Growth and melting of droplets in cold vapors

A model has been developed to investigate the growth of droplets in a supersaturated cold vapor taking into account their possible solid-liquid phase transition. It is shown that the solid-liquid phase transition is nontrivially coupled, through the energy released in attachment, to the nucleation process. The model is based on the one developed by J. Feder, K. C. Russell, J. Lothe, and G. M. Pound [Adv. Phys. 15, 111 (1966)], where the nucleation process is described as a thermal diffusion motion in a two-dimensional field of force given by the derivatives of a free-energy surface. The additional dimension accounts for droplets internal energy. The solid-liquid phase transition is introduced through a bimodal internal energy distribution in a Gaussian approximation derived from small clusters physics. The coupling between nucleation and melting results in specific nonequilibrium thermodynamical properties, exemplified in the case of water droplets. Analyzing the free-energy landscapes gives an insight into the nucleation dynamics. This landscape can be complex but generally exhibits two paths: the first one can generally be ascribed to the solid state, while the other to the liquid state. Especially at high supersaturation, the growth in the liquid state is often favored, which is not unexpected since in a supersaturated vapor the droplets can stand higher internal energy than at equilibrium. From a given critical temperature that is noticeably lower than the bulk melting temperature, nucleation may end in very large liquid droplets. These features can be qualitatively generalized to systems other than water.

Figure 1

Gaussian representation of cluster internal energy distribution just below the melting transition.

Figure 2

(Color online) (a) Nucleation path on the free-energy surface $\Delta G^0\left(u,v\right)$ at $T=243\text{K}$, $S=3$, and $\kappa =0$. The critical size is crossed here in the liquid state. (b) Contour plot of the same energy surface. The arrows follow the bottom of liquid and solid growing channels.

Figure 3

(Color online) Nucleation path on the free-energy surface $\Delta G^0\left(u,v\right)$ at $T=243\text{K}$, $S=1.6$, and $\kappa =0$. Along the minimum-energy path, the saddle point is crossed in the solid phase.

Figure 4

Liquid to solid transition size $n_T$ as a function of supersaturation for different vapor temperatures, without (solid lines) and with (dotted lines) carrier gas, respectively. Droplets are liquid below the lines and solid above. For the curves labeled $E_b=k_{B}T$, the transition size is defined as the size for which the barrier between the solid and the liquid channels becomes lower than $k_{B}T$. At 243 K are also represented the limits defined by $\Delta G_{solid}^0=\Delta G_{liquid}^0$ without (thick dashed line) and with (dashed-dotted line) carrier gas.

Figure 5

(Color online) Free-energy surface $\Delta G^0\left(u,v\right)$ at $T=257\text{K}$ and $S=1.45$. Droplets can reach infinite size in a metastable liquid state.

Figure 6

Approximate critical $S$ value from which the droplets reach “infinite” size ($n={10}^{11}$, see text) in the liquid state, plotted as a function of the vapor temperature. Two conditions are required: (a) the droplets pass the critical size in the liquid state (dashed line $a$). This limit is independent of $\kappa$. (b) The height of the barrier between liquid and solid channels at $n={10}^{11}$ is greater than $k_{B}T$. This limit is shown without buffer gas ($\kappa =0$, solid line $b$) and for a partial pressure of condensable vapor on the order of 1% ($\kappa =100$, dotted line $b^{\prime }$).

Figure 7

(Color online) Free-energy maximum $\Delta G^{\ast }$ along the minimum-energy path as a function of supersaturation near the critical value, calculated for water at 243 K and for $\kappa =0$ (no buffer gas). Three different cases are observed, depending on the value of $S$ compared with the threshold value $S_T$, for which liquid and solid free energies at the critical size are equal. (1) Dashed (blue online) line: the maximum is crossed in the solid state (see Fig. 3); (2) solid (green online) line: the maximum is determined by the barrier between solid and liquid states (see Fig. 8); (3) dashed-dotted (red online) line: the maximum is crossed in the liquid state (see Fig. 2). The dotted black line labeled “Feder” is calculated without melting transition using relation (17).

Figure 8

(Color online) Nucleation path at $T=243\text{K}$, $S=2.05$, and $\kappa =0$. In this critical case [case (2) in Fig. 7], the barrier between solid and liquid channels determines the minimum-energy path.

Figure 9

(Color online) Examples of metastable wells on the free-energy surface $\Delta G^0\left(u,v\right)$ (a) in the liquid state and (b) in the solid state.

Figure 10

Liquid to solid transition size as a function of the amount of carrier gas. $\kappa$ characterizes the number of collisions with the carrier gas between two collisions with the condensable gas.