Upon running experiments for each concentration of IPA with five trials, and processing the image frames, we begin to elucidate the details of the droplet imbibition process. As the nylon surface is hydrophobic for DI water, no absorption is observed for the water droplet regardless of the droplet size. With the increase in the IPA concentration, reduction in surface tension leads to improvement of the wettability of the solvent mixture on the nylon surface, as indicated by the apparent contact angle in Table 1. When the volume ratio of IPA exceeds 10%, the mm-sized droplet starts to imbibe into the powder substrate. However, no absorption is observed for µm-sized droplet until IPA volume ratio reaching 60%, despite of partial wetting of the IPA-water mixture on the nylon surface. Our experiments show that the critical surface tension (above which no absorption can take place for the nylon powder) is 46.25 and 26.30 mN/m for mm- and µm-droplet, respectively.
A typical absorption process for mm- and µm-droplet after landing on the nylon powder surface is shown in Fig. 4. The 2.11 mm-droplet of 60% IPA solution spreads and oscillates for 24.5 ms before reaching a stable profile. The droplet is then absorbed by the powder layer after 105 ms. The experimentally observed oscillation period is in good agreement with the capillary time τc = (ρD03/σ)0.5 = 21.3 ms. The profile of mm-sized droplet is far from the spherical cap during the initial oscillation period as shown in Fig. 4a, whereas the µm-droplet shows insignificant oscillation after impact, as shown in Fig. 4b. We also note that despite of low impact velocity (0.45 m/s) for mm-droplet there is ejection of powder particles caused by the impact of the droplet, which can be seen snapshots from 10.5 ms to 24.5 ms in Fig. 4a. Such powder ejection is not observed for µm-droplet in our experiments. However, the ejection of powder particles can occur for µm-droplet with high impact velocity, as reported by Parab et al. (Parab et al. 2019) where 30-pL droplets with U0 = 8 m/s were in their experiments.
To understand the evolution of the droplet profile during absorption in a quantitative way, the time histories of the contact angle θ and contact radius Rc are plotted in Fig. 5 and Fig. 6, respectively. For mm-droplets θ decreases nearly linearly with respect to time during absorption, whereas for µm-droplets θ varies in a narrow range from 450 to 350, as shown in Fig. 5 (a) and (b). For both for mm- and µm-droplets, Rc initially increases as the droplet spreads over the powder surface and then reduces due to absorption. We notice that the data of Rc for µm-droplets are more scattered than that of the mm-droplets of different solutions, which is likely due to the fact the small length scale for µm-droplets makes the absorption process more susceptible to non-uniformity of the morphology of the powder particles.
Our data indicate the contact line is not pinned during absorption for both mm- and µm-droplets. However, there is a concern that the particles of nylon powder may partially obstruct the view of the droplet (Susana et al. 2012). In addition, there has been a debate on whether one can assume the three-phase contact perimeter is pinned during absorption (Hapgood et al. 2002; Marmur 1988). It is conceivable that decreasing droplet height, combined with the powder particles, may partially obscure the true contact line, making the circumference appear to shrink over time. To shed more light on this question, trials were run with the camera at an angle of 25° to the plane. Figure 7 shows the absorption process of the mm-sized droplet of pure IPA solutions captured by angling the camera. The contact diameter of the droplet is clearly shown to shrink over time, meaning that the contact line is not pinned during absorption process. Figure 7 also helps explain a discrepancy we found in the image processing method. At time = 6.5 ms, the middle of the droplet forms an annular trough, but in Fig. 4 the shell integration is unable to account for this ring-shaped cavity because when observing directly from the side, the void space is not discernible. Therefore, volumes measured during the first 10 milliseconds for mm-droplets will be slightly larger than the true droplet volume.
Our image processing code calculates the droplet volume in each frame for each trial. By subtracting the current volume from the starting volume at each data point, we can infer the volume of fluid absorbed into the powder over time for mm- and µm-droplets, as shown in Fig. 8 (a) and (b), respectively. Figure 8 plots the average of five trails with standard deviation for each fluid concentration. The absorption time of mm-droplets is on the order of 100 ms. We find that 10% IPA solution takes much longer to absorb because of much larger contact angle compared to other solutions. For µm-droplets, it is no surprise that the absorption process is very fast and typically less than 1 ms. The 90% IPA solution takes longer time to absorb because the inkjet device dispenses a larger volume for 90% IPA solution than all the other IPA-water mixtures. It is also clear from Fig. 8 that the data for µm-droplets are scattered in a wider range than that for mm-droplets. As the size of the µm-droplets is not significantly larger than the pore size of the powder substrate, the imbibition process of the µm-droplets is greatly affected by the pore structure of the powders beneath the droplets. As a result, minor variation in the surface morphology of the powder substrate leads to relatively large variation in the absorption for µm-droplets.
To gain more insight into the physics governing the absorption process of different solutions for droplets of two distinctly different sizes, we nondimensionalize the data of absorbed volume vs. time using the method discussed in Section 2.3. The dimensionless absorbed volume V*=V/V0 for both mm- and µm-droplets is plotted against the dimensionless time t/tc in Fig. 9. We can see that the absorption data of different liquids approximately collapse into the power law V* ~ τα with the power exponent α = 0.856 and 0.518 for mm- and µm-droplets, respectively. The exponent α in the power law relation is markedly different for microliter and inkjet droplets, therefore there is a distinct difference in the nature of the absorption process between these two droplets. As the pore size in our powder substrate is significantly smaller than the size of our mm-droplets, the imbibition of droplets into the homogenous powder layer is inherently 3D as discussed in Section 2.2. As a result, the scaling exponent of α = 0.856 obtained from mm-droplets in our experiments is reasonably close to the theoretical α = 1.0 as predicted by Eq. (5) for the perfect radial flow in homogeneous porous media. α obtained for our study is also in good agreement with some previous experimental results (Barui et al. 2020; Liu et al. 2017; Oko et al. 2014). Since the µm-droplets dispensed by the inkjet device are not significantly larger than the pore space in the powder substrate, the imbibition process occurs primarily in the thickness direction of the substrate, dominated by viscous and capillary forces. Consequently, the power exponent of α = 0.518 measured from inkjet droplets is very close to the theoretical α = 0.5 for 1D imbibition process (i.e., Eq. (3)).