Friction of Physisorbed Nanotubes: Rolling Or Sliding?

Friction of Physisorbed Nanotubes: Rolling or Sliding?

Davide Mandelli1 and Roberto Guerra2, ∗ 1Atomistic Simulations, Italian Institute of Technology, via Morego 30, Genova 16163, Italy 2Center for Complexity and Biosystems, Department of Physics, University of Milan, via Celoria 16, 20133 Milano, Italy The structure and motion of carbon and h-BN nanotubes (NTs) deposited on graphene is inquired theoretically by simulations based on state-of-the-art interatomic force fields. Results show that any typical cylinder-over-surface approximation is essentially inaccurate. NTs tend to flatten at the interface with the substrate and upon driving they can either roll or slide depending on their size and on their relative orientation with the substrate. In the epitaxially aligned orientation we find that rolling is always the main mechanism of motion, producing a kinetic friction linearly growing with the number of walls, in turn causing an unprecedented supra-linear scaling with the contact area. A 30 degrees misalignment raises superlubric effects, making sliding favorable against rolling. The resulting rolling-to-sliding transition in misaligned NTs is explained in terms of the faceting appearing in large multi-wall tubes, which is responsible for the increased rotational stiffness. Modifying the geometrical conditions provides an additional means of drastically tailoring the frictional properties in this unique tribological system. DOI: 10.1039/D0NR01016B

I. INTRODUCTION walls12, has been reported of critical impact on their frictional and mechanical properties13. Specifically, for NTs Nanotubes (NTs), either made of carbon or hexagonal on flat surfaces, faceting can produce a fivefold increase boron-nitride (h-BN), have been investigated with enor- of the contact area, thus enhancing the adhesive forces mous interest in the last few decades due to their extraor- of the same proportion. dinary mechanical and electronic properties. Nowadays, All the above factors have a considerable influence on almost defectless NTs can be formed with lengths of 1 cm the dynamics of NTs laterally pushed over a flat surface, or more1, and precise measurements of their mechani- ultimately determining whether they will slide or roll. cal and frictional properties have started to appear in In principle, one would expect rolling motion to be hin- literature2,3, revealing a rich variety of unexpected phe- dered by the presence of facets. Hence, since only the nomena. While the empirical laws of macroscopic friction smallest MWNTs tend to keep a cylindrical shape, one are well known, the fundamental understanding of the might anticipate a rolling-to-sliding transition around the tribological mechanisms at the microscopic scale is still threshold size for faceting formation. On the other hand, under investigation from many points of view. The basic lateral friction forces generally increase with the contact difficulty of friction is intrinsic, concerning the dissipa- area, thus making rolling favorable in the macroscopic tive dynamics of systems with a large number of atoms, limit. Furthermore, changing the relative lattice orien- often involving ill-characterized sliding interfaces, which tation of nanoscale contacts can lead to gigantic effects are buried and thus experimentally hardly accessible. on their dynamical response to external forces14. These In a pioneering experiment by Falvo et al.4, both slid- considerations suggest that the apparently simple rolling- ing and rolling motion of multi-walled (MW) carbon NTs or-sliding question hides a complex interplay between on graphite were observed as a result of side pushing. many phenomena, each of them subtly depending on the Estimate of adhesion and of contact width based on NT/substrate contact geometry. Accurate simulations cylinder-on-flat models led to the explanation that lat- able to account for the NT internal degrees of freedom tice registry between the NT contact region and graphite and of many-body terms in the interatomic interactions was the cause of rolling in some cases. Due to the se- are therefore requested to shed light on the tribological vere difficulties in manipulating such small objects, other mechanisms taking place in these systems. investigations on these systems were mainly based on In this work we make use of molecular dynamics sim- atomistic simulations5–9, where to simplify the calcula- ulations to inquire theoretically the motion of carbon tions the NT was often modeled by a single-wall (SW) and h-BN DWNTs and MWNTs laterally pushed over

arXiv:2002.02253v2 [cond-mat.mes-hall] 16 Jun 2020 treated as a rigid cylinder. However, in recent years, new a graphene plane. For h-BN on graphene, the inherent evidences have recognized that SWNTs do not maintain lattice mismatch between the two materials enables us cylindrical shape when adsorbed on a surface, but rather to investigate different contact configurations, which un- tend to collapse in a “dogbone” like shape10,11. In the ravel a rich dynamical behavior. In particular, we antic- case of double-walled NTs (DWNTs) and of MWNTs in ipate that by controlling the NT size and their angular general, it has been recently shown that the interaction alignment relative to the substrate crystalline axis, one among the walls can be a source of spontaneous forma- can induce different types of motion: from pure rolling in tion of facets12. The effect of such faceting, strongly aligned geometries, to pure sliding in faceted, misaligned manifesting in large MWNTs (diameter D >∼ 10 nm) MWNTs, to a mixed rolling+sliding in collapsed mis- with small or no relative chiral angles between the NT aligned DWNTs. As discussed in the next, the resulting 2

n1 n2 D (A)˚ 15 20 27.5402 25 30 41.3103 35 40 55.0803 45 50 68.8504 55 60 82.6205 65 70 96.3906

Table I. The chiral indices n1, n2, and the nominal diameter D of the simulated (n1, n1)@(n2, n2) armchair h-BN DWNTs.

NTs are built with a boron-nitrogen bond length of ˚ √dBN = 1.442 A (honeycomb lattice constant aBN = 3dBN ≈ 2.498 A),˚ corresponding to the equilibrium value of the adopted Tersoff potential15. The lattice constant, ag, of the rigid graphene substrate is chosen so to closely reproduce the experimental lattice mismatch 16 aBN/ag ' 1.018 , while also allowing the matching of NT and graphene supercells along the NT axis direction y. In aligned contacts, the relevant zigzag y-periodicities Figure 1. (a) Perspective view and (b) cross section of some of the substrate and of the NT are equal to ag and aBN, of the double-walled and multi-walled h-BN nanotubes ph- respectively (see figure1(c), left panel). In this case, we 54 ˚ ysisorbed on graphene used in simulations. From left to right, fix ag = 55 aBN ' 2.452 A and we construct supercells we report the relaxed structures of the 65@70 DWNT and of of side Ly = 55ag = 54aBN ' 134.87 A.˚ In misaligned the 5@10, 5@10...@25 and 5@10...@45 MWNTs in the 30◦ contacts, the substrate√ y-periodicity changes to the arm- misaligned contact geometry. Carbon, boron and nitrogen chair value of 3ag (see figure1(c), right panel). In atoms are colored in gray, pink and blue, respectively. Panel this case, we set a = 2.4514 A,˚ which allows construct- (c) depicts a portion of NT/substrate interface in the aligned g √ and 30◦ misaligned configurations. ing commensurate supercells of side Ly = 10 3ag = 17aBN ' 42.46 A.˚ We note that the adopted lattice con- stants are close to the corresponding experimental values, exp ˚ exp ˚17 motion can be understood in terms of contact geometry ag ≈ 2.46 A and aBN ≈ 2.50 A . For all the considered and energetics. systems we use a graphene supercell side Lx ' 300 A,˚ Results for carbon nanotubes are provided and dis- which is large enough to avoid self-interaction of NTs cussed for comparison in SectionIIIC. with their periodic replica. TablesI andII report the ge- ometric parameters of the DW and MWNTs considered. Chiral indexes of BNNTs have been chosen so to obtain II. SYSTEM AND METHODS inter-wall distances close to the theoretical equilibrium bulk distance of ' 3.34 A˚ of the adopted inter-layer po- 18,19 Numerical modeling – The model systems studied tential . The reported nominal diameters D are for consist of armchair h-BN nanotubes of different sizes ph- the cylindrical configurations before relaxation. ysisorbed on a rigid graphene layer (see figure1(a),(b)). For brevity, in the text we label the NTs indicating a In our reference coordinates frame the graphene substrate single chiral index for each armchair wall, e.g., 15@20 ≡ lies in the (x,y) plane, while the NT axis is along y, par- (15, 15)@(20, 20). Following the discussion, the smallest allel to the zigzag axis. We consider two different contact 5@10 DWNT appears in the list of MWNTs. geometries. In “aligned” contacts, the zigzag direction of Simulation protocol – Boron-nitrogen intra-layer the graphene substrate is also oriented along y, while in interactions are modeled by a Tersoff potential as the 30◦ “misaligned” contacts, it is oriented along x (see parametrized in Ref. 27.Inter-layer interactions among figure1(c)). Periodic boundary conditions are applied the NT walls and between NT walls and graphene in x,y directions, using rectangular supercells of suit- are modeled via the many-body force-field described in ably chosen sides (Lx,Ly). Consequently, any rotation Refs. 30,31.Since interaction between second-neighbor of the NT axis (misalignment) is hindered. This choice layers is negligible19, we reduce the computational bur- was made in order to better study the main frictional den by considering only the interaction among nearest- response in the opposite ideal cases of locally commen- walls. Similarly, we only consider the interaction of surate and incommensurate interfaces. Also, periodic- graphene with the outermost wall of the physisorbed NT. ity conveniently removes any edge effect, which can play In our simulations all the NT atoms are free to move in some role in short NTs, not discussed here. every direction under the action of the external driving 3

Nw nout D (A)˚ vext =v ¯ ≈ vcm,x, so that Fext ≈ −m¯ a¯, i.e. , minus the 2 10 6.88500 total force per atom acting on the driven wall. 3 15 10.3276 Since the viscous damping is applied to all the NT 4 20 13.7701 atoms but not to its c.o.m. motion, the computed fric- 5 25 34.4252 tion results weakly dependent on the adopted γ value, 6 30 41.3103 the latter mainly determining the steady-state temper- 7 35 48.1953 ature of the sliding system. In our typical simulations, 8 40 55.0803 9 45 61.9654 which are run in the underdamped regime, we measure steady-state temperatures well below 0.01 K, indicating

Table II. The total number Nw of walls, the chiral index nout a negligible role of thermal excitations on the measured of the outermost wall, and the nominal diameter D of the friction. simulated (5, 5)@(10, 10)@(15, 15)...@(nout, nout) armchair h- To check the sensitivity of the results on the driving BN MWNTs. protocol, we have performed additional simulations in which the NT is pushed by a repulsive wall or by a repulsive tip moving at constant velocity. Since the differ- force and of the internal inter-atomic interactions. Hence, ent driving protocols produced equivalent results, in the dynamical deformations are fully taken into account. following we report only data obtained with a uniform Fully relaxed configurations are obtained via geometry external force, equation2. optimizations by numerically propagating the equation Definitions – Since F is applied uniformly to all of motion using the standard velocity-Verlet integrator ext the N atoms of the outermost tube, the instantaneous (time step ∆t = 1 fs), coupled to the FIRE energy mini- total friction force, F , is simply expressed by mization algorithm20. Simulations are stopped when the fric absolute value of the forces acting on each atom are all Ffric = NFext. (3) below the threshold value of 10−3 meV/atom. For each NT configuration, we compute the corre- The kinetic friction force, Fkinetic, is obtained by aver- sponding sliding potential energy traces. Following a aging Ffric during steady-state motion, after the initial quasi-static protocol, we perform a sequence of simula- transient dynamics has decayed, over a time window cov- tions where the NT structure relaxed at previous step is ering an integer number of oscillations of the periodic rigidly displaced along x by a small amount, ∆x = 0.1062 force traces. and 0.064 A˚ for aligned and misaligned configurations, Instead, the static friction force, Fstatic, is extracted respectively. All atomic positions are subsequently re- from the peak value of the friction force trace during laxed while fixing the x position of the NT center of mass steady-state. Equivalent results are obtained estimating (c.o.m.). The procedure is repeated until at least one full Fstatic from the maximum derivative of the potential en- period of the energy trace is obtained. Note that since in ergy traces. our setup energetics depends very poorly on y coordinate We estimate the contact area A of the fully re- – the system having a long range commensuration along laxed NT/graphene interface using as reference the ad- that direction – in the following we report energy and hesion energy per unit area calculated for an infinite h- 2 force traces only pertaining to the motion direction x. BN/graphene plane bilayer, εadh = 1.812 eV/nm and Dynamical simulations are performed propagating the 1.786 eV/nm2 for aligned and misaligned configurations, Langevin equation of motion respectively:

mi¨ri = Fi − miγ(vi − vcm) + Fextxˆ (1) E A = adh , (4) εadh where ri is the position of the i-th atom, vi and mi are its velocity and mass, Fi is the total force due to the where the adhesion energy of the NT/graphene system, chosen set of interatomic potentials and v is the NT NT/g NT g cm Eadh = E − E − E , is the difference between the c.o.m. velocity. The second term in the r.h.s. is a viscous total energy, ENT/g, of the relaxed NT/graphene system −1 drag (γ = 0.1 ps ) used to avoid system overheating. and the total energies ENT, Eg of the separate NT and Following Ref. 13, the dynamic friction force is evaluated substrate. Note that quantitatively similar A values are from the shear force required to keep the NT at constant obtained by defining the NT contact atoms as those with velocity vext = 1 m/s along x direction. To this end, we a distance smaller than 4 A˚ from the substrate, which apply an external uniform force we take as the reference threshold distance for having m¯ contact. F = (v − v¯) +mγ ¯ (¯v − v ) − m¯ a¯ (2) ext ∆t ext cm,x To describe quantitatively the rolling/sliding motion of the NT, we compute the average velocity along the x to each of the N atoms of the outermost wall of the −1 sliding direction NT, wherem ¯ = N PN 1 ,a ¯ = 1 PN Fi,x , i=1 mi N i=1 mi Ncontact 1 PN 1 X v¯ = N i=1 vi,x and ∆t = 1 fs is the numerical prop- hv i = v , (5) contact N i,x agation time step. Note that after the first time step, contact i=1 4

DWNT MWNT 15@20 45@50 # walls 12 aligned 25@30 55@60 2 4 6 8 misaligned 35@40 65@70 3 5 7 9 30 1200 9 (a) (b)

6 20 800 MW DW

3 (meV) aligned tot contact width (nm)

E 10 400 5@10 5@10@15 5@10@...@20 5@10@...@25 5@10@...@30 5@10@...@35 25@30 35@40 45@50 55@60 65@70 0 5@10@...@40 5@10@...@45 15@20 0 0 0 0.5 1 1.5 0 0.5 1 1.5 2

3 60 (c) (d)

2 40 (meV) tot

E 1 20 misaligned Figure 2. Contact width of the fully relaxed, aligned (black) ◦ 0 0 and 30 misaligned (red) NTs. 0 0.5 1 1.5 0 0.5 1

x (asub) x (asub) considering the Ncontact atoms of the NT belonging to Figure 3. Potential energy profile obtained from a quasi- the contact region defined above. The average is taken static displacement of (a),(c) DWNTs and (b),(d) MWNTs at steady-state, over a time window covering an integer for (a),(b) aligned and (c),(d) misaligned configurations. number of oscillations of the periodic force traces. By normalizing with respect to the imposed sliding velocity, vext, we obtain a scalar quantity, η = hvcontacti/vext ∈ 3

[0, 1]. A value of η = 0 is obtained when the NT atoms ) (a) aligned

sub 2 at the interface do not move (hvcontacti = 0), i.e. pure MW DW rolling motion. On the other hand, η = 1 corresponds to 1

NT contact atoms moving at full speed, i.e. pure sliding. period (a Intermediate values of 0 < η < 1 correspond to a mixed 0 rolling and sliding motion. Direct inspection of the tra- ) (b) misaligned jectories supports this classification, validating the use sub 1 of η. Detailed insights on the internal NT dynamics are MW DW obtained by monitoring the angular velocity ω of each period (a wall. Finally, we define the relevant x-periodicity of√ the 0 1.0 1.4 1.7 2.1 2.5 2.9 3.3 3.7 2.3 4.4 6.5 8.6 10.7 12.8 graphene substrate in the sliding direction asub = 3ag contact width (nm) and asub = ag, respectively for aligned and misaligned geometries (see figure1(c)). Figure 4. The periodicity of the adiabatic potential energy traces calculated for the aligned (a) and misaligned (b) configurations. III. RESULTS

A. Static properties and energetics rect geometrical consequence of the faceting of MWNTs and of the full collapse of DWNTs. Furthermore, we Upon relaxation, large-enough MWNTs form a lon- found that the contact width is in practice independent gitudinal faceting (see figure1(a),(b)) that allows ener- of the alignment of the NT axis relative to the crystal- getically favourable stacking between neighboring walls lographic directions of the substrate. This result reflects while gathering most of the curvature stress in localized the weak angular dependence of the adhesion energy in corners, between one face and the next. As predicted in twisted graphene/h-BN heterojunctions21. previous work12, we found that the number of facets is Before investigating the dynamical response to an ex- dictated by the difference in the unit cells between mating ternal force, it is interesting to consider the pure ener- layers, five in the present case. Differently, the more flex- getics obtained by an adiabatic displacement of the NTs ible DWNTs tend to collapse so to maximize adhesion (see Methods). The resulting trajectories provide infor- energy, although no full junction between the opposite mation about the type of motion in the zero-velocity parts of the internal wall occurs, which in principle could limit, while the associated energy traces carry informa- lower total energy even more. tion such as static friction and periodicity. In figure3 In all NTs considered, the contact area A and the as- we report the calculated energy traces for the consid- sociated contact width W = A/Ly grow linearly with the ered set of NTs. We note that DWNTs are charac- nominal diameter D, as reported in figure2. This is a di- terized by a smooth variation of potential energy upon 5

DWNT MWNT # walls 9 15@20 45@50 25@30 55@60 2 4 6 8 aligned 35@40 65@70 3 5 7 9 misaligned (a) (b) 1.2 9

6 0.9

MW (pN/Å) 6 y /L (pN/Å)

aligned 0.6 y static

/ L 3 F 0.3

static 3 F DW 0 0

contact width (nm) static

F 0.3 1

0 0 0 λm/2 λm 0 λm/2 λm

Figure 5. Static friction force per unit length of the NTs in contact width mod λm contact width mod λm the aligned (black) and misaligned (red) configurations. The lines are drawn to better distinguish the data sets. Figure 6. Static friction force per unit length of (a),(c) DWNTs and (b),(d) MWNTs for (a),(b) aligned and (c),(d) displacement (see figure3(a),(c)), following a sinusoidal- misaligned configurations. Data are reported as a function of the contact width W , modulo the period of the NT/graphene like trend, with periodicity depending exclusively on the interfacial moirépattern, λm. Red dashed lines are fit to alignment condition (see figure4). The adiabatic tra- 2π Fstatic(W ) = α + β sin( W + φ), from which the moirépe- λm jectories show that DWNTs advance by rolling smoothly riodicity has been estimated. over the substrate (see Supplementary Movies 1,2 †). In the case of MWNTs, the different alignment not only af- fects the periodicity (see figure4), but also results in a recent work25, for contacts of width W close to multiple completely different shape of the energy traces (see fig- integers of λm, good compensation of forces occurs that ure3(b),(d)): while aligned MWNTs show smooth (al- minimizes static friction. On the other hand, maximum though irregular) trends, misaligned MWNTs are charac- pinning to the substrate is observed for values of W that terized by a steady rise of the energy followed by a sharp are close to half odd-integers of λm. Overall, this gives drop, suggesting a different type of motion. Inspection of rise to an oscillatory behavior of the static friction force the adiabatic trajectories reveals pure rolling of aligned that can be modeled as MWNTs and mixed rolling+sliding motion accompanied 2πW by sudden backward rotational slip events of misaligned Fstatic = α + β sin + φ . (7) MWNTs (see Supplementary Movies 3,4,5,6 †). λm It is worth to note that within each of the four consid- We note that this behavior has been demonstrated so far ered groups (panels of figure3), the height of the barrier only for the static friction force opposing the sliding of opposing to motion – directly connected to the static fric- flat contacts. However, as we will show in the following, tion force – seems to vary irregularly with the NT size. our results suggest that equation7 is able to describe also In the conventional picture of friction one could expect the barriers against rolling. instead a sublinear growing of static friction with the con- 22–24 In figure6(a), we report the contact width dependence tact area , and thus with D. As reported in figure5, of the static friction force extracted from the dynamical in the present case we observe a static friction force per simulations of the aligned DWNTs, showing the expected NT length, Fstatic/Ly, of large magnitude and strongly sinusoidal trend. A fit of the data via equation7 yielded fluctuating with D for MWNTs, while of reduced inten- a value of λm ∼ 23 nm, in agreement with the nominal sity and of smoother trend for DWNTs. value of 23.36 nm predicted by equation6. The same This apparently erratic behavior can be rationalized in value of λ also provides a reasonable description of the terms of the partial compensation of lateral forces occur- m 25 static friction force measured in the aligned MWNTs (see ring in finite size incommensurate contacts , as follows. figure6(b)), with the exception of the smallest 5@10 NT Due to the inherent lattice mismatch between graphene (cross symbol). We note, however, that in the latter case and h-BN, a characteristic moirésuperstructure appears 26 the value of the contact width is not well defined due at the NT/graphene interface , whose periodicity in the to the reduced size and cylindrical shape of the NT (see x sliding direction is equal to figure1). The analysis of the type of motion (figure3), √ √ of the periodicities (figure4), and of the friction level λm = 3aBN/( 3aBN − asub), (6) √ (figure5), clearly indicates that the 5@10 NT belongs where 3aBN and asub are the relevant NT and substrate to the class of MWNTs, rather than that of DWNTs. periodicities, respectively (see Methods). As shown in Such behavior is due to the retain of the cylindrical shape 6

stead, in misaligned NTs, contact atoms always present a non-zero sliding velocity component, which is maxi- misaligned mized in the case of MWNTs (pure sliding), while be- aligned pure coming minimal for DWNTs (partial sliding). As antici- 1 ext sliding pated above, this rolling to sliding transition can be ex- >/v plained in terms of the faceting appearing in large multi- MW DW

contact wall tubes, which increases the rotational stiﬀness, thus

0 pure taxially aligned NTs, due to a maximized corrugation rolling energy which makes rolling much more convenient rather

1.0 1.4 1.7 2.1 2.5 2.9 3.3 3.7 2.3 4.4 6.5 8.6 10.7 12.8 than sliding. As an example, the energy trace obtained contact width (nm) via a rigid x-displacement (mimicking pure sliding) of the 5@10...@25 aligned MWNT yielded a static friction force about tenfold larger than that obtained from adiabatic Figure 7. The normalized velocity of the contact atoms of the rolling. driven NTs in the aligned (black) and 30◦-rotated misaligned (red) configurations. Pure rolling and pure sliding limits are Overall, in all systems investigated we observed similar marked by dashed lines. kind of motions as predicted by the adiabatic trajectories described in previous section. This indicates that the dynamics of the systems investigated is mainly dic- and the impossibility of collapsing as occurring in larger tated by the energetics of the contact, inertia plays only DWNTs. a secondary role and determines fine features of the NT Similar analysis for the 30◦ misaligned NTs (see fig- motion. ure6(c),(d)) yielded an estimate of λm ∼ 0.68 nm, close Detailed insights about the NT dynamics are gained by to the nominal value of 0.57 nm. The discrepancy can examining the dynamical friction traces Ffric and the an- be attributed to the smaller moiréperiodicity and the gular velocities ω of each wall during sliding. These quan- resulting strong oscillatory behaviour of the sinusoid in tities are plotted in figures8 and9 for few representative equation7, which makes the fit more sensitive on the DW and MWNTs, respectively. Considering at first the precise values of the contact width. The latter has been DWNTs, we observe that their smooth rolling motion is estimated based on adhesion energy arguments consid- accompanied by a regular sinusoidal trend of both Ffric ering the fully relaxed structures (see Methods). Dy- and ω, as reported in figure8. The residual sliding com- namical deformations occurring during sliding affect the ponent present in the misaligned geometry (see figure7) instantaneous contact width, contributing to the overall accounts for the reduction in the average angular veloc- uncertainty. ity when compared to the aligned case (see black curves Regarding the average static friction value, hFsi = in figure8(c),(d)). Notably, DWNTs do not roll as rigid α+β/2, we note that the always-rolling DWNT show the bodies, the inner wall rotating faster than the outermost smaller hFsi, while larger values are shown by MWNT, (see red and green curves in figure8(c),(d)). This in- especially in the case of aligned MWNT in which rolling dicates an additional frictional contribution arising from acts against faceting inducing energetically costly defor- inter-wall shear forces, which adds to the one due to the mations. Instead, internal NT deformations are mostly NT/substrate interactions. In figure9(a),(b) we report absent in misaligned MWNT where the main motion oc- the friction traces of aligned MWNTs with 3 and 6 walls. curs through sliding, yielding a reduced hFsi. Note that Despite the analysis of the contact velocity indicating the sliding degree reported in figure7, obtained from NT pure rolling of the outermost wall (see figure7), the fric- dynamics at finite velocity, as discussed in the next, is tion traces do not display a regular sinusoidal trend. The also representative of the so far discussed “motion” of origin of this peculiar behavior is found in the faceting ap- the NTs in the adiabatic limit. pearing in large NTs, which leads to periodic rearrangements of the inner walls during rolling. Specifically, for the case of the 5@10@15 NT, inspection of the trajectory B. Dynamical properties and friction reveals sudden backward rotations of the two inner walls. This is demonstrated by the negative peaks of the angular We now turn to the dynamics of NTs sliding at con- velocities reported in figure9(c), which correlate with the stant velocity atop the substrate. Upon driving, we ob- abrupt changes of the friction force. These violent events serve that all the considered epitaxially aligned NTs show disappear in systems of growing size, where the friction pure rotational motion, marked by a zero velocity of the trace becomes smoother (see figure9(b)). A clear indica- contact atoms (see figure7). We note that in all the tion of the inter-wall rearrangements occurring inside the cases rolling does not correspond to a rigid rotation of faceted MWNTs is provided by the analysis of the angu- the relaxed NTs, but rather occurs through a belt-like lar velocities reported in figure9(d), showing increasing motion of the walls so to preserve the same shape during angular velocity going from the outermost (cyan curve) advancement (see Supplementary Movies 1,3,5 †). In- to the innermost walls (red and green curves). As for the 7

aligned misaligned 40 (a) (b) 16 (a) aligned 150 misaligned 30 12 (pN) (pN/Å)

fric 100 y F

/L 8 20 50 kinetic

F 4 NT:1 0.28 (c) 0.28 (d) MW DW DWNT NT:2 0.26 0.26 0 1.0 1.4 1.7 2.1 2.5 2.9 3.3 3.7 2.3 4.4 6.5 8.6 10.7 12.8 0.24 0.24 contact width (nm) ω (rad/ns) 0.22 0.22

0.20 0.20 (b) aligned 0 0.5 1 0 1 2 12 misaligned x (asub) x (asub) 9 MW

Figure 8. DWNT – Force traces and angular velocity of the (MPa) A / 6 individual walls (average in black) during lateral pulling of the kinetic 55@60 DWNT. Left and right panels are for the aligned and F 3 DW misaligned conﬁguration, respectively. Qualitatively similar 0 results were obtained for all DWNTs considered. 14 19 23 28 34 39 44 50 31 59 88 117 145 173 contact area (nm2) 5@10@15 5@10@...@30

(a) (b) 400 600 Figure 10. Total kinetic friction force (a) per unit length and (b) per unit area of the NTs in the aligned (black) and 0 300 (pN)

fric misaligned (red) conﬁgurations. F -400 0

-800 -300 (c) (d) NT:1 aligned 20 2 1.9 3 tion of the NT, followed, and only partially compensated, 10 4 1.4 5 by a sudden backward rotational slip (panel (g), see also 0 6

ω (rad/ns) Supplementary Movie 4 †). This partial compensation is 0.9 MWNT -10 at the origin of the residual rolling component reported 0 2 4 0 2 4 in ﬁgure7 for the misaligned 5@10@15 NT. In larger x (asub) x (asub) MWNTs characterized by marked faceting, the friction

240 trace becomes more regular (see ﬁgure9(f)) and the dy- (e) 100 (f) 120 namics turns into a smooth sliding motion accompanied 50 by a periodic back and forth rocking of the NT due to (pN) 0

fric 0 F inertial eﬀects (see angular velocities in panel (h) and -120 -50 Supplementary Movie 6 †). -240 15 (g) 6 (h) NT:1 In ﬁgure 10(a) we plot the contact width dependence

misaligned 2 0 3 3 of the kinetic friction force per unit length of the con- 4 -15 0 5 sidered set of NTs, showing different size scaling for the 6 ω (rad/ns) -30 -3 MWNT cases of rolling and sliding NTs. Considering first the -6 -45 case of rolling (aligned and misaligned DWNTs, aligned 0 1 2 3 0 1 2 MWNTs), we can identify two main contributions to x (asub) x (asub) dissipation arising from NT/substrate interactions and NT internal deformations. Analysis of the static friction Figure 9. MWNT – Force traces and angular velocity of the individual walls (average in black) during lateral pulling of force – mainly dictated by NT/substrate interactions – (left panels) the 5@10@15 and of (right panels) the 5@...@30 demonstrated that the barrier against rolling follows an MWNT. Panels (a)-(d) and (e)-(h) are for the aligned and oscillatory behavior as a function of NT size, suggest- misaligned configuration, respectively. ing similar size independent contribution to kinetic friction. Internal deformations and inter-wall shear forces instead grow with NT diameter D, explaining the ob- case of the DWNTs discussed above, this phenomenon served trend. Specifically, for the case of DWNTs, only represents an additional source of frictional dissipation. one shearing interface is present, leading to linear de- A similar analysis for misaligned MWNTs is shown in pendence of Fkinetic/Ly on contact width W ∝ D, and figure9(e)-(h). The friction trace of the small 5@10@15 roughly constant kinetic friction stress, Fkinetic/A (see NT (panel (e)) displays a saw-tooth behavior, where Ffric figure 10(b)). On the other hand, in MWNT the num- increases at a constant rate and subsequently drops. In- ber of shearing walls increases with NT size and internal spection of the associated angular velocity shows that deformations become more marked due to pronounced the rising phase corresponds to an initial forward rota- faceting. Overall, this leads to supra-linear increase of 8 dissipation, as demonstrated in figure 10(b), showing in- To conclude, our results for NTs over a flat surface sug- creasing kinetic friction stress with contact area A. gest that CNT should behave similarly to BNNT, with The case of misaligned MWNTs requires distinct dis- a transition from almost pure rolling to pure sliding be- cussion, because of the pure sliding occurring in this case. tween misaligned DW and misaligned MWNT. Indeed, it is known that the increasing contact width allows better sampling of the interface incommensurability, asymptotically leading to superlubricity in the thermo- IV. DISCUSSION dynamic limit of infinite interface size27,28. This means that for stiff, shearing incommensurate contacts, as those We have simulated atomistically the induced motion considered here, sliding friction must grow – at most – of h-BN DWNTs and MWNTs on graphene, considering sublinearly with the contact area A, in agreement with the two ideally opposite cases of perfectly aligned and figure 10(b) in the case of misaligned MWNTs. Besides, maximally misaligned interface crystal lattices. oscillations in the decreasing Fkinetic/A trend are likely We show that misaligned contacts systematically dis- due to the internal rearrangements of the individual walls play a lower barrier against motion compared to the epi- taking place during motion, as discussed above. taxially aligned geometry, the latter allowing the formation of locally commensurate contact regions through in- plane strain fields26. In this respect, it has been shown C. Carbon NT results that structural lubricity effects are enhanced in misaligned configuration owing to the reduced size of the Given the fact that homogeneous graphene/graphene moirépattern and effective mutual cancellation of lat- and heterogeneous h-BN/graphene interfaces are charac- eral forces29,31. Besides, our reported static friction force terized by quantitatively similar sliding potential and ex- values indicate that faceted MWNTs experience larger foliation energy landscapes29, for the sake of comparison barriers against motion than collapsed DWNTs. More- we have performed additional dynamical simulations of over, as previously observed in the superlubric case of armchair carbon nanotubes (CNTs) on graphene in the graphitic nanoribbons25, here we show that also in the 30◦ degrees misaligned geometry. The epitaxially aligned case of rolling motion static friction oscillates with the case is less worth to investigate since CNT aligned on NT contact width with a periodicity precisely dictated graphene form perfectly commensurate interfaces, thus by the moirécharacteristic length. leading to rolling always favourable against sliding, as Kinetic friction can be interpreted from our results as already discussed above for BNNTs. sum of the separate contributions from sliding and rolling We have considered the 55@60 DWCNT and the 6- motions, which have intimately different origins. In a walls MWCNT as representative cases. The relaxation of purely sliding motion, dissipation originates at the NT these structures (see Supplementary Information Fig. S1) interface due to the substrate corrugation potential. This produces a collapsing of the DWNT and a faceting of the kind of contribution is roughly constant in the size range MWNT comparable to those in Fig.1, although faceting considered due to superlubricity effects. Conversely, in being less pronounced in the carbon/carbon interface12. a purely belt-like rolling motion the interface atoms are Dynamical simulations of the CNTs gave results equiva- at rest, while all the dissipation originates from NT in- lent to BNNT case. Namely, the DWCNT displays a belt- ternal deformations. This contribution is proportional to like motion with a small sliding component (η ' 0.1), the NT size and to the number of NT walls, so producing while no rolling is observed in the MWCNT (η = 1). The a linear growth of Fkinetic with the NT size in DWNTs, friction force and angular velocity traces of each wall as a and an unprecedented supra-linear growth in the aligned function of the CNT position during motion are also com- MWNTs. In the resulting picture, kinetic friction results parable (see Supplementary Information Fig. S2). The lower in large misaligned (sliding) and in small aligned MWCNT in particular shows a smoother dynamics when (rolling) MWNTs, with a threshold size of 7-8 walls con- compared to the corresponding BNNT, likely due to a necting the two. smaller interlayer corrugation energy. We further note Some evidence to the above results, we note, may that the traces of CNT and BNNT have the same peri- be obtainable in AFM nanomanipulation experiments32. odicities (in units of the relevant substrate lattice con- Rolling motion might conceivably be addressed by la- stant). belling a NT periphery, or indirectly by QCM experi- We note that our case must be distinguished from that ments33. Extending the initial tip-based studies4, the of telescopic sliding in which NT walls slide one relative possibility to laterally push NTs deposited at different to another3. There, friction was connected to a dynam- orientation with the substrate seems entirely open to ex- ical rearrangement of the facets13, more pronounced in perimental verification. In real – non-periodic – systems, large BNNTs due to their tendency of forming walls with the aligned geometry is expected to be energetically more similar chiral angles12, and due to enhanced LA phonon favourable for armchair NTs. More generally, the optimal scattering30. No telescopic motion is expected in our case orientation will depend on the chirality of the outermost of NT sliding/rolling over a surface, in which the relative wall6 while additional metastable orientations will ap- position among internal NT walls is mostly constant. pear, similarly to the case of physisorbed nanoribbons25. 9

Note that during nanomanipulation experiments in which in BN/graphene interfaces, even though these being lim- long NTs are pushed from one edge by a sharp tip, the re- ited to roughness heights of only ∼40 pm35, with ex- sulting motion can generally include sliding, rolling, and pected small effects on the frictional response. Finally, misalignment, exploring a multitude of interface configu- adhesion effects could perturb the relaxed configuration rations between the two ideal cases considered here. The by bending the substrate in order to maximise adhesion. results in our work can therefore serve as reference in Preliminary tests in this direction show that such per- such case of generic NT motion. turbation is limited to a small region at the edges of the interface contact, indicating that such bending would in- Roughness in graphene could also influence our ideal fluence in particular small size NTs, while larger NTs picture. For example, buckling effects are known to – closer to typical experimental sizes – would only be emerge especially in the case of applied vertical loads34. affected in a minor way. All these perturbative effects Also, rippling effects related to moiréhave been observed could be subject of future research.

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