The liquid-liquid transition in supercooled ST2 water: a comparison between umbrella sampling and well-tempered metadynamics

We investigate the metastable phase behaviour of the ST2 water model under deeply supercooled conditions. The phase behaviour is examined using umbrella sampling (US) and well-tempered metadynamics (WT-MetaD) simulations to compute the reversible free energy surface parameterized by density and bond-orientation order. We find that free energy surfaces computed with both techniques clearly show two liquid phases in coexistence, in agreement with our earlier US and grand canonical Monte Carlo calculations [Y. Liu, J. C. Palmer, A. Z. Panagiotopoulos and P. G. Debenedetti, J Chem Phys, 2012, 137, 214505; Y. Liu, A. Z. Panagiotopoulos and P. G. Debenedetti, J Chem Phys, 2009, 131, 104508]. While we demonstrate that US and WT-MetaD produce consistent results, the latter technique is estimated to be more computationally efficient by an order of magnitude. As a result, we show that WT-MetaD can be used to study the finite-size scaling behaviour of the free energy barrier separating the two liquids for systems containing 192, 300 and 400 ST2 molecules. Although our results are consistent with the expected N-2/3 scaling law, we conclude that larger systems must be examined to provide conclusive evidence of a first-order phase transition and associated second critical point.