Harrison, J.A. et al. “Atomic-Scale Simulation of Tribological and Related ” Handbook of Micro/Nanotribology. Ed. Bharat Bhushan Boca Raton: CRC Press LLC, 1999 © 1999 by CRC Press LLC 11 Atomic-Scale Simulation of Tribological and Related Phenomena Judith A. Harrison, Steven J. Stuart, and Donald W. Brenner 11.1 Introduction 11.2 Molecular Dynamics Simulations Interatomic Potentials • Thermodynamic Ensemble • Temperature Regulation 11.3 Nanometer-Scale Material Properties: Indentation, Cutting, and Adhesion Indentation of Metals • Indentation of Metals Covered by Thin Films • Indentation of Nonmetals • Cutting of Metals • Adhesion 11.4 Lubrication at the Nanometer Scale: Behavior of Thin Films Equilibrium Properties of Confined Thin Films • Behavior of Thin Films under Shear 11.5 Friction Solid Lubrication • Friction in the Presence of a Third Body • Tribochemistry 11.6 Summary Acknowledgments References 11.1 Introduction Understanding and ultimately controlling friction and wear have long been recognized as important to many areas of technology. Historical examples include the Egyptians, who had to invent new technologies to move the stones needed to build the pyramids (Dowson, 1979); Coulomb, whose fundamental studies of friction were motivated by the need to move ships easily and without wear from land into the water (Dowson, 1979); and Johnson et al. (1971), whose study of automobile windshield wipers led to a better understanding of contact mechanics, including surface energies. Today, the development of microscale © 1999 by CRC Press LLC (and soon nanoscale) machines continues to challenge our understanding of friction and wear at their most fundamental levels. Our knowledge of friction and related phenomena at the atomic scale has rapidly advanced over the last decade with the development of new and powerful experimental methods. The surface force apparatus (SFA), for example, has provided new information related to friction and lubrication for many liquid and solid systems with unprecedented resolution (Israelachvili, 1992). The friction force and atomic force microscopes (FFM and AFM) allow the frictional and mechanical properties of solids to be characterized with atomic resolution under single-asperity contact conditions (Binnig et al., 1986; Mate et al., 1987; Germann et al., 1993; Carpick and Salmeron, 1997). Other techniques, such as the quartz crystal microbalance, are also providing exciting new insights into the origin of friction (Krim et al., 1991; Krim, 1996). Taken together, the results of these studies have revolutionized the study of friction, wear, and mechanical properties, and have reshaped many of our ideas about the fundamental origins of friction. Concomitant with the development and use of these innovative experimental techniques has been the development of new theoretical methods and models. These include analytic models, large-scale molec- ular dynamics (MD) simulations, and even first-principles total-energy techniques (Zhong and Tomanek, 1990). Analytic models have had a long history in the study of friction. Beginning with the work of Tomlinson (1929) and Frenkel and Kontorova (1938), through to recent studies by McClelland and Glosli (1992), Sokoloff (1984, 1990, 1992, 1993, 1996), and others (Helman et al., 1994; Persson, 1991), these idealized models have been able to break down the complicated motions that create friction into basic components defined by quantities such as spring constants, the curvature and magnitude of potential wells, and bulk phonon frequencies. The main drawback of these approaches is that simplifying assump- tions must be made as part of these models. This means, for example, that unanticipated defect structures may be overlooked, which may strongly influence friction and wear even at the atomic level. Molecular dynamics computer simulations, which are the topic of this chapter, represents a compro- mise between analytic models and experiment. On the one hand, this method deals with approximate interatomic forces and classical dynamics (as opposed to quantum dynamics), so it has much in common with analytic models (for a comparison of analytic and simulation results, see Harrison et al., 1992c). On the other hand, simulations often reveal unanticipated events that require further analysis, so they also have much in common with experiment. Furthermore, a poor choice of simulation conditions, as in an experiment, can result in meaningless results. Because of this danger, a thorough understanding of the strengths and weaknesses of MD simulations is crucial to both successfully implementing this method and understanding the results of others. On the surface, atomistic computer simulations appear rather straightforward to carry out: given a set of initial conditions and a way of describing interatomic forces, one simply integrates classical equations of motion using one of several standard methods (Gear, 1971). Results are then obtained from the simulations through mathematical analysis of relative positions, velocities, and forces; by visual inspection of the trajectories through animated movies; or through a combination of both (Figure 11.1). However, the effective use of this method requires an understanding of many details not apparent in this simple analysis. To provide a feeling for the details that have contributed to the success of this approach in the study of adhesion, friction, and wear as well as other related areas, the next section provides a brief review of MD techniques. For a more detailed overview of MD simulations, including computer algorithms, the reader is referred to a number of other more comprehensive sources (Hoover, 1986; Heermann, 1986; Allen and Tildesley, 1987; Haile, 1992). The remainder of this chapter presents recent results from MD simulations dealing with various aspects of mechanical, frictional, and wear properties of solid surfaces and thin lubricating films. Section 11.2 summarizes several of the technical details needed to perform (or understand) an MD simulation. These range from choosing an interaction potential and thermodynamic ensemble to implementing tempera- ture controls. Section 11.3 describes simulations of the indentation of metals and nonmetals, as well as the machining of metal surfaces. The simulations discussed reveal a number of interesting phenomena and trends related to the deformation and disordering of materials at the atomic scale, some (but not all) of which have been observed at the macroscopic scale. Section 11.4 summarizes the results of © 1999 by CRC Press LLC simulations that probe the properties of liquid films confined to thicknesses on the order of atomic dimensions. These systems are becoming more important as demands for lubricating moving parts approach the nanometer scale. In these systems, fluids have a range of new properties that bear little resemblance to liquid properties on macroscopic scales. In many cases, the information obtained from these studies could not have been obtained in any other way, and is providing unique new insights into recent observations made by instruments such as the SFA. Section 11.5 discusses simulations of the tribological properties of solid surfaces. Some of the systems discussed are sliding diamond interfaces, Langmuir–Blodgett films, self-assembled monolayers, and metals. The details of several unique mecha- nisms of energy dissipation are discussed, providing just a few examples of the many ways in which the conversion of work into heat leads to friction in weakly adhering systems. In addition, simulations of molecules trapped between, or chemisorbed onto, diamond surfaces will be discussed in terms of their effects on the friction, wear, and tribology of diamond. A summary of the MD results is given in the final section. 11.2 Molecular Dynamics Simulations Atomistic computer simulations are having a major impact in many areas of the chemical, physical, material, and biological sciences. This is largely due to enormous recent increases in computer power, FIGURE 11.1 Flow chart of an MD simulation. © 1999 by CRC Press LLC increasingly clever algorithms, and recent developments in modeling interatomic interactions. This last development, in particular, has made it possible to study a wide range of systems and processes using large-scale MD simulations. Consequently, this section begins with a review of the interatomic interac- tions that have played the largest role in friction, indentation, and related simulations. For a slightly broader discussion, the reader is referred to a review by Brenner and Garrison (1989). This is followed by a brief discussion of thermodynamic ensembles and their use in different types of simulations. The section then closes with a description of several of the thermostatting techniques used to regulate the temperature during an MD simulation. This topic is particularly relevant for tribological simulations because friction and indentation do work on the system, raising its kinetic energy. 11.2.1 Interatomic Potentials Molecular dynamics simulations involve tracking the motion of atoms and molecules as a function of time. Typically, this motion is calculated by the numerical solution of a set of coupled differential equations (Gear, 1971; Heermann, 1986; Allen and Tildesley, 1987). For example, Newton’s equation of motion, (11.1) where F is the force on a particle, m is its mass, a its acceleration, v its velocity, and t is time, yield a set of 3 n (where n is the number of particles) second-order differential equations that govern the dynamics. These can be solved with finite-time-step integration methods, where time steps are on the order of 1 / 25 of a vibrational frequency (typically tenths to a few femtoseconds) (Gear, 1971). Most current simulations then integrate for a total time of picoseconds to nanoseconds. The evaluation of these equations (or any of the other forms of classical equations of motion) requires a method for obtaining the force F between atoms. Constraints on computer time generally require that the evaluation of interatomic forces not be computationally intensive. Currently, there are two approaches that are widely used. In the first, one assumes that the potential energy of the atoms can be represented as a function of their relative atomic positions. These functions are typically based on simplified interpretations of more general quantum mechanical principles, as discussed below, and usually contain some number of free parameters. The parameters are then chosen to closely reproduce some set of physical properties of the system of interest, and the forces are obtained by taking the gradient of the potential energy with respect to atomic positions. While this may sound straightforward, there are many intricacies involved in developing a useful potential energy function. For example, the parameters entering the potential energy function are usually deter- mined by a limited set of known system properties. A consequence of that is that other properties, including those that might be key in determining the outcome of a given simulation, are determined solely by the assumed functional form. For a metal, the properties to which a potential energy function might be fit might include the lattice constant, cohesive energy, elastic constants, and vacancy formation energy. Predicted properties might then include surface reconstructions, energetics of interstitial defects, and response (both elastic and plastic) to an applied load. The form of the potential is therefore crucial if the simulation is to have sufficient predictive power to be useful. The second approach, which has become more useful with the advent of powerful computers, is the calculation of interatomic forces directly from first-principles (Car and Parrinello, 1985) or semiempirical (Menon and Allen, 1986; Sankey and Allen, 1986) calculations that explicitly include electrons. The advantage of this approach is that the number of unknown parameters may be kept small, and, because the forces are based on quantum principles, they may have strong predictive properties. However, this does not guarantee that forces from a semiempirical electronic structure calculation are accurate; poorly chosen parameterizations and functional forms can still yield nonphysical results. The disadvantage of this approach is that the potentials involved are considerably more complicated, and require more computational effort, than those used in the classical approach. Longer simulation times require that Fa v ==mm d dt , © 1999 by CRC Press LLC both the system size and the timescale studied be smaller than when using more approximate methods. Thus, while this approach has been used to study the forces responsible for friction (Zhong and Tomanek, 1990), it has not yet found widespread application for the type of large-scale modeling discussed here. The simplest approach for developing a continuous potential energy function is to assume that the binding energy E b can be written as a sum over pairs of atoms, (11.2) The indexes i and j are atom labels, r ij is the scalar distance between atoms i and j, and V pair ( r ij ) is an assumed functional form for the energy. Some traditional forms for the pair term are given by (11.3) where the parameter D determines the minimum energy for pairs of atoms. Two common forms of this expression are the Morse potential ( X = e – β r ij ), and the Lennard–Jones (LJ) “12-6” potential ( X = ( σ / r ij ) 6 ). ( β and σ are arbitrary parameters that are used to fit the potential to observed properties.) The short-range exponential form for the Morse function provides a reasonable description of repulsive forces between atomic cores, while the 1/ r 6 term of the LJ potential describes the leading term in long- range dispersion forces. A compromise between these two is the “exponential-6,” or Buckingham, poten- tial. This uses an exponential function of atomic distances for the repulsion and a 1/ r 6 form for the attraction. The disadvantage of this form is that as the atomic separation approaches zero, the potential becomes infinitely attractive. For systems with significant Coulomb interactions, the approach that is usually taken is to assign each atom a fractional point charge q i . These point charges then interact with a pair potential Because the 1/ r Coulomb interactions act over distances that are long compared to atomic dimensions, simulations that include them typically must include large numbers of atoms, and often require special attention to boundary conditions (Ewald, 1921; Heyes, 1981). Other forms of pair potentials have been explored, and each has its strengths and weaknesses. However, the approximation of a pairwise-additive binding energy is so severe that in most cases no form of pair potential will adequately describe every property of a given system (an exception might be rare gases). This does not mean that pair potentials are without use — just the opposite is true! A great many general principles of many-body dynamics have been gleaned from simulations that have used pair potentials, and they will continue to find a central role in computer simulations. As discussed below, this is especially true for simulations of the properties of confined fluids. A logical extension of the pair potential is to assume that the binding energy can be written as a many- body expansion of the relative positions of the atoms (11.4) Normally, it is assumed that this series converges rapidly so that four-body and higher terms can be ignored. Several functional forms of this type have had considerable success in simulations. Stillinger and Weber (1985), for example, introduced a potential of this type for silicon that has found widespread EVr bij ji i = () > () ∑∑ pair . Vr DXX ijpair () =⋅ − () 1, Vr qq r ij ij ij pair () = . EV V V jikjilkji b body body body =+ + +… −− − ∑∑∑∑∑∑∑∑∑ 1 2 1 3 1 4 23 4 !! . © 1999 by CRC Press LLC use, an example of which is discussed in Section 11.3.3. Another example is the work of Murrell and co- workers (1984) who have developed a number of potentials of this type for different gas-phase and condensed-phase systems. A common form of the many-body expansion is a valence force field. In this approach, interatomic interactions are modeled with a Taylor series expansion in bond lengths, bond angles, and torsional angles. These force fields typically include some sort of nonbonded interaction as well. Prime examples include the molecular mechanics potentials pioneered by Allinger and co-workers (Allinger et al., 1989; Burkert and Allinger, 1982). A common variation of the valence-force approach is to assume rigid bonds, and allow only changes in bond angles. Because the angle bends generally have smaller force constants, there tend to be larger variations in angles than in bond lengths, so this approximation often gives accurate predictions for the shapes of large molecules at thermal energies. The advantage of this approximation is that because the bending modes have lower frequencies than those involving bond stretching modes, time steps may be used that are an order of magnitude larger than those required for flexible bonds, with no larger numerical errors in the total energy. One method of including many-body effects in Coulomb systems is to account for electrostatic induction interactions. Each point charge will give rise to an electric field, and will induce a dipole moment on neighboring atoms. This effect can be modeled by including terms for the atomic or molecular dipoles in the interaction potential, and solving for the values of the dipoles at each step in the dynamics simulation. An alternative method is to simulate the polarizability of a molecule by allowing the values of the point charges to change directly in response to their local environment (Streitz and Mintmire, 1994; Rick et al., 1994). The values of the charges in these simulations are determined by the method of electronegativity equalization and may either by evaluated iteratively (Streitz and Mintmire, 1994) or carried as dynamic variables in the simulation (Rick et al., 1994). Several potential energy expressions beyond the many-body expansion have been successfully devel- oped and are widely used in MD simulations. For metals, the embedded atom method (EAM) and related methods have been highly successful in reproducing a host of properties, and have opened up a range of phenomena to simulation (Finnis and Sinclair, 1984; Foiles et al., 1986; Ercollessi et al., 1986a,b). These have been especially useful in simulations of the indentation of metals, as discussed in Section 11.3. This approach is based on ideas originating from effective medium theory (Norskov and Lang, 1980; Stott and Zaremba, 1980). In this formalism, the energy of an atom interacting with surrounding atoms is approximated by the energy of the atom interacting with a homogeneous electron gas and a compensating positive background. The EAM assumes that the density of the electron gas can be approximated by a sum of electron densities from surrounding atoms, and adds a repulsive term to account for core–core interactions. Within this set of approximations, the total binding energy is given as a sum over atomic sites (11.5) where each site energy is given as a pair sum plus a contribution from a functional (called an embedding function) that, in turn, depends on the sum of electron densities at that site: (11.6) The function Φ ( r ij ) represents the core–core repulsion, F is the embedding function, and ρ ( r ij ) is the contribution to the electron density at site i from atom j . For practical applications, functional forms are assumed for the core–core repulsion, the embedding function, and the contribution of the electron densities from surrounding atoms. For a more complete description of this approach, including a much more formal derivation and discussion, the reader is referred to Sutton (1993). EE bi i = ∑ , ErFr iij j ij j = () + ()         ∑∑ 1 2 Φρ. © 1999 by CRC Press LLC For close-packed metals, it has been found that the electron densities in the solid can be adequately approximated by a pairwise sum of atomic-like electron densities. With this approximation, the com- puting time required to evaluate an EAM potential scales with the number of atoms in the same way as a pair potential. The quantitative results of the EAM, however, are dramatically better than those obtained with pair potentials. Energies and structures of solid surfaces and defects in metals, for example, match experimental results (or more-sophisticated calculations) reasonably well despite the relatively simple analytic form. Because of this, EAM potentials have been used extensively in studies of indentation. Schemes similar to EAM but based on other levels of approximation have also been developed. For example, three-body terms in the electron-density contributions have been studied for modeling a range of metallic and covalently bonded systems (Baskes, 1992). Another example is the work of DePristo and co-workers, who have studied various hierarchies of approximation within effective medium theory by including additional correction terms to Equation 11.6 (Raeker and DePristo, 1991), as have Norskov and co-workers (Jacobson et al., 1987). A potential that is similar to the EAM but based on bond orders has also been developed (Abell, 1985). Originally adapted by Tersoff (1986) to model silicon, the approach has found use in computer simula- tions of a wide range of covalently bonded systems (Tersoff, 1989; Brenner, 1989a, 1990; Khor and Das Sarma, 1988). Like the embedded atom potentials, Tersoff potentials begin by approximating the binding energy of a system as a sum over sites: (11.7) In this case, however, each site energy is given by an expression that resembles a pair potential, (11.8) The functions V A ( r ij ) and V R ( r ij ) represent pairwise-additive attractive and repulsive terms, respectively, and B ij represents an empirical bond-order expression that modulates the attractive pair potential. This bond-order term is where the many-body effects are introduced. As in the other potentials, functional forms for the bond order and the pair terms are fit to a range of properties for the systems of interest. Applications of this approach have assumed functional forms for the bond order that decrease its value as the number of nearest-neighbors of a given pair of bonded atoms increases. Physically, the attractive pair term can be envisioned as bonding due to valence electrons, with the bond order destabilizing the bond as the valence electrons are shared among more and more neighbors. Tersoff, Abell, and others have shown that this simple expression can capture a range of bond energies, bond lengths, and related properties for group IV solids. Also, if properly parameterized, the potential yields Pauling’s bond-order relations (Abell, 1985; Khor and Das Sarma, 1988; Tersoff, 1989). These properties have been shown to make this expression very powerful for predicting covalently bonded structures and, therefore, useful for predicting new phenomena through MD simulations. Although they are based on different principles, the EAM and Tersoff expressions are quite similar. In the EAM, binding energy is defined by electron densities through the embedding function. The electron densities are, in turn, defined by the arrangement of neighboring atoms. Similarly, in the Tersoff expres- sion the binding energy is defined directly by the number and arrangement of neighbors through the bond-order expression. In fact, the EAM and Tersoff approaches have been shown to be identical for simplified expressions provided that angular interactions are not used (Brenner, 1989b). Although not all potential energy expressions widely used in MD simulations have been covered, this section provides a brief introduction to those that have found the most use in the simulations of friction, wear, and related phenomena. As mentioned above, methods that explicitly incorporate semiempirical and first-principles electronic structure calculations have not been discussed. As computer speeds EE bi i = ∑ . EVrBVr i R ij ij A ij ji = () +⋅ () [] > () ∑ . © 1999 by CRC Press LLC continue to increase and as new algorithms are developed, we expect that these more exact methods will play an increasingly important role in tribological simulations. 11.2.2 Thermodynamic Ensemble When performing a MD simulation, a choice must be made as to which thermodynamic ensemble to study. These ensembles are distinguished by which thermodynamic variables are held constant over the course of the simulation. (For a broader and more rigorous treatment of ensemble averaging, the reader should consult any statistical mechanics text, (e.g., McQuarrie, 1976)). Without specific reasons to do otherwise, it is quite natural to keep the number of atoms ( N ) and the volume of the simulation cell ( V ) constant over the course of a MD simulation. In addition, for a system without energy transfer, integrating the equations of motion (Equation 11.1) will generate a trajectory over which the energy of the system ( E ) will also be conserved. A simulation of this type is thus performed in the constant- NVE, or microcanonical, ensemble. Systems undergoing sliding friction or indentation, however, require work to be performed on the system, which raises its energy and causes the temperature to increase. In a macroscopic system, the environment surrounding the region of tribological interest acts as an infinite heat sink, removing excess energy and helping to maintain a fairly constant temperature. Ideally, a sufficiently large simulation would be able to model this same behavior. But while the thousands of atoms at an atomic-scale interface are within reach of computer simulation, the O (10 23 ) atoms in the experimental apparatus are not. Thus, a thermodynamic ensemble that will more closely resemble reality will be one in which the temperature ( T ), rather than the energy, is held constant. These simulations are performed in the constant- NVT, or canonical, ensemble. A constant temperature is maintained in the canonical ensemble by using any of a large number of thermostats, many of which are described in the following section. What is often done in simulations of indentation or friction is to apply the thermostat only in a region of the simulation cell that is well removed from the interface where friction is taking place. This allows for local heating of the interface as work is done on the system, while also providing a means for efficient dissipation of excess heat. These “hybrid” NVE/NVT simulations, although not rigorously a member of any true thermodynamic ensem- ble, are very useful and quite common in tribological simulations. A particularly troublesome system for MD simulations is the nonequilibrium dynamics of confined thin films (see Section 11.4.2). In these systems, the constraint of constant atom number is not necessarily applicable. Under experimental conditions, a thin film under shear or tension is free to exchange mole- cules with a reservoir of bulk liquid molecules, and the total atom number is certainly conserved. But the number of atoms in the film itself is subject to rather dramatic changes. According to some studies, as many as half the molecules in an ultrathin film will exit the interfacial region upon a change in registry of the opposing surfaces (i.e., with no change in interfacial volume) (Schoen et al., 1989). Changes in the film particle number can be equally large under compression. The proper conserved quantity in these simulations is not the particle number N, but the chemical potential µ . During a simulation performed in the constant- µ VT, or grand canonical, ensemble, the number of atoms or molecules fluctuates to keep the chemical potential constant. A true grand canonical MD simulation is too difficult to perform for all but the simplest of liquid molecules, however, due to the difficulties associated with inserting or removing molecules at bulk densities. An alternative chosen by some authors is to mimic the experimental reservoir of bulk liquid molecules on a microscopic scale. This involves performing a constant- NVT (Wang et al., 1993a,b, 1994) or constant- NPT (Gao et al., 1997) simulation that explicitly includes a collection of molecules that are external to the interfaces (see Figure 11.14). As liquid molecules drain into or are drawn from the reservoir region, the number of particles directly between the interfaces is free to change. This method is then an approximation to the grand canonical ensemble when only a subset of the system is considered. Two drawbacks to this method are that the interface can extend infinitely (via periodic boundary conditions) in only one dimension instead of two, and also that a significant number of extraneous atoms must be carried in the simulation. © 1999 by CRC Press LLC An interesting alternative to the grand canonical ensemble is that chosen by Cushman and co-workers (Schoen et al., 1989; Curry et al., 1994). They performed a series of grand canonical Monte Carlo simulations at various points along a hypothetical sliding trajectory. These simulations were used to calculate the correct particle numbers at a fixed chemical potential, which were in turn used as inputs to nonsliding, constant- NVE MD simulations at each of the chosen trajectory points. Because the system was fully equilibrated at each step along the sliding trajectory, the sliding speed can be assumed to be infinitely slow. This offers a useful alternative to continuous MD simulations, which are currently restricted to sliding speeds of roughly 1 m/s or greater — orders of magnitude larger than most experi- mental studies. 11.2.3 Temperature Regulation As was discussed in the previous section, some method of controlling the system temperature is required in simulations involving friction or indentation. In this section, we discuss some of the many available thermostats that are used for this purpose. For a more formal discussion of heat baths and the trajectories that they produce, the reader is referred to Hoover (1986). The most straightforward method for controlling heat production is simply to rescale intermittently the atomic velocities to yield a desired temperature (Woodcock, 1971). This approach was widely used in early MD simulations, and is often effective at maintaining a given temperature during the course of a simulation. However, it has several disadvantages that have spurred the development of more-sophis- ticated methods. First, there is little formal justification. For typical system sizes, averaged quantities, such as pressure, do not correspond to those obtained from any particular thermodynamic ensemble. Second, the dynamics produced are not time reversible, again making results difficult to analyze in terms of thermodynamic ensembles. Finally, the rate and mode of heat dissipation are not determined by system properties, but instead depend on how often velocities are rescaled. This may influence dynamics that are unique to a particular system. A more-sophisticated approach to maintaining a given temperature is through Langevin dynamics. Originally used to describe Brownian motion, this method has found widespread use in MD simulations. In this approach, additional terms are added to the equations of motion, corresponding to a frictional term and a random force (Schneider and Stoll, 1978; Hoover, 1986; Kremer and Grest, 1990). The equations of motion (see Equation 11.1) are given by (11.9) where F are the forces due to the interatomic potential, the quantities m and v are the particle mass and velocity, respectively, ξ is a friction coefficient, and R ( t ) represents a random “white noise” force. The friction kernel is defined in terms of a memory function in formal applications; kernels developed for harmonic solids have been used successfully in MD simulations (Adelman and Doll, 1976; Adelman, 1980; Tully, 1980). As with any thermostat, the atom velocities are altered in the process of controlling the temperature. It is important to keep this in mind when using a thermostat, because it has the potential to perturb any dynamic properties of the system being studied. To help avoid this problem, one effective approach is to add Langevin forces only to those atoms in a region away from where the dynamics of interest occurs. In this way, coupling to a heat bath is established away from the important action, and simplified approximations for the friction term can be used without unduly influencing the dynamics produced by the interatomic forces. For heat flux via nuclear (as opposed to electronic) degrees of freedom in solids, it has been shown that a reasonable approximation for the friction coefficient ξ is 6/ π times the Debye frequency β (Adelman and Doll, 1976). Lucchese and Tully (1983) have shown that with this approxi- mation and a sufficiently large reaction zone, the vibrational modes of atoms away from the bath atoms are well described by the interatomic potential. mmRtaF v== + () ξ , [...]... film to flatten and some of the alkane molecules to wet the sides of the tip Lowering the tip farther caused drainage of the top layer of alkane molecules from underneath the tip, increased wetting of the sides of the tip, “pinning” of hexadecane molecules under the tip, and deformation of the gold substrate beneath the tip At this stage of the simulation, the force between the Ni tip and the film/Au... variable to the standard classical equations of motion that can be thought of as a large heat bath, which couples to each of the physical degrees of freedom The actual effect of the variable, however, is to scale the coordinates of either time or mass in the system The dynamics of the expanded system correspond to the microcanonical ensemble, but when projected onto only the physical degrees of freedom they... the motion of point defects The formation of defects or dislocations was driven by the need to release stored elastic energy Plots of the energy required to create a dislocation and the elastic energy vs length of the radius of the indenter cross at length values of a few nanometers In contrast, at the faster indentation rates of 10 m/s, the system did not have time to relax completely and the stored... from some level of electronic structure calculation The best way to make this choice is to understand carefully the strong points of each of these approaches, decide what one wishes to learn from the simulation, and form conclusions based on this careful understanding 11.3 Nanometer-Scale Material Properties: Indentation, Cutting, and Adhesion Understanding material properties at the nanometer scale... 1997) Substitution of the elastic constants calculated from the improved potential into Equations 11.14 and 11.15 yields values of 1367 and 1177 GPa for E(111) and E(100), respectively Recently, Sinnott et al (1997) used an sp3-hybridized carbon tip to indent a hydrogen-terminated (111) face of diamond and a thin film of amorphous carbon on (111) diamond Comparison of pictures before and after indentation... that one of the consequences of the existence of this region of inaccessible separation is that there will be significant differences between contact and noncontact AFM experiments © 1999 by CRC Press LLC FIGURE 11.14 Illustrations of atoms from an MD simulation of the orthogonal cutting of a Cu(111) substrate with a rigid cutting tool (From Belak, J and Stowers, I F (1992), in Fundamentals of Friction:... contrast, the density profiles of the © 1999 by CRC Press LLC FIGURE 11.20 Atomic configurations adopted by n-octane when confined between parallel, rigid LB substrates of various pore widths The pore widths and liquid temperatures are 1.0 nm and 297 K in (a), 1.6 nm and 297 K in (b), 1.25 nm and 297 K in (c), 1.25 nm and 250 K in (d), 1.8 nm and 297 K in (e), and 1.8 nm and 250 K in (f ) The light spheres... shear) and Poiseuille (pressure-driven) flows were examined The presence of flow had no effect on the density profile of the liquids as they were identical with the equilibrium density profiles for both types of flow and all pore widths Velocity profiles, defined as the velocity of the liquid parallel to the wall as a function of distance from the center of the pore, should be linear and parabolic for Couette and. .. contact, and hysteresis during subsequent retraction were all observed In contrast to the Ni/Au study, complete separation of the tip and substrate resulted in the transfer of a smaller number of substrate atoms to the tip when the connective neck of atoms, composed entirely of Au, was severed Because the tip was covered with Au, the interaction between the tip and the substrate was composed mostly of Au–Au... kbar) Near the melting temperature and at the slowest indentation rate, lower yield strengths (30 kbar) were observed for both amorphous and crystalline Si Thus, the simulated nanoyield strength of Si depended on structure, rate of deformation, and temperature of the sample Amorphous silicon did not show any sign of crystallization upon indentation Conversely, indentation of the crystalline silicon close . Harrison, J.A. et al. “Atomic-Scale Simulation of Tribological and Related ” Handbook of Micro/ Nanotribology. Ed. Bharat Bhushan Boca Raton: CRC Press. LLC (and soon nanoscale) machines continues to challenge our understanding of friction and wear at their most fundamental levels. Our knowledge of friction