Grain-boundary free energy via thermodynamic integration

In a previous publication by Lusk and Beale [Phys. Rev. E 69, 026117 (2004)], fluctuating cell (FC) theory was used to estimate the free energy of symmetric tilt grain boundaries in an assembly of nearly hard disks. The FC method is much faster than the more traditional thermodynamic integration, but the accuracy of the algorithm has not been assessed in association with persistent defect structures. This motivated the present work wherein the FC free energies are compared directly with the data obtained via thermodynamic integration from an Einstein crystal to an assembly of hard disks. This comparison is made over the range of possible misorientations for symmetric tilt boundaries and indicates that the FC method gives quantitatively accurate estimates for grain-boundary free energy. We also demsonstrate that the FC approximation is quantitatively accurate at determining the free-energy contribution of each particle whether in the bulk or the grain boundary. The FC calculation is about two orders of magnitude faster than a full thermodynamic integration. This approach may offer a numerically efficient means of estimating the free energy of persistent defect structures to greater accuracy than is afforded by the quasiharmonic and local harmonic approximations.