# Energy exchange in a two particles system

###### Abstract

The microscopic origin of friction forces is an important topic in science and technology which remains mostly unsolved until date. In an effort to shed some light on the possible mechanisms that could give rise to the macroscopic emergence of friction, we consider a one-dimensional Hamiltonian two particles system: one with initial velocity thrown against another which is confined in a harmonic potential. The two particles interact via a repulsive short-range Gaussian potential. From a classical description and without any ad hoc dissipation term, we study the systematic loss of translational energy of the free particle as a function of the relevant parameters of the model. This model may represent a substrate (the bounded particle)-asperity (the free particle) system. The transfer of energy from the incoming particle to the confined one can be regarded as the emerging friction force of the system. This apparently simple model can not be solved analytically. We analyze numerically how the final/initial velocity relation of the free particle depends on the masses, the elastic constant of the bounded particle, the height, and range of the interaction potential. We propose a theoretical reasoning to quantify the loss of energy of the free particle as a result of the interaction. All in all, we show a range of rich and non-trivial behavior on the effective dissipation that emerges from this simple, classical, and with no ad-hoc term model.

## I Introduction

Energy dissipation is the statistical, macroscopic manifestation of fundamental interactions among particles at the atomic scale. Generally known as friction, considerable progress has been made in the fundamental understanding of its origins Krim (2002); Bennewitz (2005); Buldum and Ciraci (1997) during the last thirty years. However, the detailed microscopic mechanisms that give rise to friction forces is still an open problem. Besides, a very important one from a practical point of view Persson (2000); Krim (2002); Makkonen (2012).

The introduction of new experimental tools made the nanometer and the atomic scales accessible to tribologists, giving rise to the field of nanotribology Holscher et al. (2008). Besides the investigation of surfaces at the atomic scale, an implicit stimulus of nanotribology is the assumption that for a better understanding of friction in macroscopic systems, the frictional behavior of a single-asperity contact should be investigated first. Our hope is that once the atomic-scale manifestations of friction at such a nanometer-sized single asperity have been clarified, macroscopic friction could be explained with the help of statistics.

The dry sliding friction between atomically flat, commensurate or incommensurate sliding surfaces is the simplest but perhaps the most fundamental type of friction in tribology Buldum and Ciraci (1997). It is indeed the simplest and most ubiquitous manifestation at a macroscopic level of the electromagnetic forces between atoms and molecules, but probably the less understood from first principles. In dry friction many interesting and complex physical phenomena are involved, like adhesion, wetting, atom exchange, elastic and plastic deformation. In principle, the understanding of these phenomena would us to control the mechanisms of friction and thus to reduce energy loss.

In the past two or three decades several theoretical models for atomic friction, based on earlier work of Tomlinson Tomlinson (1929) and Frenkel-Kontorova Kontorova and Frenkel (1938); Frenkel and Kontorova (1939); Braun and Kivshar (2004), were proposed. The advantage of such models resides in being simple but yet for retaining enough complexity to exhibit interesting behavior Buldum and Ciraci (1997); Gonçalves et al. (2004); Gonçalves et al. (2005); Fusco and Fasolino (2005); Tiwari et al. (2008); Neide et al. (2010). Such models have displayed non-trivial features, where the dissipated energy and friction force have been revealed as the result of nonlinear mechanisms. However, all those models have an ad-hoc dissipation term explicitly included. Even when the resultant friction is non-trivial, part of them is already set up from the beginning. Our idea is to start from a pure Hamiltonian description without including a dissipation term into it.

In the present contribution, we want to contribute to the understanding of the microscopic mechanisms of energy dissipation between two particles: one representing the substrate and the other representing an asperity of the adsorbate or the tip of the microscope. At the most fundamental and specific level, we expect to observe systematic loss of translational energy from the adsorbate to the vibration of the substrate. We want to check if that is possible starting from a purely classical description, i.e., without the addition of dissipative ad hoc terms.

## Ii Model

The model consists of two particles: a mass that is thrown to move past another one, which is attached to a spring (Fig. 1). The interaction between the particles is not linear and short-ranged. The motion is in one dimension, but the free particle can overcome the bounded one because the interaction potential between them has a finite maximum. As we anticipated, this seemingly unreal situation is a model simplification of a particle sliding on top of another, where we do not take into account transverse movement. The asperity is represented by a single free particle and the substrate by a harmonic oscillator. As shown in Fig. 1 the free particle is propelled from a region where it is no affected by the presence of the oscillator. As it moves it interacts locally with the oscillator. This local interaction is repulsive, with a finite maximum.

The model is described by the Hamiltonian

(1) |

where are the momentum and position of the free particle and the oscillator respectively. The masses of the two are unequal and represented by and respectively, The interaction potential is assumed to be of Gaussian type with a maximum value and equals . The Hamiltonian in Eq. (1) results in the following equations of motion

(2) | |||||

(3) |

and has five parameters. By defining dimensionless position variables and dimensionless time and writing the equations of motion in dimensionless form,

(4) | |||||

(5) |

we can reduce the number of free parameters to two, the ratio of the masses of the two objects , and a dimensionless oscillator frequency . It is therefore important to understand the dynamics of the system in terms of the relative masses and the frequency of oscillation of the oscillator. In the remainder of the paper while we work with Eqs. (2) and (3), we attempt to characterize the system behavior in terms of these two parameters. We would also like to point out that while the Gaussian nature of the potential is able to capture essential features of the interaction, it also makes the problem highly nonlinear, thereby making analytic calculations difficult.

## Iii Results

### iii.1 Rigid Oscillator

We first consider the case of a rigid oscillator. By setting and defining and as the internal and center of mass coordinate respectively in Eqs. (2) and (3) we obtain,

(6) | |||||

(7) |

Eqs. (6) and (7) can also be written in terms of conserved quantities as

(8) | |||||

(9) |

Assuming that the free particle starts far away from the interaction region and the oscillator mass is initially at rest, the initial conditions and give , , , as the initial conditions for the internal and center of mass equation respectively. Using these and the fact that we are interested in the amount of energy transferred it is sufficient to consider the situation when one of the particles has gone very far away from the interaction region. Under this assumption the potential term in Eq. (8) goes to zero and we have

(10) |

Combining Eq. (9) with Eq. (10) and first choosing the plus sign we obtain implying that the first particle continues to move in its original direction without any interaction with the second one. If we choose the negative sign we obtain and as the final velocities of the first and second particle respectively. Therefore, based on the relative values of the two masses and the initial velocity of the free particle there are three situations possible. From Eq. (8) it is easy to see, that, if the initial velocity of the free particle is larger than a critical velocity the former situation prevails under which the first particle moves without any loss of energy. If, however, the free particle’s energy is lower than this critical value then depending on whether its mass is smaller or larger than the mass of the second particle it can either get transmitted or reflected (see Fig. 2 for and ). The case and corresponds to the situation when on collision the first particle would lose its entire energy to the second one (Fig. 2 (Top)).

The particular features in the case will be discussed in the last section.

### iii.2 Non rigid oscillator

The simple relationship between the initial and final velocities can become quite complex as the vibrational degree of freedom is introduced in the the oscillator . In Fig. 3 we explore the dependence of the output velocity of the particle for non rigid oscillator for three different mass ratios and . For cases when the free particles mass is either equal to or more than the oscillator’s mass allowing for very slow oscillations changes the behavior from transmission () or stationary () to reflection for a range of initial velocities. Apart from this there exists an intermediate range of velocities, in which, as the oscillation rate increases the behavior changes drastically.

### iii.3 Soft oscillations

We first consider the case of slow oscillations and in particular try to get an estimate of the resultant velocity when the behavior changes from transmission to reflection as we perturb the system slightly from the rigid oscillator case. In Fig. 4 we show this change in behavior resulting from a slight change in the initial velocity of the free particle. For both the cases the particle interacts with the oscillator twice. However, the outcomes are completely different. In the first the free particle is reflected back with a larger energy as compared to the second where the free particle is transmitted but with lesser energy. Since the two collisions are well separated out we attempt to get an approximate estimate by extending the analysis in the previous section to two collision. It is assumed here that both the free particle and the oscillator are exactly coupled except at the instant when they are almost on top of each other, and that the energy transfer is instantaneous so that it can be described within the framework of the rigid body collision approximation.

Since the initial velocity of the free particle is lower than the critical velocity immediately after the first collision the velocities are

(11) | |||||

(12) |

Consequently, after the first collision the position of the particle and the oscillator are given as:

(13) | |||||

(14) |

The instant of the next collision is then determined by setting , which gives,

(15) |

Transfer of energy between the two is only possible if the the velocity of the internal coordinate at this time instant is less than the critical velocity . Using Eqs. (8), (11) and (14) we obtain the condition on initial velocity for which the particle’s behavior changes from transmission to reflection

(16) |

The instantaneous velocity after the second collision can be obtained from conservation of energy and momentum and is given by,

(17) | |||||

(18) |

here is used to denote the time immediately before and after the collision. We show a comparison between the approximate two collision theory and numerical results in Fig. 5. For equal masses we obtain , which is what is observed through the numerical simulations in Fig. 3. At the first collision the free particle transfers its entire energy to the oscillator which is transferred back completely to it in the second collision. This behavior persists for larger values of the oscillator frequency but the velocity from which the particle behavior changes from transmission to reflection is seen to be a function of the oscillator frequency, which is not being captured by the two collision model. Even in the case of the a heavier free particle the two collision model shows good qualitative agreement (see Fig. 5). It is however unable to provide reasonable estimate of the observed behavior as the oscillator frequency is increased primarily because the collisions can no longer be treated in isolation.

### iii.4 Energy loss calculation

The problem can be seen from a perturbation point of view, in which the free particle is scattered by the fixed one. From the observation that for large initial speeds the minimum of the energy loss occurs at small values of we set this as our perturbation parameter. Then we set in the equations 2 and 3. This gives

(19) |

Where and . For we can expand the Gaussian and keep the terms until the first order in . By expressing

(20) |

we obtain

(21) |

For small values of we can assume

(22) |

the energy conservation equation is

(23) |

The rate at which the free particle loses energy is

(24) |

The force due to the static potential is whose derivative with respect to time is . With this solution, the rate at which energy is being lost become

(25) |

The total energy loss is

(26) |

Assuming that at infinite, which is true for a pulse. From Eq. 21 we have

(27) |

Thus the energy lost by the free particle is

(28) |

Fig. 6 shows the comparison between the exact solution (continuous line) and the energy ratio obtained by numerically solving Eq. 28. F(x) is calculated by taking solution from Eq. 22 for two different initial speeds (top) and (bottom). In both cases the approximate expression shows almost the same minima (more accurate for larger speeds). The values for the energy ratio especially around the minima are different. This difference is expected since the impulse is taken from the unperturbed equations in which there is no energy loss possible. But anyway it’s a good approximation.

### iii.5 Free particle Tunneling

An interesting feature occurs when (see Fig. 7) where exists a gap in the transition between reflection and transmission where the free particle shows an unpredictable and anti-intuitive behavior. i.e., the free particle begins to reflect again after having passed the threshold of minimum energy needed to cross the fixed particle; until finally crosses a second threshold where it begins to cross again (shaded areas of the figure) and remains in this behavior as the initial energy continues to increase.

## Iv Conclusion

Our numerical study shows some expected result like that no energy absorption occurs at large values of the incoming energy. However, and despite the simplicity of the model, some non obvious results are observed too. For a certain range of incoming energy and large losses of energy are observed. The release of energy of the incoming particle to the bounded one, which can be thought as emerging friction, has non trivial relation with the parameters which describe the model, like the mass ratio and spring constant.

A more carefully of systematic study of these preliminary results is necessary, but we already envision that an extension of the simple model to an extended periodic substrate can provide an interesting inside in the macroscopic origin of friction.

## V Acknowledgments

This work was supported by the Centro Latinoamericano de Física (CLAF) and the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Brazil).

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