Experimental Investigation of the Drift and Diffusion of Small Objects

Jun 17, 2009 - Department of Chemical Engineering, Lehigh University, Bethlehem, ... According to de Gennes (see also the work of Hayakawa6 as well as...
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Experimental Investigation of the Drift and Diffusion of Small Objects on a Surface Subjected to a Bias and an External White Noise: Roles of Coulombic Friction and Hysteresis P. S. Goohpattader,† S. Mettu,† and M. K. Chaudhury* Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, USA. † These authors contributed to this work equally Received March 30, 2009. Revised Manuscript Received May 15, 2009 We study the stochastic motion of a small solid block or a small water drop on a flat solid support in the presence of an external noise and a bias. The bias is caused either by inclining the plane of the support, as is the case with the solid block, or by creating a gradient of wettability, as is the case with a water drop. Both the solid block and the water drop exhibit drifted Brownian-like motion. There are, however, differences between the motion described here and that of a classical drifted Brownian motion, in that the Coulombic friction (for solid on solid) or wetting hysteresis (for water drops on a solid) accounts for a significant resistance to motion in addition to the kinematic friction. Although the displacement distribution here is non-Gaussian, the variance of the distribution increases with time, indicating that the overall motion follows simple diffusion. The diffusivity and the mobility of the solid object are considerably lower than the values expected when the diffusion is governed by only kinematic friction. The experimental diffusivity increases with the power of the noise with an exponent of 1.61, which is close to that (1.74) of an analysis based on the Langevin equation when the Coulombic friction is taken into account in addition to the kinematic friction. The ratio of diffusivity and mobility increases slightly sublinearly with the power of the noise with an exponent of about 0.8. The experimentally observed relaxation time of the process is, however, considerably smaller than the Langevin relaxation time. When the experimental ratio of diffusivity and mobility is taken into account in the distribution function of the displacement, the later quantity becomes amenable to an analysis that is similar to the conventional fluctuation relations.

Introduction The role of Coulombic dry friction in the damping of the harmonic motion of a small solid object on a substrate or that of a pendulum has been known for quite some time.1-4 Most recently, the difference between Coulombic dry friction and kinematic friction was clearly demonstrated in an elegant experiment by Simbach and Priest,5 who studied the swinging of a pendulum, the pivot of which is a rotary variable resistor. As the pendulum swings, a Coulombic friction operates inside the variable resistor. Since the angular displacement is itself converted to voltage, its measurement as a function of time can be used to monitor the angular position of the pendulum. What is remarkable in their finding is that, unlike the usual dissipative systems where the amplitude decays exponentially with time, the amplitude decays linearly with Coulombic friction. The pendulum is clearly damped, and thus it dissipates energy, but in a way that is uniquely different from the standard kinematic friction. Recently there is a growing realization6-8 that the Coulombic friction may be an underlying mechanism of energy dissipation in granular medium. It may also play important roles in other processes, such as the stochastic motion of one object on another in the presence of vibration. de Gennes9 has recently examined the dynamics of a simple system in which a solid block rests on *Corresponding author. E-mail: [email protected]. (1) (2) (3) (4) (5) (6) (7) (8) (9)

Lapidus, I. Am. J. Phys. 1970, 38, 1360. Strobel, G. L.; Barratt, C. Am. J. Phys. 1981, 49, 500. Clark, C; Swartz, C. Phys. Teach. 1996, 34, 550. Peters, R. D.; Pritchett, T. Am. J. Phys. 1997, 65, 1067. Simbach, J. C.; Priest, J. Am. J. Phys 2005, 73, 1079. Hayakawa, H. Physica D 2005, 205, 48. Kawarada, A.; Hayakawa, H. J. Phys. Soc. Jpn. 2004, 73, 2037. Mauger, A. Physica A 2006, 367, 129. de Gennes, P.-G. J. Stat. Phys. 2005, 119, 953.

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another solid support, the latter being vibrated with white noise. The solid block is expected to undergo a random motion with a certain diffusivity, the value of which is modified from that of a kinematic friction. He worked out certain scaling laws of the diffusivity in terms of the magnitude of the Coulombic friction and the strength of the imposed external noise. According to de Gennes (see also the work of Hayakawa6 as well as that of Kawarada and Hayakawa7), the dynamics of the object on a surface subjected to an external vibration (γ(t)) and a bias (γh) is described by the modified Langevin equation:9 dV V þ ¼ γ þ γðtÞ -σðVÞΔ dt τL

ð1Þ

Here, V is the velocity of the particle, τL is the Langevin relaxation time (ratio of the mass M of the particle and its kinematic friction coefficient ζ), and γh is the external constant force divided by the mass of the object. γ(t) is the time-dependent acceleration that the object experiences from the white noise source. Ideally, it is delta correlated and has a zero mean value. If the magnitude of the static friction force is smaller than (γh + γ(t)), the object moves; otherwise it remains stuck to the surface. σ(V) is the signum function of velocity with σ(0) = 0, Δ is a measure of the Coulombic kinetic friction (force/mass). Let Γ be the root-mean-square acceleration and τc be the duration time of the pulse. For convenience, we introduce a symbol K, which is the power of noise and is equal to Γ2τc; K/2 is also the diffusivity in the velocity space. One can solve the corresponding Klein-Kramers10 form of the Fokker-Planck equation of the Langevin eq 1 or simply balance (10) Risken, H., The Fokker-Planck Equation, Methods of Solution and Applications, 2nd ed.; Springer-Verlag: Berlin, Heidelberg, and New York, 1989.

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the diffusive and convective fluxes of the probability density function (pdf). -

K DP jVj VP -Δ P¼ -γP 2 DV V τL

ð2Þ

Here, P=(P(V)) is the steady-state pdf 6-8,11 of the velocity, which can be obtained by integrating eq 2. V2 2jVjΔ 2V γþ PðVÞ ¼ Po exp K KτL K

! ð3Þ

Where, Po is normalization constant. According to eq 3, when Δ=0, the velocity distribution is Gaussian about the mean γhτL, i.e., it resembles the situation of a simple dragged Brownian 6 0, the particle with a diffusivity of KτL2/2. However, when Δ ¼ velocity pdf also has an exponential component. Let us consider the case for Δ ¼ 6 0 and γh < Δ and set the kinematic friction to zero. In this case, de Gennes9 showed that the object exhibits a diffusive motion with a value of diffusivity as ∼K3/Δ4, which is remarkably sensitive to the power of the noise in contrast to the simple kinematic situation, where diffusivity is KτL2/2. Furthermore, when the object is agitated with white noise vibration, it drifts with a velocity ∼Kγh/Δ2, which is also uniquely different from the case of pure kinematic friction for which the drift velocity at any value of K is just γτhL. Thus, while the ratio of diffusivity and mobility is MKτL/2 for the case with kinematic friction that for the dry friction is ∼M(K/Δ)2. Note that in this case of external noise, the energy is being delivered by work which is different from a typical thermal system. Thus, no fluctuation dissipation theorem is expected in these situations with external noise. Nevertheless, since the ratio of diffusivity and mobility (D/μ) varies as (K/Δ)2, an effective temperature may be defined for a dragged diffusive process with dry friction. In a somewhat more realistic situation with solid/solid interfaces, both the kinematic and solid frictions operate. In this case, eq 1 needs to be considered as a whole. We have done so with a numerical simulation of eq 1 in conjunction with an experiment that allows us to measure the diffusivity and mobility of a solid on solid with the imposition of a bias and white noise. Since the white noise term is external, we can select the noise to be Gaussian or non-Gaussian. We have experimented with both the Gaussian and truncated Levy12 (Cauchy) noises having finite powers. The subject of a Levy noise (both spatial13 and temporal14-17) driven transport of particles has been extensively discussed in the literature. The problem that we investigate is, however, somewhat different from what is usually discussed in the literature in the sense that we use a truncated Levy (here Cauchy) noise that acts against the Coulombic static friction in addition to the kinematic friction. With a truncated Levy noise, finite variance exists in both velocity and real spaces. Thus it appears that we may be able to use the same probability conservation equations (eqs 2 and 3), as is the case with the Gaussian noise, as long as the power of the Levy noise is low enough. We may expect that all the transport properties measured in terms of drift and diffusion would be identical to those obtained with a Gaussian noise, which we (11) Chaudhury, M. K.; Mettu, S. Langmuir 2008, 24, 6128. (12) Mantegna, R. N.; Stanley, H. E. Phys. Rev. Lett. 1994, 73, 2946. (13) Carmona, P. Ann. Probability 1997, 25, 1774. (14) Bao, J. D. Physica A 2005, 346, 261. (15) del-Castillo-Negrete, D.; Goncharb, V.Yu.; Chechkin, A. V. Physica A 2008, 387, 6693. (16) Dybie, B.; Gudowska-Nowak, E.; Sokolov, I. M. Phys. Rev. E 2008, 78, 011117. (17) Touchette, H.; Cohen, E. G. D. Phys. Rev. E 2007, 76, 020101(R).

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Figure 1. Schematic of the experiment is shown in the inset. A solid object or a liquid drop drifts downward on a vibrating inclined substrate by overcoming the forces of Coulombic friction or hysteresis. The probability distribution functions p(γ(t)) of the Gaussian and a truncated Cauchy noises that were used to vibrate the substrate are also shown. δ represents the width of the truncated Cauchy distribution, and γ(t) are the random acceleration pulses.

indeed find experimentally. In view of the above points, it may be asked why should we bother to consider the Levy noise to determine transport properties. We will show later that the truncated Levy noise, although yielding results like a Gaussian noise, offers certain advantages by improving the statistics in the low probability regions of the distributions, which are advantageous in certain analysis to be discussed later in the paper.

Steady Drift of a Solid Block or a Water Drop on a Surface in the Presence of Coulombic Friction or Hysteresis We investigate the following three special cases, which are related to the inertial tribometer idea pioneered by Baumberger et al.18 Here, a solid object is placed on a track, which undergoes a periodic oscillation. This transfers a shear force to the slider. By studying the dynamics of the slider, the dissipation at the solid/ solid interface can be estimated. Our experiment is almost similar to that reported in another publication19 by the same group, who studied the sliding behavior of one solid object on an inclined track by a periodic vibration. In our case, the track is submitted to white noise instead of a periodic vibration. For the most part, we study the drifted and stochastic motion of a small glass prism (∼2 g) having dimensions of ∼12 mm  12 mm  6 mm on a partially inclined glass substrate in the presence of white noise. Slight roughening (micrometer level) of the support was necessary to induce easy and uniform sliding of the glass prism over it. For very smooth surfaces, the glass prism adheres to the glass support so strongly that a very high level of vibration is needed to dislodge it. Furthermore, the prism then slides down with some random rotations about its axis. Although a translational Brownian motion in conjunction with random rotation, is, in itself, an interesting problem to study, here we keep the problem at a simple level. Slight roughening decreases the level of adhesive friction and thus eliminates some of the above problems. The experimental details are as follows11 (Figure 1). The substrate was firmly attached to an aluminum platform which was framed on a mechanical oscillator (Pasco Scientific, Model No: SF-9324). White noise that was generated using Matlab (18) Baumberger, T.; Bureau, L.; Busson, M.; Falcon, E.; Perrin, B. Rev. Sci. Instrum. 1998, 69, 2416. (19) Bureau, L.; Caroli, C.; Baumberger, T. Proc. R. Soc. London, A 2003, 459, 2787.

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Figure 2. Stochastic motion of a prism on a solid substrate subjected to truncated Cauchy noise with a power of 0.053 m2/s3.

program was fed through the sound card of the computer to the oscillator via a power amplifier (Sherwood, model no. RX4105). The whole setup was placed on a vibration isolation table (Micro-g, TMC) to eliminate the effect of ground vibration. The acceleration of the supporting aluminum plate was estimated with a calibrated accelerometer (PCB Peizotronics, model no. 353B17) driven by Signal Conditioner (PCB Peizotronics, model no. 482) and connected to an oscilloscope (Tektronix, model no. TDS 3012B). The drift velocities of the prisms were measured on the inclined plate with a low speed (30 fps) normal Sony camera (DCR-HC85 NTSC), and the stochastic motion of the prisms was monitored with a high-speed (1000 fps) Redlake Motion-Pro video camera at different powers of noise. When the position of the prism is plotted against time with the data obtained with a low speed camera providing a large field of view, an excellent straight line is obtained over a distance of 40 mm with good reproducibility, which signifies that the property of the surface is rather uniform over a significant length for meaningful measurements of velocity and other properties. The data obtained with the high speed camera were subjected to an analysis using Midas2.0 Xcitex software to obtain the instantaneous position as a function of time by tracking the edge of the prism. The solid prism moves stochastically with an average downward drift (Figure 2). Both the drift velocities and diffusivities were measured by varying the power of the noise. In order to estimate the diffusivity, displacement data for various time segments (τ) as obtained from several tracks were combined, from which a probability distribution for displacement was constructed. Each probability distribution function of displacement exhibits well-defined mean and variance. The variance increases with τ as expected of a simple diffusive process, the slope of which gives the estimation of the diffusivity according to the following relation: Æxτ2æ - Æxτæ2 =2Dτ. We also performed some experiments with small water drops with which we also analyzed the drift velocity and the displacement fluctuations. In the first experiment of this type, a small drop was placed on a hydrophobic substrate (smooth, silanized silicon wafer) that was tilted by a small angle from the horizontal.11 The stochastic motion of the drop was examined by vibrating the substrate by computer-generated white noise. Since a drop has two degrees of freedom, one of the center of mass and the other of the base, the automatic method of tracking the position of the drop as we used for the solid slider could not be adopted here. The positions needed to be tracked manually, which is a very slow process, and thus not enough data could be collected. However, analysis of at least five tracks, each lasting for 6 s, gave a sufficient amount of data for the analysis of drift and diffusion. Since the preliminary data of the drift and displacement fluctuations of the inclined drops have been reported in reference 11, they will not Langmuir 2009, 25(17), 9969–9979

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be further elaborated here except reporting some general trend of drift velocities for the purpose of comparison with other measurements. The second experiment was also with a drop. However, this time the plate was not tilted. Instead, a gradient of surface energy was prepared on a silicon wafer on which the drop was released. Usually, a drop is supposed to slide down on an inclined plane or move on a gradient surface20-22 toward the region of higher wettability, as the free energy decreases in both of these processes. However, the drop does not move spontaneously in either case because of a well-known effect known as contact angle hysteresis.23,24 However, when excited by white noise vibration, the drop moves freely on the surface. Before describing the behavior of drops with hysteresis, let us first describe the behavior of the solid prism on the inclined substrate with Coulombic friction. The prism, as is the case with the water drop, also does not slide spontaneously on an inclined plane. This is not surprising as a static Coulombic friction of magnitude μsmg cos θ operates at the interface (here, μs is the static friction coefficient, m is the mass of the prism, g is the gravitational acceleration, and θ is the angle of inclination of the plate). Usually the magnitude of the gravitational force, μsmg sin θ, has to be larger than μsmg cos θ for the cube to slide down as a result of gravity, which happens at an angle larger than a critical angle. At a smaller angle, i.e., θ < tan-1 μ, the prism can be considered to be at a “locked state” by borrowing a phrase from Risken,10 who used the above phrase for the case of a particle trapped in a potential well of a tilted ratchet. However, there is another solution to the problem, that is, if the effect of static friction vanishes completely, it would slide down with a constant velocity g sin θτL, where τL is the momentum relaxation time given as the ratio of the mass to the dynamic friction coefficient (ζ). This would be called the “running state” in the words of Risken.10 With the application of white noise with a zero mean, the prism is kicked upward and downward with random forces. However, the net motion is biased downward because of a small gravitational bias acting on it. Thus a transition occurs from the “locked” to the “running” state in which the drift velocity increases with the strength of the noise. At this point, it should be clarified that, if the imposed noise is (mathematically) white, there will always be a powerful impulse within any short duration of time that would dislodge the object. However, no external noise is perfectly white, and thus, within a short duration of time, the object may not be dislodged. The object remains stuck to the surface, until an impulse of sufficient strength arrives to dislodge it.9 With the definition of the velocity probability distribution embodied in eq 3, the average drift velocity of the prism can be obtained from the following equation: R þ¥ VPðVÞdV Vdrift ¼ R-¥ þ¥ -¥ PðVÞdV

ð4Þ

Figure 3 summarizes the drift velocities of the solid prism as a function of the power of Gaussian white noise K at three different angles of the inclination of the plate. As expected, Vdrift increases both with K and θ. These data could be analyzed with eqs 3 and 4, (20) Chaudhury, M. K.; Whitesides, G. M. Science 1992, 256, 1539. (21) Daniel, S.; Sircar, S.; Gliem, J.; Chaudhury, M. K. Langmuir 2004, 20, 4085. (22) Daniel, S.; Chaudhury, M. K.; Chen, J. C. Science 2001, 291, 633. (23) Frenkel, Y. I.; J. Exp. Theor. Phys. (USSR), 1948, 18, 659, translated into English by Berejnov, V. (http://arxiv.org/ftp/physics/papers/0503/0503051.pdf). (24) Johnson, R. E.; Dettre, R. H. Contact Angle, Wettability and Adhesion; Advances in Chemistry Series No. 43; American Chemical Society: Washington, DC, 1964; Vol. 112.

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Figure 3. (Left) Drift velocities (Vdrift) of a glass prism on a slightly roughened glass support at three different inclinations (angles are shown in the figure) and various powers (K ) of the noise. (Right) Drift velocities (Vdrift) are divided by sin θ, where θ is the angle of inclination. Open symbols correspond to Gaussian noise, and closed symbols correspond to truncated Cauchy noise. All the data roughly fall close to a single master curve.

with γh=g sin θ. From this analysis, the average values of Δ and τL are estimated to be 3.84 m/s2 and 0.067 s respectively. These values of Δ and τL were used for the numerical simulations of the drift velocity and diffusivity to be discussed below. The velocities obtained with truncated Cauchy noises of various powers are also given in these figures for comparison. Indeed the velocities obtained with both types of white noises are indistinguishable. An approximate equation that can describe the drift velocity as a function of γh, Δ, K, and τL has been reported in reference 11: Vdrift ¼

γτL 1þ ΔKτL 2

ð5Þ

All the drift velocity data can also be fitted with eq 5 quite well with slightly different numerical values of Δ (3.6 m/s2) and τL (0.03 s).

Diffusivity of Solid on Solid: Experiments and Simulations We attempted to understand how the displacement fluctuations should behave by solving eq 1 numerically. Here we briefly describe the procedure used to numerically solve the Langevin equation (eq 1) to obtain displacement (x) of the prism as a function of time (t). We used γh =g sin θ, which is the bias force acting on the prism, with θ set to 2 as in the experiments. We used the acceleration data, γ(t), obtained directly from the accelerometer, as input acceleration for the numerical analysis. The external acceleration γ(t) acting on the prism has either Gaussian or truncated Cauchy (Figure 1) probability distributions with a mean that almost approaches zero (∼0.4% of the root-meansquare (rms) value). The fact that the mean value of the acceleration is not exactly zero results from the fact that large values of acceleration are being averaged, coupled with the fact that the numbers are rounded off during the digitizing process. However, this small numerical error does not contribute appreciably in estimating the values of diffusivities, although it can lead to some error in estimating the drift velocities at very low bias. Here we focus our attention on the estimation of diffusivity. As the highspeed video recording is done in experiments at an interval (Δt) of 1 ms, data from the accelerometer is obtained at the same interval. These acceleration data are then scaled in order to match the power of noise (0.004 to 0.16 m2/s3) used in the experiments with two additional values estimated at powers of 0.3 and 0.5 m2/s3. The Langevin equation is then integrated with an integration time step of Δt=1 ms, which is equal to the video recording interval in experiments. The simulations are carried out for 50 tracks with an integration time of 10 s for each track. The values of Δ and τ used 9972 DOI: 10.1021/la901111u

Figure 4. Log-linear plot of the probability distributions of the displacement (xτ) of the prism subjected to Gaussian and truncated Cauchy noises. The data for each noise was obtained from about 132 steady state tracks, each lasting for about 2.5 s. In the lower part of the distributions, the statistics for the Cauchy noise is much better than that of a Gaussian noise, where considerable scatter is observed. The experiment is carried out at a power (K ) of 0.09 m2/s3. The displacement distributions are for τ = 0.09 s.

in the simulation are 3.84 m/s2 and 0.067 s respectively, which were obtained from the analysis of the data shown in Figure 3. Since the Coulombic friction σ(V)Δ always acts in the direction opposite to the motion of the prism, σ(V) is set as V/|V| in the simulations. When the net acceleration (|γh + γ(t)|) acting on the prism is less than the threshold acceleration (Δ) required to set the prism into motion, the prism gets stuck to the plate, and hence the velocity of the prism is set to zero. The displacement data for a given time interval τ obtained from several tracks are combined to obtain a probability distribution for displacement as is done with the experimental analysis of the displacement data. The diffusivity was estimated from the slope of the plot Æxτ2æ - Æxτæ2 versus τ. As illustrated in Figure 4, the experimentally observed pdf’s of the displacement for both the Gaussian and truncated Cauchy noises almost superimpose onto each other except for the low P(xτ) region where the statistics become somewhat poorer for the Gaussian noise compared to that obtained with a truncated Cauchy noise. Thus, these two properties of a truncated Cauchy noise that produce drift and diffusion exactly the same as that of a Gaussian noise, and the fact that it improves the statistics in the region of low probability region are quite useful to us. The features resulting from a real Levy distribution cannot be reproduced in our system, as its power diverges. For the case with a water drop, as mentioned above, there are two degrees of freedom: one of the center of mass and the other of the center of the base. Here, heavy-tailed Cauchy noise produces drastic distortion of the drop. Thus, for now, we restrict the studies of the drop dynamics with a Gaussian noise.

Relationship between Diffusivity and the Power of Noise From the experimental data of the stochastic displacement (xτ) of the prism as a function of time, the diffusivities were estimated by dividing the track to various time segments and then using the equation Æxτ2æ - Æxτæ2 = 2Dτ (Figure 5). For this analysis to be valid, τ needs to be sufficiently large in order to ensure that a steady state is reached. In the steady-state regime, the variance of the displacement is linear with time. Both theory and simulations for a prism on a solid substrate show that the diffusivity varies linearly with K in the absence of Langmuir 2009, 25(17), 9969–9979

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Figure 5. Plot of the variance Æxτ2æ - Æxτæ2 of the displacement of the prism subjected to truncated Cauchy noise of different powers. The diffusivity, which is calculated from the slope of linear plots, increases with the power of the noise. Here data are shown for three powers only. We have performed these measurements for a total of five different powers.

Figure 6. Log-log plot of diffusivity as a function of power of noise. The scaling law established from experiments (blue triangles) is D ∼ K1.61, whereas the numerical simulation (pink squares) predicts D ∼ K1.74. Diffusivity values obtained by numerical simulation in the absence of dry friction (Δ = 0) are also shown (red circles), which agree well with theoretically predicted values (blue solid line) for kinematic friction only. The experimental diffusivity (open circle) of the prism subjected to Gaussian noise with a power of 0.09 m2/s3 is close to the value obtained using Cauchy noise at the same power. The experimental diffusivity (open diamond) of a liquid drop on a gradient surface subjected to Gaussian noise with a power of 0.022 m2/s3 is also shown.

dry friction (Figure 6). The source of the noise, whether it is Gaussian or the truncated Cauchy noise, does not make any appreciable difference. When Δ 6¼ 0, the simulated diffusivity values vary as K1.74 (Figure 6). The observed exponent of K is certainly larger than that for Δ=0, but it is smaller than the value of 3 expected for a pure dry friction.9 This disagreement is expected since, as we do not have a pure dry friction situation, kinematic friction also plays a role. The experimentally observed diffusivities vary by 3 orders of magnitude (10-10 m2/s to 10-7 m2/s) with the variation of the power from 0.004 m2/s3 to 0.16 m2/s3. These values are much lower than the values obtained with Δ=0, as is found in the simulations as well. The experimental diffusivities are, however, closer to (although slightly lower than) those predicted for dry friction (Figure 6). Except for the datum obtained with the lowest power (0.004 m2/s3), the experimental diffusivity increases with K with an exponent of 1.61. This exponent is also close to that (1.74) obtained from the simulations with the Coulombic friction. When the power reaches a value as low as 0.004 m2/s3, the dynamics of the prism slows down dramatically. It is possible that, at such low power, other effects such as heterogeneities, both static and dynamic, may start to play additional roles over that of the average effect of Coulombic and kinematic friction. Langmuir 2009, 25(17), 9969–9979

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The above comparisons of the experimental diffusivities with those obtained from simulations are the main results of our paper. We now focus on the shape of the displacement distribution functions in the presence of Coulombic dry friction. Jarzynski25 pointed out that one of the remarkable features of the nonequilibrium distribution functions is that they encode various types of physical processes. For example, the equilibrium free energy changes are encoded in nonequilibrium work fluctuations. Several recent experiments26,27 exploited this idea to estimate equilibrium free energy change of biological processes from the dissipative work measurements. For a process governed by kinematic friction, the distribution function is itself Gaussian. However, in the granular media, a non-Gaussian distribution28-34 of the velocity pdf has been linked to the energy loss due to inelastic collisions. There are other cases to consider in which a threshold force of a different nature may act on a particle. An example of this involves biological cells partially adhering to a solid surface, in which case the random motion35 of the cells exhibits a stick-slip process. Another example is the condensation and growth of liquid drops on a surface that may be prone to a random motion due to various fluctuations, but its motion is inhibited by wetting hysteresis.22 Recently, some efforts have been made to transport small liquid drops on a surface by an external noise for microfluidic applications. Here too, the threshold force due to hysteresis decreases11 the mobility of the drops in a profound way. The question we ask is whether these Coulombic friction and hysteresis are encoded in the displacement distribution functions. This subject is elaborated in the following section.

Drift, Diffusion, and Displacement Fluctuation of a Water Drop on a Surface Let us consider the case of a water drop, the motion of which is arrested by hysteresis either on an inclined plane or on a gradient surface. Wetting hysteresis, like Coulombic friction, provides a threshold force, which needs to be overcome23 for a drop to slide on a surface. Hysteresis of wetting is usually measured by adding to or removing liquid from a sessile drop on a surface. In the first case, the drop grows with a contact angle on the substrate that is larger than the second case. The former is known as the advancing angle, whereas the latter is known as the receding angle;the difference being the wetting hysteresis. Different models of wetting hysteresis may be generalized in a way that various forms of surface imperfections modify the standard parabolic energy profile of a drop, thus creating metastable energy states.24 The drop can get stuck in any one of the metastable states, but is bound by two extreme saddle point energy states that correspond to the advancing and receding angles. In an ideal situation, i.e., in the absence of hysteresis, if the radius of the drop is increased or decreased beyond its equilibrium value ao, the drop is readily restored to its original position. The drop on the surface is (25) Jarzynski, C. Phys. Rev. Lett. 1997, 78, 2690. (26) Ritort, F. Poincare Semin. 2003, 2, 195. (27) Bustamante, C.; Bryant, Z.; Smith, S. B. Nature 2003, 421, 423. (28) Losert, W.; Cooper, D. G.W.; Delour, J.; Kudrolli, A.; Gollub, J. P. Chaos 1999, 9, 682. (29) (a) Olafsen, J. S.; Urbach, J. S. Phys. Rev. Lett. 1998, 81, 4369. (b) Phys. Rev. E 1999, 60, R2468. (30) Rouyer, F.; Menon, N. Phys. Rev. Lett. 2000, 85, 3676. (31) van Noije, T. P. C.; Ernst, M. H. Granular Matter 1998, 1, 57. (32) Bizon, C.; Shattuck, M. D.; Swift, J. B.; Swinney, H. L. Phys. Rev. E 1999, 60, 4340. (33) Nie, X.; Ben-Naim, E.; Chen, S. Y. Europhys. Lett. 2000, 51, 679. (34) Prevost, A.; Egolf, D. A.; Urbach, J. S. Phys. Rev. Lett. 2002, 89, 084301. (35) Selmeczi, D.; Li, L.; Pedersen, L. I. I.; Nrrelykke, S. F.; Hagedorn, P. H.; Mosler, S.; Larsen, N. B.; Cox, E. C.; Flyvbjerg, H. Eur. Phys. J. Spec. Top. 2008, 157, 1.

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Figure 7. Drift velocities of a small water drop (10-5 kg) on a hydrophobic (silanized) silicon wafer. (Left) These measurements were conducted with an inclined (10) substrate with low (a) and high (b) hysteresis surface. Hysteresis is measured here by the inclination of the hydrophobic plate when the drop just begins to slide down on its own. For panel a, this angle is 10, and for b this angle is 18. (Right) These data are obtained with an energy gradient surface (0 inclination) for which d cos θ/dx ≈ 88 m-1 θ being the contact angle of water drop, which, in this case, is position dependent. The base area of the drop is ∼5.3  10-6 m-2, and the surface tension of water is 72.8 mN/m. The experimental data are fitted with solid lines obtained using eq 5.

therefore like a spring with a spring constant Ks. The, local equilibrium of the drop is determined by the balance of the spring force and the derivative of a corrugated energy potential, which leads to eq 6: ks x ¼ ð2πUo =λÞ sin

2πx λ

ð6Þ

Here, a is the radius of the drop, and x=a - ao. Uo is the depth of the corrugated potential of the surface, and λ is its wavelength. If the spring force is larger than the maximum gradient of the potential, the drop would immediately advance or retreat to a position where it is balanced by the gradient of the potential. In the intermediate situation, multiple solutions of eq 6 exist with the maximum and minimum values of a being a ¼ ao ( ð2πUo =λks Þ

ð7Þ

The receding and advancing contact angles of the drop correspond to the maximum and minimum values of a for a given volume constraint. Now, when a drop moves on either an inclined plane or on a gradient surface, its frontal side attempts to reach the local advancing angle, whereas its rear side tries to achieve a local receding angle. This leads to a threshold force23 of the magnitude γwω(cos θr - cos θa), where γw is the surface tension of water drop, ω is its width, and θa and θr are the advancing and receding contact angles, respectively. This force must be overcome before the drop moves, and it always acts against the direction of motion of the drop. So we see that the term γwω(cos θr - cos θa)for a liquid drop is, in some way, a surrogate for the Coulombic friction force for solid/solid case. For a drop oscillating on a surface accompanied with a motion of contact line, this hysteresis force always acts against the motion of the drop. Equation 1 may thus be an adequate description (to a first order) for a liquid drop subjected to white noise and a bias with the value of Δ equal to γwω(cos θr - cos θa)/m, m being the mass of the drop. Indeed, when we perform the experiment of the drift velocity of the drop as a function of the power of noise, we observe a pattern that is similar to the solid/solid case. The drop does not move in the absence of the noise either on an inclined plane or on an energy gradient surface, even though a force g sin θ or γh=(γw/m)(d cos θ/ dx)A (A is the base area of the drop, and d cos θ/dx is the wettability gradient) acts on the drop. However, as the drop is subjected to white noise vibration, it starts to move with its velocity, increasing nearly parabolically with K (Figure 7). This 9974 DOI: 10.1021/la901111u

dependence of the drift velocity on K can be fitted well (solid lines) to eq 5. Now, we focus on the displacement fluctuation on the gradient surface. In order to make a rough comparison of the experimental and simulations results of drop displacement and its fluctuations, we plot (Figure 8) the probability distribution of xτ by nondimensionalizing it as follows: (xτ - xp)/σxτ, with σxτ being the standard deviation of xτ, and xp being the displacement value at the peak of the distribution function, which is not all that different from the mean displacement xm. First of all, we point out that the general pattern of the probability distributions of displacement as obtained experimentally is in general agreement with those obtained from the simulations with Δ > 0. The distribution is more Gaussian near the center, but then it becomes exponential as we move away from the center of the distribution. However, for these Gaussian noise-induced displacement fluctuations, an asymmetry can be noted in both experiments and simulations. Since the exponential distribution of displacement as well as its asymmetric distribution11 are signatures that a threshold force, which is the hysteresis, operates while the drop moves on a surface, it brings forth an issue of the relationship between the hysteresis and vibration itself . There are several reports36-39 in the literature that suggest that the hysteresis itself may disappear with the vibration of a drop. If this were the case, i.e., Δ=0, the displacement distribution would be symmetric and Gaussian as shown in Figure 8b. This is clearly not the situation in our case, as we obtain an asymmetric and non-Gaussian distribution. However, this observation opens up a new question as to how the displacement fluctuation behaves if the hysteresis truly disappears. Since the magnitude of the hysteresis (Δ=3.65 m/s2) for the water drop happens to be very close the value of the Coulombic threshold force for the solid/solid (Δ=3.84 m/s2) case, we have an opportunity to compare the diffusivity of the drop on the surface with that of the solid/solid case at the same power of noise. The diffusivity (8  10-09 m2/s) of the drop as obtained experimentally is indeed close to the value (4  10-09 m2/s) obtained for (36) Smith, T.; Lindberg, G. J. Colloid Interface Sci. 1978, 66, 363. (37) Andrieu, C.; Sykes, C.; Brochard, F. Langmuir 1994, 10, 2077. (38) Decker, E. L.; Garoff, S. Langmuir 1996, 12, 2100. (39) Meiron, T. S.; Marmur, A.; Saguy, I.S. J Colloid Interface Sci. 2004, 274, 637. (40) We should caution the reader that the simulation and experimental results are not in perfect agreement, as it may appear from these normalized plots. This is indeed not the case (see Figure 6). The only purpose of making these nondimensional plots is to highlight the shapes of the distributions rather than placing any emphasis on the numerical agreements.

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Figure 8. (a) Probability distribution of normalized displacement (~ xτ = (xτ - xp)/σxτ) of a water drop moving on an energy gradient surface at the following time intervals: 0.02 s (0), 0.05 s ()), 0.08 s (4), 0.12 s (O), and 0.2 s (  ). The drop is subjected to Gaussian noise with a power of 0.022 m2/s3. The blue and pink colors indicate experimental data and simulation results, respectively.40 (b) In the absence of hysteresis (Δ = 0) from the simulations, the distribution is purely Gaussian.

Figure 9. (Left) Probability distribution of normalized displacement (~ xτ=(xτ-xp)/σxτ) of the prism at the following time intervals: 0.05 s (0), 0.09 s (]), 0.13 s (4), and 0.16 s (O) with a Gaussian noise (power 0.09 m2/s3), but in the presence of a Coulombic friction. The blue and pink colors indicate experimental data and simulation results, respectively. (Right) Normalized displacement distribution with a Cauchy noise (power 0.09 m2/s3) for time intervals of 0.07 s (0), 0.09 s (]), 0.16 s (4), and 0.20 s (O). The blue and pink colors indicate experimental data and simulation results, respectively.40 In order to generate such fluctuation plots, data obtained from about 132 steady state tracks, each lasting for about 2.5 s, were combined.

the solid/solid case, as shown in Figure 6. We should also point out that this diffusivity is nicely reproduced using the Brownian dynamics simulations with wetting hysteresis replacing the dry friction.

Displacement Fluctuation of a Solid Prism on a Solid Support with Dry Friction As is the case with the displacement fluctuation for a water drop on a surface, the pdf of the displacement of the solid prism is also mainly exponential except being Gaussian toward the central region. This is indeed the case with both the truncated Cauchy and Gaussian noises, as seen with the normalized plots of displacements as summarized in Figure 9. Simulations show that the distribution is mainly exponential for Δ ¼ 6 0, but it becomes Gaussian when Δ=0 (Figure 10). Thus the phenomenon resulting from Coulombic friction (Δ 6¼ 0) is clearly encoded in the form of an exponential distribution in the experimental displacement distribution function, as is the case with the hysteresis of the water drop. Since the displacement distribution is found to have a finite variance that is linear with time, we expect that the displacement pdf to be Gaussian for the case with pure kinematic friction: 2

Pðxτ Þ ¼ Po e -ðxτ -xm Þ =ð4DτÞ Langmuir 2009, 25(17), 9969–9979

ð8Þ

Figure 10. Log-linear plot of the probability distribution of displacement (xτ) obtained using numerical simulation of modified Langevin equation (eq 1). The simulation is carried out at a power (K ) of 0.09 m2/s3 with parameters Δ=3.84 m/s2 and τL=0.067 s, which are obtained by fitting drift velocities data to eq 4. The distribution is clearly exponential. The inset shows the simulated probability distribution of displacement obtained by setting Δ=0. The displacement distribution fits well with Gaussian distribution, as shown by the blue thick solid line. The displacement distribution shown here for τ is 0.09 s.

where all the terms have their usual meanings. A work fluctuation such as equation41-45 is easily anticipated from eq 8 if we define the work as Wτ=γxτ. However, this could be a bit misleading in our case since work is being performed by both noise and gravity. The above Gaussian distribution is valid for Δ = 0 when the relaxation time is the characteristic Langevin relaxation time and the ratio of the diffusivity to mobility ( μ=Vdrift/Mγh) is D=μ ¼ MKτL =2

ð9Þ

Experimental (Figure 11) ratio of D/μ is found to increase slightly sublinearly with K as K0.8 with relaxation time scales (0.0002-0.0004 s) that are much smaller than the Langevin relaxation time (0.067 s). We expect that distribution function as shown in eq 8 needs to be multiplied by an appropriate exponential function in order to obtain a result as close as possible to the real (experimental) distributions. However, a proper theoretical treatment must also (41) Bochkov, G. N.; Kuzovlev, Y. E. Physica 1981, 106, 443. (42) Bochkov, G. N.; Kuzovlev, Y. E. Physica 1981, 106, 480. (43) Mazonka, O.; Jarzinsky, C. arXiv:cond-mat/9912121 (44) Imparato, A.; Peliti, L.; Pesce, G.; Rusciano, G.; Sasso, A. Phys. Rev. E 2007, 76, 050101. (45) van Zon, R.; Cohen, E. G. D. Phys. Rev. E 2003, 67, 046102.

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Figure 12. Kurtosis ( β) of displacement distribution as a function Figure 11. D/μ obtained for a prism subjected to truncated Cau-

of time for the prism subjected to a truncated Cauchy noise with a power of 0.09 m2/s3.

chy noise as a function of power of noise (K ).

take into account the asymmetric distribution that is observed experimentally. Some of this asymmetry definitely arises from the biased step length toward the applied force, although a significant source of the asymmetry may also come from the fact that the noise is not a true white noise, but colored with appreciable correlation that couples nonlinearly with the dry friction. We hope to explore this subject as a part of our ongoing studies. We observed that as time increases, the exponential displacement distribution, which is the signature of a dry friction, eventually evolves into a Gaussian distribution, prominently in the central region. The degree of “peakedness” of a distribution is generally measured by Kurtosis (β), which is defined by the ratio of the fourth central moment of a distribution and fourth power of standard deviation. For highly peaked distribution (e.g., exponential distribution), β is greater than 3 (Leptokurtic distribution) and is equal to 3 for Gaussian distribution. Estimation of β for different time intervals reveals that with time it approaches 3 starting from a high value (Figure 12). In the present situation, although the distribution is more like Gaussian at large time scale, the underlying effect of the dry friction is still there, which is evident from the asymmetric exponential nature of the tail region of the distribution.

Displacement Fluctuations in the Light of Conventional Fluctuation Theorems On the basis of the above observation, we consider that the experimental distribution function is represented approximately with a Gaussian function, but the values of the diffusivities and mobilities resulting from such do not satisfy D/μ=MKτL/2, as has been seen in the experiments (Figure 11). We calculate the probabilities of observing positive and negative displacement fluctuations from eq 8 and rearrange the results as follows: ! Pð þxτ Þ γxτ ¼ exp ð10Þ Pð -xτ Þ ðD=μÞ For the dry friction, as the value of D/μ is much smaller than that expected of a kinematic friction, the probability of a negative displacement fluctuation is more strongly suppressed than the case with the pure kinematic friction. We study the displacement fluctuation in terms of a scaled variable xτ = xτ/xp. As defined above, xp is the displacement value at the peak of the distribution function, whereas the mean displacement is xm. Equation 10 can be rearranged as ! Dτ Pð þxτ Þ ln ¼ xτ ð11Þ xm xp Pð -xτ Þ 9976 DOI: 10.1021/la901111u

Figure 13 shows that the fluctuation pdf of the scaled displacement obtained for the solid prism becomes steeper either with the increase of time or with the increase of the power, as expected from eq 11. All the probability data can be normalized by multiplying the function ln[P(+xτ)/P(-xτ)] by Dτ/xmxp (Figure 14). We plot this normalized function (π(xτ)), i.e., the left side of eq 11, as a function of xτ, taking the data from the measurements done either at various time segments for a constant power or at different powers corresponding to a fixed time segment. In both cases, we obtain results that superficially agree with the conventional fluctuation theorems of different varieties.41-45

Discussion The main finding of this work is that the external noise induced diffusivity of one solid on another is significantly dominated by Coulombic friction. This diffusivity is at least couple of orders of magnitude lower than that controlled by kinematic friction. A Langevin equation with an additional Coulombic dry friction term enables us to analyze the situation of both the diffusivity and the drift velocity rather satisfactorily. The drift velocity is predicted to increase sublinearly with the power of the noise, which is in agreement with the experimental observations. The experimental diffusivity values are also close to the simulated values with dry friction. While the experimental diffusivity increases with the power of the noise as K1.61, the simulations predict K1.74. Although these exponents do not represent a large discrepancy, there are differences in the numerical values of the simulated and measured diffusivities. We should be aware of some potential pitfalls related to the simulations and experiments. 1. The simulations are coarse-grained, in which the data collection and the video recordings were done at an interval of 1 ms. Some of the finer details of the stochastic dynamics may be lost in these simulations. 2. Equation 1 may be an oversimplification to account for the Coulombic friction because heterogeneity and metastable states may be involved in the real situation. 3. Heslot et al.46 studied the effect of Coulombic friction on the dynamics of an object in which they have taken into account the state and rate dependent Coulombic friction laws. Our current friction law, on the other hand, is extremely simple. Coulombic friction signifies an elastic energy dissipation during the contact and breaking of junctions. When the slider receives an external impulse of a short duration, it should accelerate or decelerate with concomitant energy dissipation of the elastic energy of the asperities.46 At high speeds, nonlinear velocity dependent Coulombic (46) Heslot, F.; Baumberger, T.; Perrin, B.; Caroli, B.; Caroli, C. Phys. Rev. E 1994, 49, 4973.

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Figure 13. (Left) Some representative probability distributions of the normalized displacement (xτ = xτ/xp) of a solid prism subjected to Cauchy noise (power 0.09 m2/s3), at different time intervals. (Right) Some representative probability distributions of the normalized displacement of a prism at different powers of Cauchy noise for a particular time interval (τ = 0.09 s). In order to obtain these distributions, data obtained from about 50 to 130 tracks were combined in order to generate good statistics. The total number of tracks is based upon the duration of each track so that a total of about 300 s of data are obtained. The solid lines represent asymmetric double sigmoidal functions that are used to fit the experimental data empirically.

Figure 14. (Left) Fluctuation plot of normalized displacement (xτ = xτ/xp) of a solid prism subjected to Cauchy noise (power 0.09 m2/s3) at different time intervals. (Right) Fluctuation plot of normalized displacement of a prism at different powers of Cauchy noise for a particular time interval (τ = 0.09 s). Here, π(xτ) is defined as (Dτ/xmxp) ln[P(+xτ)/P(-xτ)].

friction at the interface leads to higher energy dissipation. These details are not taken into account in our studies as we employ only a constant value of the Coulombic friction, Δ. Irrespective of the above-mentioned nonidealities, the theoretical predictions of eq 1 are reasonably good, at least to a first order of approximation. The general shapes of the displacement distributions, and its asymmetry in some cases, are reproduced rather well. Since we are able to measure both the diffusivity and the mobility independently, we could also test how the ratio of these quantities varies with the power of the noise. We find that this ratio D/μ varies sublinearly with the power of the noise K, with the relaxation time (0.0002-0.0004 s) being much smaller than the Langevin relaxation time of the system. One may advocate that we should seek a relationship between _ instead of D/μ versus K, D/μ and the rate of energy dissipation (q), because it is the quantity q_ that directly measures the power experienced by the slider. In order to make a rough estimate of the power dissipation, we examine the integrated form of the Langevin equation: Z t þτ 1 2 ½V ðt þ τÞ -V 2 ðtÞ ¼ g sin θ VðtÞdt 2 t KineticEnergy

Z

t þτ

þ

GravitationalWork or Potential Energy

Z

t þτ

γðtÞVðtÞdt -

t

t Vibrational Work

! V 2 ðtÞ þ ΔjVðtÞj dt τ

ð12Þ

Frictional Dissipation

The term on the left side of eq 12 is the change of the kinetic energy in going from one state to another, the average value of Langmuir 2009, 25(17), 9969–9979

which is zero. The first term on the right side of the equation is the work done by the external force, the second term is the work done by the external noise, whereas the third term in the above equation is the net energy dissipation. This term has two components: its first component is the heat generated (and dissipated) by the kinematic friction, whereas its second component is the energy dissipation due to Coulombic friction. In order to make an approximate estimate of the total energy dissipation, we consider their average values as ! V 2 ðtÞ þ ΔjVðtÞj ð13Þ q_ ≈ M τ

Æ

æ

It is easy to show that q_ is related to the power of noise as MK/2. This can be demonstrated by neglecting the kinematic friction term of eq 13 and calculating Æ|V(t)|æ using eq 4. One thus finds Æ|V (t)|æ = K/2Δ, from which we get q_ =M K/2. Equation 13 then becomes _ d ð14Þ D=μ ¼ qτ where τd is a characteristic time for energy dissipation. There is, however, a caveat in the above estimation of the rates of energy dissipation, as it relies upon the measurement of velocities with a camera that has a limited frame capture speed (1000 fps). Some of the stochastic processes occur at a faster rate than what our experimental setup can capture. Thus, this does not allow us to get an handle on the transients, or the fast acceleration/deceleration phases, which could be behind some of the interesting physics of fluctuations. In any event, the ratio D/μ is found to increase with q_ (Figure 15) roughly linearly. The positive DOI: 10.1021/la901111u

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Figure 15. D/μ varies approximately linearly with the rate of _ energy dissipation (q).

intercept of the regression line on the q_ axis suggests that a minimum power is needed before the prism starts sliding on the surface. At a power lower than the threshold, there is, of course, neither any energy dissipation nor any mobility of the slider. In the low power region, there may underlie additional interesting physics akin to jamming or glassy dynamics that may be worth pursuing in future. At this juncture, a few comments about the displacement fluctuations of the macroscopic object as studied by us are in order in the context of some previous works in the field. Characterization of systems that are driven away from equilibrium by an external force is traditionally done in the contexts of the fluctuations theorems,47-52 which deal with the entropy production in time reversible systems. These theorems consider a finite probability of the decrease of entropy in certain trajectories of a dynamic process and states that the ratio of the probabilities of positive and negative entropy production rates varies exponentially with the rate of positive entropy generation. Various attempts have been made to examine the validity of nonequilibrium fluctuation theorems experimentally and via numerical simulations. At the microscopic level, an elegant and popular experiment53-55 is to trap a colloid particle in an optical tweezer and move it with predetermined controls of parameters. What is observed is that the colloid particle moves along the direction of motion of the trap most of the time: these are the entropy-generating trajectories. Occasionally the particle exhibits reverse trajectories signifying entropy consumption. The probability distributions of these entropy generating and entropy consuming events have been found to exhibit a fluctuation-type relationship. Other experiments that successfully verified Gallavotti-Cohen (GC) theorem involve mechanical oscillators,56 measurements of the energy fluctuation57 due to flow of current in electric circuits, and stochastic motion of a pendulum56 immersed in a liquid resulting from the thermal noise. (47) Gallavotti, G.; Cohen, E. G. D. Phys. Rev. Lett. 1995, 74, 2694. (48) Gallavotti, G.; Cohen, E. G. D. J. Stat. Phys. 1995, 80, 931. (49) Evans, D. J.; Cohen, E. G. D.; Morriss, G. P. Phys. Rev. Lett. 1993, 71, 2401. (50) Gallavotti, G. J. Stat. Phys. 1996, 84, 899. (51) Evans, D. J.; Searles, D. J. Phys. Rev. E 1994, 50, 1645. (52) Kurchan, J. J. Phys. A. 1998, 31, 3719. (53) Wang, G. M.; Sevick, E. M.; Mittag, E.; Searles, D. J.; Evans, D. J. Phys. Rev. Lett. 2002, 89, 50601. (54) Carberry, D. M.; Reid, J. C.; Wang, G. M.; Sevick, E. M.; Searles, D. J.; Evans, D. J. Phys. Rev. Lett. 2004, 92, 140601. (55) Blickle, V.; Speck, T.; Helden, L.; Seifert, U.; Bechinger, C. Phys. Rev. Lett. 2006, 96, 070603. (56) Douarche, F.; Joubaud, S.; Garnier, N. B.; Petrosyan, A.; Ciliberto, S. Phys. Rev. Lett. 2006, 97, 140603. (57) Garnier, N.; Ciliberto, S. Phys. Rev. E. 2005, 71, 060101.

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There have also been significant numerical and experimental activities in verifying GC-like fluctuation theorem in systems excited by external noises. Aumatre et al.58,59 pioneered such studies, who examined numerically the power injection in a shell model of turbulence, granular gas, and a Burridge-Knopo springblock model and found that a GC-type relationship holds in all cases. More recently, Majumdar and Sood60 observed that the nonequilibrium fluctuation relation holds for sheared micellar gel in a jammed state. Feitosa and Menon61 found experimentally that the power flux fluctuation in a granular gas is generally in accord with the GC theorem as well. The energetic scaling needed to fit the power fluctuation data with the GC theorem prompted these authors to define an effective temperature of the system. Several authors,59-67 however, cautioned about the applicability of the conventional fluctuation relations to macroscopic systems in a strict thermodynamic sense, as these systems are non-Anosov and do not exhibit microreversibility. However, a fluctuation-like relation may still work59,62,63 in certain cases in the probabilistic sense of a “large deviation theorem”. So, where does our report stand in the context of the above studies? As with the granular gases, there is no microscopic reversibility in our case either. Second, the energy is delivered to the slider in terms of work and not in terms of a thermal energy. Hence, we cannot strictly analyze any of our results with the conventional entropy fluctuations theorems in any straightforward way. We feel that even converting the displacement to work is problematic, as work is being done by both the noise and gravity. Even if the displacement is negative (i.e., slider moves upward), it is not convincing that a negative work is performed in the process, it is just that powerful fluctuations push the object upward once in a while. This point can be extended further by taking into consideration an experiment published recently.67 A hydrogel rod was placed perpendicularly to an asymmetric cut of a support, and the latter was vibrated with a periodic vibration. The asymmetry in the friction rectified the vibration-induced force and led to the motion of the hydrogel. More recently, Buguin et al.68 performed an experiment (similar to that reported in ref 69), in which a coin moved on a substrate vibrated with an asymmetric waveform. In both the above experiments, where work is being done by the external vibration, velocity increased linearly with the amplitude of vibration. However, in both the experiments (Figures 4 and 2 in references 67 and 68 respectively), a threshold amplitude is observed, below which no motion takes place. This threshold force is indicative of the presence of a Coulombic friction at the interface. Now, it should also be possible to induce the motion of the hydrogel on the surface having asymmetric friction with white noise vibration with a zero mean. As there is no bias, the Ronly work done on the hydrogel rod would be of the form M t+τ t γ(t)V(t)dt, where V(t) is the (58) Auma^itre, S.; Fauve, S.; McNamara, S.; Poggi, P. Eur. Phys. J. B 2001, 19, 449. (59) Aumaıtre, S.; Farago, J.; Fauve, S.; Mc Namara, S. Eur. Phys. J. B 2004, 42, 255. (60) Majumdar, S.; Sood, A. K. Phys. Rev. Lett. 2008, 101, 078301. (61) Feitosa, K.; Menon, N. Phys. Rev. Lett. 2004, 92, 164301. (62) Farago, J. J. Stat. Phys. 2002, 107, 781. (63) Farago, J Physica A 2004, 331, 69. (64) Visco, P.; Puglisi, A.; Barrat, A.; Trizac, E.; van Wijland, F. Europhys. Lett. 2005, 72, 55. (65) Puglisi, A.; Visco, P.; Barrat, A.; Trizac, E.; van Wijland Phys. Rev. Lett. 2005, 95, 110202. (66) Visco, P.; Puglisi, A.; Barrat, A.; Trizac, E.; van Wijland, F. J. Stat. Phys. 2006, 125, 533. (67) Mahadevan, L.; Daniel, S.; Chaudhury, M. K. Proc. Natl. Acad. Sci. U.S.A. 2004, 101, 23. (68) Buguin, A.; Brochard, F.; de Gennes, P.-G. Eur. Phys. J. E 2006, 19, 31. (69) Daniel, S.; Chaudhury, M. K.; de Gennes, P.-G. Langmuir 2005, 21, 4240.

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fluctuating velocity. If the acceleration pulse and the velocity response are highly correlated, then the above work would be positive, as found by Farago,62 even though the velocity and displacement fluctuate between the negative and positive values. In any event, in the context of the current experiment, the work R γ(t)V(t)dt has to be added to that done done by vibration, M t+τ tR by the gravity, Mg sin θ t+τ t V(t)dt, in order obtain the total work done on the slider. This total work may turn out to be mainly a positive quantity with very few negative fluctuations. Any distribution that is Gaussian would comply with the fluctuation relation of the type shown in eq 10. For example, Seitaridou et al.70 recently found that the diffusion flux of colloids in small systems exhibit a Gaussian fluctuation, which is consistent with a conventional fluctuation theorem. Whether or not such a compliance with the conventional fluctuation theorems has a deeper physical significance for systems driven out of equilibrium with external noise needs further studies. At present, we hesitate to attach specific thermodynamic significance to the measured displacement fluctuations other than recognizing that it is a kind of measure of the gravitational potential energy fluctuation. However, all the terms pertaining to works performed by the noise as well as the gravitational field and the energy dissipations by all the frictional terms need to be sorted out more clearly, experimentally as well as theoretically. This is something we plan to accomplish as a part of our continuing studies by performing highly synchronized measurements of acceleration pulses and the displacement of the slider as well as its velocity.

Main Observations and Conclusions 1. A solid block or a water drop is pinned on a surface as a result of Coulombic friction or hysteresis. These objects would not usually move until the external force is larger than the Coulombic friction or hysteresis. The objects would, however, move on the substrate with a relatively small bias if they are vibrated with white noise. The drift velocity can be accounted for with a Langevin equation with Coulombic friction within the framework of the Klein-Kramers equation. (70) Seitaridou, E.; Inamdar, M. M.; Phillips, R.; Ghosh, K; Dill, K. J. Phys. Chem. B 2007, 111, 2288.

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2. In addition to studying the drift velocity, we can also study the fluctuations of the displacements related to these systems. It has been found that the probability distributions of displacements in all cases are non-Gaussian, which are adequately supported by the numerical simulation of the Langevin equation including Coulombic friction or hysteresis. The simulations also confirm that the exponential distribution of the displacement arises as a result of the presence of a threshold force (Coulombic friction or hysteresis) that needs to be overcome to initiate motion. 3. The ratio of the diffusivity to mobility varies sublinearly with the power of noise. However, the characteristic time scale observed from this analysis is much smaller than the Langevin relaxation time of the system. 4. A final and important comment is about the asymmetry of the displacement distributions of either the block or the drop in the presence of Coulombic friction or hysteresis. When the object is vibrated with white noise with Δ=0, the probability distribution is symmetrical. However, the distribution becomes asymmetric when Δ > 0, and this asymmetry increases with γh as well as with the power of the noise. Thus, the asymmetric distribution could be another signature (in addition to the predominance of the exponential distributions) of a nonequilibrium system where a threshold force such as Coulombic friction (for solids) or hysteresis (for a drop) is operative at the interface. Note Added in Proof. A recent manuscript71that is relevant to our studies provides numerical results of the work fluctuation of a Brownian particle dragged by a moving harmonic potential but subjected to a truncated Levy noise. It is evident from these calculations that for a low power Levy noise, the behavior of the particle is far from that expected of a true Levy noise of infinite power. Acknowledgment. We thank Professors H. Touchette and E. G. D. Cohen for various useful discussions. However, we are solely responsible for any mistakes in the paper. (71) Touchette, H.; Cohen, E. G. D. arXiv:0903.3869v1, 2009.

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