Calculating the free energy of 2D materials on substrates

A method was developed to calculate the free energy of 2D materials on substrates and was demonstrated by the system of graphene and {\gamma}-graphyne on copper substrate. The method works at least 3 orders faster than state-of-the-art algorithms, and the accuracy was tested by molecular dynamics simulations, showing that the precision for calculations of the internal energy achieves up to 0.03% in a temperature range from 100 to 1300K. As expected, the calculated the free energy of a graphene sheet on Cu (111) or Ni (111) surface in a temperature range up to 3000K is always smaller than the one of a {\gamma}-graphyne sheet with the same number of C atoms, which is consistent with the fact that growth of graphene on the substrates is much easier than {\gamma}-graphyne.