**7. Kinetics of nucleation**

To calculate the heterogeneous nucleation rate, it requires several atoms that contact with the surface of the nucleating agent surface. Eqs. (11) and (12) show the formula for the number of atoms and nucleation rates for heterogeneous nucleation. With minor undercooling situation, the nucleation process is happening in the cracks. To ensure that the cracks are effective, the opening of the crack should be large enough as it needs to let the solid to grow out freely without bothering the

*<sup>n</sup>*<sup>∗</sup> <sup>¼</sup> *<sup>n</sup>*1*<sup>e</sup>*

*Nhete* ¼ *f* <sup>1</sup>*C*1*e*

�Δ*Ghete*

� Δ*G* ∗ *hete*

The crystallization process for super-cooled water will start when the ice cluster solid has been formed by the nucleation. The ice starts to grow when the nucleation process has happened. Apart from it, the initial nucleus interrupts the second ice

Water forms the ices by undergoing the freezing process and this phase transition happens widely in nature. It is hard to redo the freezing process to transform water into solid with the specific crystalline structure [28]. The study of the growth of the ice layer on the subcooled surface started early in the field of geography. According to the Neumann problem, the phase change of water supposed to be uniform in the early process at the subcooled surface like ice crystal start to appear as a thin film around the surface. The ice starts thicken in the perpendicular direction uniformly. Primary nucleation will not happen simultaneously at the surface

The reaction of the ice particles in stratospheric clouds will change the halogen chemistry and at the same time, it helps to deplete the ozone layer. For the reactivity of the ice surface, it depends on its mobility or density as it is important to control the degree of the crystallinity and the structure of the ice film itself. To face this issue, ice film need to be grown well on the solid surfaces and use it as a tool to find out the reactions of the atmospheric. The structure of the ice film on the different hexagonal metal surface is sensitive to attach the energy to the metal and

For the phase change, heat and mass transfer is an important part of the onset like freezing on the subcooled surface. It is because it can be a good opportunity for the engineering application [29] such as ice formation on the cryogenic surface [31] and the ice production using scraped-surface heat exchanger [32]. For a certain process, the mass flow rate of the injected vapor is used to calculate the ice layer growth rate which is forms on the plate. Normally, the ice layer grows until it reaches the critical thickness is considered as the outcome of the process. Film-wise condensation starts to occur on the surface of the ice layer when it reaches the

The heat transfer coefficient in the scraped-surface heat exchanger without the phase change (ice growth) is three to five times smaller compared to the phase change. The torque for handle the scraper is increasing when the onset of ice growth is happening until the scrapper becomes frozen. Ice started to growth at stage II and stage III, where the ice particle from the nucleation takes place. The ice particle continuously freezes until it transforms into the ice crystal and the concentration of

*kT* (11)

*kT* (12)

radius of the solid is smaller than the critical radius [25].

*Synthesis Methods and Crystallization*

layer to growth when the latent heat is released [23].

during the onset of the freezing on the subcooled surface [29].

different structures of different transition metal surfaces [30].

**6. Ice growth**

critical thickness [31].

**54**

the mother liquor is increasing [33].

As reviewed, the part of kinetics from the rate of nucleation gives a number of the water molecule which has a potential incorporated into the ice crystal [34]. The flux of water molecule that happens be incorporated with the ice crystal must be counted for nucleation rate estimation. The flux can be written as diffusive flux (Φ) and known as the Boltzmann distribution. The equation is expressed as below.

$$
\Phi = \frac{K\_B T}{h} .e^{\frac{\Delta \mathbf{g}^\*}{K\_B T}} \tag{13}
$$

whereas h refers to the Planck's constant and Δg# is the activation energy for the transfer of a water molecule across the water-ice boundary. The Φ can also be introduced as the frequency at water molecules overcome at the activation energy [34], as for the collisions rate or the probability in breaking the bond and reassemble of molecular.

Besides, the second pre-factor stated as the Zeldovich factor (Z) is for the depletion of the cluster population caused by crystal production. This factor includes the non-equilibrium of the kinetics process which the value is between 10�<sup>2</sup> and 1, where the value 1 considers as assumed equilibrium. As a result, in describing the expression for the Φ and the pre-factor Z the rate (K) for water molecule is transformed into ice crystal can be expressed as equation below;

$$K = n\_s.4\pi r\_{crystal}^2 Z.\Phi\tag{14}$$

whereas *ns* refer to the number of molecules in jumping the crystal surface of the neighboring crystal and the 4*πr*<sup>2</sup> *crystal* is the surface are of the critical ice crystal.
