In this work we propose an extension to the analytical one-dimensional model proposed by E. Gnecco (Phys. Rev. Lett. 84:1172) to describe friction. Our model includes normal forces and the dependence with the angular direction of movement in which the object is dragged over a surface. The presence of the normal force in the model allow us to define judiciously the friction coefficient, instead of introducing it as an {\sl a posteriori} concept. We compare the analytical results with molecular dynamics simulations. The simulated model corresponds to a tip sliding over a surface. The tip is simulated as a single particle interacting with a surface through a Lennard-Jones $(6-12)$ potential. The surface is considered as consisting of a regular BCC(001) arrangement of particles interacting with each other through a Lennard-Jones $(6-12)$ potential. We investigate the system under several conditions of velocity, temperature and normal forces. Our analytical results are in very good agreement with those obtained by the simulations and with experimental results from E. Riedo (Phys. Rev. Lett. 91:084502) and Eui-Sung Yoon (Wear 259:1424-1431) as well.
Deep Dive into Velocity, temperature and normal force dependence on friction: An analytical and molecular dynamic study.
In this work we propose an extension to the analytical one-dimensional model proposed by E. Gnecco (Phys. Rev. Lett. 84:1172) to describe friction. Our model includes normal forces and the dependence with the angular direction of movement in which the object is dragged over a surface. The presence of the normal force in the model allow us to define judiciously the friction coefficient, instead of introducing it as an {\sl a posteriori} concept. We compare the analytical results with molecular dynamics simulations. The simulated model corresponds to a tip sliding over a surface. The tip is simulated as a single particle interacting with a surface through a Lennard-Jones $(6-12)$ potential. The surface is considered as consisting of a regular BCC(001) arrangement of particles interacting with each other through a Lennard-Jones $(6-12)$ potential. We investigate the system under several conditions of velocity, temperature and normal forces. Our analytical results are in very good agreemen
Understanding the origin of tribological phenomena is a fascinating and challenging enterprize. The classical point of view of the frictional phenomena, can be synthesized in the three laws of friction, valid in the macroscopic scale [1,2]:
- Friction is independent of the apparent area of contact, 2. Friction is proportional to the applied load. The ratio between the friction force and the applied load is named the coefficient of friction (µ = f L /f N ) and it is larger for static friction than for kinetic friction, 3. Kinetic friction is independent of the relative sliding velocity.
Since new tools, such as the atomic force microscopy (AFM), have made possible to examine the friction phenomenon in great detail these laws have been questioned in systems with dimensions approaching the nanometer scale. At the same time, the development of ultra fast computers have allowed to test new theories on the nano-scale friction world. Although tribology is an old science, and in spite of the efforts and progress made by scientists and engineers in the last years, tribology is still far from being a well-understood subject. In fact, it is incredible that even knowing several properties as surface energy, elastic properties and loss properties, a friction coefficient cannot be found by using an a priori calculation. Although in the macroscopic scale the friction force, f L , is independent of the relative velocity, in the nanometric scale some authors [3,4,5,6] observed that the mean value of the friction force presents a logarithmic velocity dependence. Another important result was the conclusion that friction force is proportional to the effective contact area down to the nanometer scale [7]. An analytical one-dimensional model known as Tomlison model [1] was able to explain several features of the nanoscopic friction. Using the Tomlinson model in the limit of low velocities Gneco et al [3] showed that the friction force has a logarithmic dependence with the velocity. Using the same ideas, but in the limit of higher velocities, Sang et al. [8] obtained that the friction force is proportional to | ln(v)| 2/3 , were v is the relative velocity. Using a first principle model Persson [9] was able to show that in the limit of small contact areas the result of Sang is recovered while in the limit of large contact areas the Gneco result fitted better. The aim in this work is to develop a model from first principle by extending the one-dimensional model proposed by Sang et al. [8] to three-dimensions. Based in our approach we obtain a friction coefficient which can be calculated knowing simple parameters of the model (As bound energies and the positions of the minima between atoms.). Such parameters can be obtained by using ab-initio calculations or measured experimentally by using FFM (Friction force microscopy) [3,5,10,11] or DFS (Dynamic force microscopy) [12]. We study the sliding frictional process by using two approaches:
• Developing an analytic model that considers the potential energy between a atom in the tip and the surface atoms described by the model presented by W. A. Steele [13] as a sum of pair-wise (6 -12) LJ potentials. The analytic treatment extends the one dimensional model proposed by Riedo et al. [4] including the normal force and as a consequence, the effects of adhesion energies.
• Using MD simulations by considering that the potential energy between the tip’s atom and the surface atoms is described as a sum of (6 -12) LJ potentials (See ref. [14] and references there in.). In this section we show a general picture of the nature of the kinetic friction. Consider the sliding system shown in figure (1) in which one particle of mass m is connected through a spring to a cantilever or drive. The particle experiences a total force described by the potential [1]
where
represents the harmonic spring constant complying the cantilever with the tip, V int ( q) is the surface-tip corrugated potential, q = (q x , q y , q z ) are the coordinates of the tip and r = (x, y, z) the coordinates of the support. As we are interested to study the influence of the normal force on the sliding process, let us first note that the critical state (Where we denote the drive position by r c = (x c , y c , z c ) and the particle position by q c = (q xc , q yc , q zc )) is the position where the tip jumps from a stable position in the surface to the next one. To illustrate the occurrence of the critical state, we show in the figure 2 the total potential energy of the tip as a function of q x , for two different positions of the cantilever, x < x c (full line) and x = x c (dashed line). As a matter of clarity we restrict this figure to the x direction.
From figure 2, the critical point is defined as the inflexion point of the total potential energy and mathematically it means,
The first condition, equation (3), is always satisfied at equilibrium, it states that the total force on the particle must vanish. The second conditi
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