Heterogeneous nucleation in the low-barrier regime

In simulations of the two-dimensional Ising model, we examine heterogeneous nucleation induced by a small impurity consisting of a line of $l$ fixed spins. As $l$ increases, we identify a limit of stability beyond which the metastable phase is not defined. We evaluate the free energy barrier for nucleation of the stable phase and show that, contrary to expectation, the barrier does not vanish on approach to the limit of stability. We also demonstrate that our values for the height of the barrier yield predictions for the nucleation time (from transition state theory) and the size of the critical cluster (from the nucleation theorem) that are in excellent agreement with direct measurements, even near the limit of stability.

Figure 1

Free energy barrier $g\left(m\right)$ for homogenous nucleation at $\beta J=0.65$ and $\beta H=0.05$. The symbols (circles and plus signs) give the result obtained from umbrella sampling simulations using $\kappa /J=0.01$. The solid curve is constructed from 16 separate simulations, where $M_0$ is varied from 0 to 300 in steps of 20. The data sets corresponding to each value of $M_0$ are represented by either black open circles or red plus signs. The dashed blue line gives our estimate for $g\left(m\right)$ obtained from CNT using the approach of Ryu and Cai [32, 33].

Figure 2

Example configurations of the system in the metastable phase with $\beta J=0.65$, $\beta H=+0.05$, and $l=7$ (a) when $n=n_0=21$ and (b) when $n=n^{*}=174$. Impurity sites are shown as red open squares. Up-spins ($s_i=+1$) are shown as black solid squares. White regions correspond to down-spins ($s_i=-1$).

Figure 3

(a) Free energy relative to a “bare” impurity, $G\left(n\right)-G\left(l\right)$ as a function of $n$ for impurities of size $l=3$ to 12. (b) Free energy relative to the equilibrium metastable phase, $G\left(n\right)-G_m$ versus $n$ for $l=3$ to 11. For all curves, the statistical error is less than $0.02kT$.

Figure 4

Size of the impurity cluster at the maximum ($n^{*}$) and minimum ($n_0$) of $G\left(n\right)$ as a function of impurity size $l$. Statistical errors are smaller than the symbol size.

Figure 5

(a) Comparison of the nucleation barriers $\Delta G$ and $\Delta G_0$ as a function of impurity size $l$. (b) Nucleation times ${\tau }_{\mathrm{TST}}^0$, ${\tau }_{\mathrm{TST}}$, ${\tau }_{\mathrm{MFPT}}$, and ${\tau }_{\mathrm{Sear}}$ as a function of $l$. For comparison, note that the homogeneous nucleation time when no impurity is present is $1.5\times {10}^9$ MCS [27]. For all quantities, statistical errors are smaller than the symbol size.

Figure 6

Comparison of our estimates for the excess number of up-spins in the critical cluster $\Delta n_{\mathrm{NT}}$, $\Delta n_{\mathrm{NT}}^0$, and $\Delta n_{\mathrm{sim}}$ as a function of $\beta H$, for the $l=7$ system.