Impact and freezing characteristics of deionized water droplets on cold curved surfaces Open Access

The phenomenon of droplet impact and ice formation is ubiquitous in both nature and engineering applications, encompassing fields such as aviation,^1,2 power,^3,4 construction,^5,6 and transportation.^7,8 In the aviation sector, aircraft may encounter supercooled droplets during flight, leading to icing of critical components such as wings and engines and thereby increasing flight risks.⁹ In the power industry, the icing of power transmission lines and wind turbine blades can reduce the efficiency of power transmission and may even cause structural damage.¹⁰ In the construction field, ice formation on roofs and walls can impose increased structural loads, potentially causing collapse accidents.¹¹ Therefore, an in-depth study of the dynamics of droplet impact and ice formation, the mechanisms of heat and mass transfer, and the development of control strategies are of great significance for enhancing the safety and reliability of related facilities.

When a droplet impacts a solid wall, it is subjected to various forces such as inertial forces, surface tension, and viscous forces, and undergoes complex dynamic processes such as spreading, retraction, and oscillation.^12–15 Upon impact with a surface below the droplet’s freezing point, the contact area between the droplet and the surface rapidly undergoes a phase change, with the formation of an ice layer.^16,17 The impact and freezing process is influenced by the droplet’s physical properties (e.g., temperature,¹⁸ velocity,¹⁹ and size²⁰), wall characteristics (e.g., material,²¹ roughness,²² and wettability²³), as well as environmental conditions (e.g., ambient temperature²⁴ and humidity²⁵). The process of droplet freezing is not only closely related to the dynamics of impact. but also involves complex heat and mass transfer processes.^26,27

The process of droplet impact on a low-temperature solid wall and the subsequent freezing is a complex gas–liquid–solid three-phase flow–heat transfer coupled phase change, involving a series of processes such as the fluid dynamical evolution before and after droplet impact with the wall, heat transfer between gas, solid, and liquid, and phase-change freezing of the adhering liquid film.^28–31 For instance, the diameter of freezing rain droplets ranges from 0.1 to 3 mm, and their falling velocity is between 1 and 3 m/s. When droplets impact a flat solid wall with an infinitely large aspect ratio at a certain velocity, a series of dynamic processes such as radial spreading, retraction, and stable adhesion occur.³² However, when droplets impact curved walls such as power lines or wind turbine tips with a smaller aspect ratio, the evolution of dynamic characteristics is significantly different, including circumferential and axial spreading, splitting, detachment, and liquid film adhesion.³³ In such droplet impact, there is intense flow and heat transfer behavior, ultimately leading to freezing of droplets or liquid films into ice on the solid wall.

Over the years, much systematic research has been conducted on the freezing of static suspended droplets and of droplets adhering to solid walls, and a basic consensus on droplet freezing has been established.^34,35 When droplets freeze, the latent heat of phase change between the solid and liquid phases causes the liquid phase temperature to rise slightly, accompanied by an increase in volume, with an expansion coefficient of ∼1.15–1.20.³⁶ Jin et al.³⁷ found that under natural convection conditions, the shape of the freezing droplet is essentially symmetrical, but under forced convection, the ice crystal centerline exhibits a more pronounced asymmetric distribution. Chang et al.³⁸ concluded from droplet freezing experiments on different walls that under the same cooling rate, the nucleation temperature range on superhydrophobic walls is significantly greater than on hydrophilic walls, and the freezing time of droplets on superhydrophobic walls decreases more rapidly with increasing wall temperature. A numerical study by Wang et al.^39,40 showed that the freezing rate of droplets decreases with increasing contact angle and droplet size. Antonini et al.⁴¹ and Kulinich et al.⁴² demonstrated experimentally that on low-temperature hydrophobic coatings and superhydrophobic walls, droplets are more likely to roll and avoid freezing, and that even if freezing occurs, the adhesion of ice is significantly reduced by enhanced hydrophobicity.

The impact and freezing of droplets involve transient dynamic evolution and heat transfer characteristics, and their dynamic spreading and freezing processes differ significantly from the freezing of suspended/adhered droplets.⁴³ On the basis of a molecular dynamics study, Ma et al.⁴⁴ indicated that at the nanoscale, the enhancement of viscous and interfacial effects exacerbates the impact of wall wettability on droplet contact time. However, Kong et al.,⁴⁵ in a study of freezing characteristics under different wall wettabilities, suggested that the average diffusion rate of supercooled droplets is independent of wall wettability. Shang et al.,⁴⁶ in a comparative study of static and dynamic droplet freezing behavior on superhydrophobic walls, suggested that superhydrophobic walls can delay the nucleation of ice in static droplets, but cannot inhibit the freezing of dynamic droplets. Zhu et al.⁴⁷ examined the differences between droplet impact freezing and adhered droplet freezing and found that the frozen shapes of moving droplets on hydrophilic or hydrophobic surfaces depended only on wall temperature, whereas those of adhered droplets were barely influenced by wall temperature but significantly affected by wall wettability. Bahadur et al.⁴⁸ and Jung et al.⁴⁹ demonstrated that the droplet impact freezing process is influenced by wall roughness, surface structure, chemical properties, and thermodynamic properties. Xu et al.⁵⁰ studied the dynamic spreading behavior of a single droplet impacting a cold wall and showed that the higher the impact velocity, the larger is the peak spreading diameter, and increasing the wall impact velocity can enhance the retraction after droplet spreading. Remer et al.⁵¹ investigated the effect of wall wettability on impact freezing and showed that under high-humidity and low-temperature conditions, droplets still do not freeze on superhydrophobic walls. Similarly, Mishchenko et al.⁵² also confirmed that superhydrophobic walls can significantly alter freezing conditions and effectively prevent droplet freezing adhesion. For superhydrophobic cold walls, wall temperature only affects the rebound height after droplet impact, with little influence on the peak spreading coefficient and spreading time of droplets, and superhydrophobicity can effectively suppress the instantaneous freezing deposition of droplets impacting cold walls.^53–55 Chen et al.⁵⁶ studied the dynamic characteristics of droplet impact on nanotube array cold walls, finding that at the same impact velocity, the droplet contact time increases as the temperature of the cold wall decreases, and droplets rebound without freezing when impacting a superhydrophobic cold wall at −18 °C.

Despite the research progress made to date, understanding of multiphase flow, phase-change heat transfer, and interfacial phenomena remains limited owing to the complexity of the droplet impact and freezing process on solid walls, particularly in special cases such as hydrophobic walls, low-temperature environments, and high-speed impact. Current research focuses mainly on icing behavior on flat walls. However, in practical applications, especially in the aviation and energy sectors, curved structures are more common. When droplets impact a curved wall, the presence of curvature affects spreading, retraction, and droplet freezing, making the dynamics more complex. In the present work, droplet impact dynamics and icing characteristics on curved walls are studied. Through visualization experiments, the influence of surface curvature on droplet impact and the freezing process is revealed, including the spreading, retraction, oscillation, and formation process of the ice layer. Furthermore, the effects of different initial conditions and material properties on icing dynamics are discussed, to provide a theoretical basis and technical support for anti-icing/de-icing strategies for curved structures.

Figure 1 shows the experimental visualization system for droplet impact on a curved wall, which includes a micro-injection pump, a sliding rail, a solid curved wall, and a high-speed camera. The micro-injection pump extrudes liquid from the needle at a low speed, and when the liquid’s gravity overcomes the surface tension, the droplet falls freely from the needle tip with a certain initial velocity. The initial droplet diameter can be adjusted by the micro-injection pump. The sliding rail allows for vertical and horizontal adjustment of the droplet’s falling position, horizontal adjustment of the needle position to align with the solid curved surface, and vertical adjustment to change the initial droplet velocity.^57,58 Semiconductor cooling maintains the solid wall surface at the target temperature with an accuracy of ±0.5 °C. During the experiment, a high-speed camera (Olympus i-speed TR) records the dynamic behavior of the droplet impacting the cold solid surface and the freezing process, with a frame rate of 10 000 fps, a resolution of 1024 × 800, and a recording time of 1 s.⁵⁹ The high-speed camera is tilted downward at 10° to observe the complete morphological evolution during the spreading–retraction process of the droplet.

The experimental baseline conditions were as follows: the droplet diameter was 2.1 mm, the wall surface was a solid curved surface (polished copper plate of cross-sectional diameter 9 mm), the wall temperature was −10 °C and the wall surface was hydrophilic (with a static contact angle of 73°). A hydrophobic condition was achieved by using a nano coating (silica particles), with a static contact angle of 119° for the hydrophobic wall surface. Owing to the small size of the droplet, it was treated as approximately spherical. The moment of contact between the droplet and the wall surface was defined as the initial moment, with t representing the time after contact. To minimize the influence of the surrounding environment on the experiment, the room temperature was controlled at (20 ± 0.5 °C) and the relative humidity at (60% ± 4%). To reduce errors and improve the experimental accuracy, each condition was repeated five times. The sliding rail adjustment accuracy was ±0.1 mm, and the precision of the micro-injection pump was ±0.1 μl. The experiment was conducted within a semi-enclosed acrylic chamber to reduce the impact of experimental equipment and the surrounding environment on the results. The experimental working fluid was deionized water, with the specific parameters shown in Table I.

High-speed imaging was used to record the process of droplet impact and freezing, with the captured images being processed using the i-Speed suite software. To facilitate analysis, a dimensionless number, the spreading coefficient β, was introduced. This is defined as the ratio β = D/D₀ of the spreading diameter during the droplet spreading process to the initial diameter of the droplet and characterizes the extent of spreading during the droplet impact process, as shown in Fig. 2. By varying factors such as wall temperature, impact Weber number We = ρv²l/σ (with ρ the density, v the impact velocity, l the droplet diameter, and σ the surface tension), and wall wettability, the effects on the post-impact dynamics and icing characteristics of the droplet are studied.

Temperature is an important factor affecting solid–liquid flow and heat transfer. The dynamics and icing evolution characteristics of droplet impact on solid curved surfaces were studied under two wall temperature conditions. The wall temperatures were 20 and −10 °C, with all other conditions remaining constant. Figures 3 and 4 display the dynamic characteristics of droplet impact under wall temperatures of 20 and −10 °C. From Fig. 3(a), it can be seen that when the wall temperature is 20 °C, at the initial stage of droplet impact on the solid curved surface, the upper part of the droplet still maintains a spherical shape. At the same time, the bottom begins to spread outward, with the kinetic energy of the droplet gradually transforming into surface energy for spreading. At t = 3.2 ms, the droplet spreads to its maximum along the shape of the wall, at which point the droplet morphology is characterized by a thin liquid film in the middle, with regular small liquid beads forming at the edge of the film, which then begins to retract. At t = 13 ms, two small droplets of nearly equal volume are formed on both sides of the liquid film, which continue to retract. By t = 18 ms, the retraction process is complete, but the thickness of the liquid film on both sides is larger, with a smaller thickness in the middle, and this is followed by a slight spreading of the droplet. After two spreading–contraction processes on the solid curved surface, a liquid mass is formed, which stably adheres to the wall surface under the action of surface tension. For the low-temperature wall (−10 °C), the droplet spreads along the wall surface from the impact point, with splashing occurring on both sides during the spreading process, as shown in Fig. 3(b). At t = 4 ms, the droplet has spread to its maximum diameter. As shown in Fig. 4, compared with room temperature, under low-temperature conditions, the spreading process of the droplet is more stable, the fluidity of the droplet is reduced, the flow is slower, the time required to reach the maximum spread is longer, and a phenomenon of droplet splashing on both sides occurs. This is because the liquid viscosity is related to the temperature: the lower the temperature, the stronger are the intermolecular forces between the liquid molecules and the greater is the viscosity. Increased viscosity leads to a slower flow velocity of the droplet on the wall after impact, which in turn leads to droplet splashing, resulting in a spread–splatter mode.

During the spreading process, the bottom liquid film in contact with the wall surface solidifies first. After the droplet has spread to its maximum extent, there is a slight repeat of the spreading–contraction process on the surface of the liquid film. However, as heat is transferred, eventually all the droplets on the wall surface freeze to form a relatively flat and regular thin ice layer that stably adheres to the fiber surface. Thus, under both room-temperature and low-temperature wall conditions, a decrease in temperature will slow down the spreading velocity of the droplet after its impact with the wall. Under high-We conditions, a splashing phenomena will also occur during the impact–spreading process. Moreover, the low-temperature wall will cause the bottom of the droplet to solidify on the surface first, and the retraction of the still-liquid upper droplet is not as obvious as on the room temperature wall; that is low temperature will suppress the oscillatory behavior of the droplet on the wall surface, and the height of the final stable ice film is much less than the height of the liquid film on the room-temperature wall.

The behavior of droplets and their freezing characteristics were investigated for different values of the Weber number We. Figure 5 illustrates the morphological evolution of droplet impact on a solid curved surface for We = 42, 140, and 277. It can be seen that for all these values of We, droplets undergo a process of spreading followed by retraction. When We = 42 [Fig. 5(a)], the droplet spreads slowly, with a small spreading diameter. The droplet reaches its maximum spread at t = 5.9 ms, with no small droplets forming around the periphery of the liquid film. Subsequently, under the action of surface tension, it begins to retract upward. During this process, the liquid film converges into two equally sized droplets on either side, moving toward the center, while the bottom liquid film has already frozen. Eventually, at t = 29.0 ms, the droplets above the wall surface have completely solidified, with the ice layer protruding upward in the center. When We = 140 [Fig. 5(b)], the droplet spreading diameter after impact is greater, and regular small droplets appear at the edge during the spreading process. At t = 5.1 ms, the droplet reaches its maximum spread, before beginning to retract. At t = 7.9 ms, the small droplets at the edge gradually converge to form two equally sized liquid masses on either side of the liquid film, but with a smaller volume than under the previous condition. At low temperatures, the more completely the droplet spreads, the greater is its contact area with the wall, making it more susceptible to freezing, and thus less of the droplet participates in the retraction. The droplet continues to retract while solidification occurs. At t = 23.9 ms, the droplet forms a uniform ice layer on the wall, covering a larger area of the wall with a thinner ice layer. When We = 277 [Fig. 5(c)], the droplet spreads more rapidly. At t = 1.0 ms, splashing of the droplet can be observed, which is caused by the increase in We and the decrease in temperature. At t = 4.8 ms, the droplet reaches its maximum spread. At t = 14.9 ms, owing to the low wall temperature and thin liquid film, the liquid at the edge has already solidified, and the edge of the ice layer is not smooth. As the surface liquid film slowly undergoes spreading and retraction, the droplet eventually forms a thin ice layer adhering to the wall surface.

The effect of We on the maximum spreading morphology of the droplet is shown in Fig. 6, from which it can be seen that the We has a significantly affect on the maximum spreading diameter D_max. For small We, droplets require more time to reach their D_max, the value of which is also smaller. The morphology of the droplet at D_max varies with We: when We = 42, the edge of the droplet is relatively smooth and regular, whereas when We is larger, many small droplets form at the edge of the main droplet. At We = 140, these small edge droplets are comparatively smooth, but as We increases, the liquid film edge becomes more undulating, and when We = 277, the morphology of the small droplets around the main droplet is less regular.

Thus, as We increases, the inertial force of the droplet gradually becomes dominant. After impact, the initial kinetic energy of the droplet is transformed into surface energy, promoting the spreading of the droplet. Consequently, at the same moment, the larger the value of We, the more complete is the spreading of the droplet, the larger is D_max, the faster is the spreading speed, and the shorter is the time required to reach D_max. Additionally, as the fluid velocity increases, heat transfer is accelerated, shortening the freezing time. The repeated spreading–retraction phase is not pronounced, the duration is short, and the final ice layer that is formed has an edge that is less smooth compared with that formed at lower We.

Figure 7 illustrates the variation of the droplet spreading coefficient β with time for a wall temperature T = 20 °C and Weber numbers We = 42, 140, and 277. It can be observed that when We = 42, the droplet spreads slowly between 0 and 3.4 ms, with β reaching a peak of 3.25 at t = 3.4 ms and then decreasing as the droplet immediately begins to retract. After the droplet has retracted to a certain extent, β increases slightly, then decreases, and finally stabilizes at 1.75 at 23.9 ms. When We = 140, β reaches its maximum value of 5 at t = 4.8 ms, and then follows a trend similar to the We = 42 condition, exhibiting a periodic change and eventually stabilizing at 4 at 23.9 ms. When We = 277, owing to the increased initial kinetic energy, the speed of spread of the droplet is significantly faster, as indicated by the steeper slope of the graph, with the spreading coefficient reaching a maximum value of β_max = 5.75 at t = 5.9 ms, after which it begins to decrease, and approaches a constant value of 3.25 at t = 36.6 ms.

Figure 8 represents the variation of the β over time for wall temperature T = −10 °C and We values of 42, 140, and 277, respectively. The trend of the β is consistent across different conditions, with the change in the spreading coefficient over time following a similar mode to that of impacts on the room temperature wall. However, due to the splattering phenomenon occurring after the droplet impact when We = 277, the β reaches its maximum value of 4.67 at t = 4.8 ms, which is lower than the β_max of 5.12 for We = 140.

It can be deduced that in the case of a droplet impacting a room-temperature wall, the larger the value of We, the greater is the droplet spreading velocity, the longer is the time taken to reach the maximum spread, the greater is the value of β_max, and the longer is the time taken for β to reach a constant value. However, in the case of impact on a low-temperature wall, the larger the value of We, the sooner β reaches a constant value. This is because the larger the value of We, the more completely does the droplet spread, and, owing to the lower wall temperature, the greater is the contact area with the wall, as a consequence of which more of the liquid film solidifies first, allowing for faster achievement of a stable state. Additionally, owing to the solidification of the bottom liquid film, the β_max of a droplet for a given We is larger than that in the case of a room-temperature wall. Moreover, because the viscosity of water increases with the temperature decrease when the droplet impacts a low-temperature wall, the spreading velocity of the droplet on the wall surface slows down, and hence the time taken to reach β_max is longer than on a room-temperature wall. Regardless of the wall temperature and how We changes, β will first increase, then decrease, undergo periodic changes, and finally stabilize. This is manifested externally as the droplet spreading and retracting after impacting the wall, then undergoing oscillatory spreading–retraction, and eventually reaching stability.

The effect of wall wettability on the dynamics and icing characteristics of droplet impact on solid curved surfaces was investigated by altering the wall surface properties to hydrophilic and hydrophobic (We = 277). Figure 9 illustrates the morphological evolution of droplets after impact on hydrophilic and hydrophobic walls. From Fig. 9(a), it can be seen that under hydrophilic wall conditions, a droplet spreads more readily, with the formation of irregular small droplets at its edge. At t = 4.8 ms, the droplet reaches its maximum spread, and after the spreading is hindered, it begins to retract, then spreads again, and retracts. However, owing to the lower wall temperature, the bottom of the droplet in contact with the wall after impact solidifies first, as a consequence of which the subsequent spreading–retraction phase is merely an oscillation of the fluid above the liquid film and is not pronounced. Eventually, at t = 33.0 ms, the droplet forms a layer of ice film that adheres stably to the solid surface. From Fig. 9(b), it can be observed that for a hydrophobic wall, the droplets also spreads initially, but small droplets with regular shapes form at its edge during the spreading process, which is different from the behavior on the hydrophilic wall. At t = 4.1 ms, the droplet reaches its maximum spread, at roughly the same time as on the hydrophilic wall. Because the droplet spreading diameter on the hydrophobic wall after impact is smaller than that on the hydrophilic wall, there is more liquid in the upper liquid film that has not frozen, resulting in a larger volume being involved in the retraction oscillation, a longer duration of oscillation, and a more pronounced retraction phenomenon. Similar to the case of the hydrophilic wall, the bottom liquid film solidifies first, the upper liquid film starts to solidify from the edges, and the liquid in the middle slowly solidifies during the oscillation process until the entire liquid film has completed the freezing process, with the ice layer finally stabilizing at t = 36.6 ms.

Thus, under high-We conditions, after droplet impact on a low-temperature solid curved surface, the splashing phenomenon occurs in the cases of both hydrophilic and hydrophobic surfaces, with the ultimate formation of a relatively flat and smooth ice layer that adheres to the solid surface. However, the retraction amplitude of droplets on a hydrophobic surface is larger than that on a hydrophilic surface, the splashing phenomenon is more pronounced, and the time required for freezing is slightly longer.

Figures 10 and 11 illustrate the variation of the droplet spreading coefficient with time for We = 277 and different wall wettabilities when droplets impact room-temperature and low-temperature walls, respectively. As can be seen from Fig. 10, for a room-temperature hydrophilic wall, β increases rapidly within 0–3.2 ms. At t = 5.9 ms, β reaches a peak value of 5.75, then immediately decreases, and finally stabilizes at a value of 4 after 30 ms. For a hydrophobic wall, the initial growth rate of β is essentially the same as that for a hydrophilic wall. However, after β has reached a maximum value of 6 at t = 4.1 ms, its pattern of variation is significantly different: it begins to decrease, then increases again, exhibiting oscillatory behavior, and, additionally, owing to the droplet splitting phenomenon during the retraction phase, the time taken for β to approach a constant value of 2 is longer than in the case of a hydrophilic wall.

As shown in Fig. 11, for droplet impact on a low-temperature hydrophilic wall, within 0–4.8 ms, β continuously increases, reaching a maximum value of 4.67 at t = 4.8 ms, and then decreases to 4.33 at t = 6.8 ms, remaining unchanged thereafter, indicating that the droplet has completely frozen and reached an equilibrium state. In the case of a hydrophobic wall, β also initially increases, then decreases, and finally tends to a constant value. At t = 4.1 ms, β reaches a peak value of 5, then begins to decrease, and finally stabilizes at 4.5. Thus, for a low-temperature wall, the pattern of variation of β with time is similar whether the wall is hydrophobic or hydrophilic surface, with the peak and stable values being relatively close.

Compared with the case of droplet impact on a room-temperature wall, the rate of increase of β for a low-temperature wall is slightly slower, but the time required to reach a stable state is shorter, and the final β is relatively large. Regardless of whether the wall is at room temperature or a low temperature, the β_max of a droplet on a hydrophilic wall is slightly smaller than that on a hydrophobic wall. However, the effect of wall wettability on the growth rate of β during the initial spreading stage of the droplet is not significant.

A visual experimental study of droplet impact on curved walls at room and low temperatures has been carried out, and the dynamic droplet spreading and evolution of icing during impact have been discussed. The effects of impact velocity, wall temperature, and wall wettability on droplet impact icing characteristics have been analyzed. The main conclusions are as follows:

1. When We = 277, droplets impacting both room-temperature (T = 20 °C) and low-temperature (T = −10 °C) hydrophilic walls exhibit a spreading–retracting phenomenon after impact. However, the reduction in temperature inhibits the spreading speed and oscillatory behavior of the droplets. Under high-We conditions, splashing of the droplets also occurs during spreading. In the steady state, the height of the ice film on a cold wall is much lower than the that on a room-temperature wall.

2. We has a significant impact on the dynamics of droplet impact and the freezing characteristics. In the case of droplet impact on a T = −10 °C hydrophilic wall, as We increases from 42 to 277, the droplet inertial force gradually becomes dominant. At a given moment, the larger the value of We, the more complete is the spreading of the droplet, the faster is the spreading speed, and the larger is the value of β_max. The freezing time is shorter, the spreading–retracting repeated phase is not prominent and has a short duration, and the final ice layer formed has an uneven edge.

3. At high We (We = 277), droplets impacting both hydrophilic and hydrophobic cold walls (T = −10 °C) exhibit the splashing phenomenon, and the impact results in the formation of smooth and even ice layers adhering to the solid surface. On a hydrophobic wall, the retracting amplitude after droplet spreading is slightly larger than that on a hydrophilic wall, and the time required for freezing is somewhat longer, but the wall wettability does not significantly affect the droplet behavior or the freezing characteristics.

The findings of this study enrich knowledge of the dynamics of droplet impact and freezing, providing guidance for understanding and improving processes relevant to industrial applications such as spray cooling, coating, and equipment involving droplet freezing. In the future, more detailed research needs to be conducted on the threshold conditions for droplet freezing and the freezing modes of modified surfaces, to reduce droplet adhesion and facilitate the detachment of ice layers.

Support of this work by the Hebei Key Laboratory of Geothermal Energy Utilization Technology is gratefully acknowledged.

The authors have no conflicts to disclose.

Xuanchen Liu: Project administration (equal); Validation (equal); Writing – original draft (equal). Liansheng Liu: Conceptualization (equal); Funding acquisition (equal); Supervision (equal). Ziyi Hu: Software (equal); Visualization (equal); Writing – original draft (equal). Rongji Li: Methodology (equal); Software (equal); Writing – original draft (equal). Ziyue Wang: Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.