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Dynamics of dewetting and bubble attachment to rough hydrophobic surfaces – Measurements and modelling
Minerals Engineering 85 (2016) 112–122
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Dynamics of dewetting and bubble attachment to rough hydrophobic surfaces – Measurements and modelling Jan Zawala ⇑, Dominik Kosior Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Niezapominajek 8, 30-239 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 27 April 2015 Revised 27 October 2015 Accepted 2 November 2015 Available online 7 November 2015 Keywords: Surface roughness Bubble attachment Contact line spreading Nanobubbles
a b s t r a c t The influence of solid surface roughness (hydrophobic TeflonÒ) on the timescale of the ascending air bubble (Rb = 0.74 mm) attachment and the kinetics of the spreading of the three-phase contact (TPC – gas/liquid/solid) line was studied. The moment of the rising bubble collision with a horizontal TeflonÒ plate immersed in ultrapure water was monitored using fast video recordings (4000 fps). It was shown that, depending on the solid surface roughness, the time of the TPC formation was significantly different. Similarly to our previous studies, it was shorter for higher roughnesses. Using high-frequency video acquisition, an additional factor, kinetics of the spreading of the TPC line associated with various bubble shape changes during TPC formation, could be determined. The registered attachment kinetics and bubble shape variations were very reproducible for smooth and very rough TeflonÒ surfaces, whereas for TeflonÒ of intermediate roughness, up to five different attachment scenarios were observed, with a relatively large standard deviation of time of TPC formation. Numerical calculations used for simulation of the bubble collisions with a horizontal solid wall with precisely controlled hydrodynamic boundary conditions revealed that the experimentally observed timescales of the bubble attachment and spectacular bubble shape variations can be accurately (qualitatively) reproduced for each roughness of the TeflonÒ plate studied. Good agreement between experimental and numerical data is, in our opinion, rather strong evidence for air-induced rupture of the liquid film formed between the colliding bubble and the hydrophobic solid plate. This supports the hypothesis that depending on the solid surface roughness, different amounts of air entrapped in solid surface irregularities could drastically change the solid surface hydrodynamic boundary conditions and, consequently, the kinetics of spreading and formation of the TPC. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Collisions of gas bubbles with various interfaces play an important role in many applications. For example, in flotation, a widespread physicochemical separation technique based on differences in surface properties of the minerals being separated, the effectiveness of the separation process depends on the outcome of interactions between bubbles and solid particles (Ralston, 2000). The flotation separation of ore’s useful components is based on a selective attachment of the colliding bubble with hydrophobic solid surfaces only. Interactions between a bubble and particle can be divided into three sub-processes: (i) collision, (ii) attachment (three-phase contact – TPC – formation) and (iii) detachment. In industrial flotation processes, attachment of
⇑ Corresponding author. E-mail address: [email protected] (J. Zawala). http://dx.doi.org/10.1016/j.mineng.2015.11.003 0892-6875/Ó 2015 Elsevier Ltd. All rights reserved.
mineral grains and formation of a stable bubble-grain aggregate must occur during the very short collision time. Detachment of the bubble and its bouncing results in prolongation of the attachment time and can affect the effectiveness of the flotation separation by decreasing the attachment probability. Generally, the probability of the bubble attachment to a solid particle depends mainly on the stability and kinetics of drainage of the intervening (wetting) film separating the bubble and solid surface. The liquid film drainage kinetics are governed by hydrodynamic boundary conditions at the film interfaces. To ensure a desired and controlled outcome of the flotation separation process (as well as a variety of processes involving multiphase flow), the properties of the interacting interfaces should be tuned. Among many other factors, hydrodynamic boundary conditions at both gas/liquid and solid/liquid interfaces are among the most significant. For gas/liquid interfaces in flotation, the boundary conditions are changed by adding various surface-active substances (SAS),
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whose strong adsorption capability changes the fluidity (mobility) of the interface (Levich, 1962; Dukhin et al., 1998; Ybert and di Meglio, 2000; Zhang et al., 2001; Krzan et al., 2007; Laskowski, 2010; Zawala et al., 2010). The degree of immobilization of the rising bubble surface depends on the nature (ionic, non-ionic) of the SAS and, obviously, its concentration. It was shown that depending on the surface activity of the SAS molecules, the threshold concentration needed for complete immobilization of the gas/liquid interface could vary even by several orders of magnitude (Malysa et al., 2005; Krzan et al., 2007). In the case of the solid/liquid interface changing boundary conditions are much more difficult to obtain and control. However, this effect is extremely interesting and important because the slip boundary conditions tend to reduce the relatively large surface drag coefficient (Voronov et al., 2006; Vinogradova and Belyaev, 2010) and can influence the kinetics of liquid film drainage. It is rather well established that near rough hydrophobic solid surfaces, the boundary slip could be significantly increased (Pit et al., 1999; Zhu and Granick, 2001; Baudry et al., 2001; CottinBizzone et al., 2003; Lauga and Stone, 2003; Voronov et al., 2006; Joseph et al., 2006; Vinogradova and Yakubov, 2006; Wang and Bhushan, 2010; Xu and Li, 2007). One of the most probable reasons for this effect is the presence of air either trapped in surface irregularities (Cottin-Bizzone et al., 2003; Voronov et al., 2006; Vinogradova and Belyaev, 2010) or formed spontaneously in the form of interfacial submicroscopic (nano-) bubbles at rough solid surfaces (Attard, 2003; Lauga and Stone, 2003; Krasowska et al., 2009; Craig, 2011). Attard (2003) wrote that: ‘‘. . . Such nanobubble coverings also have surprising consequences for the motion of particles in liquids or the flow of liquids next to surfaces or in capillaries. One can well anticipate a reduction in drag by such a nanobubble film, since slip obviously occurs at a fluid interface whereas stick boundary conditions are traditionally invoked in the hydrodynamic flow at solid surfaces . . .”. Despite the ongoing debate on nanobubble stability (Craig, 2011; Peng et al., 2013; Weijs and Lohse, 2013), the fact that they do exist at hydrophobic surfaces immersed in aqueous phase is rather well established (Hampton and Nguyen, 2010). Air-induced modification of hydrodynamic boundary conditions of the hydrophobic solid surfaces is very probable, especially in the case of rough surfaces. This can have extremely important implications for flotation, during which well-controlled solid surface patterning is not possible owing to the routine, industrial and large-scale character of the process (Fan et al., 2010; Calgaroto et al., 2014). In flotation, under reduced drag, the separating liquid film between a colliding bubble and a solid surface can be squeezed out faster. This can result in significant time reduction of the TPC formation (Zhou et al., 1996; Krasowska et al., 2007, 2009; Kosior et al., 2013; Ahmed, 2013; Kosior et al., 2014), which in flotation is known as induction time. This time span is the total time required for the attachment of an air bubble with a solid particle and involves the thinning and rupture of wetting film and expansion of the TPC contact line (Gu et al., 2003; Albijanic et al., 2010). This paper presents experimental and theoretical (via numerical simulation) studies on the timescale of bubble attachment to a hydrophobic solid surface and the spreading of the TPC line. Experimentally, the kinetics of bubble collision and attachment were studied in detail at model hydrophobic TeflonÒ surfaces of different roughness. In the numerical case, to reproduce the experimental observations, the boundary conditions of the hydrophobic solid surface were changed in a well-controlled (patterned) manner. The qualitative comparison between kinetics of the bubble attachment to the hydrophobic solid surface obtained experimentally and numerically was then shown. This comparison provides additional evidence supporting the hypothesis about important the role of air presence at hydrophobic rough surfaces on kinetics of the TPC line spreading and bubble attachment.
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2. Methods 2.1. Experimental approach The experimental setup used for monitoring the bubble collision with a hydrophobic solid surface was described in detail elsewhere (Zawala et al., 2007; Kosior et al., 2013). Briefly, a single bubble was formed at the capillary orifice with inner diameter 0.075 mm at the bottom of a square glass column (45 45 mm) in Milli-QÒ water. For the controlled bubble formation, a precise syringe pump (NE-1000, NewEra Pump Systems) was used. By careful adjustment of the airflow rate, it was possible to precisely control the time interval between subsequent bubble detachments. The bubble formation time was always 1.6 s, whereas the time interval between subsequent bubbles was 10–12 s. The radius of the bubble was very reproducible and was Rb = 0.74 ± 0.01 mm. The motion of the bubble was monitored and recorded using a high-speed video camera (Weinberger, SpeedCam MacroVis) with frequency 4000 fps. Sequences of the recorded frames were analysed frame by frame using image analysis software ImageJ (Abramoff et al., 2004) to determine bubble size, deformation ratio, spatial displacements and local velocities. The bubble local velocity was determined as:
u¼
s Dt
ð1Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where s ¼ ðxiþ1 xi Þ2 þ ðyiþ1 yi Þ2 , (xi, yi) and (xi+1, yi+1) are the coordinates of the subsequent positions of the bubble with respect to its bottom pole, and Dt is the time interval between subsequent bubble positions. The bubble size and deformation ratio were determined through an average of 20–40 (n) independent measurements. The bubble size was determined by means of its equivalent radius (Rb) as:
" 2 #1=3 n 1X ðdv Þi ðdh Þi Rb ¼ n i¼1 2 2
ð2Þ
where dh and dv are the horizontal and vertical axes, respectively, of the rising bubbles determined. The bubble deformation ratio (v) was calculated as:
v¼
n 1X dh n i¼1 dv i
ð3Þ
The TeflonÒ solid plate, used in our studies as a model hydrophobic surface, was positioned horizontally just beneath the water/air interface at a distance far larger than that necessary for establishment of the bubble’s terminal velocity. The terminal velocity of the bubble at the moment of collision, i.e., its impact velocity, was ut = 34.7 ± 0.2 cm/s. The value of ut determined experimentally remains in perfect agreement with corresponding values reported by other researchers for an air bubble of similar Rb rising in pure water (Duineveld, 1995; Wu and Gharib, 2002; Legendre et al., 2012). The roughness of the solid plate was modified manually by polishing the surface with sandpaper of different grid numbers—namely, 100 (Klingspor KL 375 J), 600 (S.G. Abrasives 600) and 2500 (S.G. Abrasives 2500 wet paper). For convenience, these three plates will be further referenced in the text as T100, T600 and T2500, respectively. The lateral roughness of the plates was determined on the basis of light microscopy observations as a range of lengths of surface scratches. For the T100 and T600, the roughness was equal to 80–100 lm and 40–60 lm, respectively, whereas for T2500, it was equal to 1–5 lm. The corresponding contact angles of the TeflonÒ plates, measured for Milli-QÒ water using the sessile drop technique, were 120 ± 5°, 110 ± 4° and 100 ± 3°, for T100, T600 and T2500, respectively
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(Kosior et al., 2013). To visualize the differences in roughness of the prepared TeflonÒ plates, so-called ‘‘focus-stacking microphotography” was performed. In this approach, a stack of micrographs is taken at different focal positions (aligned along the optical axis) and merged into a single, entirely focused composite image (Aguet et al., 2008). To increase the depth of field of the final image of the solid surface, a series of micrographs (20–30, 500 500 pixels) of TeflonÒ plates were taken, with the focal point set at evenly spaced intervals into the depth of field. The images in the series were then transferred to a PC and stitched together using the ImageJ Extended Depth of Field algorithm (Aguet et al., 2008). The 3D image of the solid surface was obtained on the basis of differences in the final image greyscale, which was automatically recalculated into the topography variations using the Interactive 3D Surface Plot plugin. The 3D pictures of the TeflonÒ plates obtained in this way are presented in Fig. 1. It should be noted that in the z-direction, the approach described here allows for qualitative data only, without precise information about the depth of grooves and crevices of the surface. It can be seen in Fig. 1, however, that the depth of the scratches for each TeflonÒ sample is comparable with the lateral length of irregularities (distances between peaks). All three pictures were taken under identical conditions, so Fig. 1 illustrates significant differences in the plate’s roughness. Prior to the experiments, solid plates were carefully washed with a chromic acid mixture and rinsed with large amounts of Milli-QÒ water. The glass elements of the setup (capillary, square column) were washed either with the chromic acid mixture or MucasolÒ, a commercially available cleaning liquid, and rinsed with large amounts of Milli-QÒ water. The experiments were carried out at room temperature (21 ± 1 °C). 2.2. Numerical approach Numerical simulations were performed to reproduce the bubble behaviour during its collision with the hydrophobic solid surface. Details about the selected numerical approach can be found elsewhere (Popinet, 2003, 2009; Fuster et al., 2009; Afkhami et al., 2009; Zawala and Dabros, 2013). Fig. 2A presents the scheme of the computational domain. The liquid column of height H = 100 mm and radius L = 5 mm containing a gas bubble was described by a two-dimensional, cylindrical coordinate system. The bubble radius was set to be identical to that determined during experiments (0.74 mm), and S was equal to 94.42 mm. The Dirichlet boundary conditions were applied at the column wall. For a liquid column of L = 5 mm, chosen here as a compromise between the accuracy of the results and computational time, one should be aware of the existence of a so-called wall effect. To minimize the influence of the column wall proximity on bubble motion, the boundary conditions at the side cylinder walls were assumed to
be slip. This was done to study possible ideal situations, which could be helpful in interpreting experimental observations and trends. The terminal velocity of the bubble in the numerical case was determined to be 33 cm/s. The top column surface was reconstructed using Boolean operations, and its thickness was 80 lm. At the top wall, the hydrodynamic boundary conditions were controlled as follows: (i) the degree of slip/no-slip properties were varied uniformly by gradually changing the value of fluid velocity (u) at the solid surface from 0 to 1, where 0 was related to the completely no-slip condition while 1 was assigned to the completely slip surface, or (ii) slip (u = 1) and no-slip (u = 0) areas (squares) were reconstructed alternately at the surface from small blocks with different degrees of slip, as schematically presented in Fig. 1B. The length of the slip spots (block walls) varies between 50 and 150 lm. Three different scenarios were considered: (i) the slip areas were uniformly, periodically spread at the surface (see Fig. 2B1, referenced further as scenario B1); (ii) there was only one slip spot, directly at the symmetry axis of the system (axis of symmetry of the bubble trajectory, Fig. 2B2, scenario B2); and (iii) the slip spots covered only half of the surface (Fig. 2B3, scenario B3). Depending on the scenario, the simulation was performed using either an axi-symmetrical (scenario B1 and B2) or Cartesian solver (scenario B3); the implications of this fact will be discussed further. At t = 0, the bubble was motionless (initial velocity u = 0) and spherical. After acceleration, the constant speed (terminal velocity) of the bubble was established for t > 50 ms. The contact angle of the solid surface ranged between 90° and 100°, comparable with the experimental cases. Densities of the liquid and gas were set to be 1000 and 1.3 kg/m3, respectively—i.e., on the order of water and air. Similarly, the dynamic viscosities of the liquid and gas were taken as 1 103 and 18 106 Pa s, respectively. The surface tension was assumed to be 0.0724 N/m. 3. Results and discussion 3.1. Influence of roughness on attachment time (experiment) Sequences of photos illustrating the bubble collision with T2500 and T100 surfaces are presented in Fig. 3. The time t = 0 was adjusted to the moment of the bubble’s first collision (maximum deformation during impact). In both cases, the bubble approaches the solid/water interface with a constant shape, a consequence of constant (terminal) velocity of the bubble. However, the course of bubble motion after this collision is drastically different and depends on the roughness of the solid surface. For the T2500 (smooth) surface, the bubble bounces several times before the TPC formation, whereas at the T100 (rough) surface, the TPC forms immediately during the first bubble strike. At T2500, up to 5–6 ‘‘approach-bounce” cycles, with diminishing amplitude, were
Fig. 1. Three-dimensional pictures of the (A) T100, (B) T600 and (C) T2500 surfaces, obtained using the focus-stacking microphotography approach. Units are pixels. In the xydirection, 10 pixels equals 1.2 lm.
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Fig. 2. Scheme of (A) geometry of the computational domain and (B) approach used for modification of solid surface hydrodynamic boundary conditions (slip vs. no-slip spots).
Fig. 3. Sequences of photos illustrating the bubble collision, bouncing and moment of the TPC formation for the T2500 and T100 surfaces (Dt = 1 ms).
always observed before bubble attachment. The damping of the amplitude of the bounces was caused by dissipation (viscous losses) of the kinetic energy associated with bubble motion. Note that at T2500 (smooth) surface, the TPC forms after the dissipation of virtually all kinetic energy, when the bubble is in permanent contact (practically immobile) with the solid/water interface. In contrast, for the T100 surface, the TPC was formed during the first collision, when the kinetic energy associated with the bubble motion was maximal. A quantitative description of the photo sequences from Fig. 3, with respect to variations of the local velocities of the bubble as a function of time during collision and bouncing, are presented in Fig. 4. The bubble approaches the solid surface with constant terminal velocity, which for the studied bubble size is 34.7 ± 0.2 cm/s. For T2500, after brief bubble-solid surface contact,
the rebound is observed in a manner similar to earlier publications (Tsao and Koch, 1997; Zawala et al., 2007), and the bubble’s maximum velocity was approximately 30 cm/s. Because of buoyancy and viscous losses, the bubble undergoes several approachbounce cycles with diminishing amplitude, degree of shape deformations and impact velocity of consecutive collisions (Zawala et al., 2007; Kosior et al., 2013) and eventually, after approximately 90 ms, remains in permanent contact with the solid/liquid interface. Next, the TPC is formed after approximately 130 ms, which is associated with a rapid change in the bubble’s position and, consequently, the high peak of the velocity profile. The bubble rupture and TPC formation imply that the draining liquid (water) film, separating the bubble and solid surfaces, has reached its critical thickness of rupture. For the T2500 surface, the experimentally measured values of tTPC (time span from t = 0 to the moment of
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Fig. 4. Experimentally determined variations of the local bubble velocities during collision and bouncing with T2500 surface (circles) and T100 surface (squares). For both cases, Dt = 0.25 ms.
the TPC hole formation) were very reproducible, with an average tTPC, calculated from 20 independent runs, equal to 115 ± 17 ms. The TPC was always formed after up to 5–6 approach-bounce cycles, consistent with previous studies, in which similar good reproducibility of the bubble’s attachment time to a smooth hydrophobic surface was reported (Krasowska et al., 2009; Kosior et al., 2013, 2014). Very good reproducibility was also obtained for similar experiments with the T100 surface (Fig. 1A). Here, however, tTPC was shortened drastically to only 1 ± 0.5 ms (see also Fig. 3, bottom sequence). As shown in Figs 3 and 4, immediate TPC formation at the T100 surface means that no bouncing was observed, and fast attachment of the bubble to the solid surface occurred during the first strike. This occurrence was observed in all of the experiments performed. The bubble interacting with the T600 surface produced more varied results. Here, depending on the experimental run, the TPC formed either immediately during the first strike or during the second or third collision, after a short period of bouncing. This effect is illustrated in Fig. 5, in which velocity variations of the bubble for
Fig. 5. Experimentally determined variations in the local bubble velocities during collisions with the T600 surface for four different scenarios of attachment. The moment of TPC hole formation for each scenario is marked with an asterisk (Dt = 0.25 ms).
observed attachment scenarios are presented. Here, for clarity purposes, the case of the formation of the TPC during the 3rd collision is omitted. The moment of the TPC formation is marked in Fig. 5 with asterisks. Despite the fact that liquid film ruptures during the 1st, 2nd or 3rd collision (3rd not shown in Fig. 5), four different profiles of bubble velocity variations were observed. This is a consequence of the significant discrepancy in kinetics of the TPC formation and TPC line spreading after rupture of the separating liquid film and dewetted hole formation, i.e., different attachment scenarios. This caused poor reproducibility of the determined tTPC values for the T600 surface, which is significantly lower than those of the T2500 and T100 plates. The average value of tTPC calculated from 20 independent runs was 15 ± 21 ms, with a relatively large standard deviation. Depending on the attachment scenario, the TPC could be formed either after few or approximately 40 ms, despite the fact that each experimental run was performed with the same T600 solid plate. As mentioned above, the poor reproducibility of the TPC formation in the case of T600 was a consequence of 5 different attachment scenarios. Below, sequences of photos are presented showing various bubble shape deformations at the moment of TPC formation, recorded experimentally. Fig. 6 presents the photos of the bubble at the moment of collision (t = 0) and attachment to the T600 surface, where the TPC was formed during the first collision (open circles in Fig. 5). After brief bubble-solid surface contact, the TPC hole is formed at approximately t = 1.5 ms. The TPC line then spreads, which is associated with a spectacular, ‘‘bullet-like” shape of the bubble. As can be noted from the three last pictures in the first row of Fig. 6, the hole of the TPC is formed asymmetrically, i.e., closer to the bubble edge (rim of the liquid film). Moreover, rupture of the liquid film occurs when the bubble tends to withdraw from the solid surface, i.e., wants to rebound. This bubble shape is a consequence of balance between the TPC line spreading kinetics, determining the force counteracting detachment and magnitude of the detachment force, pushing the bubble away from the solid/liquid interface (bounce). From the energy balance, the film ruptures when the kinetic energy associated with the bubble motion is almost completely transferred into the surface energy related to bubble area increase (Chesters and Hofman, 1982; Tsao and Koch, 1997; Zawala and Dabros, 2013). However, as a consequence of TPC hole formation, the capillary force acting at the TPC perimeter is strong enough to prevent bubble detachment, and the bubble withdrawal attempt is drastically impeded. The equilibrium bubble shape is established after approximately 10 ms. A similar sequence of photos of the bubble for a different experimental run at the T600 surface is presented in Fig. 7. Here, despite the fact that all experimental conditions were identical and the TPC was formed during the first contact (open triangles in Fig. 5), the manner of the bubble attachment is completely different. As seen, after collision (t = 0), the TPC line spreads very fast, and the equilibrium bubble shape is formed after approximately 2–3 ms. In contrast to the behaviour shown in Fig. 6, here, for the ‘‘spreading” scenario, there are no violent bubble shape variations. Moreover, the liquid film ruptures rather uniformly. These observations suggest that the kinetic energy associated with the bubble collision and liquid film rupture is dissipated much faster than the case described above (Fig. 6). Fig. 8 presents another photo sequence registered during experiments with the T600 surface. It was observed in some experimental runs that, during the collision, the liquid film ruptured only in one location on the solid surface (open squares in Fig. 5). This is evidenced by a clear necking formation during the backward motion of the bubble (see second row of the sequence of Fig. 8). The dimension of the spreading TPC line was, however, too small, and capillary forces were consequently not large enough to prevent
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Fig. 6. Collision and attachment of the bubble with the T600 surface. The attachment took place during the first impact with a characteristic bullet-like bubble shape at the moment of TPC formation (Dt = 0.25 ms) – ‘‘bullet-like” scenario.
Fig. 7. Collision and attachment of the bubble with the T600 surface. The attachment took place during the first impact with rapid bubble spreading at the solid/liquid interface – ‘‘spreading” scenario.
Fig. 8. Collision of the bubble with the T600 surface. The TPC hole formation, necking evolution and breakage are clearly visible. The attachment took place during the second collision, directly at the spot where the satellite bubble was left.
the bubble’s withdrawal. Moreover, after bubble withdrawal, a small satellite bubble remained at the solid surface. In the necking scenario, the TPC formation and stable bubble attachment always took place during the second collision, when the macro-bubble hit the place occupied by the satellite bubble, as described elsewhere (Krasowska et al., 2009). In fact, the TPC was formed as the consequence of coalescence of two water/air interfaces – bubble surface and surface of micro-bubble (an already dewetted spot). The qualitative data for the frequency (probability) of each scenario observed at the T600 surface are presented in Fig. 9. The ‘‘bullet-like” and ‘‘spreading” scenarios were the most frequent, whereas attachment during the 2nd collision was lower. It should be noted that the attachment scenarios observed were independent of the number of experimental runs.
Fig. 9. Statistics of the attachment scenarios observed during the bubble collision with the T600 surface (determined on the basis of 20 independent experimental runs).
3.2. Kinetics of spreading of the TPC line and attachment force Fig. 10A presents the kinetics of the TPC line spreading (dewetting) for T100, two scenarios of T600 and T2500. Experimentally measured variations in length (diameter) of the TPC line (dTPC) as
a function of time are presented there. Here, t = 0 was adjusted for the moment when the TPC hole started to form. Note that for T100 and all presented scenarios for T600, the TPC formation took place during the first collision, whereas for T2500, it occurred after
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Fig. 10. Variations of mean length of the TPC line determined for the solid surface of different roughness (A) as a function of time and (B) for t = 0 (very first moment of the TPC hole appearance).
five collisions. The spreading of the TPC line is fastest for the T100 surface; only 1.25 ms are required to establish an equilibrium TPC diameter. Moreover, the initial value of dTPC is relatively large and equals 1.62 ± 0.11 mm (see Fig 10B). The ‘‘spreading scenario” at the T600 surface is comparable with the T100 case with respect to initial dTPC values. Here, however, the TPC line spreading is a bit slower; the equilibrium dTPC is established after approximately 5 ms. For the ‘‘bullet-like” scenario, the initial value of dTPC is much lower and equals 1.27 ± 0.23 mm (Fig. 10B). Establishment of the equilibrium dTPC is also longer here (6 ms). For the T2500 surface, where the liquid film ruptured and the TPC formation occurred after dissipation of most of the kinetic energy, the bubble was captured beneath the solid surface and remained spherical, causing the initial dTPC to be the shortest (0.57 ± 0.01 mm). We can now estimate the magnitude of forces responsible for the bubble withdrawal from and attachment to the solid surface. In a simplified approach, the attachment force can be approximated by the capillary force (Fc) acting at the perimeter of the TPC hole:
F c ¼ 2pr TPC r
ð4Þ
where rTPC = 1/2 dTPC, and r is the surface tension of water. The force associated with the backward motion of the bubble (detachment force – Fd) is related to the momentum (Dp) change during the collision period (Dt) and can be written as (Zawala et al., 2013):
Dp Dt
ð5Þ
Dp ¼ m Du
ð6Þ
Fd ¼
where m is the mass of the bubble, which can be expressed as
m ¼ C m V b ql
ð7Þ
where Cm is the bubble added mass coefficient, Vb is the bubble volume, and ql is the water density; we can write that
Fd ¼
C m V b ql Du Dt
ð8Þ
The added mass coefficient of the bubble can be estimated from a simple linear expression (Klaseboer et al., 2001):
C m ¼ 0:62v 0:12
ð9Þ
where v is the bubble shape deformation ratio. Fig. 11 presents the ratio of calculated Fc and Fd values for the first bubble collision with the T100 and T600 surfaces. The tTPC values were determined exper-
Fig. 11. The ratio between attachment and detachment forces for different scenarios registered during the bubble collision with hydrophobic solid plates of different roughness.
imentally for the moment when the bubble was about to start its backward motion (approximately 1 ms after the moment of collision). The collision time (Dt) was determined to be equal to 2 ms. The Du value was taken as the bubble impact velocity (34.7 cm/s). The dashed line in Fig. 11 indicates the situation in which Fc and Fd are equal. For Fc > Fd (above the dashed line), the bubble should be attached to the solid surface; otherwise (when Fc < Fd), the bubble rebound should occur. As seen for the T100 and T600 surfaces (‘‘bullet-like” and ‘‘spreading” scenarios), Fc > Fd. Indeed, the attachment of the bubble occurred during the first collision (see Figs. 6 and 7). For the ‘‘necking” scenario, Fc Fd, and the bubble withdrew from the surface despite liquid film rupture and TPC formation (Fig. 8). For the remaining three cases, it can be seen in Fig. 11 that the Fc/Fd value was the smallest for the ‘‘bullet-like” scenario (Fc and Fd values are comparable) and increased gradually for ‘‘spreading” and T100 cases. In our opinion, the relation between the Fc and Fd values can explain different bubble shapes at the moment of the TPC hole formation. Smaller Fc (‘‘bullet-like” scenario) values cause elongation of the bubble shape during attachment and TPC line spreading because Fd is not completely dominated by the attachment force.
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Fig. 12. The moment of attachment of the bubble to the (A) T2500 surface and (B) T100 surface. The second sequence of each comparison shows corresponding numerically calculated bubble shapes for the smooth surface of respective boundary conditions.
Fig. 13. Qualitative comparison between experimentally obtained bubble images for the ‘‘spreading” scenario of attachment and corresponding numerically determined bubble shapes for the B1 surface (see Fig. 1).
3.3. Comparison between experimental and numerical results In light of the literature data (Fan et al., 2010; Calgaroto et al., 2014; Zhou et al., 1996; Krasowska et al., 2007, 2009; Kosior et al., 2013, 2014), the results presented above indicate that the TPC formation at rough solid surfaces can be air-induced. Air can be entrapped in surface grooves and crevices and also can be redistributed in the form of nano- and micro-bubbles at a solid surface immersed in aqueous phase (Krasowska et al., 2007). Moreover, air pockets can be randomly distributed at the solid/liquid interface, and the size and degree of distribution increase with increasing surface roughness (Snoswell et al., 2003; Zhou et al., 2009; Hampton and Nguyen, 2010). The results presented above support the hypothesis of an air-induced mechanism for liquid film rupture and TPC formation because The mean values of tTPC decrease with solid surface roughness. tTPC is very reproducible for T2500 and T100 surfaces (smoothest and roughest), whereas significant deviations are observed for T600, where roughness is intermediate. At the T2500 and T100 surfaces, only one attachment scenario occurs (see Fig. 3), whereas at the T600 surface, up to 5 different scenarios were registered. This confirms a more random distribution of air pockets at the T600 surface; the manner of the TPC line spreading is stochastic. The ‘‘necking” scenario shows that immediate TPC formation and fast spreading of the TPC line can be observed if the macrobubble collides with previously dewetted spots at the solid surface (fast coalescence between air/water interface). dTPC at the liquid film rupture time is larger for higher roughness. Moreover, for T600, dTPC is significantly different for different attachment scenarios (different amounts of air). This is
certainly caused by variations in the contact angle, a consequence of air entrapped in the solid surface irregularities. This air can also be redistributed in the form of submicroscopic (nano-/micro-) bubbles at the solid/water interface. The time of spreading of the TPC line to the equilibrium dTPC value is the fastest for the surface where more air is expected to be present. As stated above, the presence of air at the solid surface could modify its hydrodynamic boundary conditions. More air should induce larger hydrodynamic slip of the solid surface. In contrast, the surface with less air entrapped should behave more as a noslip condition. We decided to use this hint to model the presence of air at a hydrophobic solid surface. By proper patterning of the numerically reconstructed hydrophobic solid surface (slip/no-slip areas; see Fig. 2), we attempted to reproduce the bubble behaviour and attachment scenarios during collision with T100, T600 and T2500. Fig. 12 presents the photo sequences of the bubble collision with the T2500 (Fig. 12A) and T100 (Fig. 12B) surfaces (see also Fig. 3). The sequences of experimentally obtained photos are compared with numerically calculated outlines of the corresponding bubble shapes. The bubble behaviour at T2500 was reproduced in numerical simulations by setting the solid surface as completely no-slip. For T100, the surface was set as completely slip. Note the very good agreement between experimentally observed and numerically reproduced bubble shapes during the TPC line spreading. Fig. 13 presents the photos of the bubble colliding with the T600 surface for the ‘‘spreading” scenario (see also Fig. 7). Again, experimentally obtained photos are compared with numerically calculated bubble outlines. The numerically reconstructed solid surface was obtained for the case presented in Fig. 10 according
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Fig. 14. Qualitative comparison between experimentally obtained bubble images for ‘‘bullet-like” scenario of attachment and corresponding numerically determined bubble shapes for the B3 surface (see Fig. 1).
Fig. 15. Qualitative comparison between experimentally obtained bubble images for ‘‘necking” scenario of attachment and corresponding numerically determined bubble shapes for the B2 surface (see Fig. 1).
Fig. 16. Experimentally determined variations in the horizontal axis of the bubble during its approach to T100 (open circles) and T2500 (full squares) solid surfaces. The solid and dashed lines are simple exponential fits to the experimental points. Inset—corresponding results of numerical simulations (full slip surface—full squares, 99% no-slip surface—open circles).
to the B1 approach (see Section 2.2); i.e., the surface was created periodically with slip and no-slip blocks of a given longitudinal length (see Fig. 2B1). The agreement between experimentally observed and numerically reproduced bubble shapes during the TPC line spreading was again very good. Each characteristic and spectacular bubble shape was well reproduced in simulations. A similar comparison is presented in Fig. 14. Here, the experimentally observed ‘‘bullet-like” scenario is compared with results obtained according to numerical approach B3 (see Section 2.2), in which the Cartesian solver was applied in the calculations. Here,
the spectacular bubble shapes were also nicely reproduced, including the ‘‘bullet-like” shape. Similarly, the ‘‘necking” scenario was also accurately reproduced in numerical simulations, as shown in Fig. 15. Here, the numerical calculations were performed for numerical approach B2, with only one slip spot at the no-slip solid/liquid interface, directly at the trajectory of the geometrical centre of the bubble. As seen in the last picture of the bottom sequence (numerical data), even the formation of the small satellite bubble was reproduced in the simulations. This micro-bubble was formed at the slip area of the solid surface. Three different approaches with respect to the distribution of the slip areas at the hydrophobic solid surface were used to reproduce the experimentally observed bubble attachment scenarios at the T600 surface. For the T2500 and T100 surfaces, only one approach was sufficient. This is additional proof of a more random and stochastic distribution of air pockets at the T600 surface during its immersion into water phase, caused by intermediate surface roughness. Additional data supporting the above discussion and conclusion are presented in Fig. 16. The experimentally determined variations of the bubble’s horizontal axis (dh) are presented here for the approaching period (before the first strike). The variations in dh are normalized by the dh1 far from the wall, which is constant after the establishment of the bubble’s terminal velocity. The normalized dh variations are presented in Fig. 16 as a function of the normalized distance of the bubble’s geometrical centre to the solid surface (xc/Rb). The points are data determined on the basis of image analysis, whereas the lines are the fitted exponential regressions. The image analysis revealed that despite the experimental scatter, it is possible to notice small differences in the manner of the dh variations with the separation distance. This dependence is much stronger for the T2500 (smoothest) surface than T100 (the roughest one). In our opinion, this subtle discrep-
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ancy between the method of the approaching bubble deformation before collision with T2500 and T100 is an indication of the different hydrodynamic boundary conditions at the solid surface, determined by different amounts of entrapped air. Because of a much higher amount of air at the rough T100 surface (gas trapping mechanism – Snoswell et al., 2003; Zhou et al., 2009; Hampton and Nguyen, 2010), boundary conditions were shifted towards more slip. In contrast, the amount of interfacial air at the smooth T2500 surface was much lower, resulting in a more no-slip surface. This modifies the degree of viscous dissipation of energy in the solid surface’s vicinity and influences the bubble deformation. It should be emphasized that even for a hydrophobic solid surface of 90% no-slip (10% slip) in numerical simulations, the approaching bubble deformation was much stronger than for the completely slip solid/liquid interface. The bubble deformation pattern, therefore, can be used as a marker of different boundary conditions (amount of air) at the hydrophobic solid surface. The results of numerical calculations (presented in the inset of Fig. 16) support our hypothesis. As seen, the manner of numerically determined variations of dh of the bubble approaching numerically slip and no-slip solid/liquid interface is consistent with those determined experimentally. 4. Conclusions The bubble attachment time and kinetics of the TPC formation strongly depend on the roughness of the hydrophobic solid surface. It was observed that the time of the TPC formation can be drastically reduced when the solid surface roughness increases. Moreover, this effect is associated with different attachment scenarios and spectacular rupturing bubble shapes revealed using ultrafast video recordings. The bubble shape variations during rupture at the solid surface also strongly depend on the solid surface roughness. In our opinion, different attachment kinetics and scenarios are a consequence of different amounts of air present at topographically nonhomogeneous solid/liquid interfaces. As a consequence, the hydrodynamic boundary conditions are air-amount dependent and shift towards more slip or no-slip for T100 and T2500, respectively. This hypotheses is supported by four main findings: (i) lack of scatter of the bubble attachment time to T100 and T2500 surfaces, i.e., for two extreme cases, with respect to the degree of roughness (and entrapped air amount); (ii) strong scatter at the T600 (intermediately rough) surface; (iii) good qualitative agreement between experimentally observed and numerically determined shapes of the bubbles during their approach and rupture (TPC formation) and (iv) discrepancy between the approaching bubble deformation before collision with the T2500 (more no-slip) and T100 (more slip) surfaces. Acknowledgements The financial support from the Polish Ministry of Science and Higher Education (Iuventus Plus Project No. IP2014 053973) is acknowledged with gratitude. Partial financial support from the National Research Center (NCN Grant No. 2013/09/D/ST4/03785) is gratefully acknowledged. The study is related to the activity of the European network action COST MP1106: ‘‘Smart and green interfaces—from single bubbles and drops to industrial, environmental and biomedical applications”. The authors express their gratitude to Professor Kazimierz Malysa (Jerzy Haber ICSC PAS) for valuable comments and helpful discussion. References Abramoff, M.D., Magalhaes, P.J., Ram, S.J., 2004. Image processing with Image. J. Biophoton. Int. 11 (7), 36–42.
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Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Dynamics of dewetting and bubble attachment to rough hydrophobic surfaces – Measurements and modelling Jan Zawala ⇑, Dominik Kosior Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Niezapominajek 8, 30-239 Krakow, Poland
a r t i c l e
i n f o
Article history: Received 27 April 2015 Revised 27 October 2015 Accepted 2 November 2015 Available online 7 November 2015 Keywords: Surface roughness Bubble attachment Contact line spreading Nanobubbles
a b s t r a c t The influence of solid surface roughness (hydrophobic TeflonÒ) on the timescale of the ascending air bubble (Rb = 0.74 mm) attachment and the kinetics of the spreading of the three-phase contact (TPC – gas/liquid/solid) line was studied. The moment of the rising bubble collision with a horizontal TeflonÒ plate immersed in ultrapure water was monitored using fast video recordings (4000 fps). It was shown that, depending on the solid surface roughness, the time of the TPC formation was significantly different. Similarly to our previous studies, it was shorter for higher roughnesses. Using high-frequency video acquisition, an additional factor, kinetics of the spreading of the TPC line associated with various bubble shape changes during TPC formation, could be determined. The registered attachment kinetics and bubble shape variations were very reproducible for smooth and very rough TeflonÒ surfaces, whereas for TeflonÒ of intermediate roughness, up to five different attachment scenarios were observed, with a relatively large standard deviation of time of TPC formation. Numerical calculations used for simulation of the bubble collisions with a horizontal solid wall with precisely controlled hydrodynamic boundary conditions revealed that the experimentally observed timescales of the bubble attachment and spectacular bubble shape variations can be accurately (qualitatively) reproduced for each roughness of the TeflonÒ plate studied. Good agreement between experimental and numerical data is, in our opinion, rather strong evidence for air-induced rupture of the liquid film formed between the colliding bubble and the hydrophobic solid plate. This supports the hypothesis that depending on the solid surface roughness, different amounts of air entrapped in solid surface irregularities could drastically change the solid surface hydrodynamic boundary conditions and, consequently, the kinetics of spreading and formation of the TPC. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Collisions of gas bubbles with various interfaces play an important role in many applications. For example, in flotation, a widespread physicochemical separation technique based on differences in surface properties of the minerals being separated, the effectiveness of the separation process depends on the outcome of interactions between bubbles and solid particles (Ralston, 2000). The flotation separation of ore’s useful components is based on a selective attachment of the colliding bubble with hydrophobic solid surfaces only. Interactions between a bubble and particle can be divided into three sub-processes: (i) collision, (ii) attachment (three-phase contact – TPC – formation) and (iii) detachment. In industrial flotation processes, attachment of
⇑ Corresponding author. E-mail address: [email protected] (J. Zawala). http://dx.doi.org/10.1016/j.mineng.2015.11.003 0892-6875/Ó 2015 Elsevier Ltd. All rights reserved.
mineral grains and formation of a stable bubble-grain aggregate must occur during the very short collision time. Detachment of the bubble and its bouncing results in prolongation of the attachment time and can affect the effectiveness of the flotation separation by decreasing the attachment probability. Generally, the probability of the bubble attachment to a solid particle depends mainly on the stability and kinetics of drainage of the intervening (wetting) film separating the bubble and solid surface. The liquid film drainage kinetics are governed by hydrodynamic boundary conditions at the film interfaces. To ensure a desired and controlled outcome of the flotation separation process (as well as a variety of processes involving multiphase flow), the properties of the interacting interfaces should be tuned. Among many other factors, hydrodynamic boundary conditions at both gas/liquid and solid/liquid interfaces are among the most significant. For gas/liquid interfaces in flotation, the boundary conditions are changed by adding various surface-active substances (SAS),
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whose strong adsorption capability changes the fluidity (mobility) of the interface (Levich, 1962; Dukhin et al., 1998; Ybert and di Meglio, 2000; Zhang et al., 2001; Krzan et al., 2007; Laskowski, 2010; Zawala et al., 2010). The degree of immobilization of the rising bubble surface depends on the nature (ionic, non-ionic) of the SAS and, obviously, its concentration. It was shown that depending on the surface activity of the SAS molecules, the threshold concentration needed for complete immobilization of the gas/liquid interface could vary even by several orders of magnitude (Malysa et al., 2005; Krzan et al., 2007). In the case of the solid/liquid interface changing boundary conditions are much more difficult to obtain and control. However, this effect is extremely interesting and important because the slip boundary conditions tend to reduce the relatively large surface drag coefficient (Voronov et al., 2006; Vinogradova and Belyaev, 2010) and can influence the kinetics of liquid film drainage. It is rather well established that near rough hydrophobic solid surfaces, the boundary slip could be significantly increased (Pit et al., 1999; Zhu and Granick, 2001; Baudry et al., 2001; CottinBizzone et al., 2003; Lauga and Stone, 2003; Voronov et al., 2006; Joseph et al., 2006; Vinogradova and Yakubov, 2006; Wang and Bhushan, 2010; Xu and Li, 2007). One of the most probable reasons for this effect is the presence of air either trapped in surface irregularities (Cottin-Bizzone et al., 2003; Voronov et al., 2006; Vinogradova and Belyaev, 2010) or formed spontaneously in the form of interfacial submicroscopic (nano-) bubbles at rough solid surfaces (Attard, 2003; Lauga and Stone, 2003; Krasowska et al., 2009; Craig, 2011). Attard (2003) wrote that: ‘‘. . . Such nanobubble coverings also have surprising consequences for the motion of particles in liquids or the flow of liquids next to surfaces or in capillaries. One can well anticipate a reduction in drag by such a nanobubble film, since slip obviously occurs at a fluid interface whereas stick boundary conditions are traditionally invoked in the hydrodynamic flow at solid surfaces . . .”. Despite the ongoing debate on nanobubble stability (Craig, 2011; Peng et al., 2013; Weijs and Lohse, 2013), the fact that they do exist at hydrophobic surfaces immersed in aqueous phase is rather well established (Hampton and Nguyen, 2010). Air-induced modification of hydrodynamic boundary conditions of the hydrophobic solid surfaces is very probable, especially in the case of rough surfaces. This can have extremely important implications for flotation, during which well-controlled solid surface patterning is not possible owing to the routine, industrial and large-scale character of the process (Fan et al., 2010; Calgaroto et al., 2014). In flotation, under reduced drag, the separating liquid film between a colliding bubble and a solid surface can be squeezed out faster. This can result in significant time reduction of the TPC formation (Zhou et al., 1996; Krasowska et al., 2007, 2009; Kosior et al., 2013; Ahmed, 2013; Kosior et al., 2014), which in flotation is known as induction time. This time span is the total time required for the attachment of an air bubble with a solid particle and involves the thinning and rupture of wetting film and expansion of the TPC contact line (Gu et al., 2003; Albijanic et al., 2010). This paper presents experimental and theoretical (via numerical simulation) studies on the timescale of bubble attachment to a hydrophobic solid surface and the spreading of the TPC line. Experimentally, the kinetics of bubble collision and attachment were studied in detail at model hydrophobic TeflonÒ surfaces of different roughness. In the numerical case, to reproduce the experimental observations, the boundary conditions of the hydrophobic solid surface were changed in a well-controlled (patterned) manner. The qualitative comparison between kinetics of the bubble attachment to the hydrophobic solid surface obtained experimentally and numerically was then shown. This comparison provides additional evidence supporting the hypothesis about important the role of air presence at hydrophobic rough surfaces on kinetics of the TPC line spreading and bubble attachment.
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2. Methods 2.1. Experimental approach The experimental setup used for monitoring the bubble collision with a hydrophobic solid surface was described in detail elsewhere (Zawala et al., 2007; Kosior et al., 2013). Briefly, a single bubble was formed at the capillary orifice with inner diameter 0.075 mm at the bottom of a square glass column (45 45 mm) in Milli-QÒ water. For the controlled bubble formation, a precise syringe pump (NE-1000, NewEra Pump Systems) was used. By careful adjustment of the airflow rate, it was possible to precisely control the time interval between subsequent bubble detachments. The bubble formation time was always 1.6 s, whereas the time interval between subsequent bubbles was 10–12 s. The radius of the bubble was very reproducible and was Rb = 0.74 ± 0.01 mm. The motion of the bubble was monitored and recorded using a high-speed video camera (Weinberger, SpeedCam MacroVis) with frequency 4000 fps. Sequences of the recorded frames were analysed frame by frame using image analysis software ImageJ (Abramoff et al., 2004) to determine bubble size, deformation ratio, spatial displacements and local velocities. The bubble local velocity was determined as:
u¼
s Dt
ð1Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where s ¼ ðxiþ1 xi Þ2 þ ðyiþ1 yi Þ2 , (xi, yi) and (xi+1, yi+1) are the coordinates of the subsequent positions of the bubble with respect to its bottom pole, and Dt is the time interval between subsequent bubble positions. The bubble size and deformation ratio were determined through an average of 20–40 (n) independent measurements. The bubble size was determined by means of its equivalent radius (Rb) as:
" 2 #1=3 n 1X ðdv Þi ðdh Þi Rb ¼ n i¼1 2 2
ð2Þ
where dh and dv are the horizontal and vertical axes, respectively, of the rising bubbles determined. The bubble deformation ratio (v) was calculated as:
v¼
n 1X dh n i¼1 dv i
ð3Þ
The TeflonÒ solid plate, used in our studies as a model hydrophobic surface, was positioned horizontally just beneath the water/air interface at a distance far larger than that necessary for establishment of the bubble’s terminal velocity. The terminal velocity of the bubble at the moment of collision, i.e., its impact velocity, was ut = 34.7 ± 0.2 cm/s. The value of ut determined experimentally remains in perfect agreement with corresponding values reported by other researchers for an air bubble of similar Rb rising in pure water (Duineveld, 1995; Wu and Gharib, 2002; Legendre et al., 2012). The roughness of the solid plate was modified manually by polishing the surface with sandpaper of different grid numbers—namely, 100 (Klingspor KL 375 J), 600 (S.G. Abrasives 600) and 2500 (S.G. Abrasives 2500 wet paper). For convenience, these three plates will be further referenced in the text as T100, T600 and T2500, respectively. The lateral roughness of the plates was determined on the basis of light microscopy observations as a range of lengths of surface scratches. For the T100 and T600, the roughness was equal to 80–100 lm and 40–60 lm, respectively, whereas for T2500, it was equal to 1–5 lm. The corresponding contact angles of the TeflonÒ plates, measured for Milli-QÒ water using the sessile drop technique, were 120 ± 5°, 110 ± 4° and 100 ± 3°, for T100, T600 and T2500, respectively
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(Kosior et al., 2013). To visualize the differences in roughness of the prepared TeflonÒ plates, so-called ‘‘focus-stacking microphotography” was performed. In this approach, a stack of micrographs is taken at different focal positions (aligned along the optical axis) and merged into a single, entirely focused composite image (Aguet et al., 2008). To increase the depth of field of the final image of the solid surface, a series of micrographs (20–30, 500 500 pixels) of TeflonÒ plates were taken, with the focal point set at evenly spaced intervals into the depth of field. The images in the series were then transferred to a PC and stitched together using the ImageJ Extended Depth of Field algorithm (Aguet et al., 2008). The 3D image of the solid surface was obtained on the basis of differences in the final image greyscale, which was automatically recalculated into the topography variations using the Interactive 3D Surface Plot plugin. The 3D pictures of the TeflonÒ plates obtained in this way are presented in Fig. 1. It should be noted that in the z-direction, the approach described here allows for qualitative data only, without precise information about the depth of grooves and crevices of the surface. It can be seen in Fig. 1, however, that the depth of the scratches for each TeflonÒ sample is comparable with the lateral length of irregularities (distances between peaks). All three pictures were taken under identical conditions, so Fig. 1 illustrates significant differences in the plate’s roughness. Prior to the experiments, solid plates were carefully washed with a chromic acid mixture and rinsed with large amounts of Milli-QÒ water. The glass elements of the setup (capillary, square column) were washed either with the chromic acid mixture or MucasolÒ, a commercially available cleaning liquid, and rinsed with large amounts of Milli-QÒ water. The experiments were carried out at room temperature (21 ± 1 °C). 2.2. Numerical approach Numerical simulations were performed to reproduce the bubble behaviour during its collision with the hydrophobic solid surface. Details about the selected numerical approach can be found elsewhere (Popinet, 2003, 2009; Fuster et al., 2009; Afkhami et al., 2009; Zawala and Dabros, 2013). Fig. 2A presents the scheme of the computational domain. The liquid column of height H = 100 mm and radius L = 5 mm containing a gas bubble was described by a two-dimensional, cylindrical coordinate system. The bubble radius was set to be identical to that determined during experiments (0.74 mm), and S was equal to 94.42 mm. The Dirichlet boundary conditions were applied at the column wall. For a liquid column of L = 5 mm, chosen here as a compromise between the accuracy of the results and computational time, one should be aware of the existence of a so-called wall effect. To minimize the influence of the column wall proximity on bubble motion, the boundary conditions at the side cylinder walls were assumed to
be slip. This was done to study possible ideal situations, which could be helpful in interpreting experimental observations and trends. The terminal velocity of the bubble in the numerical case was determined to be 33 cm/s. The top column surface was reconstructed using Boolean operations, and its thickness was 80 lm. At the top wall, the hydrodynamic boundary conditions were controlled as follows: (i) the degree of slip/no-slip properties were varied uniformly by gradually changing the value of fluid velocity (u) at the solid surface from 0 to 1, where 0 was related to the completely no-slip condition while 1 was assigned to the completely slip surface, or (ii) slip (u = 1) and no-slip (u = 0) areas (squares) were reconstructed alternately at the surface from small blocks with different degrees of slip, as schematically presented in Fig. 1B. The length of the slip spots (block walls) varies between 50 and 150 lm. Three different scenarios were considered: (i) the slip areas were uniformly, periodically spread at the surface (see Fig. 2B1, referenced further as scenario B1); (ii) there was only one slip spot, directly at the symmetry axis of the system (axis of symmetry of the bubble trajectory, Fig. 2B2, scenario B2); and (iii) the slip spots covered only half of the surface (Fig. 2B3, scenario B3). Depending on the scenario, the simulation was performed using either an axi-symmetrical (scenario B1 and B2) or Cartesian solver (scenario B3); the implications of this fact will be discussed further. At t = 0, the bubble was motionless (initial velocity u = 0) and spherical. After acceleration, the constant speed (terminal velocity) of the bubble was established for t > 50 ms. The contact angle of the solid surface ranged between 90° and 100°, comparable with the experimental cases. Densities of the liquid and gas were set to be 1000 and 1.3 kg/m3, respectively—i.e., on the order of water and air. Similarly, the dynamic viscosities of the liquid and gas were taken as 1 103 and 18 106 Pa s, respectively. The surface tension was assumed to be 0.0724 N/m. 3. Results and discussion 3.1. Influence of roughness on attachment time (experiment) Sequences of photos illustrating the bubble collision with T2500 and T100 surfaces are presented in Fig. 3. The time t = 0 was adjusted to the moment of the bubble’s first collision (maximum deformation during impact). In both cases, the bubble approaches the solid/water interface with a constant shape, a consequence of constant (terminal) velocity of the bubble. However, the course of bubble motion after this collision is drastically different and depends on the roughness of the solid surface. For the T2500 (smooth) surface, the bubble bounces several times before the TPC formation, whereas at the T100 (rough) surface, the TPC forms immediately during the first bubble strike. At T2500, up to 5–6 ‘‘approach-bounce” cycles, with diminishing amplitude, were
Fig. 1. Three-dimensional pictures of the (A) T100, (B) T600 and (C) T2500 surfaces, obtained using the focus-stacking microphotography approach. Units are pixels. In the xydirection, 10 pixels equals 1.2 lm.
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Fig. 2. Scheme of (A) geometry of the computational domain and (B) approach used for modification of solid surface hydrodynamic boundary conditions (slip vs. no-slip spots).
Fig. 3. Sequences of photos illustrating the bubble collision, bouncing and moment of the TPC formation for the T2500 and T100 surfaces (Dt = 1 ms).
always observed before bubble attachment. The damping of the amplitude of the bounces was caused by dissipation (viscous losses) of the kinetic energy associated with bubble motion. Note that at T2500 (smooth) surface, the TPC forms after the dissipation of virtually all kinetic energy, when the bubble is in permanent contact (practically immobile) with the solid/water interface. In contrast, for the T100 surface, the TPC was formed during the first collision, when the kinetic energy associated with the bubble motion was maximal. A quantitative description of the photo sequences from Fig. 3, with respect to variations of the local velocities of the bubble as a function of time during collision and bouncing, are presented in Fig. 4. The bubble approaches the solid surface with constant terminal velocity, which for the studied bubble size is 34.7 ± 0.2 cm/s. For T2500, after brief bubble-solid surface contact,
the rebound is observed in a manner similar to earlier publications (Tsao and Koch, 1997; Zawala et al., 2007), and the bubble’s maximum velocity was approximately 30 cm/s. Because of buoyancy and viscous losses, the bubble undergoes several approachbounce cycles with diminishing amplitude, degree of shape deformations and impact velocity of consecutive collisions (Zawala et al., 2007; Kosior et al., 2013) and eventually, after approximately 90 ms, remains in permanent contact with the solid/liquid interface. Next, the TPC is formed after approximately 130 ms, which is associated with a rapid change in the bubble’s position and, consequently, the high peak of the velocity profile. The bubble rupture and TPC formation imply that the draining liquid (water) film, separating the bubble and solid surfaces, has reached its critical thickness of rupture. For the T2500 surface, the experimentally measured values of tTPC (time span from t = 0 to the moment of
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Fig. 4. Experimentally determined variations of the local bubble velocities during collision and bouncing with T2500 surface (circles) and T100 surface (squares). For both cases, Dt = 0.25 ms.
the TPC hole formation) were very reproducible, with an average tTPC, calculated from 20 independent runs, equal to 115 ± 17 ms. The TPC was always formed after up to 5–6 approach-bounce cycles, consistent with previous studies, in which similar good reproducibility of the bubble’s attachment time to a smooth hydrophobic surface was reported (Krasowska et al., 2009; Kosior et al., 2013, 2014). Very good reproducibility was also obtained for similar experiments with the T100 surface (Fig. 1A). Here, however, tTPC was shortened drastically to only 1 ± 0.5 ms (see also Fig. 3, bottom sequence). As shown in Figs 3 and 4, immediate TPC formation at the T100 surface means that no bouncing was observed, and fast attachment of the bubble to the solid surface occurred during the first strike. This occurrence was observed in all of the experiments performed. The bubble interacting with the T600 surface produced more varied results. Here, depending on the experimental run, the TPC formed either immediately during the first strike or during the second or third collision, after a short period of bouncing. This effect is illustrated in Fig. 5, in which velocity variations of the bubble for
Fig. 5. Experimentally determined variations in the local bubble velocities during collisions with the T600 surface for four different scenarios of attachment. The moment of TPC hole formation for each scenario is marked with an asterisk (Dt = 0.25 ms).
observed attachment scenarios are presented. Here, for clarity purposes, the case of the formation of the TPC during the 3rd collision is omitted. The moment of the TPC formation is marked in Fig. 5 with asterisks. Despite the fact that liquid film ruptures during the 1st, 2nd or 3rd collision (3rd not shown in Fig. 5), four different profiles of bubble velocity variations were observed. This is a consequence of the significant discrepancy in kinetics of the TPC formation and TPC line spreading after rupture of the separating liquid film and dewetted hole formation, i.e., different attachment scenarios. This caused poor reproducibility of the determined tTPC values for the T600 surface, which is significantly lower than those of the T2500 and T100 plates. The average value of tTPC calculated from 20 independent runs was 15 ± 21 ms, with a relatively large standard deviation. Depending on the attachment scenario, the TPC could be formed either after few or approximately 40 ms, despite the fact that each experimental run was performed with the same T600 solid plate. As mentioned above, the poor reproducibility of the TPC formation in the case of T600 was a consequence of 5 different attachment scenarios. Below, sequences of photos are presented showing various bubble shape deformations at the moment of TPC formation, recorded experimentally. Fig. 6 presents the photos of the bubble at the moment of collision (t = 0) and attachment to the T600 surface, where the TPC was formed during the first collision (open circles in Fig. 5). After brief bubble-solid surface contact, the TPC hole is formed at approximately t = 1.5 ms. The TPC line then spreads, which is associated with a spectacular, ‘‘bullet-like” shape of the bubble. As can be noted from the three last pictures in the first row of Fig. 6, the hole of the TPC is formed asymmetrically, i.e., closer to the bubble edge (rim of the liquid film). Moreover, rupture of the liquid film occurs when the bubble tends to withdraw from the solid surface, i.e., wants to rebound. This bubble shape is a consequence of balance between the TPC line spreading kinetics, determining the force counteracting detachment and magnitude of the detachment force, pushing the bubble away from the solid/liquid interface (bounce). From the energy balance, the film ruptures when the kinetic energy associated with the bubble motion is almost completely transferred into the surface energy related to bubble area increase (Chesters and Hofman, 1982; Tsao and Koch, 1997; Zawala and Dabros, 2013). However, as a consequence of TPC hole formation, the capillary force acting at the TPC perimeter is strong enough to prevent bubble detachment, and the bubble withdrawal attempt is drastically impeded. The equilibrium bubble shape is established after approximately 10 ms. A similar sequence of photos of the bubble for a different experimental run at the T600 surface is presented in Fig. 7. Here, despite the fact that all experimental conditions were identical and the TPC was formed during the first contact (open triangles in Fig. 5), the manner of the bubble attachment is completely different. As seen, after collision (t = 0), the TPC line spreads very fast, and the equilibrium bubble shape is formed after approximately 2–3 ms. In contrast to the behaviour shown in Fig. 6, here, for the ‘‘spreading” scenario, there are no violent bubble shape variations. Moreover, the liquid film ruptures rather uniformly. These observations suggest that the kinetic energy associated with the bubble collision and liquid film rupture is dissipated much faster than the case described above (Fig. 6). Fig. 8 presents another photo sequence registered during experiments with the T600 surface. It was observed in some experimental runs that, during the collision, the liquid film ruptured only in one location on the solid surface (open squares in Fig. 5). This is evidenced by a clear necking formation during the backward motion of the bubble (see second row of the sequence of Fig. 8). The dimension of the spreading TPC line was, however, too small, and capillary forces were consequently not large enough to prevent
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Fig. 6. Collision and attachment of the bubble with the T600 surface. The attachment took place during the first impact with a characteristic bullet-like bubble shape at the moment of TPC formation (Dt = 0.25 ms) – ‘‘bullet-like” scenario.
Fig. 7. Collision and attachment of the bubble with the T600 surface. The attachment took place during the first impact with rapid bubble spreading at the solid/liquid interface – ‘‘spreading” scenario.
Fig. 8. Collision of the bubble with the T600 surface. The TPC hole formation, necking evolution and breakage are clearly visible. The attachment took place during the second collision, directly at the spot where the satellite bubble was left.
the bubble’s withdrawal. Moreover, after bubble withdrawal, a small satellite bubble remained at the solid surface. In the necking scenario, the TPC formation and stable bubble attachment always took place during the second collision, when the macro-bubble hit the place occupied by the satellite bubble, as described elsewhere (Krasowska et al., 2009). In fact, the TPC was formed as the consequence of coalescence of two water/air interfaces – bubble surface and surface of micro-bubble (an already dewetted spot). The qualitative data for the frequency (probability) of each scenario observed at the T600 surface are presented in Fig. 9. The ‘‘bullet-like” and ‘‘spreading” scenarios were the most frequent, whereas attachment during the 2nd collision was lower. It should be noted that the attachment scenarios observed were independent of the number of experimental runs.
Fig. 9. Statistics of the attachment scenarios observed during the bubble collision with the T600 surface (determined on the basis of 20 independent experimental runs).
3.2. Kinetics of spreading of the TPC line and attachment force Fig. 10A presents the kinetics of the TPC line spreading (dewetting) for T100, two scenarios of T600 and T2500. Experimentally measured variations in length (diameter) of the TPC line (dTPC) as
a function of time are presented there. Here, t = 0 was adjusted for the moment when the TPC hole started to form. Note that for T100 and all presented scenarios for T600, the TPC formation took place during the first collision, whereas for T2500, it occurred after
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Fig. 10. Variations of mean length of the TPC line determined for the solid surface of different roughness (A) as a function of time and (B) for t = 0 (very first moment of the TPC hole appearance).
five collisions. The spreading of the TPC line is fastest for the T100 surface; only 1.25 ms are required to establish an equilibrium TPC diameter. Moreover, the initial value of dTPC is relatively large and equals 1.62 ± 0.11 mm (see Fig 10B). The ‘‘spreading scenario” at the T600 surface is comparable with the T100 case with respect to initial dTPC values. Here, however, the TPC line spreading is a bit slower; the equilibrium dTPC is established after approximately 5 ms. For the ‘‘bullet-like” scenario, the initial value of dTPC is much lower and equals 1.27 ± 0.23 mm (Fig. 10B). Establishment of the equilibrium dTPC is also longer here (6 ms). For the T2500 surface, where the liquid film ruptured and the TPC formation occurred after dissipation of most of the kinetic energy, the bubble was captured beneath the solid surface and remained spherical, causing the initial dTPC to be the shortest (0.57 ± 0.01 mm). We can now estimate the magnitude of forces responsible for the bubble withdrawal from and attachment to the solid surface. In a simplified approach, the attachment force can be approximated by the capillary force (Fc) acting at the perimeter of the TPC hole:
F c ¼ 2pr TPC r
ð4Þ
where rTPC = 1/2 dTPC, and r is the surface tension of water. The force associated with the backward motion of the bubble (detachment force – Fd) is related to the momentum (Dp) change during the collision period (Dt) and can be written as (Zawala et al., 2013):
Dp Dt
ð5Þ
Dp ¼ m Du
ð6Þ
Fd ¼
where m is the mass of the bubble, which can be expressed as
m ¼ C m V b ql
ð7Þ
where Cm is the bubble added mass coefficient, Vb is the bubble volume, and ql is the water density; we can write that
Fd ¼
C m V b ql Du Dt
ð8Þ
The added mass coefficient of the bubble can be estimated from a simple linear expression (Klaseboer et al., 2001):
C m ¼ 0:62v 0:12
ð9Þ
where v is the bubble shape deformation ratio. Fig. 11 presents the ratio of calculated Fc and Fd values for the first bubble collision with the T100 and T600 surfaces. The tTPC values were determined exper-
Fig. 11. The ratio between attachment and detachment forces for different scenarios registered during the bubble collision with hydrophobic solid plates of different roughness.
imentally for the moment when the bubble was about to start its backward motion (approximately 1 ms after the moment of collision). The collision time (Dt) was determined to be equal to 2 ms. The Du value was taken as the bubble impact velocity (34.7 cm/s). The dashed line in Fig. 11 indicates the situation in which Fc and Fd are equal. For Fc > Fd (above the dashed line), the bubble should be attached to the solid surface; otherwise (when Fc < Fd), the bubble rebound should occur. As seen for the T100 and T600 surfaces (‘‘bullet-like” and ‘‘spreading” scenarios), Fc > Fd. Indeed, the attachment of the bubble occurred during the first collision (see Figs. 6 and 7). For the ‘‘necking” scenario, Fc Fd, and the bubble withdrew from the surface despite liquid film rupture and TPC formation (Fig. 8). For the remaining three cases, it can be seen in Fig. 11 that the Fc/Fd value was the smallest for the ‘‘bullet-like” scenario (Fc and Fd values are comparable) and increased gradually for ‘‘spreading” and T100 cases. In our opinion, the relation between the Fc and Fd values can explain different bubble shapes at the moment of the TPC hole formation. Smaller Fc (‘‘bullet-like” scenario) values cause elongation of the bubble shape during attachment and TPC line spreading because Fd is not completely dominated by the attachment force.
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Fig. 12. The moment of attachment of the bubble to the (A) T2500 surface and (B) T100 surface. The second sequence of each comparison shows corresponding numerically calculated bubble shapes for the smooth surface of respective boundary conditions.
Fig. 13. Qualitative comparison between experimentally obtained bubble images for the ‘‘spreading” scenario of attachment and corresponding numerically determined bubble shapes for the B1 surface (see Fig. 1).
3.3. Comparison between experimental and numerical results In light of the literature data (Fan et al., 2010; Calgaroto et al., 2014; Zhou et al., 1996; Krasowska et al., 2007, 2009; Kosior et al., 2013, 2014), the results presented above indicate that the TPC formation at rough solid surfaces can be air-induced. Air can be entrapped in surface grooves and crevices and also can be redistributed in the form of nano- and micro-bubbles at a solid surface immersed in aqueous phase (Krasowska et al., 2007). Moreover, air pockets can be randomly distributed at the solid/liquid interface, and the size and degree of distribution increase with increasing surface roughness (Snoswell et al., 2003; Zhou et al., 2009; Hampton and Nguyen, 2010). The results presented above support the hypothesis of an air-induced mechanism for liquid film rupture and TPC formation because The mean values of tTPC decrease with solid surface roughness. tTPC is very reproducible for T2500 and T100 surfaces (smoothest and roughest), whereas significant deviations are observed for T600, where roughness is intermediate. At the T2500 and T100 surfaces, only one attachment scenario occurs (see Fig. 3), whereas at the T600 surface, up to 5 different scenarios were registered. This confirms a more random distribution of air pockets at the T600 surface; the manner of the TPC line spreading is stochastic. The ‘‘necking” scenario shows that immediate TPC formation and fast spreading of the TPC line can be observed if the macrobubble collides with previously dewetted spots at the solid surface (fast coalescence between air/water interface). dTPC at the liquid film rupture time is larger for higher roughness. Moreover, for T600, dTPC is significantly different for different attachment scenarios (different amounts of air). This is
certainly caused by variations in the contact angle, a consequence of air entrapped in the solid surface irregularities. This air can also be redistributed in the form of submicroscopic (nano-/micro-) bubbles at the solid/water interface. The time of spreading of the TPC line to the equilibrium dTPC value is the fastest for the surface where more air is expected to be present. As stated above, the presence of air at the solid surface could modify its hydrodynamic boundary conditions. More air should induce larger hydrodynamic slip of the solid surface. In contrast, the surface with less air entrapped should behave more as a noslip condition. We decided to use this hint to model the presence of air at a hydrophobic solid surface. By proper patterning of the numerically reconstructed hydrophobic solid surface (slip/no-slip areas; see Fig. 2), we attempted to reproduce the bubble behaviour and attachment scenarios during collision with T100, T600 and T2500. Fig. 12 presents the photo sequences of the bubble collision with the T2500 (Fig. 12A) and T100 (Fig. 12B) surfaces (see also Fig. 3). The sequences of experimentally obtained photos are compared with numerically calculated outlines of the corresponding bubble shapes. The bubble behaviour at T2500 was reproduced in numerical simulations by setting the solid surface as completely no-slip. For T100, the surface was set as completely slip. Note the very good agreement between experimentally observed and numerically reproduced bubble shapes during the TPC line spreading. Fig. 13 presents the photos of the bubble colliding with the T600 surface for the ‘‘spreading” scenario (see also Fig. 7). Again, experimentally obtained photos are compared with numerically calculated bubble outlines. The numerically reconstructed solid surface was obtained for the case presented in Fig. 10 according
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Fig. 14. Qualitative comparison between experimentally obtained bubble images for ‘‘bullet-like” scenario of attachment and corresponding numerically determined bubble shapes for the B3 surface (see Fig. 1).
Fig. 15. Qualitative comparison between experimentally obtained bubble images for ‘‘necking” scenario of attachment and corresponding numerically determined bubble shapes for the B2 surface (see Fig. 1).
Fig. 16. Experimentally determined variations in the horizontal axis of the bubble during its approach to T100 (open circles) and T2500 (full squares) solid surfaces. The solid and dashed lines are simple exponential fits to the experimental points. Inset—corresponding results of numerical simulations (full slip surface—full squares, 99% no-slip surface—open circles).
to the B1 approach (see Section 2.2); i.e., the surface was created periodically with slip and no-slip blocks of a given longitudinal length (see Fig. 2B1). The agreement between experimentally observed and numerically reproduced bubble shapes during the TPC line spreading was again very good. Each characteristic and spectacular bubble shape was well reproduced in simulations. A similar comparison is presented in Fig. 14. Here, the experimentally observed ‘‘bullet-like” scenario is compared with results obtained according to numerical approach B3 (see Section 2.2), in which the Cartesian solver was applied in the calculations. Here,
the spectacular bubble shapes were also nicely reproduced, including the ‘‘bullet-like” shape. Similarly, the ‘‘necking” scenario was also accurately reproduced in numerical simulations, as shown in Fig. 15. Here, the numerical calculations were performed for numerical approach B2, with only one slip spot at the no-slip solid/liquid interface, directly at the trajectory of the geometrical centre of the bubble. As seen in the last picture of the bottom sequence (numerical data), even the formation of the small satellite bubble was reproduced in the simulations. This micro-bubble was formed at the slip area of the solid surface. Three different approaches with respect to the distribution of the slip areas at the hydrophobic solid surface were used to reproduce the experimentally observed bubble attachment scenarios at the T600 surface. For the T2500 and T100 surfaces, only one approach was sufficient. This is additional proof of a more random and stochastic distribution of air pockets at the T600 surface during its immersion into water phase, caused by intermediate surface roughness. Additional data supporting the above discussion and conclusion are presented in Fig. 16. The experimentally determined variations of the bubble’s horizontal axis (dh) are presented here for the approaching period (before the first strike). The variations in dh are normalized by the dh1 far from the wall, which is constant after the establishment of the bubble’s terminal velocity. The normalized dh variations are presented in Fig. 16 as a function of the normalized distance of the bubble’s geometrical centre to the solid surface (xc/Rb). The points are data determined on the basis of image analysis, whereas the lines are the fitted exponential regressions. The image analysis revealed that despite the experimental scatter, it is possible to notice small differences in the manner of the dh variations with the separation distance. This dependence is much stronger for the T2500 (smoothest) surface than T100 (the roughest one). In our opinion, this subtle discrep-
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ancy between the method of the approaching bubble deformation before collision with T2500 and T100 is an indication of the different hydrodynamic boundary conditions at the solid surface, determined by different amounts of entrapped air. Because of a much higher amount of air at the rough T100 surface (gas trapping mechanism – Snoswell et al., 2003; Zhou et al., 2009; Hampton and Nguyen, 2010), boundary conditions were shifted towards more slip. In contrast, the amount of interfacial air at the smooth T2500 surface was much lower, resulting in a more no-slip surface. This modifies the degree of viscous dissipation of energy in the solid surface’s vicinity and influences the bubble deformation. It should be emphasized that even for a hydrophobic solid surface of 90% no-slip (10% slip) in numerical simulations, the approaching bubble deformation was much stronger than for the completely slip solid/liquid interface. The bubble deformation pattern, therefore, can be used as a marker of different boundary conditions (amount of air) at the hydrophobic solid surface. The results of numerical calculations (presented in the inset of Fig. 16) support our hypothesis. As seen, the manner of numerically determined variations of dh of the bubble approaching numerically slip and no-slip solid/liquid interface is consistent with those determined experimentally. 4. Conclusions The bubble attachment time and kinetics of the TPC formation strongly depend on the roughness of the hydrophobic solid surface. It was observed that the time of the TPC formation can be drastically reduced when the solid surface roughness increases. Moreover, this effect is associated with different attachment scenarios and spectacular rupturing bubble shapes revealed using ultrafast video recordings. The bubble shape variations during rupture at the solid surface also strongly depend on the solid surface roughness. In our opinion, different attachment kinetics and scenarios are a consequence of different amounts of air present at topographically nonhomogeneous solid/liquid interfaces. As a consequence, the hydrodynamic boundary conditions are air-amount dependent and shift towards more slip or no-slip for T100 and T2500, respectively. This hypotheses is supported by four main findings: (i) lack of scatter of the bubble attachment time to T100 and T2500 surfaces, i.e., for two extreme cases, with respect to the degree of roughness (and entrapped air amount); (ii) strong scatter at the T600 (intermediately rough) surface; (iii) good qualitative agreement between experimentally observed and numerically determined shapes of the bubbles during their approach and rupture (TPC formation) and (iv) discrepancy between the approaching bubble deformation before collision with the T2500 (more no-slip) and T100 (more slip) surfaces. Acknowledgements The financial support from the Polish Ministry of Science and Higher Education (Iuventus Plus Project No. IP2014 053973) is acknowledged with gratitude. Partial financial support from the National Research Center (NCN Grant No. 2013/09/D/ST4/03785) is gratefully acknowledged. The study is related to the activity of the European network action COST MP1106: ‘‘Smart and green interfaces—from single bubbles and drops to industrial, environmental and biomedical applications”. The authors express their gratitude to Professor Kazimierz Malysa (Jerzy Haber ICSC PAS) for valuable comments and helpful discussion. References Abramoff, M.D., Magalhaes, P.J., Ram, S.J., 2004. Image processing with Image. J. Biophoton. Int. 11 (7), 36–42.
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