Diffusive motion with nonlinear friction: apparently BrownianPartho S. Goohpattader and Manoj K. Chaudhury Citation: J. Chem. Phys. 133, 024702 (2010); doi: 10.1063/1.3460530 View online: http://dx.doi.org/10.1063/1.3460530 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v133/i2 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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Diffusive motion with nonlinear friction: apparently BrownianPartho S. Goohpattader and Manoj K. Chaudhurya�
Department of Chemical Engineering, Lehigh University, Bethlehem, Pennsylvania 18015, USA
�Received 22 March 2010; accepted 13 June 2010; published online 8 July 2010�
We study the diffusive motion of a small object placed on a solid support using an inertialtribometer. With an external bias and a Gaussian noise, the object slides accompanied with afluctuation of displacement that exhibits unique characteristics at different powers of the noise.While it exhibits a fluidlike motion at high powers, a stick-slip motion occurs at a low power. Belowa critical power, no motion is observed. The signature of a nonlinear friction is evident in this typeof stochastic motion both in the reduced mobility in comparison to that governed by a linearkinematic �Stokes–Einstein-like� friction and in the non-Gaussian probability distribution of thedisplacement fluctuation. As the power of the noise increases, the effect of the nonlinearity appearsto play a lesser role, so that the displacement fluctuation becomes more Gaussian. When thedistribution is exponential, it also exhibits an asymmetry with its skewness increasing with theapplied bias. A new finding of this study is that the stochastic velocities of the object are so poorlycorrelated that its diffusivity is much lower than either the linear or the nonlinear friction casesstudied by de Gennes �J. Stat. Phys. 119, 953 �2005��. The mobilities at different powers of thenoise together with the estimated variances of velocity fluctuations follow an Einstein-likerelation. © 2010 American Institute of Physics. �doi:10.1063/1.3460530�
I. INTRODUCTION
Diffusive motion induced by thermal noise has beenstudied in various chemical and physical systems. While alinear kinematic friction is the main damping force in thesetypes of motions, there are indications in such cases as DNAelectrophoresis1 in a gel and the diffusion of a colloidal par-ticle in contact with a soft microtubule2 that a nonlineardamping force may be operating. In the first case, the mobil-ity of the DNA is significantly reduced. In the latter case, therole of nonlinearity is apparent in the non-Gaussian distribu-tion of displacements having exponential tails. Non-Gaussiandisplacements have also been reported for the case of palla-dium adatoms diffusing on tungsten.3,4
While the novelty of the above examples is that they arefound in thermal systems, the signature of nonlinearity andthus the non-Gaussian distributions of displacement or en-ergy fluctuations are also evident in various athermal systemsthat include granular flow,5–10 hydrodynamic turbulence,11–15
evolution of climate,16 dusty plasma,17 and the driven motionof a liquid drop or a solid object on a surface.18–23 Whilesome of the results can be explained on the basis of a jointprobability distribution function �PDF� of the forcing andresponse functions15,24 as in the power input distribution, orwithin the framework of superstatistics25,26 as in the velocitydistribution in turbulence, there are also perceived physicalmechanisms behind some of these non-Gaussian PDFs. Ex-amples of latter cases include the inelastic collision27 and theCoulombic slip28 between particles in granular gases.
Based on the previous works of de Gennes29 as well asthat of Kawarada and Hayakawa,28 it has been shown re-cently that a non-Gaussian PDF ensues naturally when the
resistance to motion of an object is nonlinear. The nonlinear-ity may arise from a Coulombic dry friction23 for the solid-solid case, from wetting hysteresis18,19,23 for a liquid-solidcase, or �possibly� from an adhesion hysteresis related to therolling motion of a particle on a soft substrate.2 These non-linear resistances have one common unique feature that nomotion may occur when the noise pulse is smaller than thethreshold resistance, while motion occurs when a large pulserescues the object from the stuck state at a later stage.Mauger30 specifically argued that it is the non-Lipschitz con-tinuity of a resistive term in the Langevin equation that givesrise to an exponential distribution of velocity of a particle.
While there exist a plethora of examples indicating theimportance of nonlinear damping in surface diffusional dy-namics of interest to chemical physicists, systematic study islacking in which the property of the surface is modifiedchemically as well as physically to control friction, and itseffect is systematically studied to see how it modifies a sto-chastic dynamics. By extending some of our previousstudies,23 here we investigate how the stochastic behavior ofa small solid object on a solid support is influenced by anonlinear friction when it is subjected to a Gaussian whitenoise and an external bias. The long term objective of thisresearch is to implement such surface modification technolo-gies as self-assembled monolayers and chemiadsorbed poly-mers in order to control the specific and nonspecific interac-tions at surfaces and study their effects on friction anddiffusive dynamics. The current study attempts to establishthe methodology as well as the phenomenology underlyingthis approach primarily with the contact of two solid sur-faces. In one case, a smooth glass prism slides against aroughened glass support. In another case, a thin ��3.7 nm�polydimethyl siloxane �PDMS� grafted smooth glass prisma�Electronic mail: [email protected].
THE JOURNAL OF CHEMICAL PHYSICS 133, 024702 �2010�
0021-9606/2010/133�2�/024702/13/$30.00 © 2010 American Institute of Physics133, 024702-1
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slides against a PDMS grafted silicon wafer. In the first case,the nonlinearity comes from dry friction, whereas in the lat-ter case it comes from a velocity weakening kinematic fric-tion.
In order to focus our discussion, let us consider a modi-fied version of the Langevin equation as discussed by deGennes29 as well as by Kawarada and Hayakawa.28
dV
dt+
V
�L+ ��V�� = �̄ + ��t� . �1�
Here, V is the velocity, �̄ is the external force per unit mass,�L is the Langevin relaxation time �mass �m�/kinematic fric-tion coefficient ����, � is a nonlinear resistive force dividedby the mass of the object, and ��t� is the time dependentacceleration that the object experiences from external vibra-tion. It should be mentioned here that Daniel et al.20 pro-posed a coupled set of equations similar to Eq. �1� in order toformulate the motion of a liquid drop on a surface by vibra-tion, in which wetting hysteresis provides the nonlinear re-sistance. When the kinetic friction itself is nonlinear,17 theLangevin equation assumes the following form:
dV
dt+
A�V�n
m��V� = �̄ + ��t� . �2�
Here A�V�n is the nonlinear friction force, with the expo-nent n being less than unity. In general, ��t� can be periodic�symmetric or asymmetric� or stochastic. Let us consider thestochastic case here, for which the power �also called noisestrength� associated with the noise is K. In Eq. �1�, if themagnitude of � is smaller than ��̄+��t��, the object moves;otherwise, it remains stuck to the surface. For this reason, itis suitable to multiply � with a signum function ��V� whichis positive when V�0 and negative when V�0 with ��0�=0. In Eq. �2�, we do not have to consider a specific value offriction to make a demarcation between GO and STOP mo-tion of the object. In the stochastic setting, the nonlinearfriction makes the dynamics of an object governed by Eq. �1�or Eq. �2� quite different from that of a conventional Brown-ian particle. For the dry friction case, according to deGennes,29 the object exhibits a diffusive motion even in theabsence of the kinematic friction, where the variance of thedisplacement increases linearly with time, but with a diffu-sivity ��K3 /�4� that depends more strongly on the power ofthe noise than that ��K� of a normal Brownian particle. Fur-thermore, the object drifts with a velocity �K�̄ /�2 that isuniquely different from that of a free Brownian particle,where the drift velocity is simply a product of the bias �̄ andthe Langevin relaxation time �L. We will show later that thenonlinear kinematic friction, as shown in Eq. �2�, also leadsto some unusual behavior that are similar to the case of dryfriction.
Equations �1� and �2�, or their modified forms, are alsouseful in understanding how an asymmetric resistance leadsto a unidirectional drift of an object on a surface vibratedwith a symmetric periodic noise20,21 or how a symmetricfriction can induce such a motion by rectifying a periodicasymmetric waveform in the absence of any external bias.22
It also sheds light on certain types of exponential velocity
distributions observed in granular gases,28 which indeed mo-tivated Kawarada and Hayakawa to examine the conse-quences of Eq. �1� in the first place. We draw certain paral-lels of the behaviors observed in this athermal system withthose of a thermal system,31 although there are fundamentaldifferences between the two. In a thermal system, the noiseand the friction are coupled to each other, unlike the case ofan external noise. However, the provision of delivering thenoise externally in our mechanical system allows us to de-couple the origin of the noise �external� and the resistance tomotion �i.e., dry and/or kinematic friction� at the solid-solidinterface. Against this backdrop, we seek for a rudimentarymanifestation of a fluctuation-dissipation relationship �FDR�in our system, i.e., the Einstein relationship, the basic tenetof which is that the frictional constant obtained from theratio of the available vibration energy to the resulting diffu-sivity is same as that obtained from the ratio of the appliedforce to the resulting drift velocity. There are several pio-neering studies published in related fields, which will be dis-cussed in appropriate junctures later in the paper.
The instrument employed for this study is an inertialtribometer, which was first used to investigate the nature ofdry friction by Baumberger et al.32 They placed a small solidobject on a solid support, which was subjected to a biasedoscillatory force of varying amplitude. At high amplitude ofvibration, the slider ran down the supporting track. At inter-mediate amplitude, it exhibited a creeplike response. Whileat even a smaller amplitude, when no relative sliding wasdetected, the object exhibited small levels of in-phase andout-of-phase responses to the oscillatory force pointing outthe elastic and dissipative natures of the “so called” dry fric-tion. The idea of inertial tribometer was also used bySanchez et al.33 to study the spreading dynamics of a cylin-drical granular drop on a surface, excited by a periodic vi-bration. Our approach to study the motion of the solid objecton a solid support is similar to those of the above authors,except that the excitation is done with a white noise ratherthan a periodic vibration.
We study two model systems. In one case, a smoothglass prism slides against a rough glass support. In the sec-ond case, a polymer �PDMS� grafted smooth glass prismslides against a PDMS grafted silicon wafer. In the first sys-tem, which has been studied more extensively, we are able toidentify three distinct interfacial regimes: a solidlike, a flu-idlike, and a transition region characterized by a stick-slipmotion of the object. We observe that the distribution of thestochastic displacement is non-Gaussian in certain range, butis Gaussian at some other range of the power, thus indicatingthat another level of continuous transition occurs at the in-terface with increasing power of the noise even in the seem-ingly fluidized state. Although Eq. �1� describes the behaviorof this system in a general way, a potentially important newfinding about this kind of motion is that the stochastic ve-locities are poorly correlated thus leading to a much lowerdiffusivity than that predicted by de Gennes29 for a similarsystem. The second system is studied here for the main pur-pose of showing that the non-Gaussian displacement fluctua-tion can also arise when the kinematic friction itself is non-linear.
024702-2 P. S. Goohpattader and M. K. Chaudhury J. Chem. Phys. 133, 024702 �2010�
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II. THEORETICAL BACKGROUND
In the absence of any vibration, the prism on a tiltedsupport experiences two types of external forces: one is thedriving force �mg sin � for motion and the other arises fromthe static friction force �smg cos � acting parallel to thesurface opposite to the direction of motion. Here, m is themass of the prism, s is the static friction coefficient, g is thegravitational acceleration, and is the angle of inclination ofthe support. The glass prism slides on the inclined surfacewhen the gravitational force is larger than the static frictionforce; otherwise it remains stuck. The prism is rescued fromthis stuck state when it is subjected to a stochastic vibrationforce m��t�. We use Eq. �1� in which �̄ is to be identifiedwith g sin and �=sg cos . By taking into account theeffects of various forces in the system, a steady state prob-ability �P= P�V�� distribution function in velocity space canbe obtained from the integrations of Eqs. �3� and �4�. Theseequations can also be obtained from the spatially homoge-neous steady state solution of a Klein–Kramers34,35
equation.18
K
2
�P
�V+ �
�V�V
P +VP
�L= �̄P �dry friction� , �3�
K
2
�P
�V+
A�V�n
m
�V�V
P = �̄P �nonlinear kinematic friction� .
�4�
The solutions of the Eqs. �3� and �4� yield the PDF of veloc-ity fluctuation �P�V�� as follows:
P�V� = Po� exp�−V2
K�L−
2�V��K
+2V�̄
K , �5�
P�V� = Po� exp�−2A�V�1+n
m�1 + n�K+
2V�̄
K . �6�
Here, Po� and Po� are normalization constants. From Eq. �5�, itis evident that the velocity distribution has a Gaussian com-ponent due to kinematic friction and an exponential compo-nent due to Coulombic dry friction, whereas from Eq. �6�, weexpect a stretched Gaussian distribution. Equations �5� and�6� are useful in the sense that both the drift velocity and thevariance of velocity distribution can be estimated by calcu-lating the first and second moments of velocity distributionprovided that the values of �L and � �for Eq. �5��, as well asn and A �for Eq. �6�� are at our disposal. Conversely, theexperimental drift velocities obtained at different values of �̄and the power of the noise K can be used to estimate theseparameters. We used the second method to estimate �L and �as well as n and A, which were then used for further analysisand simulations.
III. EXPERIMENTAL SECTION
A solid glass prism ��1.67 g�, having dimension of�11�11�6 mm3, was used to slide against a glass plateusing the apparatus shown in Fig. 1. As with our previousstudies, some roughening of the support was necessary toinduce easy and uniform sliding of the glass prism over it.
Very smooth surfaces adhere to each other so strongly that avery high level of vibration is needed to dislodge it. Weavoided such high adhesion situations by roughening the sur-face, as our objective is to study the stochastic dynamics ofthe motion from a very low to a high power. While the mainwork of this paper focuses on the above described system,we also present some results of a study where a PDMSgrafted smooth glass prism slides against a PDMS graftedpolished silicon wafer �Fig. 2�. In the latter case, as thePDMS reduces the surface energy of the smooth surfacesconsiderably, the surfaces do not stick to each other strongly.Thus, diffusive experiments could be performed withoutroughening the surfaces.
A. Preparations of glass surfaces
A glass slide �Fisherbrand� ��9 g� having dimension of75�50�1 mm3 was grit blasted with alumina particles ��45 m� at a pressure of 90 psi for 45 s in an air fluidizedbed. The grit blasted glass surface was blown with a jet ofdry nitrogen gas followed by washing with copious amountsof de-ionized �Millipore� water. The roughened glass plateand a glass prism were sonicated first in de-ionized water andthen in acetone for 30 min each. They were sonicated againin de-ionized water for 30 min. After rinsing the plate andthe prism with de-ionized water, they were dried with nitro-gen gas. Both the glass surfaces were completely wettable byde-ionized water in the contact angle measurements, whichensures that they are free of gross organic contaminations.No debris was also evident in optical microscopic examina-tions. The roughened glass surface was examined using alaser optical profilometer �STIL Micromeasure, CHR 150-N�at different spots on the surface, each having a scanning areaof 500�500 m2, with a scanning step size of 2.5 m. Theroot mean square value of the surface height fluctuation was
Function
Generator
Amplifier
Oscilloscope
Oscillator
PrismSupport
Computer
θ
Function
Generator
Amplifier
Oscilloscope
Oscillator
PrismSupport
Computer
θθ
FIG. 1. Schematic of the experimental setup.
Glass
Rough glass plate
(a)
Glass
3.7 nm
Si wafer
Grafted
PDMS(b)
Glass
Rough glass plate
(a)Glass
Rough glass plate
GlassGlassGlass
Rough glass plate
(a)
Glass
3.7 nm
Si wafer
Grafted
PDMS(b)
Glass
3.7 nm
Si wafer
Grafted
PDMSGlassGlass
3.7 nm
Si wafer
Grafted
PDMS(b)
FIG. 2. Two test systems are shown: �a� a smooth glass prism on a roughglass support and �b� a PDMS grafted smooth glass prism on a PDMSgrafted silicon wafer.
024702-3 Diffusion with nonlinear friction J. Chem. Phys. 133, 024702 �2010�
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about �16 m, which varied slightly �within 1 m� fromspot to spot. The rectangular glass prism was prepared bycutting a borosilicate glass plate �ACE Glass, USA� using afine glass grinder. The root mean square roughness of theglass prism was about 4 nm over an area of 100 m2 mea-sured using an atomic force microscope �AFM� �DigitalInstruments, USA�. Additional details are given in thesupplementary material �SM-A�.36
B. Grafting silicon wafer and glass prism with PDMS
We used a method similar to that published in Ref. 37with a slight modification in order to graft PDMS chains onthe glass prism and the silicon wafer. Pieces �75�30�0.6 mm3� of silicon wafer �Silicon Quest International�and a glass prism of weight 1.7 g and dimensions of 11�11�6 mm3 were first cleaned in piranha solution �a mix-ture of concentrated sulfuric acid and 30% hydrogen perox-ide in 4:1 volume ratio� for 30 min. After rinsing the sampleswith copious amounts of de-ionized water �Millipore� anddrying with nitrogen gas, they were further cleaned withoxygen plasma. The roughness of the silicon and the glassprism were 0.4 and 4 nm, respectively, as evidenced from theAFM measurements. The samples were immersed intrimethylsiloxy-terminated PDMS �Gelest Inc., product code:DMS-T22, MW �9430� in a cleaned glass Petri dish. ThePetri dish was covered and kept in an oven at 100 °C for 24h. The samples were then cooled to room temperature anddipped in 99.9% pure toluene �ACS grade� for 10 min. Boththe samples were rinsed with copious amounts of flowingtoluene, after which they were dried with nitrogen gas. Usingspectroscopic ellipsometry �J. A. Woollan Co., Inc. VB-400Vase Ellipsometer� the thickness of the PDMS grafted ontosilicon wafer was estimated to be �3.7 nm. Because of thepoor contrast of the reflectivity of the glass prism and thePDMS, it was not possible to make reliable estimate of thethickness of the grafted PDMS layer on this surface. How-ever, as the methodologies used to graft PDMS were identi-cal in both cases, the thickness of PDMS on the glass surfaceshould be similar to that on the silicon wafer. The advancingand the receding contact angles of water on both the surfaceswere �110° and 103°, respectively, suggesting that their sur-face energetic properties were the same.
In a previous paper,38 we reported the thickness of thegrafted PDMS chains on surfaces where one end of polymerreacted with the surface. With the data presented in that re-port, the thickness of the grafted layer of PDMS on siliconwas about 8.7 nm for a PDMS of molecular weight compa-rable to that studied here. Here, both ends of the chain canreact with the surface; consequently, the thickness of thegrafted layer is close to half of that found previously.
The roughened glass plate or the PDMS grafted siliconwafer were firmly attached �Fig. 1� to an aluminum platform�40 g� that was mounted on a mechanical oscillator �PascoScientific, Model No: SF-9324�. Gaussian white noise wasgenerated with a waveform generator �Agilent, model33120A� and fed to the oscillator via a power amplifier�Sherwood, Model No: RX-4105�. By controlling the ampli-fication of the power amplifier, noises of different powers
were generated while keeping the pulse width constant at�40 s. The acceleration of the supporting aluminum platewas estimated with a calibrated accelerometer �PCBPeizotronics, Model No: 353B17� driven by a signal condi-tioner �PCB Peizotronics, Model No: 482� and connected toan oscilloscope �Tektronix, Model TDS 3012B�. The PDFsof these accelerations are Gaussian �Fig. 3� and their powerspectra are flat up to a total bandwidth of 10 kHz �seeSM-C�.36 The entire setup was placed on a vibration isolationtable in order to eliminate the effect of ground vibration. Thedrifted and the stochastic motion of the glass prism werecaptured with a high speed �1000 fps� Redlake Motion-Provideo camera, which was later analyzed using “MIDAS 2.0
XCITEX” software to obtain the position of the prism relativeto a fixed reference on the vibrating plate as a function oftime. All measurements were done under ambient conditions,at a temperature of 23 °C and relative humidity of 40%.
The sliding experiments with the prism on the roughenedglass were carried out at 11 different powers of the noiseranging from 0.0003 to 1.83 m2 /s3 and five different biasesby varying the angle of inclinations with a sensitive goniom-eter �CVI Melles Griot, Model No: 07 GON 006� from 1° to10° that correspond to forces ranging from 0.29 to 2.8 mN.For the case with PDMS grafted glass on a PDMS graftedsilicon wafer, the drift velocities were measured at eight dif-ferent powers. However, the detailed examination of the dis-placement fluctuations was carried out at one power �K=0.1 m2 /s3� and one bias �0.29 mN�.
We estimated the experimental error induced back-ground noise in order to ensure that our data are far above it.In order to accomplish this task, the prism was fastened tothe supporting plate with an adhesive tape, and then the platewas subjected to white noise vibrations of different powers.The position of the fixed prism with respect to a fixed refer-ence point was again analyzed using the software mentionedabove. This tracking allowed us to estimate the backgroundnoise that arose due to the errors of the measurement. Wewill show later �Fig. 13� that this background noise leads toa false diffusivity, which is nonetheless much smaller thanthe lowest diffusivity used in our analysis.
-800 -400 0 400 80010-6
10-5
10-4
10-3
10-2
10-1
γ ( m/s2 )
K=0.01 m2/s3
K=0.16 m2/s3
K=1.2 m2/s3
P(γ
)(m
/s2
)-1
FIG. 3. Probability distribution functions of Gaussian white noise obtainedfrom accelerometer at three different powers �K� as indicated inside thefigure. The solid lines represent Gaussian fit through the data.
024702-4 P. S. Goohpattader and M. K. Chaudhury J. Chem. Phys. 133, 024702 �2010�
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IV. RESULTS
A. Stochastic motion of the prism on a rough surface
The stochastic motions of the prism on the solid supportat two different powers of the noise are shown in Fig. 4,where it is evident that the prism exhibits a stick-sliplikemotion at a very low power, but a dispersive fluidlike motionat a high power. Two types of analyses have been done withthese data. First is the estimation of the drift velocity fromthe net displacement as a function of time, and second is theestimation of the diffusivity from the stochastic fluctuationsof the displacement.
B. Drift velocity and the mobility
On a roughened glass substrate, the displacement datawere taken over several tracks on different parts of the sur-face, each for certain duration of time. The prism showed anoccasional tendency to rotate as it drifted on the surface,especially at higher powers and biases. Those tracks that didnot exhibit any rotation were used for data analysis. At lowerpowers, although about ten to 12 tracks were sufficient forthe estimation of the drift velocity and the diffusivity, 25tracks were used for the data analysis. However, for thehigher powers, larger numbers of tracks ��50� were usedowing to shorter duration of time ��2 s� for each track.
A typical evolution of a displacement distribution func-tion is shown in Fig. 5 in linear scales. The general patternhere is much like the case of the propagation of a Gaussiandistribution as shown in Eq. �7�.
P�x�� = Poe−�x� − Vdrift��2/4D�. �7�
When plotted in the log-linear scales, as we will see later,these PDFs exhibit non-Gaussian �exponential or stretchedGaussian� tails in many situations, although the central partof the distribution is nearly Gaussian at longer time scales.Equation �7� suggests that the peak of the PDF moves with avelocity Vdrift and its variance broadens with �, both of whichapplies in our case. We estimate the drift velocity and thediffusivity from the gradients of the displacement and vari-ance with respect to time, respectively. With appropriate sub-stitutions, Eq. �7� can also be converted to Eq. �8�, which
represents the fluctuation of gravitational work �gravitationalpotential energy�
P�+ W��P�− W��
= exp W�
�D/�� . �8�
Here, W�=m�̄x� is the work performed by gravity, whichfluctuates with x�, and =Vdrift /m�̄ is the mobility. Accord-ing to Eq. �8�, if W� is nondimensionalized by dividing it byD /, we obtain a work fluctuation relation for this systemdriven with an external force and excited by an externalnoise. This equation states that the ratio of the probabilitiesof finding the positive and negative values of a particularvalue of work is equal to the exponential of the positivevalue �see SM-F for more details and a related reference�.36
As D / is equal to kBT in a thermal system according toEinstein equation, it is interesting to check if a similar equa-tion can be obtained by replacing kBT with an equivalentenergy scale mK�� /2 in the current athermal system.
For the prism on the roughened glass support, Vdrift in-creases sublinearly with K �Fig. 6�, while at a given value ofK, the velocity increases linearly with the bias. When thedrift velocities are normalized by dividing it with m�̄ to ob-tain a generalized mobility as a function of K, all the mobili-ties do indeed cluster nicely around a single master curve�Fig. 7�. These data are consistent with our previous report,23
although the previous studies were conducted with a smaller
06420
0.05
0.10
0.15
x(m
m)
t (s)
0 1 2
6
4
2
0
0.68 m2/s3
0.0005 m2/s3
06420
0.05
0.10
0.15
x(m
m)
t (s)
0 1 2
6
4
2
0
0.68 m2/s3
0 1 2
6
4
2
0
0.68 m2/s3
0.0005 m2/s3
FIG. 4. The trajectory of the stochastic motion of a glass prism on a glasssubstrate under the influence of applied bias �0.29 mN� and Gaussian whitenoise of power 0.0005 m2 /s3 is shown. A typical trajectory at same bias butat a high power �0.68 m2 /s3� is presented in the inset. Stick-slip motion atthe low power and smooth motion at the high power are evident.
0
4
8
12
-0.4 0 0.4 0.8 1.2
P(x
τ)
(m
m-1
)
xτ
( mm )
τ = 0.005 s
τ = 0.050 s
τ = 0.350 s
τ = 0.200 s
0
4
8
12
-0.4 0 0.4 0.8 1.2
P(x
τ)
(m
m-1
)
xτ
( mm )
τ = 0.005 s
τ = 0.050 s
τ = 0.350 s
τ = 0.200 s
τ = 0.005 s
τ = 0.050 s
τ = 0.350 s
τ = 0.200 s
FIG. 5. Probability distribution functions of the displacement of a glassprism on a rough glass support for K=0.16 m2 /s3 and m�̄=0.29 mN atdifferent time intervals � shown inside figure.
10-3 10-2
10-1
10-3
10-5
10-1 100 101
Vd
rift
(m/s
)
K (m2/s3 )
10-3 10-2
10-1
10-3
10-5
10-1 100 101
Vd
rift
(m/s
)
K (m2/s3 )
FIG. 6. Log-log plot of the drift velocities �Vdrift� of a glass prism on a glassplate as a function of the power �K� of the Gaussian noise and differentapplied biases: 0.29 ���, 0.57 ���, 1.43 ���, 2.28 ���, and 2.84 mN ���.
024702-5 Diffusion with nonlinear friction J. Chem. Phys. 133, 024702 �2010�
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variation of bias and smaller range of K. The master curve ofthe generalized mobility can be analyzed in conjunction withthe drift velocity estimated from the first moment of thePDFs shown in Eq. �5� as a function of �̄ and K, and subse-quently normalizing it by dividing it with applied bias m�̄�Fig. 7�. For the prism on the roughened glass, the values of� and �L needed to obtain the best agreement between theexperimental data and the theoretical estimates are 4.5 m /s2
and 0.06 s, respectively.
C. Numerical simulations of the motion of the prism
Having established the values of � and �L in Sec. IV C,we simulated23 the stochastic motion of the prism at twodifferent powers using the modified Langevin Eq. �1�. ��t�values as obtained from the accelerometer were used as inputacceleration for the numerical simulations. Since the dry fric-tion force always acts in the direction opposite to the motionof the prism, it is set as �V / �V� in the simulations. When thenet acceleration ��̄+��t�� acting on the prism is less than �,the velocity of the prism is set to zero. Equation �1� is inte-grated with an integration time step of �t=0.001 s, which issame as the resolution time of the high speed camera used totrack the stochastic motion.
Line II in Fig. 8 shows the trajectory of the prism that isobtained from the numerical solution of Eq. �1� at a lowpower �0.0005 m2 /s3�, in which no stick-slip motion is pre-dicted. Stick-slip motion usually is an indicator of the exis-tence of metastable energy states on a surface, which is notexplicit in a uniform dry friction used in our simulation. Wethus carried out a simulation of Eq. �1� by incorporating anadditional sinusoidal term35 H sin�2�x /L� that represents apinning potential. Here, L represents the length scale and Hdenotes the amplitude of perturbation. We surmise that �represents a background value of the static friction, whereasthe perturbation term represents defects distributed at a largerlength scale. Although the above description of friction issomewhat speculative, it provides an approximate way ofdescribing the effect of pinning defects on the motion of theprism. While the simulation is carried out with somewhatarbitrary values of H and L as 0.3 m /s2 and 17 m, respec-
tively, we note that the chosen value of L is close to thelowest jump length of the stepwise motion of the prism asreported in Fig. 4. Simulations do indeed predict a stick-slipmotion of the solid for the low power �0.0005 m2 /s3� whenthe pinning potential is considered �line I in Fig. 8�. But at ahigh power, a fluidlike motion is observed even in presenceof the same sinusoidal potential �inset of Fig. 8�. Here, oneimportant point should be noticed. As we have used the samenoise input file for both the trajectories, with or without thepinning potential, we can keep record of the nature of theimpulses. From Fig. 8, it is evident that whenever a pulse oflarge acceleration arrives from the noise field, the solid ex-hibits a big jump. The solid does not remain trapped on thesurface unless there is a pinning potential.
D. Displacement fluctuation
Figure 9 summarizes the fluctuations of the displace-ments of the prism on the solid surface corresponding to alow bias �0.29 mN� and a low power �0.04 m2 /s3� but atdifferent time intervals �; and corresponding to the same
(s/
kg)
10-4 10-3 10-2 10-1 100 101
10-2
100
101
102
10-1
K (m2/s3 )
~ K 0.8μ
FIG. 7. Log-log plot of the mobility as a function of power of the noise �K�at different biases: 0.29 ���, 0.57 ���, 1.43 ���, 2.28 ���, and 2.84 mN���. The solid line represents the mobility that is estimated by calculatingVdrift from the first moment of PDF given by Eq. �5� and dividing it by thebias. For the bias 0.29 mN, the data corresponding to the stick-slip motionof the prism are also included in this plot.
0000
0.20.20.20.2
0.40.40.40.4
0 4 80 4 80 4 80 4 8
xxxx((((m
mm
mm
mm
m))))
0.40.40.40.4
0.20.20.20.2
00000000 4444 8888
tttt ((((ssss))))
0000
0.10.10.10.1
0.20.20.20.2
0.30.30.30.3
0 2 40 2 40 2 40 2 40000 2222 44440000
0.10.10.10.1
0.20.20.20.2
0.30.30.30.3
I
II
0000
0.20.20.20.2
0.40.40.40.4
0 4 80 4 80 4 80 4 8
xxxx((((m
mm
mm
mm
m))))
0.40.40.40.4
0.20.20.20.2
00000000 4444 8888
tttt ((((ssss))))
0000
0.10.10.10.1
0.20.20.20.2
0.30.30.30.3
0 2 40 2 40 2 40 2 40000 2222 44440000
0.10.10.10.1
0.20.20.20.2
0.30.30.30.3
0000
0.10.10.10.1
0.20.20.20.2
0.30.30.30.3
0 2 40 2 40 2 40 2 40000 2222 44440000
0.10.10.10.1
0.20.20.20.2
0.30.30.30.3
I
II
FIG. 8. Line I, representing the stick-slip motion, is the simulated trajectoryof a glass prism on a glass support vibrated with the Gaussian noise ofpower 0.0005 m2 /s3 and applied bias 0.29 mN when a periodic pinningpotential is considered. Line II depicts the simulation without the sinusoidalpotential at same condition. The line in the inset is the simulated trajectorywith the sinusoidal potential at a higher power �0.01 m2 /s3� but at sameapplied bias �0.29 mN�.
-0.15 0 0.15 0.3
10-3
10-1
101
103(a)
xτ (mm)
xτ
P(
)(m
m-1
)
-0.2 0.2 0.6 110-3
10-2
10-1
100
101
102
xτ (mm)
(b)
xτ
P(
)(m
m-1
)
FIG. 9. Probability distribution function of displacement fluctuation of theglass prism on a glass substrate subjected to Gaussian white noise of power0.04 m2 /s3: �a� corresponds to a bias of 0.29 mN and different time inter-vals �: 0.005 ���, 0.05 ���, 0.20 ���, and 0.40 s ���; �b� corresponds to�=0.08 s but for different biases: 0.29 ���, 1.43 ���, and 2.84 mN ���. In�a�, the skewness value increases with � from 0.001 �for �=0.005 s� to 0.33�for �=0.40 s� and the kurtosis decreases from 3.5 ��=0.005 s� to 3.1 ��=0.40 s�. In �b�, the skewness increases with bias from 0.23 �0.29 mN� to1.12 �2.84 mN�. The kurtosis also increases with bias from 3.4 �0.29 mN� to4.5 �2.84 mN�.
024702-6 P. S. Goohpattader and M. K. Chaudhury J. Chem. Phys. 133, 024702 �2010�
Downloaded 21 Mar 2013 to 129.187.254.47. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
power as above and a fixed value of ��0.08 s� but for threebiases: 0.29, 1.43, and 2.84 mN. Evidently, all these prob-ability distributions are non-Gaussian and distinctly asym-metric with the degree of asymmetry increasing with � aswell as with the bias. In order to estimate the degree ofasymmetry, we estimated the “skewness” �S� of the displace-ment PDFs, which is defined as the ratio of the third centralmoment of a distribution to the cube of the standard devia-tion of that distribution. For symmetric distribution, skew-ness is close to zero, whereas positive and negative skewnessportray right sided or left sided asymmetry, respectively.
In order to quantify the “peakedness” of a distribution,we estimated its kurtosis � �, which is defined as the ratio ofthe fourth central moment of a distribution to the square ofthe variance. For Gaussian distribution, is close to 3whereas this value increases with the peakedness of a distri-bution reaching a value of 6 for an exponential PDF.
In Fig. 10, we replot two representative PDFs with the
data taken from Fig. 9 in nondimensional forms, which arecompared with those obtained from the numerical integrationof Eq. �1�. The simulations do indeed reproduce the non-Gaussian features of the displacement distributions alongwith the asymmetry, although the degrees of asymmetryobserved in the experiments �S=0.23 for m�̄=0.29 mN;S=1.12 for m�̄=2.84 mN� are somewhat larger than thosefound in the simulations �S=0.10 for m�̄=0.29 mN;S=0.68 for m�̄=2.84 mN�.
By contrast, the non-Gaussian nature of the displacementPDFs observed in experiments � =3.4 for m�̄=0.29 mN; =4.5 for m�̄=2.84 mN� compare well with those obtainedfrom simulations � =4.0 for m�̄=0.29 mN; =4.6 for m�̄=2.84 mN�. The displacement PDFs �Fig. 10� obtained at alower power �K=0.01 m2 /s3� as well as at a higher power�K=1.21 m2 /s3�, however, paint a somewhat different story.Here, the PDFs obtained from the experimental data showthat they are quite symmetric �S�0� and Gaussian� �3.0�. The simulations suggest that the PDF should beGaussian at the low power �K=0.01 m2 /s3, =3.1� as foundin the experiments; but it has strong exponential tails at ahigher power �K=1.21 m2 /s3, =5.3�, which disagrees withthe experimental observations. On the other hand, negligiblevalues of the skewness �S�0� obtained from simulations atboth low and high powers suggest that the displacement dis-tributions should be quite symmetric, which is consistentwith the experimental observations. Figure 11 summarizesthe experimentally obtained PDFs of the displacement fluc-tuation as a function of the power of the noise, along with thevalue of the kurtosis stamped inside each figure. The distri-bution is clearly Gaussian at a low power �as discussedabove�, but becomes non-Gaussian and asymmetric as thepower increases. It becomes Gaussian and symmetric againat a very high power. It seems that the effect of static frictionis overcome at a very high power of the noise, which mayindicate a subtle level of continuous transition occurring atthe interface with increasing K. The Gaussian distribution atthe very low power results from the insufficiency of thenumber of high acceleration pulses, i.e., the statistics is poor.This subject will be taken up again in Sec. V.
At this juncture, we would like to mention that a recentmodel on the slip avalanches39 shows that local failure stress�“pinning stress”� has to be overcome for a slip to occur.Depending on the weakening of the threshold failure stress, acontinuous transition from brittle to ductile and hardeningcan occur. This may be somewhat related to our observationshere.
xτ
~
P(
)
-6 -4 -2 0 2 4 6 8
100
10-1
10-2
10-3
10-4
(a)0.29 m N
2.84 m N
-6 -4 -2 0 2 4 6 8
100
10-1
10-2
10-3
10-4
xτ
~
P(
)
(b)0.29 m N
2.84 m N
xτ~
10-4
10-3
10-2
10-1
100
-6 -3 0 3 6
xτ
~
P(
)
(c)
xτ
~
P(
)
xτ~
10-4
10-3
10-2
10-1
100
-8 -4 0 4 8
(d)
xτ
~
P(
)
-6 -4 -2 0 2 4 6 8
100
10-1
10-2
10-3
10-4
(a)0.29 m N
2.84 m N
xτ
~
P(
)xτ
~ xτ
~
P(
)
-6 -4 -2 0 2 4 6 8
100
10-1
10-2
10-3
10-4
(a)0.29 m N
2.84 m N
0.29 m N
2.84 m N
-6 -4 -2 0 2 4 6 8
100
10-1
10-2
10-3
10-4
xτ
~
P(
)
(b)0.29 m N
2.84 m N
-6 -4 -2 0 2 4 6 8
100
10-1
10-2
10-3
10-4
xτ
~
P(
)xτ
~ xτ
~
P(
)
(b)0.29 m N
2.84 m N
0.29 m N
2.84 m N
xτ~
10-4
10-3
10-2
10-1
100
-6 -3 0 3 6
xτ
~
P(
)
(c)
xτ~
xτ~
10-4
10-3
10-2
10-1
100
-6 -3 0 3 6
xτ
~
P(
)xτ
~ xτ
~
P(
)
(c)
xτ
~
P(
)
xτ~
10-4
10-3
10-2
10-1
100
-8 -4 0 4 8
(d)
xτ
~
P(
)xτ
~ xτ
~
P(
)
xτ~
xτ~
10-4
10-3
10-2
10-1
100
-8 -4 0 4 8
(d)
FIG. 10. �a� Experimental and �b� simulated probability distributions ofdimensionless displacement fluctuations �x̃�= �x�−xp� /�x�� of a glass prismmoving on a glass plate under the influence of Gaussian noise of power0.04 m2 /s3 at different applied biases as indicated at top right corner. Here,xp corresponds to the displacement with peak probability density and �x� isthe standard deviation of the displacement distribution. The time interval �used for this plot is 0.08 s. The solid lines are obtained by fitting the ex-perimental data with asymmetric double sigmoidal functions, the centers ofwhich are bell-shaped, but have exponential tails. �c� and �d� represent PDFsof dimensionless displacement at the time intervals of 0.08 ��� and 0.35 s��� at different powers: �c� 0.01 and �d� 1.21 m2 /s3. The filled and opensymbols indicate the experimental and simulation results, respectively. Theapplied bias in all cases is 0.29 mN. It should be emphasized here that thesesimulated PDFs are not in exact numerical agreements with the experimen-tal results when plotted in terms of the absolute values of x�. The variance ofthe simulated PDF is considerably higher than that obtained experimentally.However, when plotted in the dimensionless form, it reproduces the generalfeatures of the experimental distributions.
-6 -3 0 3 6
10-4
10-3
10-2
10-1
100
-3 0 3 6 -3 0 3 -3 0 3 6
xτ
~
P(
)
6 -3 0 3 6
0.43 m2/s3 1.21 m2/s30.04 m2/s30.16 m2/s30.01 m2/s3
xτ~
2.993.263.333.29β=2.98
-6 -3 0 3 6
10-4
10-3
10-2
10-1
100
-3 0 3 6 -3 0 3 -3 0 3 6
xτ
~
P(
)
6 -3 0 3 6
0.43 m2/s30.43 m2/s3 1.21 m2/s31.21 m2/s30.04 m2/s30.04 m2/s3
0.16 m2/s30.16 m2/s30.01 m2/s3
xτ~
0.01 m2/s3
xτ~
xτ~
2.993.263.333.29β=2.98 2.993.263.333.29β=2.98
FIG. 11. PDFs of the dimensionless displacement �x̃�= �x�−xp� /�x�� of a solid prism on a solid surface subjected to Gaussian noise of different powers�indicated in the top left corner�. The applied bias is 0.29 mN. The value of the kurtosis � � is stamped inside each PDF. Skewness values of the PDFs are0.07, 0.27, 0.05, 0, and 0 for the powers 0.01, 0.04, 0.16, 0.43, and 1.21 m2 /s3, respectively.
024702-7 Diffusion with nonlinear friction J. Chem. Phys. 133, 024702 �2010�
Downloaded 21 Mar 2013 to 129.187.254.47. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
E. Diffusivity: The effect of the noise strength andbias
The experimental data of the stochastic displacement�x�� of the prism as a function of time, as obtained fromseveral tracks, were combined in order to obtain the PDF ofx� for K ranging from 0.0005 to 1.83 m2 /s3 and m�̄ rangingfrom 0.29 to 2.8 mN. These PDFs allowed the estimation ofthe diffusivities from the plot of the variance of the displace-ment �x�
2 = ��x�2 − �x� 2� versus �, using the well-known ex-
pression: D=d��x�2 � /2d�. Here we examine the time evolu-
tion of the variance of the displacement distributioncorresponding to m�̄=0.29 mN and K=0.04 m2 /s3.
The variance of the displacement fluctuation firstincreases sharply �Fig. 12� followed by a decrease at��0.02 s; it then increases linearly with time. Clearly, theprism exhibits an anomalous dispersion at short times. Thiskind of anomalous diffusive behavior at short times is repro-ducible and has been observed with other biases as well �seeSM-G, Fig. S10�.36 The diffusivity that we report in thispaper is obtained from the slope of the linear portion of thevariance of the displacement as a function of � at a longertime scale. The diffusivity data at different powers and biasesare summarized in Figs. 13 and 14.
The data summarized in Fig. 13 show that there are threedistinct transport behaviors of the prism. The apparent diffu-
sivity at the lowest power �0.0003 m2 /s3� is already sub-merged into background noise of the system. Furthermore, atthis power of the noise, no net drift of the solid object isobserved. This is like a solid phase. A phase transitionlikebehavior coupled with a stick-slip motion of the object isobserved �Fig. 13� as the power is increased from 0.0003 to0.0005 m2 /s3. The residence time in stick �solidlike� phasedecreases with the power of the noise. The frequent occur-rences of the slip motion eventually merge into the fluidlikerandom motion at higher powers �K�0.01 m2 /s3�.
The diffusivity of the prism in the fluidlike state varieswith K with an exponent of 1.6, which deviates from that�3.0� predicted by de Gennes.29 This is not surprising at firstbecause de Gennes assumed that only dry friction operates atthe interface. However, we suspect that another cause of thisdifference arises from poor correlations of displacements,which will be discussed below.
F. Estimation of diffusivity
The diffusivity governed by purely kinematic friction�D=K�L
2 /2� is in the range of 107–109 m2 /s for K rangingfrom 0.01 to 1.8 m2 /s3, which is obviously much larger thanthe experimental values ��102–106 m2 /s for same rangeof K� �Fig. 13�. By ignoring the kinematic friction, deGennes29 derived an equation for diffusivity for the dry fric-tion case as follows:
D =4
��
0
� �0
�
dtdkpk2
�p2 + k2�3exp�− K�p2 + k2�t� , �9�
where p=� /2K.Diffusivities estimated using Eq. �9� varies with K with
an exponent of 3. Furthermore, its values are in the range of103–1010 m2 /s for the range of K as above, which are stilllarger than those measured experimentally.
Clearly, our experimental data are not totally consistentwith the predictions based on the simple model of deGennes.29 Our hypothesis is that the cause for the measureddiffusivities being so much smaller than the predicted valuesis that the correlation time of the stochastic velocities ismuch smaller than either the Langevin or the dry frictiontimes ��L or ���. Indeed, using the relationship between the
0
500
1000
1500
2000
0 0.1 0.2 0.3 0.4 0.5
D= -7.0 X 10 3
D=2.4 X 10 3
γ = 0.29 mN
σ xτ2
(�m
2)
τ (s)
0
150
300
0 0.025 0.050
500
1000
1500
2000
0 0.1 0.2 0.3 0.4 0.5
D= -7.0 X 10 3
D=2.4 X 10 3
γ = 0.29 mNγ = 0.29 mN
σ xτ2
(�m
2)
τ (s)
0
150
300
0 0.025 0.050
150
300
0 0.025 0.05
FIG. 12. Variance of the displacement of the glass prism as a function oftime ���. The applied bias is 0.29 mN and the power of the Gaussian noiseis 0.04 m2 /s3. The lower inset is the enlarged view of the variance at shorttime region showing anomalous diffusivity, with even a negative diffusivity��−7000 m2 /s� in the range of ��0.021 to 0.025 s.
106
104
102
100
10-5 10-3 10-1
K (m2/s3 )
(0)
D( m
2/s
)
0 2 4
0.08
0
0.04
00.04
0 2 4
0.08
I
0
1
2
0 0.4 0.8
x(m
m)
0.4 0.8t (s)
x(m
m)
00
1
2106
104
102
100
10-5 10-3 10-1
K (m2/s3 )
(0)
D( m
2/s
)
0 2 4
0.08
0
0.04
0 2 4
0.08
0
0.04
00.04
0 2 4
0.08
00.04
0 2 4
0.08
I
0
1
2
0 0.4 0.8
x(m
m)
0.4 0.8t (s)
x(m
m)
00
1
2
0
1
2
0 0.4 0.8
x(m
m)
0.4 0.8t (s)
x(m
m)
00
1
2
FIG. 13. Log-log plot of the experimental diffusivity ��� of the glass prismas a function of the power of the noise �K� corresponding to the applied biasof 0.29 mN. Line I corresponds to the background noise �shown in rightinset� and considered as zero diffusivity �marked in bracket�. Stick-slip typemotion is observed at the powers of 0.0005 to about 0.002 m2 /s3, whereasno stick-slip motion is evident for powers ranging from 0.01 to 1.8 m2 /s3.
102
103
104
105
D( m
2/s
)
106
107
10-3 10-2 10-1 100 101
K (m2/s3 )
102
103
104
105
D( m
2/s
)
106
107
10-3 10-2 10-1 100 101
K (m2/s3 )
FIG. 14. Log-log plot of diffusivity D estimated for different powers of thenoise �K� and biases: 0.29 ���, 0.57 ���, 1.43 ���, 2.28 ���, and 2.84 mN���. Here the data corresponding to the stick-slip motion of Fig. 13 for theapplied bias 0.29 mN are not included.
024702-8 P. S. Goohpattader and M. K. Chaudhury J. Chem. Phys. 133, 024702 �2010�
Downloaded 21 Mar 2013 to 129.187.254.47. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
diffusivity and the variance of the velocity estimated fromthe second moment of Eq. �5�, at the bias of 0.29 mN, theestimated correlation time ���=�v
2 /2D� is in the range of�37–56 s, which is significantly smaller than either thetimescale of kinematic friction �L�0.06 s� or that of the dryfriction ���K /�2 �0.5–89 ms for K ranging from 0.01 to1.8 m2 /s3�. Furthermore, we notice that the power spectrumof the fluctuations of the displacements at most powers arerather flat at the time scale of the data recording �0.001 s��L or ��� �see SM-D�.36 Based on the above scenarios, weenvisage an extreme situation where the velocities are deltacorrelated, i.e., �V�t1�V�t2� =D��t1− t2�, and examine howthe diffusivities estimated from this approach compare withthose found experimentally. Here, the time integral of theabove velocity correlation function �VCF� yields the diffu-sivity in real space, just as the integral of the correlationfunction of stochastic acceleration yields the diffusivity invelocity space. Within this model, the velocity vectors �Eq.�5� or Eq. �6�� are given by the base state solution of aprobability diffusion equation �Eq. �3� or Eq. �4��, but theyare propagated completely randomly producing a stochastictrajectory.
In order to make the estimation of diffusivity, we firstdefine a characteristic displacement �jump length� obtainedfrom the following substitution: V� x̄ /�c, x̄ being the jumplength and �c is a characteristic time scale that we take to bethe pulse width of the Gaussian noise. We estimate the dif-fusivity from the trajectories simulated from this jump lengthdistributions from Eq. �10� using the method described be-low.
P�x̄� = Po� exp�−x̄2
K�L�c2 −
2�x̄��K�c
+2�̄x̄
K�c . �10�
Equation �10� is a modified version of Eq. �5�, in whichthe velocity is replaced with x̄ /�c. We generate a large matrixof jump length vectors �x̄� using Eq. �10� and randomly se-lect them to construct stochastic paths over a longer time �2s� duration. From these stochastic paths, we calculate thejump lengths for a larger time scale �0.001 s� �which alsohappens to coincide with the resolution time of the cameraused in actual experiments�. The jump lengths correspondingto 0.001 s are then randomly selected to construct stochastictrajectories of much longer duration ��50 s�. From thesesimulated trajectories, the PDF of displacement was con-structed for various values of � and the estimation of diffu-sivity was carried out in the usual way. Diffusivities wereestimated for three different cases: first by setting �=0, sec-ond by ignoring the kinematic friction, and third by consid-ering both the kinematic and static friction.
These simulations, as summarized in Fig. 15, show thatthe diffusivities, in the absence of the dry friction, are higherthan the experimental values and it varies almost linearlywith K as is the case with the correlated kinematic diffusion.The diffusivities for the case of pure dry friction and thosefor simultaneous actions of both types of frictions have com-parable values. They vary with K with an exponent �1.8,which is also close to the experimental result �1.6�. We alsoestimated the drift velocities as well as the energy dissipation
�SM-E� �Ref. 36� from the constructed trajectories when boththe kinematic and static frictions operate. These values arealso similar to those obtained from the experimental obser-vations �Figs. 15 and S8�.
G. Stochastic behavior of the PDMS grafted surfaces
For the PDMS grafted prism on a PDMS grafted siliconwafer, it was somewhat difficult to make measurements atvarious angles of inclinations as the prism slips easily on itsown at angles greater than 1°. Furthermore, here, as theprism has a greater tendency to rotate about its axis than anunmodified prism on the roughened glass, care had to betaken in order to use only those tracks that did not exhibitany rotation for data analysis. The drift and diffusivity valueswere estimated at 1° inclination for a given power�K=0.1 m2 /s3� using 45 tracks, each lasting for 4.5–5 s. Thefact that the PDMS grafted prism slides easily at very smallangle of the inclination of the supporting plate of a PDMSgrafted silicon wafer suggests that the static friction is neg-ligible here. On the other hand, the kinematic friction is non-linear as evidenced from the distributions of displacements�Fig. 16�.
10-2
10-3
10-4
10-5Vd
rift
(ca
l)(m
/s)
10-210-310-410-5
Vdrift (exp) (m/s)
(a)
101
103
105
107
D( m
2/s
)
10-3 10-1 101
K (m2/s3)10-2 100
(b)
D ~ K
D ~ K 1.8
10-2
10-3
10-4
10-5Vd
rift
(ca
l)(m
/s)
10-210-310-410-5
Vdrift (exp) (m/s)
(a) 10-2
10-3
10-4
10-5Vd
rift
(ca
l)(m
/s)
10-210-310-410-5
Vdrift (exp) (m/s)
(a)
101
103
105
107
D( m
2/s
)
10-3 10-1 101
K (m2/s3)10-2 100
(b)
D ~ K
D ~ K 1.8
101
103
105
107
D( m
2/s
)
10-3 10-1 101
K (m2/s3)10-2 100
(b)
D ~ K
D ~ K 1.8
D ~ K
D ~ K 1.8
FIG. 15. �a� Comparison between the experimental and the estimated driftvelocities �Vdrift�exp� and Vdrift�cal�� as obtained from the trajectories createdfrom the propagation of the steady state jump lengths by stitching themrandomly. �b� Log-log plot of the diffusivity as a function of power of thenoise. � represents the experimental results �ignoring the stick-slip phasesof Fig. 13�, whereas the solid line represents the diffusivities estimated fromthe trajectories created by stitching the randomized jump length as men-tioned above. The dashed and dotted lines represent the estimated diffusivi-ties after switching off the dry friction and the kinematic friction terms ofEq. �10�, respectively.
100
-3 0 3 6
τ =0.3s
xτ
~
P(
)
-6 -3 0 3 610-4
10-3
10-2
10-1
τ =0.2s
xτ
~
P(
)
10-4
10-3
10-2
10-1
τ =0.005s τ =0.1s
xτ~
xτ~
100
-3 0 3 6
τ =0.3s
xτ
~
P(
)xτ
~ xτ
~
P(
)
-6 -3 0 3 610-4
10-3
10-2
10-1
τ =0.2s
xτ
~
P(
)xτ
~ xτ
~
P(
)
10-4
10-3
10-2
10-1
τ =0.005s τ =0.1s
xτ~
xτ~
xτ~
xτ~
FIG. 16. Experimental ��� and simulated ��� probability distributions ofdimensionless displacements of a PDMS grafted glass prism moving on aPDMS grafted silicon wafer under the influence of a Gaussian noise�K=0.1 m2 /s3� and a bias of 0.29 mN at different time intervals as indi-cated inside each figure. The simulation is done using Eq. �2� with A=0.03 and n=0.4.
024702-9 Diffusion with nonlinear friction J. Chem. Phys. 133, 024702 �2010�
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The displacement PDF is quite non-Gaussian at �=0.005 s, which is supported by the high value of the kur-tosis � =3.85�. However, at longer time intervals, the valueof decreases to �3.1, indicating that the PDF is moreGaussian. The distributions are somewhat skewed as evidentfrom its values: 0.18, 0.26, 0.31, and 0.32 for �=0.005, 0.1,0.2, and 0.3 s, respectively. From a previous publication,40
we know that the PDMS grafted surfaces exhibit a linearkinematic friction at very low sliding velocity, but progres-sively become nonlinear at high velocity. In the present case,the signature of a nonlinear kinematic friction is evident inthe non-Gaussian PDF of displacement as discussed above; itis also confirmed in the sublinear profile of the mobility �Fig.17� as a function of the power of noise. Drift velocity of thePDMS grafted glass prism on the PDMS grafted silicon wa-fer were measured from the displacement of �3 cm as afunction of time using a low speed �30 fps� Sony camera�DCR HC-85 NTSC� at eight different powers of Gaussiannoise with K ranging from 0.04 to 1.2 m2 /s3 correspondingto a bias of 0.29 mN. We attempted to fit the mobilities as afunction of K with a velocity weakening form of the frictionusing an exponent of n=0.4 in Eq. �6�. The low K regime isnot fitted well with such a nonlinear function, while the fit isreasonable at values of K�0.1, i.e., starting from the valueof K used in our experiment. Using this nonlinear form ofkinematic friction, we simulated the PDFs at different valuesof � using Eq. �2�. The results summarized in Fig. 16 showgood agreements between experimental and simulated PDFs.The kurtosis for the simulated PDFs is high at short time,i.e., =4.39 for �=0.005 s. However, the value of de-creases to 3.9, 3.4, and 3.3 for �=0.1, 0.2, and 0.3 s, respec-tively. On the other hand, the simulated PDFs are more sym-metric �S=0.02, 0.11, 0.03, and �0 for �=0.005, 0.1, 0.2,and 0.3 s, respectively� than the experimental results.
From the gradient of variance of the displacement PDFwith time, we estimate the diffusivity of the PDMS graftedprism on the PDMS grafted silicon wafer as 1.7�104 m2 /s. This value is much lower than that expectedof a simple diffusion controlled by linear kinematic friction.Here too, as is the case with the prism on a roughened glass,the power spectrum of the displacement at a frequency of 1
kHz is quite flat �see SM-D, Fig. S7�.36 We thus estimatedthe diffusivity using the approach described before �Sec.IV F�, assuming that the stochastic velocities are delta corre-lated. The stochastic trajectories were simulated by stitchingrandomly selected jump lengths obtained from velocity dis-tribution given in Eq. �6� using the substitution V� x̄ /�c asbefore. The drift velocity �1.2 mm/s� of the simulated trajec-tories is almost same as the experimental value �1.1 mm/s�.The diffusivity �104 m2 /s� obtained from these simulatedtrajectories is also close to the experimental value �1.7�104 m2 /s�. The fact that the simulated diffusivity valueis slightly lower than the experimental value is expected, asin the real situation, the correlation time is finite, although itis smaller than �� or �L.
H. An Einstein-like relation
A driven stochastic system with a thermal noise can besubjected to the analysis of the fluctuations of various typesof thermodynamic parameters. As we pointed out previously�Sec. IV B�, a work fluctuationlike relationship is obtained inour system if the work values are nondimensionalized bydividing it by D /. In thermal system, as D / is equal tokBT, it is of interest to find out if a similar equation prevailsin our case. We thus explore if an Einstein-like relationship,i.e., 2D�̄ /VdriftK��=1, can be obtained for this athermal sys-tem. It would be ideal to explore if an Einstein-like relation-ship, i.e. 2D�̄ /VdriftK��=1, holds for this athermal systemwith the full knowledge of all the parameters obtained ex-perimentally. However, as the characteristic timescale �� isnot known a priori, we resort to another approach. While thediffusivity is estimated from the temporal integration of theVCF, it scales with the variance of velocity ��v
2� as D��v
2�� since the VCF usually decays exponentially with timefor both the kinematic and static friction.29 We thus test if�v
2�̄ /KVdrift is close to unity. When the variance of the veloc-ity, which is estimated from the second moment of P�V�using Eq. �5�, is plotted against KVdrift / �̄ with the data col-lected at different biases and K �Fig. 18�, they cluster nicelyaround a straight line with its slope approaching unity. Thissuggests that �v
2�̄ /KVdrift=1, which is an Einstein-like rela-tion for this system.
(
s/kg
)
K (m2/s3 )
100
101
102
10-1
10-2 10-1 100 101
(
s/kg
)
(s/
kg)
K (m2/s3 )
100
101
102
10-1
10-2 10-1 100 101
FIG. 17. The mobility of PDMS grafted glass prism on a PDMS graftedsilicon wafer as a function of the power �K� of the Gaussian noises but at aconstant bias 0.29 mN. The open squares are the experimental data. Thesolid line represents the mobilities estimated by dividing the drift velocity�first moment of the PDF given by Eq. �6� with A=0.03 and n=0.4� with theapplied bias force �m�̄�.
106
107
108
109
1010
1011
106 107 108 109 1010 1011
σv2
(μm
2/s
2)
(μ m2/s2)KVdrift
γ
FIG. 18. A plot of �v2 against KVdrift / �̄ at different biases: 0.29 ���, 0.57
���, 1.43 ���, 2.28 ���, and 2.84 mN ���.
024702-10 P. S. Goohpattader and M. K. Chaudhury J. Chem. Phys. 133, 024702 �2010�
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V. DISCUSSIONS
A. Diffusive motion of the prism and the phasetransitionlike behavior
The prism exhibits different types of drift and diffusiveresponses to external vibration. At a very low value of K, theprism does not drift. However, as K increases, it drifts withvelocities increasing with the power of the noise and theimposed bias. Overall linear response of the sliding dynam-ics confirms our previous observations23 and is consistentwith other studies published in the literature.41–43 The strongdependence of the drift velocity on the power of the noise isa departure from the standard driven Brownian system,where Vdrift is simply a product of �̄ and �L. Previously,23 wepresented an approximate expression for the drift velocity ofan object on a surface, where both the dry and kinematicfrictions operate,
Vdrift =�̄�L
1 + �2�L/K. �11�
Equation �11� indeed predicts that Vdrift increases sublin-early with K. In the absence of the dry friction, i.e., �=0, thedrift velocity is exactly same as that of Einstein’s value��̄�L�. In the presence of a finite �, Vdrift increases with Kand approaches the Einstein’s value only in the limit of K��. Equation �11� also predicts that Vdrift should increaselinearly with the bias �̄. All these predictions are consistentwith the experimental observations. The sublinear increase ofVdrift with respect to K is not only observed when dry frictionoperates, but also with a velocity weakening friction as is thecase with the surfaces grafted with PDMS chains. For thecase of a strong dry friction, i.e., when �2�L�K, Vdrift
=K�̄ /�2. This illustrates an interesting situation, namely, thata linear relation prevails between the drift velocity and thedriving force in a stochastic situation even when no kine-matic friction may be operating at the interface.
At the lowest power �0.0003 m2 /s3�, we do not detectany diffusive motion of the prism as any fluctuation exhib-ited by the prism merges with the background noise. At anintermediate level of the injected power, the glass prism ex-hibits a net drift but accompanied with intermittent stick-slipmodes. Literature is abounding with the observation of stick-slip motion in various systems of interests to tribologists andwetting specialists. The subject of tribology is beautifullysummarized in a recent paper.44 Stick-slip motion is evidentin the relaxation of the contact line of a liquid drop,19 in thedynamics of granular particles,45 migration of cells46,47 on asurface, frictional sliding between lubricated and unlubri-cated surfaces,48,49 as well as in earthquakes.50 Many of theseinstabilities are due to the competition between elastic restor-ing force and friction coupled with shear weakening of theinterface.45,51 An interesting new picture of stick-slip motionhas recently been provided,52 in which Eq. �1� is solved with-out any applied bias within the path integral framework ofOnsager and Machlup. The authors found two kinds of solu-tions referred to as direct and indirect paths. The direct op-timal path is characterized by continuous velocity and accel-eration of the slider corresponding to the slip phase. Theindirect optimal path corresponds to the sticking of the object
to the surface for some finite time. These indirect paths havebeen interpreted by the authors as leading to a stick-slip mo-tion. Our approach to explaining the stick-slip motion israther classical. By modifying Eq. �1� with a periodic pertur-bation to the background friction, we find that the slider canget trapped to the potential well until a strong accelerationtakes it to another potential well, leading to a stick-slip mo-tion. When trapped in a potential well, the slider may stillundergo a restricted diffusive motion without a net drift,53
capturing which is beyond our current experimental capabil-ity.
At this juncture, we would like to point out that the waythe static friction is idealized in Eq. �1� may not be quitecorrect, as there are indications54,55 that some small scalemotion may exist when the external bias is smaller than m�.We have seen that a nonlinear power law friction may beadequate �SM-B� �Ref. 36� in describing the drift velocityand the displacement PDFs. However, the situation may alsobe described by replacing the dry friction term with a differ-ent form of nonlinear friction as follows:
dV
dt+
V
�L+ � tanh��V� = �̄ + ��t� . �12�
Equation �12� with a high value of � indeed reproduces allthe drift velocity data as Eq. �1�. It also reproduces the ex-ponential tails of the displacement PDFs.
B. Gaussian and non-Gaussian PDF of displacementfluctuations
One new, and potentially important, observation in thiswork is that the displacement PDF is Gaussian at a lowpower, but it exhibits pronounced exponential tails at higherpowers. The Gaussian distribution at a very low power re-sults from the lack of sufficient high energy pulses, thusleading to poor statistics. As the power of the noise in-creases, statistics is improved and the displacement PDFstarts exhibiting asymmetry with exponential tails. The kur-tosis of the PDF is above the value of 3, expected for anon-Gaussian distribution, for K ranging from 0.04 to0.43 m2 /s3, but it becomes Gaussian again atK=1.21 m2 /s3. This transition from a non-Gaussian to aGaussian PDF might indicate a transition from a state gov-erned by dry friction to a state governed by a linear kine-matic friction, although the displacement correlation remainspoor as is evident in the very low diffusivity.
A non-Gaussian PDF of displacement is also evident forthe case of a PDMS grafted prism sliding against a PDMSgrafted silicon wafer, where a velocity weakening nonlinearfriction seems to operate. However, in this case, the distribu-tion is strongly non-Gaussian only at a short time scale �i.e.,0.005 s�, but it becomes more Gaussian at a longer timescale. The general features of the experimental observationsare borne out reasonably well with Langevin dynamics simu-lation with a nonlinear kinematic friction.
We anticipate that a non-Gaussian velocity distributionmay also be observed for a colloid particle undergoing aBrownian motion in weak adhesive contact with a soft sub-strate. As the particle moves, it forms new bonds in the ad-
024702-11 Diffusion with nonlinear friction J. Chem. Phys. 133, 024702 �2010�
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vancing edge, but breaks bonds at the trailing edge resultingin hysteresis of adhesion. The Fokker–Planck equation forthe motion of the colloidal particle may be of the form
�P
�t= ��L
��x��x
�P
�x+ D
�2P
�x2 . �13�
Here, � is a measure of adhesion hysteresis. The stationarysolution of the displacement is exponential. The stochasticpath of the colloidal particle for a given duration � is alsoexpected to be exponential with a suppressed diffusivity.These considerations may be relevant to the results reportedby Wang et al.2
C. Einstein-like relation
We now turn our discussion to the Einstein-like relationthat we observed in our system, which have similarities tosome of the previous studies. D’Anna et al.42 conducted anexperiment, in which a torsional pendulum was immersed ina granular medium, which was fluidized by strong agitationwith a white noise. Meanwhile, a sinusoidal torque was im-posed on the pendulum itself. These measurements allowedan estimation of the granular viscosity, which decreased withthe strength of the noise with an effective temperature thatalso scaled linearly with the power of the noise.
A different study43 focused on the behavior of a singleobject that is a small ball placed on a smooth screen whileexposing it to an upward flow of gas. The turbulence pro-duced due to the flow of the gas about the ball created arandom upward and downward motion of the ball, whichwas consistent with the Langevin dynamics with the randomforce and a frequency dependent drag satisfying a FDR.
Our study is somewhat comparable to that of Ojhaet al.43 in the sense that the system can be characterized by asingle “effective temperature.” Our studies conducted withdrift velocities and diffusivities estimated at different valuesof K and �̄ comply with an Einstein-like relation, where wefind �v
2�̄ /KVdrift=1, implying D /=m K�� /2 for the case ofan un-activated diffusion. Here �� is a characteristic velocitycorrelation timescale. However, one intriguing observation isthat the diffusivity increases with bias at a given value of Kimplying that the velocity fluctuation and/or the correlationtime increase�s� with force. This type of result may be ex-pected in the case of a thermomechanically activated diffu-sion, where diffusivity could take the form, D=DE exp��f�–Ea� /kBT�. Here, DE is the Einstein’s diffusivity�K�L
2 /2�, f is the bias which reduces the activation energyEa, and � is a length scale separating the potential minima. Ifwe develop a parallel argument to explain our current result,the molecular activation states would plausibly be related tothe metastable states on the surface due to defects with thethermal energy replaced by mK�� /2. The prediction of suchan equation is qualitatively consistent with our observationin that the diffusivity increases with both K and bias. Thisequation also predicts that all the diffusivity data would con-verge to DE at high value of K as seen in our experiments. Infact, at a value of K�8 m2 /s3, all the diffusivity data seemto merge, which also gives an estimate of DE�2�107 m2 /s at this value of K. The possibility of the dif-
fusivity being an activated process where the metastablestates provide the energy barrier and the mechanical noiseprovide the excitation is an interesting prospect. However,more studies would be needed to find out if a Kramer-liketransition may occur in such systems.
We finalize this section by reiterating the fact that notonly the magnitude of the experimental diffusivities is con-siderably smaller than that predicted by de Gennes,29 thepower law exponent �1.6� of the D-K relationship also differsfrom the prediction �3� of de Gennes. Typical methods toconstruct diffusive trajectory is to establish the base state ofthe velocity distribution function, as is given in Eq. �5� orEq. �6� and then to propagate the base state solution tempo-rally either using an Onsager Machlup or a Fokker–Planckapproach. What is striking in the current situation is that nospecial method is needed to propagate the base state vectorsin time since the velocities are almost delta correlated. Themethod provides estimation of diffusivities that are close tothe experimental values not only for the case of a slidingprism on a rough surface, but also for a PDMS grafted prismsliding on a PDMS grafted silicon wafer. The method worksbecause the correlation time of stochastic velocities is verysmall compared to �L and ��. With the above caveats, it isstill intriguing that an Einstein-like relationship is obtained.These considerations may be of importance in other athermalsystems, such as granular gas, where the noise correlationtime and frictional time scales may be comparable in somecases.
VI. CONCLUSIONS
By studying the drift and the diffusive behavior of asmall solid object on a solid substrate as a function of thestrengths of the bias and the noise, we arrive at the followingconclusions:
�1� When a nonlinear friction operates at the interface, thedisplacement distributions are non-Gaussian and asym-metric, with the asymmetry increasing with the bias.
�2� Distinct solidlike, fluidlike, and transition regions areidentified.
�3� The drift velocities at different biases together with theestimated variances of velocity fluctuations follow a ru-dimentary form of an Einstein-like relations.
ACKNOWLEDGMENTS
�Late� Professor de Gennes inspired this study. We alsothank Srinivas Mettu and Jonathan Longley for helps in vari-ous �experimental and computational� aspects of this study. Itwas a pleasure to interact with S. Granick �University ofIllinois� as well as with E.G.D. Cohen �Rockefeller Univer-sity� and his group �A. Baule and H. Touchette� at differentstages of this work. We are also grateful to the two refereesfor critical and constructive comments on the earlier versionof this manuscript.
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