3.1 Spreading and receding dynamics of SLDs

The complex freezing morphology of impacting supercooled droplet arises from the coupling between hydrodynamics and solidification. First, we investigated the spreading and receding processes of droplets impact on cold surfaces. The laws governing the dynamics characteristic parameters can provide a basis for the subsequent prediction of the freezing behaviours discussed in Sections 3.2 and 3.3.

Figure 4.

Sequence images of typical impinging-freezing processes: (a) We = 100.8, ${{\rm{T}}_{\rm{s}}}$ = −15 °C, the impinging droplet undergoes a complete receding process and freezes at the equilibrium stage, (b) We = 100.8, ${{\rm{T}}_{\rm{s}}}$ = −22 °C, the receding stage is inhibited by the growth of horizontal dendritic ice, and (c) We = 100.8, ${{\rm{T}}_{\rm{s}}}$ = −25 °C, the water film is pinned at the maximum spreading extent due to the multiple nucleation spots during the spreading stage. (The red rectangles represent that the spreading diameter has been fixed.)

Figure 4 shows typical impinging-freezing phenomena on surfaces at different temperatures. The evolutions of the three-phase contact line (TPCL) and the receding contact angle ( ${\theta _{re}}$ ) were observed from a side view. Nucleation spots and dendritic ice growth were observed from the top view. At a relatively high surface temperature (e.g., ${T_s}\; = \; - $ 15 $^\circ {\rm{C}}$ ), the liquid water film receded to an equilibrium state and subsequently froze, as shown in Fig. 4(a). As the surface temperature decreased (e.g., ${T_s}\; = - $ 22  $^\circ {\rm{C}}$ ), the receding process of the liquid film was hindered by the appearance of ‘coral’ nucleation in the contact line region. Dendritic ice grew horizontally, while the remaining liquid water continued to recede. An ice sheet was formed and fixed to the surface during the receding stage, as shown in Fig. 4(b). Lower surface temperatures led to advanced nucleation occurrence and more nucleation spots. Figure 4(c) shows that ‘fungus’ nucleation formed in the central liquid lamella during the spreading stage. The ice layer grew rapidly outwards and contacted with the surrounding rim after reaching the maximum spreading. In this regime, the subsequent receding stage became negligible.

The maximum spreading is the onset of the receding process. To investigate the receding dynamics of droplets on cold surfaces, it is crucial to analyse the influence of surface temperatures on the spreading motion. The evolutions of spreading factor ( $\beta \; = \;{D_t}\;/\;{D_0}$ ) and peak height factor ( $\alpha \; = \;{H_{p,t}}\;/\;{D_0}$ ) against dimensionless time ( ${t^*}\; = \;t{V_0}\;/\;{D_0}$ ) are summarised in Fig. 5(a) and (b), respectively. It is obvious that the maximum spreading factor ( ${\beta _{max}}$ ) increases with Weber number ( $We$ ). However, there is no significant difference in the peak height factor for the intermediate Weber numbers ( $We\; = \;$ 100.8–186.3). In addition, the results indicate that the surface temperature has a minor influence on both the maximum spreading factor and the peak height factor. This result is consistent with previously published findings [Reference Yao, Li, Tao, Yang and Zhang20, Reference Liu, Min, Zhang, Hu and Wu21]. In this study, the dimensionless parameter ( $Pr\; = \;{c_{p,l}}{\mu _l}\;/\;{k_l}$ $ \gt \;16.7$ ), where ${c_{p,l}}$ and ${k_l}$ are the specific heat capacity and thermal conductivity of liquid water, respectively. This implies that heat transfer during the spreading stage can be neglected [Reference Zhang and Liu9]. According to previous research [Reference Zhang, Liu, Min and Wu32], the thermophysical properties of water and ice used in this study are listed in Table 2.

Figure 5.

Temporal evolution of characteristic parameters during the spreading stage: (a) spreading factor, and (b) peak height factor.

Eggers et al. [Reference Eggers, Fontelos, Josserand and Zaleski33] derived an impact parameter $P\; \equiv \;WeR{e^{ - 2/5}}$ to describe the maximum spreading factor. As an extension of an earlier study by Laan et al. [Reference Laan, Bruin, Bartolo, Josserand and Bonn34], an empirical formula is developed to estimate the maximum spreading factor of SLD. The expression is given as:

(3) \begin{align}{{\rm{\beta }}_{{\rm{max}}}}{\rm{R}}{{\rm{e}}^{ - 1/5}} = {{\rm{P}}^{0.5}}/\left( {{\rm{a}} + {{\rm{P}}^{0.5}}} \right)\end{align}

where $a\; = \;$ 1.35 is a fitting constant, as shown in Fig. 6. Additionally, nucleation in the thin lamella region fails to hinder the spreading of the surrounding rim, which further explains why the maximum spreading factor is independent of the surface temperatures.

Table 2.

Thermophysical properties of water/ice

Figure 6.

The effect of impact parameter P and surface temperatures on the maximum spreading factor ${{\rm{\beta }}_{{\rm{max}}}}$ . (The insets represent the maximum spreading extent of cases We = 138.7.)

The time required for spreading to the maximum diameter is defined as the spreading time ( ${t_{sp}}$ ), which is a critical parameter for distinguishing freezing types. The experimental dimensionless spreading time ( $t_{sp}^*\; = \;{t_{sp}}{V_0}\;/\;{D_0}$ ) is shown in Fig. 7 for various surface temperatures and Weber numbers. The subcooling degree of surfaces is defined as $\Delta {T_s}\; = \;{T_m} - {T_s}$ , where ${T_m}$ is the melting temperature (0 $^\circ {\rm{C}}$ ) of water. The results indicate that $t_{sp}^*$ is independent of surface temperatures but increases with the Weber number. According to the theoretical model for the spreading velocity ( ${V_{sp}})$ proposed by Srivastava and Kondaraju (2020), the predicted formula for the dimensionless spreading time $t_{sp}^*$ is given as follow:

(4) \begin{align}t_{sp}^* = \frac{{{R_{max}}{V_0}}}{{{V_{sp}}{D_0}}} = \frac{{3{R_{max}}{V_0}}}{{4{R_0}{{\left( {V_0^2 + 4g{R_0}} \right)}^{0.5}}}}\end{align}

where $g\; = \;$ 9.81 ${{\rm{m}}^2}/{\rm{s}}$ is the acceleration owing to gravity. Substituting Equation (3) into Equation (4), the predicted values shown in Fig. 7 are obtained. The maximum deviation of 6.4% indicates that the experimental data agree well with the prediction formula. The error bars may result from slight deviations in the initial droplet diameter during the repeated experiments.

Figure 7.

The effect of Weber numbers and surface temperatures on the dimensionless spreading time ${\rm{t}}_{{\rm{sp}}}^{\rm{*}}$ . (The dashed lines represent the predicted values of ${\rm{t}}_{{\rm{sp}}}^{\rm{*}}$ .)

Shortly after reaching the maximum spread, the surface tension drives the surrounding rim toward the central lamella. Figure 8 shows the evolution of the spreading radius and the receding contact angle during the receding stage. The interval duration between the maximum spreading factor and the criterion threshold (0.97 ${\beta _{max}}$ ) is defined as the stationary contact line time ( ${t_{SCL}}$ ) [Reference Wang and Fang35, Reference Lv, Yang and Duan36]. In this region, ${\theta _{re}}$ rapidly decreases. The stationary contact line time follows a capillary-viscous scale time ${t_{SCL}}\; \propto \;{h_m}{\mu _l}/{\gamma _l}$ , where the proportional coefficient is approximately ${10^5}$ for the water film [Reference Thiévenaz, Séon and Josserand14]. The characteristic thickness ( ${h_m}$ ) is assumed to be the thickness of liquid film at the maximum spreading diameter, given by ${h_m}\; = \;2D_0^3/3D_{max}^2$ [Reference Pasandideh-Fard, Qiao, Chandra and Mostaghimi37, Reference Shen, Liu, Zhu, Tao and Wang38]. Therefore, ${t_{SCL}}$ can be approximated by the following formula (unit: ms):

(5) \begin{align}{{\rm{t}}_{{\rm{SCL}}}} = {10^5}\frac{{{{\rm{\mu }}_{\rm{l}}}}}{{{{\rm{\gamma }}_{\rm{l}}}}}\left( {{\rm{b}} + \frac{{2{\rm{D}}_0^3}}{{3{\rm{D}}_{{\rm{max}}}^2}}} \right)\;\end{align}

where $b = $ 7E-4 is a fitting constant obtained from the experimental results.

Figure 8.

The temporal evolution of spreading radius and receding contact angle during the receding stage. (The insets represent the case of ${{\rm{T}}_{\rm{s}}}{\rm{\;}} = {\rm{\;}} - $ 15 °C, ${\rm{We\;}} = {\rm{\;}}$ 100.8).

On a neutrally wetted surface, the receding process of the water film ( $t\; \gt \;{t_{sp}}\; + \;{t_{SCL}}$ ) can be divided into three stages: the inertial, capillary and spherical cap modes, as shown in Fig. 8. Bartolo et al. [Reference Bartolo, Josserand and Bonn39] defined the inertial time as ${\tau _I}\; = \;{\left( {\pi {\rho _l}D_0^3\;/\;6{\gamma _l}} \right)^{0.5}}$ . In the inertial receding mode, ${\theta _{re}}$ gradually decreases. Subsequently, as the surrounding rim disappears, capillary waves are transmitted to the TPCL, which is referred to as the capillary regime. In the capillary receding mode, ${\theta _{re}}$ oscillates slightly and remains relatively low. After that, the droplet forms a spherical-cap shape with a large ${\theta _{re}}$ , and recedes at a slow receding velocity. The receding velocity of the film is calculated from the slope of the fitting equation for the spreading radius (coefficient of determination ${R^2}\; \geqslant \;0.995$ ). These results indicate that the receding velocity remained constant during the inertial and capillary stages, which is consistent with the findings of a previous study [Reference Lv, Yang and Duan36]. In this study, the inertial-capillary receding mode is investigated.

To understand the effects of impact velocities and surface temperatures on the receding velocity, the evolution of the dimensionless spreading diameter ( ${D_t}\;/\;{D_{max}}$ ) with dimensionless time ( $t\;/\;{\tau _I}$ ) is plotted in Fig. 9(a). The results show that as the surface temperature remains constant, the receding rate ( $\dot \varepsilon \; = \;{V_{re}}\;/\;{D_{max}}$ ) of the impinging droplet is independent of Weber number, consistent with the classical receding dynamics theory [Reference Bartolo, Josserand and Bonn39]. However, when the Weber number remains constant, the receding rate ( $\dot \varepsilon $ ) decreases with the surface temperatures. In particular, when the surface temperature decreases to −25 $^\circ {\rm{C}}$ , the water film fails to reach the inertial-capillary receding mode due to the formation of the ice sheet at the bottom. Figure 9(b) suggests that the dimensionless receding contact angle ( ${\theta _{re}}\;/\;{\theta _{eq}}$ ) in the capillary-inertial regime is independent of Weber number and surface temperatures. The effect of surface temperature should be considered when predicting the receding velocity of droplets impact on cold surfaces.

Figure 9.

The effect of Weber numbers and surface temperatures on receding dynamics during the inertial-capillary receding mode: (a) the evolution of dimensionless spreading diameter, and (b) the evolution of dimensionless receding contact angle.

Based on classical theory, the receding rate ( $\dot \varepsilon $ ) follows the relationship $\dot \varepsilon \; \propto \;\tau _I^{ - 1}\sqrt {\left[ {\pi \left( {1 - cos{\theta _r}} \right)} \right]} $ in capillary-inertial mode [Reference Bartolo, Josserand and Bonn39]. The proportionality coefficient is constant for low-viscosity ( $Oh$ < 0.05) droplets impact on the same surface. However, with a decrease in the surface temperature, the proportionality coefficient gradually decreases, as shown in Fig. 10. As discussed above, the maximum spreading diameter and the receding contact angle were independent of the surface temperature. It can be inferred that the receding velocity decreases with surface temperatures. This is attributed to the fact that heat transfer between the water film and cold surfaces increased the viscosity of the droplets and enhanced the energy dissipation. By incorporating the effect of the surface temperature, the prediction formula for the receding velocity of water droplets impact on cold surfaces is modified as follow:

(6) \begin{align}\frac{{{{\rm{V}}_{re}}}}{{{D_{max}}}}\frac{{{\tau _I}}}{{\sqrt {{\rm{\pi }}\left( {1 - {\rm{cos}}{{\rm{\theta }}_{{\rm{re}}}}} \right)} }} = - 1.75{\rm{Ste}} + 0.34\;\end{align}

where $Ste$ is the Stefan number, defined as $Ste\; = \;{c_{p,i}}\Delta {T_s}\;/\;L$ . Here, ${c_{p,i}}$ denotes the heat capacity of ice and $L$ is the latent heat.

Figure 10.

The relationship between the proportional coefficient and Stefan number in the capillary-inertial mode for the same neutral wettability surface. (The insets represent the droplet at the ${\rm{t}} = {{\rm{\tau }}_{\rm{I}}}$ for the case We = 138.7.)

The experimental results indicate that the receding velocity is on the order of ${10^{ - 2}}$ ${\rm{m}}/{\rm{s}}$ , which is significantly slower than the growth velocity of horizontal ice dendritic [Reference Kong, Wang, Bian and Liu13] on an aluminum surface, given by ${{\rm{V}}_{\rm{i}}}{\rm{\;}} = {\rm{\;}}0.0149{\left( {\Delta {{\rm{T}}_{\rm{s}}}} \right)^{1.1378}}$ . This implies that once nucleation occurred, the receding process was rapidly impeded, and the frozen area is determined by the ice sheet.

3.2 The effect of impingement on final frozen area and freezing morphology

The frozen area, determined by the coupling of receding dynamics and horizontal dendritic ice growth, is crucial for determining the geometric profile of the ice droplet and its solidification time. The spreading radius at the equilibrium state is exclusively determined by the equilibrium contact angle ( ${\theta _{eq}}$ ) [Reference Attané, Girard and Morin40, Reference Wang, Wang, He, Zhang, Yang, Wang and Lee41]. At relatively high surface temperatures, the relationship between solidification and impinging dynamics is decoupled; as a result, the frozen radius ( ${r_f}$ ) can be expressed as:

(7) \begin{align}\;{r_f} = {r_0}\frac{{2sin{\theta _{eq}}}}{{\sqrt[3]{{4 - 6cos{\theta _{eq}} + 2{{\left( {cos{\theta _{eq}}} \right)}^3}}}}}\end{align}

Nucleation initiates the freezing process and the dendritic ice grows horizontally. The receding motion stops when the base of the water film is completely frozen. For both the instantaneous and delayed freezing modes, the characteristic timescale of the impinging dynamics and solidification behaviour satisfies the following:

(8) \begin{align}{t_{nu}} + {t_{i,h}} = {t_{sp}} + {t_{SCL}} + {t_{re}}\;\end{align}

where ${t_{i,h}}$ denotes the freezing time of the ice sheet, and ${t_{re}}$ is the receding time before the TPCL is fixed. Figure 11 shows the impinging characteristic times and the average nucleation time ( ${\bar t_{nu}}$ ) for different freezing modes. The receding time ${t_{re}}$ was simply estimated as ${t_{re}}{\rm{\;}} = {\rm{\;}}({r_{max}} - {r_f})\;/\;{V_{re}}$ . The average nucleation time ${\bar t_{nu}}$ is inversely proportional to the nucleation rate ( ${J_{nu}}$ ). Neglecting nucleation time under the quasi-static freezing mode, ${\bar t_{nu}}$ could be expressed as following (unit: $ms$ ):

(9) \begin{align}{\bar t_{nu}} = {J_{nu}}^{ - 1} = {\left[ {\frac{1}{n}\mathop \sum \limits_1^n \left( {\frac{1}{{{t_{nu,i}}}}} \right)} \right]^{ - 1}} = 408.9\exp\!\left( { - 0.47\Delta {T_s}} \right)\end{align}

where ${t_{nu,i}}$ denotes the measured time for the first appearance of nucleation in the $i$ th test.

Figure 11.

The relationship between impinging characteristic time and freezing characteristic time under various surface temperatures and Weber numbers. (The predicted curves of characteristic time of impinging dynamics were obtained by Equations (3)–(8).)

The statistics reveal that according to the nucleation time ( ${t_{nu}}$ ), the impinging-freezing behaviour can be categorised into three modes: instantaneous ( ${t_{nu}}\; \le \;{t_{sp}}$ ), delayed ( ${t_{sp}}\; \lt \;{t_{nu}}\; \lt {t_{sp}}\; + \;{t_{SCL}}\; + {t_{re}}$ ) and quasi-static freezing ( ${t_{nu}}\; \geqslant \;{t_{sp}}\; + \;{t_{SCL\;}} + \;{t_{re}})$ . In both the delayed and instantaneous freezing modes, the sum of the theoretical impinging dynamics times ( ${t_{sp}}\; + \;{t_{SCL}}\; + {t_{re}}$ ) is significantly longer than the sum of the freezing characteristic times ( ${t_{nu}}\; + \;{t_{i,h}}$ ). This fact means that the bottom of the liquid film can freeze before completely receding. Both earlier nucleation time and faster growth of dendritic ice contribute to an increase in frozen area ratio.

When the ‘coral’ nucleation appears in the contact line region, the spreading diameter of the water film can be expressed as: ${D_{t,nu}}\; = \;0.97{D_{max}}\; - \;2{V_{re}}\left( {{t_{nu}}\; - \;{t_{sp}}\; - \;{t_{SCL}}} \right)$ . Because the dendritic ice spreads toward the centre, the diameter of the ice sheet is given by: ${D_i}\; = \;{V_i}\left( {t\; - \;{t_{nu}}} \right)$ . At the same time, the receding distance of the remaining water film is expressed as: ${D_{re}}\; = \;{V_{re}}\left( {t\; - \;{t_{nu}}} \right)$ . When ‘fungal’ nucleation occurs in the central lamella, the dendritic ice spreads radially, whereas the surrounding rim recedes toward the centre. The diameter of the ice sheet is expressed as: ${D_i}\; = \;2{V_i}{t_i}\; = \;2{V_i}\left( {t\; - \;{t_{nu}}} \right)$ , and the receding distance of the remaining water film is expressed as: ${D_{re}}\; = \;2{V_{re}}\left( {t\; - \;{t_{nu}}} \right)$ . Regardless of the nucleation mode, the diameter of the ice sheet satisfies the following relationship:

(10) \begin{align}{D_i} + {D_{re}} = {D_{t.nu}}\end{align}

The diameter of the ice sheet ( ${D_i}$ ) and the frozen area ratio ( ${\kappa _f}$ ) for delayed and instantaneous freezing modes can be expressed as follows:

(11) \begin{align}{D_i} = \frac{{{V_i}}}{{{V_{re}} + {V_i}}}\left[ {0.97{D_{max}} - 2{V_{re}}\left( {{t_{nu}} - {t_{sp}} - {t_{SCL}}} \right)} \right]\end{align}

(12) \begin{align}{\kappa _f} = {\left( {\frac{{{V_i}}}{{{V_{re}} + {V_i}}}} \right)^2}{\left[ {0.97 - \frac{{{V_{re}}\left( {{t_{nu}} - {t_{sp}} - {t_{SCL}}} \right)}}{{{R_{max}}}}} \right]^2}\\[8pt]\nonumber\end{align}

Figure 12 compares the experimental and predicted values for the frozen area ratio ${\kappa _f}$ . The maximum deviation is 10.14%, indicating that the proposed model is validated due to the good agreement. This deviation is primarily attributed to the uncertainty in the nucleation time and the irregular shape of the ice sheets. The results demonstrate that the frozen area ratio decreases with increasing Weber number. In addition, when the $Ste\; \geqslant {\rm{\;}}0.12$ , with increasing Stefan number, the frozen area ratio increases, and the effect of Weber number weakens. Conversely, when $Ste\; \lt \;0.12$ , the frozen area ratio becomes independent of the Stefan number.

Figure 12.

Comparison of the theoretical curves and experimental results of frozen area ratio. (The ${{\rm{\kappa }}_{\rm{f}}}$ theoretical curve of the quasi-static mode is obtained by assuming ${{\rm{\theta }}_{{\rm{eq}}}} = 55.8^\circ $ ; The ${{\rm{\kappa }}_{\rm{f}}}$ theoretical curves of the delayed and instantaneous freezing modes are obtained by assuming ${{\rm{\theta }}_{{\rm{re}}}}$ =16.74 $^\circ $ .)

As the surrounding rim merges at the centre, a pointy ice droplet is formed. Otherwise, the remaining liquid water on the ice sheet fails to merge, leading to a ‘basin’ morphology. Figure 13 shows the effect of Weber numbers and Stefan numbers on the freezing morphologies of droplets impact on cold surfaces. The critical threshold for hemispherical and pancake ice droplets formation is $Ste$ = 0.12, independent of Weber number. Additionally, the critical threshold for basin and pancake ice droplets is $Ste\; = \;0.00012We\; + \;0.1623$ . As the subcooled degree of the surfaces increases, nucleation is stimulated, and the driving potential for freezing is enhanced, resulting in a transition from a pointy pancake to basin ice droplets. Moreover, an increase in Weber number leads to a larger spreading area and a reduction in the amount of liquid water available for recession on the ice sheet. Consequently, the formation of the pointy ice droplets is impeded.

Figure 13.

The phase diagram of three freezing morphologies of supercooled droplets impact on cold surfaces.

3.3 The effect of impingement on solidification time

The solidification process of the droplets is simplified to a one-dimensional heat conduction Stefan problem, involving the growth of the freezing front and the release of latent heat. The primary task is to determine the initial freezing height ( ${H_{f,0}}$ ) of an impinging droplet to calculate the subsequent solidification time.

The nucleation time is regarded as the zero-time reference point. The ice sheet fully forms at ${t_{i,h}}$ . Subsequently, the remaining ice-water mixture on the ice sheet begins to freeze and requires ${t_{i,v}}$ to complete the solidification process. The total solidification time is given by: ${t_f} = {t_{i,h}} + {t_{i,v}} \approx {t_{i,v}}$ . Figure 14 shows the solidification process and the profile evolution of hemisphere and pancake ice droplets. The results demonstrate that when the volume of the water droplets remains unchanged, a lower surface temperature increases the frozen area of the ice sheet and reduces the initial freezing height of the droplets, leading to a shorter freezing time. Additionally, random nucleation impedes receding motion, causing the asymmetrical contact angle of the ice bead. This poses a challenge in predicting the initial freezing height.

Figure 14.

Solidification process: (a) image sequences of quasi-static and delayed freezing mode, and (b) corresponding profile evolution of droplets during solidification.

In this study, the Bond number ( $Bo\; = \;4{\rho _l}gr_f^2\;/\;{\gamma _l}$ ) is 2.682 for an impinging droplet with an equilibrium contact angle ${\theta _{eq}}\; = \;$ 55.8 $^\circ $ . This indicates that the effect of gravity on the droplet shape cannot be neglected. An ellipsoidal model is employed to calculate the initial droplet profiles. The initial droplet volume can be represented as:

(13) \begin{align}\frac{4}{3}{\rm{\pi }}r_0^3 = {\rm{\pi }}a_0^2{b_0}\mathop \smallint \nolimits_{{\varphi _1}}^{\pi /2} co{s^3}\varphi \;d\varphi, \;\;{\theta _{re}} \lt \frac{\pi }{2}\end{align}

where ${\varphi _1}{\rm{\;}} = {\rm{\;arccos}}\left( {{r_f}\;/\;{a_0}} \right)$ , ${a_0}$ and ${b_0}$ are the initial semi-major and semi-minor axes of the ellipsoid, as shown in Fig. 15(a). The frozen radius of the droplets impact on the cold surface is calculated as follow:

(14) \begin{align}{r_f} = {\rm{\kappa }}_f^{0.5}\left( {{\beta _{max}}{r_0}} \right)\end{align}

Figure 15.

Schematic diagrams, experimental results and theoretical values of initial droplet height. (a) Profile schematic diagram of a sessile droplet, (b) diagram of asymmetrical droplet profile applying the two-ellipse method (two separate heights for a frozen drop, ${{\rm{H}}_{{\rm{f}},0,{\rm{l}}}}$ and ${{\rm{H}}_{{\rm{f}},0,{\rm{r}}}}$ , are identified for two different contact angles, i.e., left ( ${{\rm{\theta }}_{{\rm{f}}.{\rm{l}}}}$ ) and right ( ${{\rm{\theta }}_{{\rm{f}}.{\rm{r}}}}$ ) contact angle, respectively), and (c) contact angle of droplet ( ${{\rm{\theta }}_{{\rm{re}}}}$ ) and droplet height ( ${{\rm{H}}_{{\rm{f}},0}}$ ) at initial stage of solidification for ${\rm{We\;}} = $ 138.7. (The theoretical curves of the initial height are derived using ${{\rm{\theta }}_{\rm{f}}} = 55.8^\circ $ and ${{\rm{\theta }}_{\rm{f}}} = 16.74^\circ $ for hemisphere and pancake ice droplets, respectively.)

The slope at the tri-junction can be related to the receding contact angle ( ${\theta _{re}}$ ) by:

(15) \begin{align}tan{\theta _{re}} = \frac{{{b_0}}}{{{a_0}}}tan{\varphi _1},\;{\theta _{re}} \lt \frac{\pi }{2}\end{align}

Solutions of Equations (13)–(15) yield ${a_0}$ and ${b_0}$ , which determine the initial droplet profile under the influence of gravity. Consequently, the initial freezing height ( ${H_{f,0}}$ ) can be expressed as follow:

(16) \begin{align}{H_{f,0}} = {b_0} - {r_f}tan{\varphi _1}\end{align}

Since the ice droplet is asymmetrical, the initial freezing height of the droplet can be calculated using the two-ellipse method, as shown in Fig. 15(b).

(17) \begin{align}{H_{f,0}} = \frac{{{H_{f,0,l}} + {H_{f,0,r}}}}{2}\end{align}

where subscripts $l$ and $r$ correspond to the left and right sides of the droplet, respectively. Figure 15(c) shows the influence of the frozen area ratio ( ${\kappa _f}$ ) on the shape characteristics of droplets. The result demonstrates that the theoretical model agrees well with the experimental values.

Based on the proposed model, the predicted initial freezing heights at different Weber numbers are shown in Fig. 16. The maximum deviation is less than 10%, validating the accuracy of the proposed theoretical model.

Figure 16.

Comparison of the predicted values and experimental results of the dimensionless initial freezing height for solidification.

During the solidification stage, the ice layer grows vertically, and latent heat is released vertically toward the water and surface. According to the one-dimensional heat conduction Stefan problem, the relationship between the freezing front height ( ${H_{f,t}}$ ) and the solidification time ( ${t_f}$ ) follows:

(18) \begin{align}{H_{f,t}} = \sqrt {{\Gamma _{eff}}{t_f}} \end{align}

where ${\Gamma _{eff}}$ is the effective diffusion coefficient that determines the growth of the ice layer, which can be expressed as:

(19) \begin{align}{\Gamma _{eff}} = 2\frac{{{k_i}\left( {{T_m} - {T_s}} \right)}}{{{\rho _i}L}} = 2{\Gamma _i}Ste\end{align}

where ${\Gamma _i}$ is the diffusion coefficient of the ice layer: ${\Gamma _i} = {k_i}/\left( {{\rho _i}{c_{p,i}}} \right)$ . Subscript $i$ denotes the ice phase. Consequently, the solidification time is estimated as follow:

(20) \begin{align}{t_f} = c\frac{{H_{f,0}^2}}{{{\Gamma _{eff}}}} = c\frac{{H_{f,0}^2}}{{2{\Gamma _i}Ste}}\end{align}

where the constant ${\rm{c}} = 1.05$ , representing the volume expansion coefficient, is derived from the experimental results in this study. Only the solidification times of the hemisphere and pancake ice droplets are considered. The solidification time of ‘basin’ ice droplets is so short that ${t_{i,h}}$ cannot be ignored. By integrating Equations (13)–(20), the solidification times for various Weber numbers are predicted, as shown in Fig. 17. The proposed model accurately predicts the solidification time of the droplets impact on cold surfaces.

Figure 17.

Solidification time of impinging droplets at different Weber numbers.