Physics of droplet impact on various substrates and its current advancements in interfacial science: A review

Droplet impact is a physical phenomenon visible in everyday life, such as in the fall of raindrops on the ground and substrates, the fingering of an ink droplet, or a coffee stain. Droplet–solid surface interaction has been a topic of great interest in the diverse areas of science and technology such as medicine, aerospace, electronics, energy, and materials. When a liquid droplet interacts with a solid surface, the balance of several physical and external forces governs the dynamics of the liquid droplet motion on the surface. The physical forces are interfacial tension forces across solid–gas, solid–liquid, and liquid–gas interfaces, gravity due to the weight of the droplet and viscous forces (Mohammad Karim and Kavehpour, 2014; Meier 2000; Yilbas 2018; Bostwick 2014; Abubakar 2022; and Mohammad Karim, 2015).

Viscous forces cause resistance to the motion of the liquid droplet on the surface effected by the viscous dissipation phenomenon. Interfacial tension forces result in the stiffness of the liquid droplet in contact with the surface. Gravity leads to liquid droplet puddling and is attributed to the droplet size and the contact angle. Droplet puddling could be neglected as the gravity is approximated to be negligible since the droplet size is less than the capillary length $( σ / ( ρ g ) )$⁠, which incorporates the droplet density $(ρ)$⁠, the droplet surface tension $(σ)$⁠, and the gravitational acceleration $(g)$⁠. Besides physical forces, external forces might play key roles in droplet dynamics due to contact with a surface (solid or liquid). These external forces can be formed due to acoustic waves, the electric field, the magnetic field, dynamic pressure, and the geometric specifications of the microfluidic and nano-fluidic devices, as well as the geometric specifications of the meshed solid surface.

Droplet interaction inside another immiscible liquid also with a solid boundary has been widely applied in industries such as food processing, inkjet printing, microfluidics, and pharmaceuticals (Wibowo and Ng, 2001; Skurtys and Aguilera, 2008; Singh 2010; Zheng 2003; Suea-Ngam 2015; Singh 2020; Pruppacher and Klett, 1997; Kavehpour, 2015; Jin 2017; Choi 2012; Mashaghi 2016; Nguyen 2017; Song 2006; Teh ., 2008; Weigl 2003; Tao and Chakrabarty, 2007; and Mohammad Karim, 2019).

Droplet impact plays a significant role in a vast number of natural and technological processes such as various coating methods and printing techniques of functional and advanced smart materials (Wijshoff, 2018; Mohammad Karim 2018a, 2021a, 2021b, 2021c; and Mohammad Karim and Suszynski, 2022). Droplet impact dynamics is prominent in the dispersion and spreading of airborne viral droplets (Joung and Buie, 2015; and Bourouiba, 2021).

One of the main current notable roles of droplet impact physics research in the present pandemic time is the focus on obtaining the most effective way to fight against the spread of COVID-19. Face masks play key roles to fight against the spread of respiratory virus-laden droplets in public. This goal can be achieved by thoroughly understanding the interactions of the face masks with the incoming aerosol-based droplets (Melayil and Mitra, 2021).

There have been one and a half centuries of research in the physics of droplet impact. However, a profound understanding was not reached until two decades ago when high-speed imaging techniques enabled the unraveling of this fast physical phenomenon (Thoroddsen 2008; and Worthington, 1876). Therefore, due to the high-speed imaging and computational fluid dynamics, the study of droplet impact kinematics has remarkably increased.

Significant research in the dynamics of the droplet impact has only started to progress over the last decade. Numerous parameters including force, drag, pressure, and shear stress distributions in the event of droplet impact on a solid surface have been thoroughly studied via extensive experimental, computational, and theoretical efforts. Many factors that affect the dynamics of the droplet impact on a solid surface in various situations have also been investigated: the presence of water layers on the solid surface, air cushioning, non-spherical droplet geometry, and granular impact cratering via droplet falling (Cheng 2022; Liu 2014a; and 2014b).

The physics of droplet impact depends on the surface on which the droplet hits: solid and liquid surfaces. Predominantly, the major forces that govern the physics of the droplet impact onto a surface (solid or liquid) are inertial force, viscous force, interfacial tension forces, and gravity (Li 2021). These forces pertinently characterize the droplet impact dynamics via some non-dimensional parameters including the Reynolds number $( R e)$⁠, the Weber number $( W e)$⁠, the Ohnesorge number $( O h)$⁠, the capillary number $( C a)$⁠, the Stokes number $( S t)$⁠, the Bond number $( B o)$⁠, and the Froude number $( F r)$ (Josserand and Thoroddsen, 2016; and Moghtadernejad 2020).

Moreover, there are some additional non-dimensional parameters that could facilitate the characterization of the droplet impact dynamics: the non-dimensional time parameter $( t ∗ = U t D )$⁠, the spreading ratio $( S = d D )$⁠, the Prandtl number $( P r = μ c k T )$⁠, the Stefan number $( S t ∗ = c Δ T h e v a p )$⁠, the ratio of the thermal diffusivity of liquid and solid $( R d i f = ρ c k T ρ s c s k T , s )$⁠, the ratio of specific heat capacities $( R h c = ρ c ρ s c s )$⁠, the ratio of the total heat transferred to the droplet over the maximal heat transfer $( E h e a t ∗ = 6 Q h e a t π ρ D 3 h e v a p )$⁠, and the efficiency of evaporation $( E e v a p ∗ = 6 m e v a p π ρ D 3 )$⁠, where t denotes the time, d is the diameter of the droplet spreading, c is the droplet specific heat, k_T is the droplet thermal conductivity, $ΔT$ is the temperature change, $h e v a p$ represents the enthalpy of evaporation, $ρ s$ is the solid density, $c s$ is the solid specific heat, $k T , s$ represents the solid thermal conductivity, $Q h e a t$ is the total heat transfer to the droplet, and $m e v a p$ represents the droplet mass that is evaporated in the heat transfer process (Josserand and Thoroddsen, 2016; and Moghtadernejad 2020).

There can be more dimensionless parameters in droplet physics depending on the external forces including the electric field, the magnetic field, vibration, wind, and obstacles. Moreover, depending on the case, more non-dimensional numbers can be introduced in droplet impact physics due to chemical impurities and topographical defects on the solid surface, droplet shape, and droplet contact angle with the solid surface at the onset of impact, as well as the patterned solid surface (Josserand and Thoroddsen, 2016).

Droplet impact on solid surfaces can be divided into two categories: the dry surface and the pre-wetted surface. Experimental studies of droplet impact on dry solid surfaces have reported six physical plausible phenomena: spreading and deposition, receding and breakup, prompt splashing, corona splashing, partial rebounding, and complete rebounding as demonstrated in Fig. 1 (Yarin, 2006; Rein, 1993; Josserand and Thoroddsen, 2016; Marengo 2011; Wang 2018; Bange 2018; Guo 2014; Malla , 2017; Weisensee 2016; Zhang 2021; and Burzynski 2020). The complete rebounding is found to be highly possible for very high impact speed and both hydrophobic and superhydrophobic surfaces (Yarin, 2006; Quéré, 2005; Rioboo 2001; and Bartolo 2005). The physical behavior of the droplet after impact onto a solid surface significantly depends on the droplet's physical properties, the wettability characteristics of the solid surface, and environmental conditions (Boyer 2016; Aytouna 2010; Izbassarov and Muradoglu, 2016; Chen 2010; Deng 2013; Gauthier 2015; Xu 2005; Schutzius 2015; and Antonini 2013).

The physics of droplet–substrate interactions, after impact, shows significance in numerous applications in the current society such as pharmaceuticals, the fight against COVID-19 pandemic due to the spread of virus-laden droplets in the community, energy performance, nano-printing, bio-printing, 3D-printing, tissue engineering, the generation of smart biomaterials and functional organs, advanced flexible electronics, and high resolution additive manufacturing in healthcare and medicine. Therefore, it is critical to concentrate the focus on the physics of such phenomena, which can be very complex for droplets, substrates, and the environmental conditions.

This Review presents the previous studies of the droplet–solid surface interactions for various substrates ranging from rigid and smooth dry to flexible surfaces and from hydrophilic to superhydrophobic states. Moreover, the current advancements in the applications of this physical phenomenon are thoroughly discussed for bioprinting, smart biomaterials, printed electronics, and the struggle against COVID-19 pandemic. Furthermore, the application of machine learning for deeper physical understanding of droplet–solid surface interactions, upon impact, is also discussed. Finally, future directions of research in these physical phenomena are provided, considering the complexity of droplets, substrates, and environmental conditions, along with the current up to date advancements in technology including nanoscience, machine learning, quantum computation, and imaging techniques, such as electron microscopy.

For droplet impact on rough solid surfaces, which are patterned with posts, three widely accepted theoretical models have been proposed. These models describe the droplet configuration on the rough solid surfaces, as sketched in Fig. 2. These well-known models are Wenzel, Cassie–Baxter, and mixed wetting state as shown in Fig. 2 (Wenzel, 1936; Cassie and Baxter, 1944; and Cassie, 1948).

It is important to note that numerous wetting conditions might coexist on materials with real heterogeneity features. The variety of contact angles would be possible to recognize when the Gibbs free energy of the solid–liquid–vapor system presents multiple minimum levels (Wang et al., 2022).

As two or more nearby droplets come into contact, the droplets’ merging phenomenon happens, which is governed by a local pressure imbalance. Under this condition, droplet surface energy transforms into viscous energy dissipation and kinetic energy. Therefore, a droplet jumping event driven by multi-droplet merging is characterized in four steps (Liu 2014a; Wang 2020a; and Wasserfall 2017): a liquid bridge merges and extends (step I), speed of multi-droplet merging increases due to the presence of resistance from a non-wetting solid surface such as hydrophobic and superhydrophobic surfaces (step II), coalesced droplet jumps off the surface (step III), and finally the jumping velocity of coalesced droplet decreases (step IV). Coalescence-induced droplet jumping dynamics from a solid surface is illustrated in Fig. 2(g). Capillary, viscous, and inertia forces play key roles in characterizing the coalescence-induced droplet jumping dynamics (Wang et al., 2022; Eggers 1999; and Joseph 2012).

Droplet impact on dry smooth solid surfaces is a natural well-known phenomenon. Moreover, it has many applications in scientific and technological areas including printing, coatings, additive manufacturing, and deicing of airplanes (Yarin, 2006; Gent 2000; Bansmer, 2020; and Burzynski, 2021).

The consequence of droplet impact on a smooth dry solid surface can be characterized by two events: deposition–spreading and splashing, as sketched in Fig. 3. Splashing is a complex dynamic event that is additionally categorized into two types: corona splash and prompt splash, as demonstrated in Fig. 3. The droplet impact on a dry smooth solid surface depends on numerous factors including droplet size, impact velocity, the physical properties of the droplet, and the physical conditions of the environment such as temperature and pressure.

After impact, the droplet deposits and spreads onto the nonwetting and wetting regions of the solid surface until the kinetic energy of the droplet partly transforms to surface energy and the rest dissipates due to the viscous effect. On the hydrophobic area of the solid surface, the droplet recedes as compared to the hydrophilic region on which the droplet spreads. The droplet spreading mechanism depends on the wettability of the solid surface. Moreover, the fingering instability of the droplet rim, which happens upon impact on the smooth solid surface, depends on the wettability of the solid surface. A previous experimental study has demonstrated the comparison between droplet impact on hydrophilic and hydrophobic surfaces for water droplet impacting on a glass surface with wettability patterns, which generate hydrophilic and hydrophobic regions (Lee 2010).

Šikalo (2005) conducted experimental and numerical studies on single droplet impact dynamics onto smooth dry and partially wettable solid surfaces. They examined the droplet spreading parameter and the dynamic contact angle during droplet spreading dynamics after impact and considered the static contact angle on droplet impact dynamics (Šikalo 2005). They compared the observations and the corresponding simulations for the dynamics of water droplet impacting onto a wax surface during spreading and retraction mechanism (Šikalo 2005). They reported that empirical spreading dynamics models proposed by the Hoffman–Voinov–Tanner law (Hoffman, 1975; Voinov, 1976; and Tanner, 1979) do not reasonably describe the droplet spreading dynamics after its impact with a smooth dry solid surface due to the large capillary numbers (Šikalo 2005).

In their numerical study, they removed the hydrodynamic singularity near the moving contact line and instead considered a local force that depends on the liquid contact line velocity after droplet impact onto the solid surface (Šikalo 2005). They reported the successful validation of the simulations with the observations for predicting the droplet spreading parameter and the dynamic contact angle during droplet spreading and retraction stages (Šikalo , 2005).

Most studies in droplet impact physics onto solid surfaces have focused on the physics of droplet bouncing off from superhydrophobic surfaces. There has been less attention in the droplet bouncing mechanism off from hydrophilic and hydrophobic surfaces (Couder 2005a; and Couder 2005b).

Guan (2021) experimentally and computationally investigated the droplet bouncing off from an inclined hydrophilic solid surface. They utilized an open-source computational fluid dynamics (CFD) code to carry out the simulations for droplet rebound characteristics from an inclined hydrophilic surface. They compared the simulations with the observations of sodium dodecyl sulfate (SDS) in a de-ionized (DI) water solution droplet impacting onto the 60° inclined hydrophilic smooth solid surface. In their simulation, they considered several parameters including the droplet impact velocity, the droplet size, the static contact angle, the inclination angle, contact time, spreading time to characterize the maximum spreading parameter, and droplet motion before the onset of droplet rebound. Sahoo (2020) conducted similar studies and found that the maximum spreading parameter and spreading time increase with the inclination angle and the normal Weber number.

Various numerical studies have been attempted to study droplet impact dynamics on smooth solid surfaces (Wu and Cao, 2017). To validate the numerical models, Wu and Cao (2017) compared their simulations with their observations. They simulated the onset and development dynamics of splashing of an infinite cylindrical droplet on a smooth dry solid surface, as represented in Fig. 4(a). They applied the two-dimensional volume-of-liquid (VOF) algorithm technique (Fluent I, 2006; Davidson, 2002; Josserand and Zaleski, 2003; Schroll 2010; Coppola 2011; Josserand 2015; and Mahady 2015) for the simulation of droplet impact dynamics and splashing mechanism.

Their numerical analyses demonstrated the formation of secondary droplets in the early step of droplet spreading and evolution of splashing afterwards (Wu and Cao, 2017). Thereafter, the droplets broke up into more tertiary droplets, as illustrated in Fig. 4(a). In their simulation, they considered the shear viscosity and the surface tension in the Navier–Stokes equations alongside the general piecewise-linearly interpolated liquid–vapor interface (Wu and Cao, 2017; Fluent I, 2006; Davidson, 2002; Josserand and Zaleski, 2003; Schroll 2010; Coppola 2011; Josserand 2015; and Mahady 2015).

Their simulations were compared with experimental findings for the impact dynamics of an ethanol droplet on a dry smooth substrate (Xu 2005; and Wu and Cao, 2017). These comparisons indicated that the pressure of the gas phase has a key role on splashing dynamics (Wu and Cao, 2017).

Guan (2021) have performed a numerical study on various outcomes of the droplet impact dynamics including droplet bouncing, droplet breakup, splashing, sliding, and deposition on inclined solid surfaces with various water instability features and inclination angles. They considered different droplet impact conditions in terms of the droplet size and the static contact angle (Guan 2021). Their simulations consisted of sodium dodecyl sulfate solution droplet impact onto a hydrophilic solid surface and DI water droplet impact onto a hydrophobic solid surface. Their simulations are in close strong agreement with the experimental findings obtained from the previous study by Sahoo (2020).

Gunjal (2005) examined the mechanisms of droplet spreading, recoiling, and rebounding after impact onto smooth flat horizontal glass substrates via experimental observations with high-speed digital camera along with the computational fluid dynamics (CFD) model based on the volume-of-fluid (VOF) approach. Experimental studies were conducted for various Reynolds numbers $( 550 ≤ R e ≤ 2500)$ and Weber numbers $( 2 ≤ W e ≤ 20)$⁠. VOF-based numerical analysis characterized the droplet–glass interaction and provided insights on the droplet–glass drag forces. Their simulations demonstrate strong agreements with the experimental findings [(Fig. 4(b)].

Droplet impact onto cold solid surfaces is ubiquitous in nature and plays a key role in numerous industrial applications. Therefore, recent studies have focused on examining the role of the subcooling temperature of the solid surface on the droplet impact dynamics. Shang (2020) have experimentally studied the water droplet spreading characteristics after impact onto a smooth hydrophilic silicon surface by determining the maximum spreading parameter for the various degrees of subcooling of the silicon surface.

They considered two ranges of the Weber number: the low Weber number region and the high Weber number region (Shang 2020). They report that at the low Weber number region, the maximum spreading parameter decreases with a rise in the degree of solid surface subcooling through stronger viscous dissipation and larger capillary effect; in contrast at the high Weber number region, the maximum spreading parameter drops and then rises with an increase in the degree of solid surface subcooling (Shang 2020).

Due to fuel droplet impact on the heated surfaces, droplet breakup generates a vast number of secondary tiny droplets due to the micro-explosion process, which improves the air–fuel mixing performance in the transpiration purposes (Cen 2019). Cen (2019) have investigated the features of sputtering and the micro-explosion caused by the emulsion fuel droplets, which impact on the rough heated stainless steel surfaces for various water percentages (5% –20%) in the emulsion droplets and wide range of the heated surface temperatures (100–320 °C).

They experimentally studied the impact dynamics of the water/diesel emulsion fuel droplets on the heated stainless steel surfaces with the surface roughness of 0.4 μm using the high-speed visualization technique (Cen 2019). They found that a higher water content in the emulsion droplets and higher substrate temperature would promote sputtering and the micro-explosion in the droplet impact on the heated surface (Cen 2019). They reported that the air bubble formation happens at a specific surface temperature unless no air bubble is present during the pure emulsion fuel droplet impact on the heated surface in which no water content is present in the fuel droplet (Cen 2019).

Leidenfrost is a physical phenomenon that occurs when a droplet is brought into contact with a very hot solid surface whose temperature is larger than the boiling point of the droplet. In this condition, the droplet levitates on a vapor layer, which is formed between the droplet and the hot solid surface. The absence of the solid–liquid contact causes the free frictionless motion of the droplet on the hot solid surface (Quéré, 2013; Avedisian and Koplik, 1987; and Bernardin and Mudawar, 1999). Some studies were conducted on specifying the Leidenfrost temperature on various solid including hydrophilic and hydrophobic surfaces (Liu and Craig, 2010). Some studies were attempted on Leidenfrost vapor layer stability on patterned superhydrophobic surfaces (Vakarelski 2012).

Li (2022) have studied on the impact dynamics of viscoelastic droplets onto superhydrophobic surfaces held at a wide range of temperatures ranging from 30 °C (below the boiling point) to 240 °C (beyond the boiling point of the viscoelastic droplet and up to the Leidenfrost condition). They showed that solid surface temperature does not have any significant effect on the droplet spreading. However, an increase in the surface temperature causes a faster droplet retraction.

Zhang et al. (2021) have investigated the role of liquid viscosity on the splashing characteristics of a droplet after impacting onto a smooth solid surface. They reported that the shear viscosity of the droplet presents a reverse role on the droplet splashing mechanism: the droplet shear viscosity induces droplet splash in low viscosity situations, while it prevents droplet splash in high viscosity conditions.

They experimentally examined the droplet impact physics of four different Newtonian solutions (water, 30% glycerol–water, 50% glycerol–water, and 70% glycerol–water) on smooth solid surfaces by considering three non-dimensional parameters: the Reynolds, the Ohnesorge, and the Weber numbers. Their observations demonstrated the “spreading–splashing–spreading” mechanism as shear viscosity increases (Zhang 2021).

Driscoll (2010) studied the role of viscosity on the impact dynamics of viscous droplet onto a dry smooth solid surface in which the droplet originally spreads and it forms a thick lamella, which also spreads in comparison with the splashing dynamics of the silicone oil droplets onto a dry smooth solid surface with different viscosities (1 and 10 cSt).

Recent experimental and theoretical studies on droplet impact physics of non-Newtonian (Bingham plastic) emulsion liquids (i.e., n-decane sample, and two different combinations of water, n-decane, isoparaffinic oil, and emulsifier) onto smooth hydrophilic–lipophilic solid surfaces have revealed the outcomes of impact dynamics. This impact dynamics consisted of three physical events: spreading–deposition, prompt splash, and corona splash (Piskunov 2021).

They have examined the maximum spreading parameter dependency on the Bingham-capillary number and the non-Newtonian Reynolds number. The results have also been compared with the Newtonian droplet impact physics. Furthermore, the maximum spreading parameter obtained from the observations of the droplet impacts of Newtonian (water) and non-Newtonian (Bingham plastic) emulsions has been compared with numerous previously proposed theoretical models for single-phase liquids (Piskunov 2021).

They reported that the theoretical models could not predict the maximum spreading parameter, which were obtained from the observations of emulsion droplet impacts. Piskunov (2021) have concluded that the theoretical models need to consider the capillary forces presented at the internal liquid–liquid interfaces inside emulsion droplets and cautious measurements of the rheological characteristics of the non-Newtonian (Bingham plastic) liquids (Piskunov 2021).

They took into consideration two previous theoretical models proposed by Pasandideh-Fard (1996) and Ukiwe and Kwok (2005) and modified their models to adjust for emulsion droplets. This was accomplished through incorporating an extra surface energy parameter along the liquid–liquid interface inside the emulsion droplet in the energy conservation, the non-Newtonian Reynolds number, $R e n$⁠, and the Bingham-capillary number, $B ^$⁠. The Bingham-capillary number presents the ratio of yield stress effects to the capillary force $( B ^ = D 0 τ 0 γ L A )$⁠. The non-Newtonian Reynolds number denotes the ratio of the inertia to the viscous force $( R e = ρ D 0 n U 0 ( 2 − n ) k )$⁠, where $γ L A$ is the emulsion liquid–air interfacial tension, $D 0$ is the emulsion droplet size, $U 0$ is the emulsion droplet impact velocity, $ρ$ is the emulsion droplet density, $τ 0$ represents the yield stress, k is the consistency parameter, and n denotes the flow index.

Lee (2012) applied a very fast x-ray imaging technique to study the droplet impact dynamics onto a smooth solid surface by focusing on the dynamics of air disk entrainment and entrapment underneath the droplet after impact and visualize the geometric profile of the entrained and entrapped air layer underneath the droplet and development of the air bubbles. They illustrated three phases of the air entrapment mechanism: the receding of the air layer due to the inertia effect, the shrinkage of the air layer to form air bubbles, and the separation of tiny droplets (Lee 2012).

Similarly, Li and Thoroddsen (2015) applied a high-speed interferometry technique to observe the air layer development and air bubble formation and entrapment mechanism during the droplet impact onto a dry smooth glass surface and draw the geometric profiles of the air layer thickness underneath the droplet, as represented in Fig. 5(a). Thoroddsen (2012) applied ultra-fast optical imaging technique to visualize the early stage of the prompt micro-splashing mechanism when a droplet impacts onto a dry smooth solid surface [(Fig. 5(b)]. They observed the beginning of the droplet contact dynamics, the ejection of a thin liquid sheet moving along the substrate, the instability mechanism, the formation of tiny droplets, and the formation of the micro-scale air bubbles, and have captured the azimuthal instability, as all illustrated in Fig. 5(b) (Thoroddsen 2012; and Thoroddsen and Sakakibara, 1998).

Thoroddsen (2010) studied the viscous droplet impact onto a dry smooth glass surface by applying a high-speed visualization technique. They examined the mechanism of micro-scale air bubble formation, the air entrapment dynamics, the formation of the moving lamella on the solid surface, the lamella detachment from the liquid contact line, and the lamella breakup into tiny satellite droplets (Thoroddsen 2010). They found out that the impact of viscous droplets onto a smooth solid surface causes the edge of the formed lamella to separate from the surface and floats inside an air layer underneath the droplet (Thoroddsen 2010).

Bischofberger (2013) observed the airflow development due to the impact of a silicone oil droplet onto a rough dry glass surface and impact of water–ethanol droplet onto a smooth dry glass surface. Bischofberger (2013) compared the impact dynamics of the droplets onto the smooth and rough dry solid surfaces. They reported that roughness on the solid surface conceals the splashing process, induces an airflow due to liquid sheet spreading on the solid surface, and causes a vortex ring to form above the spreading liquid sheet. In comparison, after the impact of the droplet onto a smooth solid surface, the droplet immediately splashes and breaks into several tiny droplets (Bischofberger 2013).

Table I lists the summary of experimental, theoretical, and numerical studies conducted on the droplet impact dynamics onto dry-smooth solid surfaces. Moreover, Table I lists some of the latest studies on the droplet impact dynamics onto smooth solid surface using the data-driven analyses and the machine learning techniques.

Most droplets in daily life are very complex such as bio-inspired droplets in medicine, pharmaceutical droplets in healthcare, conductive droplets in printed electronics, and droplets with complicated physical and chemical features. Despite the complex conditions of droplets in real life problems, majority of previous studies considered Newtonian droplets. Few studies considered non-Newtonian models such as emulsions. Few physical parameters were taken into account for the droplet impact dynamics on smooth materials: droplet viscosity and substrate temperature. The complexity of the environment such as extreme temperature variation causing interfacial tension change, humidity, and air compressibility factor was not involved in the previous experimental studies for the case of droplet impact on smooth substrates.

The superhydrophobic wetting state is well known for causing a static contact angle of water droplet to be larger than 150°. Moreover, it is recognized that the superhydrophobic surfaces have low contact angle hysteresis. The contact angle determines the adhesion of the droplet on the solid surface; therefore, the lower the contact angle, the higher the adhesion. On the other hand, the contact angle hysteresis measures energy dissipation during the spreading–receding process. Energy dissipation strongly depends on the flow of the droplet on the solid surface as well as the moving contact line (solid–liquid–gas contact line) (Nosonovsky and Bhushan, 2008; Mohammad Karim, 2022a, 2022b; and 2022c). For certain droplet impact velocities, the droplet would bounce off the superhydrophobic surface in an elastic behavior (Quéré, 2005 and Bartolo 2006). These features of the superhydrophobic surfaces are ideal for biological and aeronautical applications.

Fast droplet detachment after impact on a solid surface exhibits a great significance in a wide range of technological applications including anti-icing of aircrafts, condensation, and self-cleaning purposes (Jung 2012; Mishchenko 2010; Stone, 2012; Chen ., 2011b, 2021; Blossey, 2003; Tuteja 2007; Deng 2012; Liu 2014a, 2014b; Thoroddsen 2010; Lee 2012; Wang ., 2022b; Mulroe 2017; and Cen 2019). Moreover, the condensate water droplet bouncing off from the superhydrophobic surfaces presents a significant interest in many areas including anti-fogging, electrostatic energy production, and anti-frosting purposes (Jin 2018).

A recent experimental study, conducted by Liu (2014b), demonstrated how patterned superhydrophobic solid surfaces with the lattices of submillimeter-size pillars cause an implausible bouncing mechanism. The pillars are furnished with nanoscale textures. In the bouncing mechanism, the droplets spread at the onset of the impact on the solid surface and leave the surface in a flattened, pancake configuration without retracting.

Lin (2021) demonstrated on how a compound droplet can move on a substrate. Their technique could be used for various technological purposes including the self-cleaning of coatings. They showed that surface wettability gradient significantly facilitates this phenomenon.

Condensation presents a remarkable interest in various industrial problems related to heat exchanger performance, power generation, water collection, desalinization, and numerous environmental issues (Wang and Jia, 2016; Wiedenheft 2017; McNeil 2000; Song 2018; Humplik 2011; and Lara and Holtzapple, 2011). Coalescence-induced droplet jumping off from the properly controlled nano-scale patterned superhydrophobic surfaces is due to the release of their excess surface energy of the droplets. This mechanism has shown a key role in facilitating the droplet removal off from the surfaces in micro-scale as well as a significant improvement in the dropwise heat transfer purposes in the industrial applications (Miljkovic 2013). Many experimental studies have been conducted into micro- and nano-fabrication of the superhydrophobic surfaces to enhance the performance of the condensation process.

Motivated by the agricultural plants and animals in nature, the bio-inspired advanced functional superhydrophobic surfaces have been explored for the self-removal of the droplets from the surfaces to enhance the condensation efficiency. Ju (2012) designed cone-structure spines and trichomes on the cactus stem to improve the fog collection from the surfaces. Lv (2013) studied the self-bouncing off features of the droplets from a lotus leaf.

Ghosh (2014) designed the coalescence-induced droplet bouncing off from the staggering hydrophilic–superhydrophobic hybrid wettability characteristic surfaces. This surface characteristic was affected by the capillary force motivated from the surface feature of the banana leaf. Desert beetles were shown to exhibit the multi-wettability feature via the presence of the alternating hydrophobic and hydrophilic spaces to be desired for the water collection, as presented in Fig. 6 (Parker and Lawrence, 2001).

Feng (2020) and Hou (2015) investigated the designing of the bio-inspired superhydrophobic surfaces through hydrophobic micro-/nanostructure coating with pine needles. This coating technique could enhance the directional transportation of the droplets in the condensation process for the spontaneous removal of the condensate droplets off from the surfaces. These bio-inspired surfaces were composed of alternating hydrophobic–hydrophilic regions.

Directional water droplet removal off from the bio-inspired surfaces was strongly enhanced by the Laplace pressure gradient and free interfacial energy gradient. Therefore, such surfaces have recently become the center of attention for the purpose of maximizing condensation process (Wang et al., 2022).

There are two modes of droplet bounce off from superhydrophobic surfaces: general self-jumping of small droplets and forced jumping of large stretched droplets. These modes of droplet removal off from superhydrophobic surfaces have been recently studied by designing the micro-scale patterned superhydrophobic surfaces via chemical etching and mechanical broaching (Peng 2019). They presented the coalescence-induced and forced droplet bouncing off from hierarchically micro-patterned superhydrophobic surfaces (Peng 2019). Numerous studies were attempted to design novel patterned superhydrophobic surfaces for enhancing the droplet removal process.

Wen (2018) have proposed the three-dimensional fabrication of the arrays of the copper nano-scale wires on superhydrophobic surfaces to increase the efficiency of water droplet rebound off from the surfaces to enhance water collection and heat exchange purposes (Fig. 7(a)). Jin (2018) have designed superhydrophobic surfaces with the constraint networks of the hydrophobized nano-scale cones to enhance coalescence-induced droplet jumping. Miljkovic (2013) have demonstrated a design of superhydrophobic surfaces with its proper fabrication by the silanized copper oxide posts for the great improvement of the droplet jumping performance in the heat exchange applications [(Fig. 7(b)].

In addition to micro-scale patterning, the superhydrophobic surfaces for better droplet bounce off, the combination of the micro-scale pillar networks alongside the nano-scale posts might enhance the performance of droplet jump off from the superhydrophobic surfaces for better condensation. Therefore, Lo (2019) have designed a combined wettability feature of the superhydrophobic surface, which consisted of the superhydrophobic nano-scale silicone wires and the hydrophilic micro-scale channels, as illustrated in Fig. 7(c).

Wang (2020) have studied on how to generate a copper-based extremely efficient condensation heat exchange device by the networks of copper nano-scale cones to create a low-adhesive superhydrophobic surface for facilitating droplet bouncing, as presented in Fig. 7(d). Wang (2021) designed a superhydrophobic surface with oblique networks of the nano-scale wires to avoid the penetration of the droplet vapors inside the micron spaces for enhancing micro-scale droplet jumping and improving the condensation heat transfer at large subcooling degrees.

Lu (2017) investigated the nanofabrication of the superhydrophobic surface by coating the surface with the networks of silicone nanowires. This nanofabrication technique generated more nucleation sites for coalescence-induced droplet jumping to improve the condensation performance for small and large subcooling degrees, as shown in Fig. 7(e) (Lu 2017).

Liu . (2014b) took the advantage of an advanced surface engineering for fast droplet removal from the superhydrophobic surfaces. They demonstrated a technique to pattern the superhydrophobic surfaces with the arrays of submillimeter-size pillars, which were furnished with the nano-scale posts to create a remarkable droplet bounce (Liu 2014b). In this method, droplets initially spread after impact and then detach from the superhydrophobic micro- nano-patterned surface in the flattened and the pancake geometries (Liu 2014b). This droplet detachment mechanism was induced by the conversion of capillary energy storage inside the perforated droplet into significant kinetic energy for droplet bouncing off the pillars of the superhydrophobic surface.

In this work, they considered the time for droplet spreading and the time scale for droplet detachment to be compatible (Liu 2014b). Therefore, they proposed a design of the superhydrophobic surfaces with the hybrid micro-scale and nano-scale posts in which fast droplet jumping in the pancake geometries occurs over a large span of the droplet impact speeds (Liu 2014b).

Numerous studies have been conducted to compare the simulations with the observations for droplet impact dynamics onto the superhydrophobic surfaces. Li (2021) demonstrated a numerical benchmark for impact dynamics of the micron sized water droplets on the superhydrophobic surfaces and have compared the simulations with the observations. They applied the three-dimensional incompressible Navier–Stokes formulations for the simulations and have captured the liquid free interface by the level set routine (Li 2021). Their numerical analyses were verified with their observations by differentiating the maximum spreading parameter, which was defined as the ratio of the wetted area of the solid surface to the initial size of the droplet.

Similarly, Yokoi (2011) investigated the water droplet splashing mechanism onto the dry smooth horizontal superhydrophobic surfaces with the static contact angle to be 163°. A three-dimensional numerical model was proposed that unravels the prompt splashing dynamics with the satellite tiny droplets along with the spikes. Moreover, it was reported that the advancing dynamic contact angle has a key role in droplet splashing on the superhydrophobic surfaces (Yokoi, 2011; and Tsai 2009). Comparison between his simulations and the observations, obtained by Tsai (2009), confirmed the compatibility of the numerical and the experimental studies.

A recent study by Bartolo (2006) reported that a complete water droplet bounce off from a superhydrophobic micro-patterned surface in the Cassie–Baxter wetting mode only happens for moderate impact velocities. Moreover, they reported that when the droplet impact speed increases beyond a certain limit, it generates the sticky droplets with the impalement of the droplet by the micro-scale pillars (Bartolo 2006). In contrast, when the droplet immediate impact velocity decreases, it maintains a non-bouncing and sharper non-wetting mode (Bartolo 2006).

Superhydrophobic surfaces with controllable wettability between anti-wetting and strong wetting modes can be generated under the influence of physical parameters such as the light, the temperature, the electric field, and the roughness (Feng 2004; Liu 2004; and Zhang 2004). Surfaces that are furnished with the zinc oxide and the photocatalytic oxide that become hydrophilic through exposure to ultraviolate and maintain the superhydrophobic wetting mode in dark are among the most popular ones (Feng 2004; Liu 2004; Zhang 2004; and Khojasteh 2016). Table II lists some of the experimental, the theoretical, and the numerical studies conducted on droplet impact dynamics onto the hydrophobic and the superhydrophobic surfaces.

Similar to the case of droplet impact onto smooth substrates, most previous works about droplet dynamics upon impact onto hydrophobic and superhydrophobic materials consider simple Newtonian liquid models. The complexity of the environmental conditions was not taken into account in their analyses. Almost all studies in the case of droplet impact onto superhydrophobic surfaces focus on the geometric hierarchical structures on the surface of the superhydrophobic materials.

Along with the surface roughness and the surface chemistry of the solid, another key factor in droplet impact dynamics on the solid surface is the mechanical properties of the substrates, such as elasticity and stiffness (Chen ,2016; Saiki 2007; Lester, 1961; and Blanken 2021). There have been numerous studies on the droplet impact physics on a soft material whose surface goes under deformation due to the capillary force. They reported that surface deformation is reversely related to the shear modulus of the material (Chen 2016). Solid surface deformation, due to the droplet impact, expresses an important role in various scientific areas of the technology such as contact angle, contact angle hysteresis, adhesion, condensation, frost growth, wetting characteristics, and evaporation. Table III lists various studies that have been conducted on the physics of droplet impact onto soft flexible solid materials.

Numerous studies have been attempted to unravel the physics of droplet impact on smooth soft materials. Chen (2016) experimentally studied water droplet impact dynamics onto the flexible viscoelastic solid surfaces for a wide range of the Weber numbers. They described the droplet impact mechanism onto the soft solid surfaces by evaluating the dynamic interactions between the droplet and the flexible viscoelastic material (Chen 2016).

They reported that at the small Weber number, droplet bouncing occurs after impact onto a soft material and, at the higher Weber numbers, droplet spreading and deposition are most likely to happen after impact (Chen 2016). They also outlined that at the moderate Weber number range, the air bubble entrapment inside the droplet occurs during the droplet deposition after impact onto a flexible surface and a similar behavior has been observed for droplet impact on rigid materials (Chen 2016). At the large Weber numbers, droplet bouncing off the soft viscoelastic surfaces has not been observed, while droplet partial bouncing has been captured after impact onto rigid solids and a similar droplet rebound mechanism has also been observed for elastic droplet bouncing off the superhydrophobic surfaces (Chen 2016).

They also found that the viscoelasticity of a soft solid does not affect the droplet spreading dynamics after impact onto a flexible solid surface; however, the droplet vibration after impact onto a soft viscoelastic material is controlled by the wettability characteristics of the soft viscoelastic surfaces (Chen 2016).

Lee (2016b) have studied the dynamics of the water droplet impact onto the smooth bitumen solid surfaces with distinct rheological features known as rigid and soft flexible viscoelastic. Their observations presented the dynamic stick–slip behavior of the droplet onto the soft flexible material (Lee 2016b). They compared spreading/deposition and dewetting dynamics of water droplet impact onto the rigid and the soft viscoelastic deformable substrates for various droplet impact velocities. They applied the quasi-static contact angle measurement technique and dependency of the volume ratio, which represents the ratio of the volume of the deposited droplet to the initial droplet volume on the impact speed (Lee 2016b).

Moreover, they examined various outcomes (spreading/deposition, partial bouncing, and complete bouncing) of the droplet impact physics for different impact velocities (Lee 2016b). They reported that the impact and wetting characteristics of the water droplet changes from the classical quasi-static state on the rigid materials to the dynamic unpredictable condition on the soft deformable viscoelastic substrates. They were able to predict the impact and the wetting dynamics, such as deposition and bounce, of the droplet onto the rigid substrates, while the impact and the wetting dynamics of the droplet onto the soft viscoelastic substrates were not determined due to the generation of the wetting ridges (Lee 2016b). Their observations revealed that the wetting features of the deformable viscoelastic materials along with droplet spreading dynamics should be taken into account for describing droplet impact dynamics onto the flexible viscoelastic materials (Lee 2016b).

Langley (2020) have examined the role of the modulus of elasticity of the soft elastic materials on the evolution of the entrained air layer and the air bubble entrapment underneath the droplet after impact onto the flexible substrates using ultra-fast interferometry [Figs. 8(a) and 8(b)]. They reported that the elasticity of the soft substrates delays the air compressibility process, which leads to the greater air disks compared to the size of the air disks generated after droplet impact onto the rigid substrates (Langley 2020). They concluded that the time interval over which the droplet is in contact with the solid surface plays an important role in the mass, momentum, and energy conservations for droplet impact onto the solid surface (Langley 2020).

Chen and Bertola (2017) have investigated water droplet impact dynamics onto the curved (spherical) soft flexible elastic substrates using the high-speed visualization technique to illustrate the mechanisms of droplet impact onto a curved elastic solid surface with various elasticity moduli and different surface curvatures. They conducted this study via observations and simulations for a wide range of the impact Weber numbers. They examined various parameters to determine the droplet impact mechanism onto the elastic curved surfaces: the maximum and minimum contact angles for droplet spreading, the dynamic contact angle, the area of droplet spreading, the Weber number, the curvature of the elastic curved surface, and the modulus of elasticity of the elastic substrate (Chen and Bertola, 2017).,

They determined when the curvature of the elastic surface increases, the droplet recedes more (Chen and Bertola 2017). They also outlined the dependency of the dynamic contact angle of the droplet after impact onto the curved elastic substrate on three factors: the curvature, the modulus of elasticity of the material, and the droplet impact Weber number (Chen and Bertola, 2017). Moreover, they illustrated the viscous energy dissipation process through the deformation of the flexible elastic substrate due to droplet impact by considering the curvature of the elastic soft material for the flat and the curved flexible solids.

Weisensee (2016) have revealed that the modulus of elasticity of a soft elastic surface influences on the dynamics of the droplet impact onto the soft material. They reported that the time of contact between the water droplet and the elastic superhydrophobic surface is twice as low as the contact time between the water droplet and the rigid superhydrophobic surface [Fig. 8(c)].

Considering the role of the elasticity of the soft substrate on the droplet impact mechanism, Alizadeh (2013) have demonstrated the dependency of droplet spreading and droplet retraction on viscoelastic energy dissipation. They examined the water droplet impact onto the smooth and patterned superhydrophobic surfaces with different elasticity moduli (Alizadeh 2013). They reported when the modulus of elasticity of a soft material decreases and droplet retraction reduces (Alizadeh 2013).

Moreover, they tested this phenomenon by droplet impact onto the PDMS surfaces with various elasticity moduli and onto hard silicone surfaces [Fig. 8(d)] (Alizadeh 2013). They also concluded that after impact, the droplet tends to spread more onto a soft surface with a lower modulus of elasticity (Alizadeh 2013). The influence of the roughness of soft materials on the physics of droplet impact dynamics requires significant attention in the future research studies in this field.

Almost all investigations on the droplet impact dynamics onto soft materials examine the Newtonian droplets. They did not take into consideration the non-Newtonian liquid models. Although in real world situations, most of droplets exhibit complex and non-Newtonian characteristics. The humidity of the environment was another important factor, which was not taken into account in the previous studies. Other factors for the environment that were not even considered in the previous studies are interfacial tension variation due to temperature change (the Marangoni effect) and environment compressibility.

The maximum droplet spreading is a key parameter to characterize droplet impact dynamics on a solid surface, pertinent when the splashing phenomenon does not occur. The maximum spreading parameter is defined by $β m a x= D m a x / D 0$⁠. The maximum spreading parameter represents the ratio of the droplet diameter at maximum spreading, $D m a x$⁠, to the initial droplet diameter before the onset of impact with the solid surface, $D 0$⁠. Numerous empirical and theoretical models have been proposed to predict the maximum spreading diameter of the droplet by considering the force balance between viscous, inertia, and capillary effects (Josserand and Thoroddsen, 2016).

It might not be necessary to apply droplet spreading dynamics to predict theoretically the maximum spreading parameter. Energy equation, momentum balance equation, and their integrations from the moment of droplet impact with the substrate to the instant of maximum spreading, before receding and breakup, are the key features to determine the maximum spreading parameter (Attané 2007).

Various empirical expressions have been proposed to determine the maximum spreading parameter. These empirical formulas represented the reasonable agreements with previous theoretical works, the simulations, and the observations (Ukiwe and Kwok, 2005; Yokoi 2009; Pasandideh-Fard 1996; Range and Feuillebois, 1998; Roisman 2002; Clanet 2004; Roisman, 2009; Vadillo 2009; Eggers 2010; Schroll 2010; Lagubeau 2012; Li 2013; and Seo 2015).

Ukiwe and Kwok (2005) have obtained an empirical formula for the maximum spreading factor for droplet impact on the polymer surfaces, and their model was in agreement with previous numerical models. Furthermore, Clanet (2004) demonstrated an empirical relation for the maximum spreading of a water droplet impact on a rigid solid surface by applying a mass balance equation. Clanet (2004) reported that their scaling empirical formula has been reasonably applicable with observations for droplet impact on the hydrophobic and the superhydrophobic surfaces.

Antonini (2012) studied the spreading dynamics of water droplets after impact on a dry superhydrophobic surface to determine the role of the solid surface wettability. They reported that for the modest Weber numbers, the surface wettability effects on the maximum spreading parameter and for the large Weber numbers, the surface wettability does not affect on the maximum spreading parameter as the capillary effects have been overcome by the inertia forces (Antonini 2012).

Table IV presents the list of the physical models, obtained from the experimental and the numerical results. These physical models describe the dependency of the maximum spreading parameter to the physical variables including the dynamic contact angle, the Young static contact angle, and various non-dimensional parameters, such as the Reynolds number, the Weber number, and the Ohnesorge number.

All theoretical–empirical models that describe the maximum spreading factor indicate that the maximum spreading parameter increases with the droplet impact velocity based on a power law $( β m a x ∝ U n)$⁠, where n denotes the power. It is important to note that the power, n, is different in each experimental formula. In the large Weber and Reynolds number ranges, the power $(n)$ varies between $1 / 3$ and $1 / 2$ as the inertial force is stronger than the capillary effects in droplet impact dynamics by considering the solid surface wettability (Scheller and Bousfield, 1995; Clanet 2004; Stow and Hadfield, 1981; Delplanque and Rangel, 1997; and Attané 2007). The power $(n)$ was reported to be dependent on the contact angle. Moreover, the upper and the lower values of the power $(n)$ correspond to $0 °$ and $180 °$ contact angles, respectively (Pasandideh-Fard 1996; and Mao 1997).

Printing through disbursing small droplets on solid surfaces has presented significant interest in ongoing technologies including advanced functional materials, electronics, and biomedicine (Modak 2020). Conventional inkjet printing is not suitable for the printing of liquid droplet inks, which consist of biological solutions, biopolymeric samples, microscale particles, and nanoscale particles, which are vital for such industrial purposes.

A recent study demonstrated a simple innovative method for printing to overcome this challenge, which is called the “drop-on-demand printing technique” (Modak 2020). They applied droplet impact onto a superhydrophobic sieve, which has generated the satellite-free micro-scale droplets in two ways: “cavity formation and droplet ejection” (Modak 2020). Droplet impact onto a superhydrophobic sieve resulted in the competition between dynamic pressure $( ρ U 0 2)$⁠, due to the droplet impact velocity $( U 0)$ and the density $(ρ)$⁠, and the breakthrough pressure $( 4 σ L − 1)$⁠, due to the pores of the superhydrophobic mesh, where L is the pore size of the superhydrophobic sieve (Modak 2020; Ryu 2017; and Kumar 2018).

Therefore, an array of the satellite-free micro-scale droplet ejection has been generated due to the Rayleigh instability (Shin 2011; and Lorenceau and Quéré, 2003). The schematic representation of the experimental setup, which was applied in their study, is illustrated in Fig. 9 (Modak 2020). This technique was utilized in printing for the advanced electronic applications including the flexible electronic tape printing and the large area droplet array printing with the conducting arrays of the aqueous samples of the silver inks, poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS) polymer (Fig. 9) (Modak 2020).

Droplet impact, spreading, and splashing play remarkable roles in bioprinting technology. Three-dimensional (3D) bioprinting is a leading-edge technology in biomedicine, which covers a wide range of the bio-related applications including the printing of microscale biological droplets such as DNA, cells, bacteria, and proteins. 3D bioprinting plays a key role in analyzing gene expression, printing single-cells for biological cell advanced research, and biopolymer printing. Moreover, 3D bioprinting has been applied for the printing of the live biological cells, biosensor manufacturing, stem cell assembly, artificial organ production, 3D functional organ fabrication, tissue engineering, and smart 3D functional biomaterials (Starly and Shirwaiker, 2015; Murphy and Atala, 2014; Mandrycky 2016; Dasgupta and Black, 2019; and Qu 2021). A recent study has presented the application of the same technique for a single-biological droplet bioprinting approach for red blood cell droplets over various concentrations to be printed onto a glass slide (Fig. 10) (Modak 2020).

The COVID-19 pandemic significantly influenced the public worldwide and caused loss of millions of lives over the past two years through the transmission of the coronavirus between individuals. It was realized that the respiratory viruses are transmitted in the community through droplets, which contain various respiratory pathogens including large viral loads of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which cause COVID-19 disease (Stadnytskyi 2020). The main way of contracting the virus was recognized to be through the spread of the aerosol-based coronavirus-laden respiratory droplets from the viral infected patients by sneezing, coughing, giving a small or a long speech, and exhaling in different environments with even an operating air-conditioner and a ventilation (Jayaweera 2020). Face masks were recognized to be the best approach to prevent the spread of COVID-19 in the public, as suggested by the World Health Organization (WHO) (Melayil and Mitra, 2021; and WHO, 2020).

Therefore, the flow physics, droplet–face mask interaction dynamics, interfacial phenomenon, and the adhesion characteristics of the impacting airborne virus-contained droplets with the superhydrophobic-coated face masks are extremely necessary to be thoroughly understood (Melayil and Mitra ,2021; Katre 2021; Hetherington 2021; Bhardwaj and Agrawal, 2020; Kumar , 2021; Stadnytskyi 2020; Shafaghi 2020; and Jayaweera 2020). The physics of the flow of the virus-laden respiratory droplets exhibits a pertinent important role in several aspects of the COVID-19 pandemic since it is involved in formation, aerosolization, dispersion, spreading, and deposition of the air-borne aerosols and the coronavirus-laden droplets onto the surfaces and further transmission to every individual in the public through inhalation (Fig. 11) (Mittal 2020; and Poon 2020). The worldwide response to the COVID-19 emphasized the insufficiency in the soft matter related understanding related to the interaction of the viral droplet with the human skin and the surfaces including the personal protective equipment such as the face masks and respirators to enhance the efficacy of the personal protective equipment and the disinfection techniques (Poon 2020; and Jayaweera 2020).

Recently, a combined experimental-computation effort by utilizing a machine learning technique based on the back propagation neural network topology and a complex physical process of spreading–freezing dynamics of water droplet impact onto a supercooled solid surface has been conducted (Yang 2022). In this study, they predicted spreading and icing patterns on the solid surface and, consequently, the level of surface supercooling using an artificial intelligence (AI) mode (Yang 2022). In the AI-based modeling technique, they considered six parameters to explain water droplet spreading and icing: the Reynolds number, the Weber number, the Ohnesorge number, the level of surface supercooling, the maximum spreading factor, and maximum spreading time (Yang 2022).

Impact and icing of the droplets on the cold surfaces exhibit an extensive example in the everyday life such as rain droplet impact and freeze onto the transmission lines, impact of supercooled air moisture, and water droplets onto the aircraft wings during the flight (Jin 2016; Wang, 2020; Wang 2015; Lei 2019; Yang 2022; Guo 2022; Sarma 2022; Sun 2022; Ju 2019; and Zhang and Liu, 2016). Water droplet impact onto a supercooled substrate consists of a spreading phase and a freezing stage. This is a very complicated physical phenomenon, which includes several physical events: flow of the droplet, heat transfer, and phase change (Yang 2022).

Supercooled large water droplets on the aircraft wings are a great challenge in the aerospace engineering technology and aircraft performance, which can initiate the catastrophic airplane crashes (In-Flight Icing Encounter and Crash, 2002). In the last 20 years, extensive experimental and computational efforts have been invested to unravel the physics of the supercooled large water droplet impact and the supercooled characteristics of the formed liquid film–aircraft wing interface, which substantially influence the efficiency of the airplanes and the aircraft engines (Zhang and Liu, 2016; and Federal Aviation Administration, 2014).

Current research is being conducted to understand the thermodynamics of the supercooled large droplet impact dynamics (Zhang and Liu, 2016). They experimentally studied the role of the droplet size on the thermodynamics of supercooled large droplet impact. Furthermore, they demonstrated a physical model for the unsteady heat transfer phenomenon in which the droplet size plays a key role on the heat transfer from the supercooled large droplets that are placed on the substrate; across the formed liquid film; and the substrate interface (Zhang and Liu, 2016).

After droplet impact onto a superhydrophobic surface, the chemical hydrophobicity of the surface along with the microscale/nanoscale patterned structures on the surface results in the air entrapment inside the superhydrophobic patterns. This event causes a large equilibrium advancing contact angle and a low contact angle hysteresis on the superhydrophobic surface (Rothstein, 2010; Quér,é 2008; Schellenberger 2016; Kim and Rothstein, 2017; and Kim 2018). This physical phenomenon benefits many technological applications including the anti-icing of the airplane wings, anti-icing of the wind turbine blades, and extensive drag reduction in the substrates (Rothstein, 2010; Quéré, 2008; Schellenberger 2016; Kim and Rothstein, 2017; Kim 2015; Nilsson and Rothstein, 2011; Nilsson and Rothstein, 2012; Barthlott and Neinhuis, 1997; Cao 2009; Genzer and Efimenko, 2006; Lee 2016a; Ou 2004; Daniello 2009; and Srinivasan 2013). The dynamic elasticity response of the flexible superhydrophobic surfaces after droplet impact was studied by a high-speed imaging technique and unraveled the role of the droplet reaction force on the dynamics of the flexible superhydrophobic surfaces (Kim 2018).

Droplet–solid surface interactions, upon impact, have tremendous applications in the current society such as pharmaceuticals, the fight against COVID-19 due to the spread of virus-laden droplets in public, condensation performance, nano-printing technology, bio-printing technology, 3D-printing technology, tissue engineering, 3D bio-printing technology, fabrication of smart biomaterials and functional organs, flexible electronics, and high resolution additive manufacturing in healthcare and medicine.

Heterogeneous droplet impact on a substrate has a significant application in the pharmaceutical industry. Heterogeneous droplets consist of a large number of insoluble solid particles, suspended in the aqueous solution, which are the typical liquid drugs. Moreover, the real world applications ranging from various methods of coatings, such as the spray coating of the aqueous based colloidal suspension pharmaceutical droplets, the virus particles-laden respiratory droplet impact on the substrates with complex morphology and/or chemical heterogeneity, and heterogeneous cell-laden droplets in biomaterials, require much further deepening of the knowledge in the droplet impact physics for complex droplets and complex substrates. The impact physics of droplets with colloidal suspensions and polymer solutions is an important challenge and requires further effort. Despite the fact that all current real challenges in the area of droplet impact physics involve complex droplets and complicated substrates, the majority of the studies related to this physical phenomenon has been considering the Newtonian liquids and simple substrates.

Due to the urgent need to advance in biotechnology, healthcare, medicine, pharmaceuticals, biology, energy, printed electronics, functional coatings, and smart coatings, it is crucial to emphasize more on utilizing new emerging advanced technologies including machine learning, quantum computation, and nanotechnology in imaging such as transmission electron microscopy and cryogenic electron microscopy to explore the droplet impact physics on complicated systems ranging from complex substrates surfaces in terms of morphology and chemical heterogeneity as well as droplets with complex rheology, complex interfacial properties, and complex physical properties as schematically illustrated in Fig. 12.

Environmental conditions also have important roles on the physics of droplet–solid surface interactions, upon impact. Pressure strongly effects on the strength of droplet splashing (Xu 2005; Josserand and Thoroddsen, 2016; Jian 2018; and Burzynski and Bansmer, 2019). Air can significantly enhance air layer formation and entrapment underneath the droplet after impact. It is required for the droplet to be in complete contact with the solid surface for good quality of the printings such as 3D biomaterials, smart biomaterials, tissue engineering, and printed electronics. Moreover, air compressibility can influence on the air film formation. Furthermore, the thermocapillary effect phenomenon known as the Marangoni effect can govern droplet motion on the solid surface through the interfacial tension gradient due to the environment temperature gradient. For better physical understanding of the droplet–solid surface interactions, comprehensive studies, which take into account the environmental conditions and the complex droplet–substrate interactions, will need to be conducted.

Due to the recent advancement of technologies in deep learning, the power of computation with the help of advanced collective features of the quantum states, nanoscience, and nanotechnology in imaging such as atomic force microscopy, transmission electron microscopy, and cryogenic electron microscopy, it would be highly practical to deepen the knowledge in the physics of the complex droplet impact on real substrates. Moreover, it is highly recommended that research scientists from various disciplines collaborate in this complex physical phenomenon. This complex interfacial physics requires expertise in various fields such as physics, chemistry, biology, molecular biology, genetics, mathematics, computer science, materials science, medicine, aerospace, biochemistry, physical chemistry, chemical engineering, and mechanical engineering.

Despite extreme importance for deepening knowledge in the area of complex interfacial physics, there have been only a few numerical simulations to investigate on the physics of complex droplet–solid interactions, upon impact, which is not reliable for extremely sensitive and critical applications such as tissue engineering, the fight against COVID-19, and 3D biomaterials associated with public healthcare. Therefore, it is highly important to direct the future research efforts and scientific attention to unravel the physics of the complex droplet–solid surface interactions, upon impact, via experimental techniques by harnessing the emergence of the recent advanced technologies including powerful imaging techniques such as atomic force microscopy, transmission electron microscopy, and cryogenic electron microscopy along with the advanced theoretical works by utilizing knowledge from the multi-disciplinary scientific areas for better understanding the true nature of this fascinating physical phenomenon and also to have a more precise control of droplet features for designing new advanced materials in emerging advanced technologies.

The author has no conflicts to disclose.

Alireza Mohammad Karim: Conceptualization (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.