This review summarizes recent advances in the area of tribology based on the outcome of a Lorentz Center workshop surveying various physical, chemical and mechanical phenomena across scales. Among the main themes discussed were those of rough surface representations, the breakdown of continuum theories at the nano-and micro-scales, as well as multiscale and multiphysics aspects for analytical and computational models relevant to applications spanning a variety of sectors, from automotive to biotribology and nanotechnology. Significant effort is still required to account for complementary nonlinear effects of plasticity, adhesion, friction, wear, lubrication and surface chemistry in tribological models. For each topic, we propose some research directions.

Two-dimensional heterostructures are excellent platforms to realize twistangle independent ultra-low friction due to their weak interlayer van der Waals interactions and natural lattice mismatch. However, for finite-size interfaces, the effect of domain edges on the friction process remains unclear. Here, we report on the superlubricity phenomenon and the edge pinning effect at MoS 2 /graphite and MoS 2 /h-BN van der Waals heterostructure interfaces. We find that friction coefficients of these heterostructures are below 10 -6 . Molecular dynamics simulations corroborate experiments highlighting the contribution of edges and interface steps to friction forces. Our experiments and simulations provide more information on the sliding mechanism of finite low-dimensional structures, which is vital to understand the friction process of laminar solid lubricants.

To achieve acceptable accuracy in fast-switching free energy estimates by Jarzynski equality [ Phys. Rev. Lett. 1997 , 78 , 2690 ] or Crooks fluctuation theorem [ J. Stat. Phys. 1998 , 90 , 1481 ], it is often necessary to realize a large number of externally driven trajectories. This is basically due to inefficient calculation of path-ensemble averages arising from the work dissipated during the nonequilibrium paths. We propose a computational technique, addressed to Monte Carlo simulations, to improve free energy estimates by lowering the dissipated work. The method is inspired by the dynamical freezing approach, recently developed in the context of molecular dynamics simulations [ Phys. Rev. E 2009 , 80 , 041124 ]. The idea is to limit the configurational sampling to particles of a well-established region of the sample (namely, the region where dissipation is supposed to occur), while leaving fixed (frozen) the other particles. Therefore, the method, called configurational freezing, is based on the reasonable assumption that dissipation is a local phenomenon in single-molecule nonequilibrium processes, a statement which is satisfied by most processes, including folding of biopolymers, molecular docking, alchemical transformations, etc. At variance with standard simulations, in configurational freezing simulations the computational cost is not correlated with the size of the whole system, but rather with that of the reaction site. The method is illustrated in two examples, i.e., the calculation of the water to methane relative hydration free energy and the calculation of the potential of mean force of two methane molecules in water solution as a function of their distance.

Understanding binding mechanisms between enzymes and potential inhibitors and quantifying protein-ligand affinities in terms of binding free energy is of primary importance in drug design studies. In this respect, several approaches based on molecular dynamics simulations, often combined with docking techniques, have been exploited to investigate the physicochemical properties of complexes of pharmaceutical interest. Even if the geometric properties of a modeled protein-ligand complex can be well predicted by computational methods, it is still challenging to rank with chemical accuracy a series of ligand analogues in a consistent way. In this article, we face this issue calculating relative binding free energies of a focal adhesion kinase, an important target for the development of anticancer drugs, with pyrrolopyrimidine-based ligands having different inhibitory power. To this aim, we employ steered molecular dynamics simulations combined with nonequilibrium work theorems for free energy calculations. This technique proves very powerful when a series of ligand analogues is considered, allowing one to tackle estimation of protein-ligand relative binding free energies in a reasonable time. In our cases, the calculated binding affinities are comparable with those recovered from experiments by exploiting the Michaelis-Menten mechanism with a competitive inhibitor.

In the framework of single-molecule pulling experiments, the system is typically driven out of equilibrium by a time-dependent external potential V(t) acting on a collective coordinate such that the total Hamiltonian is the sum of V(t) and the Hamiltonian in the absence of external perturbation. Nonequilibrium work theorems such as Jarzynski equality and Crooks fluctuation theorem have been devised to recover free energy differences between states of this extended system. However, one is often more interested in the potential of mean force of the unperturbed Hamiltonian, i.e., in the effective potential dictating the equilibrium distribution of the collective coordinate in the absence of the external potential. In this respect, Hummer and Szabo proposed an algorithm to estimate the desired free energy differences when pulling experiments are performed in only one direction of the process ( Proc. Natl. Acad. Sci. USA 2001, 98, 3658 ). In this paper, we present a potential of mean force estimator of the unperturbed system that exploits the work measured in both forward and backward directions of the process. The method is based on the reweighting technique of Hummer and Szabo and on the Bennett acceptance ratio. Using Brownian-dynamics simulations on a double-well free energy surface, we show that the estimator works satisfactorily in any pulling situation, from nearly equilibrium to strongly dissipative regimes. The method is also applied to the unfolding/refolding process of decaalanine, a system vastly used to illustrate and to test nonequilibrium methodologies. A thorough comparative analysis with another bidirectional potential of mean force estimator ( Minh, D. D. L.; Adib, A. B. Phys. Rev. Lett. 2008, 100, 180602 ) is also presented. The proposed approach is well-suited to recover free energy profiles from single-molecule bidirectional-pulling experiments such as those performed by optical tweezers or atomic force microscopes.

Molybdenum disulfide (MoS) is a well-known and effective lubricant that provides extremely low values of coefficient of friction. It is known that the sliding process may induce structural transformations of amorphous or disordered MoS to the crystalline phase with basal planes oriented parallel to the sliding direction, which is optimal for reducing friction. However, the key reaction parameters and conditions promoting this structural transformation are still largely unknown. We investigate, by employing reactive molecular dynamics simulations, the formation of MoS layers from an amorphous phase as a function of temperature, initial sample density, and sliding velocity. We show that the formation of ordered crystalline structures can be explained in the framework of classical nucleation theory as it predicts the conditions for their nucleation and growth. These results may have important implications in the fields of coating and thin-film deposition, tribology, and in all technological applications where a fast and effective structural transition to an ordered phase is needed.

An important limitation of nonequilibrium pulling experiments/simulations in recovering free energy differences is the poor convergence of path-ensemble averages. Therefore, a large number of fast-switching trajectories needs to achieve free energy estimates with acceptable accuracy. We propose a method to improve free energy estimates by drastically lowering the computational cost of steered molecular dynamics simulations employed to realize such trajectories. This is accomplished by generating trajectories where the particles not directly involved in the driven process are dynamically frozen. Such a freezing is dynamical rather than thermal because it is reached by a synchronous scaling of atomic masses and velocities keeping the kinetic energy of each particle unchanged. The forces between dynamically frozen particles can then be calculated rarely. Thus, it is possible to generate realizations of a process whose computational cost is not correlated with the size of the whole system, but only with that of the reaction site. The method is illustrated on a simple model system and its general applicability is discussed.

In the preceding paper of this series (Part I [P. Nicolini and D. Frezzato, J. Chem. Phys. 138, 234101 (2013)]) we have unveiled some ubiquitous features encoded in the systems of polynomial differential equations normally applied in the description of homogeneous and isothermal chemical kinetics (mass-action law). Here we proceed by investigating a deeply related feature: the appearance of so-called slow manifolds (SMs) which are low-dimensional hyper-surfaces in the neighborhood of which the slow evolution of the reacting system occurs after an initial fast transient. Indeed a geometrical definition of SM, devoid of subjectivity, "naturally" follows in terms of a specific sub-dimensional domain embedded in the peculiar region of the concentrations phase-space that in Part I we termed as "attractiveness region." Numerical inspections on simple low-dimensional model cases are presented, including the benchmark case of Davis and Skodje [J. Chem. Phys. 111, 859 (1999)] and the preliminary analysis of a simplified model mechanism of hydrogen combustion.