We investigate the stability of low-friction sliding of nanocrystal with rectangular atomic arrangement on rectangular lattices, for which analytical results can be obtained. We find that several incommensurate periodic orbits exist and are stable against thermal fluctuations and other perturbations. As incommensurate orientations lead to low corrugation, and therefore low friction, such incommensurate periodic orbits are interesting for the study of nanotribology. The analytical results compare very well with simulations of W nanocrystals on NaF(001). The geometry and high typical corrugation of substrates with square lattices increase the robustness when compared to typical hexagonal lattices, such as graphite.
1 Introduction
Friction at the atomic scale is currently actively studied [1]. One of the goals of this research is to understand whether extremely low friction can be obtained by an appropriate choice of the sliding conditions. Commensurability between the sliding lattices is one of the elements that determine friction. For a purely incommensurate infinite contact, theoretical arguments suggest that static friction should vanish [2, 3]. However, very low friction has been measured also for finite incommensurate contacts at very low velocities [4, 5] and this effect has been called superlubricity [6, 7, 8, 9]. Atomic force microscope (AFM) study [4, 5] has shown that the sliding of graphite flakes on graphite can occur with very low friction, depending strongly on the relative orientation. At the same time, rotation of the flake can lead to a rapid increase of friction and stick–slip motion, corresponding to a locking into a commensurate orientation. States of very different friction have also been observed to coexist in the sliding of nanoparticles [10] and have been attributed to contamination or amorphous surfaces.
This article examines theoretically the sliding of nanocrystals and substrates which both have different, but regular rectangular lattices. By means of a simple, analytically soluble model, we show that stable orientations exist for any size nanocrystal, and derive some general properties, showing how the stability depends on the scan line. The stable orientations are independent of the velocity and nearly independent of the corrugation, but depend only on the geometry of the substrate and contact layer of the nanocrystal. We can also estimate the energy barriers necessary to rotate the nanocrystal from a given orientation to another.
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We apply the results to W nanocrystals on an NaF(001) substrate. NaF has been studied extensively as a substrate in the context of nanotribology [12, 13] and W is commonly used for AFM tips (see, for instance Ref. [4]). Both materials have a bcc lattice structure and the (001) surfaces in contact can thus be described by one finite and one infinite square lattice. The lattice parameters of the two materials are very similar (0.31585 and 0.32668 nm, respectively), which ensures the existence of nearly commensurate orientations for small enough nanocrystal. For W, the corrugation on NaF(001) is of the order of 1 eV, a value typical for many substrate systems (see, for instance Ref. [12]).
In this article, we apply a model previously introduced to study the dynamics of hexagonal flakes on hexagonal lattices [7] to rectangular contact layers on substrates with rectangular lattices. One of the important results is that the inherent lack of robustness of the low-friction motion found in the hexagonal graphite system for particular scan lines is not present for rectangular lattices, which opens the possibility for practical applications of low-friction sliding. The analytical results of the simplified model are supported by our numerical simulations of W nanocrystal on NaF.
In Sect. 2, we introduce the notation whilst briefly reviewing the model used in Ref. [7] to describe the relation between the rotational dynamics and the friction. In Sect. 3, this model is applied to a general system consisting of a rectangular lattice and a nanocrystal with a rectangular contact layer. The analytical results are compared to numerical simulations of W nanocrystal on NaF(001) in Sect. 4 and finally the implications of our results are discussed in Sect. 5.
2 The Simplified Model of Friction and Rotation
We wish to study the dynamics of small rigid nanocrystals of atoms arranged in a regular lattice on a regular substrate, as sketched in Fig. 1. The contact layer consists of one atomic plane and lies in the x–y plane parallel to the periodic substrate. For matched lattice parameters, by changing the orientation ϕ of the nanocrystal on the substrate, the contact layer can be either commensurate or incommensurate with the substrate atoms. The incommensurate states which produce low friction can be destroyed by rotations around the z axis that lead to a locking in a commensurate orientation [5].
Fig. 1
A schematic representation of a nanocrystal on a substrate. Only the bottom layer (dark) of the nanocrystal interacts with the substrate
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Each atom of the contact layer is subjected to a periodic potential due to the substrate and to an external load force which may be applied to the nanocrystal. Additionally, the contact layer atoms experience a friction force due to dissipation of kinetic energy into phonon modes of the substrate crystal. The remaining nanocrystal atoms are further away from the substrate and we may therefore assume that they couple only to the other nanocrystal atoms. The AFM cantilever or a similar device used to investigate friction can be modelled by coupling the centre of mass of the nanocrystal harmonically with a spring with force constant c to a support moving at constant velocity vs. The friction is given by the average value of the lateral force Fs, exerted by the spring on the nanocrystal.
In three dimensions, a rigid nanocrystal is thus left with only 6 degrees of freedom: the coordinates of the centre of mass, and the orientation. Despite this, the system is still too complicated to perform the stability analysis analytically. In Ref. [7], a simple model was proposed to describe the rotation of hexagonal graphite flakes on hexagonal graphite lattices and its consequences for low-friction sliding. We briefly summarise this model and some of the results of Ref. [7] here in a general context.
A possible simplification of the system is suggested by the nature of the dynamics. We are interested in the rotation around the z axis, which affects the commensurability, and therefore the friction. Hence, the two relevant degrees of freedom are the position of the centre of mass along the scan line, x, and the orientation, ϕ. Consequently, instead of the full substrate potential V(x, y, z), we introduce a potential V(x, ϕ), one example of which is given in Fig. 2, where one can see that the corrugation felt by the nanocrystal as a whole decays quickly away from the commensurate orientations ϕ = 0, 90°. Such a model is fully described by the initial support position xs0, support velocity vs, mass M and moment of inertia I of the nanocrystal, the effective potential V(x, ϕ), and the viscous friction coefficient γ of a contact layer atom on the substrate. The parameter γ represents the fact that the substrate acts as a heat bath, removing energy provided by the driving. Altogether, this model is similar to the Tomlinson model [11], but with one additional degree of freedom which accounts for the rotational dynamics. The details of the substrate and contact layer lattice geometry, choice of scan line, and the applied external load force are accounted for in V(x, ϕ) [7].
Fig. 2
An example of the shape of V(x, ϕ) as obtained from the potential VA(x, y) in Eq. 15 at constant y and z, summed over the atoms of a nanocrystal with square contact layer of 4 × 4 unit cells (and therefore 5 × 5 atoms) with lattice parameter 0.31585 nm on a square lattice with corrugation V1 = 1 eV and V2 = V3 = V1/2 and lattice parameter 0.32668 nm. These parameters correspond to W(001) and NaF(001)
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We write the equations of motion of the simplified system as a dynamical system of first-order differential equations for the position x, velocity vx, orientation ϕ and angular velocity ω,
$$ \dot{x} = v_x, $$
(1)
$$ M \dot{v}_x = -{\frac{\partial V(x,\phi)}{\partial x}} - c (x-t v_{{\rm s}} - x_{\rm s}^0) - \gamma M v_x, $$
(2)
$$ \dot\phi = \omega, $$
(3)
$$ I \dot\omega = - {\frac{\partial V({x},\phi)}{\partial \phi}} -\gamma I \omega. $$
(4)
In a full description which includes the internal degrees of freedom of the nanocrystal, the driven motion of the centre of mass can also excite internal vibrations, leading to an additional damping, which can be incorporated in γ. Therefore, the above equations are a good approximation also for non-rigid crystals.
Some general properties of the potential energy V(x, ϕ) can be derived from the symmetries of the substrate lattice and contact layer. The translation symmetry of the substrate dictates that V(x, ϕ) must be periodic in x. A good representation of V(x, ϕ) is therefore given by
where U(ϕ) and W(ϕ) are both smooth functions that represent the average value of the potential energy and the amplitude of the modulation, respectively.
Furthermore, rotational symmetries of the substrate and contact layer lead to rotational symmetries of V(x, ϕ) of the form
$$ V(x,\phi) = V(x, \phi_{\rm symmetry}+\phi), $$
(6)
where ϕsymmetry is an angle of rotation under which the contact layer is symmetric.
These symmetries of the potential V(x, ϕ) further imply that
For the case we consider here, square lattices, we show later in Sect. 3.1 that also
$$ U(\phi) = U(-\phi), $$
(9)
$$ W(\phi) = W(-\phi). $$
(10)
This also holds for some combinations of parameters of rectangular lattices. In turn, these equations imply that U and W have extrema in ϕ = ϕ0 = 0, ϕsymmetry/2. Since the torque, given by Eq. 4, vanishes for ω = 0, ϕ = ϕ0 these conditions define a two-dimensional invariant manifold of the dynamics. If the nanocrystal is on the manifold, it will remain there, and keep its orientation. In general, there may be more invariant manifolds at other orientations, if U and W have additional extrema which coincide.
If the orientation of an invariant manifold is incommensurate, the corrugation is relatively small (see Fig. 2) and the friction force on the nanocrystal is small as well. Such incommensurate invariant manifolds are therefore interesting for the study of nanotribology.
In Fig. 3, the consequences for friction are shown for a typical system consisting of a W nanocrystal with (001) face on NaF(001) as described by simulations of the full two-dimensional system with the single-atom interaction potential of Eq. 15. The orientation of the nanocrystal remains nearly constant and the motion in the x direction exhibits stick–slip behaviour which is relatively mild for incommensurate orientations and corresponds to a relatively low-average friction force.
Fig. 3
Typical simulated trajectories of a W nanocrystal containing 4 × 4 × 4 unit cells (contact layer 5 × 5 atoms) on NaF(001) in the absence of thermal fluctuations, at y = a/2, being pulled in the x direction. The orientation remains nearly constant (a) and shows only small periodic fluctuations with period a/vs. Both the commensurate and incommensurate orientations display stick-slip behaviour in the position (b), but the friction (c) is significantly higher in the commensurate state. Further details of the simulations are described in Sect. 4
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The existence of incommensurate invariant manifolds does not necessarily mean that low friction can be observed under experimental conditions. The invariant manifolds may not be stable, namely, the nanocrystal may rotate away from it if it is at an orientation which deviates slightly from the invariant manifold. We must therefore investigate the stability and robustness of the invariant manifolds.
Consider a general potential V(x, ϕ) which has an invariant manifold at ϕ = ϕ0, i.e.
where 〈〉t denotes the time average of a quantity on a typical trajectory on the invariant manifold. If these growth rates are smaller than zero, the invariant manifold is stable. This is the case if the time average of the potential energy is at a local minimum, i.e.
The stability is thus determined by the functions U and W, and how much time the particle spends near the minima of the potential, where the cosine is negative.
In stick–slip motion, the particle spends most of its time in the minima of the potential, i.e. where the cosine is smaller than zero (see Fig. 3). In extreme cases, 〈cos〉t may be almost equal to −1. If the motion is truly superlubric, then the particle spends about the same time in the minima as it does in the maxima. If the motion is nearly superlubric, then the particle spends most of its time in the minima. Hence, for realistic cases 〈cos〉t < 0.
If the offset of the potential, U(ϕ), has a minimum at ϕ0 it contributes positively towards the stability. Similarly, if the amplitude W(ϕ) is at a maximum at ϕ0 the stability is enhanced, because the second derivative is multiplied 〈cos〉t, which is a negative number. A minimum of U and maximum of W therefore always lead to stability, whereas a maximum of U and minimum of W always lead to instability. If both are at a maximum, or both are at a minimum at ϕ0, then the stability is not directly obvious.
The representation of the potential used in this section not only gives qualitative understanding, but also allows for quantitative predictions about the dynamics, as is shown by comparison to full numerical simulations in Sect. 4.
3 U and W for Rectangular Lattices
The functions U(ϕ) and W(ϕ) determine the existence and stability of the low-friction states. We will now investigate these functions for a range of commonly occurring substrate and lattice combinations with rectangular unit cells.
Consider a two-dimensional substrate consisting of a rectangular lattice, see Fig. 4. The potential energy of an atom of the contact layer at position (X, Y) due to the presence of the substrate can be written in the general form
A top view of a general rectangular lattice (open circles) and contact layer (filled circles) with mismatch angle ϕ, lattice parameters a1, a2, b1, b2, contact layer size m, n, and the potential energy of a contact layer atom on the substrate. If a contact layer atom lies on top of a substrate atom, its potential energy is V1. If it lies directly between an atom and its nearest neighbour in the x or y direction, it has potential energy V2 or V3 respectively. If it lies in the centre of a rectangle, at equal distance from four substrate atoms, without loss of generality, we may set the potential energy to 0. The origin of the coordinate system is chosen to lie on top of a substrate atom
×
Because the potential is periodic in X with period a1, this can be rewritten in a form similar to Eq. 5,
with UA(Y) and WA(Y) the average potential energy and corrugation of a single atom travelling in the x direction at constant y = Y, respectively. They can be determined from the potential VA(x, y) in Eq. 15,
The nanocrystal is rigid, rectangular and the contact layer consists of m × n atoms. The total potential energy of the contact layer depends on the position (x, y) of the centre of mass and orientation ϕ, and can be expressed as a sum over the potential energies of the individual contact atoms,
where (Xjk, Yjk) is the position of the atom in the j-th column and k-th row of the contact layer. For a rectangular three-dimensional nanocrystal the centre of mass is directly above the centre of the contact layer, i. e.,
where Uy and Wy are the average potential energy and corrugation of a contact layer at orientation ϕ moving on a scan line along the x direction for a specific y. They can be obtained from UA and WA as,
Substituting Eq. 22 and explicitly performing the sums yields
$$ U_{y}(\phi) = m n {\frac{V_{1}+V_{2}+V_{3}}{4}} + {\frac{V_{1}+V_{2}-V_{3}}{4}} \cos\left(2 \pi {\frac{y}{a_{2}}}\right) {\frac{\sin\left(\pi n {\frac{b_{2}}{a_{2}}} \cos\phi\right) \sin\left(\pi m {\frac{b_{1}}{a_{2}}} \sin\phi\right)}{\sin\left(\pi {\frac{b_{2}}{a_{2}}} \cos\phi\right)\sin\left(\pi {\frac{b_{1}}{a_{2}}} \sin\phi\right)}}, $$
(26)
$$ \begin{aligned} W_y(\phi) = {\frac{(V_1-V_2+V_3)}{4}} {\frac{\sin\left(\pi m {\frac {b_1} {a_1}} \cos\phi\right) \sin\left(\pi n {\frac{b_2}{a_1}} \sin\phi\right)}{\sin\left(\pi {\frac{b_1}{a_1}} \cos\phi\right)\sin\left(\pi {\frac{b_2}{a_1}} \sin\phi\right)}}\\ &+{\frac{V_1-V_2-V_3}{4}} \cos\left(2 \pi {\frac{y}{a_2}}\right) {\frac{\sin\left[\pi n b_2\left( {\frac{\cos\phi}{a_2}}-{\frac{\sin\phi}{a_1}}\right)\right] \sin\left[\pi m b_1\left( {\frac{\cos\phi}{a_1}}+{\frac{\sin\phi} {a_2}}\right)\right]}{\sin\left[\pi b_2\left( {\frac{\cos\phi}{a_2}}-{\frac{\sin\phi}{a_1}}\right)\right] \sin\left[\pi b_1\left( {\frac{\cos\phi}{a_1}}+{\frac{\sin\phi}{a_2}}\right)\right]}}, \end{aligned}$$
(27)
where we have used that \(\sum_{l=0}^{d-1} \exp\{2 i x [l -(d-1)/2]\} =\hbox{sin}(dx)/\hbox{sin}(x)\)
We consider the shape of these two functions U and W for the important case of square lattices in the next section.
3.1 Square Contacts on Square Lattices
Let us consider square contact layers \((m=n\equiv d)\) of atoms arranged in a square lattice \((b_1=b_2\equiv b)\) on a substrate with a square lattice \((a_1=a_2 \equiv a, V_2=V_3).\) For this case Eqs. 26 and 27 become
$$ {w}(\phi) = {\frac{V_1}{4}} {\frac{\sin\left(\pi d {\frac{b}{a}} \cos\phi\right) \sin\left(\pi d {\frac{b}{a}} \sin\phi\right)} {\sin\left(\pi {\frac{b}{a}} \cos\phi\right) \sin\left(\pi {\frac{b}{a}} \sin\phi\right)}}. $$
(30)
$$ w_1(\phi) = {\frac{(V_1-2V_2)}{4}} {\frac{\sin\left[\pi d {\frac{b}{a}}\left(\cos\phi-\sin\phi\right)\right] \sin\left[\pi d {\frac{b}{a}}\left( \cos\phi+\sin\phi\right)\right]} {\sin\left[\pi {\frac{b}{a}}\left( \cos\phi-\sin\phi\right)\right] \sin\left[\pi {\frac{b}{a}}\left( \cos\phi+\sin\phi\right)\right]}}. $$
(31)
These equations are symmetrical under the transformation ϕ → − ϕ. The second term on the right hand side of Eq. 29, w1(ϕ), is proportional to (V1 − 2V2)/4, which is generally small, and in many cases taken to be 0 (see, for instance, the substrate potentials used in Ref. [12]). We therefore initially neglect the second term and focus on the first term and write
$$ W_y(\phi) = {w}(\phi). $$
(32)
The function w(ϕ) then completely determines U and W for all scan lines, and therefore the invariant manifolds as well as their stability. As the first term of Uy(ϕ) is constant, the extrema of Uy coincide approximately with the maxima of |Wy|, leading to invariant manifolds at the corresponding orientations,
In Fig. 5 examples of w(ϕ) are plotted for W on NaF(001) for several values of d.
Fig. 5
The function w(ϕ) which determines Uy and Wy (see Eqs. 28 and 32, and thus the invariant manifolds and their stability, is plotted for various values of d and b/a = 0.96685, corresponding to a W nanocrystal with its (001) face on NaF(001). The number of extrema increases with d. The maximum size of the extrema for d →∞ is also plotted. In general, if d is even, w(ϕ) has a minimum at 0, and if d is odd a maximum. For d = 5, besides the maxima at ϕ = 0° and ϕ = 45°, w(ϕ) has minima at \(\phi\approx 16.95^{\circ},36.25^{\circ},\) and a maximum at \(\phi\approx 28.32^{\circ}.\) For d = 10, besides the minimum at ϕ = 0° and the maximum ϕ = 45°, w(ϕ) has minima at \({\phi\approx}14.33^{\circ}, 22.98^{\circ}, 32.56^{\circ}, 40.86^{\circ}\) and maxima at \(\phi\approx 8.39^{\circ}, 19.51^{\circ}, 27.69^{\circ}, 36.34^{\circ}.\) Each of the extrema corresponds to an invariant manifold
×
The extrema of U and W give the invariant manifolds. One of the invariant manifolds is at ϕ = 0, whilst the others are at incommensurate orientations, which do not depend on the scan line y. Additionally, the nodes in the amplitude (Wy(ϕ) = 0) all occur at the same value of Uy(ϕ) = d2 (V1 + 2V2)/4. The number of such nodes can be determined from the number of zeros of the numerator, and in the range of \(\phi \in \langle 0^{\circ},45^{\circ}\rangle \) for a = b, this number is in general equal to d − 1, except when the zeros coincide, which occurs for instance when d2 can be written as the sum of the squares of two integers.
As w(ϕ) is independent of the position y of the scan line, the orientations at which invariant manifolds occur are independent of the scan line whilst their stability is controlled entirely by the prefactor cos(2π y/a) in Uy. The requirement of stability, Eq. 14, implies that each minimum of Uy(ϕ) leads to a stable invariant manifold, which in turn means that there are at least (d − 1)/2 stable incommensurate periodic orbits if a = b. If the scan line changes and the prefactor cos(2π y/a) becomes small, the amplitude term of W in Eq. 13 becomes dominant, and the invariant manifolds at the maxima of U become stable as well. If the prefactor cos(2π y/a) changes sign, Uy changes sign, so that the maxima of U becoming minima and the minima maxima. The invariant manifolds which are stable for cos(2π y/a) > 0 then become unstable for cos(2π y/a) < 0, and those which were unstable become stable. Depending on the sign and size of cos(2π y/a), there are therefore generally between (d − 1)/2 and d − 1 stable incommensurate periodic orbits. In cases of very strong stick-slip behaviour, the centre of mass spends most of the time in the potential minimum and \(\langle \hbox{cos}\rangle_t \) is almost equal to −1. In this case, the invariant manifold is stable for nearly all scan lines.
When d becomes large, the numerator in w(ϕ) oscillates rapidly. However, the denominator is independent of d and determines the size of the maxima and minima (see Fig. 5),
This function therefore estimates the robustness of the incommensurate orientations, i. e. the typical energy barrier that must be overcome to rotate from one to another nearby stable orientation,
$$ \Updelta E (\phi) \approx w_{\rm max}(\phi). $$
(35)
In the case of the graphite flakes on graphite, it was found that the incommensurate orientations are not very robust [7]. However, graphite has a very small corrugation (around 25 meV, comparable to kBT at room temperature) compared to NaF(001), which has a corrugation of around 1 eV. The typical energy barrier for W on NaF(001) is around 0.4 eV even at 45°. Consequently, the low-friction incommensurate states of this system are robust against thermal fluctuations at room temperature. Additionally, due to the much larger mass of the W atoms and large M and I, the incommensurate states are also more robust against low support velocities than in graphite.
3.2 Rectangular Lattices or Contact Areas
For a general rectangular lattice a1 ≠ a2, or a rectangular contact area, n ≠ m or b1 ≠ b2, the expressions for Uy and Wy [see Eqs. 26 and 27] cannot be simplified as done for square lattices in Sect. 3.1. Therefore, it is not straightforward to identify periodic orbits, as the extrema of these two functions do not coincide. Nevertheless, as for the case studied in Sect. 4.2, some periodic orbits likely still exist and the functions Uy, and Wy can still provide information regarding the possible locations of periodic orbits.
3.3 A Note on Hexagonal Lattices
The results presented here can be applied to determine U and W for a graphite flake subject to the potential of Ref. [5], which consists of a sum of two rectangular lattices, whilst a hexagonal flake can be written as the sum of several rectangular flakes.
Similar behaviour, including the existence of the transition scan line where cos(2π y/a) = 0 was observed for hexagonal flakes of different sizes on a hexagonal lattice [7]. The diameter for hexagonal flakes is \(2\sqrt{N/6},\) and the total number of invariant manifolds is \(2\sqrt{N/6}+1.\) It should be noted, however, that in hexagonal lattices, due to the symmetry, the corrugation vanishes for some scan lines, whereas this does not generally happen for square lattices. This low Wy in hexagonal lattices tends to occur at scan lines where the minimum in Uy at the incommensurate orientation is not very pronounced, causing the incommensurate states to become only very weakly stable [7] and not robust.
4 W Nanocrystals on NaF(001)
The results of the previous section are applicable to many systems that present square lattices, like the (001) surfaces of fcc and bcc materials. In this work, as a representative system, we have chosen a substrate which has been widely studied experimentally, NaF(001)[13]. Due to the bcc structure of the NaF, the surface atoms of the substrate are arranged in a square lattice with lattice parameter 0.32668 nm. Additionally, the corrugation of NaF, around 1 eV, is typical for many substrate systems. Unless otherwise mentioned, in this work corrugation parameters of V1 = 2 V2 = 2 V3 = 1eV will be used.
For the nanocrystal, we have used W, which is also common in experiments as a tip material. W has a bcc structure and the (001) contact layer consists of a square lattice, with lattice parameter 0.31585 nm. As the lattice parameters of the contact layer and substrate are very similar, there exist orientations at which the two lattices are nearly commensurate, as long as the contact layer is small.
This system is thus extremely suitable for studying the effects of commensurability on the friction of small nanocrystals. As the W atoms are quite heavy, the total mass and moment of inertia are large, and the motion of the nanocrystal should behave approximately one-dimensionally and the simple model described in Sects. 2 and 3 is applicable. The function w(ϕ) (Eq. 31), plotted for several cases in Fig. 5 determines the stable orientations. These orientations are mentioned in the caption for d = 5 and d = 10.
4.1 Simulations
We have performed molecular dynamics simulations of rigid W nanocrystals on NaF(001) based on the full potential VA(X, Y) in Eq. 15. The nanocrystals were coupled harmonically (spring constant c = 10N/m) to a support moving at constant velocity vs = 32.668 mm/s (or 0.1 a/ps) in the positive x direction at constant y = ys. Note that this is several orders of magnitude larger than the velocities typically used in AFM experiments, which are of the order of 1 μ/s. These values were chosen for computational reasons, but the dynamics of the system at lower velocities are represented well, as the time scales associated with the interaction between the substrate and nanocrystal are still much shorter than the time it takes the support to traverse one unit cell. For the viscous friction parameter of the substrate we have chosen the typical value of γ = 1/ps. For simplicity we have restricted ourselves to cubic nanocrystals.
In Fig. 3, several examples are shown of simulated trajectories for d = 5. The orientations are nearly constant, but may fluctuate periodically with a period equal to the time it takes the support to move one lattice spacing. These fluctuations are due to the motion in the y direction.
In Fig. 6 the results of 11739 simulations of the same system for different initial conditions and ys are shown. The final angles is plotted as a function of the scan line and initial angle, demonstrating the existence and stability of the periodic orbits at specific orientations. These orientations correspond exactly to the extrema of w(ϕ) for this system, shown in Fig. 5.
Fig. 6
A bifurcation diagram for a cubic W nanocrystal with a 5 × 5 contact layer on NaF(001) at support velocity vs = 32.668 mm/s. Simulations were run starting from initial conditions x = − a/4, y = ys, vx = vs, vy = 0, ω = 0 and a range of initial orientations between 0 and 45°, with 0.5° intervals. The final angle is plotted in black. The set of initial angles which converge to the various periodic orbits, i.e. cross sections of the basins of attraction, are plotted in different colours for each stable orientation. Apart from ϕ = 0°, 45°, stable orientations occur at \(\phi\approx16.95^{\circ},28.32^{\circ},36.25^{\circ},\) which correspond exactly to the extrema of w(ϕ) for this system, shown in Fig. 5
×
The sets of initial angles which converge towards these stable orientations are also indicated in Fig. 6. It can be seen that the orientations at 16.95° and 36.35° are more stable at ys = 0, as expected from minima of w(ϕ), whilst 0°, 28.32°, and 45°, which correspond to maxima, are, as expected, more stable at ys = a/2.
In Fig. 7, the same plot is repeated for d = 10. There are more stable orientations, and these correspond exactly to the extrema of w(ϕ) for d = 10, shown in Fig. 5. Their behaviour near ys = 0 and ys = a/2 is also exactly as expected from w(ϕ).
Fig. 7
A bifurcation diagram and basins of attraction of a system similar to the one of Fig. 6 apart from the size of the W nanocrystal, which is still cubic, but has a 10 × 10 atom contact layer. Apart from ϕ = 0°, 45°, stable orientations occur at \(\phi\approx 8.39^{\circ}, 14.33^{\circ}, 19.51^{\circ}, 22.98^{\circ}, 27.69^{\circ}, 32.56^{\circ}, 36.34^{\circ}, 40.86^{\circ}\)
×
The orbit at ϕ = 36.35° in Fig. 6 breaks down for some 0 < ys < a/4. For these scan lines the theoretical approach described in this paper predicts that this orientation is stable. However, it is impossible to explore all possible sets of initial conditions in simulations, and thus the bifurcation diagrams are necessarily incomplete. At these scan lines, the range of initial conditions chosen for the simulations does not intersect with the basin of attraction of the stable periodic orbit at ϕ = 36.35°.
The comparison of these numerical simulations to the predictions of the analytical model is exceedingly good, including the important feature that several incommensurate orbits are stable for all scan lines.
4.2 The Case V2 = V3 ≠ V1/2
Though in most cases of square lattices V2 = V1/2 (see, for instance Ref. [12]), this may not always be true. We therefore briefly consider the implications of V2 = V3 ≠ V1/2, where, in Eq. 29, the second term can no longer be neglected. In Fig. 8w(ϕ) and w1(ϕ), which determine Uy(ϕ) and Wy(ϕ) through Eqs. 28 and 29 are plotted. These two functions together determine the invariant manifolds and their stability. The extrema of U and W no longer coincide exactly, and so the location of the invariant manifolds is much less trivial than before.
Fig. 8
The functions w(ϕ) and w1(ϕ) are plotted for b/a = 0.96685, corresponding to a system of W(001) on NaF(001) for d = 5 and V2 = V1/3. w(ϕ) is the same as in Fig. 5, but Wy(ϕ) is no longer equal to it. Besides the maximum at ϕ = 0° and minimum at ϕ = 45°, Wy(ϕ) has minima at \(\phi\approx 11.73^{\circ},23.83^{\circ},\) and maxima at \(\phi\approx 17.63^{\circ}, 32.39^{\circ}\)
×
In Fig. 9, the plot of Fig. 6 is repeated for V2 = V1/3. As both w(ϕ) and Wy(ϕ) still have an extremum near \(\phi\approx17^{\circ},\) the orbit at that orientation persists, but is changed slightly. The orbits near 28° and 36° also still exist, in distorted form, because the second term in Wy, determined by \(w_1(\phi)\propto (V_1-V_2)/V_2,\) is relatively small compared to Uy(ϕ), so that Uy(ϕ) is the dominant contribution to the stability, regardless of the scan line. This is in general the case, unless V2 deviates very strongly from V1/2.
Fig. 9
The plot of Fig. 6 repeated with V2 = V1/3. Besides at ϕ = 0°, 45°, stable orientations occur at \(\phi\approx 16.89^{\circ},27.9^{\circ},36.59^{\circ}\)
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5 Conclusions
We have analytically investigated the driven nonlinear dynamics of general rectangular nanocrystals on rectangular lattices and their relation to sliding friction. We have formulated an approximate analytical model, which gives the conditions for the existence and stability of incommensurate sliding states with low friction. We show that the number of incommensurate orbits grows linearly with the diameter of the nanocrystal and that for realistic systems, several states can be robust against thermal fluctuations, change of scan line, and driving velocity. The geometry and high typical corrugation of substrates with square lattices increase the robustness compared to typical hexagonal lattices, such as graphite. Moreover, unlike hexagonal lattices, square lattices do not have a scan line where the stability becomes extremely weak. This has implications for experiments where many different scan lines are explored, and thus a nanocrystal at a stable orientation for one scan line would be forced to rotate when the apparatus switches to another scan line, as was noted for the specific case of graphite flakes on graphite in Ref. [7].
Together with the increased moment of inertia for a nanocrystal, our results suggest that it should be easier to experimentally observe incommensurate sliding for three-dimensional nanocrystals with bcc or fcc structure than for hexagonal graphite flakes.
Acknowledgments
ASW’s work is financially supported by a Veni grant of Netherlands Organisation for Scientific Research (NWO). We would like to thank Claudio Fusco for useful discussions and suggestions.
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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