tomlinson paper 2012 prl

8/16/2019 Tomlinson Paper 2012 PRL

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Temperature and Velocity Dependences in the Tomlinson/Prandtl

Model for Atomic Sliding Friction

Sergio J. Manzi1, Wilfred T. Tysoe

2 and Octavio J. Furlong

1*

1 INFAP/CONICET, Universidad.Nacional de San Luis, Ejercito de los Andes 950, 5700San Luis, Argentina

2 Department of Chemistry and Laboratory for Surface Studies, University of Wisconsin-

Milwaukee, Milwaukee, WI 53211, USA

Manuscript submitted for publication in Physical Review Letters

Keywords: Tomlinson/Prandtl Model, Monte Carlo simulations, periodic sliding potentials

PACS numbers: 68.35.Af, 07.79.Sp, 46.55.+d

* Author to whom correspondence should be addressed

Telephone: +54 266 4436151

E-mail: [email protected]


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Abstract

The Tomlinson-Prandtl model for nanoscale sliding friction is analyzed for various

sliding regimes defined by the values of the corrugation factor 2/, where E 0 is the height of the sinusoidal sliding potential, k L is the effective stiffness of the

contact, and a is the surface periodicity. For γ < 1, the friction force tends to zero,

defining a so-called superlubricious regime. The most commonly observed behavior, in

which the friction force increases monotonically with increasing sliding velocity up to a

maximum value F *, given by 1/2, is found for γ > 4.603. However,

completely different behavior is found when 1 < γ < 4.603 due to a breakdown of the

usual assumption that the height of the barrier during sliding varies as . Thevelocity and temperature dependences of the lateral force in this regime are investigated

using Monte Carlo simulations. The friction force still tends to F * at T = 0 K, but in

contrast to the behavior found when for γ > 4.603, the friction force is found to increase

with increasing temperature, reach a maximum value, and then decrease monotonically as

the temperature rises further. Unusual velocity dependences are also found in which the

friction force tends asymptotically to zero as the velocity approaches zero, but does not

increase monotonically to a maximum value as the velocity increases; instead, a

maximum friction force is found at intermediate velocities.


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Applications involving sliding interfaces at the micro- and nanoscale, such as

micro/nano-electro-mechanical systems (MEMs and NEMs), have become of great

importance for technological advances, requiring a better understanding of the frictional

phenomena down to the atomic scale [1]. The introduction of the atomic force

microscope (AFM) [2], has allowed friction phenomena under single asperity conditions

to be studied, [3,4]. The Tomlinson/Prandtl model [5,6] is extensively used to rationalize

AFM measurements of sliding friction [7]. This atomic-scale model assumes that a

harmonically coupled tip slides across a corrugated surface potential. The model is

generally analyzed using a simple sinusoidal potential to mimic the surface corrugation,

with a superimposed, moving parabolic potential for the sliding contact and can be

described as

, 2 cos 2

2

, (1)

where k L is the effective stiffness of the contact, a the surface periodicity, E 0 the potential

corrugation, x the tip position, and v the scanning velocity, so that vt becomes the time-

dependent position of the tip support, X .

The model describes the transition of the tip from one stable position to the next

when it is capable of overcoming the energy barrier ∆ E between the two stable points,

thereby giving rise to energy dissipation, resulting in a stick-slip type of motion. At T = 0

K, spontaneous sliding occurs when the combined sinusoidal+parabolic potential results

in the formation of an inflection point (∆ E = 0) that allows the tip to move spontaneously

to the next minimum in the potential, at a lateral force referred to as critical or maximum


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lateral force F *

(in the present work it is referred to only as the critical force since it will

be shown that it is not necessarily the maximum value), which is given by

2

1 , (2)

where γ (the corrugation factor) is the ratio between the strength of the tip-sample

interaction and the elastic energy of the system and is defined by

2

. (3)

At some finite temperature T , the tip transition is considered to be thermally

activated, enabling the jump to occur when ∆ E > 0. In general, this is assumed to result

in a monotonic decrease in friction force ( F L < F *) with increasing temperature and

decreasing velocity. To facilitate solution of the Tomlinson/Prandtl model, the energy

barrier is often assumed to depend on the lateral force F L as [8,9],

∆ 1 , (4)

where the parameter β [10,11] takes into account the shape of the tip-sample potential.

This assumption along with a transition-state theory analysis, yields the commonly used

relationship between friction force, temperature and sliding velocity [9],

1

ln 12 ln 1

, (5)

where v0 is a characteristic velocity given by 2/3√ , and f 0 is thetransition attempt frequency. This equation has been shown to successfully describe the

velocity and temperature dependence in sliding friction on mica and HOPG [9,12]. We

demonstrate that this equation is applicable only over a certain range of conditions due to

the assumptions involved in deriving Eq. (4) to describe ∆ E as a function of F L. A more


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detailed analysis reveals regimes where both an increase of the lateral force with T , and a

decrease with scanning velocity v, can also be described by the classical Tomlinson

model for a simple sinusoidal sliding potential.

Different regimes can be identified within the framework of the Tomlinson model

based on the values of the corrugation factor γ (Eq. (3)). It is well known that for γ ≤ 1 the

model describes a superlubricious or ultra-low friction regime [13], where friction tends

to zero since the instabilities, that under stick-slip conditions give rise to the dominant

energy dissipation mechanism, disappear. Another important range of γ values are those

at which the tip, under fully relaxed conditions ( x = 0 and X = 0), has no neighboring

minimum in the potential to jump to, so that a finite minimum force ( F min) must be

applied before a transition can occur regardless of the values of T and v. This is

illustrated in Figure 1, where the parameter values E 0 (0.317 eV), a (0.29 nm), and k L

(0.6, 2.59 and 6 N/m) have been chosen to be consistent with previous experimental

conditions [14]. This plots the positions at which the tip is at a minimum of the potential

(Eq. (1)), given by ∂V ⁄ ∂ x = 0, so at X = 0, x is given by

sin 2 . (6)

In Figure 1, the right hand side of Eq. (6) is represented by the solid black (sinusoidal)

curve, while the solid straight lines represent k L x for different values of k L. When k L <

2.59 N/m ( , k L = 0.6 N/m), there are multiple solutions to Eq. (6), indicating that

forward and backward transitions are allowed, even when the tip is fully relaxed ( x = 0, X

= 0, therefore F L = 0). However, when k L = 2.59 N/m ( ), only one possible solution

remains for each direction at x = 0, such that transitions to neighboring stable points for

k L > 2.59 N/m are not allowed since there are no stable positions to move to from x = 0


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when F L = 0 ( , k L = 6.0 N/m). The characteristic limiting value of γ, when there is

only one possible solution, can be calculated by taking the first derivative of Eq. 6, since

it is the only solution with the slopes of the two terms having the same values with

respect to the position x. This leads to

2 tan

2 4.4934… (7)

Introducing the resulting value of x (in this case 0.2074 nm) from Eq. (6), gives the value

of k L = 2.59 N/m; substituting into Eq. (3), yields characteristic value of γ = 4.603. Thus

for values of γ < 4.603, a finite lateral force has to be applied for neighboring stable

points to exist. This is illustrated in Figure 1 for the case of k L = 6.0 N/m (γ = 1.987),

where the support has to be displaced a distance X min = 0.114 nm ( F Lmin = 0.433 nN) to

induce a neighboring stable position in the displacement direction (dashed straight line).

Also shown in Figure 1, for k L = 6.0 N/m, is the maximum support displacement ( X max =

0.176 nm) before the energy barrier decreases to zero (∆ E = 0), at which point the tip

jumps regardless of the values of T and v, when a critical lateral force ( F

*

= 0.475 nN) is

obtained.

Thus, three distinct friction regimes can be defined depending on the value of γ.

For γ ≤ 1, the friction force becomes zero, producing so-called superlubricity. For 1 < γ

< 4.603, the normally assumed variation in energy barrier with lateral force (Eq. (4)) does

not apply, giving rise to unusual temperature and velocity effects, while for γ ≥ 4.603, Eq.

(5) provides a good solution to the Tomlinson/Prandtl model. This is emphasized by the

results in Figure 2 which show the calculated values of ∆ E as a function of F L for

different values of γ (solid dotted lines) where, for comparison, the results of Eq. (4) have

been included using the expression for β derived by Furlong et.al. [10] (open dotted


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lines). Although Eq. (4) does predict a monotonic decrease in ∆ E as a function of F L,

between F L = 0 and F *, the results are quite different in the range 1 < γ < 4.603. First, it is

only when a finite value of F L ( F Lmin) has been achieved that a neighboring stable position

appears and therefore a ∆ E value can be defined. The magnitude of F Lmin increases as the

parameter γ decreases from the particular value, γ = 4.603, deviating from the behavior

predicted by Eq. (4), which assumes a finite value for ∆ E starting from F L = 0. However,

F Lmin starts to show a decrease when γ approaches 1 (e.g. γ = 1.19), as it approaches the

superlubricious regime, γ ≤ 1. The corresponding values of F Lmin are indicated by the

vertical dashed lines intercepting the x-axis. Second, the decrease in ∆ E is not

monotonic, but increases with F L for higher values of F L when 1 < γ < 4.603. Since the

critical force F * is defined as that required to decrease the energy barrier ∆ E to zero, the

plot indicates that, under certain conditions, the lateral force can become larger than F *,

so that F * cannot be considered to be the maximum possible value under all conditions.

In order to emphasize this behavior, Figure 3 expands the plot in Figure 2 to show larger

values of F L. Note that, as expected, the values of F * predicted by Eq. (2) for the

different values of k L show agreement between the calculated results and those resulting

from Eq. (4). As a result, it can be concluded that the use of Eq. (4) and (5) is only valid

when γ > 4.603.

The intervening regime (when 1 < γ < 4.603) gives rise to unusual variations of F L

with T and v, that have not been previously described by the classical Tomlinson model.

In order to study the effects of T and v on the lateral force in this regime, Monte Carlo

simulations based on the one-dimensional classical Tomlinson model were performed.

This strategy has been previously used to explore velocity effects on sliding friction, and


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it has been demonstrated that it precisely reproduces the solution of the sinusoidal

Tomlinson/Prandtl model [10,15].

Figure 4 shows the simulated results of lateral force ( F L) with velocity (ln(ν)) for

different values of k L (and γ), using f 0 = 50 kHz to be consistent with previous

experimental results [14]. It can be observed that when γ > 4.603, the friction versus

velocity results are in agreement with the predictions of the semiempirical equation (Eq.

(2)), where there is a monotonic increase in friction force with velocity to a maximum

value equal to F *. However, when γ < 4.603, the maximum in lateral force appears at

intermediate velocities, after which friction decreases to the corresponding F

*

value and

similar effects have been observed experimentally [16]. In particular, when k L = 6.0 N/m

(γ = 1.987), a well-defined maximum in lateral friction is observed at ln(ν) ~ -1. In accord

with Figure 2, the value of the upper limit in lateral force ⁄ is indicated by ahorizontal dotted line. As expected, the value of F

* decreases with k L but clearly does not

correspond to the maximum lateral force that can be obtained as a function of v in the

range 1 < γ < 4.603.

Figure 5 displays Monte Carlo simulation results of lateral force versus

temperature at different velocities. All the parameters were kept the same as above,

except for f 0, which has been increased to 500 kHz, still well within the normal range of

f 0 values. As expected, the lateral force at T = 0 corresponds to F *, where based on Eq.

(5), a monotonic decrease to zero with temperature should be observed. However, under

these conditions (γ = 1.987), the simulation shows an initial increase in lateral force from

F * with temperature up to a maximum value, from which it then decreases monotonically

with temperature. Similar behavior has been observed experimentally [17]. Such effects


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have been previously ascribed to two competing processes acting at the interface: the

thermally activated formation and rupture of an ensemble of atomic contacts, each

occurring with different activation energies [18]. Here, it is shown that by properly

considering the classical one-dimensional Tomlinson model, it is possible to predict such

behavior by a single activation process. In addition, it has been shown that F * is not

necessarily the maximum force value (see Figure 4) and that under certain conditions (1 <

γ < 4.603), the lateral force increases as ∆ E increases from zero (see Figure 3). Therefore,

as the temperature increases from zero, transitions at higher values of ∆ E are favored

resulting in an increase in the lateral force at the low temperatures.

Acknowledgments

We gratefully acknowledge the National Science Foundation under grant number CMMI

0826151 and the CONICET (Argentina) for support of this work.


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Figure Captions

Figure 1: Graphic representation of Eq. (6). The right hand side of Eq. (6) is represented

by the solid black (sinusoidal) curve, while the solid dotted straight lines represent k L x for

different values of k L, using E 0 = 0.317 eV and a = 0.29 nm. The y-intercept of the dashed

straight line represents the minimum support displacement required to induce the

appearance of neighboring stable position, and the y-intercept of the dotted straight line

indicates the support displacement at which ∆ E becomes cero.

Figure 2: Calculated values of ∆ E as a function of F L for different values of k L where E 0

= 0.317 eV and a = 0.29 mm (solid dotted lines). For comparison, the results of Eq. (4)

have been included using the expression for β derived by Furlong et.al. [10] (with the

corresponding open dotted dashed lines). The vertical dashed lines represent the

minimum lateral force required to induce the appearance of a stable neighboring position

to which to jump.

Figure 3: Expanded plot of Figure 3 for larger values of F L. (k L = 0.6 N/m, solid black

line; k L = 2.59 N/m, ; k L = 4.0, ; k L = 6.0 N/m, ; k L = 8.0 N/m, ; and k L

= 10.0 N/m, ), where the results for k L = 8.0 N/m have been added for clarification

purposes. The value of F * obtained from Eq. (2) is indicated for each curve.

Figure 4: Simulated results of lateral force ( F L) with velocity (ln(ν)) for different values

of k L (γ), using f 0 = 50 kHz, E 0 = 0.317 eV, a = 0.29 nm and T = 298 K.


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Figure 5: Simulated results of lateral force ( F L) with temperature, T , for different values

of velocity (ln(ν)), using f 0 = 500 kHz, E 0 = 0.317 eV, a = 0.29 nm and k L = 6.0 N/m.

Indicated in the y-axis is corresponding critical force F *.


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Figures

Manzi et al ,Figure 1


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Manzi et al , Figure 2


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