Universal Model for the Maximum Spreading Factor of Impacting Nanodroplets: From Hydrophilic to Hydrophobic Surfaces

Spreading dynamics of a droplet upon impact with a liquid film containing solid particles

This study examines how a deionized water droplet behaves when it centrally collides with a liquid film containing TiO2 nanoparticles at low impact velocities, aiming to understand how nanoparticles affect droplet spreading, in particular its maximum spreading diameter. Typically, we found that both the spreading velocity and dynamic contact angle of the droplet would be similarly affected by increasing TiO2 nanoparticle concentration. During retraction, the droplet's dimensionless spreading diameter oscillates, with more pronounced oscillations at higher nanoparticle concentrations. Moreover, both the droplet's maximum dimensionless rebound height and dynamic contact angle show similar trends with increasing TiO2 nanoparticle concentration. Interestingly, we proved that the influence of the solid-liquid interaction (Stokes force) on the fluid during the spreading process accounts for less than 2% of the surface energy when the droplet reaches its maximum spreading diameter, indicating a negligible effect on droplet spreading. We hypothesize that the droplet's initial energy is fully converted into surface energy and viscous dissipation at maximum spreading diameter, which involves viscous dissipation both between the fluid and the solid wall surface and the fluid and solid particle surface. Based on this, we developed a model for predicting the droplet's maximum spreading diameter that includes parameters associated with the solid particles. Compared to models in the literature that do not consider the effect of solid particles, our model aligns more closely with experimental data. The results indicate that adding solid particles leads to increased viscous dissipation, which in turn reduces the droplet's maximum spreading diameter.

What Controls the Hole Formation of Nanodroplets: Hydrodynamic or Thermodynamic Instability?

Using molecular dynamics simulations, we investigate the air hole formation of water nanodroplets impacting hydrophilic to hydrophobic surfaces in the range of static contact angles from 30° to 140° with different initial surface temperatures ranging from 300 to 1000 K. We show that the hole dynamics of nanodroplets are different from those observed in millimeter-sized droplets. The hole formation can be observed on smooth surfaces for nanodroplets; however, it only occurs on nonsmooth surfaces for millimeter-sized droplets. We clarify that the hole formation of nanodroplets is triggered by a nucleated vapor bubble due to thermodynamic instability, whereas it is initiated by air bubble entrapment during impact due to hydrodynamic instability for millimeter-sized droplets. The hole formation of nanodroplets relies heavily on the surface temperature and surface wettability, because the nucleated vapor bubble more easily occurs and grows on the surface with high initial temperatures and hydrophobic surfaces. Based on the thermal stability analysis, a criterion is developed to predict the hole formation of nanodroplets, which verifies the dependence of hole formation on the surface temperature and wettability. Furthermore, we show that the ring-bouncing of nanodroplets is triggered by the nucleated vapor bubble. We clarify the reasons for the reduced contact time of nanodroplets caused by the ring-bouncing.

Diameter-Optimum Spreading for the Impinging of Water Nanodroplets on Solid Surfaces

The impinging of water nanodroplets on solid surfaces is crucial to many nanotechnologies. Through large-scale molecular dynamics simulations, the size effect on the spreading of water nanodroplets after impinging on hydrophilic, graphite, and hydrophobic surfaces under low impinging velocities has been systematically studied. The spreading rates of nanodroplets first increase and then decrease and gradually become constant with the increase of nanodroplet diameter. The nanodroplets with the diameters of 17-19 nm possess the highest spreading rates because of the combined effect of the strongest interfacial interaction and the strongest surface interaction within water molecules. The highest water molecule densities, hydrogen bond numbers, and dielectric constants of interface and surface layers mainly contribute to the lowest interface work of adhesion and surface tension values at optimal diameters. These results unveil the nonmonotonic characteristics of spreading velocity, interface work of adhesion and surface tension with nanodroplet diameter for nanodroplets on solid surfaces.

Impact of nanodroplets on cone-textured surfaces

Molecular dynamics simulations have been performed to study the dynamics of nanodroplets impacting on a flat superhydrophobic surface and surfaces covered with nanocone structures. We present a panorama of nanodroplet behaviors for a wide range of impact velocities and different cone geometrics, and develop a model to predict whether a nanodroplet impacting onto cone-textured surfaces will touch the underlying substrate during impact. The advantages and disadvantages of applying nanocone structures to the solid surface are revealed by the investigations into restitution coefficient and contact time. The effects of nanocone structures on droplet bouncing dynamics are probed using momentum analysis rather than conventional energy analysis. We further demonstrate that a single Weber number is inadequate for unifying the dynamics of macroscale and nanoscale droplets on cone-textured surfaces, and propose a combined dimensionless number to address it. The extensive findings of this study carry noteworthy implications for engineering applications, such as nanoprinting and nanomedicine on functional patterned surfaces, providing fundamental support for these technologies.

Bouncing dynamics of droplets on nanopillar-arrayed surfaces: the effect of impact position

The impact behaviors of droplets on nanostructure-arrayed surfaces are ubiquitous in nature and engineering applications. And the influence of the impact position of a droplet on its bouncing dynamics is of great significance since there is inevitable randomness in the impact position of droplets on the surface of nanostructured arrays, but the difference in the dynamics process caused by this randomness has not been recognized. Here, by using molecular dynamics simulations, the effect of impact position on the bouncing dynamics of a water droplet on nanopillar-arrayed surfaces is systematically investigated. The simulation results highlight that the impact position plays an important role in droplet dynamics after impact, especially at the retraction stage, and the effect of impact position on the bouncing behavior is highly sensitive to the impact velocity. Importantly, from the point of energy conversion, the droplet deformation and contact state jointly determine whether the droplet can bounce back or not, which reveals the mechanism of impact position effects on the bouncing behavior of the droplet. Interestingly, the effect of impact position would be weakened with an increase in the size ratio of the droplet diameter to nanopillar spacing, and this effect becomes negligible when the size ratio is greater than 5.2. These findings demonstrate the key role played by the impact position and may provide new insights into the practical application of nanostructure-arrayed surfaces.

Oblique Impacts of Nanodroplets upon Surfaces

In this work, oblique impacts of nanodroplets impacting surfaces in a wide range of impact angles (α) are investigated in detail via molecular dynamics simulations. Five outcomes are observed, including deposition, prompt splashing, break-up, separation, and shattering. With increasing impact angle, the outcomes of prompt splashing, break-up, separation, and shattering are enlarged but the one of deposition is compressed. By drawing a We_n ∼ α phase diagram, the outcome regimes and corresponding boundaries of them can be successfully identified, and the boundary between the deposition and other outcome regimes is theoretically modeled and shows good agreement with the phase diagram, where We_n is the normal impact Weber number. For further understanding of the oblique impacts, the maximum spreading factor, as the feature parameter of spreading, is investigated. Asymmetry spreading behaviors are observed, noting that β_max,∥ is always larger than β_max,⊥. β_max,⊥ is tested that it only depends on We_n with wide impact angles and could be predicted by the scaling law of β_max,⊥ = 0.7We_n^1/4. However, β_max,∥ depends on not only We_n but also impact angles. A modified model is proposed for predicting β_max,∥ as 0.7We_n^1/4 + 0.001(We_n tan² α)^3/2, which shows good agreement with data on surfaces with θ from 73 to 105° in wide We_n and α ranges.

A Conservative Level Set Approach to Non-Spherical Drop Impact in Three Dimensions

A recently developed conservative level set model, coupled with the Navier-Stokes equations, was invoked to simulate non-spherical droplet impact in three dimensions. The advection term in the conservative level set model was tackled using the traditional central difference scheme on a half-staggered grid. The pressure velocity coupling was decoupled using the projection method. The inhouse code was written in Fortran and was run with the aid of the shared memory parallelism, OpenMP. Before conducting extensive simulations, the model was tested on meshes of varied resolutions and validated against experimental works, with satisfyingly qualitative and quantitative agreement obtained. The model was then employed to predict the impact and splashing dynamics of non-spherical droplets, with the focus on the effect of the aspect ratio. An empirical correlation of the maximum spread factor was proposed. Besides, the number of satellite droplets when splashing occurs was in reasonable agreement with a theoretical model.

Effects of Nanodroplet Sizes on Wettability, Electrowetting Transition, and Spontaneous Dewetting Transition on Nanopillar-Arrayed Surfaces

In this study, the wetting and dewetting behaviors of water nanodroplets containing various molecule numbers on nanopillar-arrayed surfaces in the presence or absence of an external electric field are investigated via molecular dynamics (MD) simulations, aiming to examine whether there is a scale effect. The results show that, in the absence of an electric field, nanodroplets on coexisting Cassie/Wenzel surfaces may be in the Cassie or the Wenzel state depending on their initial states, and apparent contact angles of the Cassie or Wenzel nanodroplets increase monotonously with increasing the droplet size. Energy analysis shows that on the same coexisting Cassie/Wenzel surface, when an electric field is imposed, a small nanodroplet possesses a lower energy barrier separating the Cassie state from the Wenzel state. Therefore, the small nanodroplet is easier to collapse into the Wenzel state. Moreover, the spontaneous Wenzel-to-Cassie dewetting transition is not observed for the nanodroplets after the removal of the electric field because the Wenzel state is a globally stable energetic state. With the same pillar geometry, both the wetting transition and the dewetting transition are significantly modified for liquids with higher intrinsic contact angles. The energy barrier of the wetting transition increases for both the large and small nanodroplets, meaning that the Cassie state becomes more robust. The energy curve shows that the Wenzel state of the large nanodroplet has higher energy so that the droplet can return to the Cassie state when removing the electric field. Intriguingly, although the small Wenzel nanodroplet has lower energy in the presence of the electric field, the dewetting transition still occurs. The increased solid-liquid interfacial tension when removing the electric field is responsible for this abnormal result. The wetting and dewetting transitions follow different energy pathways, leading to a hysteresis energy loop. There exists a critical water molecule number separating the unstable/stable Wenzel configurations, above which the Cassie state is energetically favorable and the dewetting transition can occur spontaneously after removing the electric field.

Drop Impact on a Superhydrophilic Spot Surrounded by a Superhydrophobic Surface

The spatial variation in the wettability of a surface can have a significant effect on the spreading and retraction behavior of an impacting droplet and hence the overall impact dynamics. Although composite surfaces have proven applications, there is a lack of understanding of droplet impact on surfaces with a sudden jump in wettability. Here, we study the behavior of a liquid drop impacting a composite surface having a superhydrophilic (SHL) spot surrounded by a superhydrophobic (SHB) region. We find that the droplet exhibits different regimes: no-splitting, jetting, and splashing, depending upon the spot size (β_s) and the Weber number (We). At a smaller β_s, the behavior shifts from the stable to jetting regime and then to the splashing regime, with increasing We. We find that by increasing the value of β_s, one can avoid the undesirable splashing and jetting regimes and attain a stable regime even at a higher We. Our study reveals that β_s has a significant influence on the maximum spreading diameter β_max at a smaller We but a negligible effect at a higher We. We show that the dominance of capillary energy at a smaller We and viscous energy at a higher We underpins the phenomena. We employ an energy conservation approach to develop an analytical model to predict β_max on a composite SHL-SHB surface by considering the total energy of the system before the impact and at the maximum spread position. We find K = (Re^1/2/We) emerges as a key parameter in the model that accurately predicts the experimentally measured β_max. Our study reveals the existence of an inertia-viscous dominated regime at a smaller K and an inertia-capillary dominated regime at a larger K. The outcome of our study may find applications in stable and precise positioning of impacting droplets.

Head-on Collision of Two Nanodroplets on a Solid Surface: A Molecular Dynamics Simulation Study

Most researchers focus on the collision of a single droplet with a solid surface, while it is common for a droplet to collide with a sessile droplet on a solid surface in reality. This study performed the head-on collision of two nanodroplets on a solid surface using the molecular dynamics simulation method. The effects of impact velocity, interaction intensity between solid and liquid atoms, and the solid fraction of the surface on the collision process are studied with independent simulation cases. The maximum spreading factor and the dimensionless maximum spreading time are recorded and calculated to describe the collision process quantitatively. The simulation results indicate that the maximum spreading factor depends more on the solid fraction than the interaction intensity since it does not fundamentally change the wetting state of the droplet at its maximum spreading state. Because of two different effects, the maximum dimensionless spreading time decreases first and then increases with the interaction intensity, and both effects weaken with the increase of impact velocity. As the solid fraction increases, the maximum spreading factor increases significantly at high impact velocity, and the maximum dimensionless spreading time first decreases and then increases because the wetting state of the coalescent droplet at the maximum spreading moment gradually changes from the Wenzel state to the Cassie state. In general, the initial wetting state of the sessile droplet and the wetting state of the coalescent droplet at the maximum spreading moment have important effects on the maximum spreading factor and the maximum spreading time. We establish a theoretical prediction model for the maximum spreading factor on a smooth surface based on energy conservation with quite good accuracy. This research has improved our understanding of the head-on collision process of two nanodroplets on a solid surface.

Rebound Behaviors of Multiple Droplets Simultaneously Impacting a Superhydrophobic Surface

The rebound behaviors of multiple droplets simultaneously impacting a superhydrophobic surface were investigated via lattice Boltzmann method (LBM) simulations. Three rebound regions were identified, i.e., an edge-dominating region, a center-dominating region, and an independent rebound region. The occurrence of the rebound regions strongly depends on the droplet spacing and the associated Weber and Reynolds numbers. Three new rebound morphologies, i.e., a pin-shaped morphology, a downward comb-shaped morphology, and an upward comb-shaped morphology, were presented. Intriguingly, in the edge-dominating region, the central droplets experience a secondary wetting process to significantly prolong the contact time. However, in the center-dominating region, the contact time is dramatically shortened because of the strong interactions generated by the central droplets and the central ridges. These findings provide useful information for practical applications such as self-cleaning, anticorrosion, anti-icing, and so forth.

Analytical Consideration for the Maximum Spreading Factor of Liquid Droplet Impact on a Smooth Solid Surface

Based on the energy conservation approach, this study develops a universal model to predict the maximum spreading factor of liquid droplet impact on a smooth solid surface. Validated with the present simulations and experiments in the literature, this model effectively overcomes the limitation of previous models in the viscous regime and greatly reduces the computing errors from over 30% to below 6%. It is demonstrated that the underestimated maximum spreading factor by previous models results from the overestimation of viscous dissipation. By replacing the conventional model of spreading time, t_m = 8D₀/3U₀, with a more precise one, t_m = 1.47τ_iWe^-0.44, the formulation to compute the viscous dissipation of entire spreading is improved. Finally, we examine the applicability of present model in the capillary regime and good performance is also shown.

Viscoelasticity and elastocapillarity effects in the impact of drops on a repellent surface

We investigate freely expanding viscoelastic sheets. The sheets are produced by the impact of drops on a quartz plate covered with a thin layer of liquid nitrogen that suppresses shear viscous dissipation as a result of the cold Leidenfrost effect. The time evolution of the sheet is simultaneously recorded from top and side views using high-speed cameras. The investigated viscoelastic fluids are Maxwell fluids, which are characterized by low elastic moduli, and relaxation times that vary over almost two orders of magnitude, thus giving access to a large spectrum of viscoelastic and elastocapillary effects. For the purposes of comparison, Newtonian fluids, with viscosity varying over three orders of magnitude, are also investigated. In this study, dmax, the maximal expansion of the sheets, and tmax the time to reach this maximal expansion from the time at impact, are measured as a function of the impact velocity. By using a generalized damped harmonic oscillator model, we rationalize the role of capillarity, bulk elasticity and viscous dissipation in the expansion dynamics of all investigated samples. In the model, the spring constant is a combination of the surface tension and the bulk dynamic elastic modulus. The time-varying damping coefficient is associated to biaxial extensional viscous dissipation and is proportional to the dynamic loss modulus. For all samples, we find that the model reproduces accurately the experimental data for dmax and tmax.

Spreading Time of Impacting Nanodroplets

The kinematic time and maximum spreading time for the impact of nanodroplets of different types of fluids on solid surfaces with different wettability are investigated. It shows that the capillary regime still exits for the nanodroplet impact, even if viscous dissipation increases significantly when the droplet size reduces to the nanoscale. By taking into account the influence of liquid types and surface wettability, we first obtain scaling laws of the maximum spreading time for the capillary and viscous regimes. We further propose a universal scaling law by interpolating the scaling laws in the two asymptotic regimes. The universal scaling law is in excellent agreement with molecular dynamics simulations for various liquids and surface wettability.