PHYSICAL REVIEW B 99, 165412 (2019) Partial commensuration and Amontons laws of friction adapted for large atomic interfaces Behnaz Babagholami, Ali Sadeghi, * and M. Etezadifar Department of Physics, Shahid Beheshti University, G.C., Evin, 19839-63113 Tehran, Iran (Received 2 November 2018; revised manuscript received 18 February 2019; published 9 April 2019) We employ a minimal atomistic friction model to investigate the commensuration at interfaces when a large number of atoms are involved. As the normal load at the interface increases, we found a smooth transition from the ultralow friction (superlubricity) regime to the high-frictional motion via a gradual growth of the commensurate region where only a fraction of the atoms adjust their atomic arrangement according to the substrate surface potential. The center of mass of the sliding particle may lack the perfect periodicity of the conventional stick-slip motion in this partial commensuration state. The commensurate region is assigned as the effective contact, based on which the scaling behavior of friction can be explained in analogy to the classical Amontons law of friction. The revealed emergent state for large atomic sliders would be beneficial to studies on extending the fundamental frictional investigations to larger length scales. DOI: 10.1103/PhysRevB.99.165412 I. INTRODUCTION Wear of sliding solid surfaces causes plastic deformations and damages, e.g., in metallic elements of machines, and is thus of great economic relevance to modern life. Moreover, a current serious environmental concern is the energy dissi- pation due to friction at such interfaces. Although it might still seem unreachable at this point, the bottom-up approach in nanotechnology for manufacturing the products out of atomically small components is likely to help on this issue by providing a technique to design macroscopic interfaces with tunable friction just as already realized in the nanoscale decades ago (for details see below). In fact, macro- and microscopic pictures of friction [1,2] share some features and properties, whereas different laws govern the friction at the two length scales. In particular, the independence of macroscopic friction from the apparent contact area and its linear dependence on the normal load are no longer valid when ideal atomic surfaces are involved [3]. However, we show in the following that an essentially similar description of the classical laws is still applicable to the atomic scale. When a nanoparticle is dragged on a clean crystalline surface, the no-lubricant friction can be reduced or eliminated under various conditions [4–8]. Being a result of lattice mis- match and thus preventing the surfaces from getting locked into each other, this superlubric state is known as structural lubricity. The sliding is smooth with an ultra-low energy dissipation in this case. In contrast, the sliding of two com- mensurate lattices is highly dissipative due to the stick-slip motion: The surfaces remain locked and stuck together as the external shearing increases until a maximum tolerance is reached to fire a sudden slip to the next locking position, and so on. Due to such a stick-slip motion, the lateral force on the tip of an atomic force microscope (AFM) reveals a peri- odicity of the probed surface [9]. It can be explained within * ali_sadeghi@sbu.ac.ir the framework of the single-particle Prandtl-Tomlinson (PT) model [10,11]: The tip apex is assumed to be a structureless pointlike object which is dragged by a spring of stiffness K dr over the rigid surface of the sample. The interaction with the surface is described by a sinusoidal potential U cos(2π x/a), where a is the periodicity of the surface lattice. As long as the surface corrugation amplitude U (which is shown experimentally to be proportional to the normal load [5,12]) is larger than the critical value of U ∗ = K dr a 2 /4π 2 , the so-called Peierls-Nabarro barriers on the potential-energy surface (the superimposed elastic and sinusoidal contributions) prevents superlubricity by inducing stick-slip instabilities; in contrast, for U < U ∗ , the tip slides continuously and smoothly [5]. Here, we investigate the frictional behavior of a nanopar- ticle with a large contact layer over a crystalline surface as a function of the normal load. The PT model, adapted to the multiatomic case, is still adequate for addressing such deformable interfaces [13]. In the rest of this paper, we first explain the employed multiatomic model, then present and discuss the numerical results, and finally draw our conclu- sions. II. MODEL Many experimental observations are described surprisingly well with the one-dimensional PT model [5], i.e., by pro- jecting the AFM tip trajectory on the scan line. The so- called Frenkel-Kontorova-Tomlinson (FKT) model [14,15] is a multiatomic extension to the basic PT model for attacking the problems of large atomic interfaces. The particle is not pointlike but is an extended deformable layer at the interface with the substrate: a chain of N identical pseudoatoms posi- tioned equidistantly along the sliding direction and subject to the substrate potential U cos(2π x/a ). As mentioned above, the corrugation amplitude U accounts for the normal load at the interface. As illustrated in Fig. 1, N − 1 springs, each of stiffness K c and rest length a c , connect harmonically the immediate neighbors throughout the chain. Each pseudoatom 2469-9950/2019/99(16)/165412(6) 165412-1 ©2019 American Physical Society