Fast Imaging Technique to Study Drop Impact Dynamics of Non-Newtonian Fluids

The fast imaging technique can be used to quantify spreading and splashing for various impact scenarios. Figure 4(a), for instance, shows typical impact image sequences for liquid eGaIn with different oxide skin strength. By ejecting eGaIn from the same nozzle and at the same falling height, droplets with reproducible impact velocity V₀ = 1.02±0.12 m/sec and radius R₀ = 6.25±0.10 mm were generated. The left column shows the impact of an air-oxidized eGaIn drop not prewashed in acid. A long tail at the top end of the drop is formed when the fluid detaches from the nozzle. Differing from ordinary liquids, the oxide skin prevents the fluid from freely relaxing the surface energy, so this nonspherical geometry is kept unchanged during the falling stage. After the impact occurs, a thin liquid metal sheet (lamella) expands rapidly along the smooth substrate. Washing the samples in acid reduces the oxide and weakens the skin effect. The middle and right columns in Figure 4(a) show images of drops prewashed in 0.01 M and 0.2 M HCl, respectively. When the acid becomes strong enough to fully eliminate any observable skin effect, eGaIn shows no difference in spreading behavior from ordinary liquids (right column).

In order to characterize the radial expansion after impact, the spreading factor can be defined as P_m = R₀ / R_m, where the maximum spreading radius is R_m. The scaling behavior of P_m under different oxidation conditions is plotted in Figure 4(b) in a conventional way for Newtonian fluids, where Re is the Reynolds number and We* is an effective Weber number that accounts for surface stress induced by the induced skin. Here, the Reynolds number and the effective Weber number for eGaIn are defined on the scale of the entire drop. Particularly, Re = 2V₀R₀/ν where ν is the kinematic viscosity and We* = 2ρV₀² R₀ /σ_eff with ρ as the liquid density and σ_eff as the effective surface tension. ¹⁵ The data nicely collapse onto the classical scaling⁶. This suggests that the spreading of oxidized eGaIn conforms to the energy balance argument used to explain spreading for Newtonian fluids, as long as the elastic energy stored in the skin is accounted for. Generally, no splashing of eGaIn is observed since the surface tension (>400 mN/m) is much larger than in ordinary liquids.

For dense suspensions, the experiments focused on the splash onset. A nonviscous liquid was used as the solvent so that the particle Reynolds number Re_p was always larger than 400. In this regime, viscous dissipation is negligible compared to inertial effects. Figure 5 shows the splashing phase diagram for different particle densities ρ_p and radii r_p. Since the single particle dynamics dominates the impact, both Reynolds number and Weber number are defined on the single particle scale. Namely, Re_p = V₀R_p/ν and We_p = ρ_pV₀² R_p /σ, where R_p is the particle radius. Here, changing the impact velocity varies the particle Weber number We_p. For each point in the plot, the experiment was repeated for 10 times. The red hollow circles are the cases where splash is always found, and the solid blue dots correspond to the situation when no splash is found. The open green squares, however, indicate the scenarios when both splash and no splash are observed in the 10 repeats. In all cases, the transition to splashing happens at the same value of We_p ≈ 14. This is consistent with an argument that the particle-based Weber number is the relevant parameter for the splash onset¹⁶. The insets show representative images of splash and no splash situations. By comparing the results to the splashing transition of Newtonian fluids, a distinctive difference emerges. Conventionally, splashing onset for Newtonian liquids is set by the dimensionless quantity K=We^1/2Re^1/4, where Weber number, We, and Reynolds number, Re, are defined for the entire drop⁷. However, by adding particles into the liquid, an extra length scale, the particle size, is introduced into the system. As a result, in the case where suspensions are as dense as the jamming point, the dynamics of individual particle determines the splashing onset.

One of the distinctive features of dense suspensions is the lace-like structure formed in the aftermath of impact (Figure 6(a)). In order to characterize this new type of instability, the area of the opened holes is quantified through imaging analysis. First, the velocity distribution in the spreading layer can be obtained by using Particle Image Velocimetry (PIV). Then, the yellow rings in Figure 6(a) are defined with particle Weber number We_p = 10, 75, and 920, which all expand radially with time. By imaging analysis, the area of the holes and the total area between each ring are obtained as S_hole and S₀, respectively. The ratio of S_hole to S₀ is plotted against time in Figure 6(b). From the plot, it is clear that the hole-opening instability occurs mostly in the outer regime of the spreading.

Figure 1. Schematic illustration of the imaging setup. The fast camera used for this work can achieve 6,242 frames-per-second (fps) at 1,280 x 800 widescreen resolution; the maximum frame rate is 10⁶ fps at reduced resolution (128 x 8). During the experiment, the drops were slowly extruded from a nozzle by using a syringe pump. The lighting of the system is provided by two white light sources. The front and back lights are used for liquid metal and dense suspension impact, respectively.

Figure 2. (a) Typical Images taken by the camera for liquid eGaIn (left column) and a dense suspension of particles in a liquid (right column). Observation can be performed from both bottom and side. To highlight the object’s profile, the drop is illuminated in a direction perpendicular to the image plane. Specifically, for liquid eGaIn, the drop is backlit to increase the contrast at the liquid/air boundary. For dense suspensions, the sample is lit from the front, such that single particles in the drop can be distinguished. (b) An example of spatial resolution calibration at 10,000 fps. Here, there are 192 pixels across a distance of 1 cm. Thus, the spatial resolution for this figure is 1 cm/192 pixels ≈ 52 µm/pixel. Please click here to view a larger version of this figure.

Figure 3. Image Analysis. For liquid metal drops, we first threshold the images for each frame (see (a)). The average pixel value along a ring at radial position r (see the solid circle in (a)) indicates the location of the spreading boundary. Conventionally, white corresponds to zero and black to one. As a result, the plot of average pixel value (b) shows a sharp transition. The position corresponding to 0.5 gives location of the boundary, where the uncertainty is coming from the width. The moving front is the key parameter for the study of spreading. By contrast, for dense suspension impact, not only the spreading but also the splashing onset is of concern. Panel (c) shows the result from particle tracking of splashing particles, where the yellow tails attached to the particles indicate their trajectories. The plot in (d) gives the trace of particles circled in (c). Since the time step is 1/10,000 sec, the escaping velocity is constant at about 1.5 m/sec, which corresponds well to the impact velocity. Please click here to view a larger version of this figure.

Figure 4. Spreading dynamics of liquid eGaIn. (a) Typical image sequence of eGaIn drops impacting onto a glass substrate (captured by a color-sensitive fast camera. In this case, the spatial resolution is reduced to 59 µm/pixel at 7,600 fps). Drops are initially prewashed in HCl solution as indicated in the text. For all image sequences shown above, the impact velocity was kept at V₀ = 1.02±0.12 m/sec and the initial drop diameter was R₀ = 6.25±0.10 mm. (b) Capillary to viscous transition for impact behavior of eGaIn drops prewashed with different acid concentrations. The dimensionless parameter K=We*/ Re^4/5 is used to collapse all the data. Please click here to view a larger version of this figure.

Figure 5. Splash onset Weber number We_p as a function of particle radius r_p and density ρ_p. The red hollow circles are the cases where splash is always found, and the solid blue dots correspond to the situation when no splash is found in 10 successive repeats. The open green squares indicate the scenarios when both splash and no splash are observed in the 10 repeats. The inset plots are typical images of splashing and nonsplashing cases. Please click here to view a larger version of this figure.

Figure 6. Instability in suspension spreading dynamics. Panel (a) shows a typical image during the impact. During spreading, holes open between particle clusters due to the velocity gradient in the monolayer. The three yellow rings in the image indicate the radial positions corresponding to different particle Weber numbers (We_p=10, 75, 920). (b) The ratio of the area of holes (S_hole) to the total area (S₀) between each ring. S_hole/S₀ is plotted against time, t.