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To explore the existence of self-organization during friction, this paper considers the motion of all atoms in a systems consisting of an Atomic Force Microscope metal tip sliding on a metal slab. The tip and the slab are set in relative motion with constant velocity. The vibrations of individual atoms with respect to that relative motion are obtained explicitly using Molecular Dynamics with Embedded Atom Method potentials. First, we obtain signatures of Self Organized Criticality in that the stick-slip jump force probability densities are power laws with exponents in the range (0.5, 1.5) for aluminum and copper. Second, we characterize the dynamical attractor by the entropy content of the overall atomic jittering. We find that in all cases, friction minimizes the entropy and thus makes a strong case for self-organization.

Friction at the atomic scale has been rekindled as an active area of research for 20 years. Although it is well established that the laws of friction at the nanoscale differ from the well known ones for macroscopic bodies set forth centuries ago by Coulomb and Amontons [

In particular dry friction stick-slip motion has attracted particular attention as it serves as a model for fundamental macroscopic phenomena such as earthquakes [

Thus, it is pertinent to study well controlled systems to clarify the connection between friction and stick-slip, and of friction and entropy. An ideal scenario for a controlled situation consists on the production of reproducible atomic dynamics during friction. This is possible via full solution of the equations of motion of the many particle system, for well controlled external parameters such as ambient temperature and velocity. Here we present such results based on Molecular Dynamics computer simulations that model explicit stick-slip motion in dry friction between a metal sample and an Atomic Force Microscope metalic tip. This Molecular Dynamics is implemented using the code and visualization tools developed by LAMMPS at Sandia National Laboratories [

The paper is organized as follows: in

This section is divided into three subsections where we describe respectively, the Molecular Dynamics setup, the probability distribution of stick-slip avalanches, and the entropy content of the tip-sample system.

As explained in the introduction, we obtain the motion of individual atoms in the system via the program LAMMPS [

Two-dimensional cartoon of the actual three-dimensional atomic system. The speed of the tip was consistently 1m/s, and the temperature 1K. Data was collected during 1.6 ns with a time resolution of 4 fs. The light gray atoms behave as follows. The lowest three atomic planes (in the slab) were kept fixed. The upper three atomic planes (in the tip) were rigidly moved at constant velocity. The dark gray atoms are externally unrestricted. They move according to the Molecular Dynamic algorithm. For each time step, we collect the instantaneous friction force. The program is run for tips of orientations varying in 1° angular increments to simulate the rearrangement of the apex atoms during sliding.

Using MD as described in

Representative friction force versus time trace. In this case, the result corresponds to a copper tip and a copper sample at T = 1 K, with load 1 nN, tip speed 1 m/s and time resolution 4 fs.

In this case, the avalanches are in the form of stick-slip events of different sizes. We analyze the probability distribution of such events (jumps) by looking at the histograms of jumps. We will see in the Results section, that these probability distributions follow power laws with fractional exponents in all cases, necessary signatures of SOC. Specifically, we test whether:
^{−α}
_{min}, _{max}]. There must be a lower bound _{min}, to avoid divergences of Equation (1), and also because there is a resolution limit in any experiment. The upper bound _{max} is related with the necessarily finite size of the sample, which makes large events very rare and thus renders impossible any statistical analysis in that region. Thus, as is the case with many other power laws in Nature, we must search for indications of Equation (1) in a finite central range of

If SOC is indeed happening, as suggested by the power law probability distributions of the stick-slip jumps, one is immediately prompted to enquire into the nature of the attractor. In other words, in this non-equilibrium system, how this steady state is approached, and what microscopic intuition can be developed. To this end, we monitor the jittering of the atoms during sliding, and ask whether there is a global trend. Specifically, at each instant of time, we calculate the vertical displacement of each atom with respect to a fixed position, in the reference frame moving at the constant velocity equal to the externally applied sliding velocity. Then, at each time step, we compute the entropy content of the whole system. This is done by first constructing the histogram of vertical positions _{i}_{i}_{i}_{i}

Since the system is small, of nanometer dimensions, this entropy measure should provide a spatially localized characterization of the attractor.

We show first the probability of jumps histograms for a copper tip on a copper substrate, and for an aluminum tip on an aluminum substrate for various normal loads.

From the MD friction force traces (such as in _{min}, _{max}] within which the jumps

Histogram in log-log scale of friction force jumps. The probability in the vertical axes is not normalized. Aluminum on aluminum at an external normal load of 1nN. The exponent of the power law, extracted from the blue region of the data is – 1.40.

Power exponents of the stick-slip jump force probability distribution with fractional values in the range (0.5, 1.5).

Load | Material | |
---|---|---|

Aluminum | Copper | |

1.0 nN | 1.40 | 0.64 |

1.5 nN | 0.99 | 1.16 |

2.0 nN | 1.00 | 0.74 |

Histogram in log-log scale of friction force jumps. Aluminum on aluminum at an external normal load of 1.5 nN. The exponent of the power law, extracted from the blue region of the data is – 0.99.

Histogram in log-log scale of friction force jumps. Aluminum on aluminum at an external normal load of 2.0 nN. The exponent of the power law, extracted from the blue region of the data is – 1.00.

Histogram in log-log scale of friction force jumps. Copper on copper at an external normal load of 1.0 nN. The exponent of the power law, extracted from the blue region of the data is – 0.64.

Histogram in log-log scale of friction force jumps. Copper on copper at an external normal load of 1.5nN. The exponent of the power law, extracted from the blue region of the data is – 1.16.

Histogram in log-log scale of friction force jumps. Copper on copper at an external normal load of 2.0nN. The exponent of the power law, extracted from the blue region of the data is – 0.74.

With the indication of Self Organized Criticality in

Entropy of jittering as a function of time. Aluminum on aluminum at an external normal load of 1 nN. The scattered points are the actual values measured, while the continuum line is a moving average to aid the eye see the trend.

Entropy of jittering as a function of time. Aluminum on aluminum at an external normal load of 1.5 nN.

Entropy of jittering as a function of time. Aluminum on aluminum at an external normal load of 2 nN.

Entropy of jittering as a function of time. Copper on copper at an external normal load of 1 nN.

Entropy of jittering as a function of time. Copper on copper at an external normal load of 1.5 nN.

Entropy of jittering as a function of time. Copper on copper at an external normal load of 2 nN.

We have performed Molecular Dynamics computer experiments on atomic clusters to obtain the individual motion of all atoms during stick-slip dry friction motion. We considered aluminum and copper systems under various normal loads and found in all cases power law statistics which strongly suggests Self Organized Criticality. We backed up this suspicion by directly analyzing the jittering of all atoms in the system. This jittering was quantified by measuring its entropy content. In all cases studied, we found that the entropy decreases with time, which directly shows that friction tends to self organize the system. Also, the asymptotic value of the entropy at large times can be used as a label to characterize the attractor in steady state. Molecular Dynamics allows us to monitor the motion of each individual atom, in particular in the attractor state. Although this can be computed, our brain cannot comprehend such vast number of varying coordinates. However, it is as if, in steady state the atoms form a chorus that sings their entropy value, a concept simple to grasp.

Work supported by the Jay and Jeanie Schottenstein Program of Yeshiva University and by the Gamson Fund.

F.Z. conceived and designed the numerical experiments; P.C. performed the numerical experiments; F.Z. wrote the paper; P.C. read and commented on the paper and prepared the figures; P.C. and F.Z. analyzed the data. Both authors have read and approved the final published manuscript.

The authors declare no conflict of interest.