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Suppression of Droplet Breakage by Early Onset of Interfacial Instability

Hypothesis: Interfacial instabilities cause undesirable droplet breakage during impact. Such breakage affects many applications, such as printing, spraying, etc. Particle coating over a droplet can significantly change the impact process and stabilize it against breakage. This work investigates the impact dynamics of particle-coated droplets, which mostly remains unexplored. Experiments: Particle-coated droplets of different mass loading were formed using a volume addition. Then the prepared droplets were impacted on superhydrophobic surfaces, and their dynamics were recorded using a high-speed camera. Findings: We report an intriguing phenomenon where interfacial fingering instability helps suppress breakage in particle-coated droplets. This island of breakage suppression, where the droplet maintains its intactness upon impact, appears within a regime of Weber numbers where droplet breakage is inevitable. The onset of fingering instability in particle-coated droplets is observed at much lower impact energy, around two times less than the bare droplet. The instability is characterized using the rim Bond number. The instability suppresses breakage because of the higher losses associated with the formation of stable fingers. Such instability can also be seen in Leidenfrost surfaces and dust/pollen-covered surfaces, making it useful in many applications related to self-cleaning.


Introduction
Droplet breakage on impact is undesirable. It affects essential applications such as printing, spraying (e.g., pesticides), coating, bioreactors, cooling, and directional transport [1][2][3]. In printing, droplet breakage leads to the generation of undesired spots and compromises printing resolution. In the case of spraying on plants, ejected droplets are lost to the ground and cause environmental pollution [4]. Similar droplet breakage also causes pollution during fertilizer production [3]. Smaller droplets are more efficient in spreading viruses and diseases than larger ones. Thus, smaller droplet generation due to impact breakage enhances the spreading of pathogens and diseases in plants [5,6] and humans [7] onto a larger area.
Droplet impact on surfaces is studied for their implications in various applications and natural phenomena. On impact, the droplet spreads, and its kinetic energy is stored as surface energy.
It converts back into kinetic energy during rebound. While the top part of the droplet rapidly moves away from the surface, the bottom part leaves the surface slowly as adhesion delays detachment [8]. This process results in the stretching of the droplet with its stretched length ( ) larger than the maximum spread diameter ( ). Stretching increases with impact velocity. Beyond a critical Weber number, the stretching is sufficient to enable the ejection of smaller droplets from the top. Weber number is defined as = 2 0 / , where , , 0 and are density, impact velocity, diameter, and surface tension, respectively. This is driven by Rayleigh-Plateau (RP) instability. As the impact velocity increases, the stretched interface breaks down into several droplets during the lift-off phase [9]. This kind of dissociation is referred to as pinch-off. At higher impact velocities, droplet dissociates on the surface due to the onset of finger formation driven by Rayleigh-Taylor (RT) and Rayleigh-Plateau (RP) instability [10][11][12]. This type of dissociation is referred to as a receding breakup.
We report suppression of droplet breakage with the help of hydrophobic particle coating.
Particle-coated droplets are commonly known as liquid marbles (LM). The impact of LM on the superhydrophobic surface reveals a higher Weber number stability regime which is nonexistent in the bare droplet impact. Interestingly for LM, as the impact velocity increases, droplet breakage does not progress from pinch-off to receding breakup regime. The pinch-off regime is shortened, and we observe complete suppression of droplet breakage in LM for a range of higher impact velocities.
This study presents a technique for breakage suppression, which will be helpful in many applications, such as in the continuous production of mechanically stable bioreactors [13,14].
LM has also been used as a biological model [15]. Studying its large-deformation dynamics helps us understand the response of organs and cells under sudden impact conditions such as accidents [16]. Besides, the present study also helps us understand the dynamics of particlecoated curved interfaces at large deformations, which are largely unexplored. As the applications of LM in the digital microfluidics platform are increasing [17], the present paper also helps design such applications better. A similar phenomenon is observed for bare droplet impact on particle-covered surfaces and LM impact on Leidenfrost surfaces. Such scenarios help in understanding the interactions of the droplet with particles for the applications like selfcleaning and pollen dispersion [6,[18][19][20]].

Preparation of liquid marble
A polytetrafluoroethylene (PTFE) powder with an average particle diameter of 35 µm was used to prepare liquid marble (with a core liquid as DI water of 8.2 µL) by the rolling method [21].
Control of the mass loading was obtained by creating smaller liquid marble (LM) of different volumes with full surface coverage. Then the LM is placed on a superhydrophobic surface with a bare droplet. Collision of the LM with a bare droplet results in merging and forming a larger LM, as shown in Fig. 1(A). The final LM volume was 8.2 µL. Superhydrophobic surface is required to avoid rupturing of LM with lower mass loading. The geometric relation 3~2 determines the initial LM volume, where and are the surface area and the volume of the liquid drop, respectively [24]. The different volumes of LMs and water droplets used to prepare various mass loading are described in Table 1. Where and represent the volume of liquid marble and the volume of water drops. The mass loading was determined by averaging the mass of ten liquid marbles after completely drying the liquid.

Preparation of superhydrophobic surface
Superhydrophobic copper surface was prepared by the method reported previously [25,26]. A copper substrate (3 cm × 2 cm) was cleaned with acetone, isopropyl alcohol (IPA), and deionized (DI) water. This was followed by a 30 s cleaning with sulfuric acid (33% in DI water). The cleaned copper surface was then immersed in an aqueous solution of sodium hydroxide (2.5 mol/L) and ammonium persulfate (0.1 mol/L) for 20 min at room temperature.
This solution etches the copper surface and produces copper hydroxide nanowires on the surface ( Supplementary Fig. S1). The substrate is cleaned multiple times with DI water and dried with nitrogen. The substrate is dipped in Teflon solution for 10 min to get a superhydrophobic surface. Subsequently, it is dried by heating at 110 °C for 10 min. The prepared superhydrophobic surfaces show excellent repellency with a water contact angle of 171° ( Supplementary Fig. S1).   Table 1: The values of volume to be taken for particular mass loading. As described in the main text, various sizes of LM and droplets collision results in different mass loading. and represent the volume of liquid marble and water droplet, respectively. The final volume of LM used in our experiments is fixed at 8.2 μL.

Droplet impact experiments
Prepared LM with different mass loading is impacted over a superhydrophobic surface. Impact velocity was controlled by impacting the LM from different heights, and the dynamics were recorded with a high-speed camera. The recorded images were analyzed for measuring different parameters, as shown in Fig. 1(B).

The dynamic surface tension of LM during impact
For static scenarios, LM has reduced effective surface tension. As shown in Supplementary During spreading, the newly created surface is primarily a liquid surface with very few particles; thus, the change in net surface energy during the spreading process is determined by the liquid's surface tension. This scenario is analogous to the impact of droplets with slow diffusing surfactants [27,28]. Thus, as an approximation, water surface tension has been used as the dynamic surface tension in our case.
The spreading dynamics of an LM show approximately the same behavior and scaling as a bare droplet (Fig. 2). Maximum spreading diameter ( ) on the superhydrophobic surface is known to follow ~ 0 0.25 scaling [29]. LM follows approximately the same scaling

Impact outcome
The rebound of the LM follows entirely different retraction dynamics, which leads to a different outcome for the impact process. With an increase in impact velocity for a bare droplet, the outcome progresses from a no-breakage to pinch-off to receding-breakup, as shown in Fig. 3.
Interestingly, as the impact velocity increases, LM breakage does not progress from pinch-off to receding-breakup regime (Supplementary Video S1). The pinch-off regime is shortened, and we observe complete suppression of LM breakage for a range of higher impact velocities. In this zone, LM shows intactness and mechanical stability even at high . Traditionally polymer additives are used to increase viscosity for breakage suppression [30]. However, it is impossible to explain our observations through arguments of increased viscosities as the island of "no-breakage" emerges between two "dissociation" regimes. Droplets with enhanced viscosity will only show the transition from no-breakage to breakage. Our study reveals that this suppression is induced by the early onset of fingering instability in LM. In contrast, remains high for bare droplets. We attribute this reduction in and the associated recurrence of the no-breakage regime to an additional dissipation mechanism. This dissipation stems from the instability that leads to the formation of finger-like structures. ≅ . The data above the line represents higher adhesion energy, and the data below represents a higher viscous loss.

Rim instability
As the droplet reaches its maximum spread diameter, a rim of liquid is formed at the edge of the flattened droplet. At > 57, an interfacial instability sets in during the spreading phase.
As shown in Fig. 5 Finger formation is not observed for the bare droplet case at the same .  However, the finger length remains negligent in the bare droplet case. Additionally, the growth of the perturbation is not limited to the spreading phase only. The perturbations grow faster after the droplet reaches its maximum spread state (~ 2.66 ms). Compared to bare droplets, perturbation at maximum spread is higher in LM. The perturbation growth is further driven by the rapid retraction of the interface between the fingers while the tips remain nearly stationary.
For a particular LM, an increase in results in increased spread diameter and finger length, as seen in Fig. 5(D).
The formation of fingers for a bare droplet is observed at a much higher Weber number (~110).
However, the formation of fingers for bare droplets shows very distinct differences. In contrast to LM fingers, they dissociate on the surface due to receding breakup. Fig. 6(A) represents the variation of normalized finger amplitude ( 0 ⁄ ) at the maximum spread. At below 110, the LM has much higher amplitude compared to the bare droplet. However, above ~ 110, a similar trend is observed. Maximum finger length also follows the same trend, and LM shows large finger growth even at lower (Fig. 6(B)). In previous literature, the generation of fingers has been attributed to Rayleigh-plateau (RP), Rayleigh-Taylor (RT) and Kelvin-Helmholtz (KH) instabilities [11,[31][32][33].

Rim Bond number
The coupled effect of RT and RP instability has been attributed to finger formation [12]. The work has analyzed the stability of an inviscid cylindrical rim subjected to acceleration. The dispersion relation for the coupled effect is given in terms of nondimensionalized wavenumber ( ) and rim Bond number ( ) as [12] where, is the growth rate non-dimensionalized by rim capillary time ( = √ 3 8 ⁄ ).
is perturbation wavenumber non-dimensionalized by rim radius (  Finger length is determined by the growth rate of the fastest growing unstable mode. Supplementary Fig. S3 shows the theoretical growth rate for the fastest growing mode with .  The amplitude ( ) is observed to increase with the decrease in the rim Bond number. As rim width ( ) decreases with an increase in (Supplementary Fig. S5), higher impact velocities lead to lower . Mode growth time, given by = ⁄ is plotted in Fig. 7(C). is evaluated from the experimentally measured rim width ( ), and is extracted from the theoretical curve shown in Supplementary Fig. S3.   Fig. 8(A)). Similarly, the total final energy loss drops suddenly by a large magnitude in the case of LM after finger formation ( Fig. 8(B)).

Effect of viscous loss
Normalized energy loss in the pinch-off suppression regime is plotted in Fig. 9 where is the Reynolds number. Fig. 9(A) shows scaling according to Eq. (2), which agrees very well with the experiments. Despite forming prominent fingers, we do not observe its breakage into smaller droplets. As ~ , at lower , the resultant finger length is insufficient to destabilize and break into droplets. Finger breakage due to RP instability in jets [36] is possible when the ⁄ > . Fig. 9(B) represents the ratio of maximum finger amplitude ( ) and finger diameter ( ).
We found the critical ratio of / ~ 3.8 is responsible for the breakup of the fingers.
Despite prominent fingers in 57 < We < 92 regions, the / ratio for LM remains lower than the critical limit. Thus, fingers remain stable without dissociation. The growth of an unstable mode is represented by = ( ), where is the initial perturbation, is the mode-specific growth rate, and is the growth time [37]. Since the same phenomena govern the instability in both LM and bare droplet cases, the growth rate and time should remain the same for a given impact condition. This is verified experimentally in Fig   7(A) & (C). Hence, we postulated that early finger formation in LM is because of the higher initial perturbations happening during the initial phase of impact (< 1 ms). The initial perturbations result from trapped particles. During initial contact, particles get trapped between the spreading liquid and the superhydrophobic surface ( Fig. 6(A) and Fig. 10(A)). The trapped particles make the bottom surface inhomogeneous. The contact line experiences wetting discontinuity at the edge of the trapped particle region. A separate experiment was also performed to verify the role of particles trapped between the liquid and the superhydrophobic surface. A small number of particles are pre-deposited on the superhydrophobic surface. It is observed that fingering instability is triggered even for bare droplets at lower impact energies (We ~ 72), further confirming the role of initial perturbation ( Fig. 10

Conclusion
The coating of hydrophobic particles over droplets remarkably affects the stability of impacting droplets. The LM shows an anomalous stability regime at higher We (57 to 90). Interestingly, finger formation, which is usually responsible for droplet fragmentation, is helping LM to be stabilized at higher We. This study can have implications in understanding the physics behind several biological phenomena, such as disease spreading in plants through pollen-laden droplets and rupture models of cells and organs. Moreover, such mechanical stability can be applied to design better and continuous Janus LM production and splitting devices. [14,38] Besides, this paper indicates the possible way to increase the mechanical stability of fragile droplet-based chemical and biological reactors by replacing them with LM-based reactors. The phenomenon of early finger formation is not limited to LM impact on superhydrophobic surfaces. A similar finger formation can be seen on heated surfaces above Leidenfrost temperatures (Supplementary Video S4). Such scenarios also help better understand interactions between droplets and dust/pollen particles, making them extremely useful in designing self-cleaning surfaces and pollen-laden interfaces [5,6,19,20].

Data availability:
Source data are provided in this paper. All other data that support the plots within this paper are available from the corresponding author upon reasonable request.