Elucidating atomic-scale friction using molecular dynamics and specialized analysis techniques

Because all quantities associated with a given atom are known as a function of time, molecular dynamics simulations can provide unparalleled insight into dynamic processes. Many quantities calculated from simulations can be directly compared to experimental values, while others provide information not available from experiment. For example, the tilt and methyl angles of chains within a self-assembled monolayer and the amount of hydrogen in a diamond-like carbon (DLC) film are measurable in an experiment. In contrast, the atomic contact force on a single substrate atom, i.e., the force on that atom due to the tip atoms only, and the changes in hybridization of a carbon atom within a DLC film during sliding are not quantities that are currently obtainable from experiments. Herein, the computation of many quantities, including the ones discussed above, and the unique insights that they provided into compression, friction, and wear are discussed. (Some figures in this article are in colour only in the electronic version) 1. Why classical molecular dynamics? The basic setup of molecular dynamics (MD) algorithms is straightforward and has been discussed in a number of reviews [1–4]. In this technique, atoms are treated as discrete particles. The force on each atom is calculated after determining the system geometry, the positions and velocities of each atom, and the boundary conditions. In quantum mechanical methods, electronic degrees of freedom are included and calculations are based on solutions of Schrödinger’s wave equations. The inclusion of electrons and the complexity of the solutions to Schrödinger’s equation limits system sizes to tens to hundreds of atoms, depending on the level of theory. To obtain useful information for larger systems sizes, analytic approximations to the atomic potentials have been developed, which attempt to capture the underlying quantum mechanical principles. These potentials are parameterized to reproduce bond energies, force 3 Author to whom any correspondence should be addressed. constants, bond lengths, bond angles, and elastic constants that were obtained from quantum mechanical calculations and experiments. The atomic forces needed to propagate atoms in time are calculated from the derivative of the interatomic potential energy function with respect to position. The use of these potentials in classical molecular dynamics simulations has proven to be very useful in simulating a wide range of phenomena. Even with the use of classical empirical potentials, MD is limited to modeling systems comprised of tens to hundreds of thousands of atoms (more for parallel codes) and simulated times on the order of nanoseconds. Analogous experimental systems rarely, if ever, deal directly with measurements that only involve size and timescales to which MD is confined. Given this serious limitation, why is MD viewed as a useful tool? The usefulness of MD lies in the wealth of atomic-scale information it provides, information that without question will long be unavailable from experiment. This atomicscale information, if explored carefully and creatively, can 0953-8984/08/354009+15$30.00 © 2008 IOP Publishing Ltd Printed in the UK 1 J. Phys.: Condens. Matter 20 (2008) 354009 J A Harrison et al provide a unique window into understanding properties that are measured experimentally. One could make an analogy with statistical mechanics where macroscopic system properties are elucidated by a consideration of system microstates even though experimentally one would never deal directly with microstates. The wealth of atomic-scale information presents its own difficulties in that too much information is available. One of the largest concerns is how to manage, and ‘sift through’, all the information that is available, i.e., the positions, velocities, and forces on each atom at every simulation time step. Unfortunately, this is not an issue that needs to be addressed only during post-simulation analysis. It is simply not feasible to store all available information, and even if it were, it would not be practical to repeatedly process such a large data set when undertaking various analyses. Consequently, it is imperative to thoughtfully consider what information should be saved over the course of a simulation. With the duration of simulations often lasting weeks or months, rerunning simulations simply to get at information that was not originally written is frustrating, though sometimes necessary. Rarely can one anticipate a set of canned analyses that is completely satisfactory given that simulations are exploring systems that are not completely understood. Thus, the ideal is to write enough information such that the data set is flexible and manageable enough to allow for a variety of creative analysis types, including types of analyses that were not anticipated prior to running the simulations. In what follows, some of the analysis techniques that we have employed in the course of our MD simulations are reviewed and the insights they have provided into our examinations of the mechanical and tribological properties of diamond-like carbon (DLC) films and self-assembled monolayers (SAMS) are discussed. 1.1. The importance of the potential Molecular dynamics simulations require some way to evaluate the forces between atoms. Ideally, atomic interactions would be calculated from first-principles calculations. However, such calculations are computationally intensive, increasing by orders of magnitude with an increase in the number of atoms. Therefore, to obtain useful information in a reasonable amount of time, researchers have developed empirical and semiempirical approximations to atomic potentials. The choice of the potential energy function to be used in any simulation should be governed by careful consideration of the processes and materials to be modeled, and the level of detail one hopes to attain. Simple potentials, such as the Lennard-Jones (LJ), potential have been used extensively to model general behavior in rare-gas clusters [5], between polymer segments, and in tribological simulations [6, 7, 1, 8]. There are other simple potentials that make use of the LJ potential, such as the united-atom model [9] and the beadspring model [10]. For metals, the embedded-atom method (EAM) has been highly successful and has opened up a range of phenomena to simulation [11–14]. A logical extension of the pair potential is to assume that the energy can be written as a many-body expansion of the relative positions of the atoms. Tersoff was the first to use this idea to incorporate the structural chemistry of covalently bonded systems into a potential energy function [15–18]. Using the Tersoff carbon potential as a model, Brenner developed an empirical bond-order expression that described covalent bonding in hydrocarbon molecules and solid-state carbon [19]. This so-called reactive empirical bond-order potential (REBO) was originally developed to model chemical vapor deposition of diamond films, though it is now widely used in simulating many other reactive processes and it has spawned potentials for other covalent interactions [20–22]. This hydrocarbon potential was recently revised [23] so that it provides a significantly better description of bond energies, bond lengths, force constants, elastic properties [24, 25], interstitial defect energies, and surface energies for diamond. Parameters within the second-generation REBO formalism [23] have recently been developed for C–O and O–H [26] interactions as well as Si–Si [27] interactions. To model self-assembled monolayers it is necessary to include intermolecular forces between chains as well as intramolecular, or covalent, forces within the chains. The addition of long-range intermolecular forces to a potential that is capable of modeling short-range covalent bond breaking and bond forming must be carried out in a way that preserves the reactivity. For species that might form a bond if they were close enough to react, the long-range potential must be ‘turned off’. Goddard and coworkers combined long-range forces with the second-generation REBO by using a distance-based switching function [28]. The adaptive intermolecular REBO (AIREBO) potential utilizes both distance-based (Sdistance) and connectivity-based (Sconnectivity) switching functions, as well as a hypothetical bond-order switch Sbond, to smoothly interpolate between pure bonded and pure nonbonded interactions [29, 30]. The LJ potential may be either completely or partially turned off in response to the chemical environment of an interacting pair. Each one of the three switching functions may turn off the LJinteraction partially or entirely, E = (1 − Sdistance Sbond)(1 − Sconnectivity)V . (1) For each of these switches, a value of 1 is associated with turning off the LJ-interaction completely, a value of 0 is associated with a full LJ-interaction, and values in between are associated with a partial LJ-interaction. A full LJ-interaction will be included for atom pairs that are not (1, 2), (1, 3), or (1, 4) neighbors and are either beyond the cutoff distance r LJ max or have a bond order below bmin. Harrison and coworkers have used the AIREBO potential to examine the mechanical and tribological properties of self-assembled monolayers (SAMs) composed of hydrocarbon chains [31–37, 30, 38, 39]. 2. Quantification of structure and disorder In addition to chemical reactions that may be initiated by sliding, when two materials are pressed into contact, adhesion and deformation of the materials are possible. With that in mind, in both experimental and simulated systems, care should