As a rule, mean-field theories applied to a fluid that can undergo a transition from saturated vapor at density ρ υ to a liquid at density ρ yield a van der Waals loop. For example, isotherms of the chemical potential µ(T, ρ) as a function of the density ρ at a fixed temperature T less than the critical temperature T c exhibit a maximum and a minimum. Metastable and unstable parts of the van der Waals loop can be eliminated by the Maxwell construction. Van der Waals loops and the corresponding double minimum potentials are mean-field artifacts. Simulations at fixed µ = µ coex for ρ υ < ρ < ρ yield a loop, but for sufficiently large systems this loop does not resemble the van der Waals loop, and reflects interfacial effects on phase coexistence due to finite size effects. In contrast to the the van der Waals loop, all parts of the loop found in simulations are thermodynamically stable. The successive umbrella sampling algorithm is described as a convenient tool for seeing these effects. It is shown that the maximum of the loop is not the stability limit of a metastable vapor, but signifies the droplet evaporation-condensation transition. The descending part of the loop contains information on Tolman-like corrections to the surface tension, rather than describing unstable states.
A recently proposed method to obtain the surface free energy σ(R) of spherical droplets and bubbles of fluids, using a thermodynamic analysis of two-phase coexistence in finite boxes at fixed total density, is reconsidered and extended. Building on a comprehensive review of the basic thermodynamic theory, it is shown that from this analysis one can extract both the equimolar radius R(e) as well as the radius R(s) of the surface of tension. Hence the free energy barrier that needs to be overcome in nucleation events where critical droplets and bubbles are formed can be reliably estimated for the range of radii that is of physical interest. It is found that the conventional theory of nucleation, where the interface tension of planar liquid-vapor interfaces is used to predict nucleation barriers, leads to a significant overestimation, and this failure is particularly large for bubbles. Furthermore, different routes to estimate the effective radius-dependent Tolman length δ(R(s)) from simulations in the canonical ensemble are discussed. Thus we obtain an instructive exemplification of the basic quantities and relations of the thermodynamic theory of metastable droplets/bubbles using simulations. However, the simulation results for δ(R(s)) employing a truncated Lennard-Jones system suffer to some extent from unexplained finite size effects, while no such finite size effects are found in corresponding density functional calculations. The numerical results are compatible with the expectation that δ(R(s) → ∞) is slightly negative and of the order of one tenth of a Lennard-Jones diameter, but much larger systems need to be simulated to allow more precise estimates of δ(R(s) → ∞).
We report a self-adapting version of the Wang-Landau algorithm that is ideally suited for application to systems with a complicated structure of the density of states. Applications include determination of two-dimensional densities of states and high-precision numerical integration of sharply peaked functions on multidimensional integration domains.
Using the Wang-Landau algorithm we derive the full thermal order parameter probability distribution of the 4 model for various displacive degrees and temperatures and calculate the resulting free energies. We obtain high-precision data on the shape of the free-energy barrier separating states of opposite order parameter values. For order-disorder-like systems, i.e., at low displacive degree we observe phase separation below the transition temperature. A model taking into account the surface free energy related to different domain shapes, which fits the simulation data extremely well at low temperatures, is constructed. The interpretation of the results in the context of Landau or Landau-Ginzburg theory is discussed and an improved setup for simulating Landau potentials is proposed.
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