This paper presents a direct numerical simulation of solidification of a molten metal drop on a cold plate with various wettability by an axisymmetric front-tracking method. Because of the plate kept at a temperature below the fusion value of the melt, a thin solid layer forms at the plate and evolves upwards. The numerical results show that the solidifying front is almost flat except near the triple point with a high solidification rate at the beginning and final stages of solidification. Two solid-to-liquid density ratios _{sl}_{0} = 0° and 12° are considered. The presence of volume change and a non-zero growth angle results in a solidified drop with a conical shape at the top. The focusing issue is the effects of the wettability of the plate in terms of the contact angle _{0}. Increasing the contact angle in the range of 45° to 120° increases time for completing solidification, i.e., solidification time. However, it has a minor effect on the conical angle at the top of the solidified drop and the difference between the initial liquid and final solidified heights of the drop. The effects of the density ratio and growth angle are also presented.

Liquid–solid phase change process in drops sessile on a cold solid surface has been of important interest in recent years due to its wide appearance in nature and engineering problems such as water drops freezing on wind turbine blades and electric cables, and metal drops solidifying in the crystallization and atomization processes. Because of different types of solid surfaces and drop liquid materials, drops solidify from the surface at different contact angles, i.e., different wettability. Accordingly, many works related to this problem have been carried out.

Experimentally, Huang et al. [

Instead of using water, Satunkin [_{3}N_{4} for applications of solar cells. In Hariharan and Ravi’s invention [

Theoretically, Zhang et al. [

A few numerical simulations can be found in Schultz et al. [

It is evident that detailed direct numerical simulations of the solidification process of a molten metal drop (i.e., with the Prandtl number of around 0.01) under the effect of the contact angle are rarely found in the literature. This gap motivates our present study since the problem is extremely important not only in academia but also in nature and engineering applications [

_{gr}_{s}_{l}_{0} and the wetting radius _{w}_{c}_{m}_{n}_{h}_{f}_{p}_{0} and _{δ}_{f}

This equation accounts for volume change upon solidification [

The above-mentioned equations are solved by the front-tracking method combined with an interpolation technique for three phase computations [

We choose the effective radius of the drop _{0} is the volume of the initial liquid drop). The velocity scale is _{c}_{c}_{0}, density ratios _{sl}_{gl}_{gl}_{sl}_{gl}_{psl}_{pgl}

The dimensionless time and temperature are _{c}_{0} for two solid-to-liquid density ratios and two growth angles, and thus other parameters are kept constant, i.e., _{gl}_{gl}_{sl}_{psl}_{pgl}_{gl}_{0} = 1. As demonstrated in our previous works [_{0} = 1 can be negligible. These parameters correspond to a liquid drop of metals or semiconductor materials, such as silicon or germanium (i.e.,

Method validations have been carefully carried out in our previous works [

_{sl}_{sl}

Next, we consider the effects of the contact angle on the solidification process.

_{0} = 60° and _{0} = 105° with _{sl}_{gr}

Because of a wider wetted drop radius, decreasing the contact angle from 105° to 60° leads to a decrease in the drop height and thus the solidified drop height. This effect is evidently seen in _{gr}_{gr}_{gr}_{gr}

The effect of the contact angle _{0} is more clearly seen from _{f}_{d}_{0}. However, the height of the solidified drop and thus the height difference, _{d}_{0}, increase with an increase in the growth angle or with a decrease in the density ratio [

In contrast to the significant effects on the solidification time, the contact angle has a minor effect on the formation of the conical angle at the top of the solidified drop as shown in _{gr}_{gr}_{sl}

We have presented a detailed numerical work of a liquid metal drop solidifying on a cold plate by the front-tracking method. We focus on the effects of the initial shape of the drop, in terms of the contact angle _{0}, for two solid-to-liquid density ratios (_{sl}_{gr}_{sl}_{sl}_{gr}_{sl}_{gr}_{0} in the range of 45–120° show that increasing the growth angle increases the solidification time and the solidified drop height. However, the contact angle has minor effects on the height difference between the solidified drop and the liquid drop as well as the conical angle at the top of the solidified drop. Meanwhile, this conical angle increases with an increase in _{sl}_{gr}

The numerical results of the present study can be used for applications in metal atomization or crystal growth of such materials as silicon or germanium. For instance, in crystallization of molten silicon drops, with a diameter of a few millimeters, from substrates for spherical solar cells [

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.03-2017.01. We are grateful to John C. Wells at Ritsumeikan University (Japan) for facilitating our computing resources.

Cuong T. Nguyen and Duong K. Tran performed simulations and analyzed the data; Truong V. Vu wrote the paper.

The authors declare no conflict of interest.

_{3}N

_{4}substrate that repels Si melt

A molten metal drop solidifying on a cold plate: (

(_{sl}_{gr}_{sl}_{gr}

Evolution of the solidifying front with the temperature (left) and pressure (right) fields at (_{c}_{sl}_{0} = 120° and _{gr}

(_{f}_{na}_{c}

Effect of the contact angle on the solidification process for two cases _{0} = 60° and 105°: (_{f}_{sl}_{gr}

The solidified drop profiles for various contact angles, two density ratios _{sl}_{sl}_{gr}_{gr}

Effects of the contact angle, density ratio and growth angle on the solidification process: (_{f}_{d}_{0}, between the final height after complete solidification (_{d}_{0}); (