# Entropy Content During Nanometric Stick-Slip Motion

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Molecular Dynamics

**Figure 1.**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.

#### 2.2. Stick-Slip Statistics

**Figure 2.**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.

^{−α}

_{min}, J

_{max}]. There must be a lower bound J

_{min}, to avoid divergences of Equation (1), and also because there is a resolution limit in any experiment. The upper bound J

_{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 J.

#### 2.3. Tip-Sample Entropy

_{i}(t), where i labels each atom in the system, and this is done at each instant of time. From z

_{i}(t), we produce a histogram of heights from which we obtain the probability π

_{i}(t) of getting z

_{i}at time t. Finally, the entropy is computed thus:

## 3. Results

#### 3.1. Probability of Jumps

_{min}, J

_{max}] within which the jumps J are well characterized by power law probability densities. In Figure 3, Figure 4 and Figure 5 we show these results for an aluminum tip sliding on an aluminum substrate, subjected to normal loads of 1.0, 1.5 and 2.0 nN. The corresponding exponents are (the α in Equation (1)) are 1.40, 0.99, 1.00 respectively. Figure 6, Figure 7 and Figure 8 show similar results, this time for copper. The loads are also 1.0, 1.5 and 2.0 nN, while the corresponding exponents are 0.64, 1.16 and 0.74. This lack of correlation between the exponents and load and material (Table 1) has been noticed and reported in the past [20,21]. Yet, the results are consistent with the existence of power laws in all cases studied which suggests Self Organized Criticality, and begets the question: what is the attractor? That will be addressed in the next subsection.

**Figure 3.**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.

**Table 1.**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 |

**Figure 4.**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.

**Figure 5.**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.

**Figure 6.**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.

**Figure 7.**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.

**Figure 8.**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.

#### 3.2. Entropy of Jittering

**Figure 9.**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.

**Figure 10.**Entropy of jittering as a function of time. Aluminum on aluminum at an external normal load of 1.5 nN.

**Figure 11.**Entropy of jittering as a function of time. Aluminum on aluminum at an external normal load of 2 nN.

**Figure 12.**Entropy of jittering as a function of time. Copper on copper at an external normal load of 1 nN.

**Figure 13.**Entropy of jittering as a function of time. Copper on copper at an external normal load of 1.5 nN.

**Figure 14.**Entropy of jittering as a function of time. Copper on copper at an external normal load of 2 nN.

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Baumberger, T. Dry Friction Dynamics at Low Velocities. In Physics of Sliding Friction; NATO ASI Series; Springer: Dordrecht, The Netherlands, 1996; Volume 311, pp. 1–26. [Google Scholar]
- Bowden, F.P.; Tabor, D. The Friction and Lubrication of Solids; Oxford University: New York, NY, USA, 1950. [Google Scholar]
- Nosonovsky, M.; Mortazavi, V. Friction-Induced Vibrations and Self-Organization; CRC Press: Boca Raton, FL, USA, 2013. [Google Scholar]
- Trabesinger, A. Complexity. Nat. Phys.
**2012**, 8. [Google Scholar] [CrossRef] - Carpick, R.W.; Salmeron, M. Scratching the Surface: Fundamental Investigations of Tribology with Atomic Force Microscopy. Chem. Rev.
**1997**, 97, 1163–1194. [Google Scholar] [CrossRef] - Baykara, M.Z.; Schwarz, U.D. Noncontact atomic force microscopy II. Beilstein J. Nanotechnol.
**2014**, 5, 289–290. [Google Scholar] [CrossRef] [Green Version] - Miller, B.; Krim, J. Quartz Crystal Microbalance (QCM) Applications to Tribology. In Encyclopedia of Tribology; Springer: New York, NY, USA, 2013; pp. 2727–2733. [Google Scholar]
- Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, NY, USA, 2011. [Google Scholar]
- Gottlieb, S.; Sterling, T. Exascale Computing. Comput. Sci. Eng.
**2013**, 15, 12–15. [Google Scholar] - Plimpton, S.J.; Thompson, A.P. Computational Aspects of Many-body Potentials. MRS Bull.
**2012**, 37, 513–521. [Google Scholar] [CrossRef] - Becker, C.A.; Tavazza, F.; Trautt, Z.T.; Buarque de Macedo, R.A. Considerations for choosing and using force fields and interatomic potentials in materials science and engineering. Curr. Opin. Solid State Mater. Sci.
**2013**, 17, 277–283. [Google Scholar] [CrossRef] - Carlson, J.N.; Langer, J.S. Properties of earthquakes generated by fault dynamics. Phys. Rev. Lett.
**1989**, 62, 2632–2635. [Google Scholar] [CrossRef] - Carlson, J.M.; Langer, J.S.; Shaw, B.E. Dynamics of earthquake faults. Rev. Mod. Phys.
**1994**, 66, 657–670. [Google Scholar] [CrossRef] - Burridge, R.; Knopoff, L. Model and theoretical seismicity. Bull. Seismol. Soc. Am.
**1967**, 57, 3411–3471. [Google Scholar] - Carlson, J.N.; Langer, J.S.; Shaw, B.E.; Tang, C. Intrinsic properties of a Burridge-Knopoff model of an earthquake fault. Phys. Rev. A
**1991**, 44, 884–897. [Google Scholar] [CrossRef] - Fox-Rabinovich, G.S.; Gershman, I.S.; Yamamoto, K.; Biksa, A.; Veldhuis, S.C.; Beake, B.D.; Kovalev, A.I. Self-Organization during Friction in Complex Surface Engineered Tribosystems. Entropy
**2010**, 12, 275–288. [Google Scholar] [CrossRef] - Kagan, E. Turing Systems, Entropy, and Kinetic Models for Self-Healing Surfaces. Entropy
**2010**, 12, 554–569. [Google Scholar] [CrossRef] - Nosonovsky, M. Entropy in Tribology: in the Search for Applications. Entropy
**2010**, 12, 1345–1390. [Google Scholar] [CrossRef] - LAMMPS Molecular Dynamics Simulator. Available online: http://lammps.sandia.gov/ (accessed on 28 May 2014).
- Zypman, F.; Ferrante, J.; Jansen, M.; Scanlon, K.; Abel, P. Evidence of self-organized criticality in dry sliding friction. J. Phys. Cond. Matt. Lett.
**2003**, 15. [Google Scholar] [CrossRef] - Adler, M.; Ferrante, J.; Schilowitz, A.; Yablon, D.; Zypman, F. Self-organized criticality in nanotribology. In MRS Proceedings; Cambridge University Press: Cambridge, UK, 2003; Volume 782. [Google Scholar]
- Buldyrev, J.; Ferrante, F.; Zypman, F. Dry friction avalanches: Experiment and theory. Phys. Rev. E
**2006**, 74, 066110. [Google Scholar] [CrossRef] - Fleurquin, P.; Fort, H.; Kornbluth, M.; Sandler, R.; Segall, M.; Zypman, F. Negentropy generation and fractality in dry friction of polished surfaces. Entropy
**2010**, 12, 480–489. [Google Scholar] - Schneider, T.; Stoll, E. Molecular-dynamics study of a three-dimensional one-component model for distortive phase transitions. Phys. Rev. B
**1978**, 17, 1302–1323. [Google Scholar] [CrossRef] - Martens, C.C. Qualitative dynamics of generalized Langevin equations and the theory of chemical reaction rates. J. Chem. Phys.
**2002**, 116, 2516–2528. [Google Scholar] - Wang, J. Quantum Thermal Transport from Classical Molecular Dynamics. Phys. Rev. Lett.
**2007**, 99, 160601. [Google Scholar] [CrossRef] - Kantorovich, L. Generalized Langevin equation for solids I. Rigorous derivation and main properties. Phys. Rev. B
**2008**, 78, 094304. [Google Scholar] [CrossRef] - Martyna, G.J.; Tobias, D.J.; Klein, M.L. Constant pressure molecular dynamics algorithms. J. Chem. Phys.
**1994**, 101, 4177–4189. [Google Scholar] - Feller, S.E.; Zhang, Y.; Pastor, R.W.; Brooks, B.R. Constant pressure molecular dynamics simulation: The Langevin piston method. J. Chem. Phys.
**1995**, 103, 4613–4621. [Google Scholar] [CrossRef] - Daw, M.S.; Baskes, M. Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B
**1984**, 29, 6443–6453. [Google Scholar] [CrossRef] - Plimpton, S.J. Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] - Clauset, A.; Shalizi, C.R.; Newman, M.E.J. Data Analysis, Statistics and Probability. SIAM Rev.
**2009**, 51, 661–703. [Google Scholar] [CrossRef]

© 2014 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Creeger, P.; Zypman, F.
Entropy Content During Nanometric Stick-Slip Motion. *Entropy* **2014**, *16*, 3062-3073.
https://doi.org/10.3390/e16063062

**AMA Style**

Creeger P, Zypman F.
Entropy Content During Nanometric Stick-Slip Motion. *Entropy*. 2014; 16(6):3062-3073.
https://doi.org/10.3390/e16063062

**Chicago/Turabian Style**

Creeger, Paul, and Fredy Zypman.
2014. "Entropy Content During Nanometric Stick-Slip Motion" *Entropy* 16, no. 6: 3062-3073.
https://doi.org/10.3390/e16063062