Ice nucleation on carbon surface supports the classical theory for heterogeneous nucleation

The prevalence of heterogeneous nucleation in nature was explained qualitatively by the classical theory for heterogeneous nucleation established over more than 60 years ago, but the quantitative validity and the key conclusions of the theory have remained unconfirmed. Employing the forward flux sampling method and the coarse-grained water model (mW), we explicitly computed the heterogeneous ice nucleation rates in the supercooled water on a graphitic surface at various temperatures. The independently calculated ice nucleation rates were found to fit well according to the classical theory for heterogeneous nucleation. The fitting procedure further yields the estimate of the potency factor, which measures the ratio of the heterogeneous nucleation barrier to the homogeneous nucleation barrier. Remarkably, the estimated potency factor agrees quantitatively with the volumetric ratio of the critical nuclei between the heterogeneous and homogeneous nucleation. Our numerical study thus provides a strong support to the quantitative power of the theory and allows understanding ice nucleation behaviors under the most relevant freezing conditions.

Figure 1

(a) A solid critical nucleus forms at the interface between liquid and a foreign flat wall, with a solid contact angle ${\theta }_c$. The contact angle ${\theta }_c$ can be determined by the surface tensions through Eq. (3). The solid nucleus is assumed to be part of a sphere (dashed line) with a radius $r^{*}$, i.e., a spherical cap with the volume $V_{\text{cap}}=f\left({\theta }_c\right)V_{\text{sphere}}$. (b) The potency factor $f\left({\theta }_c\right)$ increases from 0 to 1 (homogeneous), as the solid contact angle ${\theta }_c$ varies from 0 to ${180}^{\circ }$.

Figure 2

Temperature dependence of ice nucleation rate (logarithm) in the mW water model for both homogeneous ice nucleation and the heterogeneous nucleation on a graphitic surface. The solid red dots and blue squares represent the calculated ice nucleation rates by using the FFS method. The dashed red and blue lines indicate the fitting on the basis of the CNT for homogeneous nucleation [Eq. (7)] and the extension for heterogeneous nucleation [Eq. (8)], respectively. The data of homogeneous nucleation were extracted from Ref. [12].

Figure 3

(a) Variation of the critical ice nucleus size with temperature for both homogeneous and heterogeneous nucleation. The critical size is estimated based on the committor probability analysis, which yields ${\lambda }_{\text{het}}^{*}$: $145\pm 5$ at 225 K, $205\pm 10$ at 230 K, $288\pm 7$ at 235 K, and $410\pm 10$ at 240 K, respectively. (See Appendixes A and B for more details.) The critical size for homogeneous nucleation was obtained in Ref. [12]. The simulation data and the fitted curves are represented by data points and solid lines, respectively. (b) A snapshot of the critical ice nucleus forming from homogeneous ice nucleation at 240 K. (c) A snapshot of the critical ice nucleus forming on graphene surface at 240 K.

Figure 4

The predicted temperature variation of (a) ice nucleation rate $R\left(T\right)$ and (b) critical size of ice nucleus ${\lambda }^{*}\left(T\right)$ in the mW water model due to the presence of a heterogeneous nucleation center with different potency factor $f$. A small potency factor $f$ indicates a strong ice nucleation efficacy, and $f=1$ corresponds to homogeneous ice nucleation. (c) The predicted variation of the critical size ${\lambda }^{*}$ for different potency factor $f$, subject to a fixed nucleation rate.

Figure 5

Calculated growth probability $P\left(\lambda \right|{\lambda }_0)$ as a function of ice nucleus size $\lambda$ for heterogeneous ice nucleation on a graphitic surface at 225 K, 230 K, 235 K, and 240 K. The interfaces ${\lambda }_i$ used in FFS simulation correspond to the abscissa of the data points.

Figure 6

Calculated committor $p_{\text{B}}$ as a function of the nucleus size $\lambda$ for heterogeneous ice nucleation on a graphitic surface at various nucleation temperatures. The dotted lines indicate the upper and lower boundaries of the calculated $p_{\text{B}}$ on the basis of the estimated error bar of $p_{\text{B}}$. The horizontal dash line corresponds to a $p_{\text{B}}=0.5$ and intersects the both boundaries of the calculated $p_{\text{B}}$, which yields the estimate of the uncertainty of ${\lambda }^{*}$.

Figure 7

The computed histogram of committor $p_{\text{B}}$ for heterogeneous ice nucleation at 230 K at the interface $\lambda =210$. The Gaussian distribution function with the same intrinsic mean and standard deviation is shown as the red line.