Biaxial extensional viscous dissipation in sheets expansion formed by impact of drops of Newtonian and non-Newtonian fluids

We investigate freely expanding liquid sheets made of either simple Newtonian fluids or solutions of high molecular water-soluble polymer chains. A sheet is produced by the impact of a drop on a quartz plate covered with a thin layer of liquid nitrogen that suppresses shear viscous dissipation thanks to an inverse Leidenfrost effect. The sheet expands radially until reaching a maximum diameter and subsequently recedes. Experiments indicate the presence of two expansion regimes: the capillary regime, where the maximum expansion is controlled by surface tension forces and does not depend on the viscosity, and the viscous regime, where the expansion is reduced with increasing viscosity. In the viscous regime, the sheet expansion for polymeric samples is strongly enhanced as compared to that of Newtonian samples with comparable zero-shear viscosity. We show that data for Newtonian and non-Newtonian fluids collapse on a unique master curve where the maximum expansion factor is plotted against the relevant effective biaxial extensional Ohnesorge number that depends on fluid density, surface tension and the biaxial extensional viscosity. For Newtonian fluids, this biaxial extensional viscosity is six times the shear viscosity. By contrast, for the non- Newtonian fluids, a characteristic Weissenberg number-dependent biaxial extensional viscosity is identified, which is in quantitative agreement with experimental and theoretical results reported in the literature for biaxial extensional flows of polymeric liquids.


INTRODUCTION
Over the last 15 years, the understanding of drop impact on solid targets has progressed considerably thanks to high-speed imaging methods [1], allowing one to observe in real time the fate of a drop upon impact under various experimental conditions and to probe a rich variety of phenomena, including dynamics of sheets in the expansion and receding regimes, spatio-temporal evolution of the thickness of the sheets [2, 3], fingering instabilities [4][5][6], fractures and production of satellite droplets [7][8][9]. Concerning the nature of the impacting drops, mainly Newtonian fluids of different viscosities have been investigated, although a few relevant studies with shear-thickening fluids [10], shear-thinning fluids [11][12][13][14][15][16][17], yield stress fluids [18] or Maxwell fluids without shear-thinning [19,20]. Most studies have been performed on drops impacting a flat surface that can be smooth or rough [21], horizontal or tilted [22], hydrophobic or hydrophilic [23]. Often, surfaces have a very large size compared to that of the drop such that the entire spreading event occurs on the target [8,14,15,[24][25][26][27], but targets of size comparable to that of the drops [2, 3,16,28,29] or drops impacting only partially a small target [30] have also been studied.
The complex interaction of a drop with a solid surface during drop collision may be removed or at least significantly reduced, by using repellent surfaces, which avoid a direct contact between the liquid sheet and the solid target. Repellent surfaces include superhydrophobic surfaces [31], hot plates above the Leidenfrost temperature [32,33] or sublimating surfaces [20,34]. Nevertheless, the fact that shear viscous dissipation can be neglected during the expansion of the sheet after impact does not mean that there is no viscous dissipation process. Indeed, biaxial extensional viscous dissipation is dominant in freely expanding sheets. Surprisingly this has never been documented to the best of our knowledge, except in a very recent paper [35], where the authors have attributed an inhibition of a drop-substrate contact during drop impact to a large increase of the extensional viscosity.
A possible reason for ignoring the biaxial extensional viscous dissipation is that for sheets expanding completely on a solid surface, viscous dissipation should be dominated by shear. For small targets however, both shear and biaxial extensional viscous dissipation processes may be relevant: this is the goal of a future publication. In the present paper, only biaxial extensional viscous dissipation is relevant as the sheet expands freely thanks to the inverse Leidenfrost effect discussed below. In this work, we investigate the expansion dynamics of free sheets of a viscoelastic thinning fluid produced upon impacting a single drop on a repellent surface in inverse Leidenfrost conditions, and compare it to the respective response of Newtonian fluids. We demonstrate that accounting for the viscous dissipation due to biaxial extensional viscosity during the expansion of the sheet is crucially important. We provide a simple approach to evaluate the biaxial extensional viscosity of thinning viscoelastic fluids, and finally propose a model to quantitatively account for the viscosity dependence of the maximum expansion of sheets. The paper is organized as follows. We first describe the materials and methods. We then show the shear rheological properties of the viscoelastic fluids of interest and their behavior upon impact on a repellent surface. Subsequently, we rationalize the results by accounting for the biaxial extensional viscosity and by means of a simple scaling model. The main conclusions are summarized in the last section.

Materials
We investigate solutions of polyethylene-oxide (PEO) of high molecular weight (M ν = 8000 kDa) from Sigma-Aldrich. Several samples with concentration C between 10 −3 wt% and 2wt% are prepared by adding PEO powder, as received, to the appropriate volume of deionized pure water, or mixtures of water and glycerol (20wt%, 35wt% and 41.66wt% glycerol), and letting the solution under stirring at T = 25 • C for 24 hours at least in the dark, until complete dissolution. Note that, in order to enhance the visualization contrast, all PEO solutions are colored using a Nigrosin dye (From sigma-Aldrich) at concentration 0.025wt%. The surface tension of high molecular weight PEO solutions is independent of polymer concentration (γ = 62mNm −1 ) [36]. Pendant drop experiments confirm that the addition of dye does not affect the surface tension of the final solutions. In addition, in order to compare the behavior of viscoelastic polymer solutions with that of simple fluids, we use two classes of Newtonian fluids, mixtures of water and glycerol and silicon oils. Silicon oils, with shear viscosities from 5.2 mPa.s to 1075 mPa.s and an average surface tension of 20 mN/m [37], are purchased from Sigma Aldrich and used as received. Mixtures of water and glycerol with concentrations ranging from 22 to 97.5 wt% glycerol are prepared, yielding shear viscosities from 1.7 mPa.s to 1910 mPa.s ( depending on the glycerol weight fraction and temperature), densities from 1.05 g/ml to 1.25 g/ml, and an average surface tension of 65 mN/m (as measured with a pendant drop set-up).

Drop impact experiments
To substantially eliminate the role of friction or adhesion with the solid surface on the impact dynamics, we work under inverse Leidenfrost conditions. This is achieved by impacting a drop at ambient temperature T (between 18.5 and 22.5 • C) on a polished quartz slide covered with a thin layer of liquid nitrogen (N 2 ) at T = −196 • C (see fig. 1). The setup is described elsewhere [20]. Upon impact of the drop, a vapor cushion forms at the liquid interface due to the evaporation of N 2 , providing a unique scenario of non-wetting slip conditions that eliminates shear viscous dissipation [34,38]. The rare cases where the impacting drop comes in direct contact with the surface, with instantaneous freezing are eliminated, so the nitrogen vapour film keeps the liquid drop separated from the surface for all the reported experiments. Before each impact, the quartz slide is first cleaned by blowing N 2 gas and then a thin layer (typical thickness 50 nm as measured by ellipsometry) of liquid N 2 is deposited on the slide. The liquid is injected from a syringe pump with a flow rate of 1 ml/min through a needle placed above the target, from the side as shown in Fig.1. The diameter of the falling drop is constant, d 0 = 3.9±0.2 mm, as measured by image analysis and confirmed from the drop mass. The drop falls from a height h = 91 cm, yielding an impact velocity v 0 = √ 2gh = 4.2 ms −1 (g is the acceleration of gravity).The drop impact is recorded from the top (Fig.2a) using a high-speed camera (Phantom V 7:3) operated at 6700 frames/s with a resolution of 800 × 600 pixels 2 . The angle between the camera axis and the horizontal plane is fixed to about 10 • . A second high-speed camera (Phantom miro M310), operated at 3200 frames/s with a resolution of 1280 × 800 pixels 2 , is eventually used simultaneously to record a side view (Fig.2b).

Image analysis
The time evolution of the sheet size is measured with ImageJ software by analyzing top view images. We first subtract the background image from the expansion movie and highlight the rim by a binary thresholding. This allows us to determine the sheets contour and measure its area A. Note that A does not include the fingers emanating from the rim of the expanding film; they may appear for low viscous samples (Fig.2), but do not develop for more viscous ones ( Fig.4). An apparent sheet diameter is π . The results are obtained by averaging for each sample the time evolution of the sheet diameter from three different experiments. Note however that corrections have to be performed for low viscosity , with d Max the maximum diameter measured from top-view images, is plotted as a function of the samples zero-shear viscosity, η 0 , in Fig.2c. We find that the fractional underestimation decreases logarithmically with η 0 , from about 22wt% for the lowest viscosity sample and vanishes for η 0 ≥ 100 mPa.s. Hence, in the following, quantitative corrections are made for the maximum expansion diameter based on the empirical logarithmic law. For samples with a shear viscosity larger than 100 mPa, the sheet remains flat and no correction is needed.

Rheology
Linear viscoelastic and steady shear viscosity measurements are performed with a MCR302 stress-controlled rheometer (Anton Paar, Austria), operating in the straincontrol mode and equipped with a stainless steel cone and plate geometry with a diameter of 50 mm, cone angle of 1 • and truncation of 101 µm. Temperature control (±0.2 • C) is achieved by means of a Peltier element. The linear viscoelastic spectra are obtained by applying a small amplitude sinusoidal strain, such that data are obtained in the linear regime (γ = 10%), varying the angular frequency, ω, from 100 to 0.01 rad/s, and measuring the storage, G (ω), and loss, G"(ω), moduli. The complex viscosity is calculated from the linear viscoelas- tic spectra as |η (ω)| = √ (G 2 (ω)+G" 2 (ω)) ω . The steady shear viscosity, η(γ) is measured by applying a ramp of steady shear rate varying from 0.01 to 1000 s −1 . Figure 3a shows the dynamic moduli as a function of oscillatory frequency for aqueous polymer solutions with various concentrations. For C > 0.6wt%, the crossover of G and G" marks a characteristic relaxation time τ 0 , which is the best estimate for the onset of the terminal regime. Figure 3c shows that τ 0 ∼ C 0.44 s. This scaling exponent is in the range that have been reported for high molecular weight PEO aqueous solutions [39]. Results for samples prepared with mixtures of water and glycerol are consistent with those obtained for pure water samples ( fig. 3c).

Shear Rheology
The zero shear viscosity, η 0 , varies by more than 5 orders of magnitude from 1 mPa.s to 10 5 mPa.s for the samples investigated. The variation of η 0 with polymer concentration, C, reveals the two expected regimes ( fig. 3d): an unentangled regime for C < 0.27 %, where the viscosity increases slowly with the polymer concentration, and an entangled regime at larger concentration, where η 0 ∝ C 4.7 , in agreement with predictions by scaling arguments based on the tube model [40]. Figure 3b depicts the complex viscosity, |η (ω)| as a function of frequency, along with the steady shear viscosity, η(γ) as a function of shear rate,γ. The nice collapse of the dynamic and steady data validates the Cox-Merz rule [41]. We find that all samples are strongly shear-thinning and that an empirical fit by means of the Cross model provides a good description of the shearthinning behavior of PEO solutions (continuous lines in fig. 3b) [42].
Here, η ∞ is the viscosity at very large shear rate that we set equal to the solvent viscosity (water or water/glycerol mixture), η 0 is the zero-shear viscosity (plotted in fig. 3d), n is the shear-thinning exponent and the parameter k is the inverse of a critical shear rate that delimitates a Newtonian regime from a shear-thinning regime. We find that the shear-thinning exponent n increases with increasing C from 0.59 to 0.85. Moreover, for all concentrations the fitting parameter k is monotonically increasing with C, similarly to the characteristic relaxation time τ 0 , and it marks the onset of shearthinning.

Drop impact experiments
Once hitting the repellent surface, the drop expands freely in air until reaching a maximum expansion. It then retracts because of surface tension. This corresponds to an an axisymmetric biaxial extensional flow . The overall behavior is illustrated in figure 4 that depicts snapshots of the drop after its impact for a PEO solution with C = 0.6% (η 0 = 628 mPa.s) and for a Newtonian sample of comparable zero-shear viscosity (η 0 = 658 mPa.s). The two samples display strikingly different behavior: the viscoelastic fluid drop expands much more than the Newtonian drop and moreover forms a thicker rim. More quantitatively, we show ( fig. 5a,b) selective raw data for the time evolution of the effective sheet diameter normalized by the original drop, d d0 , for PEO solutions at different concentrations and Newtonian liquids (here silicon oils, but water/glycerol mixtures exhibit the same behavior) with different viscosities. The origin of time is chosen at the time when the drop hits the liquid nitrogen layer. Expansion and retraction regimes are shown, yielding a bell shape for the curves. We note that the curves for the viscoelastic fluids are very symmetric as opposed to the ones for the more viscous Newtonian fluids, where a very long retraction regime is measured. These findings deserve deeper investigations in the future, but in the following, we focus on the maximum expansion diameter d Max .
For a biaxial extensional flow,during the expansion of free liquid sheets after impact on a repellent surface, the relevant viscosity is the biaxial extensional viscosity defined as η B = σrr−σzż , where˙ is the strain rate, and σ rr and σ zz are the stress tensor components in cylindrical

coordinates.
For a Newtonian fluid the constant biaxial extensional viscosity η B = η 0 B = 6η 0 , where η 0 is the zero-shear viscosity and 6 is the Trouton ratio [43]. As a first order analysis aiming at rationalizing the expansion dynamics of sheets of Newtonian and thinning fluids, we consider the maximum expansion factor, d Max /d 0 , with d 0 the initial drop diameter and plot ( fig. 6a) this quantity as a function of the biaxial extensional viscosity η 0 B . Note that the shear viscosity is measured at the room temperature of the impact experiment, this is of particular importance for water/glycerol mixtures which exhibit a strong temperature dependant viscosity [44]. Similarly to findings for Newtonian fluids impacting a small solid target [19], two regimes are observed for both polymer solutions and Newtonian fluids. At low η 0 B a capillary regime prevails, where the maximum expansion is mainly driven by a balance between surface tension forces and inertial forces, with viscous dissipation being negligible. Hence, this regime is characterized by a plateau. By contrast, in the viscous regime at higher η 0 B , d Max decreases monotonically with increasing viscosity.
To get further insight into the observed behavior, we use the normalized maximum expansion factor,d adopting the same definition as in Refs. [12,19]: where d cap Max is the maximum expansion diameter in the capillary regime (at low viscosity). This normalized quantity allows us to compare drops with different initial sizes and different surface tensions. In addition, to account for different surface tensions for different samples, data are plotted against the biaxial extensional Ohnesorge number, Oh B0 , the ratio of biaxial extensional viscous forces to inertial and surface tension forces: with ρ the sample density and γ the surface tension. Figure 6b shows the evolution ofd with Oh B0 for both polymer solutions and Newtonian fluids. The data for the two types of Newtonian samples overlap nicely onto a master curve and exhibit a capillary regime (for Oh B0 < ∼ Oh c B0 = 2) characterized by a plateau, followed by a biaxial extensional viscous dissipation regime. We find that for the thinning fluids, the onset of the viscous regime takes place at approximatively the same critical biaxial extensional Ohnesorge number Oh c B0 as for Newtonian liquids. Interestingly, however,d decreases much more gradually with Oh B0 as compared to Newtonian fluids in the viscous regime. This clearly suggests the importance of a biaxial extensional thinning of the polymer solutions in the viscous dissipation regime. In the next section, we provide a rationalization for the biaxial extensional expansion dynamics of sheets produced with Newtonian and non-Newtonian fluids.

Rationalization of biaxial extensional thinning
From the data of figure 6b, one can easily define an effective biaxial extensional thinning viscosity η shift B for polymer samples belonging to the viscous regime (Oh B0 > Oh c B0 ) by shifting horizontally the experimental data point so that they fall on the master curve found for Newtonian samples. The shifted values are discussed in

Determination of the pertinent strain rate
Experimentally, measuring properly the biaxial extensional viscosity is a challenging task, especially for relatively low viscosity fluids such as the present polymers [45][46][47][48][49][50][51]. In order to rationalize η shift B , the first step is to properly account for the deformation rate experienced by sheets during their expansion in air. Here, we provide an estimate.
The effective strain rate, defined asε = 1 d ∂d ∂t , is not constant in the expansion regime, but decreases with time and vanishes at maximum expansion ( fig. 8). Note that the film expansion is a time-dependent problem but here we focus on the maximum diameter (end of expansion process) and measure the average rate experienced by the sheet during the expansion . The average value for the strain rate in the expansion regime is then calculated as:ε Here r Max = dMax 2 is te radius of the sheet at its maximum expansion.We get an average strain rate for each solutions that it is used for the rest of the analysis. Note, however, that within experimental errors, ( 15% as shown in the inset of Fig. 8), the effective strain rate does not vary a lot with concentration. Indeed, we obtain a value of 338 ± 49 s −1 by averaging over all samples, including the PEO solutions prepared in glycerol/water mixtures.

biaxial extensional viscosity
Given the difficulty in obtaining reliable experimental data for the biaxial extensional viscosity η B (ε) with our samples, we attempt at providing reasonable estimations To this end, we rely on two pioneering experimental works for the measurements of the biaxial extensional viscosity of viscoelastic solutions [52,53], where similar scaling have been found in spite of using different techniques and different samples, i.e. wormlike micelles in Ref. [52] and concentrated polymer solutions in Ref. [47]. At low Weissenberg numbers (W i =ετ 0 < 1), i.e., for rates lower than the inverse of the terminal relaxation time, τ 0 , the biaxial extensional viscosity is independent of the rate and follows the expectation for a Newtonian fluid: η B = η 0 B = 6η 0 with η 0 the zero-shear viscosity. By contrast, when W i > 1, the biaxial extensional viscosity decreases with rate, η B ∼ε −p with a thinning exponent p = 0.5 [47]. This scaling has been also predicted by Marrucci and Ianniruberto [54] using a tube-based model for polymer melts hence, pointing out the universality of the biaxial extensional thinning behavior.
Based on the similar linear viscoelastic response for the PEO solutions and those investigated in [52,53], we expect our samples to exhibit a similar behavior for the biaxial extensional viscosity as a function of expansion rate. Thus, for each impact experiment in the viscous regime, we define an effective Weissenberg number as: W i eff = τ 0εav , where τ 0 is given by linear shear rheology measurements for the data showing crossover between G and G and extrapolated, according to the power law presented in (fig. 3c), for data without the crossover (0.5wt% and 0.6wt% );ε av is measured from experiments ( fig. 8b). The effective biaxial extensional thinning is characterized by the experimental data points (W i eff , η shift B ), where η shift B are obtained from fig. 7. We show in figure 9 the variation of the normalized effective biaxial extensional viscosity of PEO solutions η shift B /(6η 0 ) as a function of the effective Weissenberg number W i eff obtained from drop impact experiments in the viscous regime. On the same plot, we report the experimental data of Refs. [52,53]. We find a remarkable agreement between our shifted values and those from the literature, even though the expansion sheet dynamics that result from the impact of drops are non-stationary, supporting our simple approach.

Rationalization of the maximal expansion by considering the biaxial deformation
To go one step further, we provide below scaling laws to account semi-quantitatively for the dependence of the maximum expansion on the biaxial extensional viscosity ( fig.7). We restrict our analysis to Newtonian samples. Indeed, for the normalized maximum expansion factor, each non Newtonian sample in the viscous regime, can be replaced by a Newtonian sample exhibiting the same normalized maximum expansion factor a shown before.
We adopt an energy conservation balance and first consider the capillary regime, for which the initial kinetic energy is assumed to be fully converted into surface energy at the maximum expansion of the sheet: where m = ρπd 3 0 /6 is the mass of the drop and d cap Max is the diameter at maximum expansion of the sheet in the capillary regime where viscous dissipation is negligible. In the viscous regime, we need to add to the right hand side of Eq.5 a term accounting for the viscous dissipation, which here is assumed to result only from the biaxial extensional deformation: Combining Eqs.5 and 6, we obtain for the normalized maximum expansion factor,d with E B the biaxial extensional extensional energy dissipated during the process of sheet expansion. To a first approximation, E B can be written as: where V = πd 3 0 6 is the volume of the drop, σ B (ε) = η Bε , and t Max is the time to reach maximum expansion. Hence, assuming also a volume conservation with a uniform thickness sheet , Eq. 8 can be rewritten as: Using a simple first order scaling approach to calculate the strain rate, we consider ∂d ∂t ≈ dMax−d0 tMax and a geometric average between the initial and final values of the sheet diameter, d ≈ √ d 0 d Max , Eq. 9 reads: Here, (dMax−d0) 2 dMaxtMax can be considered as an apparent velocity, v app , close to the initial expansion speed of the drop upon impact. Combining Eqs. 7 and 10 together with the definition of the biaxial extensional Ohnesorge number, one predicts: and v L = γ ρdo is the typical velocity of free oscillations of the drop [55]. We find that the functional form of Eq. 11 reproduces very nicely the experimental data ( fig. 7) which implies that v app is approximatively constant. The best fit of the data (continuous red line) yields, for all concentrations, α = 0.06 ± 0.02 where the error bars are used to obtain the two envelops of the data (dashed black lines). This value is compared to the theoretical expectations for the parameter α. For Newtonian fluids, v app = 2.3 ± 1.2m/s, thus α = 00.25 ± 0.02, Overall, the fit parameter of the master curve is thus found in reasonable agreement with the ones calculated using the different experimental quantities, thus justifying the relevance of our approach.

CONCLUSION
Drop impact experiments on repellent surfaces have been performed with Newtonian fluids and shearthinning polymer solutions. Two regimes for the maximum expansion diameter of freely expanding sheets have been identified: a capillary regime, where the maximum expansion does not depend on viscosity and a viscous regime where the maximum expansion is reduced with increasing viscosity. We have demonstrated that the dominant source of viscous dissipation is the biaxial extensional deformation during sheet expansion, which consequently controls the maximum expansion of the sheets in the viscous regime. We have provided a scaling prediction of the sheet maximum expansion as a function of the biaxial extensional Ohnesorge number, in good quantitative agreement with our experimental results. For viscoelastic thinning fluids, we have proposed a simple approach to measure the biaxial extensional thinning viscosity based on the maximum expansion factor of a freely expanding sheet: the relevant characteristic thinning viscosity, is simply given by the viscosity of a Newtonian fluid with the same normalized expansion factor; it obeys the behavior of the biaxial extensional viscosity of polymeric samples as a function of the Weissenberg number in stationary conditions, providing one considers the mean biaxial extensional rate of the sheet during the expansion regime, as the relevant strain rate for the Weissenberg number. Our approach constitutes a first and crucial step toward the development of a new class of biaxial extensional rheometry tools based on drop impact experiments, but needs further investigations (in current progress) including drops of different diameters , and different impact heights .
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