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A close look at the motion of C60 on gold
Current Applied Physics 15 (2015) 1402e1411
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Current Applied Physics journal homepage: www.elsevier.com/locate/cap
A close look at the motion of C60 on gold Hossein Nejat Pishkenari*, Alireza Nemati, Ali Meghdari, Saeed Sohrabpour Center of Excellence in Design, Robotics and Automation (CEDRA), Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 May 2015 Received in revised form 4 August 2015 Accepted 4 August 2015 Available online 7 August 2015
In this paper, we have studied the motion of buckminsterfullerene (C60) on a gold surface by analyzing its potential energy and using classical molecular dynamics method. The results can be employed to investigate the motion of C60-based nanocars which have been made in recent years. For this purpose, we have studied the translational and rotational motions of C60 molecule independently. First, we have calculated the potential energy of a C60 molecule on a gold surface in different orientations and positions and employed this data to predict fullerene motion by examining its potential energy. Then we have simulated the motion of C60 at different temperatures using classical molecular dynamics methods. Specifying the regime of the motion at different temperatures is one of main goals of this paper. We have found that the rotational motion of C60 molecule on the gold substrate, was easier than its sliding (translational) motion. Also, the regime of motion of fullerene depended on temperature. The results demonstrate that three different regimes of motion, dependent on temperature, could be observed: rare jumps to adjacent cells, frequent jumps, and continuous motion. Employing the results of this paper not only helps to understand the C60 motion on the gold surface but also provides an appropriate tool for realizing motion of the thermally-driven fullerene-based nanocars. © 2015 Elsevier B.V. All rights reserved.
Keywords: Nanocar Fullerene motion Potential energy Molecular dynamics Diffusive motion
1. Introduction With the rapid development of nanorobotics, manipulation of nano-scale materials is being increasingly attractive for different technological purposes. Despite development of different transportation techniques for the nano-sized particles, these methods were poorly efficient and corresponding handling was not appropriate [1]. First, almost all of the designed nano-manipulators are several orders of magnitude larger than their payload [2]. Second, current designs can only carry few atoms or molecules simultaneously [1,2]. This is in contrast with the performance of natural nano-manipulators. In nature, atoms and molecules are transported by molecules of the same order of magnitude or even smaller than themselves. For example, Kinesin is a small protein, but it is able to transport relatively huge payloads [3]. Inspired by the natural molecular machines, several attempts have been made to synthesize nano-manipulator [4]. In recent years, James Tour et al. have synthesized several molecular machines aimed at transportation of nano-scale materials. Because of the similarity of their appearance to regular vehicles,
* Corresponding author. E-mail address: [email protected] (H. Nejat Pishkenari). http://dx.doi.org/10.1016/j.cap.2015.08.003 1567-1739/© 2015 Elsevier B.V. All rights reserved.
these machines have been called nanocar, nanotruck, etc. [2,5e10]. Due to their small sizes, a large number of nanocars can be used simultaneously to effectively transport payloads. Several types of nanocars have been made, differing in shape and number of wheels. Fullerene, p-carborane, and trans-[Ru(CeCH)2 (dppe)2] have been utilized as wheel in fabrication of nanocars [2,5,6,10,11]. To effectively use nanocars for manipulation of molecular payloads, their motion in different conditions should be studied. The motion of this class of nano-machines could be determined using experimental or analytical methods. Several types of nanocars have been imaged experimentally using scanning tunneling microscopy (STM) [12,13]. As a result, most available experimental data represents the motion of nanocars on gold surfaces. The most important experimental studies were performed by Shirai et al. [14] and Zhang et al. [13]. Though the experimental method has serious drawbacks. Beside their high cost, the imaging tools can't reveal the details of nanocars motion [15]. It is worthy to note that because of its stability and conductivity, most imaging experiments have been performed on gold surfaces. Although recently a significant progress has been made in the synthesis of different nanocars, there are only a few studies attempting to simulate and analyze the motion of nanocars. Akimov et al. [16] and Konyukhov et al. [17] simulated fullerene-based nanocars using coarse-grained molecular dynamics method. In
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their simulations, the body and the wheels of nanocars were assumed to be rigid. Also some nanocars with carborane and adamantine wheels were simulated by Kolomeisky group [18]. The first generation of synthesized nanocars were fullerenebased cars. In order to better understand the motion of fullerenebased nanocars, the motion of C60 should be studied first. Motion of C60 on graphene has been previously studied by Ejtehadi [19,20], Savin [21,22], and Jafary-Zadeh et al. [23]. In a different research Cuberes et al. [24] repositioned C60 molecules on a Cu step using the tip of an STM by controlling the movements along the step direction. Also forced motion of C60 using AFM tip on silicon was studied by Martsinovich [25] and one of recent experimental imaging of C60 by AFM has been studied by Pawlak in 2012 [26]. There are some studies about absorption of C60 on gold surface and activation energy of C60 on gold surface by Teobaldi et al. and Baxter et al. respectively [27,28]. A more closely related work is the one of Akimov et al. [29] who modeled fullerene-based nanotruck and studied charge transfer between C60 and gold substrate. Despite the progress made in the study of C60 motion, the motion regime of C60 on the gold surface, which plays a significant role in the motion of nanocars on gold substrate, is not investigated in a satisfactory manner. In this paper, we have studied the motion of C60 on a gold surface using two approaches. The first approach was calculating the potential energy of the C60 molecule and then analyzing its variation during motion. In this method potential energy of C60 was calculated in different positions and orientations and the probable motion was predicted in different conditions based on the obtained results. The second approach was using the classical molecular dynamics method. To this aim, thermallyinduced motion of C60 has been studied at different temperatures. Using the simulation results of C60 motion, may help us to predict the motion of fullerene-based nanocars in different conditions. The rest of the paper is organized as follows. In Section 2, the potential energy method will be described. Section 3 is devoted to molecular dynamics approach where more details of the C60 motion are provided and coincidence of results between two approaches are discussed. In the last section of the paper, the main conclusions are presented. 2. Potential energy approach In this section we studied the motion of C60 through analyzing variation of its potential energy. It should be noted that the potential energy of interaction between C60 and gold substrate depends on its orientation. We calculated the potential energy and
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studied its variation in four major orientations. These orientations are described below and illustrated in Fig. 1. C60 resting on a hexagon face, henceforth called Hexa-down C60 resting on a pentagon face, henceforth called Penta-down C60 resting on the edge between two hexagons faces, henceforth called Hexa-Hexa-down C60 resting on the edge between a pentagon face and a hexagon face, henceforth called Penta-Hexa-down In addition to thermal oscillations of atoms, C60 molecule rotates and slides on the surface. We have studied the rotational and translational (sliding) motions of C60, as the two main modes of C60 motion, separately. Using the results obtained by potential energy analysis, we can determine the relative ease of each mode of motion and then predict the dominant mode in different conditions. First, we focused on the sliding motion of C60. To calculate its potential energy, C60 was moved horizontally on the surface. At every position, the height of center of mass was adjusted in order to minimize its potential energy. To find the minimum of the potential energy, the line search algorithm was employed. For this purpose, minimum of the energy was computed with precision 0.01 eV/mol. It should be noted that the orientation of C60 was kept steady during this process. The potential energy of C60 on the gold surface was separately calculated for each of the four aforementioned orientations. Since C60 is assumed to be rigid, the potential energy due to the internal interactions of carbon would be constant during the simulation. Therefore only the potential energy of carbonegold interactions was considered. The Lennard-Jones potential, with the following form, was used to simulate carbonegold interactions and the potential parameters are set as ε ¼ 0.01273 eV and s ¼ 2.9943 Å [30,31]:
s 12 s6 ELJ ¼ 4ε r r
r < rcutoff
(1)
Also the cut-off radius was set to rcut-off ¼ 13 Å (rcut-off > 4 s). Furthermore, the size of the gold substrate was assumed to be sufficiently large in the horizontal directions and the thickness was set to 3 times the lattice constant or 12.23 Å. In Fig. 2, the potential energy of C60 during its sliding motion on the gold surface is illustrated for the specified orientations. As shown in Fig. 2a and b, the variation of potential energy of fullerene during sliding in the Hexa-Down orientation is more than in the Penta-Down orientation. In Fig. 3, potential energy plots for fullerene in the Hexa-Down and Penta-Down orientations are overlaid. As shown in Fig. 3 the potential energy of C60 in the Penta-Down orientation is always
Fig. 1. Orientations of C60 on gold surface (bottom view). (a) C60 on a hexagonal face (b) C60 on a pentagonal face (c) C60 on a bond between a pentagonal and a hexagonal faces (d) C60 on a bond between two adjacent hexagonal faces. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. Potential energy of fullerene during sliding motion. (a) C60 on the pentagon face. (b) C60 on hexagon face. (c) C60 on a bond between a hexagon and a pentagon face. (d) C60 on a bond between a hexagon and another hexagon face. As it can be seen, variations of energy as well as absolute value of energy are the smallest when fullerene is resting on a pentagon face.
greater than its energy in the Hexa-Down orientation at the same horizontal position. It can therefore be concluded that C60 is more stable in the Hexa-down orientation at any position. At lower temperatures, where C60 molecules have lower energy levels, they are more likely to be in Hexa-Down orientation. As can be seen in Fig. 3, C60 tends to be in the Hexa-down orientation during the sliding motion and its orientation tends to remain the same, because during sliding in the Hexa-Down orientation, C60 would reach a lower energy level relative to the Penta-Down orientation. Next, we studied the rotation of fullerene molecules using potential energy analysis. The rotation axis of C60 may be parallel with or normal to the gold surface. These two types of rotation were examined separately. Two kinds of parallel-axis rotations can be imagined: a) from the Penta-Down to Hexa-Down orientation or vice versa from the Hexa-Down to another Hexa-Down orientation. We first studied rotation of C60 from the Penta-Down to HexaDown orientation. C60 was seated in the Penta-Down orientation
Fig. 3. Variation of C60 potential energy during sliding motion in Hexa-Down and Penta-Down orientations.
and rotated slowly around a horizontal axis until it reached the Hexa-Down orientation. Employing line search algorithm, the height of the center of mass was adjusted in each step to minimize its potential energy while the horizontal position was kept constant. The potential energy of C60 as a function of rotation angle is depicted in Fig. 4. The variation of potential energy depends on the lateral position of C60 relative to the crystalline structure of the gold substrate. To examine the effect of fullerene position on the potential energy, three different positions of the surface are considered. The selected points are shown in Fig. 4. As shown in Fig. 4, the variation of potential energy completely depends on the lateral position of C60. It is worthy to note that the
Fig. 4. Variation of C60 potential energy when rotating from the Penta-Down to HexaDown orientation. Potential energy was calculated in three different positions as shown in Fig. 4. At point 1, potential energy ascends at first and then it descends. When C60 is at point 2, potential energy falls and then rises to its final value. Variation of potential energy is the least when C60 is resting on point 3.
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potential energy is always lower in the Hexa-Down orientation compared to the Penta-Down orientation. At some points, e.g. point 1, C60 should pass an energy barrier during rotation. As a result, at these points, the Penta-down orientation is locally stable configuration but not globally stable one; thus a C60 molecule tends not to change its orientation from Penta-Down to Hexa-Down at these points despite the eventual decrease in energy. At some other points, e.g. point 3 during the rotation form the Penta-Down to Hexa-down orientation, potential energy of C60 decreases sharply at first but subsequently increases to its final value. At these points, the Penta-Down orientation is an unstable configuration for fullerene molecule; C60 begins to rotate but due to loss of energy, it may not reach the Hexa-Down orientation and will stay in the middle state. At some other points, e.g. point 2, potential energy of C60 does not fluctuate considerably during rotation. The amount of energy necessary for rotation at these points is relatively small, thus rotating is more probable. The second possible type of rotation is changing the orientation from a Hexa-Down to another Hexa-Down orientation. In this type of motion, fullerene is initially placed in the Hexa-Down position and rotates around a bond to reach another Hexa-Down orientation. The height of center of mass is adjusted at each step to minimize the potential energy, while the lateral position of C60, its position on the xey plane, is kept constant. Potential energy was calculated during rotation for the three aforementioned positions (see Fig. 5). It should be noted that in this type of motion, the initial and final values of the energy are the same, since fullerene will be on the Hexa-Down orientation in both states. As shown in Fig. 6, the variation of potential energy depends on the position. At some points e.g. point 1, C60 should pass an energy peak while at some other points e.g. point 2, the energy variation is small. On the other hand, at some points e.g. point 3, the potential energy may decrease before reaching its final value. By further inspection, it can be proved that in this condition, fullerene is seated on the Hexa-Hexa-Down orientation. In Fig. 6 potential energy in Hexa-Down and Hexa-Hexa-Down orientations are plotted. At some points e.g. Point 3, as shown in Fig. 6, the Hexa-HexaDown orientation has lower energy, therefore it is more stable than the Hexa-Down orientation. In such positions, potential energy would decrease before increasing to its final value during rotation from a Hexa-Down to another Hexa-Down orientation. At some other points, e.g. point 1, Hexa-Down orientation has lower energy and it is more stable. The potential energy would increase
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Fig. 6. Variation of C60 potential energy during translational motion in Hexa-Down and Hexa-Hexa-down orientations. As it can be seen, in some points, potential energy of fullerene in Hexa-Down orientation is less than its value in Hexa-Hexa-Down one. The points which potential energy of C60 is equal at both orientations, are an appropriate (or maybe the best) position for rotating from a Hexa-Down to adjacent Hexa-Down.
and then decrease during rotation at these points. If fullerene is placed at a position where the potential energy is equal in the Hexa-Hexa-Down and Hexa-Down orientations, the variation in potential energy during rotation would be minimal. Finally, we studied the potential energy variations during rotation of C60 around the normal axis. Similar to the parallel-axis rotation, variation of the potential energy depends on the orientation and position of the C60 molecule. Fig. 7 shows the potential energy variation for the Hexa-Down and Penta-Down orientations at three aforementioned points. As shown in Figs. 4, 5 and 7, the energy variations due to rotation highly depends on C60 position on the gold substrate. The rotation is more likely to happen at the points in which the energy variation is small. Considering the results illustrated in Figs. 4 and 5, rotation of fullerene around horizontal axis mainly consists of changing the orientation from one Hexa-Down to another Hexa-Down. Since potential energy variation is smaller for the rotational motion, it can be predicted that rotation occurs easier than sliding motion. Based on the data shown in Fig. 8, it is evident that the variation of potential energy is relatively small during its rotation around the vertical axis. Thus, this type of motion occurs easier and more frequently than sliding motion and other types of rotation. Potential energy analysis is a powerful tool which enables us to predict the motion of C60 for different substrate conditions. We can use this method to predict the motion when the substrate is not smooth. This approach may help us better understand how C60 behaves near steps, vacancies, or even grain boundaries. Totally from the energy analysis approach the following conclusions may be deduced: Since potential energy variation is smaller for the rotational motion, it can be predicted that generally, rotation occurs easier than translation. Rotation around the vertical axis occurs easier and more frequently than other types of rotation. The probability of rotational motion is higher in positions with small variations of energy with respect to rotation.
3. Molecular dynamics approach Fig. 5. Variation of C60 potential energy while rotating from a Hexa-Down to an adjacent Hexa-Down orientation. Similar to rotation form a Penta-Down orientation to Hexa-Down orientation at point 1 potential energy ascends firstly and then it follows a descending behavior. In contrary with point 1, at point 2 variation of potential energy has a descendingeascending behavior. Also it is worthy to note that, variation of the potential energy is the least at point 3 compared to two other mentioned points.
In the second section of this paper, we have studied the motion of fullerene molecules on a gold surface using the classical molecular dynamics method. We have simulated the motion of C60 at different temperatures to demonstrate the effect of temperature on
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Fig. 7. Variation of C60 potential energy during rotation around vertical axis. As it is shown, variation of potential energy is very small during rotation around vertical axis. Plot b shows that the variation of potential of C60 is negligible during rotation around vertical axis when C60 is resting on a pentagon face. So we can prognosticate that rotation of C60 around vertical axis can easily happen especially when fullerene is resting on the Penta-Down orientation.
the regime of motion. The substrate is a 20 a 20 a 3 a cubic lattice having an FCC lattice structure made up of gold atoms, where a is the lattice constant of Au. The plane direction of the gold surface is set to be (001) with respect to the FCC crystalline direction. The lowest layer was assumed to be rigid, and the fullerene molecule was placed on top of the substrate. Periodic boundary conditions was used in the x and y directions. e Hoover thermostats were applied separately to the Nose substrate and C60 in order to control the temperature. AIREBO potential was used to simulate interactions of the carbon atoms in C60. Additionally, EAM potential was used for modeling the interactions among the gold atoms. The system was simulated for 8 ns. To achieve sufficiently precise results, the time step was set to 1 fs. The system was relaxed for 200,000 steps before simulation began. Simulations were performed for different temperatures ranging from 5 K to 600 K. In order to accurately model the substrate, the correct value for the lattice constant should be used. We have studied the effects of the variation of lattice constant due to temperature on the simulation results. Based on our analysis (for more details one can refer to Appendix A), the lattice constant for gold substrate is set to be 4.078 Å.
Fig. 8. C60 on gold substrate. Size of substrate was 20 a 20 a 3 a with plane direction of (001) with respect to the FCC crystalline. Boundary condition in x and y direction was periodic. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Simulation were conducted using LAMMPS software [32] and the results are visualized using VMD package [36]. AIREBO potential [33,34] is used to model the interactions among carbon atoms while EAM potential [35] is employed to model the interactions among Au atoms. It should be mentioned that both Lennard-Jones and torsion terms in the AIREBO potential are considered in our simulations. As a brief, potentials used in our simulations are listed in Table 1. As mentioned before gold surface is Au (100). We have used the NoseeHoover thermostat [37,38], to control the temperatures of the fullerene particle and gold substrate. The equations of motion are integrated employing the velocity Verlet algorithm with a timestep of 1 fs. Also for NVT ensemble, in the LAMMPS software, Tdamp parameter is set as 50 fs [39]. After running the simulations, the trajectory of fullerene molecules at different temperatures is plotted in Fig. 9. As shown in Fig. 9, the regime of motion strongly depends on the temperature. For temperatures lower than 35 K, fullerene is almost immobile. By rising the temperature to 35 K, C60 starts to move by jumping to adjacent cells. The motion at this temperature consists of occasional jumps, and thus is not continuous. The regime of motion remains the same until the temperature reaches 100 K. Further increasing the temperature would change the regime of motion from occasional jumps to frequent jumps. The frequency of jumps increases with temperature, thus fullerene could travel further on the plane. At temperatures higher than 150 K, the regime of motion becomes semi-continuous and its speed increases as the temperature rises. The continuous motion of fullerene at temperatures above 400 K is shown in Fig. 9d. At low temperatures, potential energy analysis can be used to accurately predict when the regime of motion changes. The potential energy of the C60 molecule due to the interactions between gold and carbon atoms was calculated during the simulation. Fig. 10 shows the mean and maximum potential energy as a function of the temperature. Based on the potential energy analysis elaborated in the previous section, it was determined that the potential energy of C60 should reach 0.985 eV before it could hop to an adjacent cell. Results from the molecular dynamics simulation reveal that fullerene reaches the required energy level at approximately 35 K (see Fig. 10). This conclusion demonstrates that potential energy analysis results are in good agreement with molecular dynamics simulations. Fig. 11 shows how often fullerene has enough energy to jump to an adjacent cell. It should be noted that having enough energy does not guarantee the occurrence of a jump, because C60 may not have
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Table 1 Parameters and potentials used in MD setup. Elements
Potential
CeAu AueAu CeC
Lennard-Jones (ε ¼ 0.01273 eV, s ¼ 2.9943, rcut ¼ 13 Å) EAM/Alloy [35] AIREBO (considering Lennard-Jones and torsion terms ε ¼ 0.0028 eV, s ¼ 3.4, rcut ¼ 8.5 Å) [33,34]
Fig. 9. Trajectory of fullerene on gold substrate at different temperatures for 8 ns simulation. a) C60 is almost stationary at 20 K and lower temperatures. b) C60 have frequent jumps to adjacent cells and the number of jumps increases at higher temperatures. c) Frequent jumps of C60 converted to smooth motion at 150 K and higher temperatures. d) Fullerene has a free continuous motion on the gold substrate at high temperatures.
enough horizontal speed to move to the adjacent cell. It is also evident that below 35 K, fullerene could hardly ever reach the energy level required to jump. The probability of reaching the energy level required to perform a jump increases with the temperature. At 75 K, fullerene has enough energy to move to an adjacent cell only 0.8% of the times, while at 300 K it does so approximately 51% of the times. It should be noted that a powerful jump for C60 is a jump in which potential energy between C60 and gold substrate can
Fig. 10. Average and maximum potential energy of C60 as a function of the temperature.
reach to the certain level of 0.985 eV. Potential energy analysis showed that a rotating motion around the vertical axis requires less energy than other modes of motion, thus the probability of its occurrence is higher. Fig. 12 shows the angular speed of C60 around the vertical axis at 5 K. In our
Fig. 11. Percentage of powerful jumps as a function of the temperature. The inset to the right shows a zoomed in view of the main plot for low temperatures. Number of fullerene powerful jumps increases with increasing temperature. Below 35 K temperature, C60 rarely reach the energy level required for moving to neighboring cells. It can be predicted that jumping of C60 to adjacent cells is nearly impossible below 35 K.
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Alternative parameter for describing motion of C60 is Mean Square Displacement (MSD). It could be used to describe the diffusive motion of fullerene. Then the diffusion coefficient, which is one of the best parameters to describe the random motion of C60, can be derived from the MSD. The Mean Square Displacement MSD is defined as [40]:
MSDðtÞ ¼ < jrðtÞ rð0Þj2 >
(2)
where r is the position vector of fullerene center of mass and <> represents averaging over all of particles. Diffusion constant D is defined as the slope of the MSD plot:
1 < jrðtÞ rð0Þj2 > bt
Fig. 12. Angular velocity around the vertical axis and yaw angle of fullerene at 5 K. It has been shown that C60 freely rotates around vertical axis at 5 K.
D ¼ lim
simulations, even at extremely low temperatures e.g. 5 K, fullerene spun easily around the vertical axis. As shown in Fig. 12, C60 rotated freely for a relatively long period of time between t ¼ 1 ns and t ¼ 2 ns. This is in perfect agreement with potential energy analysis results. This phenomenon can be used in design and fabrication of fullerene-based nano-bearings. As we mentioned earlier, potential energy analysis has an important role in predicting the motion on different substrates. If the substrate has defects e.g. steps, vacancies or dislocations, the energy level required for the occurrence of motion could be calculated and the temperature threshold of motion could be predicted by computing the percentage of powerful jumps, as demonstrated in Fig. 11. Potential energy analysis is one of the fastest and most practical methods for predicting the mode and regime of motion. In order to better understand the motion of fullerene on the gold substrate, the motion should be quantified. The mean speed of C60 at different temperatures might provide a better insight of the C60 motion. For this purpose, we have plotted the horizontal speed of the fullerene as a function of temperature. The horizontal root mean square speed of the fullerene molecule obtained from molecular dynamics simulations and the equipartition theorem are plotted in Fig. 13a. Fig. 13a demonstrates that the molecular dynamics results are appropriately coincident with the equipartition theorem (for more discussion on the equipartition theorem see Appendix B). As shown in Fig. 13b, similarly for rotational motion of C60, simulation results are in a good agreement with equipartition theorem. It should be noted that, the variation of vrms as a function of temperature did not provide extra information about the motion regime of C60.
where b is 6 for three-dimensional and 4 for two-dimensional motion. In this paper we are interested to study the horizontal (two-dimensional) motion of fullerene on the gold surface, thus we ignored the vibrations of fullerene in the vertical direction. To have accurate estimation of MSD, we should conduct numerous numbers of different simulations and then averaging should be done considering trajectory of all of them. Alternative method is conducting a relatively long time simulation and dividing the whole of trajectory to many individual shorter-time trajectories. Using the second approach, we have conducted a simulation for 8 ns and divided the whole trajectory of fullerene motion to sixty smaller trajectories. The mean square displacement of fullerene motion at different temperatures is plotted in Fig. 14. The MSD growth rate increases with the temperature (see Fig. 14). At low temperatures, below 35 K, the MSD increases nonlinearly with time. But at medium and high temperatures, the MSD grows nearly linear. Form Fig. 14a, we can see when the temperature rises from 20 to 35 K, a significant difference in MSD diagram will happen showing that the regime of fullerene motion changes when the temperature reaches 35 K. In addition, form Fig. 14b we can find that another change in the motion regime when the temperature rises from 100 K to 150 K. Considering Fig. 14c, it can be deduced that the fullerene motion increases nearly monotonic with temperature rising. Study of the diffusion coefficient may provide useful insight into understanding of the fullerene motion. The diffusion coefficient could be calculated using the MSD diagrams. Arrhenius analysis of the diffusion coefficient helps us better understand the changes in the motion regime of C60. The effect of temperature on the diffusion coefficient is plotted in Fig. 15. The slope of the diagram changes at 20 K and 100 K. Thus, the motion regime can be expected to change at these temperatures.
t/∞
Fig. 13. RMS of (a) C60 speed and (b) rotational speed as a function of the temperature.
(3)
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Fig. 14. The MSD of C60 at different temperatures. a) MSD of fullerene for temperature range 5K T 50 K. As shown in Fig. 15a, when temperature rises to 35 K, a distinct behavior in MSD curve can be observed. b) MSD of fullerene for temperature range 75 K T 200 K. As it can be seen from this figure, MSD curves have different behaviors when the temperature rises from 100 K to 150 K. c) MSD of C60 for temperature range 300 K T 600 K. Growth rate of MSD curve increases with temperature rising.
Fig. 15 depicts that diagram ln(D) has a minimum slope for temperatures below 20 K. Therefore in this region, motion of C60 is not sensitive to the temperature change. For temperatures between 20 K and 100 K, the slope of the diagram ln(D) has increased slightly. This increase of slope is a sign for change of the motion regime revealing more sensitivity of C60 motion to the temperature change. Raising temperature to more than 100 K makes steeper the slope of the diagram. In this range of temperature, C60 is almost free to move on the surface and ability of motion increases sharply relative to the temperature. We are highly interested in the displacement of fullerene on a gold surface, not its vibrations while trapped in a cell. For our purpose, it was more useful to count the number of lattice
constants traveled instead of mean speed. This assured us that vibrations did not enter our measurements. The rate of jumps as a function of temperature is simulated and the results are shown in Fig. 16. Using the above plot, alteration of motion regime at temperatures above 100 K can be explained. At temperatures below 100 K, fullerene may jump to adjacent cells and the rate of jumps increases with the temperature. However, this plot cannot predict changes in the motion regime below 100 K.When temperature approaches 100 K, the motion regime changes and allows molecules to jump multiple cells at once. The slope of the plot increases as a result of this change. When temperature reaches 200 K, the regime of the motion changes to a continuous motion and the speed of motion
Fig. 15. Arrhenius analysis of the diffusion coefficient of C60.
Fig. 16. Rate of C60 jumps vs. temperature.
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continued to increase with temperature. Like other computational investigations, the study presented here involves some restrictions that require attention in order to understand its applicability. As it was mentioned in the introduction section, Akimov et al. [29] studied charge transfer between gold substrate and C60. Considering the result achieved in that paper, charge transfer increases the activation energy for motion of C60. So it may be predicted that charge transfer restricts motion of C60 and reduces slightly diffusivity of C60 motion. Considering complex motion of C60 on the gold surface, it is difficult to take into account the charge transfer between gold and carbon atoms during molecular dynamics simulations. To increase accuracy in the simulation, one can consider a secondary potential between carbon and gold atoms. Secondary potential has a short range and its cutoff radius is less than 3 Å. This additional potential will model effect of electrostatic interactions among carbon and gold atoms due to charge transfer. It should be noted that for simplicity, the secondary potential is not implemented in this paper and all of the results are obtained neglecting charge transfer between C60 and gold. 4. Conclusion We studied the variation of fullerene potential energy throughout its motion on a gold surface for different orientations. We found that often the most stable state for C60 is resting on a hexagonal face. Thus, the sliding motion most likely occurs in this orientation. Rotation of C60 around the horizontal axes was easier than the sliding motion as it requires less energy. Hence, we expect rotation to occur more frequently than sliding. The least energyintensive rotation around a horizontal axes was moving from one Hexa-Down orientation to another Hexa-down orientation. However, rotation around a vertical axes required less energy than rotation around a horizontal axes. Even at extremely low temperatures like 5 K, fullerene had enough energy to rotate around a vertical axis. We studied the motion of C60 at different temperatures ranging from 5 K to 600 K using classical molecular dynamic method. We found that below 35 K, fullerene was practically motionless. Between 35 K and 100 K, C60 experienced limited mobility and could jump from one cell to an adjacent cell. Between 100 K and 200 K, C60 could jump further, yet it still did not have continuous motion. The motion became continuous at temperatures above 200 K. The motion of fullerene on gold can be compared with its motion on other materials in order to help us choose the right material for the desired application. Analytical results could also help us optimally design the experiments. Using analytical methods, the motion in many states could be predicted, thus reducing the number of required experiments to achieve a specific goal.
Fig. A1.
Fig. A1. Minimum potential energy of C60 in Hexa-down orientation as a function of temperature.
As shown in Fig. A1, the variation of the minimum energy as a function of the lattice constant is negligible. If the lattice constant is assumed to be independent from temperature and if its value at 300 K (4.078 Å) is used for all ranges of temperatures, the results would be off by less than 1%. Thus, it would not be erroneous to use a temperature-independent value for the lattice constant. However to achieve a better accuracy, we have set the lattice constant based on the temperature in the simulations. Appendix B Fig. 13 depicts that the speed of C60 increases as the temperature grows. Based on the equipartition of energy, in thermal equilibrium, energy is divided equally among all of its different forms. Therefore we expect that in the thermal equilibrium, the fullerene molecule has a horizontal translational kinetic energy 2/2 (kBT) , where kB is the Boltzmann constant. Consequently since the horizontal translational kinetic energy is equal to 1=2mðv2x þ v2y Þ, the horizontal root mean square (RMS) speed of the fullerene molecule can be calculated as
vrms
rffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2kB T A 2 2 ¼ < vx þ vy > ¼ ¼ 0:0481 T m ps
(4)
The horizontal root mean square speed of the fullerene molecule obtained from molecular dynamics simulations and the equipartition theorem are plotted in Fig. 14a. Fig. 14a demonstrates that the molecular dynamics results are appropriately coincident with the equipartition theorem. Also RMS of the fullerene rotational speed can be calculated as following:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ < u2x þ u2y þ u2z > ¼
sffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi rad 3kB T ¼ 0:0206 T 2 mr 2 ps 3
Acknowledgment
urms
The authors would like to thank the Iranian National Science Foundation (INSF) (92024635) for their financial support.
As shown in Fig. 13b, similarly for rotational motion of C60, simulation results are in a good agreement with equipartition theorem.
Appendix A Variation of coefficient of thermal expansion and lattice constant with temperature has been studied in the past years [21,41]. It has been shown that the lattice constant increases linearly with temperature, i.e. a ¼ 5.52 105T þ 4.061.We have examined the effect of this variation on the interactions between C60 and gold surface, as a proxy of the probability of motion. C60 was placed on the gold surface in the Hexa-down orientation at point 1 (see Fig. 4) and the minimum potential energy was calculated for different lattice constants. The results are shown in
(5)
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[24] M.T. Cuberes, R. Schlittler, J.K. Gimzewski, Room-temperature repositioning of individual C60 molecules at Cu steps: operation of a molecular counting device, Appl. Phys. Lett. 69 (20) (1996) 3016e3018. [25] N. Martsinovich, L. Kantorovich, Modelling the manipulation of C60 on the Si (001) surface performed with NC-AFM, Nanotechnology 20 (13) (2009) 135706. [26] R. Pawlak, et al., High-resolution imaging of C60 molecules using tuning-forkbased non-contact atomic force microscopy, J. Phys. Condens. Matter 24 (8) (2012) 084005. [27] R.J. Baxter, et al., Modelling of the adsorption of C60 on the Au (110) Surface, Chem. Phys. Chem. 5 (2) (2004) 245e248. [28] G. Teobaldi, F. Zerbetto, C60 on gold: adsorption, motion, and viscosity, Small 3 (10) (2007) 1694e1698. [29] A.V. Akimov, C. Williams, A.B. Kolomeisky, Charge transfer and chemisorption of fullerene molecules on metal surfaces: application to dynamics of nanocars, J. Phys. Chem. C 116 (25) (2012) 13816e13826. [30] W. Luedtke, U. Landman, Slip diffusion and Levy flights of an adsorbed gold nanocluster, Phys. Rev. Lett. 82 (19) (1999) 3835. [31] S. Arcidiacono, et al., Solidification of gold nanoparticles in carbon nanotubes, Phys. Rev. Lett. 94 (10) (2005) 105502. [32] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117 (1) (1995) 1e19. [33] D.W. Brenner, et al., A second-generation reactive empirical bond order (REBO) potential energy expression for hydrocarbons, J. Phys. Condens. Matter 14 (4) (2002) 783. [34] S.J. Stuart, A.B. Tutein, J.A. Harrison, A reactive potential for hydrocarbons with intermolecular interactions, J. Chem. Phys. 112 (14) (2000) 6472e6486. [35] S. Foiles, M. Baskes, M. Daw, Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys, Phys. Rev. B 33 (12) (1986) 7983. [36] W. Humphrey, A. Dalke, K. Schulten, VMD: visual molecular dynamics, J. Mol. Graph. 14 (1) (1996) 33e38. [37] W.G. Hoover, Canonical dynamics: equilibrium phase-space distributions, Phys. Rev. A 31 (3) (1985) 1695. , A unified formulation of the constant temperature molecular dy[38] S. Nose namics methods, J. Chem. Phys. 81 (1) (1984) 511e519. [39] T.-Q. Yu, et al., Measure-preserving integrators for molecular dynamics in the isothermaleisobaric ensemble derived from the Liouville operator, Chem. Phys. 370 (1) (2010) 294e305. [40] H. Qian, M.P. Sheetz, E.L. Elson, Single particle tracking. Analysis of diffusion and flow in two-dimensional systems, Biophys. J. 60 (4) (1991) 910. [41] B.N. Dutta, B. Dayal, Lattice constants and thermal expansion of gold up to 878 C by X-Ray method, Phys. Status Solidi (b) 3 (3) (1963) 473e477.
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Current Applied Physics journal homepage: www.elsevier.com/locate/cap
A close look at the motion of C60 on gold Hossein Nejat Pishkenari*, Alireza Nemati, Ali Meghdari, Saeed Sohrabpour Center of Excellence in Design, Robotics and Automation (CEDRA), Mechanical Engineering Department, Sharif University of Technology, Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 May 2015 Received in revised form 4 August 2015 Accepted 4 August 2015 Available online 7 August 2015
In this paper, we have studied the motion of buckminsterfullerene (C60) on a gold surface by analyzing its potential energy and using classical molecular dynamics method. The results can be employed to investigate the motion of C60-based nanocars which have been made in recent years. For this purpose, we have studied the translational and rotational motions of C60 molecule independently. First, we have calculated the potential energy of a C60 molecule on a gold surface in different orientations and positions and employed this data to predict fullerene motion by examining its potential energy. Then we have simulated the motion of C60 at different temperatures using classical molecular dynamics methods. Specifying the regime of the motion at different temperatures is one of main goals of this paper. We have found that the rotational motion of C60 molecule on the gold substrate, was easier than its sliding (translational) motion. Also, the regime of motion of fullerene depended on temperature. The results demonstrate that three different regimes of motion, dependent on temperature, could be observed: rare jumps to adjacent cells, frequent jumps, and continuous motion. Employing the results of this paper not only helps to understand the C60 motion on the gold surface but also provides an appropriate tool for realizing motion of the thermally-driven fullerene-based nanocars. © 2015 Elsevier B.V. All rights reserved.
Keywords: Nanocar Fullerene motion Potential energy Molecular dynamics Diffusive motion
1. Introduction With the rapid development of nanorobotics, manipulation of nano-scale materials is being increasingly attractive for different technological purposes. Despite development of different transportation techniques for the nano-sized particles, these methods were poorly efficient and corresponding handling was not appropriate [1]. First, almost all of the designed nano-manipulators are several orders of magnitude larger than their payload [2]. Second, current designs can only carry few atoms or molecules simultaneously [1,2]. This is in contrast with the performance of natural nano-manipulators. In nature, atoms and molecules are transported by molecules of the same order of magnitude or even smaller than themselves. For example, Kinesin is a small protein, but it is able to transport relatively huge payloads [3]. Inspired by the natural molecular machines, several attempts have been made to synthesize nano-manipulator [4]. In recent years, James Tour et al. have synthesized several molecular machines aimed at transportation of nano-scale materials. Because of the similarity of their appearance to regular vehicles,
* Corresponding author. E-mail address: [email protected] (H. Nejat Pishkenari). http://dx.doi.org/10.1016/j.cap.2015.08.003 1567-1739/© 2015 Elsevier B.V. All rights reserved.
these machines have been called nanocar, nanotruck, etc. [2,5e10]. Due to their small sizes, a large number of nanocars can be used simultaneously to effectively transport payloads. Several types of nanocars have been made, differing in shape and number of wheels. Fullerene, p-carborane, and trans-[Ru(CeCH)2 (dppe)2] have been utilized as wheel in fabrication of nanocars [2,5,6,10,11]. To effectively use nanocars for manipulation of molecular payloads, their motion in different conditions should be studied. The motion of this class of nano-machines could be determined using experimental or analytical methods. Several types of nanocars have been imaged experimentally using scanning tunneling microscopy (STM) [12,13]. As a result, most available experimental data represents the motion of nanocars on gold surfaces. The most important experimental studies were performed by Shirai et al. [14] and Zhang et al. [13]. Though the experimental method has serious drawbacks. Beside their high cost, the imaging tools can't reveal the details of nanocars motion [15]. It is worthy to note that because of its stability and conductivity, most imaging experiments have been performed on gold surfaces. Although recently a significant progress has been made in the synthesis of different nanocars, there are only a few studies attempting to simulate and analyze the motion of nanocars. Akimov et al. [16] and Konyukhov et al. [17] simulated fullerene-based nanocars using coarse-grained molecular dynamics method. In
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their simulations, the body and the wheels of nanocars were assumed to be rigid. Also some nanocars with carborane and adamantine wheels were simulated by Kolomeisky group [18]. The first generation of synthesized nanocars were fullerenebased cars. In order to better understand the motion of fullerenebased nanocars, the motion of C60 should be studied first. Motion of C60 on graphene has been previously studied by Ejtehadi [19,20], Savin [21,22], and Jafary-Zadeh et al. [23]. In a different research Cuberes et al. [24] repositioned C60 molecules on a Cu step using the tip of an STM by controlling the movements along the step direction. Also forced motion of C60 using AFM tip on silicon was studied by Martsinovich [25] and one of recent experimental imaging of C60 by AFM has been studied by Pawlak in 2012 [26]. There are some studies about absorption of C60 on gold surface and activation energy of C60 on gold surface by Teobaldi et al. and Baxter et al. respectively [27,28]. A more closely related work is the one of Akimov et al. [29] who modeled fullerene-based nanotruck and studied charge transfer between C60 and gold substrate. Despite the progress made in the study of C60 motion, the motion regime of C60 on the gold surface, which plays a significant role in the motion of nanocars on gold substrate, is not investigated in a satisfactory manner. In this paper, we have studied the motion of C60 on a gold surface using two approaches. The first approach was calculating the potential energy of the C60 molecule and then analyzing its variation during motion. In this method potential energy of C60 was calculated in different positions and orientations and the probable motion was predicted in different conditions based on the obtained results. The second approach was using the classical molecular dynamics method. To this aim, thermallyinduced motion of C60 has been studied at different temperatures. Using the simulation results of C60 motion, may help us to predict the motion of fullerene-based nanocars in different conditions. The rest of the paper is organized as follows. In Section 2, the potential energy method will be described. Section 3 is devoted to molecular dynamics approach where more details of the C60 motion are provided and coincidence of results between two approaches are discussed. In the last section of the paper, the main conclusions are presented. 2. Potential energy approach In this section we studied the motion of C60 through analyzing variation of its potential energy. It should be noted that the potential energy of interaction between C60 and gold substrate depends on its orientation. We calculated the potential energy and
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studied its variation in four major orientations. These orientations are described below and illustrated in Fig. 1. C60 resting on a hexagon face, henceforth called Hexa-down C60 resting on a pentagon face, henceforth called Penta-down C60 resting on the edge between two hexagons faces, henceforth called Hexa-Hexa-down C60 resting on the edge between a pentagon face and a hexagon face, henceforth called Penta-Hexa-down In addition to thermal oscillations of atoms, C60 molecule rotates and slides on the surface. We have studied the rotational and translational (sliding) motions of C60, as the two main modes of C60 motion, separately. Using the results obtained by potential energy analysis, we can determine the relative ease of each mode of motion and then predict the dominant mode in different conditions. First, we focused on the sliding motion of C60. To calculate its potential energy, C60 was moved horizontally on the surface. At every position, the height of center of mass was adjusted in order to minimize its potential energy. To find the minimum of the potential energy, the line search algorithm was employed. For this purpose, minimum of the energy was computed with precision 0.01 eV/mol. It should be noted that the orientation of C60 was kept steady during this process. The potential energy of C60 on the gold surface was separately calculated for each of the four aforementioned orientations. Since C60 is assumed to be rigid, the potential energy due to the internal interactions of carbon would be constant during the simulation. Therefore only the potential energy of carbonegold interactions was considered. The Lennard-Jones potential, with the following form, was used to simulate carbonegold interactions and the potential parameters are set as ε ¼ 0.01273 eV and s ¼ 2.9943 Å [30,31]:
s 12 s6 ELJ ¼ 4ε r r
r < rcutoff
(1)
Also the cut-off radius was set to rcut-off ¼ 13 Å (rcut-off > 4 s). Furthermore, the size of the gold substrate was assumed to be sufficiently large in the horizontal directions and the thickness was set to 3 times the lattice constant or 12.23 Å. In Fig. 2, the potential energy of C60 during its sliding motion on the gold surface is illustrated for the specified orientations. As shown in Fig. 2a and b, the variation of potential energy of fullerene during sliding in the Hexa-Down orientation is more than in the Penta-Down orientation. In Fig. 3, potential energy plots for fullerene in the Hexa-Down and Penta-Down orientations are overlaid. As shown in Fig. 3 the potential energy of C60 in the Penta-Down orientation is always
Fig. 1. Orientations of C60 on gold surface (bottom view). (a) C60 on a hexagonal face (b) C60 on a pentagonal face (c) C60 on a bond between a pentagonal and a hexagonal faces (d) C60 on a bond between two adjacent hexagonal faces. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 2. Potential energy of fullerene during sliding motion. (a) C60 on the pentagon face. (b) C60 on hexagon face. (c) C60 on a bond between a hexagon and a pentagon face. (d) C60 on a bond between a hexagon and another hexagon face. As it can be seen, variations of energy as well as absolute value of energy are the smallest when fullerene is resting on a pentagon face.
greater than its energy in the Hexa-Down orientation at the same horizontal position. It can therefore be concluded that C60 is more stable in the Hexa-down orientation at any position. At lower temperatures, where C60 molecules have lower energy levels, they are more likely to be in Hexa-Down orientation. As can be seen in Fig. 3, C60 tends to be in the Hexa-down orientation during the sliding motion and its orientation tends to remain the same, because during sliding in the Hexa-Down orientation, C60 would reach a lower energy level relative to the Penta-Down orientation. Next, we studied the rotation of fullerene molecules using potential energy analysis. The rotation axis of C60 may be parallel with or normal to the gold surface. These two types of rotation were examined separately. Two kinds of parallel-axis rotations can be imagined: a) from the Penta-Down to Hexa-Down orientation or vice versa from the Hexa-Down to another Hexa-Down orientation. We first studied rotation of C60 from the Penta-Down to HexaDown orientation. C60 was seated in the Penta-Down orientation
Fig. 3. Variation of C60 potential energy during sliding motion in Hexa-Down and Penta-Down orientations.
and rotated slowly around a horizontal axis until it reached the Hexa-Down orientation. Employing line search algorithm, the height of the center of mass was adjusted in each step to minimize its potential energy while the horizontal position was kept constant. The potential energy of C60 as a function of rotation angle is depicted in Fig. 4. The variation of potential energy depends on the lateral position of C60 relative to the crystalline structure of the gold substrate. To examine the effect of fullerene position on the potential energy, three different positions of the surface are considered. The selected points are shown in Fig. 4. As shown in Fig. 4, the variation of potential energy completely depends on the lateral position of C60. It is worthy to note that the
Fig. 4. Variation of C60 potential energy when rotating from the Penta-Down to HexaDown orientation. Potential energy was calculated in three different positions as shown in Fig. 4. At point 1, potential energy ascends at first and then it descends. When C60 is at point 2, potential energy falls and then rises to its final value. Variation of potential energy is the least when C60 is resting on point 3.
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potential energy is always lower in the Hexa-Down orientation compared to the Penta-Down orientation. At some points, e.g. point 1, C60 should pass an energy barrier during rotation. As a result, at these points, the Penta-down orientation is locally stable configuration but not globally stable one; thus a C60 molecule tends not to change its orientation from Penta-Down to Hexa-Down at these points despite the eventual decrease in energy. At some other points, e.g. point 3 during the rotation form the Penta-Down to Hexa-down orientation, potential energy of C60 decreases sharply at first but subsequently increases to its final value. At these points, the Penta-Down orientation is an unstable configuration for fullerene molecule; C60 begins to rotate but due to loss of energy, it may not reach the Hexa-Down orientation and will stay in the middle state. At some other points, e.g. point 2, potential energy of C60 does not fluctuate considerably during rotation. The amount of energy necessary for rotation at these points is relatively small, thus rotating is more probable. The second possible type of rotation is changing the orientation from a Hexa-Down to another Hexa-Down orientation. In this type of motion, fullerene is initially placed in the Hexa-Down position and rotates around a bond to reach another Hexa-Down orientation. The height of center of mass is adjusted at each step to minimize the potential energy, while the lateral position of C60, its position on the xey plane, is kept constant. Potential energy was calculated during rotation for the three aforementioned positions (see Fig. 5). It should be noted that in this type of motion, the initial and final values of the energy are the same, since fullerene will be on the Hexa-Down orientation in both states. As shown in Fig. 6, the variation of potential energy depends on the position. At some points e.g. point 1, C60 should pass an energy peak while at some other points e.g. point 2, the energy variation is small. On the other hand, at some points e.g. point 3, the potential energy may decrease before reaching its final value. By further inspection, it can be proved that in this condition, fullerene is seated on the Hexa-Hexa-Down orientation. In Fig. 6 potential energy in Hexa-Down and Hexa-Hexa-Down orientations are plotted. At some points e.g. Point 3, as shown in Fig. 6, the Hexa-HexaDown orientation has lower energy, therefore it is more stable than the Hexa-Down orientation. In such positions, potential energy would decrease before increasing to its final value during rotation from a Hexa-Down to another Hexa-Down orientation. At some other points, e.g. point 1, Hexa-Down orientation has lower energy and it is more stable. The potential energy would increase
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Fig. 6. Variation of C60 potential energy during translational motion in Hexa-Down and Hexa-Hexa-down orientations. As it can be seen, in some points, potential energy of fullerene in Hexa-Down orientation is less than its value in Hexa-Hexa-Down one. The points which potential energy of C60 is equal at both orientations, are an appropriate (or maybe the best) position for rotating from a Hexa-Down to adjacent Hexa-Down.
and then decrease during rotation at these points. If fullerene is placed at a position where the potential energy is equal in the Hexa-Hexa-Down and Hexa-Down orientations, the variation in potential energy during rotation would be minimal. Finally, we studied the potential energy variations during rotation of C60 around the normal axis. Similar to the parallel-axis rotation, variation of the potential energy depends on the orientation and position of the C60 molecule. Fig. 7 shows the potential energy variation for the Hexa-Down and Penta-Down orientations at three aforementioned points. As shown in Figs. 4, 5 and 7, the energy variations due to rotation highly depends on C60 position on the gold substrate. The rotation is more likely to happen at the points in which the energy variation is small. Considering the results illustrated in Figs. 4 and 5, rotation of fullerene around horizontal axis mainly consists of changing the orientation from one Hexa-Down to another Hexa-Down. Since potential energy variation is smaller for the rotational motion, it can be predicted that rotation occurs easier than sliding motion. Based on the data shown in Fig. 8, it is evident that the variation of potential energy is relatively small during its rotation around the vertical axis. Thus, this type of motion occurs easier and more frequently than sliding motion and other types of rotation. Potential energy analysis is a powerful tool which enables us to predict the motion of C60 for different substrate conditions. We can use this method to predict the motion when the substrate is not smooth. This approach may help us better understand how C60 behaves near steps, vacancies, or even grain boundaries. Totally from the energy analysis approach the following conclusions may be deduced: Since potential energy variation is smaller for the rotational motion, it can be predicted that generally, rotation occurs easier than translation. Rotation around the vertical axis occurs easier and more frequently than other types of rotation. The probability of rotational motion is higher in positions with small variations of energy with respect to rotation.
3. Molecular dynamics approach Fig. 5. Variation of C60 potential energy while rotating from a Hexa-Down to an adjacent Hexa-Down orientation. Similar to rotation form a Penta-Down orientation to Hexa-Down orientation at point 1 potential energy ascends firstly and then it follows a descending behavior. In contrary with point 1, at point 2 variation of potential energy has a descendingeascending behavior. Also it is worthy to note that, variation of the potential energy is the least at point 3 compared to two other mentioned points.
In the second section of this paper, we have studied the motion of fullerene molecules on a gold surface using the classical molecular dynamics method. We have simulated the motion of C60 at different temperatures to demonstrate the effect of temperature on
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Fig. 7. Variation of C60 potential energy during rotation around vertical axis. As it is shown, variation of potential energy is very small during rotation around vertical axis. Plot b shows that the variation of potential of C60 is negligible during rotation around vertical axis when C60 is resting on a pentagon face. So we can prognosticate that rotation of C60 around vertical axis can easily happen especially when fullerene is resting on the Penta-Down orientation.
the regime of motion. The substrate is a 20 a 20 a 3 a cubic lattice having an FCC lattice structure made up of gold atoms, where a is the lattice constant of Au. The plane direction of the gold surface is set to be (001) with respect to the FCC crystalline direction. The lowest layer was assumed to be rigid, and the fullerene molecule was placed on top of the substrate. Periodic boundary conditions was used in the x and y directions. e Hoover thermostats were applied separately to the Nose substrate and C60 in order to control the temperature. AIREBO potential was used to simulate interactions of the carbon atoms in C60. Additionally, EAM potential was used for modeling the interactions among the gold atoms. The system was simulated for 8 ns. To achieve sufficiently precise results, the time step was set to 1 fs. The system was relaxed for 200,000 steps before simulation began. Simulations were performed for different temperatures ranging from 5 K to 600 K. In order to accurately model the substrate, the correct value for the lattice constant should be used. We have studied the effects of the variation of lattice constant due to temperature on the simulation results. Based on our analysis (for more details one can refer to Appendix A), the lattice constant for gold substrate is set to be 4.078 Å.
Fig. 8. C60 on gold substrate. Size of substrate was 20 a 20 a 3 a with plane direction of (001) with respect to the FCC crystalline. Boundary condition in x and y direction was periodic. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Simulation were conducted using LAMMPS software [32] and the results are visualized using VMD package [36]. AIREBO potential [33,34] is used to model the interactions among carbon atoms while EAM potential [35] is employed to model the interactions among Au atoms. It should be mentioned that both Lennard-Jones and torsion terms in the AIREBO potential are considered in our simulations. As a brief, potentials used in our simulations are listed in Table 1. As mentioned before gold surface is Au (100). We have used the NoseeHoover thermostat [37,38], to control the temperatures of the fullerene particle and gold substrate. The equations of motion are integrated employing the velocity Verlet algorithm with a timestep of 1 fs. Also for NVT ensemble, in the LAMMPS software, Tdamp parameter is set as 50 fs [39]. After running the simulations, the trajectory of fullerene molecules at different temperatures is plotted in Fig. 9. As shown in Fig. 9, the regime of motion strongly depends on the temperature. For temperatures lower than 35 K, fullerene is almost immobile. By rising the temperature to 35 K, C60 starts to move by jumping to adjacent cells. The motion at this temperature consists of occasional jumps, and thus is not continuous. The regime of motion remains the same until the temperature reaches 100 K. Further increasing the temperature would change the regime of motion from occasional jumps to frequent jumps. The frequency of jumps increases with temperature, thus fullerene could travel further on the plane. At temperatures higher than 150 K, the regime of motion becomes semi-continuous and its speed increases as the temperature rises. The continuous motion of fullerene at temperatures above 400 K is shown in Fig. 9d. At low temperatures, potential energy analysis can be used to accurately predict when the regime of motion changes. The potential energy of the C60 molecule due to the interactions between gold and carbon atoms was calculated during the simulation. Fig. 10 shows the mean and maximum potential energy as a function of the temperature. Based on the potential energy analysis elaborated in the previous section, it was determined that the potential energy of C60 should reach 0.985 eV before it could hop to an adjacent cell. Results from the molecular dynamics simulation reveal that fullerene reaches the required energy level at approximately 35 K (see Fig. 10). This conclusion demonstrates that potential energy analysis results are in good agreement with molecular dynamics simulations. Fig. 11 shows how often fullerene has enough energy to jump to an adjacent cell. It should be noted that having enough energy does not guarantee the occurrence of a jump, because C60 may not have
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Table 1 Parameters and potentials used in MD setup. Elements
Potential
CeAu AueAu CeC
Lennard-Jones (ε ¼ 0.01273 eV, s ¼ 2.9943, rcut ¼ 13 Å) EAM/Alloy [35] AIREBO (considering Lennard-Jones and torsion terms ε ¼ 0.0028 eV, s ¼ 3.4, rcut ¼ 8.5 Å) [33,34]
Fig. 9. Trajectory of fullerene on gold substrate at different temperatures for 8 ns simulation. a) C60 is almost stationary at 20 K and lower temperatures. b) C60 have frequent jumps to adjacent cells and the number of jumps increases at higher temperatures. c) Frequent jumps of C60 converted to smooth motion at 150 K and higher temperatures. d) Fullerene has a free continuous motion on the gold substrate at high temperatures.
enough horizontal speed to move to the adjacent cell. It is also evident that below 35 K, fullerene could hardly ever reach the energy level required to jump. The probability of reaching the energy level required to perform a jump increases with the temperature. At 75 K, fullerene has enough energy to move to an adjacent cell only 0.8% of the times, while at 300 K it does so approximately 51% of the times. It should be noted that a powerful jump for C60 is a jump in which potential energy between C60 and gold substrate can
Fig. 10. Average and maximum potential energy of C60 as a function of the temperature.
reach to the certain level of 0.985 eV. Potential energy analysis showed that a rotating motion around the vertical axis requires less energy than other modes of motion, thus the probability of its occurrence is higher. Fig. 12 shows the angular speed of C60 around the vertical axis at 5 K. In our
Fig. 11. Percentage of powerful jumps as a function of the temperature. The inset to the right shows a zoomed in view of the main plot for low temperatures. Number of fullerene powerful jumps increases with increasing temperature. Below 35 K temperature, C60 rarely reach the energy level required for moving to neighboring cells. It can be predicted that jumping of C60 to adjacent cells is nearly impossible below 35 K.
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Alternative parameter for describing motion of C60 is Mean Square Displacement (MSD). It could be used to describe the diffusive motion of fullerene. Then the diffusion coefficient, which is one of the best parameters to describe the random motion of C60, can be derived from the MSD. The Mean Square Displacement MSD is defined as [40]:
MSDðtÞ ¼ < jrðtÞ rð0Þj2 >
(2)
where r is the position vector of fullerene center of mass and <> represents averaging over all of particles. Diffusion constant D is defined as the slope of the MSD plot:
1 < jrðtÞ rð0Þj2 > bt
Fig. 12. Angular velocity around the vertical axis and yaw angle of fullerene at 5 K. It has been shown that C60 freely rotates around vertical axis at 5 K.
D ¼ lim
simulations, even at extremely low temperatures e.g. 5 K, fullerene spun easily around the vertical axis. As shown in Fig. 12, C60 rotated freely for a relatively long period of time between t ¼ 1 ns and t ¼ 2 ns. This is in perfect agreement with potential energy analysis results. This phenomenon can be used in design and fabrication of fullerene-based nano-bearings. As we mentioned earlier, potential energy analysis has an important role in predicting the motion on different substrates. If the substrate has defects e.g. steps, vacancies or dislocations, the energy level required for the occurrence of motion could be calculated and the temperature threshold of motion could be predicted by computing the percentage of powerful jumps, as demonstrated in Fig. 11. Potential energy analysis is one of the fastest and most practical methods for predicting the mode and regime of motion. In order to better understand the motion of fullerene on the gold substrate, the motion should be quantified. The mean speed of C60 at different temperatures might provide a better insight of the C60 motion. For this purpose, we have plotted the horizontal speed of the fullerene as a function of temperature. The horizontal root mean square speed of the fullerene molecule obtained from molecular dynamics simulations and the equipartition theorem are plotted in Fig. 13a. Fig. 13a demonstrates that the molecular dynamics results are appropriately coincident with the equipartition theorem (for more discussion on the equipartition theorem see Appendix B). As shown in Fig. 13b, similarly for rotational motion of C60, simulation results are in a good agreement with equipartition theorem. It should be noted that, the variation of vrms as a function of temperature did not provide extra information about the motion regime of C60.
where b is 6 for three-dimensional and 4 for two-dimensional motion. In this paper we are interested to study the horizontal (two-dimensional) motion of fullerene on the gold surface, thus we ignored the vibrations of fullerene in the vertical direction. To have accurate estimation of MSD, we should conduct numerous numbers of different simulations and then averaging should be done considering trajectory of all of them. Alternative method is conducting a relatively long time simulation and dividing the whole of trajectory to many individual shorter-time trajectories. Using the second approach, we have conducted a simulation for 8 ns and divided the whole trajectory of fullerene motion to sixty smaller trajectories. The mean square displacement of fullerene motion at different temperatures is plotted in Fig. 14. The MSD growth rate increases with the temperature (see Fig. 14). At low temperatures, below 35 K, the MSD increases nonlinearly with time. But at medium and high temperatures, the MSD grows nearly linear. Form Fig. 14a, we can see when the temperature rises from 20 to 35 K, a significant difference in MSD diagram will happen showing that the regime of fullerene motion changes when the temperature reaches 35 K. In addition, form Fig. 14b we can find that another change in the motion regime when the temperature rises from 100 K to 150 K. Considering Fig. 14c, it can be deduced that the fullerene motion increases nearly monotonic with temperature rising. Study of the diffusion coefficient may provide useful insight into understanding of the fullerene motion. The diffusion coefficient could be calculated using the MSD diagrams. Arrhenius analysis of the diffusion coefficient helps us better understand the changes in the motion regime of C60. The effect of temperature on the diffusion coefficient is plotted in Fig. 15. The slope of the diagram changes at 20 K and 100 K. Thus, the motion regime can be expected to change at these temperatures.
t/∞
Fig. 13. RMS of (a) C60 speed and (b) rotational speed as a function of the temperature.
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Fig. 14. The MSD of C60 at different temperatures. a) MSD of fullerene for temperature range 5K T 50 K. As shown in Fig. 15a, when temperature rises to 35 K, a distinct behavior in MSD curve can be observed. b) MSD of fullerene for temperature range 75 K T 200 K. As it can be seen from this figure, MSD curves have different behaviors when the temperature rises from 100 K to 150 K. c) MSD of C60 for temperature range 300 K T 600 K. Growth rate of MSD curve increases with temperature rising.
Fig. 15 depicts that diagram ln(D) has a minimum slope for temperatures below 20 K. Therefore in this region, motion of C60 is not sensitive to the temperature change. For temperatures between 20 K and 100 K, the slope of the diagram ln(D) has increased slightly. This increase of slope is a sign for change of the motion regime revealing more sensitivity of C60 motion to the temperature change. Raising temperature to more than 100 K makes steeper the slope of the diagram. In this range of temperature, C60 is almost free to move on the surface and ability of motion increases sharply relative to the temperature. We are highly interested in the displacement of fullerene on a gold surface, not its vibrations while trapped in a cell. For our purpose, it was more useful to count the number of lattice
constants traveled instead of mean speed. This assured us that vibrations did not enter our measurements. The rate of jumps as a function of temperature is simulated and the results are shown in Fig. 16. Using the above plot, alteration of motion regime at temperatures above 100 K can be explained. At temperatures below 100 K, fullerene may jump to adjacent cells and the rate of jumps increases with the temperature. However, this plot cannot predict changes in the motion regime below 100 K.When temperature approaches 100 K, the motion regime changes and allows molecules to jump multiple cells at once. The slope of the plot increases as a result of this change. When temperature reaches 200 K, the regime of the motion changes to a continuous motion and the speed of motion
Fig. 15. Arrhenius analysis of the diffusion coefficient of C60.
Fig. 16. Rate of C60 jumps vs. temperature.
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continued to increase with temperature. Like other computational investigations, the study presented here involves some restrictions that require attention in order to understand its applicability. As it was mentioned in the introduction section, Akimov et al. [29] studied charge transfer between gold substrate and C60. Considering the result achieved in that paper, charge transfer increases the activation energy for motion of C60. So it may be predicted that charge transfer restricts motion of C60 and reduces slightly diffusivity of C60 motion. Considering complex motion of C60 on the gold surface, it is difficult to take into account the charge transfer between gold and carbon atoms during molecular dynamics simulations. To increase accuracy in the simulation, one can consider a secondary potential between carbon and gold atoms. Secondary potential has a short range and its cutoff radius is less than 3 Å. This additional potential will model effect of electrostatic interactions among carbon and gold atoms due to charge transfer. It should be noted that for simplicity, the secondary potential is not implemented in this paper and all of the results are obtained neglecting charge transfer between C60 and gold. 4. Conclusion We studied the variation of fullerene potential energy throughout its motion on a gold surface for different orientations. We found that often the most stable state for C60 is resting on a hexagonal face. Thus, the sliding motion most likely occurs in this orientation. Rotation of C60 around the horizontal axes was easier than the sliding motion as it requires less energy. Hence, we expect rotation to occur more frequently than sliding. The least energyintensive rotation around a horizontal axes was moving from one Hexa-Down orientation to another Hexa-down orientation. However, rotation around a vertical axes required less energy than rotation around a horizontal axes. Even at extremely low temperatures like 5 K, fullerene had enough energy to rotate around a vertical axis. We studied the motion of C60 at different temperatures ranging from 5 K to 600 K using classical molecular dynamic method. We found that below 35 K, fullerene was practically motionless. Between 35 K and 100 K, C60 experienced limited mobility and could jump from one cell to an adjacent cell. Between 100 K and 200 K, C60 could jump further, yet it still did not have continuous motion. The motion became continuous at temperatures above 200 K. The motion of fullerene on gold can be compared with its motion on other materials in order to help us choose the right material for the desired application. Analytical results could also help us optimally design the experiments. Using analytical methods, the motion in many states could be predicted, thus reducing the number of required experiments to achieve a specific goal.
Fig. A1.
Fig. A1. Minimum potential energy of C60 in Hexa-down orientation as a function of temperature.
As shown in Fig. A1, the variation of the minimum energy as a function of the lattice constant is negligible. If the lattice constant is assumed to be independent from temperature and if its value at 300 K (4.078 Å) is used for all ranges of temperatures, the results would be off by less than 1%. Thus, it would not be erroneous to use a temperature-independent value for the lattice constant. However to achieve a better accuracy, we have set the lattice constant based on the temperature in the simulations. Appendix B Fig. 13 depicts that the speed of C60 increases as the temperature grows. Based on the equipartition of energy, in thermal equilibrium, energy is divided equally among all of its different forms. Therefore we expect that in the thermal equilibrium, the fullerene molecule has a horizontal translational kinetic energy 2/2 (kBT) , where kB is the Boltzmann constant. Consequently since the horizontal translational kinetic energy is equal to 1=2mðv2x þ v2y Þ, the horizontal root mean square (RMS) speed of the fullerene molecule can be calculated as
vrms
rffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2kB T A 2 2 ¼ < vx þ vy > ¼ ¼ 0:0481 T m ps
(4)
The horizontal root mean square speed of the fullerene molecule obtained from molecular dynamics simulations and the equipartition theorem are plotted in Fig. 14a. Fig. 14a demonstrates that the molecular dynamics results are appropriately coincident with the equipartition theorem. Also RMS of the fullerene rotational speed can be calculated as following:
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ < u2x þ u2y þ u2z > ¼
sffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi rad 3kB T ¼ 0:0206 T 2 mr 2 ps 3
Acknowledgment
urms
The authors would like to thank the Iranian National Science Foundation (INSF) (92024635) for their financial support.
As shown in Fig. 13b, similarly for rotational motion of C60, simulation results are in a good agreement with equipartition theorem.
Appendix A Variation of coefficient of thermal expansion and lattice constant with temperature has been studied in the past years [21,41]. It has been shown that the lattice constant increases linearly with temperature, i.e. a ¼ 5.52 105T þ 4.061.We have examined the effect of this variation on the interactions between C60 and gold surface, as a proxy of the probability of motion. C60 was placed on the gold surface in the Hexa-down orientation at point 1 (see Fig. 4) and the minimum potential energy was calculated for different lattice constants. The results are shown in
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