iii
Committee
Director: Prof. Frank Dubois ‐ Université libre de Bruxelles (Belgium)
Chair: Prof. Pierre Colinet ‐ Université libre de Bruxelles (Belgium)
Members: Prof. Pierre Lambert ‐ Université libre de Bruxelles (Belgium)
Prof. Marco Marengo ‐ University of Brighton (United Kingdom)
Prof. Alidad Amirfazli ‐ York University (Canada)
Dr. Carlo Saverio Iorio ‐ Université libre de Bruxelles (Belgium)
v
Abstract
Since decades, drop impact has been a popular topic of investigation for the importance that such a phenomenology has in many different application domains. So far, the effect of micro‐particles on the drop impact morphology has been studied for a limited number of configurations and often modelled as a change in the viscosity of the carrier fluid. However, this approach has been found sometimes questionable. The aim of the thesis is to better understand the phenomenology associated with particle‐laden drop impact such as the distribution of particles in splats and to extend the number of experimental configurations for particle‐laden drop impact to occur.The impact of millimetre‐size particle‐laden drops was investigated for hydrophilic and hydrophobic substrates. The drops were dispersions of water and round, spherical and nearly iso‐dense hydrophobic particles with diameters around 200 µm and 500 µm. The substrates were transparent glass and polycarbonate plates. The impact was studied by side, bottom and angle view images in the range 148≤We≤744 and 7092≤Re≤16368.
vii
Acknowledgements
I would like to acknowledge everyone who has assisted me throughout my doctoral studies over the years.
Firstly, I would like to thank Professor Frank Dubois creating comfortable conditions for scientific exploration in the Microgravity Research Centre of ULB and for his advice in optics and care.
Secondly, I would like to express my deep gratitude to my co‐promotor, Prof. Alidad Amirfazli, for the interesting topic, professional guidance and invaluable advice in the experiments and writing. Also, I am appreciated to him for invitation to visit his research group in York University.
Thirdly, I would like to thank my scientific supervisor, Dr. Carlo Saverio Iorio, for his support the experiments, valuable discussion of results and advice in writing.
Fourthly, I would like to thank Patrick Queeckers for helping with software issues for a syringe pump and computers. Also, I would like to thank Dr. Christophe Minetti for providing software to count particles.
xi
List of Tables
Table 1.1 Summary of the work achievements in the study of particle‐laden drop impacts
13
Table 1.2 Summary of the activities 14
Table 2.1 Classification of drop impact studies on suspensions 26
Table 2.2 Parameters of the experiments 35
Table 2.3 Coefficients A and B of the linear regression Aϕ+B for maximum spreading factors. The error bars of the estimated parameters are equal to two standard deviations
48
Table 2.4 Coefficients A and B of the linear regression Aϕ+B for final spreading factors. The error bars of the estimated parameters are equal to two standard deviations
49
Table 2.5 Coefficients A and B of the linear regression Aϕ+B for maximum spreading factor caused by inertial force. The error bars of the estimated parameters are equal to two standard deviations
63
Table 2.6 Phenomena happened during drop impact in our work 64
Table 2.7 Tendency of phenomena appearance for pure water on hydrophobic substrate with the increase of the impact velocity from 1.7 to 3.7 m/s (We= (150 to 750) and Re = 7200 to 16200)
65
Table 2.8 Tendency of phenomena appearance for pure water with solid particles with the increase of the impact parameters
66
Table 2.9 Coefficients A and B of the linear regression Aϕ+B for maximum spreading factor caused by inertial force on hydrophilic and hydrophobic substrates from Tables 2.3 and 2.5. The error bars of the estimated parameters are equal to two standard deviations
69
Table 3.1 Experimental conditions, and physical characteristics of liquids and density‐ matched particles in analysed works
80
Table 3.2 Separation quality of splashing regime maps by existing and developed criterions
89
Table 4.1 Experimental conditions, and physical characteristics of liquids and particles used
99
Table 4.2 Coefficient A and B of the equation (2), estimated by least squares methods for each impact velocities U, separately. The error bars of the estimated parameters are equal to two standard errors. A is independent of U, whereas
B is not
110
Table 4.3 Parameters of the sigmoid function for the inner diameters of the ring distributions of 200 μm particles on hydrophilic substrates. The error bars of the estimated parameters are equal to two standard errors
113
xiii
List of Figures
Fig. 1.1 Applications where particle‐laden drops impact onto surfaces 2 Fig. 1.2 Possible outcomes of a drop impact onto a solid surface 4Fig. 1.3 Sessile drop on a solid substrate; represents the contact angle 6 Fig. 2.1 Possible outcomes of drop impact of liquid without particles on dry solid
substrates. The image sequences were done from top and side view relative to falling axis of a drop
23
Fig. 2.2 Forms of particle‐laden drops 25
Fig. 2.3 Schematic of the experimental apparatus used to study drop impact 33 Fig. 2.4 Image sequences show drop impact of water on the hydrophobic
polycarbonate plate at Weber number 150 (drop diameter 3.82 mm and velocity 1.68 m/s). Top and bottom row of sequences were taken by side and bottom view cameras, respectively. Time of the images was written below of them relative to the impact begin. The white crosses are the centre of the drop impacts
37
Fig. 2.5 Appearance probability, P, of temporary dry spot, drop jetting and partial rebound during the impact of water drops on the hydrophobic surfaces versus Weber number, . Error bars of Weber numbers denote two standard deviation
38
Fig. 2.6 Drop impact of water on the hydrophobic polycarbonate plate at 742
( 3.88 mm and 3.71 m/s)
39
Fig. 2.7 Drop impact of water with 200 µm particles on the hydrophobic polycarbonate plate at 151 ( 3.79 and 1.69m/s) and particle volume fraction 0.04
39
Fig. 2.8 Appearance of partial rebound versus Weber number, , and particle volume fraction, , under the drop impact of water with 200 µm particles on the hydrophobic polycarbonate plates. The dashed line separates the drop impacts with and without the partial rebound (the bottom and top part of the graph, respectively)
40
Fig. 2.10 New type of dry spots for water with 200 µm particles on the hydrophobic polycarbonate plates. (a) The dry spot due to the break of the air bubble attached to the particle and it caused the drop rupture at We=152 (D0=3.80
mm and U=1.70 m/s) and ϕ=0.07. (b) Two dry spot near the impact centre at
We=153 (D0=3.81 mm and U=1.70 m/s) and ϕ=0.05. The insets are enlarged
parts of the images. The white cross is the impact centre
41
Fig. 2.11 Splashing of water drops with 200 µm particles on the hydrophobic polycarbonate plates at (a) We=437 (D0=3.77 mm and U=2.89 m/s), ϕ=0.08
and (b) We=725 (D0=3.87 mm and U=3.67 m/s) and ϕ=0.04
42
Fig. 2.12 Appearance of the splashing versus Weber number We and particle volume fraction ϕ under the drop impact of water with 200 µm particles on the hydrophobic polycarbonate plates. The dashed line separates no splashing and splashing regimes (the left and right part of the graph, respectively)
43
Fig. 2.13 Images of the receding break‐up of water with 200 µm particles on the hydrophobic polycarbonate substrate at We=153 (D0=3.81 mm and U=1.70
m/s) and ϕ=0.05
44
Fig. 2.14 Appearance of receding break‐up versus Weber number We and particle volume fraction ϕ under the drop impact of water with 200 µm particles on the hydrophobic polycarbonate plates. The dashed line separates the drop impacts without and with the receding break‐up (the left and right part of the graph, respectively)
44
Fig. 2.15 Absence of receding break‐up for water with 200 µm particles on the hydrophobic polycarbonate substrate at We=154 (D0=3.85 mm and U=1.70
m/s) and ϕ=0.12
45
Fig. 2.16 Regime map of the drop fragmentation of water drops with 200 µm particles under impact on the hydrophobic polycarbonate substrates
45
Fig. 2.17 Spreading factor of water drop without and with 200 µm particles on hydrophobic polycarbonate surface versus time at We=152±5 and
Re=7258±170 (D0=3.83±0.06 mm and U=1.69±0.02 m/s) and different particle
volume fraction ϕ
47
Fig. 2.18 Maximum spreading factor of water with 200 µm particles on hydrophobic surface versus particle volume fraction ϕ
48
Fig. 2.19 Final spreading factor (after 1 s from impact start) of water with 200 µm particles on hydrophobic surface versus particle volume fraction. The insets are the images of the droplets after 1s from the impact: a) We=436, ϕ=0.02; b) We=156, ϕ=0.33; c) We=442, ϕ=0.17. The images are not in the same scale
49
Fig. 2.22 Singular jet appearance under drop impact of water with 500 µm particles onto hydrophobic polycarbonate substrates at ϕ=0.07 and We=158 (D0=3.95
mm and U=1.70 m/s)
51
Fig. 2.23 Lamella’s rupture of a water drop with 500 µm particles onto a hydrophobic polycarbonate substrate due to break‐up of a bubble attached to a particle and the substrate (a) or only to the substrate (b). The enlargements for side and bottom views are 2.5 and 5 times, respectively. The images (a) were done at We=157 (D0=3.84 mm and U=1.71 m/s) and ϕ=0.05, and images (b) – at
We=157 (D0=3.89 mm and U=1.71 m/s) and ϕ=0.06
52
Fig. 2.24 Regime map of the appearance of temporary dry spot and lamella rupture for water with 500 µm particles under drop impact onto hydrophobic polycarbonate substrates
53
Fig. 2.25 Regime map of splashing for drop impact of water with 500 µm particles onto hydrophobic polycarbonate substrates
53
Fig. 2.26 Splashing of water drops with 500 µm particles on the hydrophobic polycarbonate plates at We=440 (D0=3.77 mm and U=2.90 m/s) and ϕ=0.06
54
Fig. 2.27 Regime map of the receding break‐up of water with 500 µm particles under drop impact onto hydrophobic polycarbonate substrates
54
Fig. 2.28 Regime map of fragmentation for water with 500 µm particles under drop impact onto hydrophobic polycarbonate substrates
55
Fig. 2.29 Spreading factor of water drop without and with 500 µm particles on hydrophobic polycarbonate surface versus time at We=154±7 and
Re=7286±216 (D0=3.84±0.08 mm and U=1.70±0.03 m/s) and different particle
volume fraction ϕ
56
Fig. 2.30 Spreading factor of water drop without and with 500 µm particles on hydrophobic polycarbonate surface versus time at We=445±19 and
Re=12297±395 (D0=3.78±0.09 mm and U=2.91±0.03 m/s)and different
particle volume fraction ϕ
57
Fig. 2.31 Spreading factor of water drop without and with 500 µm particles on hydrophobic polycarbonate surface versus time at We=729±41 and
Re=15920±725 (D0=3.86±0.16 mm and U=3.69±0.06 m/s) and different
particle volume fraction ϕ 58 Fig. 2.32 Image sequences show drop impact of water on the hydrophilic glass plate at Weber number We=152 (drop diameter D0=3.81 mm and velocity U=1.69 m/s) 59 Fig. 2.33 Image sequences show drop impact of water with the 200 µm particles on the hydrophilic glass plate at We=157 (D0=3.89 mm and U=1.71 m/s) and ϕ=0.06 60
Fig. 2.34 Appearance of the splashing versus Weber number We and particle volume fraction ϕ under the drop impact of water with 200 µm particles on the hydrophilic glass plates. The dashed line separates no splashing and splashing regimes (the left and right part of the graph, respectively)
60
Fig. 2.35 Drop spreading of water and water with 200 µm particles on the hydrophilic glass substrates at We = 153±3 and Re=7255±116 (D=3.83±0.05 and
U=1.69±0.01) 62 Fig. 2.36 Maximum spreading factor of water with 200 µm particles on the hydrophilic glass substrates versus particle volume fraction 62 Fig. 2.37 Regime map of splashing for drop impact of water with 500 µm particles onto hydrophilic glass substrates 63
Fig. 3.1 Regime maps of splashing at glass substrates of drops with hydrophilic particles generated from data in [Nicolas, 2005]. Open and solid symbols correspond to no‐splashing and splashing, respectively. In the top of plots, the value is shown with 95.5% confidence intervals, calculated from the criterion of [Peters, 2013]
81
Fig. 3.2 (a) Schematic of splashing regime map for particle‐laden drops. (b) Separation of splashing regime map into two domains by developed criterion and splashing criterion for pure liquids (3.3)
82
Fig. 3.3 Fraction of points nonregistered by the splashing criterions (Eq. 3.3, 3.12) versus the value for splashing data from [Nicolas, 2005] for drops containing hydrophilic particles with diameter 381 or 720 µm. The criterion of Eq.3.12 was taken with 95.5% confidence interval. Total points, , in two splashing regime maps were 36 and 41, for particles 381 and 720 µm, respectively
85
Fig. 3.4 Separation of splashing regime maps at 0.180 and 0.292 for data from [Nicolas, 2005] for drops containing hydrophilic particles with diameter 720 µm. Shaded area is 95.5% confidence interval of the developed criterion (Eq. 3.12). Open and solid symbols correspond to no‐splashing and splashing, respectively.
85
Fig. 3.5 Fraction of points nonregistered by the splashing criterions (Eq. 3.3, 3.12) versus the value for splashing data from [Grishaev, 2015] for drops containing hydrophobic particles with diameter 196 or 463 µm. The criterion of Eq.3.12 was taken with 95.5% confidence interval. Total points, , in two splashing regime maps were 134 and 76, for particles 196 and 463 µm, respectively
86
Fig. 3.6 Schematic to illustrate the calculation of surface energy change due to the particle ejection ‐ from a drop interface. (1) The particle at the interface; (2) the particle ejected (splashed) from drop
87
Fig. 3.7 Fraction of points nonregistered by the splashing criterions (Eq. 3.3, 3.20) versus the value for splashing data from [Grishaev, 2015] for drops containing hydrophobic particles with diameter 196 or 463 µm. The criterion of Eq.3.20 was taken with 95.5% confidence interval. Total points, , in two splashing regime maps were 134 and 76, for particles 196 and 463 µm, respectively
88
Fig. 3.8 Regime maps of splashing for drops with hydrophilic particles impacting onto glass substrates are taken from [Nicolas, 2015]. The solid curve corresponds to the developed criterion (Eq. 3.21), and the shaded area – its 95.5% confidence interval
89
Fig. 3.9 Regime maps of splashing at hydrophilic substrates of drops containing hydrophobic particles [Grishaev, 2015]. The solid curve corresponds to the developed criterion (Eq. 3.21), and the shaded area – its 95.5% confidence interval. The splashing of drops without particles happen at 300
90
Fig. 3.10 Regime maps of splashing at hydrophobic substrates of drops containing hydrophobic particles [Grishaev, 2015]. The solid curve corresponds to the developed criterion (Eq. 3.21), and the shaded area – its 95.5% confidence interval. The splashing of drops without particles happen at 300 90 Fig. 4.1 (a) Image of 200 µm particles at the water puddle. (b) Image after brightness adjustment; the drawings explaining the contact angle measurements on the particle 100
Fig. 4.2 Set‐up used to study drop impact: a) a schematic; b) a live view. The schematic of the set‐up used to study drop impact
102
Fig. 4.3 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
103
Fig. 4.4 Pattern formation for 200 µm particles on hydrophilic substrates at 0.05,
3.83 and 1.70 m/s ( 153 and 7258). The right most
pair of images in the second row shows the capillary slow spreading of liquid once the dynamic drop impact event has ended. Time from the impact start is shown above images (t = 0 is the moment of impact)
105
Fig. 4.5 Drop spreading of pure water and water with 200 µm particles ( 0.05) onto hydrophilic substrates versus time at 3.83 mm, 1.70 m/s
( 153 and 7252) and 3.83 mm, 1.70 m/s ( 153
and 7258), respectively (colour image online)
106
Fig. 4.6 Maximum spreading factor of the drops on hydrophilic substrates versus volume fraction for drops laden with 200 µm particles for different drop impact velocities
108
Fig. 4.7 Splashing of drops laden with 200 µm particles under impact on hydrophilic substrates: a) a regime map; b) example at 447, 12352 and
0.03 108 Fig. 4.8 Comparing maximum spreading factor of pure water drops impacting onto a hydrophilic substrate versus impact velocity from different correlations. Error bars, denoting two standard deviation, are in smaller than the symbol size 111
Fig. 4.9 Image of 200 µm particle distribution in form of a ring. Blue and red 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 (red circles in the images)
Fig. 4.10 Non‐demensional internal diameter of particle (200 μm) distributions on hydrophilic substrates versus particle volume fraction for the impact velocities
1.7 m/s ( 152 4 and 7235 140), 2.9 m/s ( 443
10 and 12324 246 ) and 3.7 m/s ( 716 14 and
15724 278). Lines show the sigmoid functions Eq. (3) estimated by the least squares method
113
Fig. 4.11 Non‐dimensional inner diameter of particle (200 μm) distributions onto hydrophilic glass substrates versus the particle volume fraction for the impact
velocities 1.7 m/s ( 152 4 and 7235 140 ), 2.9 m/s
( 443 10 and 12324 246) and 3.7 m/s ( 716 14 and
15724 278). The line shows the sigmoid function calculated by the least squares method
113
Fig. 4.12 Splats on hydrophobic substrates. For pure water top row image in a pair shows the side view, whereas bottom row shows the bottom view of the drop. The images for drops with particles are taken with camera 3, see Fig. 4.2
115
Fig. 4.13 Drop fragmentation and deformation map as per various Weber number and particle volume fraction,
116
Fig. 4.14 Pattern formation for 200 µm particles on hydrophobic substrates at 0.07 and 1.7 m/s ( 152 and 7198). t = 0 is the impact moment
117
Fig. 4.15 Measurements of the initial drop diameter and the crown height at 0.07
and 1.7 m/s ( 152 and 7198)
118
Fig. 4.16 (a) Assumptions are used in the calculations of crown heights. (b) Particle packings on the segment surface are considered in the calculations. Grey lines show particle arrangements
118
Fig. 4.17 Spherical cap radius, , was determined from the measurements of the drop height, , and base diameter, , after drop impact
119
Fig. 4.18 a) Relative height of particle crown versus particle volume fraction. b) Difference between the experimental and calculated values of the crown heights
120
Fig. 4.19 Particle crowns for various volume fractions of 200 µm particles in water drops onto the hydrophobic substrates
120
Fig. A.1 Preparation scheme of hydrophilic and hydrophobic substrates for the drop impact studies 130 Fig. A.2 Images of advancing water drop on a UV/Ozone cleaned glass slides. Flow rate was 2 µl/min 131 Fig. A.3 Images of receding water drop on a UV/ozone cleaned glass slides. Flow rate was 2 µl/min 131 Fig. A.4 Contact angle hysteresis for water drop on a polycarbonate plate. Insets show the water drop under advancing and receding phases. Flow rate was 12 µl/min and the measurements of contact angle were done every 2.5 s
Fig. B.1 Schematic of the experimental set‐up for the contact angle measurements on particles. The set‐up is based on KRUSS DSA 30S
134
Fig. B.2 Images of the set‐up for contact angle measurements on particles. a) Placement of water puddle between the camera and a diffuse light source (is not shown); b) Addition of extension tubes for the increase of optical magnification 134 Fig. B.3 (a) Image of 200 µm particles at the water puddle. (b) Image after brightness adjustment; the drawings explaining the contact angle measurements on the particle 135 Fig. B.4 Scheme of the preparation of water drops with hydrophobic particles 135 Fig. B.5 Measurements of the initial drop diameter at 0.07 and 1.7 m/s
( 152 and 7198)
136
1
Introduction
Contents
1.1. General idea ... 1 1.2. Applications where particle‐laden drops impact onto surfaces ... 2 1.3. Drop impact without particles ... 3 1.4. Particle‐laden drop impact ... 7 1.5. The scope of the thesis ... 10 1.6. The framework of the thesis ... 11 1.7. Achievements and activities ... 13 1.8. References ... 151.1. General idea
Since decades, drop impact has been a popular topic of investigation for the importance that such a phenomenology has in many different application domains. As a consequence, it is possible to find a great number of references investigating the impact of pure liquid drops on substrates showing different wettability and topological structures. However, only few studies report the impact of drops seeded with particles – as is the case in many applications. In this latter case, the presence of particles is taken into account by considering an “effective” viscosity that generalize the viscosity of pure liquids when dilute dispersion are considered.left about the mechanisms underneath the morphology of drop impact, their splashing and the distribution of microparticles in splats.
The goal of this thesis is to contribute to understand such mechanisms in the case the diameters of particles lay in the micrometre range.
1.2. Applications where particle‐laden drops impact onto surfaces
The particle‐laden drop impact onto solid substrates find application in many industrial processes such as, for instance, in spray, plasma spray and drop‐on‐demand coating technologies (Fig. 1.1). Spray technologies are often used to cover solid surfaces by paints or liquid‐friction modifiers containing solid particles. In the first case, particles (pigments) are used for colouring.
In the second case, particles modify the friction coefficient between train wheels and rails to reduce their wear and squealing noise at tight curves [Suda, 2005; Eadie, 2006]. In the plasma spray coating technology, drops of partially molten powder are used to create nanostructured coatings [Fauchais, 2011]. Also, drops, containing solid particles, are used in drop‐on‐demand technologies for additive manufacturing of ceramics [Seerden, 2001; Calvert, 2001; Lewis, 2006] and electronics [Tekin, 2008; Singh, 2010]. Wide application of particle‐laden drops impels to study their impact on solid substrates. The starting point for a better understanding of the factors influencing the particle‐laden drop impact is often the characterization of the drop impact of pure liquids.
1.3. Drop impact without particles
For liquids without particles, it is known that, after an impact on a solid substrate, a drop may either bounce off from it (rebound), stick to it (deposition), or break‐down upon impact into smaller droplets (fragmentation) (Fig. 1.2). The phenomena, leading to these outcomes, are classified and form the morphology of drop impact. For example, if the detachment of small drops happens during a drop spreading, then it is named as splashing. The study of the phenomena enables to know influencing factors and determine their dependences. To find out the influencing factors, it is necessary to understand which forces act on the drop.Fig. 1.2: Possible outcomes of a drop impact onto a solid surface
force of liquid‐air interface, , could be seen as a restoring force dampening deformation. Also, with the exception of , , , the magnitudes of the other forces change during drop impact. To compare their influence on the process, the orders of their magnitudes should be dynamically evaluated and compared with each other.
The order of the force magnitudes, , , , , are determined through the initial drop parameters and liquid properties as follows.
The magnitude of gravitational force, , is equal to the drop mass times the gravitational acceleration, . The drop’s mass is determined by liquid density, , and drop volume, which is proportional to the initial drop diameter as . So, the order of the gravity force magnitude can be evaluated as
The magnitude of the inertial force, , is determined as the drop’s mass times the drop acceleration during impact. For the evaluation of the order of drop acceleration, we assume that drop velocity changes from initial to zero in a characteristic time, / . Thus, the order of the inertial force magnitude is
~ (1.2)
The viscous force appears when the liquid drop deforms under impact and it equals to the emergent shear stress multiplied by the contact area between the liquid and solid substrate. In the case of Newtonian fluids, the shear stress is equal to the dynamic viscosity of liquid, , times the shear rate. The order of the shear rate can be evaluated as the ratio between the initial drop velocity and its diameter, i.e. as / . Based on this assumption and that the contact area is ~ , the order of viscous force magnitude is equal to ~ (1.3) The surface tension force of liquid‐air interface, , is equal to the surface tension of liquid‐ air interface, , times the length of the line over which it acts. The length of the line is proportional to the initial drop diameter, so is ~ (1.4)
The surface tension forces, , , , act at the drop contact line parallel to the corresponding interfaces and determine substrate wettability. When the drop is at rest,, they are balanced (Fig. 1.3) and defined by the relationship, known as Young equation,
cos (1.5)
Fig. 1.3: Sessile drop on a solid substrate; represents the contact angle
several equilibrium contact angles on it. In this case, substrate wettability is specified by two angles: advancing, , and receding contact angles, , which can be measured under quasi‐static drop expansion and reduction. Besides the angles, the surface texture can play the role in the drop impact morphology. For example, the increase of the substrate roughness decreases the splashing threshold [Stow, 1981; Cossali, 1997; Riobbo, 2001]. So, the surface texture should be taken into account too. The ratios between the considered forces (1.1‐1.4) allow determining which forces play main and secondary roles in a drop impact. The ratios have non‐dimensional forms and are well known in fluid dynamics. If inertia, viscous and surface tension forces dominate in the drop impact, then two ratios, including them, are mainly used in the analysis of the process: Reynolds number, , , , (1.6) and Weber number, , , , (1.7) Thus, the substrate wettability, roughness and the ratios of the magnitudes of the mentioned forces determine the drop impact morphology on solid substrates. In the frame of these factors, different aspects of the drop impact morphology have been studied and understood [Yarin, 2006; Marengo, 2011].
1.4. Particle‐laden drop impact
One approach for accounting solid particles in a droplet of a Newtonian liquid is the replacement of the liquid viscosity, , by the effective viscosity, ; this latter typically increases with the growth of particle volume fraction, [Nicolas, 2005; Ueda, 2010]. The effective viscosity is used in the calculation of the Reynolds number, , which in turn determines the drop impact morphology based on pure liquid. The effective viscosity can be calculated via, for example, the equation of Krieger‐Dougherty [Nicolas, 2005]
1
0.68 . (1.8)
If the density of particles, , differs from a liquid density, , then the effective density of liquid, , is also used in the calculation of and [Bertola, 2015] and it is calculated as
1 (1.9)
Using the effective viscosity helps explain some of the observed phenomena; nevertheless its use remains questionable for both nano‐ and microparticles. The nano‐ and microparticles are particles whose overall dimensions lay in the range from 0.5 to 500 nm and from 0.5 to 500 μm, respectively.
In the case of nanoparticles, their addition to a Newtonian liquid only increases its viscosity, but also make it dependent on the shear rate, i.e. the liquid becomes non‐Newtonian and formula (1.8) is no longer applicable. For instance, the viscosity of water with 2% by weight ( 0.8%) of 20 nm silica nanoparticles is larger than that of pure water and it decreases with the increase of the shear rate; in these conditions, water behaves as a shear‐thinning liquid (Fig.8 in [Zang, 2013]).
(2004) found that the increase in viscosity caused by the addition of microparticles reduces the maximum drop contact area significantly less than in the case of the liquid without particles. Meanwhile, the addition of microparticles reduces the threshold of drop splashing [Nicolas, 2005; Peters, 2013] in contrast to the behavior of a pure liquid [Riobbo, 2001].
In the literature, the description of the impact morphology for drops containing particles was limited to only some of the phenomena typical of pure liquids. For example, the effect of nanoparticles was considered only in relation to drop rebound and droplet dynamics under its spreading onto a substrate [Zang, 2013]. Also, the influence of microparticles was considered only in relation to partial or full rebound [Ok, 2004; Ueda, 2010], deposition [Nicolas, 2005; Shen, 2009; Lee, 2014; Marston, 2013; Lubbers, 2014] and splashing [Nicolas, 2005; Peters, 2013; Marston, 2013]. Phenomena such as receding break‐up, rupture, temporary dry spot and singular jet are not considered at all.
Moreover, in many of the previous studies, the influence of solid particles is often unclear because other factors are introduced artificially. As an example, in [Shen, 2009], when analysing the drop spreading, in addition to solid particles also Arabic gum was added to the carrier fluid. The additional component contribute to change substantially the viscosity of the liquid (Table 1 in [Shen, 2009]), masking the effective influence of the dispersed particles. Sometimes, also the experimental procedure is unclear. For example, in [Lee, 2014], it is not evident the choice of the time at which the drop contact area is considered as maximum. Looking at the images’ sequence shown in the paper, the drop appears to continue spreading after the time considered as the maximum spreading time by the authors. Therefore, in these cases, the cause of changes in drop contact area is not clear.
[Nicolas, 2005]). In [Nicolas, 2005], it is suggested that this independence is still valid for dense suspension. However, the results for dense suspensions in [Peters, 2013] (Fig.3) suggest that the splashing threshold indeed decreases with the increase of the particle diameter. On the other way round, the corresponding splashing criterion for dense suspensions as in [Peters, 2013] while taking into account the diameter and density of the particles could not describe the dependence of the threshold from in the case of dilute suspensions. This example shows that in the literature results are often incomplete and contradictory and demand for the studies that are at the core of this thesis.
Besides the drop impact morphology, there is a little information about the particle distribution after an impact. For instance, after the impact of dilute suspension particles distributed in a ring or disk form [Nicolas, 2005]. In the case of dense suspension, particles formed a monolayer, in which they grouped in clusters separated by particle‐free regions [Lubbers, 2014]. The observed patterns are explained qualitatively by means of the forces acting on the particle. No analytical expression (or correlations) for maximum splat diameter or splat shape and the particle distribution after the impact, was provided.
Thus, it is questionable to use the concept of the effective viscosity when considering the drop impact of a Newtonian liquid containing solid nano‐ or microparticles. Also, the drop impact morphology of particle‐laden liquids has been considered only for the part of the phenomena typical of a pure liquid. In the case of splashing and particle distribution, there are questions to numerical description of these phenomena.
1.5. The scope of the thesis
Based on the existing knowledge gaps about the particle influence on drop impact, the following goals are set in this thesis: to know how microparticles influence on the drop impact morphology; to analyse the available data about splashing of particle‐laden drops and formulate a criterion, which in applicable for dilute and dense suspensions with both hydrophilic and hydrophobic particles; to know how particle distribution may be affected by substrate wettability, particle size, particle volume fraction, and drop impact velocities. If it is possible to find an analytical relationship for the particles’ distribution in the splat.
To achieve the set goals, we used millimetre drops. Such drops are mainly used in impact studies, so they are useful for comparative analysis. As carrier fluid, water was selected. The surface tension of water allows using substrates with different wettability, thereby allowing to cover a maximum number of possible phenomena seen for drop impact onto surfaces with different wettabilites.
The impact velocity was chosen in the range from 1.7 to 3.7 m/s (150≤We≤750 and 7100≤Re≤16400). This allowed us to examine the effect of the particles on various possible number of phenomena (e.g. splashing, deposition, partial rebound, and jetting) occurring during drop impacts with substrates. At velocities below the studied range, only drop deposition will be observed. At high impact velocities, only drop fragmentation will be observed.
studies of millimetric drops. The microparticles were hydrophobic to study the influence of their wettability on comparison with published data obtained for hydrophilic particles.
1.6. The framework of the thesis
This thesis is organized as the collection of individual articles devoted to the above mentioned goals. In Chapter 2, we consider how microparticles change the drop impact morphology. The drop impact is studied onto hydrophilic and hydrophobic substrates at different impact speeds to cover as much as possible phenomena typical for pure liquids, such as: splashing, receding break‐up, rupture, temporary dry spot, singular jet, drop partial rebound and deposition. The dependences of the phenomena are considered from particle volume fraction and their diameter.One of the findings is that the splashing appears at low impact velocity with increasing particle volume fraction and their diameter. These dependencies are not described by existing splashing criteria for pure water or dense suspensions. Therefore, in Chapter 3, we formulate a splashing criterion working for dilute and dense suspensions. The formulation is based on the analysis of the existing data from the literature integrated with ours. The obtained criterion improves upon current model in the prediction of the splashing threshold by introducing effect of the particle volume fraction and particle wettability. The criterion shows correctly the dependence of drop splashing on particle diameter, volume fraction and wettability.
the particle distribution on hydrophilic substrates and a crown height in the case of hydrophobic substrates.
After this, Chapter 5 summarizes the findings from the previous chapters. And Finally, Chapter 6 discusses the future prospects of the study of the impact of particle‐laden drops.
1.7. Achievements and activities
The main achievements of this thesis were summarized in Table 1.1. During my PhD study, I had the opportunity to collaborate to various research projects during which a paper was published in a peer‐review journal and results were presented in international conferences (Table 1.2). Table 1.1: Summary of the work achievements in the study of particle‐laden drop impactsTopic State of Art Achievement
Experimental apparatus
Registration by high‐speed camera from side‐ or bottom‐view
Registration simultaneously by high‐speed cameras from side and bottom views
Drop impact morphology
Drops composed hydrophilic particles Drop partial or full rebound, deposition and splashing
Drops composed hydrophobic particles Drop partial rebound, deposition, splashing receding break‐up, rupture, temporary dry spot and singular jet
Splashing Inapplicability of splashing criterion used for pure liquids
Criterion only for dense suspension
New criterion showing correctly the dependence from particle volume fraction, diameter and wettability
Particle distribution
Observation of ring/disk patterns and monolayers on hydrophilic substrates No analytical expression for observed patterns.
Observation of new crown‐like structures on hydrophobic substrates
Analytical expression for ring/disk patterns on hydrophilic substrates and for crown height on hydrophobic substrates
Table 1.2: Summary of the activities
Type of activity Description Year
Academic Kabov, O., Marengo, M., Legros, J.C., Chikov, S., Queeckers, P., Zaytsev, D., Rioboo, R., Aranio, L., Cheverda, V., Gluschuk, A., Biondi, F., Grishaev, V., Mameli, M.: Boiling incipience and rivulet/droplet dynamics in microgravity. 53rd ESA Parabolic Flight Campaign, Bordeaux, France (2010)
2010
Conference Grishaev, V., Lyulin, Y., Chikov, S., Kabov, O.: Anti‐wetting barrier for the CIMEX‐1 experiment. ELGRA Biennial Symposium and General Assembly "Gravity: from µ to x !", Antwerp, Belgium, Sep. 6‐9 (2011)
2011
Academic Kabov, O., Marengo, M., Araneo, L., Chikov, S., Queeckers, P., Cheverda, V., Zaytsev, D., Grishaev, V., Valdarno, L.: Stratified flows, contact line dynamics, boiling and condensation in microgravity. 55th ESA Parabolic Flight Campaign, Bordeaux, France, (2011)
2011
Conference Grishaev, V., Amirfazli, A., Kabov, O.: Prevention of spreading of the liquid with low surface tension by a micro‐groove. 8th Zsigmondy Kolloquium, March 05‐ 07, 2012, Darmstadt, Germany
2012
Paper Grishaev, V., Amirfazli, A., Chikov, A., Lyulin, Y., Kabov, O.: Study of Edge Effect to Stop Liquid Spillage for Microgravity Application. Microgravity Sci. Technol. 25, 27‐33 (2013) 2013 Academic Visiting researcher, Lassonde School of Engineering, York University, Toronto, Canada, Topic: Impact of particle laden drops onto surfaces. Sep.‐Dec. (2013) 2013
Conference Grishaev, V., Iorio, C.S., Amirfazli, A.: Impact of particle laden drops onto surfaces of various wettability, APS 66th Ann. Meeting DFD, Pittsburgh, USA, Nov. 24‐26 (2013)
2013
Conference Grishaev, V., Iorio, C. S., Amirfazli, A.: Morphology of Particle Laden Drop Impact onto a Surface, ASME 2014 4th US‐European Fluids Eng. Division Summer Meeting, Chicago, USA, Aug. 3‐7 (2014)
2014
Paper Grishaev, V., Iorio, C. S., Dubois, F., Amirfazli, A.: Morphology of particle‐laden drop impacts onto surfaces with various wettability. Submitted.
2015
Paper Grishaev, V., Iorio, C. S., Dubois, F., Amirfazli, A.: Impact of particle‐laden drops: splashing on the substrates with various wettability. Submitted.
2015
Paper Grishaev, V., Iorio, C. S., Dubois, F., Amirfazli, A.: Impact of particle‐laden drops: particle distribution on the substrates with various wettabilities. Submitted.
2015
1.8. References
Bertola, V., Haw, M. D.: Impact of concentrated colloidal suspension drops on solid surfaces. Powder Technol. 270, 412‐417 (2015) Calvert, P.: Inkjet Printing for Materials and Devices. Chem. Mater., 13, 3299‐3305 (2001) Cossali, G. E., Coghe, A., Marengo, M.: The impact of a single drop on a wetted solid surface. Exp. Fluids, 22, 463‐472 (1997) Eadie, D.T., Santoro, M.: Top‐of‐rail friction control for curve noise mitigation and corrugation rate reduction. J. Sound Vibration 293, 747‐759 (2006) Fauchais, P., Montavon, G., Lima, R. S., Marple, B. R.: Engineering a new class of thermal spray nano‐based microstructures from agglomerated nanostructured particles, suspensions and solutions: an invited review. J. Phys. D: Appl. Phys. 44, 093001 (53pp) (2011)Krieger, I. M., Dougherty, T. J.: A Mechanism for Non‐Newtonian Flow in Suspensions of Rigid Spheres. Trans. Soc. Rheol. 3, 137 (1959)
Lee, S. J., Huh, H. K., Kwon, D. H.: Energy dissipation of graphene colloidal suspension droplets impacting on solid substrates. RSC Adv. 4, 7216–7224 (2014)
Lewis, J. A., Smay, J. E., Stuecker, J., Cesarano, J.: Direct Ink Writing of Three‐Dimensional Ceramic Structures. J. Am. Ceram. Soc. 89 [12], 3599–3609 (2006)
Lubbers, L. A., Xu, Q., Wilken, S., Zhang, W. W., Jaeger, H. M.: Dense Suspension Splat: Monolayer Spreading and Hole Formation after Impact. Phys. Rev. Lett. 113, 044502 (2014)
Marston, J. O., Mansoor, M. M., Thoroddsen, S.T.: Impact of granular drops. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 88, 010201(R) (2013)
Nicolas, M.: Spreading of a drop of neutrally buoyant suspension. J. Fluid Mech. 545, 271–280 (2005)
Ok, H., Park, H., Carr, W. W., Morris, J. F., Zhu, J.: Particle‐Laden Drop Impacting on Solid Surfaces. J. Dispersion Sci. Technol. 25, 449‐456 (2004)
Peters, I. R., Xu, Q., Jaeger, H.R.: Splashing Onset in Dense Suspension Droplets. Phys. Rev. Lett. 111, 028301 (2013)
Rioboo, R., Tropea, C., Marengo, M.: Outcomes from a drop impact on solid surfaces. Atomization Sprays 11,155–165 (2001) Seerden, K.A. M., Reis, N., Evans, J. R. G., Grant, P. S., Halloran, J. W., Derby, B.: Ink‐Jet Printing of Wax‐Based Alumina Suspensions. J. Am. Ceram. Soc. 84, 2514–20 (2001) Shen, J., Liburdy, J. A., Pence, D. V., Narayanan, V.: Droplet impingement dynamics: effect of surface temperature during boiling and non‐boiling conditions. J. Phys.: Condens. Matter 21, 464133 (14pp) (2009)
Singh M., Haverinen, H. M., Dhagat, P., Jabbour, G. E.: Inkjet Printing—Process and Its Applications. Adv. Mater. 22, 673–685 (2010)
Stow, C. D., Hadfield, M. G.: An Experimental Investigation of Fluid Flow Resulting from the Impact of a Water Drop with an Unyielding Dry Surface. Proc. R. Soc. Lond. A 373, 419‐441 (1981)
Tekin, E., Smith, P. J., Schubert, U. S.: Inkjet printing as a deposition and patterning tool for polymers and inorganic particles. Soft Matter 4, 703–713 (2008)
Ueda, Y., Yokoyama, S., Nomura, M., Tsujino, R., Iguchi, M.: Bouncing behaviors of suspension liquid drops on a superhydrophobic surface. J. Vis. 13, 281–283 (2010)
Yarin, A.L.: Drop Impact Dynamics: Splashing, Spreading, Receding, Bouncing… Annu. Rev. Fluid Mech., 38,159–92 (2006)
Abstract
The aim of this work is to understand the phenomenology associated with particle‐laden drop impact. The impact of millimetre‐size drops was investigated for hydrophilic and hydrophobic substrates. The drops were dispersions of water and round, spherical and nearly iso‐dense hydrophobic particles with diameters centred around 200 µm and 500 µm. The substrates were transparent glass and polycarbonate plates. The impact was studied by side and bottom view images in the range 148≤We≤744 and 7092≤Re≤16368. The particles suppressed the appearance of singular jetting and drop partial rebound, and also caused splashing, receding break‐up and rupture. The occurrences of these phenomena depend on the impact velocity, particles’ diameter and volume fraction. The drops with 200 µm particles spread in two phases: fast and slow, caused by inertial and capillary forces, respectively. Also, the increase in volume fraction of 200 µm particle leads to a linear decrease of the maximum spreading factor caused by the inertia force on both hydrophilic and hydrophobic substrates.
2.1. Introduction
2.1.1. Drop impact morphology of pure liquids
To better understand the influence of particles on drop impact, it looks straightforward to consider as a baseline for study the case of pure liquids. The drop impact of liquid without solid additives shows a variety of phenomena: “prompt” or corona splash, receding break‐up, rupture, temporary dry spot in a lamella under drop receding, singular jet under receding, partial or complete rebound and deposition (Fig. 2.1). In the following a review of the main results as found in the literature is given.
The prompt splash is characterized by the generation of droplets directly at the contact line, while in corona splash, the formation of droplets occurs around the rim of a corona, remote from the solid surface [Rioboo, 2001]. As shown in [Xu, 2007], the prompt splash is mainly due to surface roughness while the corona splash is a result of instabilities produced by the surrounding gas. At atmospheric pressure, the onset of corona splashing was observed when [Stow, 1981; Mundo, 1995; Cossali, 1997]:
/ / (2.1)
where is a value which depends on substrate roughness, and and are Weber and Reynolds
numbers, respectively. The Weber / and Reynolds / numbers include
the main parameters of drop impact: the diameter, , and impact velocity of the drop, , the density, , viscosity, , and surface tension of liquid‐air interface, . To take into account the decrease of splashing threshold with the increase of substrate roughness, Cossali et al. (1997) proposed an empirical formula for the value of / depending on non‐dimensional roughness
parameter / :
/ 649 3.76/ . (2.2)
Vander Wal, 2006; Palacios, 2013]. This discrepancy may be related to experimental issues such as substrate cleanliness or quality of images used to assess the threshold.
The splash typically leads to drop fragmentation. However, this latter phenomenon is also characteristic of the dynamic associated with the receding breakup. The receding break‐up results from the uneven motion of the receding drop contact line (Fig. 2.1). This uneven motion often leads to finger‐like protuberances, which can tear‐off. The chance of tearing‐off becomes higher with the decrease of liquid viscosity as well as with the increase of receding contact angle of the substrate or of impact velocity [Rioboo, 2001]. Drop fragmentation could be observed also as a consequence of drop rupture. The rupture occurs due to holes formation at the impact and their subsequent growth (Fig. 2.1). This holes – often indicated as dry spots ‐ form due to the break of air bubbles trapped between the impacting droplet and the substrate [Dhiman, 2010]. The rupture depends on Reynolds and Weber numbers and substrate wettability [Dhiman, 2010]. On substrates with static contact angles 102° and 105°, the rupture have been observed to occur since 5800 and 800 [Dhiman, 2010]. At the same time, on substrates with static contact angles 47° and 160° the rupture does not happen at 5800; 11600 and 3200; 7200 [Dhiman, 2010]. Also, the rupture does not happen, when holes disappear under drop receding. In such case, they are often reported as
temporary dry spot.
The collapse of a temporary dry spot can cause the appearance of singular jets, breaking up into many small drops (Fig. 2.1). Such jets happen during the drop receding of water on superhydrophobic surfaces at 0.6 16 [Bartolo, 2006]. Sometimes, receding drops can also rebound from the solid surface. The drop rebound can be partial or complete (Fig. 2.1). The partial rebound is promoted by the increase of the drop impact velocity or the receding contact angle of the liquid on the substrate (Table 3 in [Rioboo, 2001]). If the receding contact angle is greater than 100° and 25 585, Antonini et al. (2013) observed that millimetre water drops rebound completely under impact. When impacts do not show any of the above mentioned phenomena, the outcome is often referenced to as drop deposition.
Drop deposition can happen onto surfaces with different wettability (Fig. 2.1). To characterize this process, the spreading factor is the most general criterion used. The spreading factor is the ratio of the equivalent diameter of the drop contact area with the substrate, , to the equivalent diameter of the drop before impact, . The maximum spreading factor depends on substrate wettability: it is higher on hydrophilic than on hydrophobic substrates. The substrate wettability is secondary in drop spreading dynamics at We>200 due to inertial effects overcoming capillary forces [Antonini, 2012].
So, according to the literature review, the over mentioned phenomena are mainly determined by the substrate wettability and roughness as well as Weber and Reynolds numbers under drop impact.
2.1.2. Impact of particle‐laden drops
For what concerns the kind of dispersion studied in the literature, the phenomena has been considered for three forms of drops with solid particles: liquid marbles [Aussillous, 2006; Planchette, 2012; Sivan, 2013; Zang, 2013], wet granular pellets [Fu, 2004; Fu, 2005; Reynolds, 2005] and suspensions (Fig. 2.2). Suspension drops are attractive for us owing to the possibility of changing particle concentration.
The impact of suspensions depends on the particle distribution in the drop before it. The particle distribution can be homogeneous or inhomogeneous. In the literature, the homogenity of suspensions is often considered as a consequence of a thorough ultrasonication. Particle suspensions can also be classified as dense and dilute. Suspensions are considered as dense if the particle volume fraction, , is higher than 0.5.
If we refers to the literature, dilute suspensions with nanoparticles should be considered as homogenous based on their ultrasonication. However, dilute suspension with microparticles are difficult to assess because in the reviewed papers, no clear image and/or statement is given about their homogeneity. In that case, we considered that those suspensions are most probably inhomogeneous. Drop impact studies on suspensions are summarized in Table 2.1. Dense suspensions were considered only in relation to splashing and deposition.
Fig. 2.2: Forms of particle‐laden drops
Table 2.1: Classification of drop impact studies on suspensions
Homogeneity of particle
distribution1) Particle concentration 2) Size range of particles3) References Homogeneous dense Microparticles [Peters, 2013] [Marston, 2013] [Lubbers, 2014] dilute Nanoparticles [Zang, 2013] Microparticles [Ueda, 2010] [Shen, 2009] [Lee, 2014]
Inhomogeneous dilute Micro‐ or millimetre
sized particles [Ok, 2004] [Nicolas, 2005] 1) The homogeneity of the suspensions in the references was determined by the images of drops before impact or sonication of the suspensions before drop realising or particle size and volume fraction values. 2) Particle concentration of suspension drops is dense if particle volume fraction is higher than 0.5. 3) Nanoparticles are the particles which all their dimensions lye in the range of 0.5 to 500 nm. Microparticles – 0.5 µm to 500 µm and millimetre sized particles – 0.5 mm to 5 mm. 2.1.2.1. Drop impact of homogeneous suspension
Peters et al. (2013) studied the splashing under the drop impact for monodisperse and bidisperse dense suspensions onto a glass substrate. The suspensions were dispersions of water with ZrO2 or/and glass particles with volume fraction from 0.59 to 0.65. According to the authors,
splashing onset for suspension drops was not correctly described by the Eq. (2.1) even when viscosity of liquid was substituted by effective viscosity estimated via the formula of Krieger and Dougherty. Therefore, for dense suspensions, Peters et al. (2013) proposed a splashing criterion based on an energy balance at the level of the particles in the suspension. The energy balance led to particle‐based critical Weber number
(2.3)
where and are density and radius of particles, respectively. Peters et al. (2013) found that
suspensions (suspensions contain two type of particles with different diameters). The proposed splash criterion does not take into account the particle wettability and shape. In addition, the authors did not specify the character of splashing. Nevertheless, the proposed splashing mechanism assumes that it can happen far away from the drop contact line. This latter evidence, together with a splashing threshold corresponding to 14.3 2, can be found in the data presented in [Marston, 2013].
Marston et al. (2013) investigated the spreading and splashing of 26 mm particle‐laden drops on glass surface. The suspension was a dispersion of water and sand particles with diameters of 350 µm. The particle volume fraction was 0.55. During drop splash, the speeds of particles ejected were ≈2 times higher than the impact velocity. The authors do not mention about splashing character. However, from the images of drop splashing (Fig. 1, 4a in [Marston, 2013]), we conclude that the splashing character was different from liquid without particles. The impact did not create corona and the particles detached from drop surface not only at contact line with the substrates, but also at a distance. The splashing onset was in the interval of impact velocities from 1.35 o 1.86 m/s, corresponding to the particle‐based Weber number 11.7 22.3. These results confirmed the splashing criterion (Eq. (2.3)) proposed in [Peters, 2013]. In addition, the author found that the spreading factors of the suspension grew as / at 1 similar to liquid without particles. The spreading of dense suspension was studied also in [Lubbers, 2014].
Thus, while in the spreading of dense suspension drops showed some similarity with the pure liquid case when 1, in the splashing case many differences persisted. The splashing onset could not be described by the criterion of corona splashing and it does not depend on the substrate roughness as well. To describe the splashing onset, the particle‐based critical Weber number 14 was proposed. However, it has not been checked for different particles wettability and shapes.
Differences in the particle‐laden drop impact respect ot pure liquids were also observed in the case of homogeneous dilute suspensions, containing nano‐ and micro‐particles. It is worth to note that no systematic study exist and only few occurrences have been observed. Drop impact of suspension with nanoparticles was considered on heated [Shen, 2008; Murshed, 2011] and room temperature substrates [Zang, 2013]. At room temperature, only the complete rebound and spreading dynamics were studied.
Ueda et al. (2010) studied drop division during its complete rebound on a superhydrophobic surface for water with and without calcium carbonate powder. The powder particles were sticks with diameters of ≈100 nm and lengths of ≈2 µm. The particles distributed homogeneously with content of 1% and 10% by weight (Fig.2, 3 in [Ueda, 2010]). Ueda et al. (2010) found that for 10% by weight of the particles drop division was suppressed during rebound. It was explained by viscosity increase. The results also show that 1% or 10% by weight of the particles does not suppress the rebound of aqueous suspensions at 25 and it was similar to the observations of [Zang, 2013] for 2% by weight nanoparticles to water drops at Weber numbers in the range 0 150.
The changes in the impact dynamics for homogeneous dilute suspension were also mentioned in works of [Shen, 2009] and [Lee, 2014], but many questions about the role of solid particles remain either due to the way the experiments is performed or due to the way results are elaborated. For example, in [Lee, 2014] (Fig.4) it is not clear the choice of the time at which drop contact area is considered as maximum despite the fact that it continued to increase thereafter. Therefore, in these cases, the cause of changes in drop contact area is not clear.
In the case of homogeneous suspensions, drop impacts were studied only for splashing, complete rebound and impact dynamics. Nevertheless, it is not clear how even these phenomena depend on the particle volume fraction between 0.1 to 0.5, particles’ size, wettability and shape. The drop impact was also considered in the case of inhomogeneous dilute suspensions with microparticles.
2.1.2.2. Drop impact of inhomogeneous suspensions
