Spontaneous self-dislodging of freezing water droplets and the role of wettability

Significance

Freezing of water on surfaces is ubiquitous in nature and technology; however, to control ice aggregation and rationally design icephobic surfaces, which passively inhibit ice formation and accretion, a deeper fundamental understanding of the interaction of forming ice with the underlying substrate and environment is necessary. In this work, we report the phenomenon of self-dislodging freezing water droplets, explain its physics, and develop surfaces which inhibit ice aggregation through passive ice self-removal. Successful experiments on a palette of material classes underpin the general applicability and robustness of the effect to control surface icing, and can guide further research in the field of icephobicity.

Abstract

Spontaneous removal of liquid, solidifying liquid and solid forms of matter from surfaces, is of significant importance in nature and technology, where it finds applications ranging from self-cleaning to icephobicity and to condensation systems. However, it is a great challenge to understand fundamentally the complex interaction of rapidly solidifying, typically supercooled, droplets with surfaces, and to harvest benefit from it for the design of intrinsically icephobic materials. Here we report and explain an ice removal mechanism that manifests itself simultaneously with freezing, driving gradual self-dislodging of droplets cooled via evaporation and sublimation (low environmental pressure) or convection (atmospheric pressure) from substrates. The key to successful self-dislodging is that the freezing at the droplet free surface and the droplet contact area with the substrate do not occur simultaneously: The frozen phase boundary moves inward from the droplet free surface toward the droplet–substrate interface, which remains liquid throughout most of the process and freezes last. We observe experimentally, and validate theoretically, that the inward motion of the phase boundary near the substrate drives a gradual reduction in droplet–substrate contact. Concurrently, the droplet lifts from the substrate due to its incompressibility, density differences, and the asymmetric freezing dynamics with inward solidification causing not fully frozen mass to be displaced toward the unsolidified droplet–substrate interface. Depending on surface topography and wetting conditions, we find that this can lead to full dislodging of the ice droplet from a variety of engineered substrates, rendering the latter ice-free.

The interaction of liquid water and ice with surfaces is fundamentally important in nature and many engineering applications (1–3). For decades researchers developed surfaces that repel water––so-called hydrophobicity––as these are promising for antifouling and self-cleaning applications (4–8). In addition, the rational design of surfaces that passively inhibit surface icing––so-called icephobicity––has gained attention in recent years, as ice affects the safety and performance of a broad palette of applications ranging from aircrafts and automobiles (transportation), to wind turbines and electrothermal energy storage (energy), and to power lines and roads (infrastructure) (9–13).

Recent investigations into the development of icephobic surfaces focused on ice nucleation delay by changing the surface topography and wettability (14, 15), ice nucleation prevention by droplet shedding due to self-propelled dropwise condensate coalescence on superhydrophobic surfaces (16), reduced contact time during droplet impact by droplet splitting (17) and pancake bouncing (18), as well as impact resistance to supercooled droplets (19). Further research studied the role of environmental conditions on the freezing process on surfaces (20, 21) and the freezing dynamics (22, 23). Despite intensive work at the complex intersection of nucleation thermodynamics, interfacial thermofluidics, and materials micro/nanoengineering, eventual freezing is often inevitable. Consequently, researchers also developed solutions to reduce ice adhesion, e.g., by liquid-infused surfaces (24). Recently, it was demonstrated that the physics of freezing dynamics at low environmental pressures can be exploited in mitigating icing by spontaneous levitation of water droplets during nonequilibrium freezing on microtextured, superhydrophobic surfaces, and guidelines for the design of surface textures conducive to this behavior were suggested (25).

Despite the progress mentioned above, in the vast majority of applications freezing water adheres and remains on surfaces, resulting in ice accretion, which is a highly undesirable event. Here we show and explain a freezing-driven ice-removal mechanism. We find that a freezing sessile water droplet, which is cooled primarily from its free surface, experiences a concentric inward growth of the phase boundary from the free surface toward the still unsolidified droplet–substrate interface. The key to successful self-dislodging is that the droplet–substrate interface remains a liquid throughout most of the process and freezes last. The volumetric expansion associated with the phase change, combined with the incompressibility of the droplet core, the flow permittivity of the still unsolidified droplet substrate interface, and the flow restriction imposed by the solid outer ice shell at the free surface, all result collaboratively in a displacement of the unsolidified core toward the substrate. Simultaneously, the inward phase-boundary motion near the substrate (ice–liquid–vapor contact line) drives a gradual reduction in droplet–substrate contact area (liquid–vapor–substrate contact line). We observe that the displaced core lifts the droplet away from the substrate, intrinsically prohibiting ice adhesion associated with freezing. Combined with further freezing and ice-shell growth, this can lead to a complete removal of the freezing droplet. We term this mechanism droplet self-dislodging, as the water droplet employs the freezing dynamics to remove itself from the substrate upon which, in the absence of this mechanism, it would be lodged. The broad range of investigated substrates and conditions is not capable of realizing the self-levitation or jumping behavior for droplets in contact with microtextured superhydrophobic surfaces discussed earlier (25). Experimentally, we study the effect of the surface–ice interaction, establish a thermofluidic model to predict the outcome of the freezing event, and deduce design rules for icephobic self-dislodging surfaces. We demonstrate the robustness and the general applicability of the self-dislodging process on a broad range of material groups, i.e., polymers, ceramics, and metals, ranging from smooth and hydrophilic, to nanotextured and superhydrophobic. This work gives insight into the fundamentals of freezing on surfaces, introduces a passive, self-cleaning mechanism, and can guide the further development of hydrophobic and icephobic materials.

Results and Discussion

We investigated freezing-driven self-dislodging of supercooled water droplets from engineered substrates; droplets were cooled via evaporation and froze from a supercooled state in a low-pressure, low-humidity environment [partial pressure of water $p_{\mathrm{V}}=O\left(1\hspace{0.17em}\mathrm{mbar}\right)$, Evaporative Cooling]. We performed the experiments in an environmental chamber at room temperature (Materials and Methods). Fig. 1 shows an image sequence of a freezing water droplet, from an initially supercooled state, on a smooth [root-mean-square roughness $O\left(1\hspace{0.5em}\mathrm{nm}\right)$], hydrophobic glass substrate. Fig. 1A (side view) and Fig. 1B (bottom view) are synchronized; time zero is defined as the moment when freezing starts. The freezing of the supercooled droplet proceeds in two distinct stages (26). During the first stage (recalescence), which is relatively short [duration $O\left(0.01\hspace{0.5em}\mathrm{s}\right)$], the droplet rapidly heats up adiabatically from $T=-15\pm 5\hspace{0.5em}°\mathrm{C}$ to $T=0\hspace{0.5em}°\mathrm{C}$ (equilibrium temperature, Fig. S1), resulting in a slushy mix of solid ($\varphi \hspace{0.17em}\approx$ 20 wt. %) and liquid ($\left(1-\varphi \right)\approx$ 80 wt. %) water; $\varphi$ is the solid mass divided by the total mass and depends on the degree of supercooling (14) (Recalescence Freezing). In the second, longer stage [classical freezing; duration $O\left(1\hspace{0.5em}\mathrm{s}\right)$], the remaining liquid water solidifies at $T=0°\mathrm{C}$. Fig. 1 A and B show that further droplet freezing—that is, an increase in the thickness of the outer ice shell that proceeds radially inward from the free surface ($r_{\mathrm{S}}-r_{\mathrm{F}}$)—is associated with a reduction in the droplet–substrate contact radius, $x_{\mathrm{C}}$. Here, $r_{\mathrm{S}}$ and $r_{\mathrm{F}}$ are the radial positions of the outer and inner phase boundaries, respectively. The bottom-view images reveal that the ice beyond $x_{\mathrm{C}}$ no longer contacts the substrate. The low adhesion due to the lack of contact is obvious in the end of Movie S1, where the frozen droplet slides over the substrate. Fig. 1C shows an example of what we identified as the ice shell (clear) and slushy core (opaque). Fig. 1D and Movie S1 show unequivocally that while freezing is occurring, a continuous intervening liquid layer (ILL) is maintained between freezing droplet and substrate—due to the warmer environment ($T_{\infty }\approx 23\hspace{0.5em}°\mathrm{C}$) and also substrate ($T_{sub}>0\hspace{0.5em}°\mathrm{C}$)—as identified by the strong interference patterns near the contact line. We determined the thickness of this layer to be $O\left(10\hspace{0.5em}\mathrm{\mu m}\right)$ (ILL). The simultaneous reduction in $x_{\mathrm{C}}$ and increase in $\left(r_{\mathrm{S}}-r_{\mathrm{F}}\right)$ with time $t$ is plotted in Fig. 1E. The total time for the droplet to self-dislodge from the substrate is defined as $t_{\mathrm{SD}}$.

Fig. 1.

Phenomenon of self-dislodging. Synchronized (A) side- and (B) bottom-view image sequences of a water droplet freezing—through evaporative cooling—on a hydrophobic glass substrate (Movie S1). (C) Magnified side view of a partially solidified droplet from the image sequence in A. Note the ice (transparent) and slush (opaque) regions. The red dashed lines in A indicate the outer droplet radius, $r_{\mathrm{S}}$, and the radial position of the ice–slush phase boundary, $r_{\mathrm{F}}$. The green dashed line in B indicates the droplet–substrate contact radius, $x_{\mathrm{C}}$. (D) Magnified bottom view of a partially solidified droplet from the image sequence in B. The black arrow highlights interference patterns. (E) Contact radius, $x_{\mathrm{C}}$, and ice-shell thickness, $\left(r_{\mathrm{S}}-r_{\mathrm{F}}\right)$, vs. time, $t$. The self-dislodging time, $t_{\mathrm{SD}}$, is defined from the start of freezing ($t=0$ s) until $x_{\mathrm{C}}=0$. The initial droplet volume is 5 µL. The four small circular features visible in the first image of B are vapor bubbles. [Scale bars: (A–C) 1.0 mm; (D) 0.5 mm.]

Fig. S1.

Typical recording of the droplet internal temperature $T_{\text{D}}$ (measured with a thermocouple inside the droplet at position $T_3$ in Fig. 5) and chamber pressure $p_{\infty }$ vs. time $t$. $t=0\text{s}$ is the time when the chamber pressure falls below 112.5 mbar (the upper limit of detection of the pressure sensor). The droplet is resting on a hydrophobic glass substrate.

To investigate experimentally the effect of surface wettability on the dislodging behavior, we prepared three types of glass substrates with varying advancing (${\theta }_{\mathrm{a}}^{\ast }$) and receding (${\theta }_{\mathrm{r}}^{\ast }$) contact angles: hydrophilic (${\theta }_{\mathrm{a}}^{\ast }=56\pm 3°$, ${\theta }_{\mathrm{r}}^{\ast }=31\pm 4°$), hydrophobic (${\theta }_{\mathrm{a}}^{\ast }=118\pm 3°$, ${\theta }_{\mathrm{r}}^{\ast }=93\pm 3°$), and superhydrophobic (${\theta }_{\mathrm{a}}^{\ast }=164\pm 1°$, ${\theta }_{\mathrm{r}}^{\ast }=160\pm 2°$) (Materials and Methods). As mentioned above, when supercooled droplets freeze, they undergo rapid warming, which can cause the droplet to strongly evaporate (20). On specially designed superhydrophobic microtextured surfaces, such vaporization can produce an overpressure between the droplet and substrate, which can overcome adhesion and initiate spontaneous levitation (25). In contrast, here we study droplet–surface interactions at low pressures on substrates with a wide range of wettability (from hydrophilic to superhydrophobic) and topography [from $O\left(1\hspace{0.5em}\mathrm{nm}\right)$ to $O\left(100\hspace{0.5em}\mathrm{nm}\right)$ root-mean-square roughness]. Due to their composition and structure, these substrates are unable to support high-vapor pressures and droplet self-levitation behavior, revealing dislodging as the dominant ice-shedding mechanism persisting over a broad range of wettability (Droplet–Substrate Interactions at Low Pressure). On the three substrates studied here, the water droplets froze repeatedly, but showed different dislodging performance. Fig. 2A shows an image sequence of a droplet freezing on hydrophilic glass, where we observe only partial droplet dislodging. To demonstrate this, Fig. 2B shows the droplet–substrate contact region (bottom view, through the transparent substrate) of a droplet on hydrophilic glass 1.5 s after freezing began. Here, we highlight two regions where the droplet and substrate make contact: the center (green dashed line) and peripheral (shaded red) areas. The peripheral region remained in contact with the substrate during the freezing event and hindered the removal of the droplet (Movie S2). Fig. 2C shows an image sequence of a droplet freezing and dislodging from hydrophobic glass. Fig. 2D shows the corresponding bottom view 1.5 s after the start of freezing. In contrast to the droplet on the hydrophilic substrate, the liquid–substrate contact area remained approximately circular and in one region (green dashed line), and kept reducing itself over the course of the freezing event, leading the droplet to self-dislodge completely from the substrate in a mean dislodging time of $t_{\mathrm{SD}}=2.22\pm 0.35$ s (initial droplet volume $5\hspace{0.17em}\mathrm{\mu L}$). The above highlights the important role that wettability plays in determining if droplet dislodging occurs or not. Fig. 2E shows an image sequence of a water droplet freezing on and dislodging from nanoroughened ($\approx$30-nm root-mean-square roughness), superhydrophobic glass (Movie S3). The image sequence shows that the ice beyond the contact radius $x_{\mathrm{C}}$ clearly lifts from the substrate and no longer makes contact with it. Fig. 2F shows the bottom view of a water droplet on the superhydrophobic substrate 0.2 s after the start of freezing ($t_{\mathrm{SD}}=1.11\pm 0.39$, initial droplet volume $5\hspace{0.17em}\mathrm{\mu L}$, Movie S4). The bottom-view perspective in Movie S4 also shows how during freezing the ice beyond $x_{\mathrm{C}}$ progressively moves out of focus, indicating that it is moving away from the substrate. The droplet–substrate contact region is confined to a smaller area—relative to the hydrophilic and the hydrophobic cases—due to the increased water repellency of the substrate. Also, for a constant initial droplet volume, when supercooled droplets freeze on a substrate, $t_{\mathrm{SD}}$ decreases with increasing ${\theta }_{\mathrm{a}}^{*}$ and ${\theta }_{\mathrm{r}}^{*}$. Fig. 2G presents thermographs of recalescence freezing and the self-dislodging process, allowing a noninvasive measurement of the droplet free-surface temperature, revealing its rapid increase at the moment of recalescence freezing. Of particular interest here is the subsequent surface temperature decline due to ensuing vapor sublimation (Fig. S2 and Movie S5).

Fig. 2.

Effect of substrate wettability on self-dislodging behavior. Side view of water droplets (initially 10 µL) freezing on A, hydrophilic (Movie S2), (C) hydrophobic, and (E) superhydrophobic glass (Movie S3), introducing the vertical lifting of the droplet $y$. Selected bottom views of droplets (initially 5 µL) freezing on (B) hydrophilic, (D) hydrophobic, and (F) superhydrophobic glass (Movie S4). The contact area corresponds to the region inside the green dashed line. Peripheral liquid–substrate contact area is highlighted in red. (Insets) Corresponding topographical scans by AFM. Gray scale bar for the height information is the same for all. (G) Thermographic images of a droplet self-dislodging under same conditions as C (Movie S5). [Scale bars: (A–G): 2 mm; Insets: 500 nm.]

Fig. S2.

Simulated (black) and measured (red) temperature of the sublimating droplet surface $T_{\text{S}}=T\left(r_{\text{S}},t\right)$ over time $t$. The droplet volume in the simulation is 10 µL and ${\theta }_{\text{S}}^{\ast }=96°$. The initial droplet volume in the experiment was 10 µL and hydrophobic glass was used.

To elucidate the physical mechanism responsible for the droplet self-dislodging behavior, we model the evolution of the radial position of the solid–slush phase boundary $r_{\mathrm{F}}\left(t\right)$ for the case of a solidifying sphere with an initial radius, $r_{\mathrm{S}}$. This is achieved by balancing heat removal due to sublimation from the droplet free surface, ${\dot{Q}}_{\mathrm{S}}$, with heat generation due to phase change within the droplet, ${\dot{Q}}_{\mathrm{F}}$ [one-phase Stefan problem (27)]. This model can be applied to any approximately spherical cap geometry in contact with a substrate, with an apparent contact angle ${\theta }_{\mathrm{S}}^{*}$—assuming that the droplet–substrate interface is adiabatic, as the associated heat rate ${\dot{Q}}_{\mathrm{sub}}<<{\dot{Q}}_{\mathrm{S}},{\dot{Q}}_{\mathrm{F}}$ (Fig. S3, Substrate Heat Transfer, and Spherical Cap)—allowing us to determine the radial- $a_{\mathrm{F}}\left(t\right)$, and vertical position $y\left(t\right)$, of the liquid–slush–solid–vapor quadri-junction [$s\left(t\right)=\left(a_{\mathrm{F}}\left(t\right),y\left(t\right)\right)$], Fig. 3A. Since heat is removed from the free surface, it follows that $d{r}_{\mathrm{F}}\left(t\right)/dt<0$. In addition, we assumed that $y\left(t\right)<<a_{\mathrm{F}}\left(t\right)$, which holds for low values of ${\theta }_{\mathrm{S}}^{*}$, and at ${\theta }_{\mathrm{S}}^{*}=110°$, $y\left(t\right)\approx a_{\mathrm{F}}\left(t\right)$, setting an upper limit for ${\theta }_{\mathrm{S}}^{*}$ where the above analysis is appropriate for water (Quadri-Junction Motion). The assumed spherical cap geometry of the droplet and the adiabatic droplet–substrate interface justify solving the one-dimensional heat equation (spherical coordinates) for the case of a representative, inward-solidifying full sphere ($r_{\mathrm{S}}$ determined from the volume of the spherical cap and ${\theta }_{\mathrm{S}}^{*}$), and later applying the results to the investigated droplet, as

$$\frac{\partial T}{\partial t}={\alpha }_{\text{ice}}\left[\frac{{\partial }^{2}T}{\partial r^2}+\frac{2}{r}\frac{\partial T}{\partial r}\right],$$ [1]

with the necessary initial $T\left(0,r\right)=T_{\mathrm{F}}$ (freezing temperature of water, 0 °C), and boundary conditions,

$$\begin{array}{c}T\left(t,r_{\mathrm{F}}\right)=T_{\mathrm{F}},\\ \left(1-\varphi \right)\cdot h_{\mathrm{F},\mathrm{w}}\cdot {\rho }_{\mathrm{ice}}\cdot {\dot{r}}_{\mathrm{F}}=-\left[-{\lambda }_{\mathrm{ice}}{\frac{\partial T}{\partial r}|}_{r=r_{\mathrm{F}}}\right],\end{array}$$ [2]

$${-{\lambda }_{\mathrm{ice}}\frac{\partial T}{\partial r}|}_{r=r_{\mathrm{S}}}=C\cdot h_{\mathrm{S}}\cdot T{\left(t,r_{\mathrm{S}}\right)}^{-0.5}\cdot \left(p_{\mathrm{V}}\left(t,r_{\mathrm{S}}\right)-p_{\mathrm{V}}\left(r_{\infty }\right)\right),$$ [3]

where ${\alpha }_{\mathrm{ice}}={\lambda }_{\mathrm{ice}}/\left({\rho }_{\mathrm{ice}}\cdot c_{\mathrm{ice}}\right)$, ${\rho }_{\mathrm{ice}},c_{\mathrm{ice}}$, and ${\lambda }_{\mathrm{ice}}$ are the thermal diffusivity, density, specific heat capacity, and thermal conductivity of ice, respectively, $h_{\mathrm{F},\mathrm{w}}$ and $h_{\mathrm{S}}$ are the enthalpy of fusion and sublimation of water, respectively, and $C$ is a constant from the Hertz–Knudsen equation (28) (Stefan Model). The vapor pressure of the ice–vapor interface is approximated as the saturation pressure for that ice surface temperature, $p_{\mathrm{V}}\left(t,r_{\mathrm{S}}\right)\approx p_{\mathrm{V,sat}}\left(T\left(t,r_{\mathrm{S}}\right)\right)$, while the vapor pressure at sufficiently large distance from the ice–vapor interface is $p_{\mathrm{V}}\left(r_{\infty }\right)\approx 1\hspace{0.5em}\mathrm{mbar}$ (Vapor Pressure Above the Sublimating Surface). Fig. 3B shows a plot of the solution to the above, $(r_{\mathrm{S}}-r_{\mathrm{F}})$ vs. $t$, whereby the full solution for a solidifying sphere is used only for the region of the spherical cap that is necessary for this specific case (droplet volume at the moment of freezing $4.2\hspace{0.5em}\mathrm{\mu L}$ and ${\theta }_{\mathrm{S}}^{*}=96°$) along with experimental values. The ice thickness $(r_{\mathrm{S}}-r_{\mathrm{F}})$ is scaled by $r_{\mathrm{S}}$, the total length the phase boundary traverses, and $t$ is scaled with $0.5r_{\mathrm{S}}^2/\left({\alpha }_{\mathrm{ice}}St\right)$ (Stefan time scale). Here $St=c_{\text{ice}}\cdot \mathrm{\Delta }T/h_{\mathrm{F,w}}=0.06<<1$ is the Stefan number [$c_{\mathrm{ice}}=2.1\hspace{0.5em}\mathrm{J}/\left(\mathrm{g}\hspace{0.17em}\mathrm{K}\right)$ (29), $h_{\mathrm{F},\mathrm{w}}=334\hspace{0.5em}\mathrm{J}/\mathrm{g}$ (29), and $\mathrm{\Delta }T\approx 10\hspace{0.5em}\mathrm{K}$, which is the typical average temperature difference observed in simulations and experiments between $r_{\mathrm{S}}$ and $r_{\mathrm{F}}$ (Fig. S2 and Movie S5)]. We see that $\left(r_{\mathrm{S}}-r_{\mathrm{F}}\right)$ increases approximately linearly with $t$, which is a departure from the linear one-dimensional Stefan problem, where the phase boundary progresses as $\mathrm{\propto }\sqrt{t}$ (30), and that the experiments and simulation correlate well. With the ability to predict the shape and the position of the phase boundary, we can compute experimental and theoretical values for the radial component of the quadri-junction, $s$, through the following relation for a spherical cap, $a_{\mathrm{F}}=\sqrt{r_{\mathrm{F}}^{\mathrm{2}}-r_{\mathrm{S}}^2{\cos }^2\left({\theta }_{\mathit{S}}^{\mathit{*}}\right)}$. It is instructive to compare experimental and theoretical values of $a_{\mathrm{F}}$ with experimental values of $x_{\mathrm{C}}$, as shown in Fig. 3C. We note that they correlate well in time, indicating that the position of the quadri-junction strongly influences the observed dewetting behavior. This link between $a_{\mathrm{F}}$ and $x_{\mathrm{C}}$ is to be expected due to the small thickness of the ILL and the resulting importance of capillary forces: As the quadri-junction moves radially inward (decreasing $a_{\mathrm{F}}$), a growth in the thickness of the ILL and a reduction in $x_{\mathrm{C}}$ becomes energetically favorable (ILL).

Fig. S3.

Spherical cap geometry. (A) Introduction of the variables. (B) Free-surface area $A_{\text{S}}$ (red) and contact area $A_{\text{C}}$ (blue) as a function of the contact angle ${\theta }_{\text{S}}^{\ast }$. (C) Height of the spherical cap $l_{\text{S}}$ (green), base radius $a_{\text{S}}$ (black), and radius of curvature $r_{\text{S}}$ (magenta) as a function of ${\theta }_{\text{S}}^{\ast }$. (D) Visualization of the effect of ${\theta }_{\text{S}}^{\ast }$ on the spherical cap geometry at a constant droplet volume. (E) Sketch highlighting the entrapment of slush (blue) by forming ice (gray) for large values of ${\theta }_{\text{S}}^{\ast }$.

Fig. 3.

Effect of liquid–slush–solid–vapor quadri-junction motion on droplet dewetting and dislodging behavior. (A) Schematics showing a droplet of radius $r_{\mathrm{S}}$ and contact angle ${\theta }_{\mathrm{S}}^{*}$ on a substrate (spherical cap), which is freezing due to sublimation. The droplet periphery and core consist of ice (gray) and slush (blue), respectively. The magnified region of the schematic highlights the position of the liquid–slush–solid–vapor quadri-junction, $s\left(t\right)=\left(a_{\mathrm{F}}\left(t\right),y\left(t\right)\right)$, which is defined as the phase boundary position $r_{\mathrm{F}}\left(t\right)$ near the substrate. It also highlights the ILL and its contact radius with the substrate, $x_{\mathrm{C}}$. (B) $\left(r_{\mathrm{S}}-r_{\mathrm{F}}\right)$ vs. time $t$ for theoretical (black line) and experimental cases for a freezing droplet on a surface with ${\theta }_{\mathrm{S}}^{*}=96°$ and $4.2\hspace{0.5em}\mathrm{\mu L}$ at the start of freezing (red line; four individual experiments; shaded red region shows the SD). (C) $a_{\mathrm{F}}$ vs. $t$ for theoretical (black line) and experimental cases (red line) as well as experimentally measured $x_{\mathrm{C}}$ (green line; five individual experiments; shaded green region shows the SD). (D) Plot of $s$ for experimental (blue) and theoretical (black) cases for a droplet on a surface with ${\theta }_{\mathrm{S}}^{*}=96°$ and initial volume of $10\hspace{0.5em}\mathrm{\mu L}$ (Inset). (E) Simulated (solid lines) and measured self-dislodging times $t_{\mathrm{SD}}$ as a function of ${\theta }_{\mathrm{S}}^{*}$ for two different volumes at the start of freezing: $4.1\pm 0.3\hspace{0.5em}\mu \text{L}$ (black) and $8.1\pm 0.4\hspace{0.17em}\mathrm{\mu L}$ (gray). Each point is an average of at least five individual experiments performed on as-purchased PMMA (symbol: open triangle), C₄F₈-treated PMMA (open circle), FDTS-treated PMMA (open square), FDTS-treated hydrophobic glass (solid triangle), nanotextured hydrophobic glass (solid circle), and nanotextured superhydrophobic glass (solid square); error bars indicate the SD.

As mentioned above, the key to successful self-dislodging is the inward-moving phase boundary, where the free surface solidifies first and the droplet–substrate interface freezes last. This freezing behavior differs from that in ref. 23, where the ice shell formed on the external surface was completely closed (also in the region of contact with the substrate). As a result, the freezing phase boundary in ref. 23 moved inward in a radially symmetric manner, fully enclosing and compressing the (incompressible) liquid core leading to ice-shell fracturing and an explosive shattering of the partially frozen water droplet. In the present work, we found that the frozen shell formed is not completely closed: In the contact region with the substrate the droplet remains liquid. This is responsible for the manifestation of self-dislodging. To this end, we also found that the frozen phase boundary moves inward but asymmetrically, as the base of the drop in contact with the substrate remains unfrozen and provides an exit for the also-unfrozen core of the droplet, alleviating internal stress buildup. The consequence of this though is that the droplet lifts from the substrate. In what follows below, we focus on modeling the lifting motion. During solidification, the droplet should expand due to differences in densities and the incompressibility of the solid and liquid. Here, the expansion breaks symmetry because the free interface freezes before the droplet–substrate interface; therefore, solidification causes mass to be displaced toward the unsolidified droplet–substrate interface. Since the substrate is nonwetting, and the ice–liquid–vapor contact line is pinned, the displaced mass is not expected to spread away from the droplet and wet the surface (ILL and Fig. S4). Therefore, the droplet should lift upward. Using the principle of conservation of mass, assuming that the droplet is a spherical cap and that the phase boundary progresses radially inwards (Fig. 3A), we estimate the rate of droplet lifting as

$$\frac{dy}{dt}\approx \frac{2r_{\text{F}}\cdot \left(r_{\text{F}}-r_{\text{S}}\cdot \cos \left({\theta }_{\text{S}}^{\ast }\right)\right)\cdot {\dot{r}}_{\text{F}}}{a_{\text{F}}^2}\cdot \left(\frac{\nu \left(1-\varphi \right)-\left(1-\varphi \right)}{\nu \varphi +\left(1-\varphi \right)}\right),$$ [4]

where $\nu$ is the ratio between ${\rho }_{\mathrm{ice}}$ and the density of liquid water. After integration of Eq. 4, we obtain $y\left(t\right)$, which, combined with the previous findings on $a_{\mathrm{F}}\left(\mathit{t}\right)$, yields a full solution for the quadri-junction, $s\left(t\right)$ (Quadri-Junction Motion). Fig. 3D shows theoretical and experimental values of $s$, which compare well, validating the model (Fig. S5). Such profiles are reminiscent of those obtained in freezing droplets on cold substrates, where universal pointy tips are obtained upon full solidification on top, at the center of symmetry of the drop free surface, albeit here the pointed area is inversed and is located at the bottom of the drop (22, 31). To summarize the above findings, Fig. 3E plots experimental values of $t_{\mathrm{SD}}$ vs. ${\theta }_{\mathrm{S}}^{*}$ obtained on six different substrates using two different initial droplet volumes. Also included are theoretical predictions of $t_{\mathrm{SD}}$, which is defined as the time it takes for the radial component of the quadri-junction $a_{\mathrm{F}}\left(\mathit{t}\right)$ to go from its initial value to zero. The comparison with the simulations shows good agreement for ${\theta }_{\mathrm{S}}^{*}$ between 70 and 110°, where the model is valid, and a trend of decreasing $t_{\mathrm{SD}}$ for rising ${\theta }_{\mathrm{S}}^{*}$.

Fig. S4.

Simulation result from the surface evolver program showing a side-view zoom-in of a liquid film (gray, with mesh, height ∼ 0.01 width), its interface with vapor (blue), being in contact with a hydrophilic surface on the top (green, ${\theta }_{\text{top}}^{\ast }=10°$) and a hydrophobic surface on the bottom (black, ${\theta }_{\text{bottom}}^{\ast }=105°$).

Fig. S5.

Final shape of the ice growth at the bottom of a droplet after self-dislodging as a function of the substrate contact angle ${\theta }_{\text{S}}^{\ast }$, comparing experimental results (photo) and computed shape (top left), indicating ${\theta }_{\text{S}}^{\ast }$.

Based on our analysis, we find that droplet contact line motion (dewetting) can be well-predicted with the radial quadri-junction motion; therefore, the time to complete droplet solidification at the droplet–substrate interface can be simply estimated by the Stefan time scale [$0.5a_{\mathrm{S}}^2/\left({\alpha }_{\mathrm{ice}}St\right)$], and can serve as the design rule when creating icephobic self-dislodging surfaces. For a spherical cap $a_{\mathrm{S}}\mathrm{\propto }1/{\theta }_{\mathrm{S}}^{*}$ for a given droplet volume, implying that ${\theta }_{\mathrm{S}}^{*}$ has to be maximized to minimize $t_{\mathrm{SD}}$. Furthermore, for droplet solidification to occur, one should ensure that ${\dot{Q}}_{\mathrm{sub}}<<{\dot{Q}}_{\mathrm{S}},{\dot{Q}}_{\mathrm{F}}$, which is achieved by minimizing substrate thermal conductivity, substrate thermal diffusivity, and $a_{\mathrm{S}}$ (Substrate Heat Transfer and Figs. S6 and S7). Out of the broad range of substrates self-dislodging occurs on, the above criteria enable us to determine the substrates best suited for robust and fast self-dislodging.

Fig. S6.

Heat rate from the substrate into the self-dislodging droplet ${\dot{Q}}_{\text{sub}}$ over time $t$ measured on hydrophobic PMMA (black) and hydrophobic glass (red). Symbols are individual measurements; lines are guides to the eye. $t=0$ is the moment of freezing. Five experiments each. Initial droplet size is 10 µL.

Fig. S7.

Self-dislodging time $t_{\text{SD}}$ as a function of the ingoing heat rate ${\dot{Q}}_{\text{sub}}$ on hydrophobic PMMA (black) and on hydrophobic glass (red) when placing the substrate on an O-ring (${\dot{Q}}_{\text{sub}}$, solid bar) and when placing the substrate on the heat-flux sensor (${\dot{Q}}_{\text{sub}}>0$, hatched bar). The initial droplet size is 10 µL for all experiments. There were 10 experiments for the case of ${\dot{Q}}_{\text{sub}}\sim 0$, 5 experiments for the case of ${\dot{Q}}_{\text{sub}}>0$. We ensured that the assumption of a normal distribution of our data is reasonable (Anderson–Darling, significance level of 0.05) and performed a two-sided two-sampled Student’s t test. We applied the Welch correction and indicated significant differences by asterisks (*P < 0.05 and ***P < 0.001). ${\lambda }_{\text{S},\text{PMMA}}<{\lambda }_{\text{S},\text{Glass}}$. Error bars indicate the SD.

To demonstrate the efficacy of this approach, Fig. 4A shows nine water droplets self-dislodging in one experiment from a nanotextured superhydrophobic glass substrate, satisfying the above design criteria. In addition, Fig. 4 B and C shows water droplet self-dislodging from other technically relevant materials, including stainless steel and poly(methyl methacrylate) (PMMA), designed to be sufficiently hydrophobic and thermally insulating for robust self-dislodging events. To show that self-dislodging can also occur under standard atmosphere environmental pressure conditions and basically requires that heat transfer from the droplet free surface is relatively large to ensure inward solidification, we performed experiments where the droplet was cooled by a cold gas stream at atmospheric pressure—relevant environmental conditions for ice formation. Fig. 4D shows a droplet in contact with a nanotextured superhydrophobic glass substrate—in an environment at atmospheric pressure—exposed to a cold nitrogen flow (≈−30 °C, Movie S7). We see that as the droplet freezes, the phase boundary propagates from the outside in, displacing the unsolidified core toward the substrate causing the droplet to lift, reduce substrate contact area, and fully remove from the surface, which is our definition of self-dislodging discussed in detail earlier for vacuum conditions (Fig. 4D, Insets). This underlines the importance of the self-dislodging mechanism for a wide range of applications where convective heat transfer from the droplet free surface is relevant. Therefore, we demonstrate water droplet self-dislodging on a broad range of material groups, including polymers, ceramics, and metals, both in low- and atmospheric pressure environments and expect that the identification and understanding of the phenomenon herein will pave the way to new approaches in the design of surfaces extremely resistant to ice accretion.

Fig. 4.

Robust self-dislodging on several coating compositions. (A) Top view: nine water droplets self-dislodging from superhydrophobic glass. The substrate is tilted by 2° with respect to the horizontal position (Movie S6). Fig. 2 (Inset) shows the corresponding topographical scan. (B and C) Droplet self-dislodging from hydrophobic stainless steel and FDTS-treated PMMA, respectively. (Insets) Corresponding topographical scans by AFM. (D) Self-dislodging from superhydrophobic glass at atmospheric pressure (Movie S7). The droplet is cooled by a stream of cold nitrogen. The dashed red line indicates the position of the freezing front. Magnified contact regions are shown as insets and linked by colored frames. Droplet volume: (A–C) initially 10 µL; (D) initially 5 µL. [Scale bars: (A–C): 2 mm; (D): 1 mm; (B and C, Insets) 500 nm; (D, Insets) 1 mm).] Scanning electron microscope images of the substrates are presented in Fig. S8.

Fig. S8.

Scanning electron microscope images of the three substrates used in Fig. 4: superhydrophobic glass (A and B), hydrophobic stainless steel (C and D), and FDTS-treated PMMA (E and F).

Evaporative Cooling

Exposing a water droplet to a dry low-pressure environment increases the evaporation rate. The necessary energy expended for evaporation is given by the vapor mass rate multiplied by the enthalpy of vaporization. This energy is supplied by the internal energy of the droplet. For droplets resting on hydrophobic and superhydrophobic substrates the heat flux from the substrate into the droplet is usually small. Consequently, more heat goes out than flows into the droplet, which decreases the internal energy. Pressures as low as in this study combined with thermally insulating substrates limiting ingoing heat flux can lead to significant evaporative droplet supercooling and droplet freezing in an otherwise room-temperature environment. See Fig. S1 for a typical measurement of the droplet core temperature $T_{\text{D}}$ over time $t$ when exposed to the low-pressure environment. $T_{\text{D}}$ quickly reduces over $t$, starting from approximately 20 °C down to below −10 °C. At the moment of recalescence freezing, $T_{\text{D}}$ jumps up to the equilibrium freezing temperature of 0 °C. During the second classical phase of freezing, $T_{\text{D}}$ is reduced again due to ongoing sublimation from the free surface.

Recalescence Freezing

Water droplets freeze from a supercooled state in two distinct stages. During the first stage (recalescence), which is very rapid [$O\left(0.01\text{s}\right)$], the droplet heats up quasi-adiabatically (due to short duration) from its supercooled temperature $T_{\text{D},\text{sup}}\left(-15\pm 5\hspace{0.5em}°\text{C}\right)$ to its equilibrium temperature $T_{\text{D},\text{equ}}$ (0 °C) and results in a droplet that consists of a slushy mix of solid ($\varphi \hspace{0.17em}\approx$ 20 wt. %) and liquid [($1-\varphi$) ≈ 80 wt. %] water (14). An energy balance shows that the ice content depends on $T_{\text{D},\text{equ}}$. The ratio of the mass of ice after recalescence to the total mass of the droplet is $\varphi =\left(m_{\text{ice}}/m_{\text{tot}}\right)=c_{\text{w}}\left(T_{\text{D},\text{equ}}-T_{\text{D},\text{sup}}\right)/h_{\text{F},\text{w}}$, where $\varphi \left(T_{\text{D},\text{sup}}=-15\hspace{0.5em}°\text{C}\right)\approx 0.2$.

ILL

The ILL thickness $y_{\text{ILL}}$ at the droplet–substrate interface can be estimated based on optical and heat-transfer considerations. The interference patterns in Fig. 1D set an upper limit to the possible thickness. For the used red LED with a wavelength of ${\lambda }_{\text{WL}}$ = 625 nm and a bandwidth (full width at half maximum) of $\mathrm{\Delta }{\lambda }_{\text{WL}}$ = 18 nm, the coherence length is estimated to be $L\approx {\lambda }_{\text{WL}}^2/\mathrm{\Delta }{\lambda }_{\text{WL}}\approx 22\hspace{0.17em}\mathrm{\mu }\text{m}$ (32). The computed value is in the range of what is expected for an LED. The existence of the interference patterns suggests that $y_{\text{ILL}}$ cannot be larger than $L$, so that $y_{\text{ILL},\max }\approx L$. The lower limit for the thickness we find from the interference pattern itself, where we observe typically 10 changes from constructive to destructive interference. This sets the lower limit of $y_{\text{ILL}}$ to $y_{\text{ILL},\min }={\lambda }_{\text{WL}}\cdot \left(10/4\right)\approx 2\hspace{0.17em}\mathrm{\mu }\text{m}$.

The magnitude of $y_{\text{ILL}}$ can also be estimated based on the measured heat exchanged between the droplet and the substrate during self-dislodging ${\dot{Q}}_{\text{sub}}$, the thermal conductivity of water ${\lambda }_{\text{w}}\left(0\hspace{0.5em}°\text{C}\right)=0.556\hspace{0.5em}\text{W}/\left(\text{mK}\right)$ (29), the contact area $A_{\text{C}}=\pi \cdot a_{\text{S}}^2$, the temperature inside the droplet $T_{\text{F}}=0\hspace{0.5em}°\text{C}$, and the temperature of the substrate $T_{\text{sub}}$. In the experiment used for this estimation, we observe self-dislodging from a hydrophobic PMMA substrate. The substrate is mounted on the heat-flux sensor and the temperature of the bottom of the chamber is lowered to $T_{\text{chamber}}=2\hspace{0.5em}°\text{C}$, to have repeatable self-dislodging behavior as observed in the practically adiabatic configuration, $\left|{\dot{Q}}_{\text{sub}}\right|<<\left|{\dot{Q}}_{\text{S}}\right|$ (substrate suspended). $T_{\text{sub}}$ is larger than $T_{\text{F}}=0\hspace{0.5em}°\text{C}$ but smaller than $T_{\text{chamber}}=2\hspace{0.5em}°\text{C}$, so that we assume it to be the average of the two, $T_{\text{sub}}=1\hspace{0.5em}°\text{C}$. Over the course of the self-dislodging event, we measure ${\dot{Q}}_{\text{sub}}$ to be between 0.1 and 0.01 W (Fig. S6). As an average we assume ${\dot{Q}}_{\text{sub}}\approx 0.05\hspace{0.5em}\text{W}$ and we set $a_{\text{s}}$ to a typical value of $a_{\text{s}}\approx 1\hspace{0.5em}\text{mm}$. Substituting into Fourier’s law yields $y_{\text{ILL}}={\lambda }_{\text{W}}\mathrm{\cdot }\pi \mathrm{\cdot }{a}_{\text{S}}^2\mathrm{\cdot }\left(T_{\text{sub}}-T_{\text{F}}\right)/{\dot{Q}}_{\text{sub}}\approx 35\hspace{0.17em}\mathrm{\mu }\text{m}$. Both optical and heat-transfer considerations lead to $y_{\text{ILL}}=O\left(10\hspace{0.5em}\mathrm{\mu }\text{m}\right)$.

We observe experimentally that the ice–liquid–vapor contact line moves incrementally inward. The forming steps of ice due to the incremental nature of the freezing and retraction lead to circular pinning sites, which prevent the liquid from spreading outward on the ice and the radius of the ice–liquid–vapor contact line is unable to increase. New incoming fluid from the top into the ILL will increase the volume of the ILL. The increasing volume by $\mathrm{\Delta }V$ will first slightly increase $A_{\text{C}}$ and also the liquid–vapor interface of the ILL, which results in an increase of the total surface energy of the ILL by $\mathrm{\Delta }{E}_{\text{tot}}={\gamma }_{\text{LV}}\cdot \left(\mathrm{\Delta }{A}_{\text{LV}}-\mathrm{\Delta }{A}_{\text{SL}}\cdot \cos {\theta }_{\text{S}}^{\ast }\right)$. Under the assumption that the ILL can initially be approximated by a cylindrical shape and that the incoming liquid will try to spread the ILL toward a truncated cone geometry, we can compute the $\mathrm{\Delta }{E}_{\text{tot}}$ as a function of $\mathrm{\Delta }V$ and ${\theta }_{\text{S}}^{\ast }$. We find an approximately linear relationship between adding $\mathrm{\Delta }V$ and the increase of $\mathrm{\Delta }{E}_{\text{tot}}$. The slope, which is the ratio of $\mathrm{\Delta }{E}_{\text{tot}}/\mathrm{\Delta }V$, represents an effective droplet lifting pressure from which a lifting force is obtained as $F_{\text{L}}=A_{\text{C}}\cdot \mathrm{\Delta }{E}_{\text{tot}}/\mathrm{\Delta }V$. We find that this lifting force is for ${\theta }_{\text{S}}^{\ast }>30°$ always at least two orders of magnitude larger than the droplet gravitational force $F_{\text{G}}=m_{\text{tot}}\cdot g$. For new incoming liquid, to minimize both the contact with the water repellant substrate and the liquid–vapor interface, the ILL will grow in height $y_{\text{ILL}}$ and not spread. The resulting force from capillarity dominates gravity.

We estimate the shape of the ILL from a surface energy minimization analysis with the surface evolver program (33). In the simulation, we model a fluid body, which is initially of cuboidal shape. The height of the cuboid is 1% of the width of the square base. On the bottom, the cuboid is in contact with a surface with a contact angle of 105°. On the top, the cuboid is in contact with a surface with a contact angle of 10°. We find that the ILL adapts after energy minimization the shape of a concave inverted truncated right circular cone (Fig. S4).

In the case of superhydrophobic glass substrates, we observe experimentally that the large flow rates of displaced slush toward the substrate can overwhelm the ILL and lead to flow of the liquid. Inertial forces overcome capillarity and lead to an initial increase of the droplet–substrate contact (Movie S4).

Droplet–Substrate Interactions at Low Pressure

Previously it was demonstrated that droplet freezing from a supercooled state can trigger spontaneous and sudden droplet levitation (even self-launching) in the air, the phenomenon manifesting itself for microtextured, superhydrophobic substrates (25). The effect of such ice levitation relies on a temperature increase during recalescence freezing, the associated increase (sudden boosting) of the evaporation rate, and an overpressure under the water droplet, sustained by specifically designing the surface texture to hinder drainage according to guidelines identified in ref. 25. On microtextured superhydrophobic substrates, the adhesion between the droplet and the substrate is low. In addition, the microtexture allows for the formation of a substantial overpressure (less than the saturation pressure) capable of levitating the as-forming ice above the surface. Consequently, the force from the overpressure under the droplet can overcome the adhesive forces and trigger ice levitation and removal. The removal is completed within the first phase of nonequilibrium freezing (recalescence) and takes about $O\left(0.01\hspace{0.5em}\text{s}\right)$ (25).

In full contrast, here we observe self-dislodging on a much larger variety of substrates, ranging in their wettability from hydrophilic (${\theta }_{\text{S}}^{\ast }\approx$70°) to superhydrophobic (${\theta }_{\text{S}}^{\ast }\approx$160°). While ice levitation requires superhydrophobic microtextured substrates for low adhesion and high pressure buildup, we demonstrate the self-dislodging mechanism on smooth [rms roughness $O\left(1\hspace{0.17em}\text{nm}\right)$] and nanotextured [rms roughness $O\left(100\hspace{0.17em}\text{nm}\right)$] substrates. For smooth substrates, intervening gas/air layers are practically not present, the droplet–substrate contact area is large, leading to large adhesion; therefore, levitation is not to be expected. For nanotextured superhydrophobic surfaces, despite their low droplet adhesion, levitation is not observed. We attribute this to the fact that nanotextured surfaces, by their very nature, have nanobumps and nanopits. Nanopits, due to capillary condensation, can reduce the nucleation vapor pressure locally, limiting the maximum overpressure that can build up between the droplet and substrate and trigger launching in the air. Based on classical nucleation theory, one can show that for pits with a radius of curvature near the critical radius of nucleation, the free-energy barrier to heterogeneous nucleation is proportional to the radius of curvature. A substrate like our superhydrophobic, nanotextured glass has many nanosized pits. Consequently, the free-energy barrier to heterogeneous nucleation is smaller than for superhydrophobic substrates with microscale features (e.g., micropillars). As the nucleation vapor pressure is proportional to this free-energy barrier, we can conclude that the nanoroughness can limit the maximum overpressure built up under the droplet (15). As a result, on the investigated broad range of substrates in this study, ice levitation is not possible, rendering the self-dislodging mechanism the cause of passive and process-intrinsic ice removal. In contrast to ice levitation, self-dislodging is taking place in the second stage of freezing, leading to two orders of magnitude higher removal times $\left[O\left(1\text{s}\right)\right]$.

Substrate Heat Transfer

For the experiments investigating the wettability effect of the substrate, we place the substrates on a O-ring, resulting in negligible heat conduction between the floor of the experimental chamber and the substrate (Fig. 5). Practically all heat transfer between the droplet and the substrate during the self-dislodging event is related to heat conduction at the droplet–substrate contact area. To roughly estimate the amount of heat transferred by conduction during self-dislodging, we assume the event to take 3 s, which is a conservative estimate based on our experiments. Based on this time we compute the thermal penetration depth to be ${\tau }_{\text{BSG}}=1.4\hspace{0.17em}\text{mm}$ on BSG (substrate thickness $d_{\text{BSG}}=0.2$ mm) and ${\tau }_{\text{PMMA}}=0.57$ mm on PMMA (substrate thickness $d_{\text{PMMA}}=1$ mm). We set the droplet mass to $m_{\text{D}}=5$ mg and the contact radius to $a_{\text{S}}=1$ mm. As a worst-case scenario, we define the volume of the substrate exchanging heat with the droplet to $V_{\text{ex}}=\pi \cdot {\left(\tau +a_{\text{S}}\right)}^2\cdot d$ if $\tau >d$ and $V_{\text{ex}}=\pi \cdot {\left(\tau +a_{\text{S}}\right)}^2\cdot \tau$ if $\tau <d$. We estimate the maximum change of temperature of the substrate during the self-dislodging event to be $\mathrm{\Delta }T=15\hspace{0.5em}\text{K}$, assuming that the evaporating droplet cooled down the underlying substrate to the supercooled droplet freezing temperature of −15 °C and heats it up to the droplet equilibrium temperature of 0 °C. The total amount of heat, which can be absorbed by the substrate, is then found to be $Q_{\text{sub}}=V_{\text{ex}}\cdot \rho \cdot c_{\text{p}}\cdot \mathrm{\Delta }T$, giving values of 0.1 J for BSG and 0.12 J for PMMA. The amount of heat removed by sublimation over the self-dislodging event is computed by $Q_{\text{S}}=m_{\text{D}}\cdot h_{\text{F}}=1.65\hspace{0.5em}\text{J}$. Consequently, the heat exchange between droplet and substrate, for the investigated cases, even when we estimate conservatively, is less than 8% and can be neglected in our simplified analysis.

Fig. 5.

Schematic of the experimental setup for self-dislodging visualization. (A) Configuration for side- and top-view imaging with nitrogen inlet on the left and a connection to vacuum pump on the right. The water droplet is resting on the substrate, which is on a thermally insulating O-ring. Using LED illumination and a high-speed camera, synchronized images and measurements of pressure (p), RH, heat flux, and temperature (T1–T5) in the indicated positions are recorded. (B) Configuration for high-speed bottom-view visualization through the transparent substrate with additional visualization from the side. The substrate is attached to a PMMA window with a hole. The droplet is placed on the inside of the chamber on the substrate, allowing direct optical access through the substrate. Monochromatic red LED light passes a beam splitter, is focused by an objective, interacts with the droplet substrate interface, and is collected by the high-speed camera.

When not placing the substrate on the O-ring but on the heat-flux sensor, ${\dot{Q}}_{\text{sub}}$ cannot be neglected anymore. We investigated the effect of ${\dot{Q}}_{\text{sub}}$, the substrate thermal diffusivity ${\alpha }_{\text{sub}}={\lambda }_{\text{sub}}\cdot c_{\text{p},\text{sub}}^{-1}\cdot {\rho }_{\text{sub}}^{-1}$, and thermal conductivity ${\lambda }_{\text{sub}}$ of the substrate on the phenomenon of self-dislodging. To decouple ${\alpha }_{\text{sub}}$ from other substrate properties, we used substrates from PMMA, BSG, and sapphire, which we chemically treated the same way (Materials and Methods), resulting in hydrophobic substrates with similar wettability, roughness, and geometry, but different ${\alpha }_{\text{sub}}$ spanning over two orders of magnitude (Table S1). Fig. S5 shows a summary of the experiments, which we carried out to reveal the influence of ${\dot{Q}}_{\text{sub}}$ and ${\alpha }_{\text{sub}}$ on the self-dislodging event. First, we placed the substrates on the O-ring, limiting ${\dot{Q}}_{\text{sub}}$ to a negligible amount (Fig. 5A). We observed highly repeatable droplet freezing and self-dislodging both on hydrophobic PMMA and on hydrophobic glass. We measured $t_{\text{SD}}$ for hydrophobic PMMA and hydrophobic BSG to be $3.0\pm 0.3\hspace{0.5em}\text{s}$ and $3.1\pm 0.4\hspace{0.5em}\text{s}$, respectively (mean $\pm$ SD, 10 experiments each, initial droplet volume $10\hspace{0.5em}\mathrm{\mu }\text{L}$). In contrast, we observed no freezing under the same conditions for all experiments on the hydrophobic sapphire. The droplet on the sapphire evaporated quickly with an average evaporation rate of approximately 0.1 mg s⁻¹. This shows that ${\alpha }_{\text{sub}}$ of the base material is an important parameter determining if freezing and self-dislodging is observed or not.

Table S1.

Details on the engineered substrates used in this study

Substrate name	Material	Treatment	λ, [W/(mK)]	α, m2 s−1	${\theta }_{\text{a}}^{\ast }$, °	${\theta }_{\text{r}}^{\ast }$, °
As-purchased PMMA	PMMA	—	0.19	$1.1\cdot {10}^{-7}$	86 ± 2	59 ± 2
FDTS-treated PMMA	PMMA	Sputter SiO2 + FDTS	0.19	$1.1\cdot {10}^{-7}$	119 ± 2	89 ± 2
C4F8-treated PMMA	PMMA	C4F8 plasma	0.19	$1.1\cdot {10}^{-7}$	112 ± 6	80 ± 5
Hydrophilic glass	BSG	—	1.2	$6.5\cdot {10}^{-7}$	56 ± 3	31 ± 4
Hydrophobic glass	BSG	FDTS	1.2	$6.5\cdot {10}^{-7}$	118 ± 3	93 ± 3
Nanotextured hydrophobic glass	BSG	5 min RIE + FDTS	1.2	$6.5\cdot {10}^{-7}$	129 ± 2	93 ± 4
Nanotextured superhydrophobic glass	BSG	120 min RIE + FDTS	1.2	$6.5\cdot {10}^{-7}$	164 ± 1	160 ± 2
Hydrophobic sapphire	Sapphire	Sputter SiO2 + FDTS	27.21	$1.6\cdot {10}^{-5}$	114 ± 2	96 ± 1
Hydrophobic stainless steel	Stainless steel	FDTS	15	$3.8\cdot {10}^{-6}$	146 ± 2	86 ± 3

This table reports for the different prepared substrates the name used in the document, the material, the surface treatment, the thermal conductivity, as well as the advancing and receding contact angle, averaged over five measurements giving the SDs as error. Details on the surface treatment are explained in Materials and Methods.

Second, to investigate to what extent ${\dot{Q}}_{\text{sub}}$ influences $t_{\text{SD}}$, we slightly modified the experimental setup, removing the O-ring and placing the substrate on a heat-flux sensor (Fig. 5A). As a result, ${\dot{Q}}_{\text{sub}}$ was not negligible anymore. Additionally, it was now necessary to reduce the temperature of the bottom of the environmental chamber to 2 °C, so that ${\dot{Q}}_{\text{sub}}$ from the bottom of the chamber, through the heat-flux sensor and the substrate, and into the droplet, was still small enough to allow the droplet to quickly cool down by evaporation and freeze repeatedly. With these changes applied, we found that the self-dislodging took $3.8\pm 0.2\hspace{0.5em}\text{s}$ on hydrophobic PMMA and $4.7\pm 0.9\hspace{0.5em}\text{s}$ on hydrophobic glass ($\pm$SD, five experiments each, initial droplet volume $10\hspace{0.5em}\mathrm{\mu }\text{L}$), which is significantly larger than when using the O-ring (${\dot{Q}}_{\text{sub}}\approx 0$). We present the results for ${\dot{Q}}_{\text{sub}}$ over time $t$ in Fig. S6. A nonnegligible ${\dot{Q}}_{\text{sub}}$ slows down the ice production, which in turn results in larger $t_{\text{SD}}$ (Fig. S7).

We can deduce that both ${\alpha }_{\text{S}}$ and ${\dot{Q}}_{\text{sub}}$ are important parameters for the self-dislodging performance. If ${\alpha }_{\text{S}}$ is too large as in the case of sapphire, no freezing and no self-dislodging are observed. In the case of more moderate ${\alpha }_{\text{S}}$ as for PMMA and BSG, freezing and self-dislodging occur. On materials with suitable ${\alpha }_{\text{S}}$, the level of ${\dot{Q}}_{\text{sub}}$ determines if $t_{\text{SD}}$ is small (as for ${\dot{Q}}_{\text{sub}}\approx 0$) or significantly larger (as for ${\dot{Q}}_{\text{sub}}>0$). Since for given thermodynamic conditions ${\dot{Q}}_{\text{sub}}$ can be influenced by the thermal conductivity of the material ${\lambda }_{\text{S}}$, we can conclude that an ideal surface for self-dislodging should have low ${\lambda }_{\text{S}}$ and low ${\alpha }_{\text{S}}$.

Spherical Cap

In the Stefan model, we always simulate the ice growth of an entire sphere, but only evaluate the results in the region where the spherical cap droplet effectively is located. Fig. S3A shows the geometry of a spherical cap. It illustrates with a dashed line the entire sphere, while the region of the spherical cap is of blue color. The Stefan model simulation requires the droplet volume $V_0$ and the contact angle ${\theta }_{\text{S}}^{\ast }$ as inputs. Based on these, all other geometrical aspects of the spherical cap are computed, such as radius of the simulated sphere $r_{\text{S}}={\left(3\cdot V_0\cdot {\left(\cos {\theta }_{\text{S}}^{\ast }-1\right)}^2/\pi \cdot \left(\cos {\theta }_{\text{S}}^{\ast }+1\right)\right)}^{1/3}$, the height of the spherical cap $l_{\text{S}}=r_{\text{S}}\cdot \left(1-\cos {\theta }_{\text{S}}^{\ast }\right)$, and the contact radius $a_{\text{S}}={\left(l_{\text{S}}\cdot \left(2r_{\text{S}}-l_{\text{S}}\right)\right)}^{1/2}=r_{\text{S}}\cdot \sin {\theta }_{\text{S}}^{\ast }$. The dependence of these variables on ${\theta }_{\text{S}}^{\ast }$ for constant droplet volume is plotted in Fig. S3 B–D, while Fig. S3E illustrates how ${\theta }_{\text{S}}^{\ast }$ influences the effective ice-growth geometry of the spherical cap.

Quadri-Junction Motion

To estimate the shape of the ice growth at the bottom of the droplet and with it the vertical motion of the quadri-junction, we assume the following: ${\dot{Q}}_{\text{S}}$ acts radially symmetrically and dominates ${\dot{Q}}_{\text{sub}}$ so that ${\dot{Q}}_{\text{sub}}$ is negligible. The phase boundary moves in concentrically and freezes all of the way down to the substrate. The liquid contact with the substrate $x_{\text{C}}$ equals the opening at the droplet bottom $a_{\text{F}}$, which holds for $y\ll a_{\text{F}}$.

From the geometry of the spherical cap we find a relation between the change of opening radius at the droplet bottom over time $\left(d{a}_{\text{F}}/dt\right)$ and the change of the freezing radius over time $\left(d{r}_{\text{F}}/dt\right)$. While $a_{\text{F}}=\sqrt{r_{\text{F}}^2-r_{\text{S}}^2\cdot {\cos }^2\left({\theta }_{\text{S}}^{\ast }\right)}$, the time derivative yields $\left[\left(d{a}_{\text{F}}/dt\right)=\left(r_{\text{F}}/\sqrt{r_{\text{F}}^2-r_{\text{S}}^2\cdot {\cos }^2\left({\theta }_{\text{S}}^{\ast }\right)}\right)\cdot \left(d{r}_{\text{F}}/dt\right)\right]$. Conservation of mass of the freezing droplet (neglecting mass loss due to sublimation) imposes $m_{\text{tot}}=m_{\text{shell}}+m_{\text{core}}+m_{\text{disp}}$, where $m_{\text{tot}}\hspace{0.17em},m_{\text{shell}}\hspace{0.17em},m_{\text{core}},\hspace{0.5em}\text{and}\hspace{0.5em}{m}_{\text{disp}}$ are the mass of the droplet, the ice shell, the slush core inside the ice shell, and the displaced slush pushed toward the substrate, respectively. Here $m_{\text{tot}}={\rho }_{\text{slush}}{V}_0={\rho }_{\text{ice}}{V}_{\text{shell}}+{\rho }_{\text{slush}}{V}_{\text{core}}+{\rho }_{\text{slush}}{V}_{\text{disp}}$, where ${\rho }_{\text{slush}}$ is the density of slush given as ${\rho }_{\text{slush}}=\varphi \cdot {\rho }_{\text{ice}}+\left(1-\varphi \right)\cdot {\rho }_{\text{w}}$. $V_{\text{core}}$ is the volume of the slush core inside the ice shell, $V_{\text{disp}}$ is the volume of slush displaced toward the substrate, and $V_{\text{shell}}=V_0-V_{\text{core}}$ is the volume of the ice shell. The time derivative and rearranging of the mass balance yields $\left[{\dot{V}}_{\text{disp}}={\dot{V}}_{\text{core}}\left(\left({\rho }_{\text{ice}}-{\rho }_{\text{slush}}\right)/{\rho }_{\text{slush}}\right)={\dot{V}}_{\text{core}}\cdot \left(\nu \left(1-\varphi \right)-\left(1-\varphi \right)/\nu \varphi +\left(1-\varphi \right)\right)\right]$, where $\nu$ is the density ratio of ${\rho }_{\text{ice}}$ divided by the density of liquid water, while $\varphi$ is the ratio of frozen water to total droplet mass after recalescence, $\varphi =m_{\text{ice}}/m_{\text{tot}}$. The vertical motion of the quadri-junction relates to the displaced volume as $\left[\left(dy/dt\right)=\left({\dot{V}}_{\text{disp}}/\pi \cdot a_{\text{F}}^2\right)\right]$. The initial height of the spherical cap is $l_{\text{S}}$ and the current height of the slushy core region is $l_{\text{F}}=r_{\text{F}}-\text{sign}\left(\cos \left({\theta }_{\text{S}}^{\ast }\right)\right)\cdot \sqrt{r_{\text{F}}^2-a_{\text{F}}^2}$. The total volume of the slushy core inside the initial boundaries of the spherical cap is $\left[V_{\text{core}}=\left(l_{\text{F}}^2\pi /3\right)\left(3r_{\text{F}}-l_{\text{F}}\right)\right]$. Consequently, the time derivative is ${\dot{V}}_{\text{core}}=2\pi r_{\text{F}}\cdot (r_{\text{F}}-r_{\text{S}}\cdot \cos \left({\theta }_{\text{S}}^{\ast }\right))\cdot {\dot{r}}_{\text{F}}$. Combined with our previous findings this yields Eq. 4 of the main text: $\left[\left(dy/dt\right)\approx \left(2r_{\text{F}}\cdot \left(r_{\text{F}}-r_{\text{S}}\cdot \cos \left({\theta }_{\text{S}}^{\ast }\right)\right)\cdot {\dot{r}}_{\text{F}}/a_{\text{F}}^2\right)\cdot \left(\nu \left(1-\varphi \right)-\left(1-\varphi \right)/\nu \varphi +\left(1-\varphi \right)\right)\right]$.

Summarizing all findings we see that $\left[\left(dy/d{a}_{\text{F}}\right)=\left(dy/dt\right)/\left(d{a}_{\text{F}}/dt\right)\right]$, so that $\left[\left(dy/d{a}_{\text{F}}\right)=2\cdot \left(\sqrt{a_{\text{F}}^2+{\left(r_{\text{S}}-l_{\text{S}}\right)}^2}-r_{\text{S}}\cdot \cos \left({\theta }_{\text{S}}^{\ast }\right)\right)\cdot a_{\text{F}}^{-1}\cdot \left(\nu \left(1-\varphi \right)-\left(1-\varphi \right)/\nu \varphi +\left(1-\varphi \right)\right)\right]$, which can be integrated to the final result describing the quadri-junction pathway and the ice-growth shape: $\left\{y\left(a_{\text{F}}\right)=2\cdot \left(\nu \left(1-\varphi \right)-\left(1-\varphi \right)/\nu \varphi +\left(1-\varphi \right)\right)\cdot \left[\left(\sqrt{c}/2\right)\left(\ln \left(\left|\sqrt{a_{\text{F}}^2+c}-\sqrt{c}\right|\right)-\ln \left(\left|\sqrt{a_{\text{F}}^2+c}-\sqrt{c}\right|\right)\right)-b\ln \left(a_{\text{F}}\right)+\sqrt{a_{\text{F}}^2+c}+d\right]\right\}$ with $b=r_{\text{S}}\cdot \cos \left({\theta }_{\text{S}}^{\ast }\right)$ and $c={\left(r_{\text{S}}-l_{\text{S}}\right)}^2$, and $d$ as integration constant $\left(d=\left(\left|r_{\text{S}}-l_{\text{S}}\right|/2\right)\left(\ln \left(\left|r_{\text{S}}-\left|r_{\text{S}}-l_{\text{S}}\right|\right|/\left|r_{\text{S}}+\left|r_{\text{S}}-l_{\text{S}}\right|\right|\right)\right)+r_{\text{S}}\cdot \cos \left({\theta }_{\text{S}}^{\ast }\right)\cdot \ln \left(a_{\text{S}}\right)-r_{\text{S}}\right)$.

Stefan Model

To understand the effect of substrate wettability on $t_{\text{SD}}$, we model the heat transfer of the self-dislodging event. The sublimation mass rate at $r=r_{\text{S}}$ is used as one boundary condition and computed based on the Hertz–Knudsen equation as ${\dot{m}}_{\text{S}}\left(T_{\text{S}}\right)=A_{\text{S}}\cdot B\cdot {\alpha }_{\text{S}}\cdot {\left(M_{\text{W}}/\left(2\pi \cdot R_{\text{G}}\cdot T_{\text{S}}\right)\right)}^{0.5}\left(p_{\text{V,S}}-p_{\text{S}}\left(T_{\text{S}}\right)\right)$ (28). Here $A_{\text{S}}$ is the droplet free-surface area, $B=0.75$ is the return flux coefficient (28), ${\alpha }_{\text{S}}=0.15$ is the sublimation coefficient (28), $M_{\text{W}}=18.015\hspace{0.5em}\text{g}/\mathrm{mol}$ is the molar mass of water (29), $R_{\text{G}}=8.314\hspace{0.5em}\text{J}/\left(\text{mol}\hspace{0.5em}\text{K}\right)$ is the universal gas constant, and $T_{\text{S}}=T\left(r_{\text{S}},t\right)$ is the droplet free-surface temperature. The constant $C$ used in the main text is thus $C=B\cdot {\alpha }_{\text{S}}\cdot {\left(M_{\text{W}}/\left(2\pi \cdot R_{\text{G}}\right)\right)}^{0.5}$. The vapor pressure $p_{\text{V},\text{S}}=p_{\text{V}}\left(T_{\text{S}}\right)$ above the surface is assumed to be 1 mbar (Vapor Pressure Above the Sublimating Surface). The saturation pressure is found from $T_{\text{S}}$ in Kelvin for $193.15\hspace{0.5em}\text{K}\le T_{\text{S}}\le 273.15\hspace{0.5em}\text{K}$ as $\left[p_{\text{S}}\left(T_{\text{S}}\right)=a_{\text{ps}}\cdot \exp \left(b_{\text{ps}}\cdot T_{\text{S}}/T_{\text{S}}+c_{\text{ps}}\right)\right]$, where $a_{\text{ps}}=6.1115,\hspace{0.5em}{b}_{\text{ps}}=22.542,\hspace{0.5em}\text{and}\hspace{0.5em}{c}_{\text{ps}}=273.48$ are fitting coefficients (34). We assume that $r_{\text{S}}=\text{const}.$ over the course of one experiment, neglecting the shrinkage of the droplet associated with sublimation. This is justified as the total mass loss due to sublimation during the self-dislodging event $m_{\text{S}}$ is much smaller than the total mass of the droplet $m_{\text{tot}}$.

We solve the set of equations numerically discretized in time with a time step $\mathrm{\Delta }t=0.001\hspace{0.5em}\text{s}$. To solve the problem we use a quasi-steady approximation for each time step, in which we assume the heat removal by sublimation on the outer boundary to be equal to the heat release from the ice generation at the inner boundary. Consequently, for each moment in time, the heat equation simplifies to $\left[0=\left({\partial }^{2}T/\partial r^2\right)+\left(2/r\right)\left(\partial T/\partial r\right)\right]$. We assume that the amount of sublimation dictates the phase change in the slushy region from water to ice. We write the outgoing heat rate due to sublimation as ${\dot{Q}}_{\text{S}}={\dot{m}}_{\text{S}}\cdot h_{\text{S}}=\left(1-\varphi \right)\cdot {\dot{m}}_{\text{F}}\cdot h_{\text{F},\text{w}}={\dot{Q}}_{\text{F}}$, where $\varphi =0.2$ and ${\dot{Q}}_{\text{F}}$ is the heat rate resulting from the solidification on the inside of the droplet, which corresponds to the Stefan condition. The Stefan number is $St=c_{\text{ice}}\cdot \mathrm{\Delta }T/h_{\text{F},\text{w}}=0.06\ll 1$, with $\mathrm{\Delta }T\approx 10\hspace{0.17em}\text{K}$, allowing this simplified approach. Consequently we can compute the ice production rate, ${\dot{m}}_{\text{F}}={\dot{m}}_{\text{S}}\cdot h_{\text{S}}/\left(h_{\text{F},\text{w}}\cdot \left(1-\varphi \right)\right)$. With these equations, we start the numerical solution. Based on the initial ice-production rate ${\dot{m}}_{\text{F},0}$ we compute the change of the radius of the slushy region between one time step $i-1$ and the next time step $i$ as ${\dot{m}}_{\text{F},\text{i}}=\left({\rho }_{\text{ice}}/\mathrm{\Delta }t\right)\cdot \left(4/3\right)\cdot \pi \cdot \left(r_{\text{i}-1}^3-r_{\text{i}}^3\right)$. Based on the radius of the slushy region we compute the thermal resistance of the ice shell as $\left(R_{\text{sphere}}\left(t\right)=\left(\left(1/r_{\text{S}}\right)-\left(1/r_{\text{F}}\left(t\right)\right)\right)\cdot {\left(4\pi \cdot {\lambda }_{\text{ice}}\right)}^{-1}\right)$. Based on the thermal resistance and the heat rate we compute the temperature drop across the ice shell and the temperature of the sublimating surface as $\left(T_{\text{S}}\left(t\right)=T_{\text{F}}-R_{\text{sphere}}\left(t\right)\cdot {\dot{Q}}_{\text{S}}\left(t\right)=T_{\text{F}}-\left(\left(1/r_0\right)-\left(1/r_{\text{F}}\left(t\right)\right)\right)\cdot {\left(4\pi \cdot {\lambda }_{\text{ice}}\right)}^{-1}\cdot h_{\text{S}}\cdot {\dot{m}}_{\text{S}}\left(T_{\text{S}}\left(\text{t}\right)\right)\right)$. For a more stable simulation, the computation of ${\dot{Q}}_{\text{S}}$ is based on an average of up to the last five sublimation mass rates. The computed $T_{\text{S}}\left(t\right)$ compares well with measured values (Fig. S2).

Vapor Pressure Above the Sublimating Surface

The vapor pressure at sufficiently large distance from the ice–vapor interface $p_{\text{V}}\left(r_{\infty }\right)$ is found from the chamber temperature $T\left(r_{\infty }\right)\approx 23\hspace{0.5em}°\mathrm{C}$, the corresponding saturation pressure $p_{\text{V},\text{sat}}\left(T\left(r_{\infty }\right)\right)$, and the measured $\text{RH}\approx 3\%$, giving $p_{\text{V}}\left(r_{\infty }\right)\approx 0.8$ mbar. We can also estimate an upper bound of $p_{\text{V}}\left(r_{\infty }\right)$ based on the total chamber pressure $p\left(r_{\infty }\right)$, which is in the experiments $p\left(r_{\infty }\right)=2\pm 1\hspace{0.5em}\text{mbar}$. Due to the strong vacuum pump we are using for our experiments, we can assume that $p_{\text{V}}\left(r_{\infty }\right)\le p\left(r_{\infty }\right)$. Thus, we choose $p_{\text{V},\text{S}}=1$ mbar for the simulations.

Materials and Methods

Materials.

In this study we used PMMA in thickness of 175 µm and 1 mm obtained from Schlösser GmbH, borosilicate glass (BSG) in thickness 200 µm obtained from Plan Optik AG, as well as sapphire in random orientation in thickness of 200 µm obtained from UQG Ltd. We obtained the 1.4310 stainless steel tape 10 mm wide, 0.1 mm thick, from H+S Präzisions-Folien GmbH. For the chemical functionalization we used 1H, 1H, 2H, 2H-perfluorodecyltrichlorsilane, 96% (FDTS) from Alfa Aesar GmbH, hexane anhydrous 95% from Sigma-Aldrich, isopropyl alcohol (IPA) from Thommen-Furler AG, deionized water (DIW, Merck Milli-Q direct, resistivity >18.2 MΩ cm).

Substrate Preparation.

We used the substrates in their as-purchased state, which is hydrophilic, and in their functionalized state, which is hydrophobic. To render the substrates hydrophobic, we treated the substrates with FDTS: First, we used a sputter tool (CS320S; Von Ardenne Dresden) to apply a layer of ∼10-nm silicon dioxide onto the PMMA and the sapphire samples to prepare them for the FDTS treatment. Subsequently, we cleaned and activated all samples in an oxygen plasma (7 min, 100 W; Plasma Asher Diener). We immersed the samples in a mixture of 10 µL FDTS in 20 mL hexane (BSG and sapphire: 2 min; PMMA: 1 min; stainless steel: 2 h), followed by rinsing in hexane (BSG, sapphire, and stainless steel: 1 min; PMMA: 10 s), rinsing in IPA (BSG, sapphire, and stainless steel: 1 min; PMMA: 10 s), rinsing in DIW (BSG, sapphire, and stainless steel: 1 min; PMMA: 10 s), and heating for 10 min on a hotplate (BSG, sapphire, and stainless steel: 120 °C; PMMA: 80 °C). We fabricated the nanotextured, hydrophobic, and superhydrophobic glass samples using maskless reactive ion etching (RIE) for 5 and 120 min, respectively, in an Oxford NPG80 (38 sccm Ar and 12 sccm CHF₃ gas at 200 W). We measured the root-mean-square (rms) roughness of as-purchased BSG by atomic force microscopy (AFM) in three independent scans to be $R_{\mathrm{rms}}\mathrm{=1}\mathrm{.1}\pm \mathrm{0}\mathrm{.1}\hspace{0.5em}\mathrm{nm}$ (mean ± SD). After exposure to RIE for 5 and 120 min the roughness increased to $R_{\mathrm{rms}}\mathrm{=3}\mathrm{.8}\pm \mathrm{0}\mathrm{.2}\mathrm{nm}$ and $R_{\mathrm{rms}}\mathrm{=25}\mathrm{.7}\hspace{0.17em}\pm \hspace{0.17em}\mathrm{0}\mathrm{.5}\mathrm{nm}$, respectively. The FDTS treatment was the same as for the not-etched BSG samples. We fabricated the C₄F₈-treated PMMA by exposing the PMMA to a C₄F₈ plasma in a dry etching machine for 10 min (100 sccm, Alcatel AMS 200). Table S1 lists the properties of the substrates including advancing/receding contact angles, ${\theta }_{\mathrm{a}}^{*}$, ${\theta }_{\mathrm{r}}^{*}$, and the thermal properties.

Experimental Setup.

A schematic of the environmental chamber and the data acquisition for side-view visualization is shown in Fig. 5A. In the environmental chamber we can reduce the pressure by using a vacuum pump (RZ 2.5; VACUUBRAND), create a dry environment by purging with nitrogen, and reduce for some experiments the temperature on the bottom of the aluminum chamber with the help of a Peltier element. We use a connected pressure sensor (CMR362; Pfeiffer Vacuum), a humidity sensor (IST AG LinPicco A05), and temperature sensors (PT1000 and T-type thermo elements) to record the thermodynamic conditions inside the chamber. We measured heat fluxes with a heat-flux sensor (gSKIN – XM 26 9C calibrated; GreenTEG) applying thermally conductive paste between the bottom of the chamber, the heat flux sensor, and the substrate. We synchronized all signals with the high-speed images with a National Instruments board (NI USB-6361) and the camera software PhotronFastcamViewer. We used a high-speed camera (SA1.1; Photron) at up to 2,000 frames per second and an LED front-side illumination (SL073; Advanced Illumination) to record the self-dislodging event from the side. For bottom-view visualization the experimental chamber was mounted on an inverted microscope setup using the high-speed camera, a 4× objective (Olympus), and a red fiber-coupled LED light source (M625F1; Thorlabs) for the bottom-view visualization through the transparent substrate. A sketch of the inverted microscope setup is shown in Fig. 5B. We used an additional camera (DCC1645C; Thorlabs) at 15–30 frames per second for simultaneous side-view visualization.

Experimental Procedure.

We used a pipette (20 µL; Eppendorf Research plus) to place water droplets of initial volumes between 5 and 10 µL on the substrate in the chamber. We closed the chamber, purged it with nitrogen to reach a relative humidity below 3%, and opened the valve between the environmental chamber and the vacuum pump.

For the study on the wettability effect, we placed the substrate on an O-ring limiting the heat transfer between the bottom of the chamber and the substrate to a negligible amount. We performed 10 independent experiments per substrate with an initial droplet volume of $10\hspace{0.5em}\mathrm{\mu L}$ focusing on high-quality side-view visualization and 5 independent experiments per substrate with an initial droplet volume of $5\hspace{0.5em}\mathrm{\mu L}$ focusing on the visualization of the droplet–substrate interface. For all experiments on the wettability effect, we set the absolute chamber pressure and temperature to be $p_{\infty }=2\pm 1\mathrm{mbar}$ and $T_{\infty }=23\pm 2°\mathrm{C}$, respectively, and measure the relative humidity (RH) to be $\text{RH}=3\%$, with an uncertainty of 3% RH, resulting in a low vapor pressure $p_{\mathrm{V}}$.

For the experiments at atmospheric pressure, we cooled the droplet with a stream of nitrogen vapor directed at the sessile water droplet from above. The nitrogen gas was produced by boiling liquid nitrogen (Kaltgas; KGW Isotherm). The approximate nitrogen vapor temperature impacting onto the droplet was measured with a thermocouple to be $-30\pm 5°\mathrm{C}$.

Substrate Characterization.

To determine the wettability of the substrates, we measured the apparent advancing (${\theta }_{\mathrm{a}}^{*}$) and receding (${\theta }_{\mathrm{r}}^{*}$) water contact angles by inflating (advancing) and deflating (receding) ∼10 µL of water through a flat-tipped plastic needle (GELoader Tips) using a syringe pump (New Era Pump Systems). Using a detector (DCC1645C; Thorlabs), a zoom lens (MVL7000; Thorlabs), and an LED light source, we captured images of the wetting and dewetting dynamics, which we used to measure the contact angles in ImageJ.

We measured the surface roughness with an AFM (Bruker AFM Dimension FastScan) in tapping mode in air.