Cluster free energy in the simple-cubic Ising model

Abstract

The Monte Carlo technique previously developed to compute directly the free-energy difference between two clusters containing $N+1$ and $N$ particles is applied to the three-dimensional cubic Ising model with nearest-neighbor interactions. For cluster sizes $1\ensuremath{\le}N\ensuremath{\le}19$, at two reduced temperatures ($\frac{T}{{T}_{c}}=0.4 \mathrm{and} 0.59$) it is found that down to surprisingly small sizes ($N\ensuremath{\ge}8$) the free energy ${F}_{N}$ can be written as ${F}_{N}=Na+{N}^{\frac{2}{3}}b+{N}^{\frac{1}{3}}c+d+\ensuremath{\tau}{\ensuremath{\beta}}^{\ensuremath{-}1}\mathrm{ln}N$, where $a$ and $\ensuremath{\tau}$ are given their theoretical value for bulk phases, and $b$, $c$, and $d$ can be understood, respectively, as the surface contribution, a step contribution, and a vertex contribution. A Fisher-type model (${F}_{N}=Na+{N}^{\ensuremath{\sigma}}\ensuremath{\Sigma}+\ensuremath{\tau}{\ensuremath{\beta}}^{\ensuremath{-}1}\mathrm{ln}N$) when fitted to the data yields a size- and temperature-dependent value of $\ensuremath{\sigma}$. Other proposed models are shown to depart systematically from our data. Typical values of nucleation rates for this model, as predicted from the present data, range from ${10}^{\ensuremath{-}3}$ to ${10}^{+2}$\char22{}those expected from the classical capillarity approximation of ${F}_{N}$.