Freezing of sessile droplet and frost halo formation

The freezing of a sessile droplet unveils fascinating physics, characterized by the emergence of a frost halo on the underlying substrate, the progression of the liquid-ice interface, and the formation of a cusplike morphology at the tip of the droplet. We investigate the freezing of a volatile sessile droplet, focusing on the frost halo formation, which has not been theoretically explored. The formation of the frost halo is associated with the inherent evaporation process in the early freezing stages. We observe a negative evaporation flux enveloping the droplet in the initial stages, which indicates that vapor produced during freezing condenses on the substrate close to the contact line, forming a frost halo. The condensate accumulation triggers reevaporation, resulting in a temporal shift of the frost halo region away from the contact line. Eventually, it disappears due to the diffusive nature of the water vapor far away from the droplet. We found that increasing the relative humidity increases the lifetime of the frost halo due to a substantial reduction in evaporation that prolonged the presence of net condensate on the substrate. Increasing liquid volatility increases the evaporation flux and condensation occurs closer to the droplet, as a higher amount of vapor is in the periphery of the droplet. We also found that decreasing the thermal conductivity of the substrate increases the total freezing time. The slower freezing process is accompanied by increased vaporized liquid, resulting in condensation with its concentration reaching supersaturation.

Figure 1

Different stages observed during the freezing of a water droplet on a hydrophilic polymethyl methacrylate (PMMA) substrate. Courtesy: V. Kumar; some preliminary experiments conducted in our group.

Figure 2

(a) Schematic diagram of a sessile droplet undergoing freezing on a solid substrate. Here, $S_{\infty }$ and $H_{\infty }$ are the thickness of an initial microscopic ice layer and the thickness of the precursor layer, respectively. (b) Schematic representation of the left ($m_{cl}$) and right ($m_{cr}$) edges of the condensate mass and the thickness of the halo ($d_{\mathrm{halo}}=m_{cr}-m_{cl}$) in the vicinity of the two-dimensional (2D) droplet.

Figure 3

Evolution of the freezing front, $s$ (dot-dashed lines), and shape of the droplet, $h$ (solid line), placed on a cold substrate. The rest of the dimensionless parameters are $\epsilon =0.2, \text{Ste}=0.04, T_v=0.5, A_n=6.25, D_s={\mathrm{\Lambda }}_S={\mathrm{\Lambda }}_W=V_{\rho ,r}=K=\text{Bi}=1$, and $D_w=D_v=\mathrm{\Delta }=\mathrm{\Psi }=\chi ={\text{Pe}}_v=0$. This set of parameters is similar to that of Zadražil et al. [9]. The value of the dimensionless total freezing time $\left(t_f\right)$ of the droplet is 15. Here, the evaporation is neglected.

Figure 4

(a) Variation of (a) the evaporation flux $\left(J_v\right)$ and (b) the total mass of the condensate $\left(m_c\right)$ along the substrate for different values of $t/t_f$. Temporal variation of (c) the width of the frost halo $\left(d_{\mathrm{halo}}\right)$ and (d) the left $\left(m_{cl}\right)$ and right $\left(m_{cr}\right)$ ends of the condensation halo over the substrate. The maximum width of the halo is also marked in panels (c) and (d). The rest of the dimensionless parameters are $D_w=7.5, \epsilon =0.2, \text{Ste}=1.7\times {10}^{-4}, T_v=0.2, A_n=6.25, D_g=2, D_s=0.9, {\mathrm{\Lambda }}_S=3.82, {\mathrm{\Lambda }}_W=191, K=8\times {10}^{-4}, \text{Bi}=2.29$, and $D_v=4.85\times {10}^{-6}$. The value of the dimensionless total freezing time $\left(t_f\right)$ of the droplet is 1220, respectively.

Figure 5

Variation of the vapor concentration $\left({\rho }_v{|}_h\right)$ along the substrate at different values of $t/t_f$ for (a) $V_{\rho ,r}=0.60$ and (b) $V_{\rho ,r}=0.85$. The values of the rest of the dimensionless parameters are $\text{Ste}=1.22\times {10}^{-3}, T_v=0, A_n=6.25, D_g=2, D_s=0.9, {\mathrm{\Lambda }}_S=3.89, {\mathrm{\Lambda }}_W=0.33, \chi =0.01, K=8\times {10}^{-4}, \text{Bi}=0.16, D_w=15, \epsilon =0.2, D_v=1.65\times {10}^{-6}, \mathrm{\Delta }={10}^{-4}, \mathrm{\Psi }=0.94$, and ${\text{Pe}}_v=1$. The dotted line represents the maximum vapor concentration attainable at the liquid-gas interface.

Figure 6

Variation of the evaporative flux $\left(J_v\right)$ along the substrate for (a) $V_{\rho ,r}=0.35$, (b) $V_{\rho ,r}=0.60$, (c) $V_{\rho ,r}=0.85$, and (d) $V_{\rho ,r}=0.95$. The rest of the dimensionless parameters are the same as Fig. 5. The dotted line is plotted to distinguish the positive and negative values of the evaporation flux. The values of $t_f$ for $V_{\rho ,r}=0.35$, 0.60, 0.85, and 0.95 are $t_f=4473$, 4700, 5121, and 5550, respectively.

Figure 7

Temporal variation of (a) the width of the frost halo $\left(d_{\mathrm{halo}}\right)$ and (b) the left $\left(m_{cl}\right)$ and right $\left(m_{cr}\right)$ ends of the condensation halo over the substrate for different values of the vapor density ratio $\left(V_{\rho ,r}\right)$. In panel (b), the solid and dashed lines represent the variation of $m_{cl}$ and $m_{cr}$, respectively. The rest of the dimensionless parameters are the same as Fig. 5.

Figure 8

(a) Comparison of the evolution of the freezing front obtained from our simulation with the experimental result of Sebilleau et al. [44]. Here, the results shown by lines are from our simulations with different values of the relative humidity for a droplet with a contact angle of $14.3^{\circ }$. The symbols are associated with the experimental results of Sebilleau et al. [44]. (b) Variation of the freezing front with ${\left(\alpha t\right)}^{1/2}$ for different values of the contact angle for $\text{RH}=0.7$. The values of the dimensionless parameters are $\text{Ste}=1.22\times {10}^{-3}, T_v=1, A_n=6.25, D_g=2, D_s=0.9, {\mathrm{\Lambda }}_S=3.89, {\mathrm{\Lambda }}_W=698, \chi =0.01, K=8\times {10}^{-4}, \text{Bi}=0.16, D_w=15, V_{\rho ,r}=0.70, \epsilon =0.2, D_v=1.65\times {10}^{-6}, \mathrm{\Delta }={10}^{-4}, \mathrm{\Psi }=0.94$, and ${\text{Pe}}_v=1$.

Figure 9

Evolution of the freezing front, $s$ (dot-dashed lines), and shape of the droplet, $h$ (solid line), for (a) $D_v={10}^{-4}$ and (b) $D_v={10}^{-2}$. The values of the rest of the dimensionless parameters are $\text{Ste}=1.22\times {10}^{-3}, T_v=0, A_n=6.25, D_g=2, D_s=0.9, {\mathrm{\Lambda }}_S=3.89, {\mathrm{\Lambda }}_W=0.33, \chi =0.01, V_{\rho ,r}=0.85, K=8\times {10}^{-4}, \text{Bi}=0.16, D_w=15, \epsilon =0.2, \mathrm{\Delta }={10}^{-4}, \mathrm{\Psi }=0.94$, and ${\text{Pe}}_v=1$. The values of $t_f$ for $D_v={10}^{-4}$ and ${10}^{-2}$ are $t_f=4210$ and 634, respectively.

Figure 10

Variation of the evaporative flux $\left(J_v\right)$ along the substrate for (a) $D_v={10}^{-4}$ and (b) $D_v={10}^{-2}$. Temporal variation of (c) the width of the frost halo $\left(d_{\mathrm{halo}}\right)$, (d) the left $\left(m_{cl}\right)$ and right $\left(m_{cr}\right)$ ends of the condensation halo over the substrate, and (e) the location of maximum condensation flux $\left(x_{\mathrm{halo}}\right)$ for different values of $D_v$. In panel (d), the solid and dashed lines represent the variation of $m_{cl}$ and $m_{cr}$, respectively. The rest of the dimensionless parameters are the same as Fig. 9. The values of $t_f$ for $D_v={10}^{-4}, {10}^{-3}$, and ${10}^{-2}$ are $t_f=4210$, 2318, and 634, respectively.

Figure 11

Evolution of the freezing front, $s$ (dot-dashed lines), and shape of the droplet, $h$ (solid line), for (a) ${\mathrm{\Lambda }}_W=1$ and (b) ${\mathrm{\Lambda }}_W=500$. The values of the rest of the dimensionless parameters are $\text{Ste}=1.22\times {10}^{-3}, T_v=0, A_n=6.25, D_g=2, D_s=0.9, \chi =0.01, {\mathrm{\Lambda }}_S=3.89, V_{\rho ,r}=0.85, K=8\times {10}^{-4}, \text{Bi}=0.16, D_w=15, \epsilon =0.2, D_v=1.65\times {10}^{-6}, \mathrm{\Delta }={10}^{-4}, \mathrm{\Psi }=0.94$, and ${\text{Pe}}_v=1$.

Figure 12

Variation of the evaporative flux along the substrate for (a) ${\mathrm{\Lambda }}_W=1$ and (b) ${\mathrm{\Lambda }}_W=500$. Temporal variation of (c) the width of the frost halo $\left(d_{\mathrm{halo}}\right)$, (d) the left $\left(m_{cl}\right)$ and right $\left(m_{cr}\right)$ ends of the condensation halo over the substrate, and (e) the location of maximum condensation flux $\left(x_{\mathrm{halo}}\right)$ for different values of ${\mathrm{\Lambda }}_W$. In panel (d), the solid and dashed lines represent the variation of $m_{cl}$ and $m_{cr}$, respectively. The rest of the dimensionless parameters are the same as Fig. 11. The values of $t_f$ for ${\mathrm{\Lambda }}_W=1$, 10, 100, and 500 are $t_f=4240$, 1400, 352, and 240, respectively.

Figure 13

Variation of the mass of ice layer, $m_{\mathrm{ice}}$, with ${\log }_{10}(t/t_f)$ for different values of ${\mathrm{\Lambda }}_W$. The rest of the dimensionless parameters are the same as Fig. 11.

Figure 14

Variation of the evaporative flux along the substrate for (a) $T_v=0$ and (b) $T_v=1$. Temporal variation of (c) the width of the frost halo $\left(d_{\mathrm{halo}}\right)$, (d) the left $\left(m_{cl}\right)$ and right $\left(m_{cr}\right)$ ends of the condensation halo over the substrate, and (e) variation of the mass of ice layer, $m_{\mathrm{ice}}$, with ${\log }_{10}(t/t_f)$ for different values of $T_v$. In panel (d), the solid and dashed lines represent the variation of $m_{cl}$ and $m_{cr}$, respectively. The values of the rest of the dimensionless parameters are $\text{Ste}=1.22\times {10}^{-3}, A_n=6.25, D_g=2, D_s=0.9, {\mathrm{\Lambda }}_S=3.89, {\mathrm{\Lambda }}_W=0.33, \chi =0.01, V_{\rho ,r}=0.85, K=8\times {10}^{-4}, \text{Bi}=0.16, D_w=15, \epsilon =0.2, D_v=1.65\times {10}^{-6}, \mathrm{\Delta }={10}^{-4}, \mathrm{\Psi }=0.94$, and ${\text{Pe}}_v=1$. The values of $t_f$ for $T_v=0$, 0.5, and 1 are $t_f=5121$, 7919, and 19060, respectively.

Figure 15

Evolution of the freezing front, $s$ (dot-dashed lines), and magnified shape of the upper part of the droplet, $h$ (solid line), placed on a cold substrate. The results obtained using 2001, 4801, and 8001 grid points were indistinguishable. The rest of the dimensionless parameters are $V_{\rho ,r}=0.85, \chi =0.23, \mathrm{\Delta }={10}^{-4}, \mathrm{\Psi }=0.14, {\text{Pe}}_v=1.0, D_w=7.5, \epsilon =0.2, \text{Ste}=1.7\times {10}^{-4}, T_v=0.2, A_n=6.25, D_g=2, D_s=0.9, {\mathrm{\Lambda }}_S=3.82, {\mathrm{\Lambda }}_W=191, K=8\times {10}^{-4}, \text{Bi}=2.29$, and $D_v=4.85\times {10}^{-6}$. The values of the dimensionless total freezing time $\left(t_f\right)$ of the droplet is 1220, respectively.

Figure 16

Figure showing the estimation of the tip angle for a typical set of parameters ($\text{Ste}=1.22\times {10}^{-3}, T_v=1, A_n=6.25, D_g=2, D_s=0.9, {\mathrm{\Lambda }}_S=3.89, {\mathrm{\Lambda }}_W=698, \chi =0.01, K=8\times {10}^{-4}, \text{Bi}=0.16, D_w=15, V_{\rho ,r}=0.70, \epsilon =0.2, D_v=1.65\times {10}^{-6}, \mathrm{\Delta }={10}^{-4}, \mathrm{\Psi }=0.94$, and ${\text{Pe}}_v=1$).