk , we determine that the nature of the contacts does not change with a change in δf . Instead, a simple change in the number of contacts is enough to describe the data on adjusting δf after establishing good contact between the tip and substrate. In Figure 4 we show at three different oscillation amplitudes, the friction force as a function of the number of contacts, Nc , between the tip and substrate. The contact number was obtained from fits to the BP model; for each amplitude we find a linear dependence on the friction with contact number.

To predict the attempt frequency, ω = 6.3 MHz extracted from the fits, BP 20 proposed to consider a filament with length h and contact area A that is dragged along a surface. The expected attempt frequency of contact breaking is thus given by the resonance frequency of the filament in contact with the surface, which for the situation described is ωres ≈ (3.52/h2 )(EI/ρA)1/2 (section 25, ref. 36 ). Using the tungsten filament diameter 2R = 125 µm, Young’s modulus E = 411 GPa, area moment of inertia I = πR4 /4 and the length of the

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filament used here h = 420 µm, we find ωres ≈ 3 MHz in reasonable agreement with the value extracted from the velocity-dependent frictional data. Here, the dominant sliding velocity is imposed by the oscillation of the tuning fork, which vibrates at a frequency two orders of magnitude smaller than the expected attempt frequency – this separation of scales justifies the consideration of the tip oscillation frequency rather than that of the tuning fork for the attempt frequency. As mentioned above, we have been able to fit all data sets without adjusting the attempt frequency (orange points in Figures 3c)) except for the cases of weakest interaction (grey points). To test the hypothesis that the attempt frequency for our nano-scale friction experiment is set by the entire frictional probe, rather than that of the individual asperity as suggested by BP for the macro-scale, we have performed similar experiments to those presented above using shorter independent tips with similar tip radii. We verified that the dynamic friction data display the same scaling crossover displayed in Figures 2 and 3a), with details found in the Supporting Information. Furthermore the best fitting {kk , x∗ , Nc } are similar within a factor of order unity compared to those for the tip described above. Importantly, a significantly different ω is necessary. For a shorter tip of length 220 µm we find ω = 50 ± 4 MHz. A suitable choice of the frictional probe geometry can thus be used to tune the velocity dependence of the friction passively; active suppression of friction was recently achieved by Socoliuc et al. 5 using similar principles. While not necessary for the validity of the analysis presented above, we now turn to a discussion of the possible nature of the nanoscopic contacts. In contrast to the suggestions of BP for pure solid/solid contacts, we have found above a constant kk ≈ 0.005 N/m, whereas for e.g. metal/metal contacts with high loads, the contact stiffness is expected to be of order 105 N/m. As described above, however, we measure nanonewton friction forces which do not damage the sample. While the SAMs used here have relatively low surface energy (advancing water contact angles of ca. 110 ◦ ) it is well known that tungsten 37,38 and other relatively high surface

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energy materials 39 exposed to air will always have at least a monolayer of water adsorbed under ambient conditions. Motivated by this observation, and inspired by recent experiments demonstrating the existence of capillary bridges between high-surface-energy AFM tips and high energy surfaces 24,25,40,41 , we tentatively suggest that the friction observed here is due to the formation and breakup of nanoscale capillary bridges between our SAMs and the tungsten tip. To make a simple estimate of the expected spring constant for such a capillary bridge, we make an idealized assumption 24 of a right cylinder. Under the assumption of small deviations, it is easy to estimate a spring constant kbridge ∼ γ/2 with the dimensionless prefactor accounting for the geometry of the capillary bridge. 42 Taking the surface tension of water as γ/2 ≈ 0.035 N/m, we again find reasonable agreement —within factors of order unity— with the experimentally determined model parameter kk = 0.0056 ± 0.003 N/m. Lastly, we find that the elongation at break satisfies x∗ = 0.58 ± 0.02 nm for the higher δf data as shown in Figure 3d). We do not have a definitive explanation for this value, yet we propose two possibilities. First we note that the density of grafted chains in our SAMs is roughly 4 nm−2 as evaluated from our X-ray reflectivity data (Figure 1a.ii)) and following refs. 28,43 . Thus, a translation over one molecular distance may be enough to break these supposed capillary bridges. Alternatively, at ambient conditions (relative humidity in our laboratory is ca. 40%), the Kelvin radius, found by balancing the capillary pressure and the vapour pressure 44 , is given by rk = γΩ[kb T ln(p/psat )]−1 ≈ −0.87 nm, where Ω is the molecular volume and p/psat is the relative humidity. Thus if the capillary bridges as proposed exist, we may expect them to have typical dimension of order the Kelvin radius and they would break at comparable elongations. Conclusion We have measured the friction between metal tips with radius of several hundred nanometers and solid self-assembled alkyl silane monolayers. Measurements were made using quartz tuning forks operating at resonance with the ability to resolve the conservative and dissipative forces. The friction/velocity relation scales linearly at low velocity and crosses over into a logarithmic scaling at higher velocity. These observations are consistent

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with a recent multi-contact friction model, with the experimental data furthermore being in excellent quantitative agreement. Despite using a solid probe and a solid substrate, the model parameters are consistent with an interpretation that the weak friction forces measured arise from nanoscale capillary bridges forming between the solid contacts which break after elongations on the order of one nanometer. Finally, the two principal conclusions of our study are the following. Practically, the velocity dependence of the friction is sensitive to the geometry of the tip which interacts with the substrate, offering a new method to tune the friction at the nanoscale. Fundamentally, the magnitude of the friction is dependent on the number of contacts between the two solids facing one another, crossing the bridge between frictional phenomena at the macroscale and at the nanoscale.

Acknowledgement The authors acknowledge funding from the European Union’s H2020 Framework Programme / ERC Starting Grant agreement number 637748 - NanoSOFT. JDM was supported by LabEX ENS-ICFP: ANR-10-LABX-0010/ANR-10-IDEX-0001-02 PSL.

Supporting Information Available The following files are available free of charge. • JDMetal_supp.pdf: contains technical descriptions of the experiments, model and fits, as well as calibrations of the lateral stiffness, friction data from other tips, and raw X-ray reflectivity data.

References (1) Lessel, M.; Loskill, P.; Hausen, F.; Gosvami, N.; Bennewitz, R.; Jacobs, K. Physical Review Letters 2013, 111, 035502. 12


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(16) Bennewitz, R. Fundamentals of Friction and Wear on the Nanoscale; Springer, 2015; Chapter 1: Friction Force Microscopy. (17) Persson, B. Physical Review B 1995, 51, 13568–13585. (18) Braun, O.; Peyrard, M. Physical Review Letters 2008, 100, 125501. (19) Braun, O.; Peyrard, M. Physical Review E 2010, 82, 036117. (20) Braun, O.; Peyrard, M. Physical Review E 2011, 83, 046129. (21) Giessibl, F. Applied Physics Letters 1998, 73, 3956–3958. (22) Giessibl, F. Reviews of Modern Physics 2003, 75, 949–983. (23) Niguès, A.; Siria, A.; Vincent, P.; Poncharal, P.; Bocquet, L. Nature Materials 2014, 13, 688–693. (24) Lee, M.; Kim, B.; Kim, J.; Jhe, W. Nature Communications 2015, 6, 7359. (25) Kim, B.; Soyoung Kwon, S.; Lee, M.; Kim, Q.; An, S.; Jhe, W. Proceedings of the National Academy of Sciences 2015, 112, 15619–15623. (26) Schreiber, F. Progress in Surface Science 2000, 65, 151. (27) Genzer, J.; Bhat, R. Langmuir 2008, 24, 2294. (28) Lessel, M.; Bäumchen, O.; Klos, M.; Hähl, H.; Fetzer, R.; Seemann, R.; Paulus, M.; Jacobs, K. Surface and Interface Analysis 2015, 47, 557. (29) Lestelius, M.; Engquist, I.; Tengvall, P.; Chaudhury, M.; Liedberg, B. Colloids and Surfaces B: Biointerfaces 1999, 15, 57. (30) McGraw, J.; Chan, T.; Maurer, S.; Salez, T.; Benzaquen, M.; Raphaël, E.; Brinkmann, M.; Jacobs, K. Proceedings of the National Academy of Sciences 2016, 113, 1168–1173. 14


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(31) Davitt, K.; Pettersen, M.; Rolley, E. Langmuir 2013, 29, 6884. (32) Ju, B.-F.; Chen, Y.-L.; Ge, Y. Review of Scientific Instruments 2011, 82, 013707. (33) Karrai, K.; Grober, R. Applied Physics Letters 1995, 66, 1842–1844. (34) Comtet, J.; Chatte, G.; Antoine Niguès, A.; Bocquet, L.; Siria, A.; Colin, A. Nature Communications 2017, 8, 15663. (35) We note that an equivalent fit could be obtained by increasing the effective temperature – as may occur due to instrumental noise 45,46 – and the contact spring constant by a constant factor, while decreasing the number of contacts by the same factor; that is F (αkk , αkb T, Nc /α) = F (kk , kb T, Nc ) for constant α. We note furthermore that in our calculations of the friction, we have taken the distribution of elongations at contact breaking to be the one suggested by BP in Equation 8 of ref. 20 with n = 1; the functional dependence of FD (v) does not depend sensitively on this choice, although the value of C0 may change. (36) Landau, L.; Lifshitz, E. Theory of Elasticity; Pergamon Press, 1970. (37) Hill, R.; Jacobs, P. Nature 1957, 180, 1117–1118. (38) Hill, R. Vacuum 1961, 11, 260–271. (39) Asay, D.; Kim, S. The Journal of Chemical Physics 2006, 124, 174712. (40) Choe, H.; Hong, M.-H.; Seo, Y.; Lee, K.; Kim, G.; Cho, Y.; Ihm, J.; Jhe, W. Physical Review Letters 2005, 95 . (41) Khan, S. H.; Matei, G.; Patil, S.; Hoffmann, P. M. Physical Review Letters 2010, 105 . (42) The geometry of any nanoscale capillary bridge that may be formed in our experiment will clearly deviate from this idealized right-cylinder calculation, yet the scaling is expected to hold. 15


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(43) McGraw, J. D.; Klos, M.; Bridet, A.; Hähl, H.; Paulus, M.; Castillo, J. M.; Horsch, M.; Jacobs, K. The Journal of Chemical Physics 2017, 146, 203326. (44) Israelachvili, J. Intermolecular and Surface Forces, 3rd ed.; Academic Press, 2011. (45) Dong, Y.; Gao, H.; Martini, A.; Egberts, P. Physical Review E 2014, 90 . (46) Liu, X.-Z.; Ye, Z.; Dong, Y.; Egberts, P.; Carpick, R.; Martini, A. Physical Review Letters 2015, 114 .

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a.i)

PA

Nano Letters ng uni

lock-in, phase-locked loop

fork

t

a.ii)

1 μm

tip substrate

a.ii)

5Å DTS

z-piezo

ACS Paragon Plus Environment z [Å]

1 2 3 4 5 6 7 8 9

Page 18 of 22

SiO2 Si

10 0.1

ρe [Å-3]

6 5 4 3 2 1

V [mV]

δf [Hz]

4 b) 4 c) = 0.15 nmNano Letters a = 4 nm Page 19 ofa22 0.15 3 3 1 0.1 2 2 2 3 4 0.05 51 1 6 0 7 0 ACS Paragon Plus0Environment 8 40 80 0 20 40 60 9 0 10 t [s] t [s]

6

a)

5 FD,ref [nN]

1 2 3 4 5 6 7 8 9 10 11

Nano Letters 10

1

Page 20 of 22

10 0

4

10 -1 10 -1 6

3 2

b) 10 0

10 1

4

1

2

0 0

c) 0 ACS Paragon Plus Environment 5 10 10 -1 10 0 10 1 a [nm] a [nm]

20 21 of2.00 Page 22

Nano Letters

1.67 1.19 0.86 0.54 0.21

a)

ω [Hz]

x∗ [nm]

Nc kk [N/m]

FD [nN]

15 1 2 10 3 4 5 5 6 0 7 0 2 4 6 8 10 8 a [nm] 9 ×10 7 10 c) 11 50 b) 5 12 13 0 14 1 15 25 d) 16 17 18 ACS Paragon Plus Environment 0 0.5 19 0 2.5 5 0 1 2 20 2ktf δf /f0 [N/m] δf [Hz] 21

20

Nano Letters

Page 22 of 22

FD [nN]

15 1 2 3 10 4 5 5 6 7 8 0 0 9 10

10 nm 4 nm 1 nm

ACS Paragon Plus Environment 2000

4000 6000 Nc

8000

Contact Dependence and Velocity Crossover in ... - ACS Publications

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