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A Classical Molecular Dynamics Study of the Effect of the Atomic Force Microscope Tip Shape, Size and Deformation on the Tribological Properties of the Graphene/Au(111) Interface

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19 December 2023

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Abstract
Atomic force microscopes are used, beside their principal function as surface imaging tools, in surface manipulation and measurement of interfacial properties. In particular, they can be modified to measure lateral friction forces that occur during the sliding of the tip against the underlying substrate. However, the shape, size, and deformation of the tips profoundly affect the measurements in a manner that is difficult to predict. In this work, we investigate the contribution of these effect to the magnitude of the lateral forces during sliding. The surface substrate is chosen to be few-layer AB-stacked graphene surface whereas the tip is initially constructed from face-centered cubic gold. In order to separate the effect of deformation from shape, rigid tips of three different shapes were considered first, namely a cone, a pyramid and a hemisphere. The shape was seen to dictate all aspects of the interface during sliding, from temperature dependence to stick-slip behavior. Deformation was investigated next by comparing a rigid hemispherical tip to one of identical shape and size but with all but the top three layers of atoms free to move. The deformation, as also verified by an indentation analysis, occurs by means of the lower layers collapsing on the upper ones, thereby increasing the contact area. This collapse mitigates the friction force and decreases it with respect to the rigid tip for the same vertical distance. Finally, the size effect is studied by means of calculating the friction forces for a much larger hemispherical tip whose atoms are free to move. In this case, the deformation is found to be much smaller but the stick-slip behavior is much more clearly seen.
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Subject: Chemistry and Materials Science  -   Theoretical Chemistry

1. Introduction

Beyond its use as an imaging tool, the atomic force microscope (AFM) is also often utilized in characterization and manipulation of surfaces and surface adsorbates. Use of an AFM tip for surface indentation [1], surface scratching [2] and displacement of nanoclusters on a surface [3] has now become routine. Additionally, interfacial properties such as the energy of adhesion can also be determined via the pull-off force experienced by the tip as it is being moved away from a surface [4]. Among the various such uses of an AFM tip, a prominent one is the friction force microscopy (FFM) [5] where the tip is modified to measure lateral forces at an interface of two materials in relative motion. This function enables the understanding and identification of low-friction interfaces and lubricants. For instance, in the recent years, two-dimensional materials have emerged as excellent lubricants with very low friction [6,7,8,9].
The foundations of the atomic scale description of sliding friction dates back to the late 1920’s. The earliest model developed simultaneously by Prandtl [10] and Tomlinson [11] is a simplified representation of an AFM tip sliding on a surface where the cantilever and the tip have been replaced by two point masses connected by a spring. The substrate over which the tip slides, on the other hand, is described by a sinusoidal potential energy surface (PES). This simple model reveals the characteristic stick-slip motion where the tip is periodically trapped in the pits of the PES and is released when the elastic energy build-up in the spring is large enough to overcome the barrier. Since the Prandtl-Tomlinson model was proposed, there have been more sophisticated models developed over the years, most notably by Frenkel, Tomlinson, and Kontorova [12], Sokoloff [13,14], and Zhong and Tomanek [15]. However, the stick-slip behavior has always been a common theme in all models proposed. The chemical composition and the physical morphology of the tip, which is largely ignored in these simple models, has proven to be a crucial element for understanding and controlling interfacial phenomena via an AFM [16]. Friction investigations often include a mapping of the interfacial PES, however, the morphology of the PES depends on the properties of both the components making up the interface. For instance, the stick-slip motion depends on both the material that the AFM tip is made of and the surface [17]. Ko et al. [18] showed that tips made out of metal alloys show different shear and adhesion behavior during sliding depending on whether the alloy is miscible or immiscible. Dong et al. [19] observed, through AFM measurements and MD simulations that the size and the shape of the tip affects the lateral forces when moving up and down surface steps of graphite. Furthermore, the size and the shape of the actual contact area also depends upon the vertical load applied to the AFM tip, which in turn affects the PES [20]. Further complications may occur due to the presence of an interfacial fluid, such as water or a lubricant [21] as well as wear and material consumption due to friction [22].
Due to the aforementioned difficulties associated with direct imaging, computational tools such as density functional theory and molecular dynamics (MD) have become indispensable in understanding the atomistic picture underneath the operation of the AFM tip [1,16]. Among such studies, some notable examples are as follows. In the MD simulation by Korayem et al. [23], the manipulation of a Au nanoparticle by a Ag tip on a Ag substrate in the absence and presence of a water meniscus was studied. Jacobs et al. [4] used a continuum model along with MD simulations and experimental TEM data to understand the effect of surface roughness on the pull-off force of H-saturated diamond tips. Hu et al. [24] studied, via MD simulations, the nanoscale wear of a silicon dioxide tip sliding on a copper substrate. Reischl et al. [25] used simulated AFM images generated by postprocessing of MD data to understand whether a carbon nanotube tip can resolve individual point defects on a surface covered with explicit water. Kopycinska-Müller et al. [26] demostrated that a Si tip will undergo a phase change under load. Xu et al. [16] conducted MD simulations of a 10 nm-radius melt-quenched Pt tip sliding on a Au(111) surface. They noted that in the presence of neighboring grains, irregular stick slip occurs whereas if the tip is a grain-free single crystal, the stick-slip is much more regular. Xu et al. [27] studied, using MD, the stick-slip signature of the amorphous silica tip on the Au(111) surface. A small and a large tip were prepared and used for recording lateral forces. For the small tip, a high friction regime appears for loads larger than a threshold value, which is attributed to surface defects.
In this paper, molecular dynamics simulations were used to explore the effect of size, shape and deformation of AFM tips in the determination of the sliding friction force. To study the effect of shape, Au tips of three different geometries, namely a cone, a hemisphere and a pyramid were chosen and the friction forces while sliding on the same few-layer graphene substrate were compared. For the dependence on load, three different heights were chosen and average friction forces were calculated. Rigid and plastically deformable hemispherical tips were then compared to understand the effect of deformation. In addition to friction force calculations, indentation analyses were conducted as well. Two different sizes were studied for the hemispherical tip in order understand size dependence.

2. Materials and Methods

The MD simulations presented in this paper were performed using the open-source software suite Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [28]. Bonded and non-bonded Au-Au and C-C were modeled using the Embedded Atom Model (EAM) [29], and Adaptive Intermolecular Reactive Empirical Bond Order (AIREBO) [30] force fields, respectively. The non-bonded interaction between Au and C is described using the Lennard-Jones 12-6 potential [31], where the interaction potential energy between atom i and j is described by
E ( r i j ) = 4 ε σ 6 r i j 6 − σ 12 r i j 12
where r i j is the interatomic distance while ε and σ are the model parameters that are related to the bond energy and bond length, respectively. For this work, σ and ϵ were taken from previous work [32] to be 3.003 Å and 0.0341 e V , respectively. Atoms that are further apart from one another than 15 Å were excluded from the calculation and considered as noninteracting.
In our MD simulations, the equations of motion were integrated using the velocity Verlet algorithm [33] with a time step of 1 fs. Temperature control was achieved via the Berendsen thermostat [34]. The surface was modeled by a slab of four ABAB-stacked graphene layers where periodic boundary conditions were imposed on the plane of the surface. Throughout the simulations, the bottom two layers were fixed while the top two layers were allowed to move according to the equations of motion. The indentation study, however, was conducted on a six-layer graphene substrate with two bottommost layers fixed.
The shape effect study was conducted by employing tips of three geometric shapes, namely a cone, a pyramid and a hemisphere as represented by the first three models in Figure 1(a). During the sliding simulations, the Au atoms in all three tips were kept at rigid positions with respect to one another to exclude deformation effects. The instantaneous lateral force profiles and average forces were compared for three different tip heights (roughly corresponding to three loads) and three different temperatures (T=1 K, 100 K, and 300 K). Next, deformation was included by relaxing the constraints to all but the top three layers of the hemispherical tip (leftmost tip in Figure 1(a)) and allowing the lower atoms to move during the simulations. The numbers of Au atoms in the conical, pyramidal, and hemispherical tips are 71, 285, and 372, respectively. With a 13 Å radius, the nonrigid tip contains 198 mobile Au atoms. To further elucidate the deformation, an indentation simulation, where the nonrigid hemispherical tip was steadily lowered on to the surface by about 3.5 Å without sliding was conducted. Finally, to explore the size effect, a larger hemisphere with a radius of 30 Å was also built where only the top three layers were taken to be fixed. This tip includes 2966 Au atoms, of which 1816 are mobile Figure( Figure 1(b)). All tips were created from the face-centered cubic phase of Au. For the smaller tips, a 120 Å× 123 Å underlying graphene substrate with 22736 C atoms was used (Figure 1(c)). For the largest tip, the substrate surface area was extended to approximately 221 Å× 221 Å with 74880 C atoms.
During the sliding simulations, lateral friction forces were calculated by means of pulling the tips along the surface at a predetermined height with a constant velocity and recording the component of the forces in the opposite direction to the motion. The average of the magnitude of the lateral forces was then calculated and reported for comparison. To understand the effect of increasing load, sliding simulations were conducted at different heights. In this context, the height h is defined to be the distance between the topmost fixed layer of the tip and the bottommost fixed layer of the surface. For the fixed tips, three heights h 1 , h 2 , and h 3 Å were considered, which correspond to a distances of 2.0 Å, 2.5 Å and 3.0 Å from the lowest point of the tip to the top of the unperturbed surface, respectively. Since each rigid tip consists of a different number of layers, h 1 , h 2 , and h 3 are all different for the three tips, which are listed in Table 1.For the nonrigid hemispherical tip, we consider three different distances of h 4 = 24.27 Å, h 5 = 24.77 Å, and h 6 = 25.27 Å, however, the translation of these distance to the bottom of tip-to-surface distance is ambiguous due to deformations. In order to compare sliding along different directions, all tips except for the large hemisphere tips was dragged along the x axis (moving parallel to the C-C bonds) and the y axis (moving perpendicular to the C-C bonds) for each load and temperature.
The MD protocol followed in the preparation and production stages of our calculations can be summarized as follows: Before any sliding or indentation simulations were initiated, the mobile atoms of the substrate was gradually heated from 1 K to the target temperature in an NVT ensemble. The system was then kept at an NVT ensemble at the target temperature for 1 ns. After this thermalization stage was complete, the tip was brought closer to the substrate slowly. A further thermalization step was conducted before the sliding or indentation process at the target temperature. Afterwards, sliding/indentation was initiated at the prescribed speeds while continuing to control the temperature within an NVT ensemble. As a test, in one of the sliding simulations, the thermostat was turned off and the system was allowed to evolve in an NVE ensemble. The results were seen to be unaffected and therefore the NVT ensemble was employed in all simulations.
During sliding, all tips were along on the substrate for 10 Å with a velocity of 10 cm/s except for the large hemispherical tip, which moved at a speed of 1 m/s. All sliding simulations were carried on for a total of 10 ns. In the indentation simulation, the tip was displaced down the z axis (normal to the surface).

3. Results

Rigid AFM tips in the shape of a pyramid, a cone, and a hemisphere were pulled along the x and y directions as defined in Figure 1(c) at three different heights of h 1 , h 2 , and h 3 with a speed of 10 cm/sec at temperatures of 1 K, 100 K, and 300 K. Average vertical loads ( F N ) corresponding to these three heights and for all temperatures for all rigid tips are listed in Table 2 in nN. The largest distance ( h 3 ) results in negative loads, which correspond to the case of the surface lifting upwards towards the tip due to van der Waals forces. The magnitude of the vertical load is seen to increase with temperature for all shapes and heights but for a few exceptions. There is no perceptible dependence on sliding direction.
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Figure 1. Smaller (a), and larger (b) models for the AFM tip. The smaller tips are pyramidal, conical, hemispherical from left to right. The smaller rigid hemispherical tip on the graphene substrate is shown in (c). Yellow and red spheres represent Au atoms. Red spheres, in particular, correspond to atoms that are fixed in two hemispherical tips. Magenta and green spheres represent free and fixed carbon atoms in the substrate.
Figure 1. Smaller (a), and larger (b) models for the AFM tip. The smaller tips are pyramidal, conical, hemispherical from left to right. The smaller rigid hemispherical tip on the graphene substrate is shown in (c). Yellow and red spheres represent Au atoms. Red spheres, in particular, correspond to atoms that are fixed in two hemispherical tips. Magenta and green spheres represent free and fixed carbon atoms in the substrate.
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The horizontal friction forces measured during sliding as a function of distance are shown for the rigid AFM tips with three different shapes along x and y directions, and for the three temperatures involved (T=1 K, 100 K, 300 K) are shown in Figures S1, S2, and S3 of the Supplementary Material (SM). From these results, it is immediately obvious that the shape of the tip makes a very significant difference in friction measurements. Starting first with the common features in the T = 1 K results in Figure S1, the energy barriers for the tip to overcome that occur along the y direction (i.e. against C-C bonds) are much larger than those along the x direction for all tip shapes but the piramid. This difference is particularly pronounced for the conical tip. The maximum force encountered while sliding along the y direction is approximately ten times the force along the x axis. For the pyramid, the vertical force displays a double-peak character along the y axis, which is not seen in any of the other tips studied here. For this tip, only a single atom at the edge of the tip is in direct contact with the surface. As a result, the lateral friction force profile is a good representation of the actual potential energy surface along the x and y directions of the underlying substrate. The single-atom at the end of the tip is positioned such that it actually passes over atoms in a maximum-overlap configuration, and sliding starts at a hexagonal hollow site of the surface. This means that the double peak feature corresponds to the single tip sliding over one atom and then the next at either end of a C-C bond. Finally the hemisphere displays the well-known stick-slip pattern for the two smallest vertical distances while sliding along the y-axis. This is also seen to a certain extent for the cone but is much more pronounced for the hemisphere. In an actual sliding experiment, the stick-slip occurs due to both the elastic properties of the tip and the substrate [7], whereas in the present case of rigid tips, we only observe the effect of the substrates. At T = 100 K, the periodic structure of the lateral force with well-defined peaks is mostly lost for the cone and the hemisphere for sliding along the x-direction due to increased thermal motion of the atoms. In all other cases, the sinusoidal pattern is maintained with some noise. At T = 300 K, the noise increases but the pattern from T = 100 K is maintained for all tips in both directions. For instance, for the hemisphere, the stick-slip pattern is persistent at all temperatures along the y-direction.
Next, we present and compare our results for average total lateral forces for all tips and temperatures with the goal of understanding whether the differences seen in the sliding characteristics for the different tips are still present when averaged over the total sliding distance. The average friction force for all heights, temperatures and shapes are given in Table 3 and trends are shown as plots in Figure 2.
Unlike the average vertical load, the friction behavior shows significant variation across the parameter space. A common observed trend is the increase of friction with increasing temperature. For the cone, the average lateral force along the x and y directions is nearly identical at all heights and all temperatures. For the pyramid and the hemisphere, however, the y-direction yields lower lateral forces.
Next, we consider the effect of deformation. For this part of our investigation, we consider a hemispherical tip with only the top three layers fixed while all the atoms in the remaining layers are free to move . Figure 3 shows the completely rigid hemispherical tip described above along with a tip with only the top three layers fixed. The height of the tips in this figure correspond to h 5 as defined in the Methods section. The sizable deformation of the nonrigid tip is already seen clearly for this size. The initial five layers that taper off down the tip have now merged into four evenly sized layers increasing the contact area between the tip and the surface. As a result, the inward depression of the top surface layers visible for the rigid tip has disappeared for the nonrigid tip.
The lateral forces measured as a function of sliding distance along x and y at T = 300 K and three heights h 4 , h 5 , and h 6 are shown in Figure 4. Interestingly, the behavior of the rigid and nonrigid tips are very similar as far as the lateral forces are concerned (compare Figure 4 with Figure S3 of the SM). The stick-slip pattern is more pronounced for the nonrigid tip in comparison to the rigid one. Evidently the added elasticity of the tip will contribute to the build up of the forces in the stick phase, increasing abruptness of the slip.
The average vertical loads ( F N ) and average lateral friction forces ( F f ) at heights h 4 , h 5 , and h 6 are shown in Table 4 for a temperature of T = 300 K . Although the heights are not the same for the rigid and the nonrigid tips, the overall magnitudes of the lateral friction forces are similar. For a hemispherical tip of this size then the effect of the deformation is moderate.
In order to further understand the deformation of a tip under heavier loads, we conducted an MD simulation at 300 K in which the hemispherical tip approaches a six-layer graphene surface from the top. We then monitor the deformation via the mean square displacement of the graphene layers. An animation of this motion is provided as a part of the SM. Different stages of the tip during its descent onto the surface are reported in Figure 5. During the approach, atoms on the outer layer of the tip are seen to climb up the tip along the sides. The upward displacement of the Au atoms in groups occurs suddenly as seen by jumps in the means square displacement of the graphene layers seen in Figure 5. In this figure, the mean square displament from equilibrium for the top four mobile layers of graphene are shown. As the tip presses onto the surface, all layers first are compressed downward slightly and continously. At around 3.1 Å of tip displacement, the top layer experiences a sudden displacement of more than 1 Å, which corresponds to the displacement of the bottom layer of the tip in the upward direction. The second and third layers from the top also experience a displacement at this point, but by a much smaller amount. Just before the tip reaches a displacement of 3.5 Å,the top layer experiences another sudden displacement by a small amount, by this time the large displacement is also seen in the second and third layers as well. Throughout this motion the fourth layer remains mostly unperturbed.
Finally, in Figure 6(a), the largest tip with three top layers fixed is shown during sliding at a distance of approximately 1 Å between the bottom of the tip and the top graphene layer. At this distance, the vertical load as a function of sliding distance along the y direction is shown in Figure 6(b) with an average value of 4.92 nN. The lateral friction force as a function of sliding distance, on the other hand, is shown in Figure 6(c) with an average of 2.2 nN.
At this distance, the deformation in the tip is much smaller than the smaller tip. The upward motion of the atoms is much smaller and most importantly, the stick slip motion is much more visible.

4. Discussion

In this work, we have explored, through finite temperature classical molecular dynamics, the effect of the shape, size, and deformation of the tip on the AFM reading during sliding of the tip on a surface. Due to the desirable properties of layered materials as lubricants, we have used a few-layer graphene substrate as the surface and Au tips of various shapes and sizes.
In order to separate the effect of the shape from that of tip deformation, we have first considered three completely rigid tips of different shapes, namely a cone, a pyramid, and a hemisphere. The vertical load, sliding direction, and surface temperature have also been included as parameters of the problem. The shape is seen to effect all aspects of the AFM readings. Some notable observations can be summarized as follows: The conical tip, terminating in a single atom, shows identical behavior along x and y directions. The rigid hemisphere, on the other hand, shows three-to-four times more resistance to sliding along the y-axis than the x-axis. For the pyramidal tip, directional divergence of friction force behavior is moderate. The signature stick-slip behavior of atomic scale friction is more or less prominent depending on the shape of the tip, as detailed in the Results section.
The effect of deformation is explored using two hemispherical tips, one with only three layers fixed from the top and one completely rigid. The two tips start out identical but during sliding at different distances, the nonrigid tip experiences significant deformation. The mode of the deformation is the upwards motion of the lower layers of the tip and a flattening of the curved surface, thereby increasing the contact area. A comparison between the behavior of the rigid and nonrigid tip reveals a smaller average load and smaller average friction force for similar distances between the top layer of the tip and the top surface layer. This is, of course, due to the freedom of the lower layers that are free to move upwards and mitigate the vertical load.
The details of the deformation has further been analyzed by means of an indentation experiment. The same nonrigid hemispherical tip was monitored as it approaches a slightly thicker few-layer graphene surface. The mean-square displacement of the graphene layers, which was monitored throughout the experiment reveals that deformation occurs through sudden jumps of several atoms together that relieves the underlying graphene causing an upward jump. While the topmost layer of the graphene layer is effected the most, as the distance between the tip and the graphene surface decreases, the second and third layers are equally affected.
Finally, the size of the tip has also been explored. During sliding, the larger hemisphere exhibits a much smaller degree of deformation. At the same time, the stick-slip pattern is clearer than the smaller hemisphere.

Author Contributions

Conceptualization, C. M., D. T., and H. U.; formal analysis, C.M.; writing—original draft preparation, C. M., D. T., and H. U.; writing—review and editing, D. T. and H. U.; visualization, C. M.; supervision, H. U.; project administration, H. U.; funding acquisition, H. U., D.T. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by TUBITAK, The Scientific and Technological Research Council of Turkey (Grant No: 115F493). The computational resources were provided by the National Center for High Performance Computing (UHeM).

Data Availability Statement

Data is available upon request.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

Abbreviations
The following abbreviations are used in this manuscript:
PES Potential energy surface
LAMMPS Large-scale Atomic/Molecular Massively Parallel Simulator
EAM Embedded Atom Model
AIREBO Adaptive Intermolecular Reactive Empirical Bond Order
MD Molecular Dynamics

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Figure 2. Trends in average friction force as a function of temperature for three different rigid tips (top to bottom:cone, pyramid, hemisphere) for three different heights ( h 1 red lines, h 2 blue lines, h 3 green) and for sliding in two different directions (solid circles for x, crosses for y). The speed of sliding is 10 cm/s.
Figure 2. Trends in average friction force as a function of temperature for three different rigid tips (top to bottom:cone, pyramid, hemisphere) for three different heights ( h 1 red lines, h 2 blue lines, h 3 green) and for sliding in two different directions (solid circles for x, crosses for y). The speed of sliding is 10 cm/s.
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Figure 3. Left panel: Hemispherical tip with three top layers fixed, with the rest of the atoms free to move. Right panel: Completely rigid hemispherical tip. Red and yellow spheres represent the fixed and free Au atoms. Green and magenta spheres represent fixed and free carbon atom layers.
Figure 3. Left panel: Hemispherical tip with three top layers fixed, with the rest of the atoms free to move. Right panel: Completely rigid hemispherical tip. Red and yellow spheres represent the fixed and free Au atoms. Green and magenta spheres represent fixed and free carbon atom layers.
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Figure 4. The lateral friction force along the x-direction (top panels) and the y-direction (bottom panels) for a flexible (nonrigid) tip sliding with a speed of 10 cm/s at T = 300 K .
Figure 4. The lateral friction force along the x-direction (top panels) and the y-direction (bottom panels) for a flexible (nonrigid) tip sliding with a speed of 10 cm/s at T = 300 K .
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Figure 5. Top panels: Hemispherical tip in an indentation sequence descending onto a 6-layer graphene surface. Bottom panel: Mean square displacement of atoms on each of the top four graphene layers. Red and yellow spheres represent the fixed and free Au atoms. Green and magenta spheres represent fixed and free carbon atoms.
Figure 5. Top panels: Hemispherical tip in an indentation sequence descending onto a 6-layer graphene surface. Bottom panel: Mean square displacement of atoms on each of the top four graphene layers. Red and yellow spheres represent the fixed and free Au atoms. Green and magenta spheres represent fixed and free carbon atoms.
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Figure 6. (a) Large hemispherical tip sliding at speed 1 m/sec and at a temperature of 300 K with the closest distance between the tip and the surface being approximately1 Å. (b) Average vertical load as a function of sliding distance. (c) Average friction force as a function of sliding distance. Red and yellow spheres represent the fixed and free Au atoms. Green and magenta spheres represent fixed and free carbon atoms.
Figure 6. (a) Large hemispherical tip sliding at speed 1 m/sec and at a temperature of 300 K with the closest distance between the tip and the surface being approximately1 Å. (b) Average vertical load as a function of sliding distance. (c) Average friction force as a function of sliding distance. Red and yellow spheres represent the fixed and free Au atoms. Green and magenta spheres represent fixed and free carbon atoms.
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Table 1. Tip heights h 1 , h 2 and h 3 (in Å) for the three rigid tips. The definition of these heights are given in the text.
Table 1. Tip heights h 1 , h 2 and h 3 (in Å) for the three rigid tips. The definition of these heights are given in the text.
Heights Pyramidal Conical Hemispherical
h 1 28.36 24.27 26.35
h 2 28.86 24.77 26.85
h 3 29.36 25.27 27.35
Table 2. Average vertical load, F N , in nN for the three rigid tips at T=1 K, 100 K, 300 K for three heights and two sliding directions. h 1 , h 2 and h 3 are defined in Table 1. The sliding speed is 10 cm/sec.
Table 2. Average vertical load, F N , in nN for the three rigid tips at T=1 K, 100 K, 300 K for three heights and two sliding directions. h 1 , h 2 and h 3 are defined in Table 1. The sliding speed is 10 cm/sec.
Along x Along y
Pyramid 1 K 100 K 300 K 1 K 100 K 300 K
h 1 2.521 2.636 2.829 2.525 2.643 2.833
h 2 0.264 0.348 0.510 0.282 0.362 0.515
h 3 -0.904 -0.856 -0.756 -0.878 -0.830 -0.723
Cone 1 K 100 K 300 K 1 K 100 K 300 K
h 1 8.072 8.300 8.727 8.034 8.281 8.714
h 2 2.90 2.466 2.799 2.252 2.420 2.738
h 3 -0.782 -0.664 -0.401 -0.815 -0.688 -0.427
Hemisphere 1 K 100 K 300 K 1 K 100 K 300 K
h 1 16.697 17.253 18.270 16.567 17.111 18.131
h 2 2.116 2.526 3.27 2.210 2.619 3.379
h 3 -4.010 -4.590 -4.886 -3.908 -4.496 -4.792
Table 3. Average lateral friction force, F f , in nN for the three rigid tips at T=1 K, 100 K, 300 K for three heights and two sliding directions. h 1 , h 2 and h 3 are defined in the text. The sliding speed is 10 cm/sec.
Table 3. Average lateral friction force, F f , in nN for the three rigid tips at T=1 K, 100 K, 300 K for three heights and two sliding directions. h 1 , h 2 and h 3 are defined in the text. The sliding speed is 10 cm/sec.
Along x Along y
Pyramid 1K 100K 300K 1K 100K 300K
h 1 0.0960 0.1312 0.1883 0.0811 0.1276 0.1953
h 2 0.0824 0.0933 0.1165 0.0646 0.0822 0.1103
h 3 0.0384 0.0427 0.0519 0.0303 0.0359 0.0464
Cone 1K 100K 300K 1K 100K 300K
h 1 0.0164 0.1240 0.2246 0.2815 0.3031 0.3537
h 2 0.0121 0.0641 0.1175 0.1744 0.1831 0.2085
h 3 0.0160 0.0286 0.0509 0.0896 0.0951 0.1079
Hemisphere 1K 100K 300K 1K 100K 300K
h 1 0.0638 0.1979 0.3481 1.1776 1.0498 1.1520
h 2 0.0573 0.1179 0.2042 0.6938 0.7159 0.7563
h 3 0.0242 0.0597 0.1095 0.4146 0.4277 0.4564
Table 4. Average loads, F N , and lateral friction, F f , forces in nN for the nonrigid hemisphere shape at three distances and at a temperature of T = 300 K .
Table 4. Average loads, F N , and lateral friction, F f , forces in nN for the nonrigid hemisphere shape at three distances and at a temperature of T = 300 K .
Along x Along y
Height [Å] F N F f F N F f
h 4 5.780 0.305 6.843 1.175
h 5 1.669 0.254 1.731 1.086
h 6 -2.299 -0.221 3.012 0.689
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Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
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