Impact of particle-laden drops: Particle distribution on the substrate

Journal of Colloid and Interface Science 490 (2017) 108–118

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Regular Article

Impact of particle-laden drops: Particle distribution on the substrate Viktor Grishaev a,b, Carlo Saverio Iorio b, Frank Dubois b, A. Amirfazli c,⇑ a

Center for Design, Manufacturing and Materials, Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Building 3, Moscow 143026, Russia Service de Chimie-Physique EP, CP165-62, Université Libre de Bruxelles, 50 Av. F.D. Roosevelt 1050, Brussels, Belgium c Department of Mechanical Engineering, York University, Toronto, ON M3J 1P3, Canada b

g r a p h i c a l a b s t r a c t

a r t i c l e

i n f o

Article history: Received 6 September 2016 Revised 8 November 2016 Accepted 9 November 2016 Available online 10 November 2016 Keywords: Drop impact Suspension Particle distribution Splat Wettability Volume fraction

a b s t r a c t The splat morphology after the impact of suspension drops on hydrophilic (glass) and hydrophobic (polycarbonate) substrates was investigated. The suspensions were mixtures of water and spherical hydrophobic particles with diameter of 200 lm or 500 lm. The impact was studied by side, bottom and angled view images. At Reynolds and Weber numbers in the range 150 6 We 6 750 and 7100 6 Re 6 16,400, the particles distributed in a monolayer on the hydrophilic substrates. It was found that the 200 lm particles self-arranged as rings or disks on the hydrophilic substrates. On hydrophobic substrates, many particles were at the air-water interface and 200 lm formed a crown-like structure. The current study for impact of particle-laden drops shows that the morphology of splats depends on the substrate wettability, the particle size and impact velocity. We developed correlations for the inner and outer diameter of the particle distribution on the hydrophilic substrates, and for the crown height on hydrophobic substrates. The proposed correlations capture the character of the particle distributions after drop impact that depends on particle volume fraction, the wettability of both particles and the substrate, and the dimensionless numbers such as Reynolds and Weber. Ó 2016 Published by Elsevier Inc.

1. Introduction Many technologies are associated with the impact of particleladen drops such as additive manufacturing [1–3], printed ⇑ Corresponding author. E-mail addresses: [email protected] (V. Grishaev), [email protected] (C.S. Iorio), [email protected] (F. Dubois), [email protected] (A. Amirfazli). http://dx.doi.org/10.1016/j.jcis.2016.11.038 0021-9797/Ó 2016 Published by Elsevier Inc.

electronics [4,5], plasma coating technology [6] and spraying of liquid friction modifiers [7,8]. For these technologies, it is crucial to understand the role that particles and substrates play on drop impact phenomena as well as the fate of particles after impact. It is well known that for pure liquids, the impact morphology depends on Weber We ¼ qD0 U 2 =r and Reynolds Re ¼ qD0 U=l numbers. These non-dimensional numbers include the main

V. Grishaev et al. / Journal of Colloid and Interface Science 490 (2017) 108–118

parameters of the process: the drop diameter, D0 ; drop impact velocity, U; drop density, q; drop viscosity, l, and surface tension of liquid–air interface, r. There are also many other parameters such as the substrate roughness, ambient pressure, etc. to affect drop impact (details can be found in Refs. [9] and [10]). For particle-laden drops, significant changes occur depending on the particles’ size, volume fraction, /, and the substrate and particle wettability. Particle-laden drops can be in the form of liquid marbles and suspensions. Liquid marbles are liquid drops covered fully by particles in contrast to suspension drops where particles are locating inside of drops as well as drop surfaces. In this paper, we will study suspension drops, and henceforth will focus the discussion on such systems. Addition of particles can suppress the appearance of singular jet when drop retracts after impact on hydrophobic substrates [11] as well as partial drop rebound [11,12]; drop break-up during rebound from superhydrophobic substrates can also be suppressed in particle-laden drops [13] so as rebound [14]. Also, it was reported that particles can lead to drop splashing on hydrophilic [11,15,16] and hydrophobic [11] substrates. The splashing of suspensions can happen far away from the drop contact line, a phenomenon remarkably different from prompt or corona splash for pure drops [11]. Furthermore, the addition of particles can lead to drop fragmentation as a result of receding break-up or rupture of the drop’s lamella [11]. Particles not only lead to the suppression or the appearance of new phenomena in drop impact, but can also change the drop spreading. It has been shown that the addition of particles can reduce the maximum drop spreading factor, Dmax =D0 , where Dmax is the maximum diameter of the drop contact area during spreading [11,12,14,15,17]. Nicolas [15] proposed that it could be explained by effective viscosity but it can be questionable for systems where distinct particles exist (see below). The effective viscosity was calculated by the Krieger-Dougherty model, and was used in the estimation of maximum spreading factor [15]. The maximum spreading factor was found assuming that kinetic energy is dissipated mostly by viscous forces at Re  1 and We  1. For particle volume fractions much smaller than random close packing (which was taken as 0.68), the formula for maximum spreading factor was:

Dmax  D0




1=4 Re ½1  0:69  / 12

ð1Þ

The used value for the random close packing (/ ¼ 0:68) is questionable (see for example, Jaeger and Nagel [18]). Nevertheless, this may not be important, because Eq. (1) was obtained using the particle volume fraction much smaller than random close packing. Nicolas [15] found that this formula works for hydrophilic glass substrates in the range of Reynolds and Weber numbers: 79 < Re < 6000 and 10 < We < 370, respectively. However, Eq. (1) was not applicable to high Reynolds and Weber numbers 6000 < Re < 10000 and 370 < We < 1276. At these conditions, the maximum spreading factor increased with an increase of particle volume fraction. This observation is contradictory to the effect of an increase in viscosity that should result in reduced spreading; it was explained, however, by the non-circular shape of the splats caused due to a distorted contact line, or the drop break-up. Nevertheless, it seems that the effective viscosity is not quite able to explain the changes in the maximum spreading factor. Concerning splat morphology and particles’ arrangement in the splat, limited studies have been done. The limited studies that exist have examined the splat morphology for dense (particle volume fraction / P 0:5) and dilute suspensions (/ < 0:5), separately. The splat morphology for dense suspension drops on hydrophilic glass substrates was studied in the work of Lubbers et al. [19].

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The suspensions were mixtures of water or silicone oils seeded with hydrophilic particles of zirconium dioxide (average diameter: 250 lm). It was found that the suspension drops rapidly expanded to a monolayer at Weber numbers equal or greater than 1862 while the particles grouped into clusters separated by particlefree regions, i.e. a mesostructure was formed. The development of the mesostructure was quantitatively explained by using models deduced from the balance of forces acting on the individual particles. The forces acting on moving particles were viscous drag and surface tension of deformed air–liquid interface near particles. No analytical expression (or correlations) for the maximum splat diameter or splat shape, and the particle distribution after the impact, was provided. In the case of dilute suspensions, the splat morphology and the particle distribution have been considered in [15], where a ring distribution was seen for particles with diameter, dp , of 380 lm, and 720 lm at Reynolds numbers 3184 < Re < 3513, and Re  5000, respectively; on the contrary, for Re < 807 and particles with dp ¼ 640 lm, a disk-like distribution of particles was observed. The occurrence of the ring distribution was explained in [15] by the liquid oscillations (explained as the liquid movement towards the centre during recoil), which became larger as Reynolds number increased. When the disk or ring distribution was observed, the particles were absent near the drop contact line at a dimensionless distance from the centre of the drop impact (a distance divided by the equivalent radius of a drop contact area with a substrate) larger than 0:7. The absence of particles was explained by thin liquid film in this region, which prevented the particles moving closer to the contact line. There was no analytical relationship provided for the particle distributions. To summarize, although a number of valuable works on splats of suspensions with hydrophilic particles on hydrophilic substrates exist, no correlations for the particle distributions or analytical expressions exist. Therefore, the following questions remain: how particle distribution may be affected by substrate wettability, particle size, particle volume fraction, and drop impact velocities. Another question is that whether or not one can find an analytical relationship for the particles’ distribution in the splat. This experimental study aims to investigate mainly dilute suspensions that are less studied. Furthermore, it focuses on the hydrophobic particles that are scarcely studied to date. The idea is to gain a first impression of overall behavior of particles and their distribution in the splats upon impact of a particle-laden drop onto surfaces of different wettabilities.

2. Material and methods Main parameters of the experiments are presented in Table 1. For drop generation, water (deionized reagent grade III, Acros Organics) and a dispersion of polyethylene particles in water were used. The density and dynamic viscosity of water at room temperature were considered to be equal to 1 g/cm3 and 0.890 mPa s, respectively. Surface tension of water was measured by the pendant drop method (drop shape analyser DSA30S, KRUSS) and found as 72:8 mN/m at room temperature. Blue polyethylene spherical particles with diameter 180–210 lm (BLPMS-1.00 180–212 lm, Cospheric) and 425–500 lm (BLPMS-1.00 425–500 lm, Cospheric) were used in preparing the complex drops. The single particle wettability was characterized by measuring the contact angle of particles as they floated at the air-water interface. The procedure to conduct the measurements was to form a puddle of DI water on a polycarbonate plate with an approximate size of 70  70  71 (W  7L  7H) mm. A camera and the diffused light source of the drop shape analyzer (KRUSS DSA30S) allowed for implementing a shadowgraphy technique. The particles were

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Table 1 Experimental conditions, and physical characteristics of liquids and particles used. Liquid Liquid density, q, (g/cm3) Liquid viscosity, l (mPa s) Surface tension of liquid-air interface,

r (mN/m)

Water 1 0.890 72.8

Particles Particle density, qp (g/cm3)

Polyethylene spheres 0.99–1.01

Particle volume fraction, / Particle diameter, dp (lm) Static contact angle of water on a particle, hp (°) Drop diameter, D0 (mm) Drop velocity, U (m/s)

0–0.40 180–212 & 425–500 91 ± 5 & 95 ± 7 3.8 ± 0.1 1.7; 2.9; 3.7

Substrate Advancing contact angle of water, ha (°) Receding contact angle of water, hr (°)

Glass & Polycarbonate <5 & 102 ± 1 <5 & 79 ± 2

poured onto the centre of the puddle at the focal plane of the camera. The optical magnification of the DSA30S system was increased by using an extension tube 85 mm long. An image of a 200 lm particle on water surface is shown in Figs. 1SA and SB (in page 2 of the Supporting Material) before and after brightness adjustment, respectively. The brightness was adjusted to facilitate identifying the particle edge and the airwater interface. The contact angle of water on the particle was determined as follows. A circle was fitted onto the image of the particle above the air-water interface; a line was also passed through air-water interface. The air-water interface can be considered flat due to small Bond number ðBo < 102  1Þ which is given by Bo ¼ ðq  qa Þr2 g=c, where qa is the density of air, r is the radius of a particle and g is gravitational acceleration. In the intersection of the line and the circle, a tangent to the circle was placed. The angle between the tangent line and the liquid-air interface was taken as a measure of the contact angle of water with the particle as shown in Fig. 1SB. The contact angle of water based on eight measurements were 91 5 and 95 7 for the 200 lm and 500 lm particles, respectively. The confidence intervals of the contact angles overlapped, so statistically the average values of the contact angles are indistinguishable. The dispersions were prepared as follows: the particles were poured into a vessel containing water; the vessel was closed and shaken by hand, then the complex liquid were collected by the syringe from the bottom of the vessel. The details of drop generation can be found in the Supporting Material of a recent paper by us [11]. No surfactant was used for drop preparations and particles were both on the drop surface and inside of the drop before its impact on a substrate. We did multiple experiments. In each case, particles happen to be inside or in different parts of the surface as the drop fell. The way that the particles distribute varied from test to test. Considering this situation and seeing the results being the same, one can conclude that the dispersions tested on average behaves like a homogeneous one. In our experiments, the particle volume fraction varied from one experiment to another. The particle volume fraction was determined by dividing the number of particles in a drop by the volume of the drop. The volume of the drop was known during dispensing, and the number of particles were determined after drop impact and when all liquid evaporated. The dried layer of particles was imaged and the particles in the image were automatically counted by a program written in Visual C++. When particles agglomerate in a multilayer on hydrophobic substrates, before the count, the agglomerate was disrupted by wetting with HFE-7100 (from 3M), then flattened to one layer and dried again. HFE-7100 allowed

for wetting and break-up of the agglomerations easily due to its low surface tension (r ¼ 13:6 mN/m). We did 58 and 39 experiments for 200 lm particles on hydrophilic and hydrophobic substrates, respectively. For 500 lm particles, we did 24 and 20 experiments on hydrophilic and hydrophobic substrates, respectively. The details for the number of experiments at each We is provided in Table S1 in the Supporting Material. To allow observation of impact phenomenon from side and bottom views, we used transparent hydrophilic and hydrophobic substrates. As hydrophilic substrates, clean borosilicate glass slides (Nexterion slides glass B, Schott) with a size of 70  70  1 (W  L  H) mm were used. The glass slides were cleaned in a UV/ozone system (PSD-UV4, Novoscan). For hydrophobic substrates, polycarbonate plates (769-8720, RS components) with the same size as glass as used. Before experiments, electrostatic charges, possibly present in the substrates, were removed using ionized air generated by an anti-static gun (Milty Pro Zerostat 3, Armourhome). Wettability of the substrates was characterized by measuring the advancing and receding contact angles of deionized water using the drop shape analyze system DSA30S (KRUSS). For the glass slides, advancing and receding angles were less than 5 . For the polycarbonate sheets, advancing contact angle was 102 1 , and receding contact angle was 79 2 . The set-up for studying drop impact and particle distribution in splats is schematically shown in Fig. S2 (in page 3 of the Supporting Material). The drop impact was studied from side and bottom views by two monochrome high-speed cameras CL600  2 (Optronis) and EoSens MC1362 (Mikrotron), respectively. Cameras’ recording speed was 2500 fps and the exposure time equal to 6 ls. The spatial resolutions for side and bottom view cameras were 0.036 and 0.057 mm/pixel, respectively. The third camera (Samsung ST150F) was used to record the splats at an angle, after 1 s from the moment of the drop impact. 3. Results and discussion Intuitively it is easy to understand that the inertial force is an important factor determining the particle distributions on substrates after impacts. However, our experiments show that inertia is not the only important factor. Others such as substrate and particle wettabilities, particle size and volume fraction also play a role. In our earlier study [11] we showed that substrate wettability plays an important factor in the impact morphology, so to facilitate the discussion we provide our analysis of particle distribution on the substrate, under two separate headings of hydrophilic and hydrophobic substrates. In each section we will also further discuss the effects of particle size and volume fraction, for each substrate type. 3.1. Hydrophilic surfaces 3.1.1. Splat morphology The images of typical splats after drop impact onto the hydrophilic substrates with and without particles are shown in Fig. 1. The drops of pure water formed round thin splats. If the 200 lm or 500 lm particles were present in the drop, then they were distributed in the form of a monolayer over hydrophilic substrates. The pattern of the 500 lm particles was random for / 6 0:13 (see Fig. 1). The 200 lm particles formed either a disk or a ring (Fig. 1). The particle distribution is considered in the form of a ring when vast majority of particles are within a delimited annular region.

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Fig. 1. Bottom view images of splats on hydrophilic substrates after 1 s from drop impact. The white solid line and cross show drop contact line, and the point of the drop impact, respectively.

The observation of the ring or disk pattern for 200 lm particles depended on the particle volume fraction. Thus, at / ¼ 0:05 we observed a ring (the second column of Fig. 1), and at / ¼ 0:25 – a disk (the third column of Fig. 1). So, the increase of the particle volume fraction led to the change of the particle pattern from ring to disk. At the same time, the increase of the drop impact velocity from 1:7 to 3:7 m/s did not alter the shape of the particle distributions, but increased the outer splat diameter (first and second rows in Fig. 1). Formation of a monolayer of particles was qualitatively similar to the results of Lubbers et al. [19] for dense suspensions of water or silicone oil with a 250 lm hydrophilic particles at We > 1862. However, we also observed a ring or a disk pattern for 200 lm particles unlike Lubbers et al. [19]. The absence of the ring/disk pattern in Lubbers et al. [19] can be explained by the fact that they studied the impact of non-spherical drops (see Fig. 1 in [19]). The asymmetric particle distribution before an impact led to asymmetric splat morphology. The observations of the ring or disk distributions were similar to the results for suspensions of water and glycerol with 640 lm particles, or salt water with 380 lm or 720 lm particles in [15]; however, it was not clear in [15], if particles formed a monolayer, or not. In the next sections we will provide, first, the mechanistic reasons for the observed pattern and explain its physical basis; then using such insight we endeavour to formulate a number of empirical correlations to describe the observed patterns in a mathematical form. Such formulae can be useful to determine the morphology of particle distribution knowing the impact conditions and the degree of particle loading in a drop. 3.1.2. Formation of the ring patterns As an example of ring formation, let’s first examine the image sequence of a water drop with 200 lm particles impacting onto a hydrophilic substrate at U ¼ 1:7 m/s and / ¼ 0:05 (Fig. 2). Seen from Fig. 2, at the beginning, drop rapidly spreads over the surface from the impact point (demarcated with a white cross in bottom view images). During fast spreading, most of the particles were carried by the liquid inertia and spread accordingly. The velocity of these particles was such that after 2 ms almost all of them

reached the contact line and then moved with it. After 12.8 ms, splat was composed of a rim made of particles and liquid in its centre. In the subsequent stage, the surface tension force causes excess liquid at the rim to move back towards the drop centre away from the contact line. But compared to inertia force during spearing phase, the flow induced by the surface tension force is weaker and will not be able to carry the particles back, so they remained at the outline of the splat in the form of a ring. Small disturbances of the air-water interface decayed with time, while the liquid spread slowly outwards (right frame of second row in Fig. 2, corresponding to 923.2 ms). Thus, we observed two stages of spreading of water with 200 lm particles: a fast dynamic stage within the first 12.8 ms, followed by a slower one from 12.8 ms to 1 s. The ring formed during the dynamic stage. The first stage, i.e. spreading of the drop is due to the action of inertial forces, and the second statge is due to the capillary forces, i.e. surface wicking. The slow (capillary stage) spreading was not observed for pure water drops. This is clearly seen in time dependences of the spreading factor for pure water and water with 200 lm particles (Fig. 3A). The spreading factor is the ratio of the equivalent diameter of a drop contact area with the substrate, D, to drop diameter before impact, D0 . The equivalent diameter of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi drop contact area, D, is defined by the formula D ¼ 4  A=p, where A is drop contact area determined from the bottom view images. The drop diameter before impact, D0 , was measured by the exprespffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sion D0 ¼ 4  A0 =p, where A0 is the area of the drop from side view image before impact. Relative errors of measured D and D0 were 0.5% and 1%, respectively. The capillary spreading of water with 200 lm particles is attributed to the presence of particles near the drop contact line at the end of the inertial spreading phase. A particle near the contact line distorts the liquid-air interface around itself (see Fig. S3A in the Supporting Material). The contact angle of liquid on the substrate is higher than equilibrium contact angle (Fig. S3A) due to constraint wetting facilitated by the small particle, and the dynamic condition at the end of the inertial spreading. Once the dynamic phase is over, since the equilibrium contact angle at the substrate is low, the surface forces act to adjust the liquid contact angle to its low value, hence lead to the further liquid spreading (Fig. S3B in page 4 of the Supporting Material).

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Fig. 2. Pattern formed for water drops with 200 lm particles on hydrophilic substrates at / ¼ 0:05, D0 ¼ 3:83 mm and U ¼ 1:70 m/s (We ¼ 153 and Re ¼ 7258). The last two images in the second row show the beginning and the end of the capillary spreading. Time from the start of impact is shown above images (t ¼ 0 is the moment of impact).

The capillary spreading did not expand the outer edge of the particle distribution. As it can be seen from the splat image in Fig. 3B and C the red circle, bounding particles, has the same diameter from 12.8 ms to 923.2 ms. This is understandable again due to small inertia created by the surface forces that is not able to move the particles. The diameter of the circle relative to initial drop diameter, Dout =D0 , is used for describing the ring/disk distribution. The area of the circle matched to the drop contact area (the region bounded by the blue line in Fig. 3B) with the substrate after the inertial spreading. Therefore, the maximum spreading factor caused by inertia determines the Dout =D0 for 200 lm particles. The capillary spreading was observed for dense suspension drops of octadecene with 0.64 lm particles impacting onto hydrophilic glass substrate at We = 26–262 [20,21]. However, for such suspensions the dynamic stage was not reported. For suspension drops of water and glycerol with 640 lm particles, or suspension drops of salty water and 380 lm or 720 lm particles, impacting a glass substrate, the opposite situation was reported: only dynamic stage was mentioned by Nicolas [15]. So as this study is a first to consider both stages of spreading, one can conclude that observed pattern of particles is due to the operative forces in the first stage of spreading, and the second stage of spreading only concerns further expansion of the liquid on a hydrophilic substrate. We attribute observing both dynamics and capillary spreading to both particle hydrophobicity and smaller size. The particle hydrophobicity leads to the fact that they are located at the air-water interface during spreading phase. The combined

circumstances of particles residing at the interface and small particle size, reduces the possibility of their contact with the substrate, and thus the friction between the two. Therefore, the particles move easier when liquid spreads. When the liquid recedes, the particles touch the substrate/surface because of thinning of lamella as liquid drains/recoils; so the particles contact the substrates and remain still on the substrate due to friction between particles and substrate. This friction cannot be overcome by the low flow inertia induced by the surface tension during the retraction of excess liquid from rim towards the centre. Nicolas [15] explained the formation of particle distribution in the form of a ring, with the oscillations of the liquid after the dynamic stage. Examining videos, we did not observe any such oscillations. In the next section, we presented a set of mathematical formulae to describe particle distribution. The idea here is to see if by making some simple assumptions one can describe the particle formation as a function of particle volume fraction, drop size, and impact conditions to parameters such covered area by particles, the shape of covered area, and the surface density of monolayer of particles.

3.1.3. Predictive correlations for particle patterns As mentioned above, the outer diameter of the ring (Dout ) is determined by the inertial forces during the dynamic stage of spreading. Therefore, we consider the maximum spreading factor and examine the effect of variables such as particle volume fraction (/), and drop impact velocity (U) on Dout .

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Dmax ¼A/þB D0

ð2Þ

where A and B are constants. The coefficients A and B for each impact velocity, U, are shown in Table 2. As can be seen from Table 2, the 95% confidence intervals of coefficients A are overlapped for different U, in contrast to coefficients B. So, it was concluded that coefficient A was independent of U and equal to 5:6 0:6, and B was changing with U. It is worth mentioning that, on hydrophobic substrates, the maximum spreading factor of water with 200 lm particles also decreased linearly with increasing particle volume fraction [11]. Unlike the hydrophilic substrates, the coefficient A decreased with the increase of the impact velocity from 1.7 to 2.9 m/s [11]. The coefficient B corresponds to the maximum spreading factor seen for pure liquids. In literature, there are many correlations describing how much a pure liquid spreads [15,22–32]. Analyzing all of the correlations (see pp. 6–7 of the Supporting Material) it was found that the following three correlations closely describes our data for pure water and hence can be used to estimate B. Eq. (3) is from [25] and has the form of:


4
2
 3 We Dmax Dmax 1 þ ð1  cos ha Þ  We þ 4 ¼ 0 2 Re D0 3 D0

ð3Þ

where ha is the advancing contact angle; Eq. (4) is from [28]:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dmax u We þ 12 u   ¼t D0 ffiffiffiffi 3ð1  cos ha Þ þ 4 pWe

ð4Þ

Re

And Eq. (5) is from [15]: Fig. 3. (A) Spreading of pure water and water with 200 lm particles (/ ¼ 0:05) after impact onto hydrophilic substrates versus time; D0 ¼ 3:83 mm, U ¼ 1:70 m/s (We ¼ 153 and Re ¼ 7252). The splats of the water drop with 200 lm are shown in the insets (B) and (C) after 12.8 ms and 923.2 ms from the impact start, respectively. In the insets, the blue line indicates the drop contact with the substrates and red circle – the outer diameter of the particle distribution. (D) Maximum spreading factor of the drops on hydrophilic substrates versus volume fraction (/) for drops laden with 200 lm particles for different drop impact velocities, U. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

We measured the maximum spreading factor for the particle volume fraction from 0 to 0:4, and the impact velocities from 1:7 to 3:7 m/s, corresponding to 148 6 We 6 744 and 7092 6 Re 6 16368 (Fig. 3D). In this range, sometimes splashing was observed (Fig. S4A of the Supporting Material), i.e. 1–3 particles or drops with the diameter 200 lm were ejected (Fig. S4B). The velocities of the detached parts were measured from bottom view images and were no more than 2:4  U. Such velocities and volumes of detached parts could cause minor relative changes in the maximum spreading factor of the remaining drop (less than 103, see the Supporting Material). So, the maximum spreading factor was also measured for the splashing drops and this data was included in our analysis. The decrease of the maximum spreading factor with the addition of the particles can be caused by three factors: a decrease in water volume fraction in a drop as particle concentration is increased (i.e. less liquid is available); also as particle concentration increases, it is natural to expect that the viscous dissipation between liquid and particles to increase (leading to a decrease in spreading factor). Finally, the energy dissipation through frictional losses between particles and a substrate, is to increase when particle concentration increases, again leading to reduction in maximum spreading factor [11]. In Fig. 3D, a clear linear dependence for the maximum spreading factor on the particle volume fraction can be seen. As such, one can write


1=4 Dmax Re ¼ 12a D0

ð5Þ

where a is a dimensionless number and was taken to be equal to 1. In using Eqs. (3) and (4), the advancing contact angle value was taken to be zero, since on the hydrophilic glass substrates advancing contact angle was less than 5 . Therefore, the Eqs. (3) and (4) take the form of



1=4 Dmax 2 12 Re 1 þ ¼ 9 We D0 and

Dmax ¼ D0

ð6Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi We þ 12 pffiffiffiffiffiffi Re 4  We

ð7Þ

respectively. Values of the maximum spreading factor for pure water, calculated according to the formulae (5)–(7) are shown in Fig. 4 in comparison with experimental data for pure water in this work. As it can be seen, the correlation (5) from [15] gives the closest value to the experimental data. Therefore, it was chosen to estimate the coefficient B. The coefficient B depends only on the Reynolds number as dissipation is mainly due viscous forces (note that in our tests Weber numbers are much greater than 1, so the surface energy is less of a factor during spreading).

Table 2 Coefficients A and B for the Eq. (2), estimated by the least squares methods for each impact velocity, U. The errors are equal to two standard deviations. A is independent of U, whereas B is not. The last row shows the average value of A for the three impact velocities. U (m/s)

A

B

1.7 2.9 3.7

5.1 ± 0.5 6.2 ± 0.6 5.5 ± 0.5 5.6 ± 0.6

4.94 ± 0.04 5.76 ± 0.04 6.11 ± 0.06

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Fig. 4. Comparing maximum spreading factor of pure water drops impacting onto a glass substrate versus impact velocity from different correlations. Error bars, denoting two standard deviation are smaller than the symbol size.

By combining Eqs. (2) and (5) as well as data from Table 2, Eq. (8) can be constructed as:


1=4 Dout Dmax Re ¼ ¼  5:6  / 12 D0 D0

ð8Þ

It should be noted that a general correlation for maximum spreading factor of particle-laden drops should include particle diameter. This is so since if particles are much smaller than what we used in this study, the surface tension induced flow during liquid retraction (stage 1), or subsequent capillary spreading, may have sufficient inertia to carry the particles. For example, the results of Yoo and Kim [33] cannot be explained by Eq. (8) as they did not observe any changes in the drop impact dynamics for pure octanol and octanol with 10% particles. However, they used particles that were two orders of magnitude smaller than ours (2 lm), and had test conditions that had a much smaller We number (We  6). To completely describe the particle distribution in the splats, we examined the dependence of inner diameter of particle patterns (Dinn ) in form of a ring, on parameters such as /. The inner diameter of a ring, Dinn , was measured manually from splat images by drawing a circle delimiting the inner contour with its centre being the point of the drop impact (see Fig. 5A). The inner diameter of the particle distribution (Dinn ) relative to the initial drop diameter, Dinn =D0 , depended on the volume fraction of 200 lm particles (Fig. S6 of the Supporting Material). The transition from the ring to disk distribution when the inner diameter of the distribution is zero was characterized by a step change for all three drop impact velocities (Fig. S6). To describe this behavior, we used a sigmoid function for the relative internal diameter in the form of

Dinn b1 ¼ D0 1 þ eb2 ð/b3 Þ

ð9Þ

where b1 , b2 and b3 are constant coefficients. The nonlinear regression analysis showed that the sigmoid function, calculated by least squares, fit the data with R2 -values ranging from 0:74 to 0:99 (Fig. S6). The values of the parameters of the sigmoid functions are given in Table 3. As it can be seen from Table 3, at the impact velocities of 1:7 and 3:7 m/s, the corresponding coefficients of sigmoid functions are not different from each other within their confidence intervals. Therefore, these coefficients may not depend on the drop impact velocity. On this basis, these coefficients were calculated by using the least squares method for fitting a sigmoid to data from all three velocities. The result of the regression analysis is shown in the last row of Table 3. Fig. 5B shows the data and the fitted sigmoid function. The resulting sigmoid function fit the data with R2 -value equal to 0:73. Thus, the relative internal diameter of the ring distribution can be described by the equation below

Fig. 5. (A) Image of 200 lm particle distribution in the form of a ring on a glass substrate. Blue and yellow circles show outer and inner diameters of the ring, respectively. Note that particle distribution is considered in the form of a ring as vast majority of particles are within the delimited ring (between blue and yellow circles in the images). (B) Relative inner diameter of particle (200 lm) distributions over hydrophilic glass substrates versus the particle volume fraction for the impact velocities U ¼ 1:7 m/s (We ¼ 152 4 and Re ¼ 7235 140), 2:9 m/s (We ¼ 443 10 and Re ¼ 12324 246) and 3:7 m/s (We ¼ 716 14 and Re ¼ 15724 278). The line shows the sigmoid function found by the least squares method. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 3 Parameters of the sigmoid function (Eq. (9)) for the inner diameters of the ring distributions of 200 lm particles on hydrophilic substrates at different drop impact velocities, U. The last row presents the parameters calculated jointly for three velocities. The error bars of the estimated parameters are equal to two standard errors. U (m/s)

b1

b2

b3

1.7 2.9 3.7 1.7 & 2.9 & 3.7

3.6 ± 0.2 3.0 ± 0.2 3.0 ± 0.8 3.6 ± 0.6

192 ± 256 623a 36 ± 22 31 ± 20

0.07 ± 0.2 0.20 ± 14 0.11 ± 0.2 0.11 ± 0.02

a The error is very large due to the absence of data in the transition region (for / from 0.18 to 0.2 in Fig. S6B).

Dinn 3:6 ¼ 1 þ e31ð/0:11Þ D0

ð10Þ

The Eqs. (8) and (10) were found for when the particle volume fraction is less than 0:4. Obviously, they will be meaningless for when / equals to zero because there are no particles. In the range 0 < / < 0:4, the developed equations also allow calculating the ring width, the area covered by the particles, and the surface density for the particles. To find the width of the ring shaped distribution of particles (w), Eq. (11) can be written by using Eqs. (8) and (10), as follows:

V. Grishaev et al. / Journal of Colloid and Interface Science 490 (2017) 108–118

w ¼ ðDout  Dinn Þ=2 ¼

D0 2

"


1=4 Re 3:6  5:6  /  12 1 þ e31ð/0:11Þ

#

ð11Þ The area covered by the particles (Ap ) can also be found using Eqs. (8) and (10) as follows:

2 3 !2

14 2 D2out D2inn D20 4 Re 3:6 5 Ap ¼ p  5:6  /  p ¼p 12 1 þ e31ð/0:11Þ 4 4 4 ð12Þ Knowing the area, Ap , the particle volume fraction, /, and diameter, dp , and the fact that they spread out as a monolayer, one can also determine a surface density of particles (qs ) as: /D30

qs ¼

d3p

2

p d4p

Ap

2 3 !2
2 1
14 /D30 4 Re 3:6 5 ¼  5:6  /  12 1 þ e31ð/0:11Þ dp ð13Þ

3.2. Hydrophobic surfaces 3.2.1. Splat morphology Splats on hydrophobic substrates were very different from those on hydrophilic substrates. Fig. 6 shows the typical splats on hydrophobic surfaces after 1 s from the drop impact for pure water and water with 200 lm or 500 lm particles. In the case of water drops with 500 lm particles, the type of splats and particle distribution had a random character for / < 0:31. For water drops with 200 lm particles, most of the particles have accumulated near the drop contact line and formed a ‘‘crown”. It happened due to particle hydrophobicity, their neutral buoyancy and drop spreading in the lamella under impact. With increasing volume fraction, 200 lm particles completely covered the air-water interface, and at concentrations greater than 0.15 they deformed the drop (see Fig. 6). The increase of impact velocity led to an increase in the probability of drop fragmentation, while the crown remained in the main drop at concentrations below 0.15 (Fig. 6). The regime map of the fragmentation and deformation for water drops with 200 lm particles as a function of particle volume fraction and Weber number is shown in Fig. S7 of the Supporting Material. The formation of the crown when there was no drop fragmentation or deformation after an impact is considered next.

115

3.2.2. Crown formation Fig. 7 shows the crown formation for the 200 lm particles during the drop impact onto a hydrophobic substrate. The spreading of the drop on the substrate led to the appearance of the particles at the air-water interface. Most of the particles reached the drop contact line and continued to spread outward with contact line, as in the case of hydrophilic substrates (see Fig. 7). After spreading, the drop began to recoil (the images after 6.8 ms in Fig. 7). Particles also retracted. Note that for hydrophobic surfaces the flow induced by surface tension during the retraction phase is stronger than the case for hydrophilic surfaces, so there is enough inertia to carry the particles as the liquid retracts. The final drop shape was hemispherical with most of the particles remaining at the air-water interface due to particle hydrophobicity and their neutral buoyancy. Note that migration of particles to the drop interface took place after extend period of time compared to preimpact time. In a second slower process, the particles near drop apex moved along the air-water interface towards the drop contact line. It can be seen by comparison of side-view drop images at 200 ms and 951.6 ms in Fig. 7. This movement is likely due to a combination of gravity, buoyancy and capillary forces. As a result of the movement, particles are collected in the region near the contact line and formed a pattern similar to a ‘‘crown”. The final splat is shown in the inset of the last frame. Obviously for cases where / is high (see / ¼ 0:25 in Fig. 6) the second slower process described does not occur. Below we will attempt to describe the geometry of the observed crown. The idea here is to see if by making some simple assumptions about how particles position themselves at the interface one can describe the particle formation as a function of particle volume fraction and drop size. To construct a model for the crown height, the following assumptions were made: a particle-laden drop has the shape of a spherical cap with radius, r (Fig. 8A). The radius of the spherical cap, r, was determined from measurements of the drop base diameter, D, and its height, H, using formula

r¼

ðD=2Þ2 þ H2 2H

ð14Þ

On the surface of this cap, 200 lm particles are half-submerged and evenly distributed near the drop base in the spherical segment with height, h. Particles distribution on a spherical segment was considered as either of hexagonal, square, or snub square packaging (Fig. 8B). The crown height, h, can be found from the equation for the area of a spherical segment. Area of the spherical segment, S, is found from:

Fig. 6. Splats on hydrophobic substrates. For pure water, top row image in a pair, shows the side view, whereas bottom image in the pair, shows the bottom view of the drop. The images for drops with particles are taken with camera 3, see Fig. 2.

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V. Grishaev et al. / Journal of Colloid and Interface Science 490 (2017) 108–118

Fig. 7. Pattern formation for 200 lm particles on hydrophobic substrates at / ¼ 0:07 and U ¼ 1:7 m/s (We ¼ 152 and Re ¼ 7198). t ¼ 0 is the moment of impact. In each row, top image shows the side view whereas bottom image is the view from bottom.

Fig. 8. (A) Geometrical parameters used in the calculations of the crown height. (B) Various particle packings on the surface that were considered.

S ¼ 2prh

ð15Þ

The area, S, can be determined from the density of the particles on the spherical segment, n, the area of the particle cross section, and the number of particles, N p , in a drop, i.e.,

1 dp 1 d D3 p D30 p Np ¼ p p 30 / ¼ / n 4 n 4 dp 4n dp 2

S¼

2

Setting Eq. (15) equal to Eq. (16), h can be found as:

ð16Þ

V. Grishaev et al. / Journal of Colloid and Interface Science 490 (2017) 108–118

h¼

1 D30 / 8n rdp

ð17Þ

and, accordingly, the relative height, g, as:

g¼

h D20 ¼ / D0 8nrdp

ð18Þ

For the hexagonal, square, and snub square packings, the partipffiffiffi pffiffiffi cle densities were n ¼ 3p=6, n ¼ p=4, and n ¼ ð2  3Þp, respectively. Consideration of particle packing on a planar surface rather than on a curved surface is possible due to the fact that the particle radius (rp ¼ 0:1 mm) is much less than the drop radius (r  2 mm). The results of measurements for crown height and its calculations based on Eq. (18) for different particle volume fractions, /, are shown in Fig. 9A (for crown height measurements see SI). As can be seen, the calculations qualitatively describe the observed increase of the crown height with the increase of the particle volume fraction. Analysis of the relative difference between the measured and calculated value of g indicates that the predictions mainly underestimates the crown height (Fig. 9B). When / is less than 0.10, the difference between the measured and calculated values of the height can reach about 60%. Such a large difference at low concentrations is attributed to errors in the measurement of the crown height and the inapplicability of the assumption about a uniform particle distribution in a crown (see Fig. 9C). So, the correlation (18) allows describing the trend for the relative crown height as particle volume fraction, / changes. In the range of / from 0.09 to 0.15 Eq. (18) can be good (the relative error is less than 15%). For / from 0 to 0.09, it provides only the order of magnitude because the particle packing is not ideal, i.e. as shown in Fig. 9C. For / higher than 0.15, the correlation is not applicable obviously due to a drop deformation by particles. With our results we showed that inertia is not the only factor determining the particle distribution. We showed that substrate wettability is a very important factor in this first study of its kind. For hydrophilic substrates, 200 lm particles occupied drop periphery in form a ring and with increasing particle volume fraction they occupied the entire splat footprint. On hydrophobic substrates, 200 lm particles formed crown.

117

4. Conclusions In many technologies, impact of particle-laden drops is observed and to improve such processes a knowledge of particle distribution in splats after the impact is needed. For example, for the additive manufacturing of composite materials, it can be important to know possible ways of creating ordered particle distributions on substrates. Up to now, particle distribution was studied on hydrophilic substrates with hydrophilic particles [15,19,33]. In this paper, drop impact onto both hydrophilic and hydrophobic substrates were first experimentally studied for the case of water with suspended hydrophobic particles of 200 and 500 lm diameters. It was found that 200 lm particles form an ordered distribution on both types of substrates. On hydrophilic substrates, they had a ring or disk distribution, and on hydrophobic – they formed crowns covering the drop (the particles are positioned at the liquid-air interface). Previously, disk or ring distribution were observed for 380 lm and 640 lm hydrophilic particles [15], but in our work, the nature of distribution formation was different. In our case, the formation of particle distributions happened during inertial phase of drop spreading and at maximum spreading the particles were at the drop contact line. The presence of the particles at the drop contact line caused further slow spreading of water under the action of capillary forces (we termed this a second stage of spreading as capillary spreading phase). The increase of the impact velocity did not affect the patterns for 200 lm particles on the hydrophilic substrates; while on the hydrophobic substrates it led to the drop break-up. We also for the first time provided mathematical formulae to describe the particle distribution after drop impact considering the diameter of particles and their volume fraction, the wettability of particles and the substrate, and also dimensionless numbers such as Reynolds and Weber used to describe the dynamics of drop impact. Acknowledgments We would like to thank Dr. Christophe Minetti (ULB) for providing the software of particle counting and Patrick Queeckers (ULB)

Fig. 9. (A) Relative height of particle crown versus particle volume fraction. (B) Difference between the experimental and calculated values of the crown height. (C) Particle crowns for various volume fractions of 200 lm particles in water drops onto the hydrophobic substrates. At low / values, none of the packings in the model can hold.

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for helping with software issues for the syringe pump and computers. Also, the authors acknowledge funding from Transport Canada (Clean Rail program), the Natural Science and Engineering Research Council of Canada (NSERC) and from the Belgian Federal Science Policy Office (BELSPO).

Appendix A. Supplementary material The estimations of change in the maximum spreading factor caused by splashing are supplied as the Supporting Material. Also, in the Supporting Material you can find the comparison of correlations [15,22–32] for maximum spreading factor with our data for pure water on hydrophilic substrates. This material is available free of charge via the Internet. Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10. 1016/j.jcis.2016.11.038.

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