Rate-state friction in microelectromechanical systems interfaces: Experiment and theory

Friction in microelectromechanical systems (MEMS) is of great interest because it can dominate the mechanical response of structures in which inertial forces are small. Generally, multiple asperities are in contact when MEMS surfaces rub. While sliding tests performed with atomic force microscopes provide information about frictional behavior, the number of asperities in contact is nominally one or at most a few.^1,2 To measure realistic MEMS interfaces directly, several studies have used in-situ platforms^3–7 to gain information on static and dynamic friction. The studies provide information on the relative effectiveness of different coatings in lowering friction and in providing resistance to wear. Static and dynamic coefficients of friction (⁠$μs$ and $μd$⁠, respectively), assumed to be constants, have been reported for polysilicon,^5–7 single crystal silicon,⁴ and organic monolayer-coated polysilicon.³ These in-situ test platforms typically employ comb drive actuators to provide tangential force to the rubbing interface. Such actuators deliver relatively low forces (≲10 μN) over a relatively small travel range (≲20 μm). Furthermore, they inherently control load rather than displacement. As such, they are not ideal for studying stick-slip phenomena, which requires a pulling actuator that can deliver high forces over a large travel range while controlling displacement.

At the macroscale, the prototypical test system is a constant velocity puller, a spring, and a mass-loaded block as illustrated in Figure 1 schematic. The spring force is the product of the spring constant $k$ and the spring elongation $e$⁠, which is the difference between the puller position $xp$ and the frictional block position $x$⁠. The elongation $e$ = 0 corresponds to zero spring force.

From this system, a rich picture on stick-slip and steady sliding friction has been advanced. A formalism known as rate-state theory^8–11 has been developed for interfaces contacting at multiple asperities. One widely used model is the Ruina-Dieterich slip law, which gives the instantaneous friction coefficient $μ$ as a function of the state $ϕ$ (contact age, s) and instantaneous block velocity $ẋ$

The parameter $μ0$ is the friction coefficient for $ẋ$ = $ϕ$ = 0, while $vc$⁠, $dc$⁠, $A$⁠, and $B$ are constants. The “dot” indicates the derivative with respect to time. Unity is added to the logarithms to regularize them (to prevent divergence at $ẋ=0$ or $ϕ=0$⁠). Both $A$ (velocity strengthening) and $B$ (static aging) coefficients are taken to be positive. The parameter $vc$ normalizes $ẋ$⁠, while $dc$ represents a memory length that is associated with sliding and is on the order of the diameter of contacting asperities.¹² This law has been shown experimentally to apply to a wide range of materials and experimental setups in dry conditions, including Bristol board¹³ and rock on rock.^14,15 It can also be derived theoretically assuming thermally activated motion due to a periodic pinning potential biased by the external load.^13,16,17

The form of the differential equation that governs the state's evolution with time is also crucial. One such form is the Dieterich-Ruina aging law

where $dc$ is a memory length. This equation allows for re-strengthening of stationary contacts (for $ẋ=0$⁠, $ϕ̇=1$⁠, meaning contact age increases with time) and an exponential approach to a steady-state value at constant pulling speed $vp$ (for $ẋ=vp$⁠, $ϕss=dc/vp$⁠, where the ss subscript means steady state). Equations (1) and (2) are coupled. In one simple example, it is seen that the static friction coefficient scales logarithmically with time when the block is at rest $(μs∼ln(1+tvc/dc))$⁠. In another, if the block moves at constant velocity, $ϕ$ and hence $μ$ attain steady-state values.

In related work, Israelachivili and coworkers^18–20 studied stick-slip friction of spherical and chain molecule boundary lubricants using the surface forces apparatus (SFA). They found that the films underwent transitions between solid-like and liquid-like states, causing stick-slip motion. Transitions between these states were governed by characteristic melting and freezing rates, both of which exhibited load and temperature dependence. Stop-start experiments were performed where the block was first slid at a constant velocity and then stopped at a time taken to be $t=0$⁠. The block was then pulled again after a wait time $t$⁠. For $t<τ$ (a characteristic solidification nucleation time), the film remained liquid-like and no stiction spike was observed, while for $t>τ$⁠, the film became solid-like and a stiction spike was observed. The transition from zero stiction to a large stiction spike was abrupt (Fig. 15 in Ref. 20). They found that for increasing normal load, $τ$ increased.

Motivated by the SFA experiments, Carlson and Batista²¹ proposed a set of equations similar to the Ruina-Dieterich slip law and Dieterich-Ruina aging law. Accordingly,

In this phase transition model (PTM), the friction force $F0$ is composed of state component $θ$ (N) and a velocity component $βẋ$ (⁠$β$-Ns/m). The state of the interface is bounded by $θmin$ and $θmax$⁠, corresponding to fully liquid-like and fully solid-like, respectively. Transitions from liquid-like to solid-like (“freezing”) are governed by the parameter $τ$ (N s), while the reverse (“melting”) is governed by $α$ (1/m). With this model, they were able to capture the important characteristics of the stop-start experiments reported by Israelachivili and coworkers.

There are several important points of comparison between the models. Most importantly, the physical assumptions between the models are completely different. In the Ruina-Dieterich model, the multi-asperity surfaces have a small real contact area. This area may increase continuously with time but it never saturates, as the real area remains small in comparison to the nominal area. In fact, Dieterich and Kilgore¹² showed the real contact area was at most several percent of the nominal area. In the PTM, the key idea is the contact area remains fixed, but the aging of the third body, the film, causes the changes in frictional behavior. This film has well defined limits corresponding to fully solid-like and fully liquid-like states.

With these differences in physical assumptions, it is not surprising that the corresponding models have differences in how the state component evolves with time or distance. In the PTM, the state component of friction is bounded, unlike in the Dieterich-Ruina aging law, where the contact age and thus friction coefficient increase indefinitely. For a block at rest, the friction force follows a logistic (sigmoidal) curve for the PTM compared to a logarithmic curve. The parameter $α$ that controls melting in the PTM is analogous to 1/$dc$ in the Ruina-Dieterich aging law. Although analogous, $α$ was used in the context of phase change after a sliding length in the SFA experiments, where there is contact in a single asperity interface over a large contact area. Estimating values (i) change from static to kinetic friction $|Δμ|$ = 0.5, (ii) normal load $Nb$ = 30 mN, (iii) spring constant $k$ = 10³ N/m, and (iv) non-dimensionalized $α*=30$⁠, the characteristic slip distance²¹ 1/$α$ = $ΔμNb/kα*$ is 0.5 μm. This corresponds to a distance much less than the contact length (many tens of μm) in the SFA. On the other hand, $dc$ is a memory length associated with contact diameter in a multi-asperity interface, and it is assumed that fresh sets of asperities are coming into contact as sliding occurs. Hence, aging is associated with the time of contact rather than the phase of the interface material. Another important point in the PTM model is that $F0$ represents only a friction force—no normal load force dependence is explicitly incorporated.

In this paper, we propose and evaluate a MEMS friction test platform that resembles the prototypical mass-spring system shown in Figure 1. A stepper motor that provides long-range travel, a wide range of step frequencies, and high pulling force is implemented. Fixed-guided beams provide spring force, while electrostatics are used to provide normal load to the sliding block. Two different monolayer boundary lubricants have been applied to the devices. Due to surface roughness, a multi-asperity interface is being examined. If frictional aging is observed with this platform, it is it due to the increased real contact area, the aging of the third body, or some combination of both. We will examine this issue in Sec. VI.

An important difference from the macroscale test, intrinsic to the microscale, must be noted. In many experimental macroscale systems, the “constant” velocity puller is actually a stepper motor. However, steady sliding can still be achieved provided that the stepping period is smaller than the inertial response time, $mb/k/π,$ of the mass-spring system. The microscale actuator implemented here is also a stepper motor with step size $Δxp$ operating at step frequency $fp$⁠. At the microscale, although $Nb$ can be made to range widely, the block mass $mb$ cannot easily be changed. The microscale block mass is ∼10⁹ times smaller than the macroscale block, and its inertia is insufficient for the steady sliding to occur at the average velocities $Δxpfp$ currently attained by the puller. Hence, we report below on stick slip, but not on steady sliding behavior.

Besides exhibiting static aging, during stick-slip cycles, the microscale platform reveals characteristic slip lengths $δ$ that depend on $Nb$ and $k$⁠. The lengths cannot be explained with constant values of $μs$ and $μd$⁠. With the assumption of full contact rejuvenation after slip events, parameter values for static aging are extracted using an exponential model, which fits the data better than a logarithmic or logistic model. A numerical state-space model, a hybrid of the rate-state models described above, is then developed to simulate the full stick-slip sequence, including the individual steps of the stepper motor. The rejuvenation assumption is revisited and is seen to be justified. Furthermore, the model satisfactorily predicts the unexpected observation that the slip event load drop, equal to $kδ$⁠, depends on $k$⁠.

The device was implemented using the Sandia Ultra-planar Multi-level MEMS Technology (SUMMiT V^TM) fabrication process.²² This process was developed by Sandia National Labs and uses five levels of polysilicon that are separated by sacrificial oxide layers. The platform consisting of these components is called the nanofrictor. The applied normal load on the block $Nb$ and the puller (or stepper motor) step frequency $fp$ can be varied in a given test platform. On adjacent platforms on the same chip, the spring stiffness $k$ has been changed. However, puller step size $Δxp$ is constant in all experiments. The average puller velocity is $v¯p$⁠, which equals $fpΔxp$⁠. An optical image of the test system is shown in Figure 2, where the stepper motor, spring, and friction block comprise the main system components. These are overviewed next and subsequently described in more detail.

To map system response, it is desirable to design a test system in which the parameters $v¯p$⁠, $mb$⁠, and $k$ can be varied over a wide range. The puller frequency $fp$ is easily varied. With surface micromachining, it is difficult to vary $mb$ by a large amount because the layers are thin. However, one major effect of $mb$ is simply that it is used to change the normal force, $Nb$⁠. Varying $Nb$ over a large range can be easily accomplished with electrostatic loading. It is also easy to vary $k$ by design because it is proportional to (⁠$w/l$⁠)³, where $w$ and $l$ are the width and length of an individual spring element (see Figure 2).

The puller must be able to supply force over a wide range in order to sustain a friction force $F0$ = $μNb$⁠. We conservatively expect 0.1 < $μ$ < 0.9 from previous studies,^3,23 while $Nb$ can range from nN to mN. Hence, the actuator should supply force up to the milliNewton scale. Another desirable actuator feature is long-range travel. This makes it possible to overcome $F0$ when the spring is soft (low $k$⁠), or the friction block is heavily loaded (high $Nb$⁠). Long-range travel also enables friction testing over a large travel range, which results in many stick-slip events. Hence, information on friction reproducibility can be gained from a single run. Finally, the actuator motion should be reversible, so that many tests can be performed. Typical comb drives used in MEMS provide relatively weak forces (tens of μN) and small travel ranges (less than 20 μm).²⁴ Thermal actuators can provide high forces (many mN), but their travel range is typically less than 10 μm.²⁵ Scratch drive actuators can achieve high forces and long range motion, but only move in one direction,²⁶ which would allow just a single test. A suitable choice is the nanotractor, a stepper motor that provides mN forces over a reversible $±$100 μm travel range at step frequencies $fp$ ranging from arbitrarily low up to 80 kHz.²⁷ The nanotractor devices used in this study took 46 nm steps and successfully ran with $fp$ up to 1 kHz, with some data being taken at 10 kHz.

The springs consist of parallel sets of fixed-guided beams in series. For each device, the spring stiffness was fixed by the folded-beam width $w$⁠. The beams have length $l$ = 189 μm, thickness 2.25 μm, and widths varying from 1.64 μm to 7.23 μm. These widths take into account the edge bias of 0.1 μm per edge. The variations produce spring constants of 0.028, 0.24, 2.13, and 20.3 μN/μm.

When deciding on platform variations in $k$⁠, limitations of the travel range and spatial resolution must be considered. If the springs are too soft, both the elongation amplitude and the period of stick-slip oscillations become too large. This means that the applied force $ke$ would not exceed the friction force, so it would not be measured. On the other hand, if $e$ is too small to be measurable, information on the friction force cannot be gained. It was conservatively assumed that slip events of 90 nm could be resolved experimentally. For a load of $Nb$ = 235 μN, $k$ = 2.13 μN/μm, $μ$ = 0.9, $e$ = 100 μm is found. For the same $k$⁠, but $Nb$ = 1.92 μN and $μ$ = 0.1, $e$ = 0.09 μm is found. This means that friction should be observable over at least two decades in $Nb$ for a single spring constant. Such calculations were performed to select $k$ values. A full set of design guidelines is shown in Table I.

The criteria in Table I can be visualized in the $Nb$–$k$ plane as shown in Figure 3. Allowable combinations of spring stiffness and normal force are shown in a shaded region with the upper and lower diagonal boundaries occurring as a result of the actuator range limit and resolution limit, respectively. Spring stiffness coefficients were selected according to the four dashed vertical lines to allow friction testing over a wide range of normal loads.

The friction block is schematically represented in Figure 4. Four equipotential electrodes (orange regions) run below the block and beam configuration (blue region). The nanotractor, the spring, and the block are electrically connected to each other and are electrically grounded. When no voltage is applied to the block electrodes, the block is suspended above the substrate as shown in Figure 4(b). Above a critical pull-in voltage, the three friction feet (hashed purple regions) contact the thin green polysilicon friction countersurfaces, causing friction when the block moves horizontally in Figure 4(a). Each foot is approximately 100 μm long and 3 μm wide. There are three feet, and more precisely the total apparent contact area is 846 μm².

The friction block and underlying actuation electrodes function as parallel plate electrodes. The four sections (hashed blue regions in Figure 4(a)) result in an overlap area of 14 700 μm². With no voltage applied, the block feet are 2.1 μm above the substrate and the block itself is 3.7 μm above the actuation plates. This means the block gap $gb$ is 1.6 μm when the feet contact the substrate. Excluding fringing field effects (a good approximation here), the electrostatic force between the two parallel plates is given by Eq. (5), where $Fe$ is the electrostatic force, $ϵair$ is the permittivity of air, $A$ is the area of a plate, $Vb$ is the voltage between the plates, and $gb$ is the gap between the plates

The total normal force on the block (⁠$Nb$⁠) is the sum of the electrostatic force $Fe$⁠, the out-of-plane spring force (⁠$kzgb$⁠), the gravitational force (⁠$mbg$⁠), and adhesion force (⁠$Fa$⁠). Accordingly,

The out-of-plane spring constant $kz$ was calculated to be 0.030 μN/μm using finite element analysis (FEA) software. Using this estimate, a force of 0.048 μN is required to close the gap, equivalent to an applied voltage of $Vb$ = 1.54 V (with $gb$ = 1.6 μm). Therefore, at low voltages, this effect is significant (at $Vb$ = 5 V, it reduces the normal force by ∼10%) and it is important to include $kzgb$ in Eq. (6). The weight of the block and beams, $mbg$⁠, causes the gap to close by only ∼3% (estimated by FEA), and thus is neglected for all applied voltages. Adhesion forces can be estimated by determining the offset in the friction force versus normal force curve as done in Ref. 28. We neglect adhesion forces here as the data below do not show a strong effect. That is, friction scales linearly with normal force (as defined by Eq. (6) with $Fa=0$⁠) without an offset.

By varying $Vb$⁠, forces over many orders of magnitude can be produced. While $Vb$ = 55.02 V (⁠$Nb$ = 60.8 μN) is the maximum value for data analyzed in detail below, the block has exhibited slip-stick cycles over a range of 1.7 V < $Vb$ < 150 V. This indicates an $Nb$ range from 10 nN to 0.45 mN (4 ½ decades) and a corresponding apparent pressure range in the friction feet from 12 Pa to 0.54 MPa.

In addition to the energy lost due to friction between the feet and substrate, there is another dissipation mechanism to consider—air damping. This is primarily due to the small, constant gap between the block and substrate as the block slides. Corwin and de Boer²⁸ found that a first-order Couette calculation gave a quality factor that was within 20% of the experimental results for a similar gap between plates and an electrode. The damping coefficient $c$ is a function of the viscosity of the surrounding fluid $η$⁠, the block plate area $Ab$⁠, and the gap between the block plate and the substrate $gb$⁠, as given below:

Here, η = 1.98 × 10⁻⁸kg/m s (air) and A_b = 28,830 μm², so c = 3.57 × 10⁻⁷ N s/m. At f_p = 10 kHz, the nanotractor has a peak average speed of 460 μm/s corresponding to an average damping force of 0.16 nN. During an inertial slip event of the block, however, the instantaneous speed is much greater than the average speed. The natural frequency of the system ranges from 1.4 kHz to 37.6 kHz for the lowest spring stiffness and highest spring stiffness, respectively. Assuming a slip event is a free vibration of amplitude 10 μm (the largest reported in these trials), the maximum viscous damping force then ranges from 0.03 μN to 0.98 μN corresponding to damping ratio of ζ = 0.056 to 0.002. The parameter ζ is defined as the ratio of the actual damping in the system to the critical damping. At critical damping (ζ = 1), there would be no oscillations in the response as the system transitions from underdamped to overdamped. Friction in these experiments causes the system to oscillate for only a single half-cycle, corresponding to an overdamped (ζ > 1) condition for all reported loads. This is an important point that will be elaborated in Sec. V B. Hence, the fluid damping force remains small in comparison to the frictional force on the feet even during the slip events.

After fabrication, devices were released in HF acid and two different monolayers were applied—FOTAS (CF₃(CF₂)₅(CH₂)₂Si(N(CH₃)₂)₃)²⁹ and OTS (CH₃(CH₂)₁₇SiCl₃).³⁰ These monolayer coatings are commonly applied in MEMS to reduce frictional and adhesive forces. The majority of successful stick-slip tests came from devices with k = 2.13 and 20.3 μN/μm, with pulling frequencies from 10 to 10 000 Hz, with N_b ranging from 0.98 to 60 μN.

The block and actuator were both held at ground potential, while actuation signals were applied to electrodes under the nanotractor clamps and the plate, and a DC voltage was applied to the friction block. Hence, a total of five electrical probes were required. Voltages of 150 V were applied to the nanotractor plates and clamps (see Ref. 27 for details of the timing diagram). A given clamp slides easily when its voltage is reduced to near zero. Here, the nanotractor clamp voltages were reduced to 1 V rather than 0 V to prevent out of plane vibrations. Otherwise jumping in and out of contact of the nanotractor clamps occurred due to the out-of-plane force provided by the folded beam suspension that guides its travel. The signals were supplied by a computer using a PCI analog output card (National Instruments PCI-6733 - 16 bit resolution, 1 MS/s update rate) and through amplifiers (Tegam 2350). A halogen illuminator (Dolan-Jenner MI-150) was used to illuminate the chip, which was imaged by an optical microscope using a 20X Mitutoyo long working distance objective (NA = 0.42), enabling a sufficiently large field of view to observe the displacement gauges over the full travel length. The trials were recorded with a high-speed camera (Vision Research Phantom v7.0) at up to 10 000 fps. The microscope was also equipped with interferometric capability³¹ in order to observed out-of-plane deflections. Figure 5 shows a schematic of the probe station experimental setup.

For load-controlled tests, the step frequency f_p was fixed while the block normal load $Nb$ was varied. In frequency-controlled tests, $Nb$ was held fixed while f_p was varied.

During a given test run, parameters f_p, $Nb$⁠, and k were held constant while the actuator travelled over a distance of 20–100 μm. The high-speed camera's frame rate was scaled with f_p to ensure multiple images were taken per actuator step (usually 2). The illumination was always set at the maximum level, and the frame exposure time was minimized to obtain the least amount of motion blur.

Displacement data were extracted from each frame in recorded trials using sub-pixel pattern matching with a bilinear interpolation scheme.³² This method allows high resolution in displacement changes. It compares an object feature connected to the spring (or to the nanotractor) to a reference feature fixed to the substrate. The object features used can be seen in the middle of the spring in Figure 2. The success of the pattern-matching algorithm depends on the exposure length, noise, and other factors in the experimental setup. An accuracy of pixel/10 using the pattern-matching algorithm was routinely achieved. With a 20× objective and 250 mm projector lens, the optical resolution was 0.88 μm/pixel. This means we could extract position data with $≈$90 nm resolution. In fact, the individual 46 nm steps of the actuator were routinely resolved. A 50× objective was used when additional resolution was required at the expense of a smaller field of view. The product $e⋅k$ gives the friction force $F0$⁠, which was then divided by $Nb$ to give the friction coefficient, $μ$⁠.

Preliminary tests were conducted to ensure that the block remained flat while translating horizontally. Using interferometry, it was found that applying voltages to the block electrodes of $Vb$ ≥ 1.7 V resulted in no observable tilting of the block as evidenced by stable fringes when viewing under interferometric imaging. Thus, we expect the nominal pressure distribution over the friction feet to be highly uniform; i.e., the friction feet are not ploughing on edge but are sliding with the bottom face fully in contact with the substrate.

Also of concern was the behavior of the stepper motor under different loads. Ideally, the stepper motor step size should be independent of the friction due to the block, even at high $Nb$⁠. Figure 6 shows the stepper motor behavior at $Nb$ = 0.98 and 60.8 μN (⁠$Vb$ = 7.15 V and 55.02 V, respectively). No dependence of $Δxp$ on $Nb$ or f_p is observed.

Results from approximately 50 different devices (30 FOTAS, 20 OTS) and 200 different trial runs are reported in Sec. IV.

We first present the results of tests with constant $Nb$⁠, f_p = 100 Hz, and $k$ = 2.13 μN/μm. Results from four representative OTS devices are shown by the solid black lines in Figure 7. As might be expected, the mean peak elongation, $e¯peak$⁠, increases with increasing $Nb$⁠. The slip distance $δ$ (the peak-to-valley amplitude of the slip events) also increases with $Nb$⁠, and after a slip event, the spring remains stretched by a significant amount, increasingly so at higher normal forces. At fixed $Nb$⁠, there is variability in the magnitude of the friction force, $kepeak$⁠, from one slip event to the next. This is presumably due to variation along the path due to differences in surface topography. Similar results are seen both experimentally and in simulations (e.g., Ref. 33) for in single asperity contacts.

Especially for small $Nb$⁠, many slip events occur. Rather than determining the mean peak elongation $e¯peak$ and slip distance $δ¯$ by analyzing each slip event, we used the distribution of instantaneous spring elongations. If the elongation vs. time plot is a perfect saw-tooth curve, elongation data $e$ will fall in a uniform distribution of width δ. This is because the inertial slip time of the block is short compared to the time the spring is stretched. The slip distance δ is equal to the distribution width and is related to the standard deviation $σu$ of the uniform distribution by $δ$ = $σu12$⁠.³⁴ The peaks and valleys of the data (corresponding to maximum and minimum friction force, respectively) are given $e¯±δ/2$⁠, where $e¯$ is the time average of the elongation data.

Thus, from calculating the mean and standard deviation of the instantaneous spring elongation, $δ¯$ and $e¯peak$ were estimated without identifying individual slip events. This indirect method was validated by identifying individual peaks by eye and recording the associated slip distances—good agreement was found. As an example of the results, at $Nb$ = 60.8 μN in Fig. 7, a slip length of $δ$ = 7.37 μm is indicated. However, subslips are sometimes seen as well. The algorithm just described yields a value of $δ$ = 7.02 μm.

Repeating the load tests at different f_p values for each coating, it was observed that $e¯peak$ decreases with f_p. In Figure 8, the quantity $k⋅e¯peak$ is plotted versus $Nb$⁠, where $k$ = 2.13 μN/μm. The slope is the static friction coefficient, $μs$⁠. It can be observed (1) that a linear fit is appropriate as stated by Amontons' First Law (the force of friction is directly proportional to the applied load), (2) that the OTS coating has a lower friction coefficient than the FOTAS coating, and (3) that increasing f_p lowers $μs$⁠. For the sake of clarity, error bars are not shown in Figure 8, but are approximately 7% of $k⋅e¯peak$⁠.

Figure 9 shows the representative results of puller frequency-controlled tests at constant $Nb$ with $k$ = 2.13 μN/μm. Here, the normal load is $Nb$ = 0.98 μN for both FOTAS and OTS coatings. As the step frequency increases, $e¯peak$ decreases, corresponding to a lower $μs$⁠. This effect was observed for both coatings, with a maximum change in friction coefficients of 0.31 and 0.17 for FOTAS and OTS, respectively.

One unanticipated result is that the friction coefficients for both FOTAS and OTS here were higher than those reported in literature. Static friction coefficients for FOTAS are around 0.33 (Ref. 28) compared to our results of 0.4–0.8 depending on f_p. For OTS, reported values are around 0.1 (Refs. 3, 27, and 35) compared to our results of 0.3–0.6 depending on f_p. We can eliminate adhesion as a possible explanation as the lines in Figure 8 pass close to the origin, meaning that there is no significant tangential frictional force when there is no applied load. In addition, there is no reason to believe that we have smoother surfaces, which may increase the real contact area and friction coefficient. Using AFM, an average roughness of ∼5 nm was found compared to 5–7 nm in Ref. 28. The $μ$ values depend on calibration of voltages, gaps, and pixel lengths—all were carefully checked. The friction coefficient is not a material property—the system setup and methodology differ in these comparisons, making the disagreement less surprising. Although $μ$ is higher than commonly reported, it is seen that it is higher for FOTAS than OTS by a factor of ∼2, which is typical. We assume that some combination of the differences in system setup and the extended period of time between the coating application and device testing accounts for the higher friction. Possibly, while storing the chips in a gel pack^TM (over a year), compounds from the gel migrated and/or physisorbed over the surfaces, increasing the friction coefficient. A similar aging effect was seen in Ref. 36 for alkanethiol monolayers on a gold substrate.

In the microscale case, there are important points to note. First, the main component of the normal force is not the weight of the block but the applied electrostatic load. For increasing loads, the inertia remains constant, unlike most macroscale experiments (e.g., Ref. 13), where small weights were added to adjust the block mass so that the inertia and normal force scale together. Second, the ratio $k/mb$ is very large in the microscale experiments, making the inertial response time, i.e., the slip time in a stick-slip cycle, much less. Using the layout of the masks and knowledge of the SUMMiT V process²² (which gives the layer thicknesses and edge loss due to wet etching), we estimate an effective mass $mb$ = 3.64 × 10⁻¹⁰kg. This calculation lumps the mass of the block, load spring, and guide beams together and is weighted by their relative motions for a given change in spring elongation. For $k$ = 2.13 μN/μm, the inertial response time $τi≈π⋅mb/k$ is approximately 41 μs. This is less than the time resolution of the high-speed camera (100 μs at 10,000 fps), and therefore inertial events (i.e., the slip events) cannot be captured in detail, unlike the macroscale experiments in Ref. 37.

When the inertial time of the block is much shorter than the time between actuator steps, the block cannot slide steadily. Instead, there are two phases of the motion of the block, forming a stick-slip cycle. During the stick phase, the block is at rest, and according to rate-state friction theory, the contacts can age. The static friction coefficient increases with time and meanwhile, the spring is also being elongated. When the maximum spring force equals the friction force (⁠$epeakk=μsNb$⁠), the block slips. During the slip phase, the contacts are rejuvenated, lowering the friction coefficient near to the initial value of the stick phase. We will investigate the stick and slip phases in detail.

The transition from steady sliding to stick slip, examined in many macroscale works with a similar puller/spring/friction block system,^8,11,13 cannot be studied here because there is no steady sliding as the block inertia is so small. Likewise, we cannot study the time for re-solidification to occur after stopping the block after steady sliding, as done by Israelchavili and coworkers in several publications.^18–20,38 However, static aging effects are observed. Therefore, we can examine whether the measured aging effects at one set of conditions (for example, a single curve in Figure 7) are commensurate with the data taken at different $fp$⁠, $k$⁠, and $Nb$⁠. That is, does an aging model explain slip lengths and measured values of $μs$ for different values of these parameters?

In Sec. I, two coupled models for static aging were introduced (Eqs. (1)–(4)). We first model the aging behavior measured above and then compare it to those models. We find that aging in our systems follows an exponential rather than a logistic or logarithmic curve. Then, we use the slip lengths to gain insight into the velocity dependence of friction.

The block is at rest between slips. For a given slip length $δ$⁠, step frequency $fp$⁠, and step size $Δxp$⁠, the rest time (when contacts age) is given by

Equation (8) assumes an average slip length $δ$⁠, and therefore the rest time $trest$ is an average as well. In Figure 7, one can see some subslips as well as variation in the slip lengths. After a subslip, the rest time until the next slip event is significantly less than if the block slipped the full distance. However, it is still useful to consider the average rest time when characterizing the system, as our goal is to determine the average parameter values in a friction model.

In Figure 10, we recast Figure 9 in terms of rest time, now showing data compiled from 10 devices from FOTAS and five devices from OTS coatings. The solid lines represent exponential fits of the form

Values for $μmin$⁠, $μmax$ and $τ$ are given in Table II, with a correlation coefficient $R2$ > 0.97 for both coatings.

In Figure 8, we showed that the friction force scales linearly with load. That is, $F0=μNb$⁠. In general, $μ$ may have both rate and state components, as in Eqs. (1)–(4). We will consider the instantaneous value of $μ$ to be a function of a state variable $ψ$ (unitless) representing the state component of $μ$⁠, and instantaneous velocity such that $μ=ψ+f(ẋ)$⁠. The rate term in this equation will be examined in Sec. V C.

Motivated by the exponential fits in Figure 10, we propose $ψ$ evolves with time according to

Thus, $dψ/dt$ = $μmax/τ$ for $ψ$ = 0 and $dψ/dt$ = 0 for $ψ$ = $μmax$⁠. If the block sits at rest with some initial $ψ0$⁠, the analytical solution to Eq. (10) is

where $ψ(0)=ψ0$ and $ψ(∞)=μmax$⁠. The given form simulates the charging of a capacitor (note that the abscissa in Figure 10 is plotted on a log scale) and can be interpreted as follows. After a slip event, new contact pairs are established. Some small contact age is retained, giving rise to a friction $μ$ = $ψ0$ which is nearly equal to (but larger than) $μmin$⁠. Aging happens quickly at first. To lower their free energy, monolayer molecule tail groups located on previously noncontacting asperities reconfigure when they encounter a fresh counterface. The reconfiguration rate is initially high, then decreases as the number of monolayer tail groups that have yet to reconfigure decreases. The rapid reduction occurs because the head groups are covalently attached to the substrate,^29,30 hence the tail groups cannot stretch very far.

Figure 11 illustrates how this rate evolves with time for both coatings. Solid lines are theoretical exponential fits to the experimental data for FOTAS and OTS. Dashed lines are representative logistic and logarithmic fits that correspond to the alternative models described Sec. I. For the exponential fits, we see that FOTAS evolves faster than OTS, and the change in friction coefficient (area under the curve) is larger for FOTAS than OTS. Neither a logistic²¹ nor a logarithmic⁹ fit matches experimental results. A logistic fit (dashed line in Figure 11) is not appropriate here because the experimental data show a monotonically decreasing rate. Logistic curves allow for some maximum reconfiguration rate at nonzero time—the frictional aging rate is maximized at some $θ$ between $θmin$ and $θmax$ in Eq. (4). A logarithmic fit (dashed-dotted line in Figure 11) captures the initially decreasing reconfiguration rate, but not the shape of this decrease; the curve is concave up rather than down.

Figure 12 illustrates how the static coefficient evolves with time from a minimum value for FOTAS (blue) or OTS (green) with parameters values for $μmax$⁠, $μmin$⁠, and $τ$ given in Table II. For a given $fp$⁠, $k$⁠, and $Nb$⁠, the normalized spring force $ke/Nb$ (dashed black lines in Figure 12) increases at a rate proportional to $kfpΔxp/Nb$⁠. Eventually, an actuator step is taken where the spring force exceeds the friction force. This point, corresponding to $e=epeak$ and $μ=μs$⁠, is when the block will slip. Decreasing $fp$⁠, $k$⁠, $Δxp$⁠, or increasing $Nb$ has a similar effect on the rest time of the block. That is, increasing $fp$ from 10 Hz to 100 Hz should produce $epeak$ and $μs$ changes similar to those when increasing $k$ or reducing $Nb$ tenfold. One caveat to this simplified interpretation is the initial spring force will not necessarily begin at zero, but at some starting elongation $e0=epeak−δ$⁠. This offsets the spring force lines in Figure 12 vertically and will be examined in more detail in the next section.

The slip phase can be viewed as an energy dissipation mechanism that occurs when the frictional force is no longer able to hold the block in place due to the spring elongation reaching some peak value, $epeak$⁠. The energy dissipated by the block due to the relaxation of the spring is given by

During the slip event, the friction force is the main dissipation mechanism, as the viscous damping is too small to be significant. Assuming the friction coefficient takes on some effective $μ̃d$ over the slip

Here, $lslip$ is the total path length of the block corresponding to a single slip. For the underdamped case, $lslip>δ$⁠, while for the overdamped case, $lslip=δ$⁠. Only the latter case, corresponding to one half oscillation cycle with e > 0, was observed in our experiments. The conditions under which this can occur are given in the Appendix. We also obtained experimental evidence. Namely, for the lowest spring constant, the natural frequency was 1.4 kHz. In this case, we could resolve the slip events at $Nb$ = 0.98 μN while capturing video at the maximum framerate 10 000 fps. No full oscillations were observed. The damping ratio $ζ$ is proportional to 1/$k$⁠, so for higher spring constants we can safely assume that the system remains overdamped.

As discussed in the Appendix, the parameter that identifies the number of oscillatory half cycles is the ratio of the static friction coefficient just before the slip to friction coefficient during the slip, given here as $r=ψ/μ̃d.$ For $r$ < 3, the block will slip for only a half oscillation, corresponding to the overdamped case. To obtain an upper bound on $r$⁠, we can assume $ψ=μmax$ and $μ̃d=μmin$⁠. Using the values in Table II, $r$ = 1.64 and 1.43 for FOTAS and OTS, respectively. These values are both well below $r$ < 3. A half oscillation could also occur in which the block stops with $e$ < 0. However, this was never observed in the experiments, as can be seen in Figure 7.

Equating (12) and (13) yields an analytic expression for the slip length, given by

Equation (14) predicts the force drop $kδ$ is proportional to $Nb$ and is independent of $k$ assuming $μs$ and $μ̃d$ are constants independent of $k$ and $Nb$⁠. This may be the case if $μs≈μmax$ (⁠$ψ$ saturates for $trest>4τ$⁠) and $μ̃d$ has no significant velocity dependence for the range of experimental parameters. In Sec. V C, we will revisit these assumptions.

During the slip event, the increase in contact strength due to memory effects is quickly lost. Both the Dieterich-Ruina aging law and PTM have similar forms that capture this effect. Considering a variation of Eq. (2) with a nonzero minimum state variable (ψ = μ_min), we choose

for the memory loss effect in our model, where $ψ$ represents a nondimensional contact strength. If we approximate the velocity of the block as some constant over the slip $ṽslip$⁠, the $ψ$ decays according to

We note that the quantity $Δtṽslip=δ$⁠, where $Δt$ is the duration of the slip event, so the minimum contact age is given by

Thus, if the slip length is long enough, almost all of the memory effects are lost. This model assumes the lowest value of kinetic friction is μ_min. The average diameter of asperities in contact was found to be ∼20 nm, as estimated by Ref. 27. It was found by obtaining a surface profile using AFM from devices fabricated in the SUMMiT V^TM process and applying the Greenwood-Williamson model.³⁹ As we are unsure of the exact relationship between average asperity diameter and memory length scale, we treat $dc$ as a free parameter within reason. A value of $dc$ = 20 nm is used in the subsequent analysis. The slip lengths are relatively insensitive to variation $dc$ provided $dc$ remains much smaller than the slip length. This also justifies the assumption of contact rejuvenation during slip events.

Combining the equations given above, we have a friction force scales linearly with load and has both rate and state components given by Eq. (18). The form of the rate dependence $f(ẋ)$ was chosen to be identical to Eq. (1)

From Eqs. (10) and (15), $ψ$ evolves with time according to

Hence Eqs. (18) and (19) are the final form of the coupled equations we used for simulations below. The block's equation of motion is given by

where $sgn(ẋ)$ is +1 for $ẋ$ > 0 and −1 for $ẋ$ < 0. The instantaneous value of the friction force $Ffric$ is bounded by the force required to start sliding from rest, $F0$⁠. When the block is in motion, $Ffric=F0$⁠. When the block is at rest $Ffric≤F0$⁠, such that the net force on block is minimized. That is, the frictional force will cause the block to remain at rest until a force $F0$ is insufficient to keep it from accelerating. To implement this behavior, a smooth function was used in order to avoid numerical instabilities, similar to Eq. (19) in Ref. 12 by Lim and Chen.

For our system, the puller moves in discrete steps, so $xp$ can be represented as the sum of step functions. The time for the actuator to take a step (∼1 μs) is much shorter than the inertial response of the block, so this approximation is reasonable. Because the inertial events cannot be resolved, we have used $A$ as a free parameter to fit the experimental slip lengths, as seen below. The parameter $A$ has been attributed to an instantaneous viscous response and occurs for both dry conditions (e.g., granite, steel, and wood surfaces¹²) and with thin films (e.g., squalane³⁸ and poly(dimethylsiloxane) (PDMS)⁴⁰).

The state-space model allows for variation of any parameter, allowing for comparison to experiments. State equations were integrated in time using an implicit integration scheme designed for stiff systems. Solution times were on the order of seconds to hours depending on parameter choices. Solution convergence was tested by successively limiting the maximum step size and comparing key outputs, such as the maximum velocity during a slip.

An example output for $Nb$ = 5 μN, $k$ = 2.13 μN/μm, and $fp$ = 100 Hz using the FOTAS parameters in Table II is shown in Figure 13. The state-space results display expected characteristics. For example, from Figures 13(a) and 13(b), the block is mostly at rest (⁠$ẋ$ = 0) with short slip events reaching some maximum velocity, in this case 2 cm/s. From Figure 13(d), any gain in $ψ$ is quickly lost during a slip event, meaning the state after a slip starts from the minimum value $ψ0≈μmin$ each time. Varying the system parameters within reason does not invalidate this assumption. From Figures 13(c) and 13(e), once the spring force exceeds the friction force, the block slips, as expected.

As previously mentioned, $A$ was treated as a free parameter. Increasing $A$ decreases the slip lengths at a given normal load, as energy can be dissipated more quickly. In Figure 14, it is assumed that aging has saturated such that $μs≈μmax$⁠, which is reasonable for k = 2.13 μN/μm and $fp$ = 100 Hz. The simulated slip lengths for $A$ = 0.000, 0.010, 0.015, and 0.020 are shown in red with $k$ = 2.13 μN/μm and $fp$ = 100 Hz. The experimental results (black triangles) are fit reasonably well with $A$ = 0.015. The results could also be fit with $A$ = 0.000 and $μ̃d=0.66$⁠. However, this is larger than $μmin=0.48$⁠, the $μ$ value that would be expected for large slip lengths as in Figure 14 ( > 1 μm). The analytical solution (black X's) given by Eq. (14) is approximated as $δ≈2Nb(μmax−μmin)/k$⁠, with the assumption that $μ̃d≈μmin$⁠. These points lie almost exactly on the $A$ = 0.000 simulation line, but well above the data. For OTS devices of the same pulling frequency and stiffness, the best fit occurs when $A$ = 0.005.

Using the same values of $A$⁠, simulations and experiments can be compared for $k$ = 20.3 μN/μm and $k$ = 2.13 μN/μm devices, as shown in Figure 15. Experimental results for both FOTAS and OTS show there is a decrease in the force drop $kδ$ for $k$ = 20.3 μN/μm compared to $k$ = 2.13 μN/μm at a given load. This trend is captured in the simulations as well. There are two competing effects to consider when evaluating $(μs−μ̃d)$ in Eq. (14). For increasing $k$⁠, the maximum velocity of the block decreases. To first order, this is because for the same starting spring force, the force drops off more quickly during a slip if k is larger. As an example from the simulations, for FOTAS parameters with $k$ = 2.13 μN/μm, $Nb$ = 30 μN, and $fp$ = 100 Hz, the maximum block velocity is 15 cm/s. For the same conditions except $k$ = 20.3 μN/μm, the maximum block velocity is 1.7 cm/s. This means $μ̃d$ decreases with increasing $k$⁠, as the rate component of $μ$ becomes smaller (i.e., there is less velocity strengthening). This effect works to increase $kδ$ with $k$⁠. However, for increasing $k$⁠, $trest$ also decreases. For decreasing $trest$⁠, $μs$ decreases as given by Eq. (9). If this rest time becomes small enough (⁠$trest<4τ)$⁠, $μs$ will be significantly less than $μmax$⁠. This effect works to decrease $kδ$ with $k$⁠. The latter effect dominates, resulting in a net decrease in $kδ$ with $k$⁠.

We can also reproduce the dependence of $μs$ on $fp$ using the state-space simulations. In Figure 9, simulation data are overlaid on the experimental data, and there is reasonable agreement between the two for both FOTAS and OTS.

Finally, it is observed in Figure 7 that Eqs. (18) and (19), as represented by the dashed red lines, capture the actual slip-stick cycles relatively well. They cannot capture local differences in topography, which explains why the data deviate from the model to a small degree. While Figure 7 is only for one value of $fp$⁠, similar agreement is obtained for higher and lower $fp$ values, which exhibit distinctly different force drops, $kδ$⁠.

There are clearly aging effects in the MEMS nanofrictor system for both monolayer coatings. We have attempted to capture this effect with a hybrid model. The parameter scaling in the model is in good agreement with experimental results, indicating the model is reasonable.

Although we can capture the aging effects empirically, there remains the question of the physical process responsible for this increase in friction coefficient with time. Is it the quality of contacts, quantity of contacts, or some combination of both that is the responsible mechanism for aging and velocity strengthening? The change in friction coefficient cannot be due to the increase in real area alone, as the real area would have to increase by a factor of two—this is unlikely for silicon, which exhibits little creep at room temperature. The most convincing evidence is the clear saturation of friction coefficient shown in Figure 10. This leads us to speculate it is the aging of the third body, the organic monolayer coating that is primarily responsible for aging in our system. This physical picture corresponds to the PTM model that has well defined limits on the friction coefficient as described in Sec. I. However, the nature of the aging is different in that it initiates immediately, rather than requiring a nucleation time. Unlike the logarithmic law, the aging is limited. We suggest that this can be explained by the tail groups that can minimize their energy by bonding across a newly formed interface, but are limited in doing so because their head groups are covalently bound to the substrate.

In SFA experiments, a critical nucleation time was required to change from liquid-like to solid-like.^18–20 This would produce an abrupt decrease in $μs$ in Figure 9 at some critical $fp$⁠, as the monolayers would not have time to solidify. Instead, we see that $μs$ decreases steadily with $fp$⁠. As mentioned in Sec. I, there is a key difference between multi asperity and SFA experiments. In multi-asperity experiments, the block is moving to a fresh set of contacts each time after a slip, unlike SFA, where the characteristic slip distance is less than the contact length. With this in mind, it is not surprising that aging in this system differs from Eqs. (3) and (4), which have been proposed specifically to capture the characteristics of SFA experiments.

We have taken a memory length $dc$ to be equal to the asperity contact diameter, but the true value may be much less, even on the order of the monolayer chain length. In our model, $dc$ can be reduced with little effect on the simulations, as a reduction in $dc$ increases the rate of reduction of $ψ$ during a slip, an already fast process. The time it takes to decrease $μ$ from some initial value just before slip to $μmin+A ln(1+ṽslip/vc)$ would compose a smaller fraction of the total slip time. This makes $μ̃d=μmin+A ln(1+ṽslip/vc)$ an even better approximation.

Although the values of $μ$ are higher than expected for FOTAS and OTS, their ratio of ∼2 agrees with previous literature. This can be understood in the sense that FOTAS chains, containing only 8 C atoms are not as well ordered as OTS chains which contain 18 C atoms. Hence, when FOTAS molecules encounter an FOTAS counterface, their driving force for ordering can be expected to be higher. This is reflected in Figure 11, where $dμ/dt$ is greater for FOTAS than OTS. On the other hand, the time constant $τ$ is approximately the same for the two coatings. This could indicate that the reconfiguration is limited because the lubricants are chemically bound to the surface. Likewise, the velocity strengthening coefficient is greater for FOTAS (⁠$A$ = 0.015) than OTS (⁠$A$ = 0.005), which again may correlate with a higher driving force for restructuring in FOTAS than OTS when asperities come into contact.

We have designed, fabricated, and tested a true 1-D microscale sick-slip friction platform. With this platform, we are able to vary actuator frequency, normal force, and spring constant over a wide range. Two coatings, OTS and FOTAS, were applied the device to gain a measure of control over static and dynamic friction.

Tests performed in air revealed a static friction coefficient that exponentially approaches a maximum during times when the block is a rest. This is similar to the PTM model, where the third body ages while the real contact area remains the same. However, there is a distinct difference that the aging initiates immediately. FOTAS devices aged faster than OTS devices, and the change in static friction coefficient was greater for FOTAS than OTS. This required the development of a modified aging law. We propose this aging law is due to an initially fast reconfiguration of organic monolayers on asperities that have just made contact.

Using new coupled equations (18) and (19), we developed a robust numerical state-space simulation that allows for parameter variations of both inherent system parameters (⁠$τ$⁠, $A$⁠, $dc$⁠, $vc$⁠, $μmax$⁠, $μmin$⁠, and $Δxp$⁠) and experimental parameters (⁠$k$⁠, $fp$⁠, and $Nb$⁠). It was found that the assumption of contact rejuvenation used to characterize $μs$ could be justified. The inclusion of the velocity strengthening parameter $A$ reduced the simulated slip lengths, bringing them in line with experimental results. Experimentally, it was also observed that the $kδ$ versus $Nb$ data depended on $k$⁠, whereas in a simple model, there should be no such dependence. This was explained by a stronger dependence of aging on $k$ than of velocity strengthening on $ẋ$⁠. The rate of aging and velocity strengthening was seen to be less for OTS than FOTAS. This may correlate with the longer OTS chains, which already have a large degree of interchain bonding and therefore there is only a lesser need to reconfigure across the interface when new contacts are made.

This work is funded by the National Science Foundation under NSF collaborative Grant Nos. CMMI-1030322 and CMMI-1030278. S.S.S. acknowledges the NSF Graduate Research Fellowship Program (GRFP) for support. We also would like to thank Siddharth S. Hazra for his work in preliminary testing. The nanofrictor devices were fabricated by the staff in the Microelectronics Development Laboratory at Sandia National Labs in Albuquerque, NM.

Consider a mass-spring system (⁠$m$⁠, $k$⁠) subjected to a frictional force $F0=μNb$⁠. The variable $x$ represents the position of the bock and $x=0$ corresponds to the unstretched spring. In the simplest model, $μ=μs$ when the block's velocity $ẋ=0$ and $μ=μd$ for $ẋ≠0$⁠. For initial conditions $x(0)=x0$ and $ẋ(0)=0$⁠, the block will oscillate at the natural frequency $ω02=k/m$ with a linear decay in amplitude of slope $−2μdNbω0/πk$⁠.⁴¹ At the critical points, where $ẋ=0$⁠, $tn=nπ/ω0$⁠, where $n$ is an integer representing the number of half cycles completed. The corresponding block positions $xn*$ lie on the oscillation envelope E(t) as shown in Fig. 16 and have a magnitude given by

The block will come to a stop when (i) its velocity is zero (⁠$x=xn*)$ and (ii) the spring force (⁠$kx)$ is insufficient to overcome to maximum friction force (⁠$μdNb)$⁠. Note we have used $μd$ rather than $μs$ here as the block is just coming to rest and the contacts have not yet had time to age. Accordingly,

Then, $n$ represents the smallest integer to make the above inequality true. Now, we consider the case where $x0=μsNb/k$⁠, that is $x0$ is such that the maximum static friction force is just equal to the spring force. (This represents the typical stretch of the spring upon initiation of sliding. It is only possible to attain larger $x0$ values by holding the block in place with some force larger than $μsNb$⁠). With this substitution, we can rewrite the inequality as follows, defining $r=μs/μd$⁠:

Thus for $μs/μd=r<3$⁠, the block will only complete one half oscillation.

Let us further consider the case where the dynamic friction force has a velocity strengthening term, given by $F0=μdNb=(μmin+A|ẋ|)Nb$⁠. The coefficient $A$ is positive, so more energy is dissipated for a given length than the case when $A=0$⁠. Thus, we can say, for cases of half oscillations, the slip length decreases as $A$ increases, approaching $δ=(μs−μk)Nb /k$⁠.

For simulation purposes, it is helpful to nondimensionalize the system. We define $t*=tk/m$⁠, $x*=kx/Nb$⁠, $F0*=F0/Nb$⁠, and $A*=ANb/km$⁠.

The equations of motion and friction force are then

In the first simulation, we will fix $μmin=0.1$ and $A*=0$ and vary $μs$ such that $r=μs/μmin=$ 6, 3.4, 2.6, or 1.4. This reduces the initial position according to $x*(0)=rμmin$⁠. Figure 16 shows the block displacement with time for this setup. As expected, $r<3$ is the criterion that must be met such that only one half-oscillation will occur. Outside this region, the amplitude envelope, given by $E*(t*)=$$x*(0)−2μmint*/π$ and shown as black dotted lines of constant slope, predicts $ẋ*=0$⁠. That is, for $r$ > 3, $ẋ*=0$ values lie on $E*(t*)$⁠. If $ẋ*=0$ lies inside the red dashed lines, $|x*|<μmin$⁠, and the block comes to rest. The agreement between the analytical solution and the numerical simulation validates the latter.

Now, we examine the effect of introducing $A*$⁠. We fix $μmin=0.1$ and vary $μs$ such that $r=μs/μmin=$ 6, 3, or 1.4. For these values of $r$ and $A*=0, n=$ 3, 1, and 1 half-cycles, respectively. Increasing $A*$ to $A*=0.1$ results in faster energy dissipation and a change in the slip length $δ$⁠. For cases of a single half-oscillation, $δ$ is always less than that in the $A*=0$ case, as shown in Figure 17. For the $r=6$ case, the number of half oscillations is decreased from 3 to 2. Increasing $A*$ to $A*=0.2$ produces even shorter slip lengths in half-oscillation case.