ABSTRACT
Initial cap formation is an important process of carbon nanotube (CNT) growth where a hexagonal carbon network is lifted off from the catalyst surface. In this study, free energy surface (FES) of initial cap formation in the CNT growth is investigated by metadynamics simulation. A two-dimensional collective variable (CV) space is newly developed to examine the complicated formation process of the cap structure, which consists of the formation of a hexagonal carbon network and lift-off of the network from the catalyst surface. States before and after the lift-off of the carbon network are clearly distinguished in the two-dimensional FES. Therefore, free energy difference before and after the lift-off can be directly derived from the two-dimensional FES. It was revealed that the cap structure is stable at a high temperature due to the entropy effect, while the carbon network covering the catalyst surface is energetically stable. The new insight in this study is achieved owing to metadynamics simulation in conjunction with a newly developed two-dimensional CV space since it is impossible to explore FES for such complicated processes in the framework of conventional molecular dynamics simulation.
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.
