# Liquid encapsulation in a freezing sessile drop

Sijia Lyu, Xun Zhu, Dominique Legendre, Chao Sun

► To cite this version:

Sijia Lyu, Xun Zhu, Dominique Legendre, Chao Sun. Liquid encapsulation in a freezing sessile drop. Droplet, 2023, 2 (4), 10.1002/dro2.90. hal-04837690

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# Liquid encapsulation in a freezing sessile drop

Sijia Lyu<fnref n="1" /> ID

Xun Zhu<fnref n="1" />

Dominique Legendre<fnref n="2" />

Chao Sun<fnref n="1" />,<fnref n="3" />

## Correspondence

Chao Sun, Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China.
Email: chaosun@tsinghua.edu.cn

## Funding information

Natural Science Foundation of China, Grant/Award Numbers: 11988102, 91852202, 11861131005; New Cornerstone Science Foundation; China Postdoctoral Science Foundation, Grant/Award Numbers: 2022T150068, 2021M700507

## Abstract

During the solidification of a sessile drop, the effect of heat exchange from the gaseous environmental medium is generally ignored. However, by combining experimental observations, direct numerical simulations, and a theoretical model, we have demonstrated that the environmental medium, particularly one with high thermal conductivity such as a liquid, has nonnegligible heat exchange with both the drop and the substrate, leading to accelerated cooling of the outer surface of the sessile drop. Consequently, it causes alterations in the geometry of the freezing front and ultimately results in the formation of a solidified shell that encloses the drop. Furthermore, the encapsulated liquid continues to solidify, which induces volume change and consequently changes the final outcome of the freezing process. This study highlights the importance of considering the properties of the environmental medium and provides novel strategies to manipulate the freezing rate and reshape the morphology of the solidified drop.

## INTRODUCTION

Solidification of a sessile drop is very common and significant in nature and industry. Especially in the cold winter season, accumulating frozen drops on substrates may cause various adverse effects, such as fracture electrical wires,<fnref n="1" /> aviation safety hazards,<fnref n="2" /> and added load on wind turbines.<fnref n="3" /> On the other hand, solidification of molten material drops is pivotal in advanced manufacturing industries, such as thermal spraying,⁴ three-dimensional (3D) printing,⁵ and laser beam welding.⁶ Different solidification rates and final frozen morphologies could cause variations in the structure and strength of materials.

Many factors can influence the freezing dynamics of drops. These influencing factors are mainly divided into three categories. The first category is related to the characteristics of drops, such as size,⁷,⁸ physical properties of the material,⁹,¹⁰ initial configuration,¹¹,¹² and

impact velocity.¹³⁻¹⁵ The second category is related to the substrate, such as substrate temperature¹⁶,¹⁷ and wettability.¹⁸⁻²² The third category is related to the environmental medium, which is generally ignored. The common stationary gaseous environmental medium is usually regarded as adiabatic due to very low thermal conductivity.¹⁰ However, when the environmental medium is flowing, changes in the heat transfer process and the geometry of the drop result in different final frozen morphologies.²³⁻²⁶ Besides, if water vapor exists in the gaseous environmental medium, it may condense on the freezing drop surface and the released latent heat could affect the freezing front kinetics, the tip angle of a frozen drop, and even induce the growth of ice crystals at the drop tip.²⁷⁻³⁰ In addition, the presence of a neighboring drop induces nonuniform cooling at the droplet-gas interface of a sessile drop, which may cause its asymmetrical solidification.³¹ From these investigations, it is clear that even a little

<footnote n="1">¹Center for Combustion Energy, Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing, China</footnote>

<footnote n="2">²Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS-Toulouse, France</footnote>

<footnote n="3">³Department of Engineering Mechanics, School of Aerospace Engineering, Tsinghua University, Beijing, China</footnote>

<footnote>This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.
© 2023 The Authors. Droplet published by Jilin University and John Wiley & Sons Australia, Ltd.</footnote>
heat transfer from the surrounding medium can obviously affect the freezing dynamics of a sessile drop. When a sessile drop is deposited within a liquid or gel environmental medium with higher conductivity or when the temperature difference between a drop and the environmental medium is very high, the heat transfer between the drop and the environmental medium becomes nonnegligible. In this paper, we will show an intriguing effect of the heat exchange with the environmental medium on the freezing dynamics of a sessile drop by combining experiments, simulations, and a theoretical model.

## RESULTS AND DISCUSSION

**External freezing dynamics:** In experiments, we deposit water drops within gaseous and liquid environmental media with typical different thermal conductivities. A simple sketch of the experiment setup and some cross-sectional characteristic parameters of the drop is shown in Figure 1a. Using a microliter-pipette, degassed and deionized water drops with a constant volume $V_l = 10 \mu L$ are deposited on the substrate. In Figure 1b, the gaseous environmental medium of low thermal conductivity is nitrogen. The specific thermal conductivity ratio of the environmental medium to the solidified phase, $\lambda_e/\lambda_s$, is 0.01. In addition, nitrogen can avoid the effect of water vapor condensation. When the substrate temperature $T_w$ approaches -9°C, a water drop starts to freeze. From the side view, the front light illuminates the ice part. Visible freezing fronts are shown as the orange solid lines. During solidification, the drop outer surface and front geometry always remain symmetric. Figure 1c shows that the external contour of the completely solidified drop is smooth and symmetric, with a tip angle of 136°, which is in agreement with previous observations.10,27 In Figure 1d, the liquid environmental medium of high thermal conductivity is silicone oil. The conductivity ratio $\lambda_e/\lambda_s$ is changed to 0.06. The initial height of the oil bath is 2 mm. A water drop starts to freeze on the substrate at

$T_w = -10°C$. The backlight illuminates the water part and makes it easy to identify the freezing front. For the environmental medium of high thermal conductivity, the freezing dynamics of a sessile drop change dramatically. First, on comparing the second column in Figure 1b,d, it is clear that both visible freezing fronts are symmetric, but the curvature of the freezing front near the triple line is larger in the environmental medium with a high conductivity ratio $\lambda_e/\lambda_s$. Second, on comparing the second and third images in Figure 1d, it is clear that a dark ice crystal structure appears near the top of the drop, which is indicated by a blue arrow. This ice structure is nearly immobile (a more complete process is shown in Supporting Information: Movie S2), which means that it could limit part of the drop's outer surface. In addition, as a green arrow indicates, a clear ice shell appears on the drop's outer surface, and the ice in the middle is cloudy. The divergence of the backlight transmittance may result from the sequence of solidification. The first frozen ice is the cleanest and purest, which pushes air bubbles and impurities out32,33 and causes the remaining ice to become cloudy. Thus, the experimental result indicates that the outer surface of the drop may freeze before part of the internal water trapped inside the ice shell. These special ice structures are not found for a freezing sessile drop in nitrogen. Third, the drop's outer profile in the environmental medium with a high conductivity ratio breaks the symmetry, as shown in the third and fourth images in Figure 1d. A nonsmooth asymmetric frozen drop finally forms (shown in Figure 1e), which is totally different from the result in the case of nitrogen (shown in Figure 1c).

**Encapsulated drop:** To precisely explain the different phenomena observed in different environmental media, a detailed investigation of the freezing process in the drop is required. For this purpose, we perform well-controlled direct numerical simulations (DNS) of the Navier-Stokes equations and the solidification process using the in-house developed code named JADIM.34-36 This numerical method can accurately simulate the quasi-equilibrium isotropic solidification process without protrusions or dendritic crystal structures. For more

<figure><img src="image_2.png" /><figcaption>**FIGURE 1** Experimental investigations of a freezing sessile water drop in different environmental media. (a) Simple sketch of the experimental setup and some cross-sectional parameters. (b) A water drop freezes in the gaseous environmental medium (nitrogen) with a conductivity ratio $\lambda_e/\lambda_s = 0.01$. See Supporting Information: Movie S1. (c) A water drop freezes in the liquid environmental medium (silicone oil) with $\lambda_e/\lambda_s = 0.06$. See Supporting Information: Movie S2. External contours of two completely solidified drops are compared in (c, e). Visible freezing fronts are shown as orange solid lines. A fixed ice crystal structure and an ice shell are indicated by a blue arrow and a green arrow, respectively.</figcaption></figure>
details on the numerical method and validation, please refer to our previous paper.³⁶

We systematically investigate an axisymmetric sessile water drop that freezes in different environmental media with different thermal conductivities. Figure 2 shows the evolution of freezing drops on the cross-section. The initial geometry of a drop is a spherical cap with a radius R = 1mm. The effect of the contact angle on drop-freezing dynamics has been studied in our previous work.³⁶ Here, we set the contact angle at θ = 90°. At the initial moment t = 0s, the temperature of the drop Tᵢᵢ and that of the environmental medium Tᵢₖ are equal at 20°C. The cold wall temperature Tᵣ is always maintained at -20°C. The Stefan number Ste = Cₚs(Tᵣ - Tᵣ)⁄L is 0.12, where Cₚs is the specific heat capacity of ice, Tᵣ is the freezing point of water, and L is the latent heat of solidification. The color in Figure 2 shows the temperature distribution, and the color map shows the temperature range. Red and black lines, respectively, represent liquid-solid interfaces and outer surfaces of drops.

In Figure 2a, the environmental medium, the liquid phase, and the solidified phase are nitrogen, water, and ice, and their thermal conductivities are λe = 0.026 W/(m K), λl = 0.555 W/(m K), and λs = 2.187 W/(m K), respectively. Other physical properties are shown in Supporting Information. As freezing starts, the drop and the environmental medium are meanwhile cooled down by the cold substrate. Once the liquid temperature is lower than its freezing point Tf = 0°C, the liquid phase is transformed into the solid phase. In our

simulations, for water drop solidification, the drop volume expansion is induced by the density difference,³⁷ and the deformation of the liquid phase is controlled by the balance of surface tension and gravity. These two conditions change the drop's outer profile.³⁶ Thus, on comparing the leftmost and the rightmost images in Figure 2a, it is clear that the height of the frozen drop obviously increases and a tip appears at the top of the drop. These observations have been confirmed in experiments.¹⁰,³⁸,³⁹ Another characteristic is the curved shape of the freezing front.⁴⁰,⁴¹ Because the conductivity ratio λe/λs is 0.01 and the temperature difference between the drop surface and the environmental medium is not large, most of the released latent heat is transferred through the solidified phase to the cold substrate,⁴² with an isothermal freezing front nearly perpendicular to the surface of the drop.¹⁰ Finally, the freezing front clearly converges to the drop apex, resulting in tip formation. The freezing time in this case is 3.6 s.

However, when the conductivity of the environmental medium λe further increases, the freezing dynamics of a sessile drop are significantly impacted. Figure 2b,c shows results when the conductivity ratio is increased to λe/λs = 0.06 and 0.59. The simulations clearly reveal the formation of a liquid in the frozen drop. As the conductivity ratio increases, the solidification rate of the triple line increases, resulting in a change in the shape of the icing front and encapsulation of the liquid phase when the drop's outer surface is fully solidified. A small encapsulated liquid formation is shown in

<figure><img src="image_3.png" /><figcaption>**FIGURE 2** Numerical investigations of a freezing sessile water drop in environmental media with different thermal conductivities. (a-c) Images show the evolution of a freezing drop on the cross-section. Red and black lines represent freezing fronts and outer surfaces of drops, respectively. The thermal conductivity ratio of the environmental medium to the solidified phase λe/λs = 0.01, 0.06, and 0.59. The third column shows the precise moment the drop's outer surface is completely solidified. The drop height at this moment is defined as hₐ. The color of the background shows the temperature distribution, and the temperature range is shown in the color map. The numerical solidification processes of (a) and (c) are shown in Supporting Information: Movies S3 and S4. Scale bars in white: 0.5 mm.</figcaption></figure>
Figure 2b, while a much larger volume of liquid is encapsulated in Figure 2c. We define the corresponding encapsulation time as $t_b$ and $h_i = h_b - h_c$ as the height of the encapsulated liquid, where $h_b$ and $h_c$ are the freezing front heights at the triple line and the central axis, respectively (see Figure 1a). Note that the geometry of the drop after encapsulation is fixed in the simulation because the ice is considered perfectly rigid, while it may deform and crack in the experiments as shown in Figure 1d,e and described in the following.

Time evolution of the freezing front height in Figure 2a,c is compared in Figure 3a. For the case of low conductivity ratio $\lambda_e/\lambda_s = 0.01$, the central height $h_c$ increases at a slower rate than $h_b$ in the beginning, but its growth rate rapidly increases toward the end. Finally, $h_c$ and $h_b$ nearly meanwhile reach the drop apex. However, for the case of high conductivity ratios such as $\lambda_e/\lambda_s = 0.59$, due to the cooling effect of the environmental medium, the height of the triple line increases much faster. Just after 1 s, the triple line reaches the drop apex, while the evolution of $h_c$ is similar to that of low conductivity

ratios. Thus, a large height difference $h_i$ occurs at $t = 1$ s, which indicates that a large amount of water remains in the encapsulated drop.

The characteristic parameters for ice shell formation on the drop surface, that is, the encapsulation time $t_b$ for the surface ice shell to reach the drop apex and the corresponding drop height $h_a$ (see Figure 2c), are plotted as functions of $\lambda_e/\lambda_s$ in Figure 3b,c. The height is normalized by the drop initial height $h_d = 1$ mm and the time is normalized using the characteristic thermal diffusion time of the liquid phase, $t_d = h_d^2/\alpha_l = 7.58$ s, where $\alpha_l$ is the thermal diffusivity of water. As $\lambda_e/\lambda_s$ increases, $t_b/t_d$ decreases, and $h_a/h_d$ decreases slightly, except for the lowest two ratios. This means that an environmental medium with higher conductivity may promote a faster encapsulation process and lower apex height of the encapsulated drop.

**Freezing front:** In addition, the morphology of the freezing front dramatically changes as $\lambda_e/\lambda_s$ increases, especially the curvature near the triple line. Thus, we define the front-to-interface angle in the solid phase as $\gamma$ (see Figure 4a). As shown in Figure 4b, for $\lambda_e/\lambda_s = 0.59$, the angle $\gamma$ decreases to less than 90° and decreases with time, which is clearly different from the common stable gaseous environmental medium ($\lambda_e/\lambda_s = 0.01$, almost adiabatic$^{42}$) with $\gamma \approx 90^{\circ}$.10,36

The difference in the freezing front morphology may be due to the use of an environmental medium with high thermal conductivity, which has nonnegligible heat exchange with the drop. To further study the heat exchange of the environmental medium, the distribution of the heat transfer rate is calculated by numerical simulations. For a sessile freezing drop, there are three kinds of external interfaces (see Figure 4a), including the ice-substrate interface, the ice-environmental medium interface, and the water-environmental medium interface. The heat transfer rate between the base of the droplet and the cold substrate is $Q_1$. The heat transferred to the environmental medium through the ice-environmental medium interface and the liquid-environmental medium interface is represented by $Q_2$ and $Q_3$, respectively. In simulations, since the density of the environmental medium and the surface tension between the liquid phase and the environmental medium are both constant, there is no thermal or Marangoni convection in the environmental medium.$^{43}$ We mainly consider the heat conduction between the environmental medium and the drop. According to Fourier's law of heat conduction, the absolute value of the heat transfer rate is calculated by $Q = \left| -\int \lambda \frac{\partial T}{\partial n} dA \right|$, where $dA$ is the local area and $\frac{\partial T}{\partial n}$ is the rate of temperature change in the direction perpendicular to the local surface. The interface area of a 3D drop is calculated according to the axisymmetric assumption.

The time evolution of quantitative heat transfer rates through different interfaces is plotted in Figure 4c. For a low conductivity ratio $\lambda_e/\lambda_s = 0.01$, $Q_1$ is always considerably higher than both $Q_2$ and $Q_3$, suggesting that almost the entire heat exchange of the drop is directly transferred to the cold substrate through the ice-substrate interface, which is in agreement with the previous work.$^{42}$ Since the thermal resistance increases with the height of the freezing front, $Q_1$ reduces with time. However, for a high conductivity ratio $\lambda_e/\lambda_s = 0.59$, $Q_2$ and $Q_3$ increase considerably. Especially close to the encapsulation time $t_b$, $Q_1$ and $Q_2$ have the same magnitude; thus,

<figure><img src="image_4.png" /><figcaption>**FIGURE 3** Quantitative effect of the environmental medium's conductivity on drop solidification. (a) Time evolution of $h_b$ and $h_c$ for cases with $\lambda_e/\lambda_s = 0.01$ and 0.59. (b, c) The dimensionless encapsulation time $t_b/t_d$ and the dimensionless final height $h_a/h_d$ as functions of $\lambda_e/\lambda_s$. The characteristic length of the drop is its initial height $h_d = 1$ mm, and the characteristic time is the thermal diffusion time of the liquid phase, $t_d = h_d^2/\alpha_l$, where $\alpha_l$ is the thermal diffusivity of water.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**FIGURE 4** Geometry of the freezing front. (a) On the cross-sectional diagram of a drop, the front-to-interface angle in the solid phase is defined as γ. The heat transferred to the substrate through the base of the droplet is Q₁. The heat transferred to the environmental medium through the ice-environmental medium interface and the liquid-environmental medium interface is represented by Q₂ and Q₃, respectively. (b, c) Time evolution of γ and heat transfer rates for λₑ/λₛ = 0.01 and 0.59.</figcaption></figure>

the heat transfer rate between the environmental medium and the drop is important. This heat transfer from the environmental medium contributes to an increase in the freezing rate on the drop surface, which obviously changes the morphology of the freezing front.

**Explanation of experimental phenomena:** These numerical results provide a full picture of the icing process on changing the external conductivity and allow us to explain the experimental results. For the liquid environmental medium with high thermal conductivity, it is first cooled by the cold substrate and then becomes another important cold source. Then, the environmental medium cools down the drop outer surface, which increases the freezing rate of the triple line, changes the geometry of the freezing front with the formation of an ice shell at the drop surface, and induces liquid encapsulation inside the drop. Special ice structures in Figure 1d conform that a solidified shell appears on the drop outer surface and encapsulates the drop. For a sessile water drop freezing on a cold substrate, the solidification starts from the bottom to the top of the drop, and the unfrozen liquid can be pushed upward before the encapsulation time t_b. After the drop is encapsulated by the ice shell, the internal water within the shell continues to solidify and expand, which may induce a high pressure inside the drop.⁴⁴ However, unlike an inward freezing water drop, which has a relatively uniform thick ice shell and finally explodes,⁴⁴ the ice shell on the top of the sessile drop is very thin (see the third image in Figure 2c), which can be easily overcome by the high pressure of the internal liquid. Thus, the internal liquid breaks through the ice shell near the top of the drop, flows, and freezes outside of the ice shell. Finally, a nonsmooth and asymmetric frozen drop may form (shown in Figure 1e). However, the deformation and rupture of the solidified phase are not considered in simulations, and simulations are stopped at the encapsulation time t_b.

**Liquid encapsulation model:** From the above research, we find that when a solidified shell limits the outer surface of a freezing sessile drop, due to the volume change of the encapsulated liquid because of solidification, the geometry of this drop may become totally different. Thus, liquid encapsulation is a very important characteristic of these freezing drops. To quantitatively analyze and predict the formation and geometric features of liquid encapsulation, a phenomenological

model is developed as detailed in Supporting Information. Considering that the cooling process at the central axis of the drop is almost not impacted by the external medium (see Figure 3a), most of the released latent heat is transferred to the cold substrate through the solidified phase:

$$ \rho_s L \frac{dh_c}{dt} = \lambda_s \frac{\Delta T}{h_c}, \quad (1) $$

where ρ_s is the density of the solidified phase, L is the latent heat of solidification, and ΔT is the temperature difference between the freezing front and the cold substrate. Thus, the time evolution of the central freezing front height is given by

$$ h_c = \left( \frac{2\lambda_s \Delta T t}{\rho_s L} \right)^{\frac{1}{2}}. \quad (2) $$

The evolution of the front position at the drop surface l_b (see Figure 1a) is compared with the front propagation at the drop center as l_b = fh_c, where f is the acceleration function of the freezing process at the drop surface due to the geometry of the drop outer surface and the cooling effect induced by the environmental medium. Ignoring the expansion of the drop, when the triple line reaches the drop apex, l_b = (θπR/180), the encapsulation time is

$$ t_b = \frac{\rho_s L}{2\lambda_s \Delta T} \left( \frac{\theta\pi R}{180f} \right)^2. \quad (3) $$

At this moment, if the central freezing front has not reached the drop apex, an ice shell forms on the drop's outer surface and the encapsulated liquid is formed inside the drop. The liquid's height h_i at the central axis normalized by the drop's initial height h_d is

$$ \frac{h_i}{h_d} = \frac{h_d - h_c}{h_d} = 1 - \frac{\theta\pi}{180f(1 - \cos\theta)}, \quad (4) $$

where h_d = R(1 - cos θ). According to the discussion in Supporting Information, f depends on the contact angle θ of the drop and the conductivity ratio λₑ/λₛ. The above-reported numerical results yield a
<figure><img src="image_6.png" /><figcaption>FIGURE 5 Dimensionless height of the liquid at the central axis $h_i/h_d$ versus the conductivity ratio $\lambda_e/\lambda_s$. The open circles correspond to experimental data, the closed triangles correspond to numerical data, and the green line represents the theoretical data of the model. DNS, direct numerical simulation; EXP, experimental.</figcaption></figure>

formula of the acceleration function, $f(\theta, \lambda_e/\lambda_s) = \frac{\theta\pi}{180(1 - \cos\theta)}(1 + 2.4^{\lambda_e/\lambda_s})$ (see the corresponding plot in Supporting Information). Using this acceleration function and Equation (4), a predictive relationship between the dimensionless height of the liquid $h_i/h_d$ and the conductivity ratio $\lambda_e/\lambda_s$ is

$$ \frac{h_i}{h_d} = 1 - \frac{1}{1 + 2.4^{\lambda_e/\lambda_s}}, \quad (5) $$

which could be used to provisionally extrapolate data for numerical simulations.

As shown in Figure 5, Equation (5) (green line) optimally represents the characteristics of numerical (DNS) and experimental (EXP) results. As $\lambda_e/\lambda_s$ increases, the dimensionless height of the liquid $h_i/h_d$ increases and a decreasing slope indicates a decreasing growth rate. Thus, for a freezing drop, when the thermal conductivity ratio of the environmental medium to the solidified phase is large enough, a liquid may be encapsulated in the drop. Due to the density difference between the liquid phase and the solidified phase, the liquid could expand or shrink as solidification, resulting in the appearance of a crack or a cavity in the drop.

## CONCLUSION

In summary, we have shown that heat transfer from the environmental medium to a sessile drop can drastically influence the drop-freezing dynamics. Environmental medium with non-negligible heat exchange with the drop and plate can be cooled by a cold substrate and becomes a new cooling source. The coupling effect of different cooling sources increases the solidification rate of the triple line and modifies the morphology of the freezing front, and consequently, a solid shell can form on the drop's outer surface with the formation of an encapsulated liquid. As the freezing process continues, the liquid will expand or shrink, which can considerably influence the final geometry of a freezing drop. For a freezing sessile water drop, the encapsulated liquid could expand

and induce high pressure within the drop. The internal liquid breaks through the weakest point of the ice shell, flows, and solidifies on the ice shell, which results in a nonsmooth asymmetric frozen drop. This study builds our understanding of the importance of an environmental medium, and presents new strategies to change the freezing rate and reshape the morphology of the solidified drop, which offers new possibilities to manipulate solidification in anti-icing and additive manufacturing.

## METHODS

**Experimental settings:** The environmental medium is added to a transparent cubic cell with a width of 84 mm. The cold substrate is made of copper with a nickel coating. A circulating bath (PolyScience PP15R-40) is used to cool down the substrate temperature. Since the surface tension of water and different environmental media is different, the wettability of the substrate is modified to obtain similar initial drop morphology with a contact angle higher than 90°. In silicone oil, the contact angle $\theta$ of the water drop on the nickel coating is around 128°. In nitrogen, a hydrophobic spray (SOFT99 SF-04172) is used to change the wettability of the substrate and the contact angle $\theta$ of the water drop is set to 111°.

**Numerical settings:** Numerical parameters are set as follows. The following boundary conditions are considered: a no-slip condition and a constant temperature on the bottom wall, an outlet and adiabatic condition on the top wall, a symmetry condition on the left boundary (the symmetry axis), and a no-slip boundary and adiabatic condition on the right wall. The domain extensions along the radial and vertical directions are both 2 mm, corresponding to twice the drop radius. The spatial and temporal resolutions are 2 µm and 1 µs, respectively, considering the grid and time convergence reported in our previous paper.36

## ACKNOWLEDGMENTS

We thank Mingjiu Ni, Lin Shi, Feng He, Yuanyuan Duan, Shuhong Liu, and Mingbo Li for fruitful discussions. We acknowledge Lei Yi for helping us with the experimental setup. This work was supported by the Natural Science Foundation of China under Grant Nos. 11988102, 91852202, and 11861131005, the New Cornerstone Science Foundation through the XPLORER PRIZE, and the China Postdoctoral Science Foundation (Grant Nos. 2022T150068 and 2021M700507).

## CONFLICT OF INTEREST STATEMENT

The authors declare no conflict of interest.

## ORCID

Sijia Lyu http://orcid.org/0000-0002-4546-7419

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## SUPPORTING INFORMATION

Additional supporting information can be found online in the Supporting Information section at the end of this article.

**How to cite this article:** Lyu S, Zhu X, Legendre D, Sun C. Liquid encapsulation in a freezing sessile drop. *Droplet.* 2023;2:e90. doi:10.1002/dro2.90