Experimental investigation of particle–droplet–substrate interaction

The impact of droplets on non-fixed spherical particles placed on a plane polymethyl methacrylate (PMMA) substrate is investigated. This interaction is a highly abstracted level of a high-pressure spray cleaning process. Water droplets in a diameter range between 0.68 and 1.66mm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.66\,{\mathrm{mm}}$$\end{document} and spherical particles (PMMA) with a diameter of 1.55 mm are used. The droplet velocity range of 1.05≤vd≤2.0m/s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1.05 \le v_{\mathrm{d}} \le 2.0\,{\mathrm{m/s}}$$\end{document} results in a Weber number range of 13≤We≤94\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$13\,\le \,{\mathrm{We}}\,\le \,94$$\end{document}. The particle-droplet-substrate interactions are investigated for different Weber numbers, droplet-to-particle diameter ratios and eccentricities. Different droplet impact scenarios are identified: A—Lift-off during initial recoil; B—Lift-off during a later recoil; C—No Lift-off, deposition of the droplet and D—No Lift-off, wetting of the substrate. The behavior of the particle-droplet-substrate interaction is determined depending on Weber number and particle-to-droplet diameter. Additionally, the analysis of the eccentricity in relation to the lift-off behavior shows that the lift-off height and duration increases with the centrality. The investigation of temporal change of the half-spread and contact angle results in a criterion for the point in time at which the lift-off takes place. Finally, a simplified analytical model is provided quantifying the probability of the lift-off of the particle-droplet system after the interaction. Experimental and analytical results are used to create a map of occurring impact regimes in terms of the particle-droplet configuration.


Introduction
High-pressure spray cleaning of surfaces contaminated with particles and particle agglomerates has a wide variety of applications, e.g. food industry (Gibson et al. 1999;Gerhards et al. 2019). The cleaning process is characterized by the breakup of jets into single droplets which is a complex phenomena depending on the nozzle geometry, pressure and the surrounding environment (Lin and Reitz 1998;Lefebvre 2017). This results in a droplet size distribution with varying velocities impacting the contaminated surface. The investigation of droplet impact onto dry clean flat surfaces shows multiple phenomena, such as droplet spreading, liquid film formation or primary and secondary splashing (Kalantari and Tropea 2007). To obtain a fundamental understanding of the phenomena occurring during the cleaning process, the particle-droplet-substrate interaction is the first important step.
The impact of droplets on planar surfaces is investigated extensively in the literature and summarized by e.g. Josserand and Thoroddsen (2016). Bakshi et al. (2007) examine the impact of water, glycerine-water solution and iso-propanol droplets onto a fixed spherical particle (stainless steel, diameters 3.2 − 15 mm ) with a droplet diameter of 2.4 − 2.6 mm . The temporal and spatial variation of the film thickness on the spherical particle is measured. Three distinct temporal phases of the film dynamics are determined based on the experimental results: (1) the initial drop deformation phase, (2) the inertia dominated phase and (3) the viscosity dominated phase. The influence of the Reynolds number of the droplet and the particle-to-droplet diameter ratio on the dynamics of the film flow on the surface of the particle are investigated. Hardalupas et al. (1999) report on experimental investigations on the impact of monodisperse water-ethanol-glycerol droplets onto a fixed spherical target. The droplet diameter range is between 160 and 230 μm with a velocity of 6 ≤ v d ≤ 13 m∕s . The target material is stainless steel. They observe the retraction of the liquid crown at low droplet impact velocity and the disintegration from cusps located on the crown rim at high impact velocities. Nils Janssen, Jana R. Fetzer, Jannis Grewing, Sebastian Burgmann and Uwe Janoske have contributed equally to this work.
An increase in sphere curvature promotes the onset of splashing. Head-on collisions between water droplets and fixed spherical targets are investigated by Charalampous and Hardalupas (2017). The water droplet diameter range is 170 − 280 μm with a velocity of 6 − 11 m∕s resulting in a Weber number regime of 92 ≤ We ≤ 1015 , chosen sphere diameters are 500, 1000 and 2000 μm . This results in droplet-to-particle diameter ratios of 0.09 < Φ < 0.55 . Three different types of droplet-particle collisions can be observed. Two of them are deposition and splashing. Both phenomena can also be observed when droplets collide with flat surfaces. A third type, unique to droplet-particle collisions, is the overpass type, where a stable crown is formed that propagates around the particle and crosses the equatorial plane without breaking up. The droplet overpass is only observed when the droplet diameter is comparable to the particle diameter. Pawar et al. (2016) analyse the impact of 2.9 mm diameter water droplets onto fixed spherical particles with diameters of 2.5 or 4.0 mm experimentally. The collision outcomes are classified as follows: (1) agglomeration, the whole particle is engulfed by the droplet, (2) stretching separation, only part of the droplet attaches to the particle, the formation of satellite droplets is observed. The transition between agglomeration and stretching seems to scale inversely with the Weber number. Water droplet collisions onto fixed spherical particles are experimentally examined by Banitabaei and Amirfazli (2017) focusing on the effect of droplet impact velocity and wettability of the particle surface on the collision phenomena. A droplet-to-particle diameter ratio of approximately 1.75 is investigated, in a droplet Weber number range of 0.1 < We < 1146 . Target particles are glass beads with a diameter of 2 ± 0.01 mm . Using the silanization method, different levels of wettability namely contact angles of Θ = 70 • , 90 • and 118 • are created. They conclude that for droplet impact at various impact velocities onto a hydrophilic particle the droplet is neither disintegrated nor stretched enough to form a liquid film after impact. However, the droplet follows the particle curvature and slightly deforms around it in the entire studied velocity range. Mitra et al. (2017) analyse experimentally and numerically the collision between a single droplet (diameter: 2.48 − 2.61 mm ) and a fixed hydrophobic thermally conductive particle (brass, diameter 3 mm ). Examined Weber numbers range from 0.9 to 47.1. Both, the cold and hot states of the sphere are investigated and evaluated with high-speed imaging. Two different outcomes are observed in the cold state: (1) deposition of the droplet onto the particle at low Weber numbers and (2) a complete wetting of the particle surface through the motion of a spreading lamella with a thick peripheral rim at higher Weber numbers. Mitra and Evans (2018) examine experimentally and numerically the surface wetting behavior of a spatially fixed spherical brass particle (diameter 10 mm ) with and without heat transfer for different water droplet impact Weber numbers ranging from 4 to 104 with a droplet diameter of 2.0 ± 0.1 mm . Mitra and Evans (2018) show for interactions in the cold state ( 20 • C) that the droplet exhibits oscillatory interfacial motion comprising of periodic spreading and recoiling motion. Mitra et al. (2013Mitra et al. ( , 2016, investigate experimentally and numerically the impact of water, isopropanol and acetone and isopropyl alcohol droplets onto a highly thermally conductive spherical surface in the temperature range between 20 and 350 • C. Dubrovsky et al. (1992) perform an experimental investigation on the interaction of droplets (0.65 − 1.05 mm) with fixed solid particles (3.2 − 8.0 mm) . Droplet impact velocities range from 7 to 10 m/s. A collision of a fastmoving small droplet with a large solid particle will result in droplet breakup followed by the formation of a certain number of liquid fragments. They found an increase in the coalescence parameter as the particle to droplet diameter ratio is increased. Ge and Fan (2007) investigate the fixed particle-droplet collision mechanics including film-boiling evaporation on a 3.2 or 5.5 mm diameter brass particle. The acetone droplet has a diameter of 1.8 − 2.1 mm . In the film boiling regime, the droplet undergoes a spreading, recoiling and rebounding process. In addition to droplet impact onto spatially fixed dry particles, the impact of droplets onto wetted spheres is experimentally investigated by e.g. Liang et al. (2014). Mitra et al. (2015) investigate the impact of a particle onto a droplet placed on a larger particle. The impact of droplets on fixed cylinders is analysed experimentally by Villermaux andBossa (2011), Juarez et al. (2012), Hung and Yao (1999) and Rozhkov et al. (2002). Sechenyh and Amirfazli (2016) investigate the impact of a droplet onto a particle in mid-air. At similar Weber numbers, compared to experiments with a fixed particle, two kinds of particles with different wettability are used. The formation of liquid lamellar or ligaments depends on the wetting behavior.
The literature review indicates no previous experimental work investigating the impact of a single droplet onto a nonfixed spherical particle placed on a substrate which will be discussed in the following sections in respect of the wetting behavior i.e. half-spread and contact angle and motion of the particle-droplet system for impacts with different eccentricities. An analytical model is presented to identify cases showing a lift-off of the droplet-particle system and a map of occurring impact regimes in terms of Weber number and droplet-to-particle diameter ratio is given.

Materials and experimental setup
A schematic of the experimental setup used to analyse the particle-droplet-substrate interaction is illustrated in Fig. 1. Table 1 lists the utilized equipment and the abbreviations used in Fig. 1. The generation of the droplet (D) is realised by a pressure transducer (PT) combined with a cartridge (C) and a dosing needle (DN). The particle (P) is placed on the substrate (S) below the dosing needle at an adjustable distance h (in this study: 20 mm and 50 mm ). Both cartridge and substrate are adjustable in the horizontal plane (x-and y-direction) to ensure a variable impact position. Two synchronized high-speed cameras (HS 1/2) record the process. The cameras are aligned at a 90 • angle. Strobe lights (SL 1/2) scattered by diffusers (D 1/2) provide the necessary exposure. The droplet generation, the recording and the exposure are controlled by a synchronizer (Sync) and a computer (PC). Demineralized water is used as fluid for all the experiments with the physical properties of the droplets at 20 • C (density d = 998 kg∕m 3 , surface tension d = 0.073 N∕m and viscosity d = 10 −3 Pa s ). Three different droplet sizes ( 0.69 ± 0.05, 0.99 ± 0.03 and 1.55 ± 0.02 mm ) are generated with the droplet generator. Chosen target particles consist of PMMA with diameters of 1.55 ± 0.007 mm and a density p = 1200 kg∕m 3 .

Methodology
In the experiments, the particle is placed using tweezers on the substrate and positioned using positioning device PD1, see Fig. 1. Before each experiment, the substrate is cleaned and dried using demineralized water and pressurized air. Additionally the substate is cleaned with iso-propanol. The particle-droplet-substrate interaction is recorded with a frame rate of 20,000fps with an exposure time of 49 μs . The image resolution is 400 × 512 px . Considering the zoom factor of the macro lenses a resolution of 73 px∕mm is achieved. Between 80 and 100 experiments with identical particle and droplet parameters provide one series of experiments. In total, six series of experiments with different parameter configurations are conducted.
For each of the impacts the eccentricity of the particledroplet-substrate interaction, the offset between the centers of gravity (CG) of particle and droplet, are determined as shown in Fig. 2a leading to the dimensionless centrality of the impact C I :  Δx describes the offset of the droplet CG to the particle CG measured with HS1 whereas Δy describes the offset of the two CGs measured with HS2. The sum of the deviations from the center is related to the arithmetic diameter of droplet d d and particle d p . The surface tension forces F can be derived from the measured contact and half-spread angle between the particle and the droplet. Following from Fig. 2 (right), the vertical component of the surface tension force which is the resulting vertical force F z , leads to the lift-off is given in Eq. 2. The behavior, while the angle passes through its minimum, can be seen as a critical point for the lift-off characterization, see Sect. 4.2.
The experimental results are evaluated using a set of dimensionless parameters. The Weber number We (ratio of the inertial forces to the surface tension forces of the droplet) and the Capillary number Ca (ratio of the viscous forces to the surface tension and inertial forces) are defined as with the density d and viscosity d of the droplet, surface tension d and the diameter of the droplet d d . The velocity of the droplet v d is the velocity before the collision which (1) is determined based on the image processing described in Sect. 3.1. Furthermore, the Reynolds number which is the ratio of inertia forces to viscous forces can be expressed by Weber and capillary number: The set of non-dimensional parameters is concluded by the geometric ratio Φ of particle diameter d p and droplet diameter d d and the centrality C I , defined in Eq. 1. Table 2 summarizes the range of the dimensionless parameters covered in this study.

Image processing
The parameters of the droplet-substrate-interaction are determined individually for each experiment by analyzing the single images from the high-speed cameras. The images of both cameras are processed using the library OpenCv of the programming language Python. The image Schematic explanation of determination of the centrality (left) and the acting forces of the droplet impact (right), where D is the droplet, P is the particle, M1/2 are the macro lenses of the cameras d p and d p are the diameters of particle and droplet, Δx and Δy describe the offset of the CG of a particle and CG of a droplet, T P is the tan-gent to the particle, F p is the weight force, F is the surface tension force, F Z is the vertical component of the surface tension force, is the contact angle between droplet and particle, is the half-spread angle of the fluid and is the angle between the vertical component and surface tension force processing is divided into five parts: The import of the raw image, the determination of the droplet parameters before the impact, the determination of particle parameters before and during the impact, the determination of the liftoff parameters and the determination of particle-droplet system parameters, like contact or half-spread angle. The following list and Fig. 3 describe the steps of the image processing.
• The greyscale image is transformed into a binary black and white image by a threshold which is based on a sensitivity analysis of the possible threshold range. The images are depicted in 8 bit grey scale. The pixels of the contour of the droplet-particle system are stored in a list. • The CG of the droplet ( CG d ) before impact is determined by a function for the calculation of the center of a contour. The diameter of the droplet is determined by the maximum distance of white pixels in x-and y-direction. The velocity is determined by the displacement of CG d between sequential images and the frame rate of the highspeed camera. Droplet velocity and diameter are both averaged from three images before the impact, to reduce the influence of the oscillation of the droplet surface.
• The determination of the particle diameter is similar to the droplet diameter, but only with one image before impact. • To determine the CG of the particle ( CG p ) for all images of an experiment it is necessary to mask its shape before the impact with an ideal circle (gray circle in the figure). For later images the position of this circle is adjusted to accurately map the bottom rows of the particle (red pixels in the figure). CG p is set equal to the CG of this mapped circle. • The CG of the particle-droplet system ( CG s ) after the impact is determined similarly to the CG d before the impact. The x-value the centroid is used to split the image into a left and a right part. • An overlap of the masked circle and the detected contour of the particle-droplet system is used to detect the pixel of the three-phase-point (3PP). • To quantify the movement of the particle-droplet system, the flight height and time of flight are tracked. The code calculates the current lift-off state by identifying the number of pixels between the bottom of the particle and the substrate (bottom edge of the image). If there is at least one pixel inbetween, lift-off state is detected. The time of flight is determined by summing up the number of images with lift-off state and multiplication with the inverse of the frame rate. • The half-spread angle is determined based on the coordinates of the CG p and the two 3PPs. • The contact angles are calculated between the tangent of the particle surface and the tangent of the droplet contour at the 3PPs. The tangent of the particle is determined by the grey circle and the 3PP using a python function. The tangent of the droplet is determined by the 3PP and the coordinates of a pixel of the droplet contour near the 3PP. The selection of the second pixel is subject to a case distinction depending on the wetted particle area and some test studies. • All calculated values are averaged based on both camera views.
The used methods of image processing are validated by a comparison between calculated and manually measured values for contact angle and half-spread angle in the wetting phase and the reached height h max for six experiments of three different series. The contact angle has an average deviation of 4.6 % with a standard deviation SD of 4.00. the half-spread angle has a deviation of 7.25 % with SD 10.45 and the maximum reached height h max has a deviation of 0.02 % with a SD of 0.04. The manually determined values are based on raw images. For the determination of the reached height the deviation is quiet small. The determination of the angles is a bit less accurate but still suitable.

Results and discussion
The results are presented in the following sections starting with a detailed explanation of the different phenomena observed in the experiments. In a next step, the influence of the eccentricity of the impact is analysed. Then, the characterizing angles between droplet and particle at the point in time of the lift-off are investigated. An analytical model is presented, quantifying the probability for lift-off and finally a map of the observed impact regimes in terms of the configuration of the particle-droplet-substrate interaction is given.

Impact phenomena
The impact of droplets on non-fixed particles placed on a substrate shows four different phenomena depending on the Weber number We, the droplet-to-particle diameter ratio Φ and the centrality C I of the impact. Figure 4 shows the phenomena exemplarily for a chosen set of parameters in each region, whereas two cases (A and B) show a lift-off. A-Lift-off during initial recoil is shown in Fig. 4a. The droplet has a Weber number of We = 34.56 and collides on a spherical particle with an eccentricity of the impact C I = 5.43 % . The size ratio of the droplet to the particle is Φ = 1.1 . The droplet touches the particle at time t 0 = 0 ms and wets the surface of the particle nearly symmetrically. After 1.4 ms the droplet covers almost half of the surface of the sphere without having any contact with the substrate. The slight offset of 5.43 % initiates a rolling movement of the particle in horizontal direction, which increases the asymmetry of the particle-droplet system. Due to the surface tension, a contraction of the droplet can be observed. The surface tension forces during the contraction of the droplet lead to a lift-off of the particle at t = 2.3 ms , which is followed by a period of approx. 15 ms where the particle is in flight. During the flight phase, the droplet bounces several times on the substrate. The particle-droplet system turns and the fluid finally touches the substrate at 18.5 ms , which is due to the rolling movement at the beginning of the process. As the droplet touches the substrate, surface tension forces lead to a wetting of the surface. The particle is fixed to the substrate and the movement of the particle stops. Fig. 4b. A droplet with a Weber number We = 18.41 impinges on a spherical particle with an eccentricity of the impact C I = 4.16 % . The size ratio of the droplet to the particle is Φ = 0.64 . After the impact of the droplet, a symmetrical wetting of the surface can be observed as in case A. Due to the smaller diameter ratio an oscillation of the droplet on the particle which is still in contact with the substrate can be seen. A rolling movement of the sphere can be observed and the oscillating droplet leads after 3.7 ms to a small lift-off of the particle. The movement stops if the droplet touches the substrate followed by wetting of the surface and pinning of the particle.

B-Lift-off during a later recoil is shown in
C-No lift-off, deposition of droplet on the particle is the next observed phenomenon shown in Fig. 4c where a droplet with a Weber number We = 35.36 impinges on a spherical particle with an eccentricity of the impact C I = 8.36% . The size ratio of the droplet to the particle is Φ = 0.43 . The behavior of the droplet motion on the particle is very similar to case B. Due to the smaller diameter ratio, only an oscillation of the droplet on the surface of the particle can be seen without any lift-off or rolling of the particle. Fig. 4d. An impact with Weber number of We = 94.6 and an eccentricity of the impact C I = 1.52 % is shown. The size ratio of the droplet to the particle is Φ = 1.1 . As the Weber number is much larger than in case A, the entire particle surface is covered by the droplet after the impact. During the covering, the droplet forms the characteristical peripheral rim. The droplet touches the substrate and leads to a maximum wetting of the surface. Small magnitude surface waves of the droplet, which were previously described by Mitra et al. (2017), can be observed. The particle is still enclosed in the droplet. At the end of the process, the particle partially Page 7 of 13 44 dewets and swims in the droplet. The observed phenomena C and D are similar to previous investigated droplet-particle collisions for fixed particles, e.g. reported by Hardalupas et al. (1999) or Mitra et al. (2017). Figure 5 shows a map of the individual regimes for the four described impact phenomena, based on the conducted experiments. The hatched areas represent the known parameter range of the different regimes. The map represents experiments with a certain centrality ( C I ≤ 10% ). The two phenomena in which a lift-off of the particle is observed (A and B) delimit a region which has a minimum required droplet-to-particle diameter ratio ( Φ > 0.5 ). The regime of phenomenon B, forms the transition zone between A and C. For a parameter combination with lower values of Φ , the phenomenon C occurs. On the one hand, the force applied to the particle to pull it up is too low. On the other hand, there is not enough fluid to overflow the particle and to wet the substrate. It is assumed that the region in which C occurs extends into a range of higher Weber number for small Φ for the investigated parameter range. Furthermore, there seems to be a maximum allowable Weber number for which lift-off can occur because the droplets with Φ > 0.5 would otherwise have too much energy, overflow the particle and wet the substrate directly (phenomenon D). For the regime of D it is assumed that there is a lower limit of Φ above which only wetting of the substrate occurs for all Weber numbers. For the boundary of the regime of D it is assumed that for decreasing Φ larger Weber numbers are necessary.

D-No lift-off, wetting of substrate is shown in
Further investigations are required for a more detailed assignment of the blank areas. This is done within the scope of this work with an analytical model to differentiate between lift-off and no lift-off, see Sect. 4.3.

Analysis of the interaction
As shown qualitatively in Sect. 4.1, the motion of the particle-droplet system depends on the dimensionless number Weber number We, the droplet-to-particle diameter ratio Φ and the eccentricity C I of the impact. In the following section, the motion of the particle-droplet system is analysed in detail. Regarding the given x, y, z coordinate system, the wetting of the particle after the impact described by contact angles as well as the length of the wetted area is considered. Especially the influence of the eccentricity of the impact C I is investigated. In the case of lift-off (A and B), an exact central impact leads to a motion only in z-direction. The motion of the particle-droplet system in the x-or y-direction increases with rising eccentricity. In order to compare different parameter combinations, the maximum height h * and the time of flight t * of the impact are taken into account in the dimensionless form shown in Eqs. 7 and 8 which is based o n a n e n e r g y b a l a n c e ( 1 2 m d v 2 d = m p gh max with m = 6 d 3 and p ≈ d and = Figure 6I shows the dimensionless height h * (left) and the time of flight t * (right) as a function of the centrality C I for a total of 92 experiments in the range 49.43 < We < 51.42 and 0.62 < Φ < 0.66 , (resp. in Fig. 6II 31.32 < We < 34.56 and 1.04 < Φ < 1.12 with 76 experiments). It can be seen that the particle motion is subjected to fluctuations. Unavoidable slight pressure variations during droplet generation, resulting in unstable oscillations of the generated droplet, as well as irregularities in the sphericity of the particles, or slight impurity of the particles are reasons for the deviations of the observed particle motion between experiments with identical parameter configurations. In the case of the calculation of time of flight the fluctuation increases for particles that often bounces on the substrate and reaches only small heights. For this reason, a large number of experiments were performed to identify trends as a function of eccentricity. The conclusion that can be drawn is that regardless of whether height or time is considered, more motion is observed with less eccentricity. Depending on the experimental parameters We and Φ there is a limit in the eccentricity for which no lift-off occurs. In this case, the drop slightly touches the particle and directly wets the substrate without exerting any recoil on the particle.
In the following, the behavior of the droplet geometry during the impact is analyzed. Figure 7 shows the wetting behavior of the particle by the droplet after the impact, based on the three angle contact angle, vertical angle and fullspread angle (see Fig. 2right) as a function of time for three parameter combinations representing the regimes of A and B where lift-off takes place (I and II are series of experiments Fig. 5 Map of impact regimes of the phenomena A-D in in terms of Weber number We and droplet-to-particle diameter ratio Φ appertain to the regime of A and III appertains to the regime of B).
The point in time at which the lift-off takes place shows some deviation between the different experimental configurations. In theory, the droplet exerts the maximum vertical force on the particle when = 0 . This is the critical point if lift-off takes place during the recoil of the droplet. If the particle droplet configuration is non-critical and benefits lift-off, the lift-off can start former. Considering the first experiment ( Fig. 7I) with We = 52.13 and Φ = 0.66 the lift-off takes place directly after passing the minimum of the angle . The configuration is near the critical conditions, which can be seen also in the maximum reached height of the lift-off which is very low. In comparison the second shown experiment (Fig. 7II) is non-critical, the lift-off starts before the minimum of and the maximum reached height is greater. Both of these experiments appertain to the regime of A, because the lift-off is during the first recoil of the droplet. The third shown experiment demonstrates a lift-off during a later recoil of the droplet on the particle as described in Sect. 4.1 phenomenon B. Due to the necessary rotation of the particle which has to start first, the lift-off do not start after passing the first local minimum of (which is also the global minimum in the process), but it takes place after passing one of the later local minima. A distinction between critical and non-critical lift-off would be pointless, since in regime of B all lift-offs can be seen as critical.
All lift-offs that take place before or during the passing of the first (global) minimum of the angle can be assigned to the regime of A. The determination of whether phenomenon A occurs for a given particle-droplet configuration must therefore take place at the time when min( ) applies.

Analytical model for lift-off
As discussed in Sect. 4.1, the lift-off of the particle-droplet system from the substrate is determined by the surface tension force of the droplet and the gravitational force of the combined particle-droplet system. Hence the ratio Ω of the surface force F and the gravitational force F G gives an indication if a particle-droplet system lifts off the substrate for a given droplet-to-particle diameter ratio and droplet impact Weber number. In the model, a central impact is assumed. Furthermore, the spreading of the lamella is assumed to be symmetrical and the droplet at maximum spread has the shape of a spherical cap on the particle (Attané et al. 2007). Figure 8 depicts a schematic of the droplet impact and subsequent film spreading. The top circle indicates the undeformed droplet before impact with a diameter of d d and impact velocity v d . In gray, the droplet contour at maximum spread is outlined with a contact angle . P TP indicates the three-phase point which is defined by the half-spread angle , the arc length of the wetted surface of the particle L p respectively. With the assumption of symmetrical spreading, Fig. 6 Dimensionless maximum height h * and dimensionless time of flight t * for impacts with different eccentricity C I for two series of experiments (I and II) Fig. 7 Contact angle , fullspread angle 2 and resulting vertical angle for different droplet-particle-substrate interactions. The analysis of case II, provides valid values only for times < 4 ms central impact and the formation of a spherical cap on the particle the surface tension force is defined as where r * is the radius of the wetted surface. The gravitational force of the particle-droplet system is given by where g is the acceleration due to gravity and are the mass of particle and droplet. The densities of droplet and particle are d and p . Hence the force ratio Ω in the z-direction is given as For critical lift-off, the force in z-direction is at its maximum, so the angle is equal to zero. The equation simplifies to: As seen in Fig. 8 the radius of the wetted surface is a function of the half-spread angle and the radius of the particle r p = d p ∕2 , hence and m p = . .
The half-spread angle can be modeled according to Khurana et al. (2019) as a function of droplet-to-particle diameter ratio Φ and the Weber, Reynolds and Capillary number: Equation 15 is solved numerically to obtain the half-spread angle which is used in the ratio of the forces. The calculated value corresponds to the angle at maximum wetting during the droplet impact. The experimentally observed half-spread angles are comparable to Khurana's studies. Now Eq. 12 can be solved to identify regions with lift-off ( Ω > 1 ) resp. no lift-off ( Ω ≤ 1 ). The model is not applicable to particle-droplet-substrate interactions where substrate wetting occurs during the initial impact and recoil of the droplet, cf. Sect. 4.1. The result of the analytical model is an assignment of the interaction behavior (lift-off or no lift-off) as function of the droplet-to-particle diameter ratio Φ and the Weber number We . After a validation of the model by a comparison between the experimental and the analytical results, the new information about the lift-off behavior can supplement the map of impact regimes (cf. Fig 5). Areas where the model predicts lift-off, i.e. Ω > 1 can be found in the grey-colored area which is enclosed by the black line. The resulting map of impact regimes is shown in Fig. 9, including the comparison between experiments and model. Experimental results where a lift-off can be observed are shown with green circular data points. Situations with no lift-off are indicated by red rectangular data points. For droplet-to-particle diameter ratios larger than 0.5 the experimental and numerical results show the same trends. The regime of phenomenon A, visualized by the hatched area, is connected to the predicted parameter combinations for which lift-off occurs ( Ω > 1).
In in the transition zone between regime of B and C ( Φ < 0.5 and We < 40 ), some discrepancies between model and experimental results appear, which can be related to small inaccuracies in the experiments and the simplifying assumptions of the model. In this area, the model cannot provide any new information on the boundaries of the impact regimes.

Conclusion
The impacts of droplets on non-fixed particles was analysed experimentally. The Weber number We, particle to droplet diameter ratio Φ and the eccentricity C I of (14) sin = 2r * d p .
We 12

Fig. 8
Schematic of the droplet impact and subsequent film spreading at maximum spread of the droplet on the particle surface the impact were varied within the ranges 13 ≤ We ≤ 94 , 0.4 ≤ Φ ≤ 1.1 and 0% ≤ C I < 100 % . The investigations focused on the occurring phenomena of the collision between water droplets and spherical particles loosely placed onto a substrate. The findings are as follows: 1. Different impact phenomena were detected depending on droplet-to-particle diameter ratio Φ and Weber number We : A-Lift-off during initial recoil of the droplet, B-Lift-off during later recoil of the droplet, C-No lift-off, deposition of the droplet on the particle, and D-No lift-off, wetting of the substrate. The phenomena C and D were previously reported also for droplet impact on fixed spherical particles, e.g. by Charalampous and Hardalupas (2017). A map showing the regimes of the different phenomena depending on the experimental parameters is given for impacts with sufficient centralities. 2. It has been shown that for a given Weber number and droplet-to-particle diameter ratio the characteristic behavior of the particle-droplet system depends on the eccentricity of the impact C I . If the requirements for liftoff, considering We and Φ are given, the reached height and the time of flight of the particle-droplet motion are rising with increasing centricity. From a certain level of decentrality, there is no longer any lift-off of the particle-droplet system (see Fig. 6). The assumption in the analytical model, that a perfect head-on collision is given for critical lift-offs is confirmed.
3. The lift-off during the first recoil of the droplet (A) of the particle-droplet system at critical Weber number We and droplet-to-particle diameter ratio Φ configurations takes place at the maximum amount of the vertical component of the surface tension force. This is given when the angle , which can be determined by the contact angle and the half-spread angle , is at its minimum (see Fig. 7). This finding is used in the analytical model, simplified to = 0 , for the determination of the value of Ω 4. A simplified analytical model to determine the behavior of a given particle-droplet configuration has been developed. The model is based on the ratio between surface tension force of the droplet and the gravitational force of the particle-droplet system, called Ω . This model indicates that the vertical component of the surface tension force has to be higher than the gravitational force for the particle-droplet system to induce a lift-off from the substrate. The results of the model can be used as supplementary boundaries in the map of impact regimes, see Fig. 9.
Author Contributions G, F and J carried out the experiments. J, F and B developed the theory. J, G and J designed the model and the computational framework and analysed the data. All authors discussed the results and contributed to the final manuscript. Fig. 9 Map of impact regimes in terms of Weber number We and droplet-to-particle diameter ratio Φ including experimental and numerical results, where A is Lift-off during initial recoil, B is Lift-off during a later recoil, C is No lift-off, deposition of droplet and D is No lift-off, wetting of substrate. The gray area indicates parameter combinations with predicted lift-off by the analytical model