Effect of wettability on the impact force of water drops falling on flat solid surfaces

The phenomenon of drops impacting solid surface is extensively observed in rainfall,^1,2 pesticide spraying,^3,4 ink-jet painting,⁵ and spray cooling.⁶ The wettability of the substrate plays a significant role in the dynamics of drop impact on solid surfaces, leading to different outcomes such as deposition, rebounding, or splashing.^7–13 Hydrophilic surfaces prompt spreading and result in larger maximum spreading diameters, which is desirable in spray coating and ink-printing applications. Conversely, superhydrophobic surfaces possess intrinsic features that enable drop repellency, inspiring their applications in anti-icing,¹⁴ self-cleaning,¹⁵ anti-corrosion,¹⁶ and reducing the spread of respiratory disease.¹⁷ The impact force governs drop deformation and can even result in substrate erosion or wetting transitions on superhydrophobic surfaces.^18,19 To gain insights into the underlying physics, it is crucial to investigate the transient impact force of falling drops.

Early studies on the impact forces of falling drops primarily focused on hydrophilic surfaces.^20–27 When a water drop impacts a hydrophilic surface, a distinctive peak with a rapidly increasing stage followed by a relatively slowly decreasing stage can be observed in the impact force curve.^22,23 In the regime dominated by inertia with Re > 200 and We > 68, the peak force is approximately expressed as F ≈ 0.83ρ_lV₀²D₀², which is independent of the Reynolds number Re = ρ_lV₀D₀/μ_l and the Weber number We = ρ_lV₀²D₀/γ, where ρ_l, V₀, and D₀ are the mass density, the initial impact velocity, and the drop diameter, respectively; μ_l is the dynamic viscosity; and γ is the surface tension. In the case of Re < 200, however, viscosity plays a significant role in the impact process, and the normalized peak force F increases with decreasing Re.^24,25 It has been demonstrated that, in the regime dominated by inertia, the normalized temporal evolution of the impact force exhibits self-similarity and collapses into a master curve.^24–26 Furthermore, the initial evolution of the impact force before the peak force follows a square-root scaling law,²⁵ and the later evolution of the impact force after the peak force follows an exponential decay.²⁶ 

In the recent years, there has been growing interest among researchers in studying the impact force of drops falling on superhydrophobic surfaces.^28–31 Zhang et al.²⁸ conducted comprehensive experimental, numerical, and theoretical investigations on the impact force of water drops falling on a superhydrophobic surface with 1 < We < 400 and 800 < Re < 10⁵. They revealed the presence of two prominent peaks in the temporal evolution of the impact force. The first peak force is attributed to the momentum change during the initial impact, while the second arises from the momentum change resulting from the converging flow during rebound, accompanied by a Worthington jet. By analyzing the characteristics of the peak forces and the morphology of the impacting drops, they identified four distinct regimes in terms of We: capillary, singular jet, inertial, and splashing. In the inertial regime, the value of the second peak can be estimated using the recoiling velocity, jet velocity and spreading diameter at the instant of the second peak. Interestingly, they found that while there is no significant difference in the first peaks observed when drops fall on hydrophilic surfaces (with an equivalent contact angle θ = 40° ± 4°) and superhydrophobic surfaces (θ = 165° ± 1°), the second peak only occurs on superhydrophobic surfaces and is nearly absent on hydrophilic surfaces.

Though these previous studies have primarily examined the impact forces of drops falling on hydrophilic and superhydrophobic surfaces, the influence of surface wettability on the impact forces (especially the second peak) has been rarely explored. In this study, we aim to fill this gap and investigate the impact force in relation to surface wettability. We will first describe the experimental and numerical methods. Then, the evolution of the impact force on surfaces with different levels of wettability, along with the corresponding geometric relations, will be illustrated. We perform scaling analysis to rationalize these relations in the inertial regime (30 < We < 100). Finally, we will summarize our main findings.

Figure 1(a) illustrates the schematics of the experimental setup. A water drop with a diameter of D₀ impacts a substrate at an initial velocity of V₀. The normal impact force is measured by a highly sensitive piezoelectric transducer, while the deformation of the drop shape is simultaneously recorded by a high-speed camera in a side-view. The camera is triggered to record the drop morphology when the impact force exceeds 1 mN. For the present experiments, de-ionized water is used, which has the mass density ρ_l = 1000 kg/m³, surface tension γ = 0.073 N/m, and dynamic viscosity μ_l = 1.0 mN m. The drop diameter is D₀ = 2.05 mm, and the falling height is adjusted to vary the impact velocity. A hydrophilic surface (advancing and receding contact angles are 47° ± 2° and 13° ± 2°, respectively) and a superhydrophobic surface (advancing and receding contact angles are 167° ± 2° and 154° ± 2°, respectively) are compared in the experiment. More details of the experimental method can be found in the supplementary material and the work of Zhang et al.²⁸ 

Figure 1(b) illustrates the domain of the simulation, which is axisymmetric with respect to the z axis. The top and outer parts of the domain are set as pressure outlet boundaries, with zero pressure. At the bottom of the domain, a no-slip boundary condition is applied. The drop is initially generated as a sphere with a diameter of 2.05 mm, positioned at 0.1 mm above the substrate. To achieve different Weber numbers, the initial velocity is varied. In the simulations, the physical properties of water are the same as those in the experiments. The mass density and dynamic viscosity of air are set as ρ_a = 1.20 kg/m³ and μ_a = 0.0181 mN m, respectively. Gravitational acceleration is included with a value of 9.81 m/s². Contact angle hysteresis is not considered, and a user-specified contact angle θ is set for each simulation. In other words, the advancing and receding contact angles in each simulation are set to be identical. The first peak of the impact force is independent of the contact angle. The second peak is a result of the redirection of liquid flow during the retraction phase of the drop, and therefore, it is influenced by the receding contact angle rather than the advancing contact angle. To compare the numerical and experimental results depicted in Fig. 2 and verify the accuracy of our simulations, the experimental values of the receding contact angles are assigned to simulate the entire impinging process [i.e., θ = 13° for Figs. 2(a) and 2(c) and θ = 154° for Figs. 2(b) and 2(d)]. It is important to emphasize that in all of our simulations, the assigned contact angle θ corresponds to the receding contact angle expected in real experiments. The simulations are conducted using the OpenFOAM computational fluid dynamics (CFD) software, and further details are given in the supplementary material.

In this section, we first revisit the temporal evolutions of the impact force and shape deformation of drops colliding on hydrophilic and superhydrophobic surfaces to comprehensively understand the impact dynamics. Subsequently, we examine the influence of surface wettability on the two peak forces and their corresponding instants. In addition, we elucidate the mechanisms behind the transient force and drop morphologies with varying contact angles. In this study, our primary focus is to examine the influence of surface wettability on impact forces. Therefore, we specifically investigate drop impacts in the inertial regime (30 < We < 100). Within this regime, the Ohnesorge number (⁠ $O h = μ l / ρ l D 0 γ$⁠) remains constant at 0.0026. Moreover, by estimating the Froude number (⁠ $F r = V 0 / g D 0$⁠) for a drop with a diameter of D₀ = 2.05 mm and an impact velocity of V₀ = 1 m/s, we find Fr = 69. These estimations suggest that the influence of viscous and gravitational forces can be neglected. Here, g represents the acceleration of gravity. Moreover, the influence of contact line pinning is not considered in the simulations.

Figure 2(a) presents both the experimental and numerical results of the temporal evolution of the impact force for a drop with We = 50 (D₀ = 2.05 mm and V₀ = 1.33 m/s) on the hydrophilic surfaces. For this case, Fig. 2(c) illustrates the representative snapshots of the drop shape, with the top row depicting the experimental images and the bottom row displaying the numerical simulations. In each snapshot of the numerical results, the left part gives the pressure field and the right part displays the velocity field. The white velocity vectors with the snapshots indicate the internal flow, with the vector size indicating the magnitude of velocity. During the initial impact, the transient force rapidly increases and reaches the first peak F₁ = 6.16 mN at t₁ = 0.40 ms when the spreading diameter approaches the drop diameter D₀ (as seen in the snapshot at t = 0.40 ms). Subsequently, the impact force gradually decreases as the spreading diameter continues to increase. Throughout the initial impact and the subsequent spreading regime, the velocity vectors shift from the vertical direction to the horizontal. As the spreading diameter reaches its maximum, the impact force approximates 0 mN. Finally, the drop recedes and undergoes oscillation until it settles as a deposited drop on the surface. The receding and oscillation are relatively gentle, and the impact force remains near 0 mN. A notable characteristic observed during drop impact on hydrophilic surfaces is the presence of only one peak force in the force profile.^22–26 

Rich impact dynamic phenomena are observed when a drop collides with a superhydrophobic surface.²⁸  Figure 2(b) illustrates the experimental and numerical impact forces of a drop falling on superhydrophobic surfaces at We = 50 (D₀ = 2.05 mm and V₀ = 1.33 m/s), accompanied by representative drop shapes shown in Fig. 2(d). During the initial impact and subsequent spreading, the temporal evolution of the impact force and drop shape on the superhydrophobic surface is similar to that on the hydrophilic surface. The first peak (F₁ = 6.19 mN at t₁ = 0.40 ms) of the drop on the superhydrophobic surface also undergoes a sudden increase followed by a decrease. Simultaneously, the liquid flux switches from the vertical direction to the horizontal as the spreading diameter approaches its maximum, D_m. At the moment of maximum diameter, t_m, the drop becomes a thin liquid film in the center and thick rims along the edges, as depicted in the numerical snapshot at t = 2.3 ms. Subsequently, the drop retracts due to the capillary force. During the early stage of retraction, the drop maintains the configuration of a thin film bounded by a thick rim, with the impact force remaining close to 0 mN. As the receding mass converges toward the center, the velocity vector shifts from the horizontal direction to the vertical (as observed in the snapshot at t = 4.9 ms), leading to the second peak force F₂ = 2.76 mN at t₂ = 4.85 ms, accompanied by an upward Worthington jet. As the drop rebounds on the substrate, the impact force gradually decreases to 0 mN, and ultimately the drop detaches from the surface. Figure 2 demonstrates that for both the impact processes on hydrophilic and superhydrophobic surfaces, the simulations accurately predict the characteristics of force evolution and drop deformation. Thus, it is reasonable to believe that our numerical simulations can also accurately predict the impact dynamics of drops falling on surfaces with other wettabilities.

Figure 3(a) compares the transient force profiles of drops falling on surfaces with different contact angles ranging from 13° to 180° at We = 50 (D₀ = 2.05 mm and V₀ = 1.33 m/s). It is illustrated that all force profiles exhibit a consistent master curve for the first peak, including the increasing and the decreasing stages, in agreement with previous research.²⁸ The overlapping profiles indicate that the first peak is insensitive to the surface wettability. The notable characteristic of these force profiles is the second peak, which diminishes with decreasing contact angle. At θ = 13°, no second peak is discernible. In the sequel, hence, our focus is on the contact angles ranging from 40° to 180°. Additionally, t₂ increases with decreasing contact angle. These characteristics of the second peak force arise from the effect of surface wettability. During retraction, different wettabilities affect the force balance in the radial direction, resulting in varying retracting velocities of the surrounding rim and subsequent changes in momentum rate (i.e., transient impact force) when the flow converges at the drop center. As depicted in Fig. 3(b), higher contact angles correspond to a higher retraction velocity and increased pressure at the drop center at t₂. Further details regarding the influence of surface wettability on the experimental force profiles can be found in the supplementary material.

Figure 4(a) presents the instants of the first peak, t₁, normalized by the inertial-capillary timescale τ_γ = (ρ_lD₀³/γ)^1/2, as a function of the Weber number for different contact angles. It is illustrated that the normalized instant of the first peak is unaffected by wettability but influenced by the Weber number, i.e., t₁/τ_γ ∼ We^−1/2 (red solid line). This scaling law can be understood by considering that the drop undergoes momentum changes over the vertical length D₀ at velocity V₀, implying t₁ ∼ D₀/V₀. Moreover, Fig. 4(b) illustrates the first peak force F₁, normalized by ρ_lV₀²D₀², as a function of θ for different Weber numbers. It is observed that the normalized peak force remains unaffected by surface wettability and We, i.e., F₁/(ρ_lV₀²D₀²) ≈ 0.81 (black solid line). At t₁, the spreading diameter approximates D₀ without considering the spreading lamella. Thus, the peak force can be estimated by the dynamic pressure ρ_lV₀² over the contact area D₀², i.e., F₁ ∼ ρ_lV₀²D₀² (Refs. 23, 28, and 30).

For drops on surfaces with varying wettabilities, the second peak forces arise from the momentum changes occurring when the horizontally converging liquid redirects vertically. This momentum change is closely related to the kinematics of the impacting drops. To further understand the physical mechanisms behind the second peak force, we will initially examine the kinematics of retracting drops at t₂. Figure 5(a) illustrates the drop morphology on the surface with θ = 80° at t₂, characterized by the spread diameter D₂, drop height h₂ (see the supplementary material for the measurement of h₂), horizontal receding velocity U_t2, jet diameter D_j, and jet velocity U_j. Importantly, it is noted that these physical quantities are specific to the instance of t₂, which distinguishes our work from previous investigations that examined the average values or characteristic values in the entire retracting stage.^11,32–35

In summary, we have combined numerical simulations and scaling arguments to investigate the influence of wettability on the impact force of a falling drop on flat surfaces under various Weber numbers and contact angles. It is revealed that the momentum changes during the initial impact leads to the first peak force F₁ ∼ ρ_lV₀²D₀², and the normalized instant of the first peak follows t₁/τ_γ ∼ We^−1/2, which are insensitive to surface wettability. During the retracting regime, the normalized second peak force can be expressed as $F 2 / ( ρ l V 0 2 D 0 2 ) ∼ U ¯ j U ¯ t 2 / D ¯ 2$⁠, reminiscent of the momentum change in the converging flow from the horizontal direction to the vertical on superhydrophobic surfaces during drop impingement.²⁸ Here, we extend this discovery to a wider range of contact angles (40° < θ < 180°) for inertial impact (30 < We < 100). Furthermore, at the instant of the second peak force t₂, simpler scaling relations closely linked to the surface wettability are established, including t₂/τ_γ ∼ (1 − cos  θ)^−1/2, h₂/D₀ ∼ (1 − cos  θ)^1/2, D₂/D₀ ∼ (1 − cos  θ)^−1/4, and $U t 2 / γ / ( ρ l h 2 ) ∼ ( 1 − cos θ ) 1 / 2$⁠. Notably, these relations remain independent of the Weber number. These findings distinguish our work from previous investigations that primarily focused on the impact force of drop impingement on hydrophilic or superhydrophobic surfaces or the morphological evolution of impinging drops.

It is emphasized that our investigations primarily focus on the inertial regime (30 < We < 100) and rely on scaling arguments to understand the main mechanisms. As a result, other factors such as viscous effect, contact line pinning, gravitational force and the complex inner fluid field within the drop have been ignored in this study, which could account for the deviations observed in some experimental data. To achieve a comprehensive understanding, future research should encompass a more extensive survey considering these impact factors. Furthermore, we have observed that as the contact angle decreases, the magnitude of the second peak force diminishes until it reaches zero. It is plausible that viscosity and gravitational force play important roles.¹⁰ Exploring the threshold of the second peak force in relation to surface wettability and its connection to rebounding is also an interesting topic for future effort. Finally, while Eq. (3.6) effectively characterize the second peak force, our ultimate objective is to express the second peak force as an explicit function of We and θ. This would provide a deeper understanding of the drop impact force, along with the underlying physics.

See the supplementary material for details of the experimental setup, numerical method, mesh size independence, experimental impact force on surfaces with different wettabilities, relationship between Dm and D2, and measurement of the height h2 (PDF).

This study was supported by the National Natural Science Foundation of China (Grant Nos. 11902179, 12172189, 52111540269, and 11921002) and the Tsinghua University Initiative Scientific Research Program (Grant No. 20221080070).

The authors have no conflicts to disclose.

Bin Zhang and Chen Ma contributed equally to this work.

Bin Zhang: Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Chen Ma: Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). Huanlei Zhao: Data curation (equal); Visualization (equal); Writing – review & editing (equal). Yinggang Zhao: Data curation (equal); Validation (equal); Writing – review & editing (equal). Pengfei Hao: Resources (equal); Validation (equal); Writing – review & editing (equal). Xi-Qiao Feng: Funding acquisition (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Cunjing Lv: Conceptualization (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (lead); Resources (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.