Abstract: Atomistic simulation methods for the quantification of free energies are in wide use. These methods operate by sampling the probability density of a system along a small set of suitable collective variables (CVs), which is, in turn, expressed in the form of a free energy surface (FES). This definition of the FES can capture the relative stability of metastable states but not that of the transition state because the barrier height is not invariant to the choice of CVs. Free energy barriers therefore cannot be consistently computed from the FES. Here, we present a simple approach to calculate the gauge correction necessary to eliminate this inconsistency. Using our procedure, the standard FES as well as its gauge-corrected counterpart can be obtained by reweighing the same simulated trajectory at little additional cost. We apply the method to a number of systems—a particle solvated in a Lennard-Jones fluid, a Diels–Alder reaction, and crystallization of liquid sodium—to demonstrate its ability to produce consistent free energy barriers that correctly capture the kinetics of chemical or physical transformations, and discuss the additional demands it puts on the chosen CVs. Because the FES can be converged at relatively short (sub-ns) time scales, a free energy-based description of reaction kinetics is a particularly attractive option to study chemical processes at more expensive quantum mechanical levels of theory.

Abstract: The free energy surface (FES) for carbon segregation from nickel nanoparticles is obtained from advanced molecular dynamics simulations. A suitable reaction coordinate is developed that can distinguish dissolved carbon atoms from segregated dimers, chains and junctions on the nanoparticle surface. Because of the typically long segregation time scale (up to ms), metadynamics simulations along the developed reaction coordinate are used to construct FES over a wide range of temperatures and carbon concentrations. The FES revealed the relative stability of different stages in the segregation process, and free energy barriers and rates of the individual steps could then be calculated and decomposed into enthalpic and entropic contributions. As the carbon concentration in the nickel nanoparticle increases, segregated carbon becomes more stable in terms of both enthalpy and entropy. The activation free energy of the reaction also decreases with the increase of carbon concentration, which can be mainly attributed to entropic effects. These insights and the methodology developed to obtain them improve our understanding of carbon segregation process across materials science in general, and the nucleation and growth of carbon nanotube in particular.