Dynamic wetting of dilute polymer solutions: The case of impacting droplets

Advances in Colloid and Interface Science 193–194 (2013) 1–11

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Dynamic wetting of dilute polymer solutions: The case of impacting droplets V. Bertola ⁎ University of Liverpool, School of Engineering, Brownlow Hill, Liverpool L69 3GH, UK

a r t i c l e

i n f o

Available online 23 March 2013 Keywords: Dynamic wetting Dilute polymer solutions Drop impact

a b s t r a c t The moving contact line of a dilute polymer solution that advances over, or recedes from a solid substrate, is a fundamental problem of fluid dynamics with important practical applications. In particular, the case of droplets impacting on hydrophobic surfaces received much attention in the recent past. Experiments show that while the advancing motion proceeds as with Newtonian liquids, recession is severely inhibited. This phenomenon was initially understood as an effect of elongational viscosity, which was believed to cause large energy dissipation in the fluid. Later on, a hydrodynamic mechanism was proposed to suggest that the slowing down of the contact line is due to non-Newtonian normal stresses generated near the moving droplet edge. Recent experiments however ruled out the role of elongational viscosity, showing that the fluid velocity measured inside the droplet during retraction is the same in water drops and polymer solution drops. Direct visualization of fluorescently stained λ-DNA molecules showed that polymer molecules are stretched perpendicularly to the contact line as the drop edge sweeps the substrate, which suggests an effective friction arises locally at the drop edge, causing the contact line to slow down. © 2013 Elsevier B.V. All rights reserved.

Contents 1. 2. 3. 4.

Introduction . . . . . . . . . . . . . . . . . . . Dilute polymer solutions . . . . . . . . . . . . . Early approaches . . . . . . . . . . . . . . . . . Recent results . . . . . . . . . . . . . . . . . . 4.1. Dynamic contact angle measurements . . . . 4.2. Velocity field inside impacting droplets . . . 4.3. Polymer dynamics near the contact line during 5. Conclusions . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . drop retraction . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction The effect of polymer additives on static and dynamic wetting of liquid drops is relevant to several industrial processes, including coating or cleaning surfaces, inkjet printing, additive manufacturing, enhanced oil recovery, drug delivery, and many others. Despite its importance, this subject was not systematically investigated until about 30 years ago, when it was identified as an emerging issue in soft matter physics [1]. Since then, wetting phenomena in polymer solutions received growing attention, both from the experimental [2–7] and from the theoretical ⁎ Tel.: +44 1517944804. E-mail address: [email protected]. 0001-8686/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cis.2013.03.001

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1 3 4 7 7 8 9 9 11 11

point of view [8–10]. In this context, a milestone discovery was the observation that a receding meniscus of a dilute polymer solution is able to stretch partially adsorbed polymer molecules on the wetted surface [11]. This led to the development of various DNA stretching techniques, such as molecular combing, spin-stretching, and air-blowing [12]. An outstanding example of rapid wetting/dewetting occurs during drop impact on solid substrates, where the contact line advances and retracts with timescales of the order of 10 −2 s. The impact morphology of droplets of Newtonian liquid onto solid, dry surfaces, is well-known [13–17]. Upon impact, the liquid spreads on the surface taking the form of a disk; for low impact velocity, the disk thickness is approximately uniform, while for higher impact velocities the disk is composed of a thin central part (often called “lamella”) surrounded by a circular

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rim. This initial spreading stage is typically very fast (≈5 ms). After the drop has reached maximum spreading, two qualitatively different outcomes are possible. If the initial kinetic energy exceeds a threshold value capillary forces are insufficient to maintain the integrity of the drop, which disintegrates into smaller satellite droplets jetting out of its outermost perimeter (splashing). If splashing does not occur, the drop is allowed to retract under the action of capillary forces, which tend to minimize the contact with the surface; in some cases, retraction is so fast that the liquid rises in the middle forming a Worthington jet, which may subsequently result in the complete rebound of the drop from the surface. Impacts onto smooth and chemically homogeneous surfaces, for low or moderate impact kinetic energy, are controlled by three key factors: inertia, viscous dissipation and interfacial energy [13,17]. During the initial stages of impact with the surface, the vertical inertia of the falling drop is converted into the horizontal motion of the fluid, and as the drop spreads kinetic energy is partly stored as surface energy. This balance is characterized by the Weber number, We = ρvi2D0/σ, where ρ and σ are the fluid density and surface tension, respectively, D0 is the equilibrium drop diameter, and vi the normal impact velocity. As the fluid spreads across the surface, the kinetic energy of the fluid is partly dissipated by viscous forces in the fluid, which is described using the Reynolds number, Re = ρviD0/μ, where μ is the fluid viscosity. This is sometimes used in combination with the Weber number to yield the Ohnesorge number, Oh = We 1/2/Re. Finally, the retraction stage is governed by the balance between interfacial energy and viscous

dissipation, expressed by the Capillary number, Ca = μvr/σ, where vr is the retraction velocity. While there exists a significant volume of literature about single drop impacts of simple (Newtonian) fluids, the number of works about fluids with complex microstructure (polymer melts or solutions, gels, pastes, foams and emulsions, etc.) is comparatively very small. However, these fluids are frequently used in common applications, such as painting, food processing, and many others. Moreover, with a better understanding of liquid microstructures, industries have realized that working fluids can be tailored specifically to optimize existing industrial processes, by altering their formulation (e.g. by means of chemical additives) in such a way as to change one or more physical properties. The study of polymer solution drops received considerable attention after the accidental discovery that small amounts (of the order of 100 ppm) of Polyethylene Oxide (PEO) can reduce the tendency of drops to rebound after impacting on low surface energy (hydrophobic) surfaces, which can be exploited to control many spray applications and in particular the distribution of agrochemicals [18–20]. This phenomenon is illustrated in Fig. 1, which compares the impact morphologies of two drops, one of de-ionized water and one of a 200 ppm PEO solution in the same water, impacting with the same velocity on a Parafilm-M surface (equilibrium contact angle with water: ~105°). After the initial inertial spreading, which is similar for both drops, water drops exhibit fast recoil (~30 ms) under the action of surface forces, which evolves into almost complete dewetting and rebound

Fig. 1. Impact morphology of a water (top) and a 200 ppm PEO solution drop (bottom) on a Parafilm-M surface.

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on the impact surface. On the contrary, the recoil of polymer solution drops is very slow, and terminates in a sessile equilibrium state only after several seconds. Despite the obvious relationship with dynamic wetting phenomena, the different impact morphologies were initially attributed to the bulk rheological properties of polymer solutions, which hindered the correct physical understanding of this phenomenon until recently. The aim of this paper is to present a review of the literature about the impact of dilute polymer solution drops, form early works to the most recent developments, with the purpose of retracing the sometimes tortuous path towards a correct interpretation of its phenomenology, explaining at the same time some mistakes made along the way. In particular, the paper is organized as follows: Section 2 presents an overview of dilute polymer solutions and their main features; Section 3 describes the early approaches, and the reasons of their incorrectness; and Section 4 presents the results of three recent, independent experiments that allow one to explain the impact morphology of dilute polymer solutions in terms of wetting. Finally, Section 5 draws some concluding remarks, and addresses the most recent research trends.

The linear coefficient [η] is called the intrinsic viscosity, and can be obtained as the slope in the relative viscosity vs. concentration plot in the low concentration limit:

2. Dilute polymer solutions

c ¼

c→0

ð1Þ

η−ηs : cηs

ð2Þ

The intrinsic viscosity is a quantity characteristic of a polymer, which increases with the size of the polymer molecule (i.e., the molecular weight MW). Experimentally, it is expressed by Mark–Houwink equation: α

½η ¼ K M M W :

ð3Þ

The coefficients KM and α are different from polymer to polymer and can depend on the solvent as well. In particular, for polyethylene oxide solutions in water KM = 0.0125 and α = 0.78, while for polyacrylamide solutions in water KM = 0.068 and α = 0.66 [23,24]. The reciprocal of the intrinsic viscosity is often used to represent the overlap concentration of a given polymer: 

The fluids considered in this paper are solutions of long-chain polymers in a Newtonian solvent (e.g., water) [21,22]. The macroscopic physical properties of these solutions are strongly dependent on the polymer concentration, as illustrated in Fig. 2. For low concentrations, the average distance among polymer molecules is larger than their size, so that they do not interact with one another: polymers exhibit a random coil conformation and can be described as spherical particles suspended in the fluid (dilute regime). As the polymer concentration grows, the average distance between polymer molecules reduces until the monomers placed on the external shell of one coil can interact with monomers of other coils. This happens for a critical value of concentration (the overlap concentration), and marks the beginning of the semi-dilute regime. In the semi-dilute regime, polymer chains become randomly entangled and can be no longer described as coils. Moreover, their mobility is greatly reduced, which causes drastic changes in the solution properties as compared with dilute solutions. The viscosity of polymer solutions, η, is higher than that of the pure solvent, ηs. However, when the concentration is sufficiently low, their difference is small, and one can express the relative viscosity (i.e., the ratio between the solution viscosity and the solvent viscosity), in the form of a Taylor's series, truncated to first order:

η ≈1 þ ½ηc: ηs

½η ¼ lim

1 : ½η

ð4Þ

This means that a polymer solution at the overlap concentration is about twice as viscous as the pure solvent. The adsorption dynamics of these macromolecules on the free surface of the solution also varies from polymer to polymer. However, because of the high molecular weight hence of the large size of polymer molecules, in general this process is very slow, and may take several hours to reach equilibrium [25]. This is important when studying the behavior of dilute polymer solutions at short time scales, such as in the case of drop impact phenomena, because the surface properties (e.g., the surface tension and the apparent contact angle) of the solution can be considered identical to those of the pure solvent. When polymer solutions flow, the conformation of polymer chains depends on the competition between Brownian effects and hydrodynamic effects. At low deformation rates, Brownian motion is strong enough to offset the viscous pull of the solvent, therefore polymer chains remain close to their equilibrium random coil conformation. At high deformation rates, hydrodynamic forces overcome Brownian motion, and coils are deformed and oriented in the flow direction. It must be remarked that the chain conformation is also affected by the solvent quality: in a good solvent, interactions between monomers and the solvent molecules cause polymer coils to expand, and vice-versa in a poor solvent. However, this analysis is beyond the scope of this paper, which focuses essentially on dilute solutions of long-chain flexible polymers in good solvents. When molecules are stretched, and then the applied stress is removed, the molecular chains relax to their equilibrium conformation,

Fig. 2. Solution regimes of flexible polymers in a Newtonian solvent. The overlap concentration, c*, is defined in Eq. (4).

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however relaxation is not instantaneous. The time constant associated with relaxation, called the relaxation time, is a property of the fluid, and can vary from a fraction of a second to a few minutes. More precisely, polymer solutions are characterized by a spectrum of relaxation times, associated with different relaxation modes. For fluids with long relaxation times, elastic effects are observed easily because the stresses relax so slowly; for fluids with short relaxation times, elasticity cannot usually be observed but it can be generated provided the deformation rate is high enough. The relaxation time of polymer solutions can be obtained from models describing the polymer chain dynamics, such as the Rouse model, suitable for unentangled concentrated polymer solutions, and the Zimm model, which gives a good description of the dynamics of unentangled dilute polymer solutions [21,22,27,28]. In practice, one can use empirical correlations taking into account the effect on the relaxation time of different parameters. In particular, the relaxation time of dilute PEO solutions can be calculated as [26]:

pffiffiffi −3 −11 2 τ ¼ 1:82  10 ½η−2:9  10 ½η−0:51 c exp −0:0004T : ð5Þ Fig. 3 shows the relaxation time of dilute PEO solutions as a function of the polymer concentration, for different temperatures. As expected, the relaxation time hence the fluid viscoelasticity grows with the polymer concentration; one can also observe a strong dependence on temperature, which increases the mobility of polymer chains reducing dramatically the relaxation time. Dilute solutions are of particular importance in many engineering applications, because due to the low polymer concentration their viscosity is very similar to that of the pure solvent, although the solution exhibits viscoelastic properties. In viscoelastic fluids, part of the deformation energy is stored as elastic energy, and released with a certain delay depending on the relaxation time of the fluid. Viscoelasticity is the cause of many phenomena, including creep (the time-dependent strain resulting from a constant applied stress), stress relaxation resulting from a steady deformation, the Weissenberg rod-climbing effect due to nonzero normal stress difference, and many others [27,28]. The basic feature that essentially all viscoelastic fluids share is the occurrence of elastic stress effects: when the shear rate is sufficiently strong, the forces along the normals of a little cubical fluid element are different in different directions, unlike what happens for a Newtonian fluid where the pressure is isotropic. From the microscopic point of view, this behavior is usually related to conformational rearrangements of the macromolecules, which tend

to stretch and align parallel to streamlines under the action of hydrodynamic forces. The entropic tendency of polymers that are stretched by the flow to recover their equilibrium chain conformation generates an additional elastic tension along the streamlines, the macroscopic effect of which is a difference in stress between the flow direction and the direction normal to it [28]. Thus, the non isotropic principal components of the stress tensor allow one to characterize the elasticity of polymer solutions and melts through normal stress differences: in particular, the first normal stress difference, N1 = τxx − τyy (where x is the streamline direction), is the only significant elastic effect in most cases, and has a negative value to account for the elastic tension along streamlines. For the well-known Oldroyd-B fluid model [28,29] in steady-state shear flow, the first normal stress difference is a quadratic function of the shear rate, γ: 2

N1 ¼ Ψ1 γ_ :

ð6Þ

The dissipation of energy associated to the process of stretching and relaxation of macromolecules is described by introducing the concept of elongational (or extensional) viscosity, the ratio of the first normal stress difference to the rate of elongation of the fluid: ηE ¼

τ xx −τyy : εxx

ð7Þ

For a Newtonian incompressible fluid, one can easily verify that the elongational viscosity is three times the shear viscosity. For a polymer solution the ratio ηE/η, also known as the Trouton ratio [30], can be of the order of 10 3–10 4. Quantitative measurements of elongational viscosity are not easy [31], especially for dilute polymer solutions in low-viscosity solvents, because they require the creation of a steady-state elongational flow. This is difficult to achieve in practice because it is not possible for a volume of fluid to stretch to infinity, since it will get thinner and thinner and eventually break-up. Furthermore, the stiffness of polymer chains is not constant, but grows as they approach the maximum elongation: thus, the instantaneous values of elongational viscosity are not constant during measurements. Measurements that reasonably approach steady state have been obtained for certain polymer solutions by means of the filamentstretching technique [32,33]. Unfortunately, this technique is only applicable to relatively viscous liquids, as the filament breaks up too rapidly for low-viscosity samples. The alternative stagnation point devices, such as the opposed nozzle rheometer [34], do offer a stationary flow, but the residence time of a polymer chain in the elongational flow field is typically quite short, has large statistical fluctuations and depends on the rate of elongation. Therefore, a steady-state value for the elongational viscosity is very hard to obtain. At a molecular level, the energy dissipation mechanism for elongational flows can be explained in terms of the interaction between the additive and the surrounding fluid, which is essentially due to hydrogen bonds between water molecules and monomers. Thus, when the polymer is coiled, the only monomers affected by the interaction are those located in the external shell, and the polymer molecule behaves like a spherical particle advected by the flow. As the velocity gradient increases the polymer stretches, and therefore more of its monomers become affected by the interaction with the fluid, increasing molecular friction and hence viscous dissipation. 3. Early approaches

Fig. 3. Relaxation time of dilute PEO solutions as a function of the polymer concentration, for different temperatures, calculated using Eq. (5).

The effect of flexible polymers on drop impact was first investigated at Rhodia (now part of Solvay Group) [18,19], with the aim to improve the performance of agrochemical formulations. In particular, it was discovered that very tiny amounts (of the order of 100 ppm, or

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0.01% in mass) of high-molecular weight polyethylene oxide (PEO) added to water improve dramatically the adhesion of drops to smooth hydrophobic surfaces, such as waxy plant leaves, whereas drops of pure water bounce off. A similar result had been previously obtained using certain surfactant additives, such as sodium dodecyl sulfate (SDS), that reduced the dynamic surface tension of the fluid [35,36]. However, this method of preventing drop rebound was found not to be well suited for agricultural spray applications, because lowering the surface tension creates smaller droplets out of the spray nozzle that can easily drift off target. Furthermore, a lower surface tension requires less energy to disintegrate the drop upon impact, thus making worse the problem of splashing. The adsorption dynamics of PEO molecules at the air–liquid interface of dilute solutions is very slow [25], therefore this type of additive does not have such unwanted side effects. The same phenomenon was described independently in another work [37], the main focus of which was to study the influence of fluid elasticity on the onset of splash. This work showed that dilute polymer solution drops exhibit splash and that the splash threshold increases with increasing Rouse relaxation time. To explain the behavior of dilute PEO solution drops upon impact, it was initially suggested that the critical property is the high elongational viscosity provided by the polymer, which dampens the drop retraction after impact hence prevents droplet rebound [38]. The elongational viscosity of the solutions was obtained in two ways: using an opposed-jet extensional rheometer (Rheometrics RFX), and fitting the FENE-P constitutive equation [28] to data obtained in shear flow with a custom made cone and plate geometry rheometer, which allowed access to the high shear rates that are necessary to measure the elastic response of the dilute polymer solutions. When the elongational viscosity is used instead of the shear viscosity to calculate the capillary number, the relationship between the capillary number and the recoil velocity of polymer solution drops has the same qualitative trend as in the case of Newtonian drops of different viscosity (water–glycerol solutions). The characteristic elongation rate of the retracting drops was obtained from the rim velocity (about 30 cm/s) divided by the thickness of the stretched droplet (estimated as 0.1 mm), which led to typical elongation rates of 3000 s −1. However, this approach suffers from two rather obvious flaws. Firstly, the value of the rim velocity used in this calculation corresponds, according to the authors, to the minimum recoil velocity required for drop bouncing; indeed, the recoil velocity of polymer solution drops is much slower (using the data shown in Fig. 1 of Ref. [38] yields a recoil velocity of 3.6 cm/s). Secondly, since the stretched drop thickness is perpendicular to the rim velocity, the quantity calculated by these authors is, by definition, a shear rate and not an elongation rate. Another problem, of different nature, concerns the elongational viscosity data. The measurement of elongational viscosity in dilute polymer solutions is discussed extensively in another paper from the same research group [39]. This paper reports the elongational viscosity of the same PEO solutions considered in [38], measured with the same instrument; thus, the two sets of data should be in close agreement. However, Fig. 4 demonstrates this is not the case, because while some pairs of data are identical, others show significant differences, up to 100%. Considering the large uncertainty associated with these measurements, it seems difficult to use them in support of any theoretical argument. Despite these flaws, and the lack of direct evidence of a cause–effect relationship between elongational viscosity and the behavior of polymer solution drops impacting on hydrophobic surfaces, this approach was echoed in several works [40–44], and its rapidly growing popularity gained acknowledgment in the reference literature [17]. A careful experimental study was carried out to establish the independent influence of dynamic surface tension and elongational viscosity on drop impact [41]. In particular, fluids with a similar and

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Fig. 4. (A) Elongational viscosity measurements on dilute PEO solutions, obtained with the same measurement technique, reported in two different papers. (B) Direct comparison of data shows several pairs with a difference around 100% (discontinuous line). Re-plotted from Refs. [38] and [39].

constant low viscosity and surface tension were constructed to isolate the effect of elasticity. The Trouton ratio, expressing the elongational viscosity of the fluid, was correlated by a power-law fit to the nondimensional height of the drop above the surface, and by a linear fit to the recoil velocity of the drop. Based on these correlations, it was concluded that the increase in extensional viscosity obtained with polymer additives is responsible for the suppression of recoil of drops impacting a Parafilm-M surface. Another work [43] puts the accent on the dissipative nature of extensional viscosity, rather than on its relationship with the fluid elasticity. This work suggested that the fluid stretches during both the inertial expansion of the drop and the subsequent retraction, unfolding and deforming the high molecular weight polymer molecules. This deformation drains energy out of the drop, so it can no longer bounce off from the surface. In other words, drops do not rebound because the initial kinetic energy is dissipated by the high elongational viscosity. Although these contributions seemed to reinforce the initial understanding of the phenomenon in terms of a consequence of the fluid elongational viscosity, they also suggested further objections. If the

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dissipation mechanism is active already during the inertial spreading of the drop, on should observe a reduction of the maximum spreading diameter in polymer solution drops, while experimental data presented in the same works indicate this quantity is not affected by the polymer additive. More in general, a strong correlation between two quantities (in this case, the recoil velocity and the elongational viscosity) does not necessarily imply that there is a cause–effect relationship between them, because they could be different consequences of something else. A study of drops impacting on small targets [45] demonstrated that polymer additives do not change the retraction velocity. In these experiments, the influence of the substrate was removed by using a small target, and the polymeric lamellae retracted in the same manner as the water lamella. These findings proved that the polymeric additives do not have any effect on the bulk elongational deformations of the drop but they have a great influence on the interaction of the lamella with the substrate at the retraction stage if the impact happens on a plane, smooth, solid substrate. This suggested that the substrate influence is predominant for retarding the lamella retraction with polymeric solutions. The polymeric additives likely oppose a large resistance to the motion of the contact line hence they influence the dynamic contact angle. Another method to completely remove the influence of the substrate is heating the impact surface to create a thin vapor film between the drop and the substrate, which is known as “dynamic Leidenfrost phenomenon” [46]. Experiments on Leidenfrost drops of dilute polymer solutions brought further evidence against the elongational viscosity argument [47]. In particular, measurements of the maximum spreading diameter (Fig. 5a) showed that polymer additives cause only a slight reduction of the maximum spreading diameter and of the retraction velocity. Because in these experiments wetting effects are absent or negligible, one must conclude that the retraction velocity reduction observed in drops containing flexible polymers impacting on solid surfaces is due to the drop-surface interaction rather than to an increased energy dissipation connected to the elongational viscosity of the fluid. Furthermore, measurements of the maximum bouncing height of Leidenfrost drops (Fig. 5b), which are indicative of the fraction of the initial kinetic energy which is not dissipated during impact, suggested that in some cases polymer additives indeed reduce instead of increasing the overall energy dissipation. This result is consistent with recent direct numerical simulations showing a reduction in small-scale convective motions in dilute polymer solutions hence in bulk viscous dissipation [48]. Therefore, one must conclude that the real cause of this phenomenon is to be sought in the dynamic wetting behavior of dilute polymer solutions, and not in some bulk property of the fluid such as the elongational viscosity. The contradictions emerging from the initial experiments led to reconsider the initial understanding, and to propose alternative approaches explaining the slowing down of the contact line by polymers in terms of wetting. In particular, it was suggested that the contact line dynamics is ruled by the competition between the surface tension that drives the retraction and the elastic normal stresses that counter it [49]. This work reports experiments with aqueous solutions of polyacrylamide (molecular weight Mw = 15 × 10 6 amu), and poly-ethylene oxide (molecular weight Mw = 4 × 10 6 amu and Mw = 8 × 10 6 amu), with concentrations up to 5 g/l. Using an equation that generalizes the lubrication theory for thin films, and accounting for capillarity and normal stresses in addition to shear stresses, it is found that for large contact angles the retraction velocity, vr, assumed to be identical to the average velocity of the fluid over the drop thickness, is related to the normal stress coefficient, Ψ, and to the contact angle, θ, according to the following equation:
 4Ψv2 r σ cosθ− cosθeq ¼ lm

ð8Þ

Fig. 5. Maximum spreading diameter (A) and maximum bouncing height (B) of bouncing Leidenfrost drops of water (open symbols) and a 200 ppm PEO solution (filled symbols). From Ref. [47].

where θeq is the equilibrium contact angle, and lm the microscopic cut-off height of the drop in the neighborhood of the contact line. Unfortunately, this approach is not correct for several reasons. First of all, while the slowing down of the contact line is observed for drops of dilute polymer solutions (i.e. when the polymer concentration is less than the overlap concentration), Eq. (8) has been validated with experimental data based on solutions with polymer concentrations well above the overlap threshold. In fact, using Eq. (4) one can find that the overlap concentration for the polyacrylamide solutions is 0.27 g/l, while those of the polyethylene oxide are respectively 0.57 g/l and 0.33 g/l. Since increasing the polymer concentration of a solution above the overlap value corresponds to a marked increase of viscosity, the slow retraction observed in these experiments might be simply due to a higher shear viscosity of the solution; however no comparison with Newtonian fluids of similar viscosity is provided. Moreover, dilute polymer solution do not exhibit appreciable normal stresses in the range of shear rates observed during the retraction stage, when “an upper bound shear rate can be approximated by (VRET/h) with h the height of the pancake droplet at the onset of retraction” [49]. Using the values obtained from typical experiments as described above (VRET ≈ 3.6 cm/s, h ≈ 0.1 mm), one finds an upper bound shear rate γ_ ≈ 360 s−1, however in these conditions dilute solutions do not

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exhibit appreciable normal stresses, as one can clearly see from data reported in Ref. [49] (re-plotted in Fig. 6) as well as in another paper from the same group [39]. Thus, normal stresses cannot be invoked to explain the behavior of these fluids. In the authors' intentions, Eq. (8) represents “the balance between the driving capillary force and the dissipation by normal stresses in the wedge”. In other words, the only normal stresses effect considered in their model is a dissipative force opposing to the capillary force, while the elastic component is neglected. However, the tension along streamlines generated by normal stresses, net of all dissipations, would result into an elastic force pulling the contact line, in the same direction as the capillary force. This can be understood in terms of elementary continuum mechanics. Therefore, the macroscopic effect of normal stresses should be an acceleration of drop retraction, which is not observed in experiments. The derivation of Eq. (8) also relies on the assumption that when a microscopic thickness lm is reached, the contact angle is equal to the equilibrium contact angle, despite the system is clearly out of equilibrium. This cut-off length is not known a priori, therefore Eq. (8) cannot be used to obtain a quantitative estimation of the retraction velocity, but is used instead to extract values of lm from the fit to experimental data. To verify the correctness of this approach, such microscopic lengths are then compared with the lengths of the fully extended polymer chains, and found to be of the same order of magnitude. However, this implicitly suggests that polymer chains are thought to stretch in the vertical direction, although the main velocity component, which should provide the stretching action, is parallel to the impact surface. Therefore, this conclusion is in contrast with the rest of the paper. Moreover, if it was true that polymer chains stretch in the vertical direction near the contact line, they should increase the contact angle with respect to the case without polymer (Newtonian fluid). This does not correspond to the experimental situation, where the apparent contact angle measured for dilute polymer solutions is much smaller than in the case of Newtonian fluids [50]. The above considerations demonstrate the approach proposed in Ref. [49] does not provide a reasonable explanation for the peculiar behavior observed in dilute polymer solution drops, and probably is not even useful in case of semi-dilute or concentrated solutions, which do exhibit normal stresses in the range of shear rates of interest unlike dilute solutions. Errare humanum est, perseverare diabolicum. 4. Recent results 4.1. Dynamic contact angle measurements To understand the effect of polymer additives on impacting droplets in terms of wetting, invaluable insight can be gained from the study of the

Fig. 6. The first normal stress difference of polyacrylamide solutions with concentrations of 0.2 g/l (○), 0.5 g/l (□), 1 g/l (◊), 2 g/l (×) and 5 g/l (+). Reprinted from Ref. [49]. Copyright (2007) by The American Physical Society.

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apparent dynamic contact angle during drop retraction [50]. This quantity was surprisingly ignored in all previous studies [37,38,40–44,49]. The qualitative behavior of the receding contact angle is displayed in Fig. 7, which shows side views of a water drop and a 200 ppm PEO solution drop, both impacting on a Teflon surface (equilibrium contact angle: 119° ± 2°) from a height of 6 cm (We ≈ 55). Images with a resolution of 576 × 288 pixels (typical spatial resolution: 17 μm/pixel) were captured at 8000 frames per second. These pictures suggest that the polymer additive significantly reduces the dynamic contact angle with respect to pure water, however the change is localized in the wedge near the contact line, as if the contact line was pinned on the surface, while the bulk fluid seems to flow back towards the drop center without encountering the same resistance. Measurements of the base diameter and of the apparent dynamic contact angle obtained from digital image processing are shown in Fig. 8. The base diameter of water drops (Fig. 8a) grows and decreases approximately at the same rate, and becomes equal to zero at the moment of drop rebound, about 50 ms after impact. After rebound, it reaches the final equilibrium value after a few oscillations, typically in a very short time (20–30 ms). The apparent contact angle has an oscillatory behavior around the equilibrium value, with the exception of a discontinuity in correspondence of the drop rebound. Throughout the drop retraction phase, its value remains above ~ 70°. In polymer solution drops (Fig. 8b), these quantities are remarkably different. The base diameter initially grows at the same rate as in water drops, and reaches a maximum approximately of the same magnitude, however the retraction phase is much slower, and the base diameter takes several seconds to reach the equilibrium value. The retraction phase is characterized by stick–slip dynamics of the drop edge, which causes sharp peaks visible on the base diameter plot. Unlike in the case of water drops, the dynamic contact angle initially decreases to a very small value (~10°), and then grows slowly as the drop retracts, with small oscillations about the mean growth trend. The local minima of these oscillations correspond to the stick–slip peaks on the base diameter plot. During the approach to the equilibrium value, the dynamic contact angle of dilute polymer solution drops remains smaller than the contact angle measured with water drops under the same experimental conditions. A simple interpretation of this behavior can be given in terms of the Young–Laplace force balance: a small contact angle corresponds to a large horizontal component of the liquid–vapor interfacial force that drives the drop retraction. Thus, since the contact angles observed during the retraction of polymer solution drops are significantly smaller than those observed in drops of pure water, one can conclude that the receding movement of the contact line of polymer solution drops requires a larger driving force than in the case of water. It must be remarked that, strictly speaking, the Young–Laplace equation should not be applicable even if the radial velocity is zero because the system is out of equilibrium, however this approach is still justified because the timescale of the phenomenon is still much longer than molecular timescales (~ 10 −7 ÷ 10 −9 s [51,52]). Because the advancing contact angle (during drop spreading) is similar for all drops, one can also conclude that polymer solution drops show larger contact angle hysteresis. Contact angle hysteresis around the equilibrium value is generally understood in terms of roughness and/or chemical heterogeneity of the surface [1]. However, more recently it has been proposed the contact angle hysteresis may be caused by a liquid film left behind the contact line during retraction [53–55]. Since both drops of pure water and those of polymer solution impact on identical surfaces, the difference observed in the contact angle hysteresis cannot be interpreted in terms of surface roughness or chemical heterogeneity. Thus, it can be argued that the polymer additive changes either the chemical structure of the surface, or the properties of the liquid film left behind the contact line during retraction, or both.

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Fig. 7. Side views of a water drop and a polymer solution drop (D = ~3 mm) impacting on a Teflon surface from a height of 6 cm (We ≈ 55).

In conclusion, the analysis of base diameter and contact angle data shows that the receding movement of the contact line in dilute polymer solution drops is driven by a larger force than in water drops. This suggests that the receding contact line of dilute polymer solution drops encounters a stronger resistance to movement, which might be due to a force arising outside the drop, in a thin liquid layer left behind the contact line.

Fig. 8. Base diameter and apparent dynamic contact angle of (A) water drops and (B) 200 ppm PEO solution drops impacting on a Teflon surface (equilibrium contact angle: 119°) from a height of 6 cm (We ≈ 55).

4.2. Velocity field inside impacting droplets Most studies on the impact dynamics of dilute polymer solution drops attribute the slowing down of the receding contact line to some bulk property of the fluid (in particular, the elongational viscosity and the normal stresses), as described in detail above. To get a deeper insight about such bulk effects, one can investigate the velocity field inside impacting droplet, obtaining a direct experimental picture of the effect of polymers. Recent particle velocimetry measurements inside impacting drops [56–58] showed that the local velocities measured during expansion and retraction are similar for the drops of polymer solution and for those of pure water. In these experiments, drops were seeded with fluorescent colloids (2 μm diameter), with a concentration of about 0.001 wt.%. Drops were released from a height of 100 mm, corresponding to an impact velocity of about 1.4 m/s. Movies for particle velocimetry were collected at 2000 fps, and illumination was provided by a pulsed laser at a frequency of 8 kHz, resulting in each colloid being exposed four times in each frame along its radial trajectory. The plane of focus was set to a height of about 10 μm above the impact surface. A linear fit to each sequence of four images of a same particle was extrapolated back to the point of intersection, to estimate the position of the drop center and corresponding radial distance of each particle, within a reasonable error. Velocity at each radial position (i.e., the Eulerian velocity field) was calculated as the distance between two images of the same particle divided by the time interval between two laser pulses. These experiments demonstrated that the velocity fields in the two drops are similar both qualitatively and quantitatively, during the inertial expansion as well as during the drop retraction. For example, Fig. 9 shows that the presence of polymer molecules does not induce significant changes in the velocity magnitude at different radial positions in the expanding drop. The only noticeable difference is the smaller error bar of measurements in the polymer solution drops, which indicates a smaller amplitude of velocity fluctuations. The velocity gradient in the fluid during drop spreading, obtained from the slope of these velocity profiles, is also similar; this quantity gives an indication of the rate of deformation of fluid elements within the drop hence the effects of the elongational viscosity. During retraction, radial velocity gradients are small (≪ 1000 s −1) in both cases, and the fluid elements are in compression rather than extension, making the stretching of molecules in the drop interior unlikely. A comparison of the fluid velocity in the bulk of the droplet during retraction with the velocity extracted from macroscopic observations of the contact line shows a dramatic difference between water and PEO drops. Fig. 10 shows that the velocity of the contact line for droplets of pure water at the onset of retraction is similar to that of the bulk fluid. By contrast, the motion of the contact line for PEO drops is one order of magnitude slower than that of the corresponding

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Fig. 10. Velocities measured in the drop at the beginning of the retraction stage. While the fluid velocity (open symbols) is similar for water and polymer solution drops, the contact line velocity (filled symbols) is rodersi of magnitude different. Data re-plotted from Ref. [56].

Fig. 9. Velocity profiles inside impacting droplets of (A) water and (B) a 200 ppm PEO solution, measured at different times after impact on a fluorinated surface with equilibrium contact angles of about 105° from a height of 10 cm [58].

bulk velocity measurements, and further confirms that the difference between the behaviors of the two fluids occurs only at the droplet edge. 4.3. Polymer dynamics near the contact line during drop retraction The nature of the dissipative phenomena arising near the contact line during drop retraction was revealed by visualization experiments aiming at the direct observation of the contact line morphology at microscopic scale, and of the polymer conformation [56,57]. Fig. 11 shows three images of the retracting contact line, at different magnifications, comparing drops of pure water with drops of a 200 ppm PEO solution, both impacting on a Parafilm surface from a height of 6 cm (We ≈ 55). At all magnifications, the polymer solution drop exhibits a more irregular contact line, whereas the water drop has a smooth edge; in particular, the highest magnification image of the polymer solution drop clearly shows a finger protruding in the direction opposite to the contact line movement, exactly as if it was pulled out by an external resisting force; in addition, one can observe several microscopic protrusions. These fingers and protrusions persist for several frames at a time before disappearing. To study the dynamics of polymer molecules during drop retraction, hence shedding light on their effect on the velocity of the receding contact line, fluorescent λ-DNA was added to the impacting drop and observed through an optical microscope equipped with image intensifier [56,57]. λ-DNA is a linear biopolymer with a random coil conformation, with a diameter of about 1.4 μm, although its stretched length is about 22 μm [59] and thus visible using a fluorescent microscope. Using a drop height of 20 mm and focusing about 2/3 of the way to the maximum spreading radius, the retraction of a droplet

was observed. Movies of DNA solutions were collected at a frame rate of 1000 fps with an exposure time of 400 μs. After the passing of the contact line, stretched DNA molecules can be observed on the substrate, oriented in the direction perpendicular to the contact line, as shown in Fig. 12. This bears strong similarities with other DNA stretching methods, such as molecular combing or air blowing techniques [12]. In these techniques DNA molecules are stretched using combination of hydrodynamic and surface forces arising when a liquid meniscus moves on a solid surface. For example, in molecular combing such meniscus is created by slowly pulling out a plate from a solution containing DNA. The same conditions occur when an impacting droplet retracts on the target surface after maximum spreading, the only difference being that this process is orders of magnitude faster than molecular combing, where the typical velocity of the meniscus is 0.2 mm/s. This mechanism is illustrated qualitatively in Fig. 13. The ensemble of polymer molecules stretching as the drop edge sweeps the surface provide the dissipative force necessary to slow down the displacement of the contact line. This can be interpreted, from a macroscopic point of view, as an additional, dissipative force acting on the contact line and opposed to its movement, or an effective contact line friction. This also explains the reduction of the dynamic contact angle observed in experiments: to overcome the action of polymer molecules on the contact line, the horizontal component of the surface force driving the droplet retraction must be larger than in a Newtonian fluid, therefore the apparent dynamic contact angle must be smaller. 5. Conclusions The impact of dilute polymer solution drops on hydrophobic surfaces represents an interesting case of dynamic wetting of complex liquids, which stimulated the debate within the scientific community, often raising controversial interpretations. After the initial discovery that very small amounts of polymer additives cause a dramatic reduction of the receding contact line, which prevents drop rebound on hydrophobic surfaces, attempts to explain this phenomenon as an effect of the elongational viscosity of the fluid became rapidly popular. However, although experiments showed there is an empirical correlation between the velocity reduction of the contact line and the elongational viscosity of the fluid, it was

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Fig. 11. Detail of the contact line during drop retraction on a Parafilm surface (equilibrium contact angle: 105°) after impact from a height of 6 cm (We ≈ 55), at different magnifications.

not possible to demonstrate also the existence of a cause–effect relationship between them. According to another theory, the cause for the contact line velocity reduction was to be sought in a dissipative effect of normal stresses arising in the non-Newtonian droplets. A careful analysis of such theory, unfortunately, reveals a number of serious flaws. More recently, the combination of different experimental results led to propose a more realistic picture of the physical mechanism underpinning the dynamic wetting of dilute polymer solution droplets impacting on hydrophobic surfaces. In particular, these results demonstrated that the receding contact line is slowed down by a force, in the direction opposed to the contact line movement, which arises in the liquid film left behind the drop edge during retraction. This force is due to the collective action of polymer molecules that stretch as the contact line moves on the surface, in a similar fashion to molecular combing.

Fig. 12. Image of the impact surface after the transit of the receding contact line of a drop containing fluorescent DNA mlecules, showing stretched DNA molecole uniformly distributed and parallel to the contact line [56]. The white reference bar corresponds to the length of a fully stretched DNA molecule. Copyright (2010) by The American Physical Society.

It is important to remark that this approach, where the main action of polymer molecules on the contact line is located outside the drop, is substantially different from previous theories, which attempted to explain the peculiar behavior of dilute polymer solution drops in terms of bulk properties of the fluid (either the elongational viscosity or the normal stresses), therefore sought the physical origin of the phenomenon within the volume of the droplet. The wide credit obtained by the elongational viscosity and normal stresses theories, despite their obvious flaws, within the scientific community, deserves some further comments. In particular, in one case there seem to be inconsistencies between sets of data used in support of the theory [38,39], while in another case data obtained from a different physical system were used [49]. It is also important to emphasize once more the distinction between causes and effects. The elongational viscosity, normal stresses, and the slowing down of

Fig. 13. Tentative model for the dissipative force arising at the contact line during drop retraction: as the meniscus recedes, polymer molecules in the liquid wedge are stretched by molecular combing.

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the contact line, are all consequences of polymer molecules stretching in dilute polymer solutions, however there are no cause–effect relationships among them. Finally, one can identify at least two issues that require further investigations. The first arises when comparing the polymer molecule stretching behind the contact line and molecular combing experiments, where one end of the polymer chain is grafted on the surface; thus, it appears necessary to study the polymer–solvent–surface interactions at short timescales in order to understand how the polymer chain attaches in one or more pints to the surface. The second concerns the role of the film left behind by the receding contact line, which is often neglected in wetting theories of simple liquids, but seems to be of fundamental importance in case of complex fluids. Acknowledgments This work would not have been possible without the invaluable contribution of Dr. Michael Smith. Financial support from the Engineering and Physical Sciences Research Council, United Kingdom (EP/E005950/1) is also gratefully acknowledged. References [1] de Gennes P-G. Wetting: statics and dynamics. Rev Mod Phys 1985;57:827–63. [2] Nieh S-Y, Ybarra RM, Neogi P. Wetting kinetics of polymer solutions. Experimental observations. Macromolecules 1996;29:320–5. [3] Fondecave R, Brochard Wyart F. Wetting laws for polymer solutions. Europhys Lett 1997;37:115–20. [4] Ybarra RM, Neogi P, MacElroy JMD. Osmotic stresses and wetting by polymer solutions. Ind Eng Chem Res 1998;37:427–34. [5] Ghannam MT. Wetting behavior of aqueous solutions of polyacrylamide over polyethylene substrate. J Chem Eng Data 2002;47:274–7. [6] Kajiya T, Nishitani E, Yamaue T, Doi M. Piling-to-buckling transition in the drying process of polymer solution drop on substrate having a large contact angle. Phys Rev E 2006;73:011601. [7] Poulard C, Damman P. Control of spreading and drying of a polymer solution from Marangoni flows. Europhys Lett 2007;80:64001. [8] Dolinnyi AI. Transition from incomplete to complete wetting of solid surface in polymer–solvent system: 1. The influence of chain length and force of attraction between chain units and the wall. Colloid J 2000;62:273–83. [9] Dolinnyi AI. Transition from incomplete to complete wetting of solid surface in polymer–solvent system: 2. The regime of strong adsorption. Colloid J 2001;63: 540–9. [10] Dolinnyi AI. Transition from incomplete to complete wetting of solid surface in polymer–solvent system: 3. The regime of weak adsorption. Colloid J 2001;63: 687–98. [11] Bensimon A, Simon A, Chiffaudel A, Croquette V, Heslot F, Bensimon D. Alignment and sensitive detection of DNA by a moving interface. Science 1994;265:2096–8. [12] Kim JH, Shi W, Larson RG. Methods of stretching DNA molecules using flow fields. Langmuir 2007;23:755–64. [13] Rein M. Phenomena of liquid-drop impact on solid and liquid surfaces. Fluid Dyn Res 1993;12:61–93. [14] Chandra S, Avedisian CT. On the collision of a droplet with a solid surface. Proc R Soc Lond A Math Phys Eng Sci 1991;432:13–41. [15] Rioboo R, Tropea C, Marengo M. Outcomes from a drop impact on solid surfaces. Atomization Sprays 2001;11:155–66. [16] Rioboo R, Marengo M, Tropea C. Time evolution of liquid drop impact onto solid, dry surfaces. Exp Fluids 2002;33:112–24. [17] Yarin AL. Drop impact dynamics: splashing, spreading, receding, bouncing. Annu Rev Fluid Mech 2006;38:159–92. [18] Bergeron V, Martin J-Y, Vovelle L. Interaction of droplets with a surface: impact and adhesion. Fifth International Symposium on Adjuvants and Agrochemicals, 17–21 August, Memphis, Tennese, USA; 1998. [19] Bergeron V, Martin J-Y, Vovelle L. Utilisation de polymères comme agents anti-rebond dans des formulations mises en oeuvre en milieux aqueux. French Patent Application 9810471 (1998). International Extension PCT/FR99/02002; 1999. [20] Vovelle L, Bergeron V, Martin JY. Use of polymers as sticking agents. World Patent WO 0008926; 2000. [21] Teraoka I. Polymer solutions: an introduction to physical properties. New York: John Wiley & Sons, Inc.; 2002. [22] Yamakawa H. Modern theory of polymer solutions. New York: Harper & Row; 1971. [23] Beiley FE, Koleske JV. Poly(ethylene oxide). New York: Academic Press; 1976. [24] Cheremisinoff NP. Handbook of engineering polymeric materials. New York: Marcel Dekker; 1997.

11

[25] Glass JE. Adsorption characteristics of water-soluble polymers. II poly(ethylene oxide) at the aqueous–air interface. J Phys Chem 1968;72:4459–67. [26] Kalashnikov VN, Askarov AN. Relaxation time of elastic stresses in liquids with small additions of soluble polymers of high molecular weights. J Eng Phys Thermophys 1989;57:874–8. [27] Phan-Thien N. Understanding viscoelasticity. Berlin Heidelberg: Springer-Verlag; 2002. [28] Bird RB, Armstrong RC, Hassager O. Dynamics of polymeric liquids. New York: Wiley; 1987. [29] Joseph DD. Fluid dynamics of viscoelastic liquids. New York: Springer-Verlag; 1990. [30] Trouton FT. On the coefficient of viscous traction and its relation to that of viscosity. Proc R Soc Lond A 1906;77:426–40. [31] Macosko CW. Rheology principles, measurements and applications. Wiley-VCH; 1994. [32] Sridhar T, Tirtaatmadja V, Nguyen DA, Gupta RK. Measurement of extensional viscosity of polymer solutions. J Non-Newtonian Fluid Mech 1991;40:271–80. [33] Bazilevskii AV, Entov VM, Rozhkov AN. Breakup of an Oldroyd liquid bridge as a method for testing the rheological properties of polymer solutions. Polym Sci Ser A 2001;43:716–26. [34] Fuller GG, Cathey CA, Hubbard B, Zebrowski BE. Extensional viscosity measurements for low-viscosity fluids. J Rheol 1987;31:235–49. [35] Zhang XG, Basaran OA. Dynamic surface tension effects in impact of a drop with a solid surface. J Colloid Interface Sci 1997;187:166–78. [36] Mourougou-Candoni N, Prunet-Foch B, Legay F, Vignes-Adler M, Wong K. Influence of dynamic surface tension on the spreading of surfactant solution droplets impacting onto a low-surface-energy solid substrate. J Colloid Interface Sci 1997;192:129–41. [37] Boger DV, Crooks R. Influence of fluid elasticity on drops impacting on dry surfaces. J Rheol 2000;44:973–96. [38] Bergeron V, Bonn D, Martin J-Y, Vovelle L. Controlling droplet deposition with polymer additives. Nature 2000;405:772–5. [39] Lindner A, Vermant J, Bonn D. How to obtain the elongational viscosity of dilute polymer solutions? Physica A 2003;319:125–33. [40] Bergeron V, Quéré́ D. Water droplets make an impact. Phys World 2001;14: 27–31. [41] Crooks R, Cooper-White J, Boger DV. The role of dynamic surface tension and elasticity on the dynamics of drop impact. Chem Eng Sci 2001;56:5575–92. [42] Cooper-White J, Crooks R, Boger DV. A drop impact study of worm-like viscoelastic surfactant solutions. Colloids Surf A Physicochem Eng Aspect 2002;210: 105–23. [43] Bergeron V. Designing intelligent fluids for controlling spray applications. C R Phys 2003;4:211–9. [44] Williams PA, English RJ, Blanchard RL, Rose SA, Lyons L, Whitehead M. The influence of the extensional viscosity of very low concentrations of high molecular mass water-soluble polymers on atomization and droplet impact. Pest Manag Sci 2008;64:497–504. [45] Rozhkov A, Prunet-Foch B, Vignes-Adler M. Impact of drops of polymer solutions on small targets. Phys Fluids 2003;15:2006–19. [46] Rein M. Interactions between drops and hot surfaces. In: Rein M, editor. Drop-surface interactions, CISM courses and lectures no. 456. New York: Springer; 2003. p. 5–89. [47] Bertola V. An experimental study of bouncing Leidenfrost drops: comparison between Newtonian and viscoelastic liquids.Int J Heat Mass Transfer 2009;52: 1786–93 [Corrigendum, International Journal of Heat and Mass Transfer 58: 652–653 (2013)]. [48] Boffetta G, Mazzino A, Musacchio S, Vozella L. Polymer heat transport enhancement in thermal convection: the case of Rayleigh–Taylor turbulence. Phys Rev Lett 2010;104:184501. [49] Bartolo D, Boudaoud A, Narcy G, Bonn D. Dynamics of non-Newtonian droplets. Phys Rev Lett 2007;99:174502. [50] Bertola V. The effect of polymer additives on the apparent dynamic contact angle of impacting drops. Colloids Surf A Physicochem Eng Asp 2010;363:135–40. [51] Barnes AC, Neilson GW, Enderby JE. The structure and dynamics of aqueous solutions containing complex molecules. J Mol Liq 1995;65–66:99–106. [52] Borodin O, Smith GD. Molecular dynamics simulations of poly(ethylene oxide)/LiI melts. 2. Dynamic properties. Macromolecules 2000;33:2273–83. [53] Chibowski E. Surface free energy of a solid from contact angle hysteresis. Adv Colloid Interface Sci 2003;103:149–72. [54] Chibowski E. Surface free energy and wettability of silyl layers on silicon determined from contact angle hysteresis. Adv Colloid Interface Sci 2005;113:121–31. [55] Chibowski E. On some relations between advancing, receding and Young's contact angles. Adv Colloid Interface Sci 2007;133:51–9. [56] Smith MI, Bertola V. Effect of polymer additives on the wetting of impacting droplets. Phys Rev Lett 2010;104:154502. [57] Smith MI, Bertola V. The anti-rebound effect of flexible polymers on impacting drops. Proc. 23rd European Conference on Liquid Atomization and Spray Systems, Brno, Czech Republic; September 6–8 2010. [58] Smith MI, Bertola V. Particle velocimetry inside Newtonian and non-Newtonian droplets impacting a hydrophobic surface. Exp Fluids 2011;50:1385–91. [59] Smith DE, Chu S. Response of flexible polymers to a sudden elongational flow. Science 1998;281:1335–40.