Comparative Study of Dimensionality and Symmetry Breaking on Nanoscale Friction in the Prandtl–Tomlinson Model with Varying Effective Stiffness

Abstract

Friction Force Microscopy (FFM) measurements on NaCl immersed in ethanol display an increase of the effective contact stiffness with the applied load. This stiffness is estimated from the measured local contact interaction of the tip with the NaCl surface and the Prandtl–Tomlinson (PT) parameter, which reflects the relation between the corrugation stiffness and the effective contact stiffness. Different from FFM measurements in ultrahigh vacuum, for measurements in ethanol surroundings the PT parameters showed a maximum with the applied load. We incorporated this measured load-dependent effective stiffness together with the load-dependent amplitude of the corrugation energy into simulations based on the PT model, and studied its effect on the lateral friction for symmetric 1D and 2D potentials, as well as for an asymmetric 1D potentials. The simulations reproduced the experimentally observed non-monotonous behavior of the PT parameter, and enabled a glimpse on the relation of the characteristic observables (mean maximal slip forces and stiffness) with respect to their governing parameters (corrugation energy, effective stiffness). In all, apart from large deviations from symmetry in the interaction potential, the PT parameter provides a reliable estimate for nanoscale friction over periodic surfaces.

Graphic Abstract

Introduction

The field of nanotribology witnessed a tremendous development in recent decades with the technological advent of measuring techniques that enabled to directly approach and study contacts at the single asperity level with increasing resolution and accuracy. One of these approaches, friction force microscopy (FFM), utilizes the atomic force microscope (AFM) for measuring friction forces with sub-nanometer resolution [1,2,3]. In FFM, the frictional interaction is recorded between the sharp tip of the AFM cantilever and a surface of interest, as they slide against each other. The AFM is used to manipulate the position of the support of a cantilever, and scan the surface at different rates and various loadings. During a scan, the tip bends and releases with respect to its position along the atomic-scale periodicity of the sample (for ordered surfaces), and produces a stick-slip pattern in the lateral force signal. Studies on nanoscale friction have shown that several properties influence this dynamics, such as contact area [4,5,6,7,8,9], sliding velocity [10,11,12,13,14,15,16], temperature [11, 17,18,19,20,21,22,23], anisotropy, symmetry and dimensionality [13, 24,25,26,27,28,29,30,31,32,33], and the applied normal load [4, 5, 23, 26, 33,34,35,36,37,38,39,40,41,42,43].

Efficiently capturing the fundamentals of nanoscale friction, the Prandtl–Tomlinson (PT) model [44,45,46] is used to understand and interpret FFM measurements by describing the sliding interaction between two surfaces in contact at the single asperity level. Within this description, the AFM cantilever’s tip is considered as a point-mass particle, which is dragged by a spring (characterized by an effective stiffness) at some velocity across a periodic interaction (corrugation) potential. Through this portrayal the recorded stick-slip dynamics reflects instabilities during the motion of the tip between consecutive minima in the tip–surface corrugation interaction potential. Accordingly, it adequately expresses various situations and behaviors observed in nanoscale friction [10, 13, 14, 19, 21, 22, 36, 38, 47,48,49]. Although the PT model was criticized for lacking some behaviors manifested by chemical bonding [50], oversimplifying subcritical damping [51], and energy dissipation [52], it still provides a highly useful generic framework, whose underlying principles encompass the basic understanding of nanoscale friction [53].

Essentially, the PT model describes the interaction between two surfaces through their corrugation and elastic interactions. The interplay between these contributions can account for the nature of the friction dynamics (continuous or sharp transitions). A dimensionless parameter, known as the PT parameter, η, was defined as the ratio between the amplitude of the corrugation and elastic interactions [54, 55], and is often used to describes its stability.

FFM measurements on NaCl surface in ultrahigh vacuum (UHV) with a silicon tip showed that at low range of normal loads (up to 6 nN), the mean slip forces increased with the applied load, while the effective stiffness showed small variation around 1 nN/nm [36]. As a result, η increases with the applied load to a value of ~ 5. Going to higher values of normal loads (13–91 nN) under the similar conditions, η linearly grew to 14.5 [56]. Measured in liquid (ethanol) surrounding, the lateral interaction between NaCl surface and silicon tip showed a different behavior, as the effective stiffness increased with the applied normal loads (up to ~ 60 nN), and the maximal value calculated for η was ~ 14 [57]. Comparison between FFM measurements and simulations under UHV and liquid surrounding concluded a close proximity in the friction signals [58], and corrugation energy amplitudes [57]. While these two properties are related to the applied load, the increase of the effective stiffness observed at high loads in liquid surroundings [57] may indicate that the applied load may be manifested also in the elastic term within the PT model.

Previous work on the interaction between NaCl surface and a silicon tip in ethanol using FFM [57] showed an increase of both corrugation energy and effective stiffness with the applied load. PT parameter, calculated from the experimental data (mean slip forces and stick force-position slopes), deviated from the PT parameter which was calculated from the reconstructed parameters (corrugation amplitude and effective stiffness). These differences were speculated to result from the possibility that the actual contour of the surface potential may differ from the ideal symmetric periodic potential presented by the PT model, and also from the possibility of the one-dimensional simplification of the two-dimensional motion of the tip. Here, with the intention to better understand these differences, as expressed through the PT parameter and the effective contact stiffness, we observe an unexpected non-monotonous behavior of η with the applied normal load, as it increases to a maximal value and then slowly decreases. The resulting effective stiffness shows an increase with the applied load, as previously reported [57].

Based on these experimental observations, we perform FFM simulations within the framework of the PT model, and explore the effect of the increasing effective stiffness (with the applied load) with respect to the dimensionality (1D and 2D potentials) on the friction dynamics. Additionally, since the corrugation potential landscapes may exhibit contours that deviate from the ideal sinusoidal description posed by the PT model, we looked into the effect of symmetry breaking in the 1D potentials on the simulated frictional signal. The FFM simulations recreate the non-monotonous behavior with the applied load with minor differences between the 1D, 2D, and small symmetry breaking potentials. These differences were examined by means of the relations between the simulated characteristics with respect to their underlying origins (that were used in the simulations). Large deviations from symmetry showed very different and unexpected features.

Methodology

FFM Measurements of NaCl in Ethanol

NaCl crystal (American Educational Cleavable Halite Mineral) was freshly cleaved along the {100} plane, and glued to a pre-cleaned metal disc with Tempfix adhesive (Ted Pella, inc.). The sample was immediately immersed with ethanol (99.9%, Romical), and placed in the AFM (Cypher-ES, Asylum Research, Oxford Instruments) measuring chamber. This procedure was performed under ambient conditions with a relative humidity (RH) of ~ 35%. At this RH, water molecules are adsorbed to some extent on the NaCl surface; however, it was shown through preferential adsorbance of water on NaCl step edges that above a characteristic value of 40% a water monolayer is formed and initiates the process of ion solvation [59], displaying a sharp transition in surface conductivity [60]. The measurements were performed in ethanol, in which the solubility of NaCl is very limited, and were shown to produce FFM measurements with lattice resolution without damaging the NaCl surfaces [57]. FFM friction loops (with 0.01 gain, at room temperature) measurements on the NaCl surface were performed with silicon cantilevers (SNL-10D Bruker) that were calibrated with the wedge method [61, 62], providing a normal spring constant of 0.05 ± 0.01 N/m, inverse lever sensitivity of 47.07 ± 3.43 nm/V and a wedge conversion factor (lateral sensitivity) of 135.4 ± 27.9 nN/V (equivalent to lateral spring constant of 38.95 ± 5.55 N/m). The scan direction of the FFM (displaying lattice stick-slip) was perpendicular to the cantilever body length direction at a scanning velocity of 60 nm/s collecting at least 100 cycles per normal load. From the friction traces, we measured a distance between slip events of 0.584 ± 0.05 nm, corresponding with the lattice constant of NaCl (a = 0.564 nm).

FFM Numerical Simulations

In order to explore the effect on the normal force on atomic-scale friction dynamics according to our FFM measurements, we numerically solve the Langevin equation of motion (presented below for 1D)

$$M_{eff} \ddot{x}\left( t \right) = - \gamma \dot{x}\left( t \right) - \nabla E\left( {x,F_{N} } \right) + \Gamma \left( t \right).$$
(1)

Being a momentum balance, this equation describes the motion along the x scanning coordinate of the sliding cantilever tip, with an effective mass Meff, over the corrugated surface. γ is the damping coefficient that characterizes the dissipation of the kinetic energy, and Γ(t) is white noise correlated stochastic term with zero mean and \(\left\langle {\Gamma \left( t \right)\Gamma \left( {t^{\prime}} \right)} \right\rangle = {2}\gamma k_{B} T\delta \left( {t{-}t^{\prime}} \right)\) according to the fluctuation-dissipation theorem with kB being Boltzmann's constant, and T is the absolute temperature. The overall potential of the system, E, according to the PT model is defined as a superposition between the corrugation interaction potential and an elastic term:

$$E\left( {x,t} \right) = U_{interaction} \left( x \right) + U_{elastic} \left( {x,t} \right).$$
(2)

The elastic interaction term is approximated as a harmonic spring, \(U_{elastic} \left( {x,t} \right) = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} K_{eff} \left( {x{-}X_{s} } \right)^{{2}}\), where Keff is an effective spring constant, accounting for the contributions of stiffness of the cantilever and the shear at the tip–surface contact [63]. \(X_{s} = Vt\) is the position of the support, where V is the scanning velocity in the lateral direction, and t is the time.

The interaction potential between the tip apex and the sample is given as symmetric periodic function

$$U_{{{\text{interaction}}}}^{1D} \left( {x,F_{N} } \right) = - U_{0} \left( {F_{N} } \right)\cos \left( {\frac{2\pi x}{a}} \right),$$
(3)

where \(U_{0} \left( {F_{N} } \right)\) is the amplitude of the corrugation potential at a given normal load FN, and a is the lattice periodicity. As previously mentioned, the applied normal load is related to the amplitude of the corrugation potential [35, 36, 41, 57, 64, 65]. Following our previous work [57], we quantify this dependency by approximating a (single periodic) local minimum and its consecutive maximum in the interaction potential, which can be written in terms of a reduced bias field [14, 66]:

$$U_{0} \left( {F_{N} } \right) = \varepsilon \left( {1 - \frac{{F_{N} }}{{F_{0} }}} \right)^{\nu }.$$
(4)

This ansatz predicts an increase in the corrugation amplitude with the applied normal load for a repulsive interaction force F0, defined as the force at which the interaction energy amplitude U0 vanishes. ε is the value of the interaction potential amplitude in the absence of external normal load, and ν is a general empiric power.

We extended the simulation for 2D PT model, using the approach introduced by Hölscher et al. [25], which proved highly useful in describing FFM on alkali halide surfaces [30, 31]:

$$U_{{{\text{interaction}}}}^{2D} \left( {x,y,F_{N} } \right) = - U_{0} \left( {F_{N} } \right)\cos \left( {\frac{2\pi x}{a}} \right)\cos \left( {\frac{2\pi y}{a}} \right).$$
(5)

Now, in addition to Eq. (1) the dynamics is now coupled to another equivalent equation for the y-coordinate, where now both the xy coordinates describe the position of the tip along the plane of the interaction potential via

$$\begin{gathered} M_{eff} \ddot{x} = - \gamma \dot{x} - \frac{{\partial E\left( {x,y,F_{N} } \right)}}{\partial x} + \Gamma \left( t \right) \hfill \\ M_{eff} \ddot{y} = - \gamma \dot{y} - \frac{{\partial E\left( {x,y,F_{N} } \right)}}{\partial y} + \Gamma \left( t \right) \hfill \\ \end{gathered}.$$
(6)

Deviations from the symmetric description of the 1D PT model were introduced using asymmetric periodic potentials. For the simulations we assumed that the effective stiffness on the x-axis is equal to that on the y-axis [30]. The general description of these potentials is given by [67, 68]:

$$U_{{{\text{interaction}}}}^{ASym} \left( {x,F_{N} } \right) = \frac{{\alpha \beta U_{0} \left( {F_{N} } \right)\sin \left( {\frac{\pi x}{a}} \right)\cos \left( {\frac{\pi x}{a}} \right)}}{{1 + \beta \cos^{2} \left( {\frac{\pi x}{a}} \right)}}.$$
(7)

We distinguish between two symmetry breakings: Large asymmetry (LAsym), where αL = –0.9 and βL = –0.85, and small asymmetry (SAsym) with αS = 0.9 and βS = –0.25. These values were used to provide the desired shapes of the asymmetric potentials: SAsym, with a small deviation from the symmetric periodic potential and a skew to the right, and LAsym with a large deviation from the symmetric periodic potential and a skew to the left.

For the FFM simulations, we used the following values that were previously reported for NaCl in ethanol [57]: ε = 0.0143 eV, F0 = – 0.245 nN and υ = 1.084. Notice that according to Eq. (4), negative value of F0 makes the amplitude of the interaction potential grow with the applied load, and suggests a mild interaction at negative loads. The fitted υ ~ 1 indicates a linear relation of \(U_{0} \left( {F_{N} } \right)\) for the NaCl-Si in ethanol system. Additional values were taken as Meff = 10-12 kg [30], V = 60 nm/s, a = 0.564 nm. \(K_{eff} \left( {F_{N} } \right)\) was taken as a power-law fit to the measured data, as described below in the next section. The critical damping coefficient, γ, representing the dissipation rate, is associated to the natural vibrational frequency via \(\gamma = \left( {{2}/M_{eff} } \right)\left( {K_{eff} M_{eff} } \right)^{\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }\) [25, 30, 69]. The lateral forces traces were calculated along the scanning (x) direction (which is set out by y for the 2D potential), according to Hooke's law, \(F_{L} = K_{eff} \left[ {x\left( t \right) \, {-}X_{S} } \right]\) [30, 56]. For both experimental and simulated data, errors for measurable sizes (Fmax, Kexp, etc.) were taken as the standard deviations of their distributions, while errors for calculated parameters (η, Keff, U0, etc.) were calculated using the variance formula (propagation of error).

Results and Discussion

We measured friction loops of NaCl in ethanol at a scanning velocity of 60 nm/s, under normal loads with the range FN = 2.3–58.6 nN. Figure 1a shows three exemplary friction loops measured under normal loads of 3.4, 19.8, and 56.3 nN (light-blue, crimson, and black, respectively). The hysteresis between the forward and backward traces of the loops increases with the applied load, and is accompanied with an increase in the maximal slip forces. The slope of the lateral force traces, \(K_{{\exp}} = dF_{L} /{\text{d}}x\), also increases with the applied load, and provides an estimate of the contact stiffness [63]. Figure 1b plots the mean local stiffness \(\left\langle {K_{{\exp}} } \right\rangle\), calculated from the force loops that were measured under different loads, with the mean maximal slip lateral forces, \(\left\langle {F_{{\max}} } \right\rangle\) (with respect to the same applied loads) in its inset.

Fig. 1
figure1

FFM measurements of NaCl in ethanol at 60 nm/s. a Three representative friction loops measured at three different normal loads. b Mean experimental stiffness, measured by the slope of the lateral force in the stick phase as a function of the normal load. The inset shows the mean lateral slip forces as a function of the normal loads. c PT parameter, and d effective stiffness as a function of the applied normal load

The PT parameter is defined as the relation between the corrugation stiffness and the effective contact stiffness, \(\eta = ({2}\pi/a)^{{2}} U_{0} /K_{eff}\) [54, 55]. Stick-slip is expected when the tip-sample interaction is stronger than the contact effective stiffness, i.e., η ≥ 1, and continuous sliding in the opposing situation, η < 1. Additionally, η is also used as a measure to distinguish between single to multiple slip regimens [36, 56, 69, 70]. Within the PT model, for a symmetric periodic (sinusoidal) interaction potential, η can also be calculated directly from the experimentally measured parameters, i.e., from the mean lateral slip forces, \(\left\langle {F_{{\max}} } \right\rangle\), and \(\left\langle {K_{{\exp}} } \right\rangle\) [36] according to

$$\eta = \frac{{2\pi \left\langle {F_{\max } } \right\rangle }}{{\left\langle {K_{{\exp}} } \right\rangle a}} - 1.$$
(8)

When U0 and Keff are in the same order of magnitude, the following correction was proposed to relate the measured mean local stiffness with the effective stiffness [54]:

$$K_{eff} = \left( {1 + \frac{1}{\eta }} \right)\left\langle {K_{{\exp}} } \right\rangle.$$
(9)

At high η, where the corrugation interaction is larger than the effective stiffness, the experimentally assessed stiffness, \(\left\langle {K_{{\exp}} } \right\rangle\) is expected to approach the effective stiffness of the contact, Keff [63].

The values of the PT parameter were calculated with Eq. (8) using the measured \(\left\langle {K_{{\exp}} } \right\rangle\) and \(\left\langle {F_{{\max}} } \right\rangle\), and are presented in Fig. 1c as a function of the applied load, with a spline interpolated dashed line to assess its general trend. Interestingly, the PT parameter rapidly grows to a maximal value of 13.6, and then, above a normal load of ~ 10 nN, slowly decays to ~ 7. Keff was then calculated with Eq. (9), and is plotted as a function of the applied load in Fig. 1d. The effective contact stiffness increases with the applied load, and was empirically fitted with the following power law (dashed line):

$$K_{eff} \left( {F_{N} } \right)\sim K_{0} + K_{1} F_{N}^{k}$$
(10)

with the fitted parameters K0 = 0.95 nN/nm, K1 = 0.012 nN1–k/nm, and k = 1.69. This trend is different than the one reported for FFM NaCl measurements in UHV, where η showed an increase with the applied load, and Keff was nearly constant [36]. When \(\left\langle {F_{{\max}} } \right\rangle\) overwhelms the measured increase in \(\left\langle {K_{{\exp}} } \right\rangle\) (i.e., the corrugation energy increases faster than the effective stiffness), then η is expected to increase (as reported for NaCl in UHV conditions). Here, however, \(\left\langle {F_{{\max}} } \right\rangle\) grows faster than \(\left\langle {K_{{\exp}} } \right\rangle\) below ~10 nN, while above it the trends switch, and \(\left\langle {K_{{\exp}} } \right\rangle\) increases faster than \(\left\langle {K_{{\exp}} } \right\rangle\).

Based on the above, the presence of the solvent appears to affect the effective contact stiffness through the applied load (in addition to the known impact of the load on the amplitude of the interaction energy). We chose to focus on data that were collected at room temperature, at a single constant scanning velocity, such that the amplitude of the corrugation interaction potential and the effective stiffness can be affected by effects of symmetry and dimensionality [13, 25,26,27, 30,31,32,33] through the applied normal load [26, 33, 42, 43]. Consequently, the 1D interpretation of 2D motion of the tip, or some deviation from the symmetry of the sine periodic interaction energy ascribed by the PT model, can potentially influence the behaviors of U0 and Keff. We therefore set to explore the manifestation of the varying \(K_{eff} \left( {F_{N} } \right)\) as described by Eq. (10) within the 1D and 2D PT framework using stochastic numerical calculations that simulate the friction experienced in FFM (see methods for detail). Additionally, we looked into the possibility of large and small symmetry breaking using two asymmetric 1D potentials. Figure 2 shows all of these four interaction potentials.

Fig. 2
figure2

Interaction potentials used in the FFM simulations. a One-dimensional symmetric potential, as given by Eqs. (3) and (4) for several normal loads. b One-dimensional symmetric (solid blue) and two asymmetric potentials, showing small (SAsym, dotted red) and large (LAsym, scattered orange) deviations from the 1D symmetric (sinusoidal) potential, calculated at FN = 15 nN. The asymmetric potentials are given by Eqs. (7) and (4). c Two-dimensional potential calculated at FN = 15 nN given by Eqs. (5) and (4)

Using the potentials shown in Fig. 2, we numerically solved Eqs. (1) and (2) for the 1D potentials and Eqs. (6) and (2) for the 2D potential surface, while including the varying effective stiffness as given by Eq. (10). Figure 3a shows representative simulated FFM traces for the 1D potentials at 15 nN, all displaying stick-slip arrangement. Two exemplary 2D trajectories calculated at two different scan lines (YS) are plotted in Fig. 2b. While the calculated lateral force projections of these 2D trajectories on the XS coordinate display stick-slip pattern (not shown here), their motion along the 2D topology of the surface of the interaction potential follows minimum energy paths, exhibiting zigzag motion, as the “tip” locally slips sideways to a neighboring minimum [30, 31, 71].

Fig. 3
figure3

FFM simulations with Keff(FN) and four interaction potentials. a Three simulated force traces with the one-dimensional potentials (1D, blue marks the sinusoidal symmetric potential, SAsym, red, marks the potential with small deviation from symmetry, and LAsym, orange, marks the potential with large deviations from symmetry). b Plot of two simulated trajectories on the two-dimensional surface, advancing in zigzag that corresponds with the stick-slip motion of the tip. c Mean maximal slip forces, and d. Mean local contact stiffness (evaluated from the stiffness of the force traces in the stick phase, \(K_{{\exp}} = dF_{L} /dx\)) for the four interaction potentials used in the simulations as a function of the normal load

We collected the maximal slip forces (Fmax) and the slopes at the stick phases (Kexp) from all the simulated traces at every normal load, and plotted their averages and standard deviations in Figs. 3c and 4c, respectively. For all the potentials used in the FFM simulations, we see an increase of \(\left\langle {F_{{\max}} } \right\rangle\) and \(\left\langle {K_{{\exp}} } \right\rangle\) with the normal load. First distinction can be made between the symmetric 1D and 2D potentials. The dynamics over the 1D potential produce higher mean lateral forces compared to the ones measured on the 2D surface, as its minimum energy paths are forced to follow the contour of the potential [37]. In the 2D model, the asperity can evade the maxima in the potential's surface (as can be seen in Fig. 3b), resulting with lower lateral forces [13, 25, 30]. The 2D zigzag motion and the lowering values of \(\left\langle {F_{{\max}} } \right\rangle\) with the applied load in 2D simulations compared to the 1D simulations have been previously reported and discussed [13, 24, 25, 30]. Here we also notice that \(\left\langle {K_{{\exp}} } \right\rangle\) attains close values for both 1D and 2D potentials at normal loads below a certain normal load (40 nN for the parameters used here), while above this value the 2D stiffness lags behind the 1D stiffness. Unlike the differences in \(\left\langle {F_{{\max}} } \right\rangle\), which are apparent from low loads, the differences in stiffnesses between the 1D and 2D potentials begin at relatively high load, at which the 2D trajectories are forced more intensely to the nearest minimum, resulting with a non-uniform trend of \(\left\langle {K_{{\exp}} } \right\rangle\) with the load. Considering the deviation from the 1D sinusoidal symmetry, we notice that the simulations based on the potential with the small symmetry breaking produced higher lateral slip forces compared to the 1D symmetric model, while the large asymmetry results with a considerably lower slip forces. This difference is not a consequence of the amount of deviation from symmetry, but rather from the skew of the potential and its orientation. The small asymmetry potential tilts to the left, thus creating a large local uphill stiffness, while the large asymmetry has a strong incline to the left, and exhibits an opposite effect. Furthermore, when combined with the elastic term (Eq. (2)), a saddle point is formed in the transition region of the overall potential, which constitutes the flattening of the lateral force (at lower values compared to the other 1D models) in the vicinity to the transition where the slip occurs (Fig. 3a, orange trace) [68]. A similar tendency, yet to smaller extent is observed for \(\left\langle {K_{{\exp}} } \right\rangle\).

Fig. 4
figure4

Characteristics of load-dependent effective stiffness and corrugation amplitude on simulated lateral friction for four interaction potentials. a PT parameter. b Comparison between the ratio of the estimated corrugation according to the PT model, \(\left\langle {F_{{\max}} } \right\rangle a/({2}\pi )\), and U0 used in the simulations as a function of the load. c Variation of the local stiffness with respect to the effective stiffness with the PT parameter. d Comparison between the ratio of the estimated effective stiffness by its definition (Eq. (9)), \(({1} + {1}/\eta )\left\langle {K_{{\exp}} } \right\rangle\), and Keff used in the simulations as a function of the load

Next we calculate η. Eq. (8) was used for the 1D and 2D sinusoidal potential based simulations. However, in the cases of the asymmetric potentials, the definition given by Eq. (8) becomes inadequate, and the PT parameter can then be evaluated by \(\eta = {-}\left( {d^{{2}} U_{interaction} /dx} \right)|_{{\max}} /K_{eff}\) [6]. This means that η was not evaluated for the asymmetric potentials with "experimental" parameters, but rather directly from their actual potentials and Keff values that were used in the simulations (unlike the 1D and 2D sinusoidal potentials that were estimated from the simulation's output). This intends to compare η, which is “experimentally inaccessible” of the asymmetric potentials to those assessed from the simulations that are based on the PT model, and observe their proximity (or remoteness). The calculated η for the four cases are plotted in Fig. 4a, all displaying the same general maximum non-monotonous behavior with the applied load, in concert with experimentally calculated η from the NaCl measurements in ethanol (Fig. 1c). Nevertheless, the specific nature of each case study is reflected in the individual behavior of its corresponding η. Dwelling first on the dimensionality differences, it can be seen that at normal loads below 60 nN, η2D < η1D, consistent with the behavior observed for their \(\left\langle {F_{{\max}} } \right\rangle\) and \(\left\langle {K_{{\exp}} } \right\rangle\). However, above 60 nN they coincide (within the error range). This may indicate that under sufficiently high loading, the dimensionality has very little influence on the ratio between the corrugation and the contact stiffness. The small deviation asymmetric potential produces higher values of η compared to the symmetric potential up to a normal load of 60 nN above which they overlap. η of the large asymmetric potential grows about twice above all the rest as the ratio between its corrugations to its effective stiffness is considerably larger with respect to other potentials due to its unique deformed contour. Yet, as in the other cases its corrugation stiffness (reflected by its second derivative at the maximum) grows faster than its effective stiffness up to a maximum at 25 nN, from which they reverse their courses.

If we hypothetically consider the values of the simulation variables (U0(FN) and Keff(FN)) as the realistic or absolute true values, we can potentially learn how they influence their phonotypical simulated observables (\(\left\langle {F_{{\max}} } \right\rangle\),\(\left\langle {K_{{\exp}} } \right\rangle\),η) with respect to the different case studies presented above. Starting with the simulated \(\left\langle {F_{{\max}} } \right\rangle\), we used the definition \(\it U_{{0,{\text{obs}}}} \equiv \left\langle {F_{max} } \right\rangle a/({2}\pi)\) as a proxy for the corrugation amplitude based on the PT model [36]. Normalizing the latter with U0, one can see in Fig. 4b that for the 1D sinusoidal potential this ratio is growing asymptotically towards one with the normal load, yet not reaching it. The distance from equality between U0,obs and U0 can be associated with the effect of temperature, as one would expect \(\left\langle {F_{{\max}} } \right\rangle\) to increase when the temperature lowers, and assumes a value that will correspond to the actual barrier height U0 [11, 13]. The effect of dimensionality amplifies the reduction in this ratio, which also grows with the applied load, but to considerably lower values compared to the 1D potential (as \(\left\langle {F_{{\max}} } \right\rangle\) are lesser for the 2D, as can be seen in Fig. 3c). One would expect that this estimate will reflect more realistically the averaging of the experimental measurements shown in Fig. 1. Seeing this ratio for the asymmetric potentials, an additional complexity of deviation from symmetry is added, which in turn can affect the friction dynamics. The sharp (small) asymmetry breaking shows an overshoot of the \(U_{{0,{\text{obs}}}} /U_{0}\) ratio for FN > ~20 nN, while the more distorted (large) asymmetry displays a constrained behavior with very low values.

The measured stiffness with respect to the effective stiffness stresses out the effect of deviation from the sinusoidal symmetry in Fig. 4c. The 1D and 2D follow by definition the η/(η + 1) line, as described by Eq. (9). Within the range of the simulated normal loads (up to 100 nN), they remain within η < 12, and do not reach the region where \(\left\langle {K_{exp} } \right\rangle = K_{eff}\) although it attains ~ 93 % of it. While the sharp (small) asymmetric potential underestimates η/(η + 1) to a small extent, the large asymmetric deviation is quite large, and unlike the other cases, its evolution with η cannot be predicted, as it increases with η for FN < 20 nN and then decrease at higher loads. We defined \(K_{eff,obs} \equiv ({1} + {1}/\eta )\left\langle {K_{exp} } \right\rangle\) as the observed effective stiffness (assessed from the simulations output), and plotted it with respect to the Keff that was used for the simulations (Fig. 1d) for the four case studies as a function of the normal load in Fig. 4d. For the 1D symmetric case, the observed effective stiffness is higher than its underlying value; however, as the normal load increases Keff,obs becomes slightly smaller than Keff (within the range of ~10% beneath Keff). This behavior is a direct outcome of the force dependency of Keff with the normal load. Should Keff display a small variation with the load (as in the reported case for NaCl in UHV [36], η would increase even at high loads (with the increase of \(\left\langle {F_{{\max}} } \right\rangle\)) [56], and Keff,obs is expected to rapidly approach Keff. The 2D simulations show a similar behavior with a wider spread, ranging from within ~25% above Keff to ~20% below it. Below FN = 40 nN, Keff,obs is mostly dominated by η (〈Kexp for 1D and 2D are almost identical), and above it the lower values of \(\left\langle {K_{{\exp}} } \right\rangle\) become more influential (where η becomes similar to that of the 1D simulations with the increase of FN). The small asymmetry simulations produce \(K_{eff,obs} /K_{eff}\) closer to one, with a slight decrease with the normal load. This behavior is rather expected, since this potential has a sharper slope compared to the symmetric potential (Figs. 2b and 3d), while having a similar corrugation amplitude. The opposing asymmetric extremity shows a similar decaying trend, as all the other potentials; however, the range of its values is noticeably lower, going down to 60% underneath Keff. The moderate change of \(\left\langle {F_{{\max}} } \right\rangle\) with respect to \(\left\langle {K_{{\exp}} } \right\rangle\) for this large asymmetric case (Fig. 3c and d), particularly with respect to their U0 and Keff (Figs 4b and 4d), results with the very different behavior of η shown in Fig. 4a.

Conclusion

In this work, we observe experientially that in the presence of ethanol, the interpolated trend of the PT parameter, η, based on its calculated values for NaCl, grows to a maximum and then decreases, different from its reported behavior in UHV conditions. Consequently, we see a considerate increase of the effective contact stiffness with the applied load. This reveals the impact of the surrounding on nanoscale friction. η is influenced by both the mean slip forces (\(\left\langle {F_{{\max}} } \right\rangle\), related to the amplitude of the corrugation potential) and the mean local stiffness (\(\left\langle {K_{{\exp}} } \right\rangle\), determined by the contour of the potential) at a given normal load. At constant temperature and scanning velocity, these parameters are expected to be affected by the dimensionality and the contour of the underlying interaction potential. We therefore implemented the experimentally observed increase of the effective stiffness into PT model-based FFM simulations. This supplements the general understanding that the effect of the normal load is typically introduced through the amplitude of the corrugation interaction potential. The FFM simulations served as a “toy model” to rationalize the effect of normal load, dimensionality and deviations from symmetry in FFM.

From the numerical FFM simulations, where both the corrugation amplitude and the effective stiffness vary with the applied load in a similar fashion to that observed experimentally, we reproduce the non-monotonous behavior of the PT parameter and resulting effective stiffness. Although the effect of the variations in mean slip forces and local stiffnesses for the different simulated scenarios is present, it appears to be less significant compared to the variation in the increasing effective stiffness. Studying the effect of dimensionality and deviation from the symmetric sinusoidal description of the PT model, we see that apart from the extreme case of large deviation from symmetry, these effects indeed manifest themselves in the friction dynamics to some extent, however conserve the main governing trends. The possibility of moderate separation of the interaction potential from a symmetric sine shape indicates that the PT model still may serve as a relatively good approximation for the contour of the surface corrugation potential.

The observed increase effective stiffness with the normal load raises questions regarding its origin. FFM measurements and molecular dynamics simulations on graphene/graphite and metals in the presence of water surroundings concluded that at high normal loads the water had insignificant effect of friction [58, 72]; however, the presence of water was shown to affect the load-dependent friction hysteresis on graphene [39]. Yet, the question whether this effect is specific to the system measured here or may be of more general nature for the interaction between other non-polar liquids with ionic crystals requires further investigation. It is most probable the superposition between the nature of the surfaces (alkali halides, metals, etc.) and their surrounding (UHV, air, N2, liquids) impacts the friction in unexpected ways through mechanisms of heat release and possible lubrication at the single molecule level. A key to understand this interaction most probably lays through variations in the dissipation coefficient [13, 56, 69, 73, 74] (and its manifestation in slip length) that was regarded here as constant, and serves as subject for a different work. Nanoscale friction in liquid surrounding opens a myriad of questions that need to be more thoroughly investigated with various sliding contacts, environments, and reinforcement with detailed molecular simulations.

References

  1. 1.

    Mate, C.M., McClelland, G.M., Erlandsson, R., Chiang, S.: Atomic-scale friction of a tungsten tip on a graphite surface. Phys. Rev. Lett. 59, 1942–1945 (1987)

    CAS  Article  Google Scholar 

  2. 2.

    Bhushan, B., Israelachvili, J.N., Landman, U.: Nanotribology: friction, wear and lubrication at the atomic scale. Nature 374, 607–616 (1995)

    CAS  Google Scholar 

  3. 3.

    Carpick, R.W., Salmeron, M.: Scratching the Surface: fundamental investigations of tribology with atomic force microscopy. Chem. Rev. 97, 1163–1194 (1997)

    CAS  Article  Google Scholar 

  4. 4.

    Carpick, R.W., Ogletree, D.F., Salmeron, M.: A general equation for fitting contact area and friction vs load measurements. J. Colloid Interface Sci. 211, 395–400 (1999)

    CAS  Article  Google Scholar 

  5. 5.

    Mo, Y., Turner, K.T., Szlufarska, I.: Friction laws at the nanoscale. Nature 457, 1116–1119 (2009)

    CAS  Article  Google Scholar 

  6. 6.

    Müser, M.H.: Velocity dependence of kinetic friction in the Prandtl-Tomlinson model. Phys. Rev. B 84, 125419 (2011)

    Article  CAS  Google Scholar 

  7. 7.

    Dietzel, D., Feldmann, M., Schwarz, U.D., Fuchs, H., Schirmeisen, A.: Scaling laws of structural lubricity. Phys. Rev. Lett 111, 235502 (2013)

    Article  CAS  Google Scholar 

  8. 8.

    Cihan, E., İpek, S., Durgun, E., Baykara, M.Z.: Structural lubricity under ambient conditions. Nat. Commun. 7, 12055 (2016)

    CAS  Article  Google Scholar 

  9. 9.

    Dong, Y., Li, Q., Martini, A.: Molecular dynamics simulation of atomic friction: a review and guide. J. Vac. Sci. Technol. 31, 030801 (2013)

    Article  CAS  Google Scholar 

  10. 10.

    Gnecco, E., Bennewitz, R., Gyalog, T., Loppacher, C., Bammerlin, M., Meyer, E., et al.: Velocity dependence of atomic friction. Phys. Rev. Lett. 84, 1172–1175 (2000)

    CAS  Article  Google Scholar 

  11. 11.

    Sang, Y., Dubé, M., Grant, M.: Thermal effects on atomic friction. Phys. Rev. Lett. 87, 174301 (2001)

    CAS  Article  Google Scholar 

  12. 12.

    Dudko, O.K., Filippov, A.E., Klafter, J., Urbakh, M.: Dynamic force spectroscopy: a Fokker-Planck approach. Chem. Phys. Lett. 352, 499–504 (2002)

    CAS  Article  Google Scholar 

  13. 13.

    Fusco, C., Fasolino, A.: Velocity dependence of atomic-scale friction: a comparative study of the one- and two-dimensional tomlinson model. Phys. Rev. B 71, 045413 (2005)

    Article  CAS  Google Scholar 

  14. 14.

    Evstigneev, M., Schirmeisen, A., Jansen, L., Fuchs, H., Reimann, P.: Force Dependence of Transition Rates in Atomic Friction. Phys. Rev. Lett. 97, 240601 (2006)

    Article  CAS  Google Scholar 

  15. 15.

    Apostoli, C., Giusti, G., Ciccoianni, J., Riva, G., Capozza, R., Woulache, R.L., et al.: Velocity dependence of sliding friction on a crystalline surface. Beilstein J. Nanotechnol. 8, 2186–2199 (2017)

    CAS  Article  Google Scholar 

  16. 16.

    Iglesias, M.L., Gonçalves, S.: Sliding and dry friction: prandtl-tomlinson athermal model revisited. Braz. J. Phys. 48, 585–591 (2018)

    Article  Google Scholar 

  17. 17.

    Szoszkiewicz, R., Riedo, E.: Nanoscopic friction as a probe of local phase transitions. Appl. Phys. Lett. 87, 033105 (2005)

    Article  CAS  Google Scholar 

  18. 18.

    Schirmeisen, A., Jansen, L., Hölscher, H., Fuchs, H.: Temperature dependence of point contact friction on silicon. Appl. Phys. Lett. 88, 123108 (2006)

    Article  CAS  Google Scholar 

  19. 19.

    Krylov, S.Y., Frenken, J.W.M.: The crucial role of temperature in atomic scale friction. J. Phys. Condens. Matter. 20, 354003 (2008)

    Article  CAS  Google Scholar 

  20. 20.

    Barel, I., Urbakh, M., Jansen, L., Schirmeisen, A.: Multibond dynamics of nanoscale friction: the role of temperature. Phys. Rev. Lett. 104, 066104 (2010)

    Article  CAS  Google Scholar 

  21. 21.

    Jansen, L., Hölscher, H., Fuchs, H., Schirmeisen, A.: Temperature dependence of atomic-scale stick-slip friction. Phys. Rev. Lett. 104, 256101 (2010)

    Article  CAS  Google Scholar 

  22. 22.

    Mazo, J.J., Dietzel, D., Schirmeisen, A., Vilhena, J.G., Gnecco, E.: Time strengthening of crystal nanocontacts. Phys. Rev. Lett. 118, 246101 (2017)

    Article  Google Scholar 

  23. 23.

    Ouyang, W., Cheng, Y., Ma, M., Urbakh, M.: Load-velocity-temperature relationship in frictional response of microscopic contacts. J Mech Phys Solids 137, 103880 (2020)

    Article  Google Scholar 

  24. 24.

    Fujisawa, S., Kishi, E., Sugawara, Y., Morita, S.: Atomic-scale friction observed with a two-dimensional frictional-force microscope. Phys. Rev. B 51, 7849–7857 (1995a)

    CAS  Article  Google Scholar 

  25. 25.

    Hölscher, H., Schwarz, U.D., Wiesendanger, R.: Modelling of the scan process in lateral force microscopy. Surf. Sci. 375, 395–402 (1997)

    Article  Google Scholar 

  26. 26.

    Fujisawa, S., Yokoyama, K., Sugawara, Y., Morita, S.: Analysis of experimental load dependence of two-dimensional atomic-scale friction. Phys. Rev. B 58, 4909–4916 (1998)

    CAS  Article  Google Scholar 

  27. 27.

    Müser, M.H.: Structural lubricity: role of dimension and symmetry. Europhys. Lett. (EPL) 66, 97–103 (2004)

    Article  CAS  Google Scholar 

  28. 28.

    Gao, G., Cannara, R.J., Carpick, R.W., Harrison, J.A.: Atomic-scale friction on diamond: a comparison of different sliding directions on (001) and (111) surfaces Using MD and AFM. Langmuir 23, 5394–5405 (2007)

    CAS  Article  Google Scholar 

  29. 29.

    Karino, W., Shindo, H.: Frictional force microscopic detection of anisotropy at NaCl (100), (110) and (111) surfaces. Tribol. Int. 40, 1568–1573 (2007)

    CAS  Article  Google Scholar 

  30. 30.

    Steiner, P., Roth, R., Gnecco, E., Baratoff, A., Maier, S., Glatzel, T., et al.: Two-dimensional simulation of superlubricity on NaCl and highly oriented pyrolytic graphite. Phys. Rev. B 79, 045414 (2009)

    Article  CAS  Google Scholar 

  31. 31.

    Fajardo, O.Y., Gnecco, E., Mazo, J.J.: Anisotropy effects and friction maps in the framework of the 2d PT model. Physica B Condens. Matter. 455, 44–48 (2014)

    CAS  Article  Google Scholar 

  32. 32.

    Takoutsing, C.S., Djuidjé Kenmoé, G., Kofané, T.C.: Effects of anisotropy and substrate shape on atomic friction force in two-dimensional model. Tribol. Lett. 65, 107 (2017)

    Article  CAS  Google Scholar 

  33. 33.

    Dagdeviren, O.E.: Exploring load, velocity, and surface disorder dependence of friction with one-dimensional and two-dimensional models. Nanotechnology 29, 315704 (2018)

    Article  CAS  Google Scholar 

  34. 34.

    Fujisawa, S., Kishi, E., Sugawara, Y., Morita, S.: Load dependence of two-dimensional atomic-scale friction. Phys. Rev. B 52, 5302–5305 (1995b)

    CAS  Article  Google Scholar 

  35. 35.

    Zaloj, V., Urbakh, M., Klafter, J.: Modifying friction by manipulating normal response to lateral motion. Phys. Rev. Lett. 82, 4823–4826 (1999)

    CAS  Article  Google Scholar 

  36. 36.

    Socoliuc, A., Bennewitz, R., Gnecco, E., Meyer, E.: Transition from stick-slip to continuous sliding in atomic friction: entering a new regime of ultralow friction. Phys. Rev. Lett. 92, 134301 (2004)

    CAS  Article  Google Scholar 

  37. 37.

    Fusco, C., Fasolino, A.: Power-law load dependence of atomic friction. Appl. Phys. Lett. 84, 699–701 (2004)

    CAS  Article  Google Scholar 

  38. 38.

    Pawlak, R., Ouyang, W., Filippov, A.E., Kalikhman-Razvozov, L., Kawai, S., Glatzel, T., et al.: Single-molecule tribology: force microscopy manipulation of a porphyrin derivative on a copper surface. ACS Nano. 10, 713–722 (2016)

    CAS  Article  Google Scholar 

  39. 39.

    Ye, Z., Egberts, P., Han, G.H., Johnson, A.T.C., Carpick, R.W., Martini, A.: Load-dependent friction hysteresis on graphene. ACS Nano. 10, 5161–5168 (2016)

    CAS  Article  Google Scholar 

  40. 40.

    Sharp, T.A., Pastewka, L., Lignères, V.L., Robbins, M.O.: Scale- and load-dependent friction in commensurate sphere-on-flat contacts. Phys. Rev. B 96, 155436 (2017)

    Article  Google Scholar 

  41. 41.

    Agmon, L., Shahar, I., Birodker, B.-E., Skuratovsky, S., Jopp, J., Berkovich, R.: Application of static disorder approach to friction force microscopy of catalyst nanoparticles to estimate corrugation energy amplitudes. J. Phys. Chem. C 123, 3032–3038 (2019)

    CAS  Article  Google Scholar 

  42. 42.

    Fessler, G., Sadeghi, A., Glatzel, T., Goedecker, S., Meyer, E.: Atomic friction: anisotropy and asymmetry effects. Tribol. Lett. 67, 59 (2019)

    Article  Google Scholar 

  43. 43.

    Cao, X.A., Gan, X., Lang, H., Yu, K., Ding, S., Peng, Y., et al.: Anisotropic nanofriction on MoS2 with different thicknesses. Tribol. Int. 134, 308–316 (2019)

    CAS  Article  Google Scholar 

  44. 44.

    Prandtl, L.: Ein Gedankenmodell zur kinetischen Theorie der festen Körper. Z. Angew. Math. Mech. 8, 85–106 (1928)

    Article  Google Scholar 

  45. 45.

    Tomlinson, G.A.C.V.I.: A molecular theory of friction. Lond. Edinb. Dubl. Phil. Mag. 7, 905–939 (1929)

    CAS  Article  Google Scholar 

  46. 46.

    Popov, V.L., Gray, J.A.T.: Prandtl-Tomlinson model: history and applications in friction, plasticity, and nanotechnologies. Z. Angew. Math. Mech. 92, 683–708 (2012)

    Article  Google Scholar 

  47. 47.

    Hölscher, H., Raberg, W., Schwarz, U.D., Hasbach, A., Wandelt, K., Wiesendanger, R.: Imaging of sub-unit-cell structures in the contact mode of the scanning force microscope. Phys. Rev. B 59, 1661–1664 (1999)

    Article  Google Scholar 

  48. 48.

    Schirmeisen, A., Jansen, L., Fuchs, H.: Tip-jump statistics of stick-slip friction. Phys. Rev. B 71, 245403 (2005)

    Article  CAS  Google Scholar 

  49. 49.

    Hölscher, H., Ebeling, D., Schwarz, U.D.: Friction at Atomic-Scale Surface Steps: Experiment and Theory. Phys Rev Lett 101, 246105 (2008)

    Article  CAS  Google Scholar 

  50. 50.

    Vanossi, A., Manini, N., Urbakh, M., Zapperi, S., Tosatti, E.: Colloquium: modeling friction: from nanoscale to mesoscale. Rev. Mod. Phys. 85, 529–552 (2013)

    CAS  Article  Google Scholar 

  51. 51.

    Krylov, S.Y., Frenken, J.W.M.: The problem of critical damping in nanofriction. Colloid J. 74, 569–572 (2012)

    CAS  Article  Google Scholar 

  52. 52.

    Krylov, S.Y., van Baarle, D.W., Beck, M.E.S., Frenken, J.W.M.: Why do we “feel” atoms in nano-scale friction? Colloid J. 79, 81–86 (2017)

    CAS  Article  Google Scholar 

  53. 53.

    Schwarz, U.D., Hölscher, H.: Exploring and explaining friction with the Prandtl-Tomlinson model. ACS Nano. 10, 38–41 (2016)

    CAS  Article  Google Scholar 

  54. 54.

    Gnecco, E., Bennewitz, R., Gyalog, T., Meyer, E.: Friction experiments on the nanometre scale. J Phys Condens Matter 13, R619–R642 (2001)

    CAS  Article  Google Scholar 

  55. 55.

    Bennewitz, R., Gnecco, E., Gyalog, T., Meyer, E.: Atomic friction studies on well-defined surfaces. Tribol. Lett. 10, 51–56 (2001)

    CAS  Article  Google Scholar 

  56. 56.

    Roth, R., Glatzel, T., Steiner, P., Gnecco, E., Baratoff, A., Meyer, E.: Multiple slips in atomic-scale friction: an indicator for the lateral contact damping. Tribol. Lett. 39, 63–69 (2010)

    CAS  Article  Google Scholar 

  57. 57.

    Agmon, L., Shahar, I., Yosufov, D., Pimentel, C., Pina, C.M., Gnecco, E., et al.: Estimation of interaction energy and contact stiffness in atomic-scale sliding on a model sodium chloride surface in ethanol. Sci. Rep. 8, 4681 (2018)

    Article  CAS  Google Scholar 

  58. 58.

    Vilhena, J.G., Pimentel, C., Pedraz, P., Luo, F., Serena, P.A., Pina, C.M., et al.: Atomic-scale sliding friction on graphene in water. ACS Nano. 10, 4288–4293 (2016)

    CAS  Article  Google Scholar 

  59. 59.

    Verdaguer, A., Sacha, G.M., Luna, M., Ogletree, D.F., Salmeron, M.: Initial stages of water adsorption on NaCl (100) studied by scanning polarization force microscopy. J. Chem. Phys. 123, 124703 (2005)

    Article  CAS  Google Scholar 

  60. 60.

    Shimizu, T.K., Maier, S., Verdaguer, A., Velasco-Velez, J.-J., Salmeron, M.: Water at surfaces and interfaces: from molecules to ice and bulk liquid. Prog. Surf. Sci. 93, 87–107 (2018)

    CAS  Article  Google Scholar 

  61. 61.

    Ogletree, D.F., Carpick, R.W., Salmeron, M.: Calibration of frictional forces in atomic force microscopy. Rev. Sci. Instrum. 67, 3298–3306 (1996)

    CAS  Article  Google Scholar 

  62. 62.

    Varenberg, M., Etsion, I., Halperin, G.: An improved wedge calibration method for lateral force in atomic force microscopy. Rev. Sci. Instrum. 74, 3362–3367 (2003)

    CAS  Article  Google Scholar 

  63. 63.

    Carpick, R.W., Ogletree, D.F., Salmeron, M.: Lateral stiffness: A new nanomechanical measurement for the determination of shear strengths with friction force microscopy. Appl Phys Lett 70, 1548–1550 (1997)

    CAS  Article  Google Scholar 

  64. 64.

    Zhong, W., Tománek, D.: First-principles theory of atomic-scale friction. Phys. Rev. Lett. 64, 3054–3057 (1990)

    CAS  Article  Google Scholar 

  65. 65.

    Riedo, E., Gnecco, E., Bennewitz, R., Meyer, E., Brune, H.: Interaction potential and hopping dynamics governing sliding friction. Phys. Rev. Lett. 91, 084502 (2003)

    CAS  Article  Google Scholar 

  66. 66.

    Garg, A.: Escape-field distribution for escape from a metastable potential well subject to a steadily increasing bias field. Phys. Rev. B 51, 15592–15595 (1995)

    CAS  Article  Google Scholar 

  67. 67.

    Pesz, K., Gabryś, B.J., Bartkiewicz, S.J.: Analytical solution for the Feynman ratchet. Phys. Rev. E 66, 061103 (2002)

    Article  CAS  Google Scholar 

  68. 68.

    Berkovich, R., Klafter, J., Urbakh, M.: Analyzing friction forces with the Jarzynski equality. J. Phys. Condens. Matter. 20, 354008 (2008)

    Article  CAS  Google Scholar 

  69. 69.

    Gnecco, E., Roth, R., Baratoff, A.: Analytical expressions for the kinetic friction in the Prandtl-Tomlinson model. Phys. Rev. B 86, 035443 (2012)

    Article  CAS  Google Scholar 

  70. 70.

    Medyanik, S.N., Liu, W.K., Sung, I.-H., Carpick, R.W.: Predictions and observations of multiple slip modes in atomic-scale friction. Phys. Rev. Lett. 97, 136106 (2006)

    Article  CAS  Google Scholar 

  71. 71.

    Gyalog, T., Bammerlin, M., Lüthi, R., Meyer, E., Thomas, H.: Mechanism of atomic friction. EPL 31, 269–274 (1995)

    CAS  Article  Google Scholar 

  72. 72.

    Lim, Y., Park, H., Caron, A.: Investigation on the role of interfacial water on the tribology between graphite and metals. RSC Adv. 9, 7285–7291 (2019)

    CAS  Article  Google Scholar 

  73. 73.

    Cammarata, A., Nicolini, P., Simonovic, K., Ukraintsev, E., Polcar, T.: Atomic-scale design of friction and energy dissipation. Phys. Rev. B 99, 094309 (2019)

    CAS  Article  Google Scholar 

  74. 74.

    van Baarle, D.W., Krylov, S.Y., Beck, M.E.S., Frenken, J.W.M.: On the non-trivial origin of atomic-scale patterns in friction force microscopy. Tribol. Lett. 67, 15 (2018)

    Article  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge the support from the I-CORE Program of the Planning and Budgeting Committee and The Israel Science Foundatisson (Grant No. 152/11).

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Skuratovsky, S., Agmon, L. & Berkovich, R. Comparative Study of Dimensionality and Symmetry Breaking on Nanoscale Friction in the Prandtl–Tomlinson Model with Varying Effective Stiffness. Tribol Lett 68, 113 (2020). https://doi.org/10.1007/s11249-020-01355-0

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Keywords

  • AFM
  • Effective stiffness
  • Friction force microscopy
  • Interaction potential
  • Langevin simulations
  • Nanotribology
  • Prandtl–Tomlinson model