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 in- terface 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 re- sulting 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 fric- tional 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 ˚ pdBN = 1:442 A (honeycomb lattice constant aBN = 3dBN ≈ 2:498 A),˚ corresponding to the equilibrium value of the adopted Tersoff potential15. The lattice con- stant, 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 substratep 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 p 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 Section IIIC. 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.