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Predictive Framework for the Spreading of Liquid Drops and the Formation of Liquid Marbles on Hydrophobic Particle Bed Arul Mozhi Devan Padmanathan, Apoorva Sneha Ravi, Hema Choudhary, Subramanyan Namboodiri Varanakkottu, and Sameer Vishvanath Dalvi Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b00698 • Publication Date (Web): 30 Apr 2019 Downloaded from http://pubs.acs.org on May 1, 2019

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Predictive Framework for the Spreading of Liquid Drops and the Formation of Liquid Marbles on Hydrophobic Particle Bed Arul Mozhi Devan Padmanathan 1#, Apoorva Sneha Ravi 1# , Hema Choudhary 1, Subramanyan Namboodiri Varanakkottu 2 and Sameer V. Dalvi 1* 1 Chemical

Engineering, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar - 382355, Gujarat, India

2Department

of Physics, National Institute of Technology Calicut, Kozhikode - 673601, Kerala, India

ABSTRACT In this work, we have developed a model to describe the behavior of liquid drops upon impaction on hydrophobic particle bed and verified it experimentally. Polytetrafluoroethylene (PTFE) particles were used to coat drops of water, aqueous solutions of glycerol (20, 40 and 60 v/v %) and ethanol (5 and 12 v/v %). The experiments were conducted for Weber number (We) ranging from 8 to 130 and Reynolds number (Re) ranging from 370 to 4460. The bed porosity was varied from 0.8-0.6. The experimental values of 𝛽max (ratio of the diameter at the maximum spreading condition to the initial drop diameter) were estimated from the time lapsed images captured using a high-speed camera. The theoretical 𝛽max was estimated by making energy balances on the liquid drop. The proposed model accounts for the energy losses due to viscous dissipation and crater formation along with a change in kinetic energy and surface energy. A good agreement was obtained between the experimental 𝛽max and the theoretical 𝛽max estimated. The proposed model yielded least % Absolute Average Relative Deviation (%AARD) of 5.5  4.3 when compared with other models available in the literature. Further, it was found that the liquid drops impacting on particle bed are completely coated with PTFE particles with 𝛽max values greater than 2. 1 ACS Paragon Plus Environment

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KEYWORDS: Liquid Marble, Solid liquid interface, viscous dissipation, maximum spread, crater formation, kinetic energy, energy balance, drop spreading # Both

authors have contributed equally.

* To whom correspondence should be addressed: Email: [email protected], Phone: 091-79-3241-9529

2 ACS Paragon Plus Environment

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Introduction Liquid drops when released from a certain height on a bed of hydrophobic particles get encapsulated in a layer of particles. Such liquid drops coated with particles are called as Liquid Marbles (LM). David and Quere[1] reported formation of LM for the first time when they rolled a small drop of liquid in hydrophobic lycopodium powder leading to its encapsulation. This powder encapsulated liquid drop was perfectly non-wetting and could move easily by rolling. Since then there has been an increase in research activities focused on LMs. The hydrophobic shell of LMs provides a protective cover to the liquid drop making LMs selectively permeable to gases

[2]

and restricts contact of outside solid or liquid with the liquid inside LM. Reports

available in the literature demonstrate that LM can be used as effective micro reactors biological applications

[5-9, 11-12],

for 3-D cell culturing

[11]

and as gas sensors

[10]

[3, 4]

for

due to their

selective permeability to gases. Also, magnetic nanoparticles can be used to produce magnetic LMs

[13]

which display superb resistance to compression

closed controllably using a magnet

[15]

[14].

Magnetic LMs can be opened or

which allows for easy analysis of droplet contents. Other

innovative applications of LMs include encapsulation of pressure sensitive adhesive by hard shelled nanoparticles

[16],

and heterogeneous catalytic reduction

[17].

LMs can easily be

manipulated by varying the pH [21], through acoustic levitation [22], by magnetic control [23, 25] and light.

[26-27]

LMs have therefore been increasingly attracting the attention of researchers all over

the world. [16, 18-20] LMs can be synthesized by: (i) rolling or shaking of drops on a hydrophobic powder bed[28] (ii) mixing with an optimal liquid-particle ratio[29] and (iii) releasing liquid drops on a hydrophobic powder bed from a certain height.[30,31] The latter method can be easily used for a large scale production of uniformly coated LM of the same size.


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In this work, we have studied the dynamics of LM formation by releasing liquid drops on hydrophobic particle beds from different heights. When a liquid drop falls from a height on a hydrophobic particle bed, four different outcomes are possible. These outcomes are shown in Figure 1. Liquid drops can transform into a partial coated LM (Figure 1a), a completely coated LM (Figure 1b), a deformed LM (Figure 1c) or shatter to form daughter droplets (Figure 1d).

a

b

c

d

Figure 1. Snapshots of different outcomes of drop impact on hydrophobic particle bed (a) partially coated liquid marble (b) completely coated liquid marble (c) deformed marble (d) shattering of drop to yield daughter droplets Several studies have been published on formation of liquid marbles.[28-37] These reports have provided a framework for successful LM formation[31,37] by developing an understanding of the 4 ACS Paragon Plus Environment

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influence of parameters such as hydrophobic particle size[37,38], drop diameter (Do)

[37],

surface

energy[36-39], viscosity (𝜇𝐿)[34] and initial drop kinetic energy (𝐸𝐾𝑖)[30-32] on LM formation. An important parameter that has been central to understanding the dynamics of marble formation is the non-dimensional maximum spread of the liquid drop, βmax which is the ratio of the diameter (Dm) of the drop at maximum spread and the initial diameter of the drop (Do). The diameter of the drop at the maximum spreading (Dm) is an essential parameter to understand the interplay of several forces acting on the droplet namely, inertia, capillarity, surface roughness and viscosity. Understanding and controlling droplet spreading is crucial in many applications such as spray coating[40], pesticide deposition on plant leaves[41], inkjet printing[42], bio-array design[43] etc. Precision in agricultural spray applications involves increasing spreading diameter and decreasing droplet rebound to enhance drop retention on the leaf surface and to combat environmental pollution. The crops have an inherent defense mechanism against pests since leaves have a hydrophobic waxy surface but this also causes low retention of pesticides Several researchers have used polymeric additives charged polyelectrolytes

[46],

[45],

and vesicular surfactant

surfactants

[47]

[44],

[44].

solutions of oppositely

to enhance droplet deposition on the

leaves. Similarly, successful inkjet printing needs precise deposition of ink by control of its spreading diameter before its drying. Interaction of ink droplet with the solid surface direct indication of the quality of inkjet printing

[48].

Lim et al.

[49]

[48-50]

is a

used picoliter drops of water

and ethylene glycol to understand spreading and evaporation processes on silicon substrate at room temperature and at 40°C. In case of Forensics, understanding the spreading dynamics of blood droplet helps gaining more information about the crime. Smith et al. [51] studied dynamics of blood droplet to reconstruct crime scenes. In another recent paper, Smith et al.


[52]

studied

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blood drop deformation onto different non-porous surfaces and were able to define a splashing threshold for blood droplets as a function of impact velocity and roughness of different surfaces. While there are several studies available in the literature[28-37] which provide insight to the dependence of droplet spreading behavior and LM formation on surface energy, viscous dissipation and kinetic energy, contradicting observations have been reported regarding limiting values of Weber number (We) corresponding to LM formation for a given particle size and drop diameter. This inconsistency in the reported literature results mainly from the use of an empirical approach to understand LM formation. Eshtiaghi et al. (2009)

[32]

demonstrated that an increase

in kinetic energy during drop impact experiments increases the particle coverage of liquid drops. Marston et al.(2010)[30] found that maximum spread of liquid drops on glass beads scaled with We1/5 for conditions of Bond numbers (Bo) less than 1 whereas for Bo>1, this scaling represented only an upper limit. Nefzauoi and Skurtys (2010) [34] found that maximum spread of water + surfactant solution scales as We1/4 whereas for liquids with higher surface tension (~70 mJ/m2) the maximum spread scales as We1/5. Supakar et al. (2016, 2017) [31, 53] found this scaling law to be We1/3 for liquid spread over glass beads. Zhao et al. (2015)

[54]

found the spreading of

liquid drops over soda lime glass beads to scale as We1/3. Zhao et al.(2017)[55] found that the maximum drop spread scaled as We1/4 for lower particle sizes (where capillary regime dominates) and it scaled as Re1/5 for higher particle size (where viscous regime dominates). Thus, different authors proposed different scaling laws and it appears that using only Weber number (We) may not help explain the drop spreading behavior completely. Moreover, despite the significance of βmax, no comprehensive model has been reported till date which estimates the βmax when a liquid drop is released from a certain height on a bed of hydrophobic particles. There are several models available in the literature which predict βmax,


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however these models have been developed for a liquid drop impacting on hard and nondeformable surfaces such as stainless steel,[56] glass[57,58] etc. Most of these models are mainly based on energy balances

[56-63]

on a liquid drop before the impact and at the maximum spread

condition. Apart from energy balance models, momentum balance models

[64-67]

have also been

used where impact parameters such as impact velocity (𝑈𝑜), drop diameter (𝐷𝑜), liquid viscosity(

(

𝜇𝐿), liquid density (𝜌𝐿), surface tension (𝛾𝐿𝑉), Reynolds number 𝑅𝑒 =

(

number 𝑊𝑒 =

𝜌𝐿 ∗ 𝑈2𝑜 ∗ 𝐷𝑜 𝛾𝐿𝑉

) were used to predict β

max.

) and Weber

𝜌 𝐿 ∗ 𝑈𝑜 ∗ 𝐷 𝑜 𝜇𝐿

It has been shown by Wildeman et al.

[63]

that momentum balances predict βmax well at lower viscosity ranges whereas at higher viscosity ranges energy balance models work better. One of the earliest models proposed by O.G. Engel [58]

where energy balances were made on a water drop impacting on glass plate covered with a

filter paper that was previously wet with acidified starch and potassium iodide solutions. Since then several researchers have made efforts to improve these models by considering the effect of viscous dissipation [68]

[56-63],

slip conditions

[63],

surface wettability

[68]

and atmospheric conditions

on the drop impact dynamics and the maximum spread (βmax). However, these authors have

given contradicting accounts of the extent to which these parameters affect the drop spreading. Healy et al.

[69]

and Ukiwe et al.

[60]

compared various models available in the literature for

prediction of βmax. Healy et al.[69] used these models[56,70-74] for the prediction of βmax and compared the predicted βmax values with the experimental βmax values from the literature.[74-78] On the other hand, Ukiwe et al.[60] used these models[59,65,70,71] to estimate βmax and compared it with their experimental studies on the impact of water and formamide drops on poly(methyl methacrylate) (PMMA), poly(methyl methacrylate/n-butylmethacrylate) (P(MMA/nBMA)) and poly(n-butyl methacrylate) (PnBMA) surfaces. Healy et al. model

[70]

[69]

found the Kurabayashi-Yang

yields the least % error of 9.9 ± 10.3 against experimental data retrieved from the 7 ACS Paragon Plus Environment

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literature whereas Ukiwe et al. [60] found that a modified Pasadideh-Fard[59] model yields least % error of 5.09 ± 5.05 with their experiments. Recent reports have also proposed models for the estimation of βmax mainly for the impact of liquids such as blue inkjet printer ink,[61] ethanol, water and glycerol[62] on hard surfaces such as glass, steel, aluminium and paraffin/thermoplastic film[62]. It should be noted that these models were developed for the impact of a liquid drop on flat and rigid surfaces. To the best of our knowledge, there are no models which can predict βmax values for an impact of a liquid drop on the deformable particle bed. This work, therefore aims at presenting a quantitative theoretical model for the prediction of the maximum spread of liquid drop during the LM formation on the hydrophobic particle bed which deforms upon drop impaction. The model has been developed by making energy balances on the drop impacting a hydrophobic particle bed. At lower impact velocities, the drop spreads and takes up the top layer of particles on the bed whereas at higher impact velocities, it leaves a hemispherical cavity in the bed which is called crater

[79-80]

(see Figure 2). The effect of surface

deformation on the maximum spread is considered by estimating energy loss due to crater formation

[54]

in addition to viscous and kinetic losses. The values of βmax predicted from the

proposed model have been validated against the experimental data. Liquid marble formation experiments were conducted using six different liquids (water, 20% glycerol, 40% glycerol, 60% glycerol, 5% ethanol, and 12 % ethanol) of varying viscosity and surface tension. Also, PTFE particles with two different particle sizes (1µm and 35 µm) and particle beds with three different bed porosities (ranging from 0.6 to 0.8) were used, and liquid drops were released from different heights. Further, to adapt models reported in the literature in order to predict spreading of liquid drops on the deformable particle bed system, these models where modified to include the energy


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loss due to crater formation (𝑬𝒄𝒅) and their predictions were compared with the experimental βmax estimated in this work.

Figure 2. Crater formation when a 40% glycerol drop (Do = 4mm) impacts onto a PTFE particle bed with porosity  = 0.72 and with impact velocity Uo = 1.25 m/s. Experimental setup Materials PTFE particles of two different sizes, 35 and 1 µm in diameter and glycerol (>99%) were purchased from Sigma-Aldrich Pvt. Ltd, India and Spectrochem Pvt. Ltd, India, respectively. The 2.5 cc syringe and 24-gauge needle used to produce liquid drops were purchased from BD Biosciences, India. Particle size and size distribution of PTFE particles were estimated using a Beckman coulter Laser Diffraction Particle Size Analyzer (LS 13 320). The mean size of 35 µm PTFE was found to be 33.9 ± 24.58 𝜇m with D10/D50/D90 to be 10.46/ 28.99/57.94 𝜇m whereas the mean size of 1𝜇m PTFE particles was found to be 1.65𝜇m with D10/D50/D90

to

be 0.63/

1.17/3.29 𝜇m. The size distribution of these particles is shown in Figs. S1(a) and (c) and the corresponding SEM images have been presented in Fig. S1(b) and (d).


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Figure 3. Schematic of the experimental setup for liquid marble formation a. Light source b. Particle bed c. Photron high speed camera d. Syringe pump e. Height gauge with needle Liquid Marble formation The experimental setup consists of a light source, a high-speed camera (FASTCAM Mini UX100 type 200K-C-8G with 4000 fps and 1280 x 1024 pixel size), a particle bed, a height gauge and a syringe pump (Figure 3). Similar set-up was used by Kwok et al. (2005) for drop impaction studies on solid surfaces. In this work, six different liquids were used namely, pure water, water with 20% glycerol, water with 40% glycerol, water with 60% glycerol, water with 5% ethanol and water with 12 % ethanol (all liquid compositions are on the volume basis). About 0.05% w/v of methylene blue (Sigma-Aldrich Pvt. Ltd, India) was added in these liquids to make sure that the contrast in images (such as the ones shown in Figure 1) taken by Dino capture 2.0 camera is visible. Drops were dispensed from the 24-gauge needle connected to a 2.5 cc syringe placed on the syringe pump (Mode NE-300, Just Infusion syringe pump, New Era Pump Systems, Inc., India). The needle was attached to an adjustable height holder. The liquid drops were released 10 ACS Paragon Plus Environment

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from various heights ranging from 2 to 13 cm for each of the six liquid mixtures onto the particle bed. A summary of the impact conditions while performing the experiment is given in Table 1. The drop diameter was measured using ImageJ software and was also verified by comparing the volume of a drop to the volume of a sphere of diameter 𝐷0 (see Table 1). Further, the drop oscillations (in terms of variation in drop size) during its flight once released from a particular height were also quantified by measuring drop diameter during its flight (See Fig. S2 and Table S1). The drop diameter was found to vary about 2-7 % from the mean drop diameter. To analyze the effect of packing of particles in bed on liquid marble formation, the loosely packed bed of particles was subjected to three different levels of compressions i.e. the gauge pressure of 0.153 kPa, 0.311 kPa and 0.623 kPa (Table 2). The corresponding bed porosities were calculated by measuring the volume occupied by PTFE particles in a known volume of a petri dish (see Table 2). Each experiment was repeated five times for a combination of a height, liquid and bed porosity. Various physicochemical properties of liquid mixtures used at 25 °C are listed in Table 3. Surface tensions and viscosities were measured using Kruss Drop Shape Analyzer (DSA-25) and Modular compact Rheometer (Model no. MCR 302, SN81285303) respectively. Contact angles of liquids over PTFE was measured by placing different liquid drop gently over PTFE surface prepared by attaching PTFE particles to microscopic glass slide using adhesive tape. Apparent contact angle of liquid drops on PTFE surface were captured using One Attension Theta Optical Tensiometer (Model no. C204A). Full range of contact angle is listed in Table 3. Particle bed was prepared by filling PTFE particles in a circular petri dish (depth = 5.45 mm and length = 25.5 mm).


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Wetting transitions Aurbach et al.

[81]

and Bormashenko et al.

[82]

reported that addition of solvent to the liquid

marble made of water induces wetting transition from Cassie to Wenzel state. Authors found that addition of ethanol above critical concentration of 20% (v/v) resulted in a wetting transition from Wenzel to Cassie state. In this work, concentration of ethanol used was well below 20% and hence chances of wetting transition are negligible. However, to verify if any wetting transitions occur, the experiments were performed by adding ethanol (5 and 12%) and glycerol (20, 40 and 60%) to liquid marbles made of water and the contact angles were measured (See Fig. S3 and Table S2). It can be observed that there are no significant changes in contact angles in case of glycerol (the contact angles changed by only about 2-8o). In case of ethanol however, the contact angle changed by about 10 – 16o. It should also be noted that the wetting transitions reported by Bormashenko[82] and Aurbach et al.[81] are for addition of a solvent to liquid marble and not for an aqueous solution drop impacting the particle surface where the contact angles could be different from the ones recorded during solvent addition in already prepared liquid marbles. The contact angles obtained for drops settled on the powder beds can be found out by placing drops of aqueous solutions directly on a particle bed. These contact angles are reported in Table 3 and it can be seen that these values of contact angles are different than the one presented in Table S2.


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Table 1. Summary of drop impact conditions during experiments Liquid

Drop diameter D (mm)

Impact velocity Uo (m/s)

Weber number (We)

Reynolds number (Re)

Bond number (Bo)

35 μm

1 μm

35 μm

1 μm

35 μm

1 μm

35 μm

1 μm

Water

2.93

0.4-1.54

0.54-1.29

8 - 116

13-72

1287 - 4460

1672 - 3981

1.17

1.32

20% Glycerol

2.89

0.6 - 1.54

0.54-1.29

17 - 107

14 - 80

1121 - 2993

1145 - 2653

1.28

1.43

40% Glycerol

2.76

0.6 - 1.6

0.54-1.21

19 - 116

16 - 81

1496 - 1761

699 - 1563

1.33

1.68

60% Glycerol

2.73

0.6 - 1.7

0.54-1.29

20 - 131

17 - 96

369 - 943

361 - 859

1.37

1.74

5% Ethanol

2.87

0.44-1.25

0.63-1.33

8-63

16 - 70

1233 - 3487

1743 - 3698

1.12

1.12

12% Ethanol

2.81

0.44-1.25

0.63-1.33

10-79

20 - 88

919 - 2599

1300 - 2757

1.38

1.38


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Table 2. Particle bed compression and corresponding particle bed porosities Pressure (kPa)

Bed porosity (𝜱) 35 μm

1 μm

0.153

0.72

0.80

0.311

0.69

0.79

0.623

0.61

0.77


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Table 3. Physicochemical properties of liquids used to make LM at 25°C Liquid

Density Viscosity

Surface

(kg/m3)

tension

(Pa.S)

Contact angle (Ө) with PTFE at different bed porosities 35 μm

1 μm

(mN/m) 𝜱 = 0.72

𝜱 = 0.69

𝜱 = 0.61

𝜱 = 0.8

𝜱 = 0.79

𝜱 = 0.77

Water

991.6

0.001

0.0712

139.5

146.80

147.5

125.4

127.2

129.0

20% Glycerol

1041.4

0.0015

0.0661

137.3

144.3

147.4

126.4

127.5

127.7

40% Glycerol

1114.6

0.0027

0.0626

134.5

136.9

155.4

127.4

129.7

129.9

60% Glycerol

1138.8

0.0053

0.0605

131.8

133.7

156.9

129.9

130.3

131.3

5% Ethanol

991.6

0.00102

0.0711

126.7

126.9

129.4

127.2

129.3

130.5

12% Ethanol

979.5

0.0013

0.055

124.3

125.1

127.8

119.2

123.4

124.8


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Modeling Prediction of max The process of liquid marble formation was considered to consist of following three stages: (a) release of a liquid drop from a particular height (b) impact of drop on the surface of the particle bed and its spreading on the bed and (c) retraction of the liquid disc and rebounding to form a liquid marble. Figure 4 schematically illustrates these stages. (a)

(b)

Do

Z Dm

Stage a

Stage b (c) Do

Stage c

Figure 4. Three stages of drop impaction process during liquid marble formation (a) release of a liquid drop from known height, (b) impact of drop on the bed particle surface, its spreading on the bed of particles and (c) retraction of the liquid disc and rebounding Model assumptions Following assumptions were made while developing the model to describe dynamics of drop impact on the particle bed:


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1. The energy loss due to drag force on the liquid drop during its fall through the air is negligible. 2. The shape of the drop during maximum spread can be assumed to be cylindrical.[59] 3. Half of the initial kinetic energy is dissipated in internal circulation (1/2-rule)

[63]

during

impact. Wildeman et al.[63] had simulated the drop impact using the fluid solver “Gerris” for different We and Re number combination and found from the energy budget calculated from the simulation that half of the initial kinetic energy is lost and not converted to surface energy for a drop impacting with We between 30 < We < 3000. 4. The contact angle during impact is 180 degrees because of the air gap between the substrate and the drop. [61, 63] Many researchers have suggested using the Young’s contact angle [60] and advancing contact angle [56, 59 and 83] drop impact study. However, using 180° as contact angle is appropriate because of the presence of air gap between the drop and the substrate just after the impact. A study by Klyuzhin et al. [84] verified the air gap theory by changing the air pressure around the impacting drop. Stage a: Release of a drop from a particular height and just before the impact on hydrophobic particles When released from a particular height, a liquid drop possesses kinetic energy and surface energy (Fig. 4a). Therefore, the total energy (E) of the drop just before the impact is, 𝐸𝑖𝑚𝑝𝑎𝑐𝑡 = 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑑𝑟𝑜𝑝𝑙𝑒𝑡 (𝐸𝐾𝑖) + 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝑒𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑙𝑖𝑞𝑢𝑖𝑑 (𝐸𝑆𝑖)

…(1)

where, 𝐸𝐾𝑖 = Potential energy of a drop held at height Z = mgZ and 𝐸𝑆𝑖 = 𝐴𝐷𝑟𝑜𝑝𝑙𝑒𝑡𝛾𝐿𝑉 = 𝜋𝐷20𝛾𝑙𝑣 Therefore, 𝐸𝑖𝑚𝑝𝑎𝑐𝑡 = mgZ + 𝜋𝐷20𝛾𝑙𝑣

…(2)

where, m is the mass of the liquid drop, 𝛾𝑙𝑣 is the vapor-liquid interfacial energy and 𝐷0 is the diameter of a liquid drop. 17 ACS Paragon Plus Environment

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Page 18 of 42

Stage b: Impact of liquid drop on the particle bed results in maximum spreading of liquid drop in the form of a disc At this stage liquid drop expands to a maximum spread upon impaction on the particle bed (Fig. 4b). Kinetic energy of the drop becomes zero at this moment. The only energy possessed by the liquid disc at this stage is its surface energy. The flattened drop can be assumed to be a cylinder as widely assumed in literature [57, 59, 60-63] with height h, and diameter Dm. Surface energy of liquid disc at maximum spread (𝐸𝑆𝑚𝑎𝑥) = Surface energy of liquid-air interface + Surface energy of a liquid in contact with the solid surface Therefore, 𝐸𝑆𝑚𝑎𝑥 =

𝜋𝐷2𝑚 4

𝛾𝐿𝑉 +

𝜋𝐷2𝑚 4

(𝛾𝑆𝐿 ― 𝛾𝑆𝑉)

…(3)

As per Young’s equation[85]: 𝛾𝐿𝑉𝑐𝑜𝑠𝜃 = 𝛾𝑆𝑉 ― 𝛾𝑆𝐿

…(4)

Substituting Young’s equation in equation 3 𝐸𝑆𝑚𝑎𝑥 =

𝜋𝐷2𝑚 4

…(5) where,

(1 ― 𝑐𝑜𝑠𝜃)𝛾𝐿𝑉

𝐷𝑚 is the diameter of a disc at maximum spreading and 𝜃 is the contact angle between liquid and particle bed (assumed to be 180o due to presence of air gap beneath the drop).

Energy loss due to viscous dissipation (𝑬𝒗𝒅) Energy lost due to viscous dissipation during maximum spreading (𝑬𝒗𝒅) of a liquid drop can be estimated using the following equation 𝑡

𝐸𝑣𝑑 = ∫0𝑚𝑎𝑥∫( )𝜏𝑑Ω𝑑𝑡 = 𝜏Ω𝑡𝑐 Ω

…(6) where,

𝜏 is the mean value of the viscous dissipation energy per unit time per unit volume and is equal to 𝜏 = 𝜇

2

(𝑉ℎ) . Also Ω = 𝜋4𝐷2𝑚ℎ which is the volume of the liquid disc where viscous dissipation 18 ACS Paragon Plus Environment

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Langmuir

2𝐷3𝑜

takes place (𝛿 > ℎ).[57] Here h is the thickness of the liquid disc and is given by 3𝐷2 whereas 𝛿 is 𝑚

thickness of a liquid layer to which momentum can diffuse and is given by time taken by the drop to spread maximum and is given by

2𝐷0 𝑅𝑒

. Further, 𝑡𝑐 is the

8𝐷0

3𝑉 .

After substitution of all the above parameters in Equation (6) and further simplifying, [57] we get: 𝐸𝑣𝑑 =

6 𝛽4𝑚 𝑅𝑒

…(7)

𝐸𝐾𝑖

𝛽𝑚 is ratio of liquid drop diameter at maximum spread (𝐷𝑚) to initial diameter (𝐷0).

Energy loss due to crater formation (𝑬𝒗𝒅) As explained earlier, impact of a liquid drop on the particle bed results in the formation of a cavity termed as a crater. The depth of this crater depends on the impact velocity, volume of the drop, porosity of the bed and respective densities of the liquid and particles. To consider the energy loss due to crater formation, an equation proposed by Zhao et al. [54] was adapted. Zhao et al.

[54]

developed this equation after analyzing the data available in literature for situations

ranging from rain drop impact on soil to the impact of asteroid strikes on planets. They calculated the energy lost in the formation of a crater during impact using the following equation, 𝐸𝑐𝑑 = 𝜑𝜌𝑝𝑉𝑐𝑔𝑑𝑐

…(8)

where, 𝑑𝑐 is crater depth, 𝜑 is volume fraction of the bed, 𝜌𝑝 is the density of the particles, 𝑉𝑐 is the volume of the crater, g is the acceleration due to gravity.


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Page 20 of 42

Further, it was found that the ratio between the crater depth (𝑑𝑐 ) and diameter (𝐷𝑐 ) is a constant with a value of 0.2. Therefore, …(9)

𝑑𝑐 = 𝛼𝐷𝑐 where, 𝛼 = 0.2 and 𝐷𝑐 =

𝜋 𝜌𝑃𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 (8𝛼2Φ 𝜌 )

―1 6

[(𝜌𝑔)

―1 6

1

1

]

…(10)

𝐷03 𝐾𝐸6

Substituting equation (9) & (10) in equation (8), one can obtain 𝐸𝑐𝑑 = 𝑘

1/3

𝐵𝑜 (𝑊𝑒 )

…(11)

𝐸𝐾𝑖

where, 𝑘 = 0.39

1 3

( ) is the crater dissipation constant, 𝐸 𝜑𝜌𝑝

𝐾𝑖

𝜌𝑤

is the initial kinetic energy

possessed by the drop.

Energy loss due to internal flows in the drop (𝑬𝑯 𝒅) As explained in the model assumptions, the impact of the drop on the substrate sets in the internal flow which consumes a part of the kinetic energy. This loss is treated as a head loss (𝐸𝐻 𝑑) because of internal flows and is called the 1/2-rule [63]. 1

𝐸𝐻 𝑑 = 2𝐸𝐾𝑖

…(12)

Energy balance equation By conservation of energy between Stage a and Stage b (see Figure 4) 𝐸𝐾𝑖 + 𝐸𝑆𝑖 = 𝐸𝑆𝑚𝑎𝑥 + 𝐸𝑣𝑑 + 𝐸𝑐𝑑 + 𝐸𝐻 𝑑 𝐸𝐾𝑖 + 𝜋𝐷20 𝛾𝐿𝑉 =

𝜋𝐷2𝑚 4

(1 ― 𝑐𝑜𝑠𝜃)𝛾𝐿𝑉 +

…(13) 6.3 𝛽4𝑚 𝑅𝑒

𝐸𝐾𝑖 + 𝑘

1/3

𝐵𝑜 (𝑊𝑒 )

𝐸𝐾𝑖 +

Rearranging,


1

2𝐸𝐾𝑖

…(14)

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Langmuir

(

𝐸𝐾𝑖 1 ―

6.3 𝛽4𝑚 𝑅𝑒

―𝑘

1 3

( ) ― 1 2) +𝜋𝐷20 𝛾𝐿𝑉 = 𝐵𝑜 𝑊𝑒

𝜋𝐷2𝑚 4

(1 ― 𝑐𝑜𝑠𝜃)𝛾𝐿𝑉

…(15)

Dividing by 𝜋𝐷20 𝛾𝐿𝑉 𝑊𝑒 12

(

1―

6.3 𝛽4𝑚 𝑅𝑒

―𝑘

1 3

( ) 𝐵𝑜 𝑊𝑒

―

1

)

2 +1 =

𝛽2𝑚 4

(1 ― 𝑐𝑜𝑠𝜃)

…(16)

Here, the contact angle is taken to be 180 degrees [61, 63] as explained in the model assumptions. Further simplifying equation (16) we get, 𝟔.𝟑𝟔 𝑹𝒆

𝜷𝟒𝒎

+

[ ] 𝟔 𝑾𝒆

𝜷𝟐𝒎

―

[

𝟏𝟐 𝑾𝒆

𝟏 𝟐

𝟏 𝟑

( ) ]=𝟎

+ ―𝒌

𝑩𝒐 𝑾𝒆

…(17)

Equation (17) can now be used to predict the values of βmax for a given set of conditions.

Stage c: Retraction of liquid after maximum spreading After the maximum spread, the drop starts retracting back mainly due to surface tension. It keeps retracting back until it attains its initial spherical shape (Fig. 4c). The liquid loses some energy due to viscous dissipation during retraction of the drop. The energy content of a drop at this stage can be given as, Surface energy at maximum spreading (at stage b) – Energy lost due to viscous dissipation during retraction where, energy lost due to viscous dissipation during retraction (𝐸𝑣2 𝑑 ) of a liquid drop can be estimated empirically[57] by 0.63 2.3 𝐸𝑆𝑖 𝐸𝑣2 𝑑 = 0.12 𝛽𝑚 (1 ― 𝑐𝑜𝑠𝜃)

…(18)

where, 𝐸𝑆𝑖 is the initial surface energy of the drop


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Page 22 of 42

The energy remaining with the drop after the energy lost due to viscous dissipation during ∗ retraction is the excess rebound energy, 𝐸𝐸𝑅𝐸 . The dimensionless excess energy (reported by

Mao et al.) can be defined as, ∗ = 𝐸𝐸𝑅𝐸

𝐸𝑆𝑚𝑎𝑥 ― 𝐸𝑣2 𝑑 ― 𝐸𝑆𝑓

…(19) where,

𝐸𝑆𝑓

𝐸𝑆𝑚𝑎𝑥 is the surface energy of the drop at maximum spread and 𝐸𝑆𝑓is the surface energy of the drop at final state after coating – Stage c (see Fig. 4c) The surface energy (𝐸𝑆𝑓) for a fully coated drop at Stage c 𝐸𝑆𝑓 = 𝜋𝐷2𝑜(𝛾𝑆𝑉 ― 𝛾𝐿𝑉𝑐𝑜𝑠𝜃)

…(20)

βmax value for complete coating of liquid drops The surface area of a spherical drop of diameter (Do) at rest is, 𝐴𝑜 = 𝜋𝐷20

…(21)

The interfacial area between the drop and the substrate at maximum spread condition is, 𝐴𝑚𝑎𝑥 =

𝜋𝐷2𝑚

…(22)

4

If we assume that the particles readily adsorb at the liquid surface during impact of a liquid drop on a particle bed then the interfacial area between the drop and the substrate at maximum spread condition must at least be equal to the surface area of the drop of size Do for complete coating. The interfacial surface area of a drop at the maximum spread in contact with the bed of solid particles can be given as

𝜋𝐷2𝑚 4

where Dm is the diameter at the maximum spread condition.

Therefore, one can write, 𝜋𝐷2𝑚 4

≥ 𝜋𝐷20

…(23)


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Langmuir

Simplifying the above equation one can show,

𝐷2𝑚 𝐷20

≥ 4 or βmax ≥ 2. Thus, βmax = 2 is a

boundary condition for completely coated LM and below this limit only partially coated LM can be formed. Results and Discussions Experimental observation of droplet impact on PTFE particle bed (a)H= 1.5 cm

t= 0 ms

t=7.25 ms

t=16.25 ms

t= 200 ms

t= 15 ms

t= 233 ms

t= 8 ms

t= 100 ms

(b)H=5.5 cm

t=0 ms

t= 4.25 ms (c)H=11.5 cm

t=0 ms

t= 3.5 ms

Figure 5. Sequential images of 3.1 mm diameter water drop impacting on PTFE 35 μm particle bed when released from different heights (A) H = 1.5 cm (B) H = 5.5 cm and (C) H = 11.5 cm. [ T y p e a q

[ T y p e a

23

ACS Paragon Plus Environment

q

[ T y p e

[ T y p e

a

a

q

q

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b

a

t= 0 ms

Page 24 of 42

d

c

t= 4.75 ms

t=17.25 ms

t=21.13 ms

Figure 6. Side view images of 5% ethanol drop spread on PTFE 35 μm bed of volume fraction ø =0.61. The images represent a sequence of events from drop impact, spreading, jet formation and formation of a completely coated liquid marble.

Figure 5 shows a sequence of images for a water drop (Do = 3.1 mm) impacting on PTFE particle beds with the bed porosity (ø) of 0.61. The drops were released from different heights of 1.5 cm, 5.5 cm and 8.5 cm with corresponding velocities of 0.54 m/s, 1.04 m/s and 1.3 m/s. The four image- sequence shows the drop just before impact, at the maximum spread, during retraction and at the recoiled state (at rest). A significant difference in the nature of the drop spread can be observed with the change in height. The drop spreads less at lower impact velocities (such as 0.54 m/s) whereas at higher impact velocities (such as 1.3 m/s) the spread is large with more finger-like protrusions. These finger-like protrusions observed at higher impact velocities result into unsymmetrical retraction pattern which ultimately leads to shattering and the formation of daughter droplets. Figure 6 shows the side-view images for the impact of 5% ethanol on the PTFE particles which again show typical steps involved in the liquid marble formation such as


Page 25 of 42

droplet spread (Fig. 6b), drop rebound through jet formation (Fig. 6c) and completely coated liquid marble (Fig. 6d).

Maximum Spread (βmax) vs Weber Number (We)

10.0

Experimental βmax

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Langmuir

Water_35 μm 20% Glycerol_35 μm 40% Glycerol_35 μm

Slope~0.3

60% Glycerol_35 μm 5% Ethanol_35 μm 12% Ethanol_35 μm Water_1 μm 20% Glycerol_1 μm 40% Glycerol_1 μm 60% Glycerol_1 μm 5% ethanol_1 μm 12% Ethanol_1 μm

1.0 1

10

We

100

1000

Figure 7. Normalized maximum spread (βmax) as a function of Weber number (We). The values of the maximum spread (βmax) were estimated for several combinations of liquids, bed porosities and release heights. Figure 7 shows a variation in the nature of the experimentally estimated maximum spread (βmax) vs We. The effect of viscosity on βmax can be clearly observed from Figure 7. Further, it is clear that βmax does not scale with We1/4[64] or We reported in literature. These reports

[67, 86, 87]

1/2

as widely

had computed βmax at different We for drop impact


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on rigid surfaces. Supakar et al.

[31]

analyzed drop impact behavior on particle bed surfaces and

concluded the power law exponent between βmax and We to be 0.32. From the data provided in Figure 7, βmax was found to scale with ~We0.3 which is in coherence with the value reported by Supakar et al. [31] Models Predictions for βmax Figure 8(a) shows the match between the experimentally estimated βmax and βmax predicted by the proposed model (equation 17). The model has an excellent agreement with the experimental data with % AARD of 5.5 (Table 4). The agreement is better for higher βmax values compared to lower βmax values. This limitation for very low impact velocities (< 1m/s) is attributed to the assumption of a flat cylindrical drop at maximum spread condition. [57] At low velocities, drop would not spread significantly resulting into a quite thick rim of the drop spread. Therefore, for very low impact velocities, the surface energy of the rim of the cylindrical drop is significant which results in these slight deviations since the model has neglected the surface energy of a rim. However, adding the term for the surface energy of the rim of the cylindrical drop at maximum spread complicates the expression to find βmax. As the deviation because of this assumption is limited, one might choose to add the energy term for the rim when dealing predominantly with drop impact velocity lesser than 1 m/s.


Page 26 of 42

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Langmuir

Table 4. Comparison of % AARD for model proposed in this work and its variations S. No.

Models

% AARD

1

Proposed model without 1/2-rule

24.77

2

Proposed model without crater energy dissipation

8.03

3

Proposed model without air gap theory

7.77

4

Proposed model (equation 17)

5.5


Langmuir

a) Proposed model %AARD= 5.5

3.5

3

Theoretical βmax

Theoretical βmax

b) %AARD= 8.03

3.5

3

2.5

2.5 2

2

1.5

1.5

1

1 1

1.5

2

2.5

3

1

3.5

1.5

c)%AARD= 24.77

3.5

2

2.5

3

3.5

3

3.5

Experimental βmax

Experimental βmax

3.5

3

Theoretical βmax

Theoretical βmax

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

Page 28 of 42

2.5

d) %AARD= 7.7

3

2.5

2

1.5

2

1.5

1

1 1

1.5

2

2.5

3

3.5

1

Experimental βmax

1.5

2

2.5

Experimental βmax

Figure 8. Comparison between experimental βmax plotted against the βmax calculated. a) Improved model. b) Without including the energy loss due to crater energy (equation 22) c) Without including the 1/2-rule (equation 24). d) Without 180° as the contact angle – air gap theory (equation 25). The dotted lines correspond to 10% AARD which have been added to enhance the visual recognition of the deviation 28 ACS Paragon Plus Environment

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Langmuir

The model proposed in this work estimates energy losses due to viscous dissipation, internal flows in the drop upon impaction on the particle bed and crater formation. Further, the model also assumes that the contact angle at the time of impaction to be 180°. These assumptions significantly reduce the % AARD for the model predictions as shown in Table 4. Below, we describe how exclusion of these terms from the proposed model affects the overall predictions and increase %AARD in predictions. Model without energy loss due to crater formation The energy balance equation (13) excluding the term for crater energy loss becomes, 𝐸𝐾𝑖 + 𝐸𝑆𝑖 = 𝐸𝑆𝑚𝑎𝑥 + 𝐸𝑣𝑑 + 𝐸𝐻 𝑑

…(21)

The corresponding equation to calculate βmax is 6.36 𝑅𝑒

𝛽4𝑚 +

12 1 + 2] = 0 [𝑊𝑒6 ]𝛽2𝑚 ― [𝑊𝑒

…(22)

Figure 8(b) shows the match between the experimental βmax and the βmax calculated using the proposed model but without accounting for the crater energy losses. It was found that the % AARD in this case is 8.03 (Table 4) with more values of βmax lying beyond the 10% error range. Thus, it is clear that the crater formation plays a significant role in the drop spreading phenomenon and the energy loss due to crater formation has to be considered while analyzing the impact dynamics. Model without 1/2-rule The energy balance (equation 13) excluding the term for the loss of kinetic energy due to internal circulations (1/2 rule) becomes, 𝐸𝐾𝑖 + 𝐸𝑆𝑖 = 𝐸𝑆𝑚𝑎𝑥 + 𝐸𝑣𝑑 + 𝐸𝑐𝑑

…(23)

The corresponding equation to calculate βmax is


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6.36 𝑅𝑒

𝛽4𝑚

+

[ ] 6 𝑊𝑒

𝛽2𝑚

―

[

12 𝑊𝑒

1 3

( )]=0

+1―𝑘

𝐵𝑜 𝑊𝑒

Page 30 of 42

…(24)

Figure 8(c) shows that without the 1/2-rule, the proposed model cannot predict βmax accurately. None of the calculated βmax falls within the 10% error range with all βmax values being overestimated. Consequently, the % AARD increases drastically to 24.77. Since the losses due to internal flow are not considered in this model without 1/2-rule, the drop has more energy with it and spreads to a greater extent. Thus, the βmax values predicted by this model are overestimated.

Model without considering air gap between the liquid drop and the substrate during impaction Further, the model also included 180° as the contact angle. If we exclude this from the energy balance, the corresponding equation to calculate βmax is 6.36 𝑅𝑒

𝛽4𝑚

+

[

3 𝑊𝑒(1

― 𝑐𝑜𝑠𝜃)

]

𝛽2𝑚

―

[

12 𝑊𝑒

1 2

1 3

( )]=0

+ ―𝑘

𝐵𝑜 𝑊𝑒

…(25)

Figure 8(d) shows that the exclusion of the air gap theory increases the % AARD to 7.7. From Figures 8(a)-(d) and Table 4, it is clear that including the energy losses due to crater formation, and internal flows in the drop ( ½ rule), and use of 180° as contact angle during drop impaction are the reasons why the proposed model depicts the real-life drop impact phenomenon better than other literature models. Comparison with literature models Different models available in the literature for spreading of liquid drops on solid surfaces were used for prediction of βmax for spreading of liquid drops on powder bed. Further, these models were modified to include crater energy dissipation term so that these models could be adapted to


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Langmuir

powder bed surfaces. The βmax predictions made by each of the original models and the models modified to include crater energy loss term were compared with the model developed in this work (Equation 17). Table 5 summarizes % AARD obtained for all the original literature reported models, literature models modified by adding crater energy loss terms and the model developed in this work. Fig. S4 presents the comparison between βmax values predicted by the models and the experimental βmax. It is evident from Table 5 and Fig. S4 that the % AARD is the least for the model proposed in this work as compared to all other models reported in the literature (with and without inclusion of crater energy loss). The inclusion of crater energy, ½rule, air gap theory and viscous dissipation energy facilitates the accurate estimation of the energy losses and hence the accurate predictions of βmax by the proposed model.

Table 5. % Absolute Average Relative Deviation for different models SN

Models

Methodology

% AARD w/o crater energy

with crater energy

1

Pasandideh-Ford et al., 1996

Energy balance

15.91

11.51

2

Mao et al., 1997

Energy balance

7.09

5.90

3

Ukiwe et al., 2004

Energy balance

11.4

7.14

4

Roisman et al., 2009

Semi-empirical model using momentum balances

21.47

-

5

Visser et al., 2012

Energy balance

9.82

6.86

6

Lee et al., 2016

Energy balance

16.76

12.33

7

Wildeman et al., 2016

Energy balance

32.38

28.62

8

Proposed model

Energy balance

8.03

5.5


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Condition for complete coating of liquid drops with particles: βmax vs Excess energy ∗ Figure 9 shows correlation between the Excess energy (𝐸𝐸𝑅𝐸 ) calculated using equation (19) and ∗ the experimentally estimated βmax. Interestingly, the 𝐸𝐸𝑅𝐸 values cross over from negative to ∗ positive at about βmax equal to 2. It is evident from Figure 9 that negative 𝐸𝐸𝑅𝐸 corresponding to ∗ the partially coated liquid marbles whereas the positive values of 𝐸𝐸𝑅𝐸 correspond to completely ∗ coated liquid marbles or shattering of liquid drops. As such 𝐸𝐸𝑅𝐸 signifies the excess energy that

the drop possesses after the maximum spread and retraction with respect to a completely coated ∗ liquid marble. Therefore, negative values 𝐸𝐸𝑅𝐸 indicate lower energy content of the drop after ∗ spreading and retraction. On the other hand, positive 𝐸𝐸𝑅𝐸 values indicate higher energy content

of the drop when compared to a completely coated liquid marble condition. Such a high energy can result either in a completely coated drops due to subsequent spreading and retraction or shattering of the liquid drop. We have already shown in Section 3.2 that the value of βmax of 2 can be shown to correspond to the completely coated liquid drop condition. The plot shown in Fig. 9 indicates clearly that the ∗ parameters 𝐸𝐸𝑅𝐸 and βmax can be used to identify the outcome of drop impact experiment. The ∗ negative values of 𝐸𝐸𝑅𝐸 and βmax < 2 indicate the partially coated LM condition whereas positive ∗ values of 𝐸𝐸𝑅𝐸 and > 2 indicate either a completely coated LM formation or shattering of the

liquid drop. Further, it is evident from Figure 9 that very high values of βmax and high values of ∗ mostly correspond to shattering of the drop upon impact. 𝐸𝐸𝑅𝐸


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2.0 Partially coated Completely coated Shattering

1.0 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

E*ERE Figure 9. Excess rebound energy (E*ERE) as a function of βmax.

Conclusions In this work, a model has been developed to predict the spreading behavior of a liquid drop upon impaction on a bed of hydrophobic particles. The theoretically predicted βmax were compared with the experimentally obtained βmax. The model developed in this work predicted βmax more accurately as evident from a % AARD of 5.5. The % AARD for other models available in literature was as high as 32.38. The models available in the literature were mainly developed for ,the spread of the drop on flat and hard surfaces and hence cannot accurately capture the drop impaction behavior on powder bed. However, the model developed in this work adapts very well to the powder beds by considering the air gap between the liquid drop and the substrate and the 33 ACS Paragon Plus Environment

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energy lost due to the formation of crater on particle bed, the viscous dissipation and internal ∗ circulations in the drop. Further, it was found that the βmax and excess energy (𝐸𝐸𝑅𝐸 ) can be used

for qualitative prediction of the outcome of the liquid marble impact experiment. The negative ∗ values of 𝐸𝐸𝑅𝐸 and βmax < 2 indicate the partially coated LM condition whereas positive values of ∗ and > 2 indicate either a completely coated LM formation or shattering of the liquid drop. 𝐸𝐸𝑅𝐸

Supplementary Information Figures: Characterization of PTFE particles used for liquid marble formation, Aspect ratio vs time plots to understand the oscillations observed in a liquid droplet during its free fall, Images taken to observe wetting transitions of water marbles after addition of glycerol to yield a final concentration of 60% Glycerol in the marble. Tables: Percentage deviation in droplet diameter from its average diameter during its flight when released from different heights Summary of change in contact angle observed in a water liquid marble coated with 2 different sizes of PTFE particles, namely 35 and 1 μm, Comparison of the proposed model with literature models: Pasandideh-Fard (P-F) Model, Mao Model, Ukiwe Model, Roisman Model, Visser Model, Lee Model and Wildeman model

Acknowledgements: The authors gratefully acknowledge the financial support from Ministry of Human Resources and Development (MHRD), Government of India and Indian Institute of Technology Gandhinagar (IITGN) to carry out this work. Dr. Subramanyan Namboodiri Varanakkottu greatly acknowledge the funding from the Department of Science and Technology, Ministry of Science and Technology, India through an INSPIRE faculty (Faculty award 2016/DST/INSPIRE/04/2015/000544). The authors thank Dr.Rochish M. Thaokar of Chemical


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Engineering at IIT Bombay for providing access to a PHANTOM high-speed camera. Authors would also like to thank Thamarasseril Vijayan Vinay, Pradip Singha and Mohit Singh for helping with the experimental setup.

Notations: D0 : Initial Drop diameter Dm : Maximum drop spreading diameter Greek Letters: 𝛾𝐿𝑉 : Liquid air interfacial tension (J/m2) 𝛾𝑆𝑉 : Solid air interfacial tension (J/m2) 𝛾𝑆𝐿 : Solid liquid interfacial tension (J/m2) µ : Viscosity(kg/(s·m)) 𝜌 : Density (Kg/m3)

Abbreviations: PTFE: Polytetrafluoroethylene AARD: Absolute Average Relative Deviation

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Table of Contents Graphic

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Shattering

2.0

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Completely coated LM

1.0 -0.6

-0.4

-0.2

0

0.2 E*ERE


0.4

0.6

0.8

Predictive Framework for the Spreading of Liquid ... - ACS Publications

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