Numerical tests of nucleation theories for the Ising models

The classical nucleation theory (CNT) is tested systematically by computer simulations of the two-dimensional (2D) and three-dimensional (3D) Ising models with a Glauber-type spin flip dynamics. While previous studies suggested potential problems with CNT, our numerical results show that the fundamental assumption of CNT is correct. In particular, the Becker-Döring theory accurately predicts the nucleation rate if the correct droplet free energy function is provided as input. This validates the coarse graining of the system into a one dimensional Markov chain with the largest droplet size as the reaction coordinate. Furthermore, in the 2D Ising model, the droplet free energy predicted by CNT matches numerical results very well, after a logarithmic correction term from Langer’s field theory and a constant correction term are added. But significant discrepancies are found between the numerical results and existing theories on the magnitude of the logarithmic correction term in the 3D Ising model. Our analysis underscores the importance of correctly accounting for the temperature dependence of surface energy when comparing numerical results and nucleation theories.

Figure 1

(Color online) Effective surface free energy ${\sigma }_{\text{eff}}$ as a function of temperature for the 2D Ising model from analytic expression [21]. The free energy of the surface parallel to the sides of the squares, ${\sigma }_{\left(10\right)}$, is also plotted for comparison.

Figure 2

The probability $P\left({\lambda }_i\mid {\lambda }_0\right)$ (solid line) of reaching interface ${\lambda }_i$ from ${\lambda }_0$ and average committor probability $\overline{P_B}\left({\lambda }_i\right)$ (circles) over interface ${\lambda }_i$ at $\left(k_{B}T,h\right)=\left(1.5,0.05\right)$ for the 2D Ising model. The 50% committor point is marked by ${}^{\ast }$.

Figure 3

(Color online) (a) Droplet free energy $F\left(n\right)$ obtained by U.S. at $k_{B}T=1.5$ and $h=0.05$ in the 2D Ising model. (b) Fluctuation of droplet size $⟨\Delta n^2\left(t\right)⟩$ as a function of time.

Figure 4

(Color online) The nucleation rate $I$ computed by FFS (open symbols) and Becker-Döring theory with U.S. free energies (filled symbols) in the (a) 2D and (c) 3D Ising models. The ratio between nucleation rates obtained by FFS and Becker-Döring theory at different temperatures in the (b) 2D and (d) 3D Ising models. The symbols in (b) and (d) match those defined in (a) and (c), respectively.

Figure 5

(Color online) (a) For 2D Ising model, the critical droplet size $n$ obtained from FFS (filled symbols) and umbrella sampling (open symbols). $n_c$ predicted by Becker-Döring theory (dotted line) and by field theoretic equation (solid line) are plotted for comparison. (b) For 3D Ising model, the critical droplet size $n$ obtained from FFS (filled symbols) and umbrella sampling (open symbols).

Figure 6

(Color online) (a) Histogram of committor probability in an ensemble of spin configurations with $n=496$ for the 2D Ising model at $k_{B}T=1.5$ and $h=0.05$. Representative droplets are also shown, with black and white squares corresponding to $+1$ and $-1$ spins, respectively. (b) Histogram of committor probability in an ensemble of spin configurations with $n=524$ for the 3D Ising model at $k_{B}T=2.20$ and $h=0.40$.

Figure 7

(Color online) (a) Droplet free energy curve $F\left(n\right)$ of the 2D Ising model at $k_{B}T=1.5$ and $h=0.05$ obtained by U.S. (circles) is compared with Eq. (6) (solid line) and Eq. (9) (dashed line). Logarithmic correction term $\frac{5}{4}{k}_{B}T\text{ln}n$ (dot-dashed line) and the constant term $d$ (dotted line) are also drawn for comparison. (b) Magnified view of (a) near $n=0$, together with the results from analytic expressions (squares) available for $n\le 17$ (see Appendix ).

Figure 8

(Color online) (a) Droplet free energy $F\left(n\right)$ of the 3D Ising model at $k_{B}T=2.40$ and $h=0$ obtained by U.S. (circles) is compared with Eq. (30) (solid line) and Eq. (16) (dots). Logarithmic term $\tau k_{B}T\text{ln}n$ is also plotted (dot-dashed line). The difference in predictions by classical expression Eq. (16) and field theory Eq. (30) are very small compared to $F\left(n\right)$ itself and cannot be observed at this scale. (b) Magnified view of (a) near $n=0$, together with the analytic solution of small droplets (squares, see Appendix ) and the exponential correction term (dashed line).

Figure 9

(Color online) (a) Surface free energies of the 3D Ising model as functions of temperature. Circles are fitted values of ${\sigma }_{\text{eff}}$ from Eq. (30), dashed line is the expected behavior of ${\sigma }_{\text{eff}}$ over a wider range of temperature, and solid line is the free energy of the (100) surface [44]. Numerically fitted values of ${\sigma }_{\text{eff}}$ from Heermann et al. [29] are plotted as $+$. (b) $\tau$ values that give the best fit to the free energy data from U.S. $\tau$ can be roughly described by a linear function of $T$ shown as a straight line. No abrupt change is observed near the roughening temperature $T_R$.

Figure 10

(a) The pre-exponential factor $f_c^{+}\Gamma$ in 2D computed from Monte Carlo and U.S. (b) The ratio between the attachment rate $f_c^{+}$ in 2D computed by Monte Carlo and that predicted by Eq. (13). (a) The pre-exponential factor $f_c^{+}\Gamma$ in 3D computed from Monte Carlo and U.S. (b) The ratio between the attachment rate $f_c^{+}$ in 3D computed by Monte Carlo and that predicted by Eq. (20).

Figure 11

(Color online) (a) Droplet free energy as a function of droplet size $n$ at $h=0.1$ and different $k_{B}T$ for the 2D Ising model. The critical droplet free energy is marked by circles. (b) Critical droplet free energy (circles) from (a) as a function of $k_{B}T$ for the 2D Ising model. The solid line is a linear fit of the data, and the dashed line is the prediction of Eq. (11). (c) Droplet free energy as a function of droplet size $n$ at $h=0.45$ and different $k_{B}T$ for the 3D Ising model. (d) Critical droplet free energy from (c) as a function of $k_{B}T$ for the 3D Ising model. The solid line is a linear fit of the data, and the dashed line is the prediction of Eq. (18).

Figure 12

Droplets in (a) 2D and (b) 3D Ising models randomly chosen from FFS simulations at different $\left(T,h\right)$ conditions. $n$ is the size of the droplet.