Kinetic nanofriction: a mechanism transition from quasi-continuous to ballistic-like Brownian regime
© Jafary-Zadeh et al; licensee Springer. 2012
Received: 21 October 2011
Accepted: 21 February 2012
Published: 21 February 2012
Surface diffusion of mobile adsorbates is not only the key to control the rate of dynamical processes on solid surfaces, e.g. epitaxial growth, but also of fundamental importance for recent technological applications, such as nanoscale electro-mechanical, tribological, and surface probing devices. Though several possible regimes of surface diffusion have been suggested, the nanoscale surface Brownian motion, especially in the technologically important low friction regimes, remains largely unexplored. Using molecular dynamics simulations, we show for the first time, that a C60 admolecule on a graphene substrate exhibits two distinct regimes of nanoscale Brownian motion: a quasi-continuous and a ballistic-like. A crossover between these two regimes is realized by changing the temperature of the system. We reveal that the underlying physical origin for this crossover is a mechanism transition of kinetic nanofriction arising from distinctive ways of interaction between the admolecule and the graphene substrate in these two regimes due to the temperature change. Our findings provide insight into surface mass transport and kinetic friction control at the nanoscale.
Atoms, molecules, and nanoparticles are the basic building blocks for many applications in nanotribology and nanomachines including nano-electro-mechanical systems [1–5]. When a bottom-up approach is used, one often has to manipulate these blocks through positioning, packing, and moving them on a surface. Meanwhile, at finite temperature, a building block on a surface may undergo thermally-driven diffusive motion , in which it interacts with its surrounding atoms and experiences kinetic friction. Therefore, there is an intrinsic connection between kinetic friction and surface diffusion at the atomic scale, which has recently attracted considerable attention [7, 8].
Due to the scientific and technological importance of surface diffusion, a great deal of effort has been devoted to understand the microscopic mechanisms by which adsorbates move on a surface . In systems with strong potential energy barriers and at low temperature, surface diffusion occurs through a series of uncorrelated random jumps between neighboring adsorption sites as described by transition-state theory; while at extremely high temperatures, a crossover from the thermal activated jump regime to the high-temperature Brownian motion regime was theoretically described . The Langevin equation (LE) of motion for an isolated adsorbate is a remarkably successful model of surface diffusion. It characterizes diffusion by two phenomenological parameters: the strength of the potential energy barrier (E a ), and the kinetic friction coefficient (η), which indicates the rate at which energy is transferred between the adsorbate and the surface. Supposing independency between E a and η, solutions of the LE has found out four distinct regimes of surface diffusion [9, 11]:
• Regime I (single jumps), where the E a is strong (comparing to the thermal energy (k B T) of the adparticle), and the η is high, the adsorbate mainly resides inside local minima of the potential energy surface (PES). This adsorbate moves by hopping from one minimum to a neighboring minimum. The diffusion rate tends to follow Arrhenius behavior and is well described by the transition-state theory.
• Regime II (multiple jumps), where the E a is strong, and the η is low, hopping with long jumps is the dominant feature. The adsorbate may hop from one minimum to a distant local minimum, flying over sites. In the limit of extremely low friction, the microscopic motion is stick-slip, and the trajectory might be characterized by Lévy flight .
• Regime III (quasi-continuous Brownian motion), where the E a is weak, and the η is high, the adsorbate moves continuously without being confined to a single local minimum of the PES. In this case, the adsorbate motion is similar to that of a Brownian particle in a high-friction liquid .
• Regime IV (ballistic-like Brownian motion), where the E a is weak, and the η is low. The adsorbate moves continuously and travels in linear trajectories at the picosecond time scale, which resembles the directional motion of a projectile. In this case, the adsorbate motion is similar to that of a Brownian particle in a low-friction liquid.
Most of the experimental observations of surface diffusion (especially in the chemisorbed systems with high energy barriers) have been classically described and characterized by the transition-state theory and the single-hop model , though at elevated temperatures, multiple jumps are also observed. Characteristics of extremely long jumps (Lévy flight) were observed in systems like gold-cluster/graphite  and graphene-flake/graphene . Observation of adsorbates undergoing Brownian motion is relatively rare, even though this behavior is theoretically expected at extremely high temperatures . The first observation of high-friction Brownian motion (Regime III) was in the benzene/graphite system . Nevertheless, to the best of our knowledge, we are unaware of any definite observation of ballistic-like Brownian motion (Regime IV) in a realistic system.
Despite a great deal of experimental and theoretical effort that has been devoted to the study of surface diffusion , many issues associated with surface diffusion in the systems with low energy barriers and/or low friction are still not well understood . Therefore, it is necessary to investigate adsorbate/substrate systems with shallow potential energy surface and low friction so as to discover realistic systems which exhibit nanoscale ballistic-like surface Brownian motion. To this end, we considered the C60/graphene system which is important in current nanoscience research [16, 17]. Miura et al. showed that a C60 monolayer between the graphite plates exhibits ultra-low friction. Recently, Neek-Amal et al. studied the diffusion of C60 on graphene at room temperature and reported a shallow potential energy barrier. Hence, from the scientific point of view, the C60/graphene system can be considered as an ideal model system to study basic principles of Brownian motion at the nanoscale, and this system is also a promising candidate to exhibit ballistic-like Brownian motion due to its low energy barriers and low surface friction. From the application point of view, fullerenes have been widely considered as a promising material for superlubricity and nanobearing applications [20, 21].
In the present work, we use molecular dynamics (MD) simulations to study the motion of an isolated C60 molecule on a graphene substrate, focusing on identifying different surface diffusion regimes, their crossover, and their underlying mechanisms. MD simulation is a standard tool to study surface diffusion and have reached a level of accuracy that can often be compared with experimental results . Perhaps more importantly, MD simulations can often be used in scenarios where cannot be reached by experimental techniques. Here, we show, for the first time, that the C60/graphene system not only exhibits both Regimes III and IV, but also reveals a crossover between them by simply increasing the temperature of the system.
Method and model
MD method and interatomic potential
The first term, a slightly modified version of reactive empirical bond order (REBO) , is capable to handle short-range interactions (distance between atoms, r < 2 Å) as well as 3-body and 4-body interactions with nearest neighbor atoms in hydrocarbon systems. The second term takes into account the long-range interactions (2 < r < cutoff) using the standard Lennard-Jones potential with a cutoff of 12 Å. The third term is an explicit 4-body potential that describes various dihedral angles in hydrocarbon system. It is noteworthy that the REBO potential itself is an application of the Tersoff potential [27, 28] in hydrocarbon systems.
In our simulations, the graphene substrate has the dimensions of 100 by 100 Å, consisting of about 3,770 atoms (see Figure 1a). Periodic boundary conditions were applied along the in-plane directions.
Calculation model and data analysis methods
A C60 molecule was positioned on the top of the graphene layer in such a way that one of its hexagon faces was parallel to the graphene. The edges of this hexagon were parallel to the edges of the graphene hexagons, and later on, this configuration is referred as 'Hex.-In Phase'.
In order to study the dynamics of the system, the microcanonical ensemble was selected. The simulations were performed in the temperature range of 5 to 200 K.
At the beginning of each simulation, energy minimization was performed to relax the atomic positions of the system. The Polak-Ribiere version of the conjugate gradient method implemented in the LAMMPS code was used for the energy minimization. After energy minimization, the velocities of the C60 molecule and the graphene atoms were assigned following Maxwell-Boltzmann distribution at the desired temperature.
The time step of the Verlet integration algorithm was chosen as 1 fs. Each trajectory calculation was proceeded by a thermal equilibration of 50,000 integration steps (50 ps), followed by a run for 10 ns to extract data for diffusion and friction analysis.
where < > Nt is the ensemble or time averaging over the trajectories, and X CM is the two-dimensional position vector of the C60 center of mass, t 0 is the time origin, and t is the elapsed time from the time origin.
where m is the mass of C60, we can calculate the friction coefficient of the C60/graphene system over the temperature range from 25 to 200 K.
Results and discussion
Because of the importance of rotational degrees of freedom (DOF) in this system, it is not possible to completely understand the dynamics of the diffusing C60 using only the Langevin model. To this end, we examine the interplay between translational and rotational kinetic energies of C60 molecule during its motion on the surface. Figure 3a, b shows the distinct energy conversion patterns between rotational and translational modes of C60 motion at 50 (in Regime III) and 200 K (in Regime IV), respectively. It can be seen that in Regime III, that is, at the low temperature regime, the energy transfer occurs with a higher frequency comparing to that in Regime IV, that is, the high temperature regime. This pattern suggests that in Regime III, the energy corrugation of the surface (corresponding to the PES) plays an important role in the 'push-pull' of the energy between translational and rotational DOF, and the anti-correlation between these DOF occurs with a higher frequency (see Figure 3a). In contrast, in Regime IV, the overall kinetic energy of the C60 is high compared to the shallow PES, and the C60 receives extra kinetic energy from the high energy thermally excited graphene atoms in the form of instant kicks. In this regime, the role of the PES is negligible in the 'push-pull' of the energy between translational and rotational DOF. When the high speed C60 moves over the graphene surface, it occasionally collides with the surface thermal corrugations. Due to such collisions, the energy is exchanged between the C60 and the graphene, as well as between the C60 translational and rotational DOF. Generally, such collisions do not lie on the C60 center of mass and, thus, create rotational torques. As a result, translational energy is converted into rotational energy. On the other hand, rotating C60 may also hit another thermal bump of the surface and pull kinetic energy back into the translation mode. This process is repeated during motion and exhibits as a clear anti-correlation between translational and rotational kinetic energies of the C60 (see Figure 3b). The anti-correlation between translational and rotational DOF at high temperatures resembles the 'ballistic nanofriction' process recently described by Guerra et al. in the gold-cluster/graphite system . The mechanism of ballistic nanofriction in their work appears to be similar to the mechanism of motion in Regime IV reported here in two ways: first, it also exhibits a clear anti-correlation between rotational and translational kinetic energies; second, their damping mechanism is governed by the thermal corrugation, and not the potential corrugation of the substrate. Nevertheless, there is a fundamental difference between our and their work. In their work, the ballistic regime was achieved by applying a large instantaneous external force to the gold cluster to generate an initial kick. Hence, the gold particle is not in thermal equilibrium, and the linear-response theory and the Einstein's theory of Brownian motion are no longer applicable to their motion.
In the present work, the temperature is limited to 200 K. We have performed simulations with temperatures above 200 K. At these higher temperatures, the C60 molecule receives stronger instant kicks from the graphene through thermal fluctuations. Consequently, the C60 can break the potential energy barrier, causing desorption of the C60 from the graphene.
The contour plot of the energy surface of Figure 4f is presented in Figure 4g, in which a path (indicated by the white arrow) parallel to the  crystallographic direction of the graphene is illustrated. This path indicates a smooth diffusive pathway with a negligible energy barrier of approximately 4 meV. Therefore, one could expect that the trajectories of the C60 molecule must be confined in this minimum energy path. However, this is not the case in the temperature range studied in the current work. According to Figure 4h, i, when a C60 molecule faces an energy barrier, the molecule can overcome it by rotating to another configuration with an even lower barrier. We illustrate such scenario using Figure 4h: the C60 in the Hex.-In Phase orientation may move from point 1 to point 5 along the  direction, where it has to overcome an energy barrier of about 0.024 eV. However, at point 2, it can partially tilt to the Line-In Phase orientation (see Figure 4d) and move to point 3 and then to point 4 by crossing a lower energy barrier. After passing point 4, the C60 can tilt back to the Hex.-In Phase orientation and continue its way along  direction to point 5. Energetically, this whole process is more favorable. Therefore, we conclude that the rotational degrees of freedom of C60 together with its faceted shape offer various possible paths on the graphene substrate with low energy barriers. Consequently, there is no preferable diffusion path for C60 on graphene in Regime III. It should be noted that chemical modification of graphene, which is widely used to control the electronic properties of graphene, may have significant effects on the PES of the C60/graphene system and cause a drastic change in the diffusive behavior of the C60 molecule.
The thermally-induced motion of C60 on the graphene substrate with a shallow potential energy surface was investigated. We found that the C60/graphene system exhibits two distinct regimes of nanoscale surface Brownian motion. For the first one, the C60 molecule exhibits a quasi-continuous Brownian motion (Regime III) in the temperature range of 25-75 K. For the second one, the C60 molecule follows a ballistic-like Brownian motion (Regime IV) at temperatures above 75 K. Moreover, these two regimes of Brownian motion imply the existence of two distinct mechanisms of nanoscale kinetic friction, which are responsible for the exchange of energy between the C60 molecule and the graphene substrate. In Regime III, the PES and the facets of C60 molecule play a dominant role in the exchange of energy between C60 and the substrate. In contrast, the thermal corrugation of the graphene plays a dominant role in Regime IV. The crossover between these two regimes arises from the change in the system temperature. Since there is an intrinsic connection between surface diffusion and kinetic friction at the atomic scale, the present findings not only provide insights into controlling the surface mass transport and nanofriction, but also guidelines for experimentalists to observe and characterize the intriguing diffusive regimes in the C60/graphene system and to explore for new materials for nanoscale electro-mechanical applications.
The authors are grateful to Dr. Mark Jhon for the useful discussions and insightful comments on the manuscript.
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