MSc Physics and Astronomy
Advanced Matter and Energy Physics
Master Thesis
The Effect of the Dynamic Surface Tension on Droplet
Impact and Deposition
by
Hanne Hoffman
12362476(UvA)
2651829(VU)
60 ECTS
March 2020 - January 2021
Supervisors:
prof. dr. D. Bonn
R. Sijs MSc
Second examiner:
prof. dr. Noushine Shahidzadeh
Abstract
Droplets impacting on a surface are key to a wide range of applications such as spray deposition in pesticide spraying and inkjet printing. Yet, a full understanding of how they spread and retract is still lacking. Surfactants are often added to improve spreading and coverage of aqueous solutions, resulting in variations of the surface tension at time scales beyond the reach of conventional dynamic surface tension measurement methods. Here we study the impact dynamics of aqueous surfactant solutions on hydrophobic surfaces at millisecond time scales. We find that the spreading and retraction of droplets cannot be adequately described by the equilibrium surface tension. Instead, it is the dynamic surface tension that is the important parameter to describe this. We infer the dynamic surface tension in the first milliseconds after impact from the spreading dynamics of droplets, a timescale that conventional methods cannot reach. ’Slow’ surfactants that take a lot of time to reach newly created droplet surface, only slightly decrease or even increase the surface tension, while ’fast’ surfactants, on the other hand, allow efficient wetting of aqueous solutions on hydrophobic surfaces. Our findings allow for the first time to tailor surfactants for efficient drop deposition or spray application.
Acknowledgements
I would like to thank my supervisors, Rick Sijs en Daniel Bonn, for their help and guidance during my thesis and for giving me the opportunity to write an article about my research, from which I have learned a lot. I would also like to thank Thijs de Goede for his help with everything camera-related and Paul Kolpakov for his tutorials and help in the lab.
Furthermore, I would like to thank Catrien, for all the online and offline coffee breaks we have had over the past year and George, for putting up with me the many days working from home in the same room together.
Contents
1 Introduction 4
2 Theory 6
2.1 Influence of surfactants on the surface tension . . . 6
2.2 Dynamic surface tension in droplet impact . . . 7
3 Materials and methods 9 3.1 Surfactant solutions . . . 9
3.2 Droplet impact measurements . . . 9
3.3 How to obtain the dynamic surface tension . . . 11
3.4 Calibration with water . . . 11
4 Results and discussion 13 4.1 Bubble pressure measurement . . . 13
4.2 Droplet spreading and retraction . . . 13
4.3 Results for the dynamic surface tension . . . 14
4.4 Retraction rate . . . 15
5 Conclusions 18 6 Further Research 19 6.1 First Results for Follow-Up Research . . . 19
6.2 Outlook . . . 22
Appendices 25
A Silanization procedure of glass slides 25
B Bubble pressure tensiometer fits 25
C Effective dynamic surface tension for other velocities 26
1
Introduction
The dynamics of droplets of complex fluids impacting on surfaces is of paramount importance for many appli-cations. For example, in pesticide spraying, it is not unusual that over 50% of applied pesticide spray misses its target due to effects such as bounce-off and drift, ending up as environmental pollution[1]. For inkjet printing, the droplets need to stick and spread to obtain the best possible print[2]. Other processes where droplet impact is a crucial element are de-icing[3], food conservation[4], spray painting [5], spray cooling[6], waterproofing textiles[7] and forensic research[8]; even for virus transmission in aerosols, the details of droplet impact are important[9]. Despite these wide-ranging applications, a full understanding of the droplet spread-ing and retraction dynamics of complex fluids is missspread-ing. This is especially true when surfactants are added to the aqueous solutions to improve spreading and coverage of the droplets, complicating the fluid dynamics. Surfactants are surface active agents that are widely employed to influence the interaction between a droplet and a surface; by lowering the surface tension of a liquid, an impacting droplet can spread more easily on a surface and exhibits a decreased retraction rate. This increases the chances of the droplet ’sticking’ to a hydrophobic surface instead of bouncing off. Inspired by the challenge of minimizing soil pollution due to pesticide spraying, we provide a new approach to studying the impact dynamics of aqueous surfactant solutions on hydrophobic surfaces, using insights into the impact process at millisecond timescales to draw conclusions about macroscopic parameters of importance to spraying applications.
t = 0 ms t = 2 ms t = 5 ms t = 10 ms (a) Water Slow surfactant Fast surfactant 0.05 0.10 0.50 1 5 10 2 4 6 8 concentration (CMC) Improvement (coverage surfactant /coverage water )
Coverage improvement after spraying broccoli plants
(b)
Figure 2: Droplet rebound and coverage on plant leaves. a Snapshots of a water droplet bouncing off a hydrophobic cabbage leaf. b Relative improvement of the liquid coverage of broccoli leaves as a function of surfactant concentration. Coverage is determined from an added fluorescent, which lights up, thus indicating where the leaves are covered. The deposition improves with higher concentrations of surfactant. Even though the equilibrium surface tension of the ’slow’ surfactant (22 mN/m, red squares) is lower, the ’fast’ surfactant (32 mN/m, green diamonds) gives a better coverage.
Fig. 2a shows how a water droplet bounces off a hydrophobic surface, here a cabbage leave, which is a generic observation in pesticide spraying. On the one hand, plant surfaces are notoriously hydrophobic; on the other hand, there has been a trend towards spraying with larger droplets to avoid spray drift[10]. Larger droplet sizes lead to an increased chance of bounce-off, resulting in a relatively low coverage of the leaves with active ingredients and an increase of leakage into the soil[11]. The improvement in coverage after adding different types of surfactants to a water spray, in an experiment resembling pesticide spraying, is shown in Fig. 2b. This experiment is explained in more detail in chapter 6, where the experiment was repeated using a model surface instead of plant leaves. The results of Fig. 2b show that there is a very significant difference in coverage after spraying between surfactants: surprisingly, the surfactant with the lower equilibrium surface tension results in a poorer coverage of a hydrophobic surface after spraying. It thus follows that the equilibrium surface tension is insufficient to predict the droplet behavior.
The goal of this thesis is to determine what type of surfactant is best for increasing plant coverage. To do this we measure the dynamic surface tension of surfactant solutions directly from droplet impact. This is a relevant parameter during droplet impact that determines the dynamics of the impact and whether or not a droplet sticks or bounces of the surface. This parameter will be linked to the retraction rate of the droplets and eventually to the coverage of a parafilm surface after spraying. The main part (ch. 1 - 5) of this thesis is based on a paper written about this research on obtaining the dynamic surface tension from droplet impact. At the time of writing, this paper has been submitted for the second time to Physical Review Fluids. This is followed by first results of the link between coverage after spraying and the dynamic surface tension of surfactant solutions.
This thesis is structured as follows: In chapter 2 an introduction to surfactants and the dynamic surface tension is given, accompanied by an overview of relevant literature. Chapter 3 follows with a description of the used materials and methods. After this the results and discussion can be found in chapter 4 and a conclusion is drawn in chapter 5. In chapter 6 first results of linking the coverage of parafilm to the dynamic surface tension are given, followed by an outlook on future research.
2
Theory
2.1
Influence of surfactants on the surface tension
Within a body of fluid, there are attractive forces between molecules. A molecule inside the fluid will be equally attracted towards all sides. On the surface of the fluid, for example at the fluid-air interface, a molecule will only be attracted in the direction of the rest of the fluid. This is illustrated in Fig. 3. The net effect of these forces is that the surface is pulled inwards. This inwards-puling force is called the surface tension σ and has the unit N/m. Due to this force, droplets tend to be spherical; any other shape has more surface area and therefor costs more energy[12].
Figure 3: Illustration of the intermolecular forces in a fluid.
Surfactants, or surface active agents, are so-called ’amphiphilic’ molecules: molecules with one hydropho-bic (repelled by water) and one hydrophilic (attracted to water) side. When dissolved in water, it is ener-getically favorable for these molecules to be on the surface, where the hydrophobic side is pointed outward, as far away as possible from water molecules. When the concentration of surfactant is sufficiently increased, the entire surface will be covered with surfactant molecules and when more surfactants are added micelles will form: structures of surfactant molecules with their hydrophobic parts directed inwards, for example spherical or cylindrical. The concentration at which the surface is completely filled with surfactant is called the critical micelle concentration (CMC)[13]. An illustration of surfactant molecules in water is given in Fig. 4.
Figure 4: Illustration of surfactant molecules at different concentrations in a fluid.
By adsorbing at the air-water interface, surfactants lower the surface tension of water from 72 mN/m to a minimum that is generally between 40 and 15 mN/m[14]. When the solution is in rest and all surfactant molecules have either adsorbed to the surface or formed micelles, this is called the equilibrium surface tension[15]. Surfactants only lower the surface tension due to their presence at the surface. Because of this, the minimal surface tension is reached at the critical micelle concentration, it’s value depending on the type of surfactant. Adding more surfactant will not lower the surface tension any further. For concentrations below CMC not the entire surface will be filled with surfactants and the resulting surface tension will be somewhere between the surface tension of water (72mN/m) and this minimum. A typical example of an
equilibrium surface tension measurement is shown in Fig. 5. The CMC value is the point where the surface tension no longer decreases with concentration, around 10−3 mol/dm3in this figure.
Figure 5: A typical measurement of the equilibrium surface tension σ as a function of surfactant concentration. This figure was
adapted from [16]. A glucamide surfactant was used for this measurement. The CMC value of this surfactant is around 10−3
mol/dm3.
2.2
Dynamic surface tension in droplet impact
The important property governing the physics of impact, retraction and rebound of surfactant solutions is the dynamic surface tension (DST). The impact and expansion of the impacting drop creates a large amount of fresh surface in a very short time, and this new surface is not necessarily covered with surfactants. The surface tension shortly after creating the new surface is higher than the equilibrium surface tension with surfactants. The surface tension can be close to the surface tension of pure water just after impact, and then decreases in time, with a characteristic time scale of the diffusion of the surfactant molecules from the bulk to the surface. An empirical formula by Hua and Rosen[17] is often used to describe the dynamic surface tension of the droplet after creating new surface. This formula has been shown to fit experimental data from the bubble pressure tensiometer (BPT) well[18, 16]. However, Zhang and Basaran[19] show that there are competing processes caused by the DST during the spreading of droplets with surfactants: On the one hand the sur-factant lowers the surface tension, leading to more spreading when the sursur-factant is faster. On the other hand stresses due to an uneven distribution of the surfactant molecules on the surface decrease spreading by Marangoni effects. Crooks, Cooper-White and Boger[20] show that the recoil behaviour of droplets con-taining low concentrations of surfactant does not correlate with the dynamic surface tension from the BPT, due to the Marangoni stresses causes by a heterogeneous surfactant concentration at the surface.
For simple fluids without surfactant Bartolo, Josserand and Bonn[21] show that the retraction rate and recoil behaviour of an impacting droplet is independent of impact velocity, but does depend on the surface tension of the droplet. When surfactants are present, Aytouna et al.[15] found that the expansion and retrac-tion dynamics of droplets does depend on the dynamic surface tension; they make a distincretrac-tion between fast
and slow surfactants. Their data suggest that the characteristic time τ with which the surfactant molecules diffuse from the bulk towards the surface determines whether a surfactant is ’fast’ (τ ∼ 1 ms) or ’slow’ (τ ∼ 20 ms) compared to the dynamics of the drop itself. This characteristic time decreases with increasing concentration and diffusion coefficient of the surfactant, the latter being a molecular property given roughly by the size of the surfactant molecules.
All this suggests that it is the dynamic surface tension that determines the interaction of the drop with the surface on short timescales. However, the typical time scale for droplet spreading is in the range of 2-3 ms[18], and current techniques for measuring the dynamic surface tension are unable to measure on this short timescale. The problem we address here is how to obtain the dynamic surface tension on such short timescales. We will show below that one can infer the dynamic surface tensions of surfactant solutions in the first milliseconds after impact from the spreading dynamics of droplets themselves. This can subsequently be connected to the retraction rate which is an important indicator of droplet rebound [21].
3
Materials and methods
3.1
Surfactant solutions
In this paper we focus on two surfactant types: trisiloxane and alcohol terpenes, designated as slow and fast surfactant, respectively, as defined by Aytouna et al.[15]. Solutions were prepared in concentrations of 0.1, 0.5, 1, 5, and 10 times the critical micelle concentration (CMC) for both surfactants. For the fast surfactant, additional concentrations of 0.01 and 0.05 times the CMC were prepared. The trisiloxane and alcohol terpenes are used as a basis for this article, but also AOT, CTAB, Glucamide and Polysorbate have been measured to verify validity. The equilibrium surface tensions and CMC values of all surfactants are summarized in Tab. 1. All surfactants have a density ρ of 0.997 kg/L and dynamic viscosity µ of 1 mPa s, equal to the values of water.
Surfactant σeq(mN/m) CMC (g/L) Alcohol Terpenes [15] 32 10 AOT [15][22] 32 1 CTAB [22] 37 0.36 Glucamide 26 1 Polysorbate 20 [22] 34 0.06 Trisiloxane 22 0.1
Table 1: Surfactants and their properties.
3.2
Droplet impact measurements
In order to investigate the influence of the dynamic surface tension on droplet impact, droplets were re-leased from a syringe through a needle with a diameter of 0.5 mm. The resulting droplet diameters varied between 1.8 and 2.2 mm. The impact velocity (v) of the droplets was varied between 1.2 and 2.1 m/s by adjusting the height of the syringe. A schematic drawing of the setup is shown in Fig. 6a and a photo showing the setup in Fig. 6b. The impact of the surfactant droplets has been studied using hydrophobic silanised glass slides (contact angle ∼ 100◦, the silanization procedure can be found in Appendix A) and was
recorded with a high-speed camera (Phantom Miro M310) with a frame rate of 8100 frames per second and a spacial resolution of ∼ 50 pixels per mm. All uncertainties were obtained by calculating the standard devia-tion between at least five repetidevia-tions. Wherever the uncertainty bar is not shown, it is smaller than point size. The diameter of the droplet (D(t)) was determined by measuring the maximum diameter of the droplet in each image of the image sequence. A typical example of such an image sequence is shown in Fig. 7a. Using the maximum diameter is different from other authors, who use the diameter at the contact line [21]. In Fig. 7b we compare the diameters obtained by using both methods for a typical water droplet impact at v = 1.4 m/s. We find that the maximum diameter (blue circles) and contact line diameter (red squares) are identical during spreading (0-3 ms) and retraction (3-10 ms). There is a deviation between the two diameters between 10 and 20 ms after impact, as the droplet almost detaches from the surface due to a high retraction rates (see inset Figure 7b). However, this deviation has no influence on the analysis of droplet spreading, as the maximum spreading diameter and contact line diameter are identical at maximum spreading (t = 3 ms; see inset Figure 7b), which is the diameter used to determine the dynamic surface tension. The two diameters overlap until right before the minimum, so that the retraction rates obtained from both diameter profiles are also identical.
(a) (b)
Figure 6: a A schematic drawing of the setup. b Picture of the setup with 1) an automatic dripping system 2) a syringe, 3) a light source, 4) a diffuser and 5) a silanised glass plate
t=0 t=1 Spreading t=3 t=7 t=9 Retraction t=15 t=18 t=30 Relaxation t=38 (a) Dmax Dcontact 0 5 10 15 20 25 30 35 1 2 3 4 5 6 t(ms) D /D 0
Droplet spreading and retraction
t=15 t=3
(b)
Figure 7: a Image sequence of a water droplet impact showing the spreading (first row), retraction (second row) and relaxation
(third row) of the droplet. Timestamps are in ms. b Normalized droplet diameter D/D0 as determined using the maximum
diameter (red squares) and the contact line diameter (blue circles) as a function of spreading time. The impact velocity of the droplet is 1.4 m/s.
3.3
How to obtain the dynamic surface tension
Our millisecond time-scale measurements of the droplet spreading and subsequent retraction allow us to gain insight into the dynamic surface tension at the relevant time scale. This is done along two lines of analysis. First, we use the maximum spreading diameter Dmax of the drop compared to the original diameter
D0. This parameter follows from a balance between the inertial forces that drive droplet expansion, and the
viscous and capillary forces that resist it. The resulting maximum spreading ratio Dmax/D0 is then given
by [8]: Dmax D0 = Re1/5 √ W eRe−2/5 1.24 +√W eRe−2/5, (1) where W e = ρv2D
0/σef f is the Weber number and Re = ρvD0/µ the Reynolds number, both functions of
density ρ, impact velocity v, initial diameter D0, effective surface tension σef f and dynamic viscosity µ. For
a surfactant solution, the capillary force resisting the expansion will be given by the instantaneous value of the dynamic surface tension. The effects of this time-dependent surface tension are reflected in an effective surface tension σef f. To first order, we approximate the effective surface tension over the spreading process
(t ranging from 0 (moment of impact) to tmax(moment of maximum spreading)) by the surface tension at
tmax, so σef f ≈ σ(t = tmax).
A second method to determine the dynamic surface tension at maximum spreading is by using the timescale of the spreading. The time from the start of the impact to the maximal spreading tmax was
analyzed by Bartolo, Josserand, and Bonn [21], who found the following equation for the inertial timescale ti of a droplet impact in the inertial regime [23]:
ti= C
r ρR3
i
σ , (2)
where C is a proportionality constant, ρ is the density of the fluid, Ri the droplet radius and σ the
sur-face tension of the fluid, here again taken at the moment of maximum spreading. Note that this time of spreading is predicted to be independent of the impact velocity. Although the viscous forces are small in the experiments presented here, they do affect the dynamics and therefore the proportionality constant C in Eq. 2. Therefore, we establish the proportionality constant experimentally using water, as all the surfactant solutions used here have the same viscosity as water. This is shown in Sec. 3.4.
In order to connect our measurements of the dynamic surface tension to the equilibrium value, we use the empirical formula by Hua and Rosen [17]
σ(t) = σ∞+
σ0− σ∞
1 + (τt)n, (3)
where σ∞ is the equilibrium surface tension of the surfactant, σ0 the surface tension of water (72 mN/m),
τ a characteristic time for the surfactant molecules to reach the droplet surface and n a fit parameter that we fix here at a value of 1 [24]. Using a bubble pressure tensiometer (BPT) to measure the surface tension over a time range of 15-16000 ms, we can use the above equation to extrapolate σ to t = tmax(on the order
of a few ms).
3.4
Calibration with water
In order to obtain the dynamic surface tension from the time of spreading (ti), the proportionality constant
in Eq. 2 was determined from the spreading of water droplets, where the surface tension is constant at 72 mN/m. A proportionality constant of 0.67 was obtained. Note that for impact velocities below 1 m/s, the
ti method does not work, as the surface tension that obtained from the ti method for water is found to be
much lower than the expected value. This could be explained by the fact that the droplet inertia does not dominate the viscous effects during droplet retraction at these low impact velocities. Although the droplet spreading ratio Dmax/D0 is dependent on the surface wettability at low impact velocities [25, 26, 27], the
surface wettability does not seem to affect droplet spreading enough to significantly influence the calculation of the surface tension from this method at the measured impact velocities. This is likely due to the fact that the influence of surface wettability is minimal for hydrophobic surfaces [27]. These results show that both the spreading ratio and ti methods could be used to measure the surface tension of the liquid during droplet
4
Results and discussion
4.1
Bubble pressure measurement
Water Slow surfactant Fast surfactant Extrapolation water
Extrapolation slow surfactant Extrapolation fast surfactant
1 10 100 1000 104 0 20 40 60 80 100 t(ms) σ (mN / m ) Extrapolation BPT 10 CMC
Figure 8: Surface tension as determined from bubble pressure tensiometer measurements as a function of surface age (time that has passed since the creation of the surface) for 10 CMC concentrations of the fast (green diamonds) and slow (red squares) surfactants and water (blue circles, constant at 72mN/m). The dashed lines are the best fits of Eq. 3 to the experimental data. These fits are used to extrapolate the dynamic surface tension of the liquids to the typical time of spreading of a droplet, which is 3 ms (grey dotted line). Fit parameters σ0(mN/m), σ∞(mN/m), τ (s), n: σ0 = 72, σ∞ = 19.4, τ = 62.1, n = 7.51 ∗ 10−1
(slow), σ0= 72, σ∞= 18.3, τ = 9.83 ∗ 10−8, n = 5.13 ∗ 10−2(fast).
For comparison, the dynamic surface tension was measured with a Kr¨uss Bubble Pressure Tensiometer BP50. This apparatus allowed us to measure the dynamic surface tension of the solutions for a minimum surface age of 15 ms and measuring up to a surface age of 16000 ms. Fig. 8 shows the measured time dependencies of the dynamic surface tension with the bubble pressure tensiometer for water and the fast and slow surfactants fit tot Eq. 3. The concentration of both surfactant are at 10 CMC. The results for the other concentrations can be found in Appendix B. These measurements show that the surface tension of the fast surfactant (green diamonds) are well below the slow surfactant (red squares), despite the equilibrium surface tension being higher. These results are calibrated to the constant surface tension of water of 72 mN/m (blue circles). As the typical time of spreading of a droplet (≈ 3 ms) is outside the measurement range of the bubble pressure tensiometer, we extrapolate to this timescale by fitting Eq. 3 to the experimental data (dotted lines in Fig. 8).
4.2
Droplet spreading and retraction
Fig. 9 shows how droplets of water, a fast-surfactant (mix of alcohol terpenes) and a slow-surfactant (trisilox-ane) solution typically behave during impact on a hydrophobic (silanised) glass slide. The droplet sizes are similar to what is used in pesticide spraying nowadays, and also similar to the situation of Fig. 2b where we sprayed broccoli leaves. The surface is a model hydrophobic surface with a contact angle of ∼ 100◦. We observe that the droplet containing the ’slow’ surfactant retracts faster than water, especially when the im-pact speed is high. As a result, this droplet is more prone to bounce-off. This is also visible in the snapshots of the droplet impact (Fig. 9a) and in the evolution of the droplet diameter as function of time (Fig. 9b). The retraction rate of the ’fast’ surfactant, on the other hand, is lower than that of water, and the droplet
t=0 t=3 t=13 t=21 t=0 t=3 t=16 t=21 t=0 t=3 t=13 t=21 (a) Slow surfactant Fast surfactant 0.01 0.10 1 10 0.0 0.5 1.0 1.5 Concentration (CMC) ϵ /ϵ water
Retraction Rates compared to water
(c) Water Slow surfactant Fast surfactant 0 5 10 15 20 0.5 1.0 1.5 2.0 2.5 3.0 3.5 t(ms) D /D 0
Droplet spreading and retraction
(b)
Figure 9: Droplet spreading and retraction. a Snapshots of typical droplet spreading for water (in blue), slow surfactant solution (0.1 CMC; in red) and fast surfactant (0.1 CMC; in green) impacting at a speed of 1.9 m/s. Timestamps are in ms. Maximum spreading is reached at 3 ms, equilibrium around 21 ms. Conventional surface tension measurement techniques do not reach these time scales. b Normalized droplet diameter D/D0as a function of time for the same droplets as shown in a. c
Retraction rate ˙ normalized by the retraction rate of water ( ˙water= 50 s−1at an impact speed of 1.9 m/s, as a function of
surfactant concentration.
retracts less: this surfactant does a good job in improving deposition, as indeed observed in Fig. 2b. These findings agree with those of Zhang and Basaran [19], who showed that the retraction of a droplet containing a slow surfactant can be faster than for water and attributed this to Marangoni stresses caused by uneven distribution of the surfactant molecules along the surface.
4.3
Results for the dynamic surface tension
The surface tension at maximum spreading as calculated using both the maximum diameter (Eq. 1) and time of spreading (Eq. 2) are shown in Fig. 10a for an impact speed of 1.6 m/s and different surfactant concentrations. It can be seen that both methodologies yield the same results and that they are in good agreement with the results from the bubble pressure extrapolation for this impact velocity. However, other impact velocities yield effective dynamic surface tensions that are slightly below (low v) or above (high v) the bubble pressure extrapolation (see Fig. 12 and Appendix C) This tells us that the bubble pressure mea-surement does not give us the full information about the surface tension at droplet impact, since it cannot take into account impact velocity, which our new methods of obtaining the DST can do.
Fig. 10 shows that the effective dynamic surface tension of the slow surfactant reaches values even higher than the surface tension of water (72 mN/m). That the effective surface tension values for the surfactant solutions can be higher than that of water is likely due to Marangoni stresses[19], resulting from an uneven distribution of surfactant molecules along the surface. Similar results of the effective dynamic surface ten-sion are observed for a range of other surfactants; dioctyl sulfosuccinate sodium salt (AOT), cetrimonium bromide (CTAB), glucamide and polysorbate 20 as can be seen in Fig. 10b. Similar to Fig. 10a, the obtained surface tensions of the higher concentrations are lower than the ones of the lower concentrations and the
Extrapolation BPT slow surf Extrapolation BPT fast surf Slow surfactant from Dmax/D0
Fast surfactant from D/D0 Slow surfactant from ti Fast surfactant from ti
0.01 0.10 1 10 0 20 40 60 80 100 concentration (CMC) σ (mN / m )
Surface tension at tmaxfor v = 1.6 m/s
(a)
Extrapolation BPT AOT Extrapolation BPT CTAB Extrapolation BPT Glucamide Extrapolation BPT Polysorbate 20 AOT from Dmax/D0
CTAB from D/D0 Glucamide from Dmax/D0 Polysorbate 20 from D/D0 AOT from ti CTAB from ti Glucamide from ti Polysorbate 20 from ti 0.05 0.10 0.50 1 5 10 0 20 40 60 80 concentration (CMC) σ (mN / m )
Surface tension at tmaxfor other surfactants
(b)
Figure 10: Effective dynamic surface tension at maximum spreading. Dynamic surface tension σ calculated using the two methods described in the main text as a function of fast (green diamonds) and slow (red squares) surfactant concentration for a medium impact velocity v = 1.6 m/s. The dotted lines are extrapolations based on BPT measurements (see Appendix B) and Eq. 3. d Dynamic surface tension at maximum spreading as a function of surfactant concentration for other surfactants (AOT, CTAB, glucamide and polysorbate 20) as calculated using the two methods described in the text. Impact velocity v = 1.6 m/s. The dotted lines are extrapolations based on BPT measurements and Eq. 3.
extrapolation of the bubble pressure tensiometer gives similar surface tensions at medium impact speed of 1.6 m/s. A difference between fast (AOT, glucamide) and slow (CTAB, polysorbate 20) surfactants can be observed.
The influence of the impact velocity on the dynamic surface tension of the droplets at maximum spreading can be observed in Fig. 12: the surface tension, determined with the droplet spreading ratio (solid symbols) and ti(open symbols), increase with increasing impact velocity. The results are compared to an extrapolation
of the BPT measurements (dotted line and white circles), which is independent of impact velocity. The difference in time of spreading between the different impact velocities (the time that we use to calculate the dynamic surface tension using Eq. 3) is so small that it does not translate into a significant variation in surface tension from the BPT. From the impact dynamics is to see that the effective surface tension depends on the impact velocity. The difference can be explained from the fact that the dynamic surface tension at maximum spreading results from a competition between the rate of formation of new droplet surface and the rate at which the surfactants arrive at the interface from the liquid bulk. So formally Dmaxfollows from
σ(t) with t between 0 and tmax and not from σ(tmax); however it turns out that the latter is a good and
easily implementable approximation. Higher droplet impact velocities imply faster droplet spreading, and hence a higher rate of surface area formation, leading to a smaller decrease in surface tension for the same concentration of surfactant. It therefore follows that our method is more accurate than the BPT.
4.4
Retraction rate
We now extend our analysis to t > tmax, which is when the droplet retracts. The retraction phenomenon
is particularly relevant to spraying applications, as it was found that above a critical retraction speed, bounce-off occurs, with the droplet (partially) disconnecting from the surface after impact [1]. Therefore, the retraction rate is an indicator of bounce-off, and hence of poor coverage during spraying. How a droplet
Fit to extrapolation from BPT fast Fast surfactant from D/D0 Fast surfactant from ti Extrapol. fast surfactant from BPT
1.0 1.2 1.4 1.6 1.8 2.0 2.2 20 30 40 50 60 v (m/s) σ (mN / m )
Surface tension at tmaxfast surfactant 1 CMC
Figure 11
Figure 12: a Dynamic surface tension at maximum spreading as a function of droplet impact velocities for a solution of fast surfactant with a concentration of 1 CMC. The dotted line is the extrapolation of BPT measurements using Eq.3.
retracts after impact is strongly influenced by the surface tension. The retraction rate has been previously defined as [21] ˙ = Vret/Rmax∝ t−1i = r σ ρR3 i , (4)
where Vret is the maximum slope during retraction, Rmax is the maximum radius of the droplet during
spreading and σ, ρ and Ri are the surface tension, density and initial diameter of the droplet respectively.
Fig. 13a shows the retraction rate ˙ of the droplets containing fast and slow surfactant as function of the dynamic surface tension at the moment the droplet has spread to its maximum diameter Dmax. Note
that the retraction rate above which bounce-off is expected to occur, ∼100 mm/s, is not reached in these experiments. We find that above a critical value of σ(tmax), the droplet retraction rates roughly follow the
predicted√σ dependence. The retraction rate can be made dimensionless by looking at ˙τiversus the inverse
Ohnesorge number (Oh = µ /√ρσRi). This is shown in Fig. 13b. A linear fit through the data is shown,
which is the predicted behaviour of the dimensionless retraction rate in the capillary regime by Bartolo, Josserand and Bonn[21]. The uncertainties of the data are still quite large and additional research over a larger range would be necessary to quantitatively confirm this exact relation. However, the most important message to take from Fig. 13 is that higher dynamic surface tensions lead to higher retraction rates. At high concentrations of the fast surfactant and hence low surface tensions, the droplet retraction rate reaches a minimum. As a result these droplets are least likely to bounce-off.
Because the retraction rate is a leading parameter that determines droplet bounce-off, the observed dy-namic surface tensions explain why the coverage of hydrophobic surfaces with fast surfactants is higher than with slow surfactants. In line with our observations, Aytouna et al.[15] reported bounce-off on hydrophobic parafilms for a slow surfactant and no bounce-off for a fast surfactant with the same σeq.
Slow surfactant Fast surfactant Water 20 40 60 80 100 0 20 40 60 80 100 σ (mN/m) ϵ ( s -1)
Retraction rate vs. surface tension at tmax
-242.6 + 35.21 σ0.5 (a) Slow surfactant Fast surfactant Water 5 6 7 8 9 10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Oh-1 ϵ τ i
Normalized retraction rate vs. the Ohnesorge number
-0.47 + 0.072 Oh-1
(b)
Figure 13: a Retraction rate ˙ of droplets for 1.7 < v < 2.1 m/s plotted against surface tension calculated from Dmax/D0. The
dotted line is the prediction by Eq. 4. b Dimensionless retraction rate versus 1/Oh. The dotted line is the best fit to the data ∝ 1/Oh.
find the optimum surfactant properties and concentrations to influence this process. Progress in preventing droplet rebound could for example reduce the use of expensive or contaminating chemicals and their leakage into the environment or increase precision in applications like high-performance inkjet printing. In other processes, like forensic research, a refined understanding of this process can be used to get a better insight in situations where droplet impact is crucial for reconstructions.
5
Conclusions
We used two methods to obtain the dynamic surface tension of impacting droplets at the impact timescale. In this article we focus on two different surfactant types: one ’fast’ surfactant and one ’slow’ surfactant. At similar concentrations, the equilibrium surface tension for the ’slow’ surfactant is lower than for the ’fast’ surfactant. Our measurements quantitatively reveal that for the first stage of the droplet impact process the dynamic surface tension of a solution with a ’fast’ surfactant is much lower than that of a solution containing a ’slow’ surfactant. We attribute this to the inability of ’slow’ surfactant molecules to diffuse to the droplet surface at these short time scales. Measurements for a range of other surfactants yield very similar results.
When we vary the droplet impact velocity we find small variations in the dynamic surface tension eval-uated at Dmax. This can be explained by the fact that higher droplet impact velocities result in faster
droplet spreading, and hence a higher rate of surface area formation, leading to a smaller decrease in surface tension for the same concentration of surfactant. With the conventional BPT measurement these variations with impact velocity cannot be measured. Because we measure the surface tension directly from droplet im-pact for different imim-pact velocities, our new methods yield more adequate results than the BPT measurement. We find a relation between the dynamic surface tension and the retraction rate of the impacting droplets that is in agreement with literature[21]. Because the retraction rate is a leading parameter that determines droplet bounce-off, the observed dynamic surface tensions explain why the coverage of hydrophobic surfaces with fast surfactants is higher than with slow surfactants.
In conclusion, our measurements provide a first experimental verification of previous predictions, such as of the relation between long time-scale surface tension and dynamic surface tension at the moment of maximum droplet spreading, and of the relation between droplet retraction rate and dynamic surface tension.
6
Further Research
I have made a start on follow-up research by looking closer into the coverage after spraying with surfactant solutions. The results of these first experiments are reported in this chapter, followed by a chapter containing an outlook on future research.
6.1
First Results for Follow-Up Research
The experiments determining the coverage improvement (coverage of the solution compared to the coverage with water) of plants after spraying as in Fig. 2b were done on real plant leaves, which vary in shape and composition from plant to plant. In order to get a more reproducible result of the coverage, the same exper-iment was done with the plants replaced by parafilm surfaces; an ideal hydrophobic surface. A schematic drawing of the setup is shown in Fig. 14a. A pressure tank is filled with a solution of water, surfactant and fluorescent. This is connected to a nozzle that is attached to a sledge. A weight is connected to the sledge, so that when it is released, the sledge will slide parallel to the surface, where a square parafilm sheet with the size of 10 cm is placed. For a short time the sample is sprayed with the solution from the pressure tank, mimicking the what happens in pesticide spraying. When the parafilm sheet is dry, a photo is taken under ultraviolet light as shown in Fig. 14b. The sample lights up where droplets have dried up, leaving behind fluorescent. The same photo after analysis is shown in Fig. 14c, where the parts covered with fluorescent are now white. From the picture, the coverage of the sheet can be extracted using the definition coverage = Atotal/Acovered, where Atotal is the total area of the sheet and Acovered the area of the sheet
covered with fluorescent. Two different setups were used, one where the parafilm sample was placed parallel to the spray installation, another one where the parafilm sample was tilted 60 degrees with respect to the spray installation.
The results of the coverage improvement are compared to the effective surface tension σef f as measured
from droplet impact experiments as shown in Fig 10a. The results for different surfactants are shown in Fig 15. For both configurations, but especially for the tilted surface the coverage improvement achieved by the surfactants seems to decrease linearly with their effective surface tension. This confirms the presumption that the coverage is governed by the dynamic surface tension. The medium velocity of 1.6 m/s was taken, as this velocity was also taken to compare the results of the BPT with in Fig. 12. The results for the high and low velocity can be found in Appendix D. Measurements were done for the fast and two different slow surfactants (trisiloxane and polysorbate 20).
In order to check that it is indeed the dynamic surface tension that determines the droplet deposition after spraying, the coverage improvement was also measured with the same experiment for water-ethanol solutions. The surface tension of these solutions is constant under deformations of the surface and can be tuned by changing the fraction of ethanol. The surface tension for different water-ethanol mixtures is shown in Tab. 2. ethanol wt% σeq(mN/m) 0 72.0 10 48.1 20 38.0 40 30.2 100 21.8
Table 2: Surface tension of water-ethanol mixtures sorted by ethanol weight percentage (wt%)[27]
The resulting coverage improvement can be seen in Fig. 15 for a flat and a tilted surface. If the surface tension at maximum spreading is leading, the results of the coverage improvement with the same experiment
(a)
(b) (c)
Figure 14: a Setup of the spray coverage experiment. When the sledge with the nozzle is released, the weight is pulled down by gravity, causing the nozzle to slide over the sample, spraying it whilst moving over it, mimicking pesticide spraying. Two configurations were used: one with the parafilm sample placed horizontally (θ = 0◦)and the other one with the parafilm sample
under an angle θ = 60◦. b Resulting photo of the parafilm sample under ultraviolet light on a red background. c The same
photo after analysis with fluorescent-covered area in white.
for ethanol mixtures would be expected on the same line as the surfactant results. However, the ethanol mixtures can be seen to a significantly lower results of the coverage improvement than would have been expected, especially for the flat surface. For the tilted surface the 10 wt% ethanol solution results in the expected coverage improvement, but the solutions with higher ethanol concentrations give lower coverage improvements than expected. There are three possible explanations for this discrepancy. The first explana-tion is that the droplet size distribuexplana-tion of the ethanol mixtures has altered enough to change the coverage due to the change in liquid properties. The second explanation is that the fluorescent that was used in order to determine the coverage does not work properly with ethanol solutions. The last possible explana-tion is that the surface tension at maximum spreading is not the important parameter to determine coverage. Starting with the first explanation: measurements of the droplet size distribution have showed that aver-age droplet sizes of the ethanol solutions are slightly lower than the one of water. This does not explain the observed difference: smaller droplet size is expected to decrease rebound and therefor increase the coverage after spraying. The addition of fluorescent was shown not to have influenced this distribution. The droplet size distributions are shown in Fig. 16. However, an explanation of the unexpected results is that the cover-age is not properly represented by the fluorescent residue of the dried parafilm samples for ethanol solutions. When equal size droplets of water and ethanol solutions with fluorescent were deposited onto parafilm, a picture of both the just deposited and dried out droplets shows that the used fluorescence residue of tap
Water Slow Slow2 Fast Ethanol 20 40 60 80 100 0 1 2 3 4 5 6 σeff(mN/m) Coverage Improvement
On flat parafilm surface compared to σefffor v =1.6m/s
(a) Water Slow Slow2 Fast Ethanol 20 40 60 80 100 0 1 2 3 4 5 6 σeff(mN/m) Coverage Improvement
On tilted parafilm surface compared to σefffor v =1.6m/s
(b)
Figure 15: Coverage improvement on parafilm surface for velocities a 1.6 m/s on flat surface b 1.6 m/s on 60◦tilted surface.
Water, 365 μm 10% Ethanol, 345 μm 20% Ethanol, 328 μm 40% Ethanol, 330 μm 0 500 1000 1500 2000 0 2 4 6 8 Droplet Size (μm) % V
Droplet Size Distributions Water-Ethanol mixtures
Figure 16: Droplet size distributions of water-ethanol mixtures. The average droplet sizes are given in the legend.
water represents the coverage of the droplet, but this is not the case for distilled water and ethanol. The results are shown in Fig. 17.
One important thing that is not taken into account is that when the equilibrium surface tension is much below the DST, slow spreading after impact will increase the coverage. Taking this into account it would be able to obtain the coverage directly after impact, possibly resulting in an even better linear relationship
Figure 17: Fluorescent residues of ethanol, tap water and distilled water droplets on parafilm.
between the coverage improvement and the dynamic surface tension. Furthermore, the average impact ve-locity in the coverage experiment is estimated to be > 4m/s, so outside the range of the droplet impact experiments. However, from Fig 19 one can expect that an increase in impact velocity will only lead to a shift in DST, but the linear relationship conserved. To create a better comparison, the droplet experiment should be done at the corresponding impact velocity to obtain the correct DST.
In conclusion, the first results of these spray coverage experiments indicate a relationship between the DST from droplet impact and the coverage improvement after spraying, as was expected from Fig. 2b: a lower DST leads to a higher coverage improvement. However, long-timescale spreading is not taken into account and the impact velocities of the two experiments do not coincide. For more reliable results these two problems should be resolved.
6.2
Outlook
In future research the coverage results could be properly compared to the results with ethanol mixtures, when a better way to measure spray coverage of ethanol solutions has been found. Additionally, the slow spreading caused by the transition from the dynamic surface tension at droplet impact to the equilibrium surface tension can be calculated, so that it is possible to look at these two different aspects of the coverage separately: the amount of fluid sticking to the surface and the spreading of the droplets.
Another possibility would be to do droplet impact experiments on parafilm , that has a higher contact angle than silanised glass, making it more probable that droplet bounce off, to compare the chances at rebound of different surfactant solutions to the chances of rebound of ethanol solutions. This could confirm that droplet rebound is determined by the DST at maximum spreading
It would also be interesting further investigate the relationship between the dynamic surface tension and the retraction rate. Comparing these with different surfaces, like parafilm and metal, could lead to a more generalized connection between these two droplet impact characteristics.
Although still leaving plenty of questions unanswered, the results of this research can hopefully be used as a next step towards a better understanding of the dynamics of droplet impact with complex fluids. The new measurement methods of the dynamic surface tension presented in this paper can be used to categorize or design new surfactants in order to find the best surfactant for each application, like pesticide spraying. It is my hope that with this improvement of additives for pesticide spraying future soil pollution can be reduced to a minimum, leading to a more sustainable world.
References
[1] Vance Bergeron, Daniel Bonn, Jean Yves Martin, and Louis Vovelle. Controlling droplet deposition with polymer additives. Nature, 405(6788):772–775, 2000.
[2] David R. Karsa, Boris Zhmud, and Fredrik Tiberg. Surfactants in ink-jet inks. Surfactants in Polymers, Coatings, Inks and Adhesives, 1(5):211–244, 2020.
[3] Yong Han Yeong, Rafael Mudafort, Adam Steele, Ilker Bayer, Eric Loth, and Guillaume De Combarieu. Water droplet impact dynamics at icing conditions with and without superhydrophobicity. In 4th AIAA Atmospheric and Space Environments Conference, page 3134, 2012.
[4] R. Andrade, O. Skurtys, and F. Osorio. Drop impact behavior on food using spray coating: Fundamen-tals and applications. Food Research International, 54(1):397 – 405, 2013.
[5] Gary N Carlton. A model to estimate a worker’s exposure to spray paint mists. Technical report, University of North Carolina, 1996.
[6] Jungho Kim. Spray cooling heat transfer: The state of the art. International Journal of Heat and Fluid Flow, 28(4):753 – 767, 2007. Including Special Issue of Conference on Modelling Fluid Flow (CMFF’06), Budapest.
[7] David S. Vernez, Pierre-Olivier Droz, Catherine Lazor-Blanchet, and Sylvain Jaques. Characterizing emission and breathing-zone concentrations following exposure cases to fluororesin-based waterproofing spray mists. Journal of Occupational and Environmental Hygiene, 1(9):582–592, 2004. PMID: 15559330. [8] Nick Laan, Karla G. De Bruin, Denis Bartolo, Christophe Josserand, and Daniel Bonn. Maximum
diameter of impacting liquid droplets. Physical Review Applied, 2(4), 2014.
[9] G. Aernout Somsen, Cees van Rijn, Stefan Kooij, Reinout A. Bem, and Daniel Bonn. Small droplet aerosols in poorly ventilated spaces and SARS-CoV-2 transmission. The Lancet Respiratory Medicine, 8(7):658–659, 2020.
[10] Greg R. Kruger, Robert N. Klein, Clyde L. Ogg, and Bruno C. Vieira. Spray Drift of Pesticides. University of Nebraska Extension Publications, i(September):G1773, 2007.
[11] David Nuyttens, M. Schampheleire, Pieter Verboven, Eva Brusselman, and Donald Dekeyser. Droplet size and velocity characteristics of agricultural sprays. TRANSACTIONS OF THE ASABE, 52:1471– 1480, 09 2009.
[12] Ira M. Cohen and Pijush K. Kundu. Introduction. In Fluid Mechanics, pages 8–9. Academic Press, Amsterdam, NL, 4th edition edition, 2008.
[13] Thomas A. Witten and Philip A. Pincus. Structured Fluids : Polymers, Colloids, Surfactants. OUP Oxford, 2010.
[14] Randal M. Hill. Silicone (siloxane) surfactants. In Robert A. Meyers, editor, Encyclopedia of Physical Science and Technology (Third Edition), pages 793 – 804. Academic Press, New York, third edition edition, 2003.
[15] Mounir Aytouna, Denis Bartolo, Gerard Wegdam, Daniel Bonn, and Salima Rafa¨ı. Impact dynamics of surfactant laden drops: Dynamic surface tension effects. Experiments in Fluids, 48(1):49–57, 2010. [16] J. Eastoe and J. S. Dalton. Dynamic surface tension and adsorption mechanisms of surfactants at the
air-water interface, 2000.
[17] Xi Yuan Hua and Milton J. Rosen. Dynamic surface tension of aqueous surfactant solutions. I. Basic paremeters. Journal of Colloid And Interface Science, 1988.
[18] N. Mourougou-Candoni, B. Prunet-Foch, F. Legay, M. Vignes-Adler, and K. Wong. Influence of dynamic surface tension on the spreading of surfactant solution droplets impacting onto a low-surface-energy solid substrate. Journal of Colloid and Interface Science, 192(1):129–141, 1997.
[19] Xiaoguang Zhang and Osman A. Basaran. Dynamic surface tension effects in impact of a drop with a solid surface. Journal of Colloid and Interface Science, 1997.
[20] Regan Crooks, Justin Cooper-White, and David V. Boger. The role of dynamic surface tension and elasticity on the dynamics of drop impact. Chemical Engineering Science, 56(19):5575–5592, 2001. [21] Denis Bartolo, Christophe Josserand, and Daniel Bonn. Retraction dynamics of aqueous drops upon
impact on non-wetting surfaces. Journal of Fluid Mechanics, 545:329–338, 2005.
[22] R Sijs, S Kooij, and D Bonn. How surfactants influence the drop size in sprays. arXiv preprint arXiv:1907.09723, 2019.
[23] D. Richart, C. Clanet, and D. Qu´er´e. Contact time of a bouncing drop. Nature, 417:811, 2002. [24] Rick Sijs and Daniel Bonn. The effect of adjuvants on spray droplet size from hydraulic nozzles. Pest
Management Science, 76:3487–3494, 2020.
[25] J. B. Lee, N. Laan, K. G. de Bruin, G. Skantzaris, N. Shahidzadeh, D. Derome, J. Carmeliet, and D. Bonn. Universal rescaling of drop impact on smooth and rough surfaces. Journal of Fluid Mechanics, 786:R4, 2015.
[26] Sander Wildeman, Claas Willem Visser, Chao Sun, and Detlef Lohse. On the spreading of impacting drops. Journal of fluid mechanics, 805:636–655, 2016.
[27] Thijs C de Goede, Karla G de Bruin, Noushine Shahidzadeh, and Daniel Bonn. Predicting the maximum spreading of a liquid drop impacting on a solid surface: Effect of surface tension and entrapped air layer. Physical Review Fluids, 4(5):053602, 2019.
Appendices
A
Silanization procedure of glass slides
1. Mix Toluene + Trichlorooctylsilane (Volume conc. 1%), stir until it’s a clear solution 2. Pour solution into a large petri dish (glass) and cover with2 a large glass lid
3. Blow of dust (air gun)
4. Plasma treatment for 30 sec (cleaning process)
5. Place the glass slides into solution, cover with a lid and leave it for 15min 6. Then rinse the glass slides with isopropanol (generous amount)
7. Store slides away from dust and any other impurities 8. Use the solution only once per batch
B
Bubble pressure tensiometer fits
Water 0.1 CMC 0.5 CMC 1 CMC 5 CMC 10 CMC 1 10 100 1000 104 105 0 20 40 60 80 t(ms) σ (mN / m )
Extrapolation BPT Slow Surfactant
(a) Water 0.1 CMC 0.5 CMC 1 CMC 5 CMC 10 CMC 0.01 CMC 0.05 CMC 1 10 100 1000 104 105 0 20 40 60 80 t(ms) σ (mN / m )
Extrapolation BPT Fast Surfactant
(b)
Figure 18: Bubble pressure tensiometer data and fits to Eq. 3 for concentrations a slow surfactant b fast surfactant compared to water (blue circles, constant at 72mN/m). The dashed lines are the best fits of Eq. 3 to the experimental data. These fits are used to extrapolate the dynamic surface tension of the liquids to the typical time of spreading of a droplet, which is 3 ms (grey dotted line). Fit parameters σ0(mN/m), σ∞(mN/m), τ (s), n: σ0 = 72 for all fits,a σ∞= 68.1, τ = 5.56 ∗ 103, n = 1.21
(0.1 CMC), σ∞ = 1.00, τ = 1.01 ∗ 104, n = 8.22 ∗ 10−1 (0.5 CMC), σ∞ = 17.5, τ = 2.22 ∗ 103, n = 8.12∗10−1 (1 CMC), σ∞ = 19.0, τ = 1.63 ∗ 102, n = 7.38 ∗ 10−1 (5 CMC), σ∞ = 19.4, τ = 62.1, n = 7.51 ∗ 10−1 (10 CMC). b σ∞ = 57.7, τ = 1.14 ∗ 104, n = 3.7∗10−1. (0.01 CMC), σ ∞= 41.5, τ = 7.01 ∗ 102, n = 3.10 ∗ 10−1(0.05 CMC), σ∞= 35.2, τ = 2.53, n = 0.2 (0.1 CMC), σ∞ = 22.5, τ = 1.4 ∗ 10−6, n = 4.74 ∗ 10−2 (0.5 CMC), σ∞ = 1.00, τ = 1.04 ∗ 102, n = 3.76 ∗ 10−2 (1 CMC), σ∞= 17.4, τ = 3.12 ∗ 10−7, n = 4.39 ∗ 10−2(5 CMC), σ∞= 18.3, τ = 9.83 ∗ 10−8, n = 5.13 ∗ 10−2(10 CMC).
C
Effective dynamic surface tension for other velocities
Extrapolation BPT slow surf Extrapolation BPT fast surf Slow surfactant from Dmax/D0
Fast surfactant from D/D0 Slow surfactant from ti Fast surfactant from ti
0.01 0.10 1 10 0 20 40 60 80 100 concentration (CMC) σ (mN / m )
Surface tension at tmaxfor v = 1.2 m/s
(a)
Extrapolation BPT slow surf Extrapolation BPT fast surf Slow surfactant from Dmax/D0
Fast surfactant from D/D0 Slow surfactant from ti Fast surfactant from ti
0.01 0.10 1 10 0 20 40 60 80 100 concentration (CMC) σ (mN / m )
Surface tension at tmaxfor v = 2.1 m/s
(b)
Figure 19: Effective dynamic surface tension at maximum spreading. Dynamic surface tension σ calculated using the two methods described in the main text as a function of fast (green diamonds) and slow (red squares) surfactant concentration for a low impact velocity v = 1.2 m/s and b high impact velocity v = 2.1 m/s. The dotted lines are extrapolations based on BPT measurements (see Suppl. Inform.) and Eq. 3.
D
Coverage improvement for other velocities
Water Slow Slow2 Fast Ethanol 20 40 60 80 100 0 1 2 3 4 5 6 σeff(mN/m) Coverage ImprovementOn flat parafilm surface compared to σefffor v =1.2m/s
(a) Water Slow Slow2 Fast Ethanol 20 40 60 80 100 0 1 2 3 4 5 6 σeff(mN/m) Coverage Improvement
On tilted parafilm surface compared to σefffor v =1.2m/s
(b) Water Slow Slow2 Fast Ethanol 20 40 60 80 100 0 1 2 3 4 5 6 σeff(mN/m) Coverage Improvement
On flat parafilm surface compared to σefffor v =2.1m/s
(c) Water Slow Slow2 Fast Ethanol 20 40 60 80 100 0 1 2 3 4 5 6 σeff(mN/m) Coverage Improvement
On tilted parafilm surface compared to σefffor v =2.1m/s
(d)
Figure 20: Coverage improvement on parafilm surface for velocities a 1.2 m/s on flat surface b 1.2 m/s on 60◦tilted surface c 2.1 m/s on flat surface d 2.1 m/s on 60◦tilted surface
