Lifting a sessile oil drop from a superamphiphobic surface with an impacting one

Approach

In our experiments, a sessile oil drop is gently positioned on a superamphiphobic surface and then impacted with a second identical oil drop (Fig. 1A). The superamphiphobic surface is composed of a ~20-μm-thick layer of templated candle soot (10, 47). Candle soot consists of a porous network of 50± 20–nm–sized carbon nanobeads. Making use of chemical vapor deposition of tetraethyl orthosilicate catalyzed by ammonia, a ~25-nm-thick layer of silica is deposited over the porous nanostructures to increase the mechanical stability of the fragile network (Fig. 1A-i and fig. S1). The soot-templated silica network is fluorinated with trichloroperfluoroctylsilane to lower the surface energy, producing a superamphiphobic surface that repels water and most oils. As a model oil, we use hexadecane for its low surface tension, low volatility, homogeneous properties, and Newtonian behavior. A drop of hexadecane (Fig. 1A-ii) exhibits an apparent contact angle of Θ^app = 164 ° ± 1°, an apparent receding contact angle of Θrapp=158°±3°, and an apparent advancing contact angle of Θaapp≈180° (48), as determined by confocal microscopy (Fig. 1A-iii and figs. S2 and S3). Low lateral adhesion of hexadecane is confirmed by measuring a low roll-off angle of α = 3 ° ± 2° (49).

For our drop impact studies, a sessile drop of hexadecane is gently placed on this superamphiphobic surface with a needle connected to a syringe pump (dosing rate, 2 ml/hour). When gravity exceeds the drop-needle adhesion, the drop releases from the needle; this results in a drop volume of V ≈ 3 μl (Fig. 1A). This volume corresponds to a Bond number of 0.3 (Bo = ρ_lgR²/γ, where ρ_l is the density of the liquid, g is the gravitational acceleration, and R is the radius of a spherical droplet of identical volume). The Bond number relates gravitational forces to surface forces, reflecting how gravity affects the shape of the sessile drop. Note that a low Bond number implies that a spherical cap can describe the drop. However, it does not provide insight on whether the drop passes the Cassie-to-Wenzel transition. Still, the shape of the drop is important as it forms the initial condition for the numerical simulation. To calculate and confirm this shape numerically, we solved the Young-Laplace equation−∂P′∂Xi+(κ−Δρ Bo Z)δsni=0(1)

In Eq. 1, P′ refers to the reduced pressure [as defined in (50)], X_i refers to the coordinate system unit vector, Bo Z is the gravitational potential, κ is the curvature of the liquid-gas interface, Δρ is the normalized density difference across this interface (nondimensionalized with ρ_l), δ_s is the Kronecker delta function (1 at the interface and 0 otherwise), and n_i is the unit vector normal to the interface. Note that all equations in this manuscript are written using the Cartesian tensor notation. The shape of the drop is calculated by solving Eq. 1 and matches well with the experiments (Fig. 1A-ii).

The control parameters of the drop collision, determining the outcome, are the Weber number (We), which is related to the impact velocity (U₀), and the impact parameter (χ), which describes the offset from head-on alignment of the two colliding drops. The impact velocity, U₀, is controlled by positioning the needle to a defined height (Fig. 1A). The corresponding Weber number, We=ρlU02R2/γ, compares fluid inertia and surface tension, where ρ_l = 770 kg/m³ is the density of the hexadecane and γ = 27.5 mN/m is the surface tension. In our experiments, the Weber number ranges from 0.02 to 9. The substrate is then translated laterally to position the drop in the X and Y directions. At an identical dosing rate, a second drop is released with an identical volume, V ≈ 3 μl, and impacts the sessile drop. Two high-speed cameras are perpendicularly positioned to capture the dynamics of the drops in the X, Y, and Z directions. The impact parameter of the two drops is given by the ratio [χ = d/(2R)], where d is the horizontal difference of the center of masses of the impacting drop and the sessile drop (Fig. 1B). Although we cannot exactly predict the impact parameter beforehand, the two-camera system allows us to precisely measure the offset from head-on alignment by image analysis. χ = 0 describes a perfect head-on collision, whereas χ = 1 corresponds to the situation when the two drops merely brush each other (d = 2R).

Experimental observations

When varying the offset from head-on alignment χ and the Weber number We, six outcomes for the impact dynamics are observed, termed cases I to VI (Fig. 2). The column A images are taken just as the collision starts (t = 0 ms) and are used to quantify the impact parameter χ. Column B is at the point of maximum sessile drop compression, and column C demonstrates the shape of both drop just before they separate or coalesce. Column D illustrates the overall outcome of the collision event. We first consider the outcomes at We ≈ 1.5 while varying χ. For a near-zero χ, case I is observed, which is a head-on collision (Fig. 2, movies S1 to S3, and fig. S4). During impact, both drops deform and spread radially and, as a result, show axial compression. The kinetic energy of the system is transferred to the surface energies of both deformed drops. Moving forward in time, both drops start to retract. The sessile drop transfers energy back to the impacting drop in the form of kinetic energy. Upon completion of the collision, the impacting drop bounces off while the sessile drop stays on the substrate. The sessile drop oscillates, hinting that it retains a part of the energy gained during impact. For a slightly higher offset, χ ≲ 0.15, case II is observed (movies S4 to S6 and fig. S5). The initial collision is similar to case I in that the drops collide, followed by vertical compression and lateral spreading. However, unlike case I, the deformations are no longer symmetric, and the sessile drop also lifts off the surface. The displacement for either drop with respect to the center of mass of the initial sessile drop is in opposing lateral directions. Further increasing of the offset from head-on alignment to χ ≲ 0.5, the impacting drop glides over the sessile drop and rolls on the substrate, as illustrated by case III (Fig. 2, χ = 0.24; movies S7 to S9; and fig. S6). Unlike cases I and II, no rebound of the impacting drop is observed. Instead, the sessile drop lifts off the surface. As the impact parameter is increased even further (χ > 0.5, case IV), the impacting drop still rolls over the sessile drop (movies S10 to S12 and fig. S7). However, during retraction, the impacting drop rebounds from the surface while the sessile drop moves along the surface.

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In the above cases I to IV, the Weber numbers were kept constant at We ∼ 1.5, while the offset was varied. However, the outcome of the impact event also varies with the Weber number. To provide a better intuition on how both χ and We affect the observed outcomes, we plot our data as a phase diagram (Fig. 3). When the Weber number is increased above We ≥ 6, regardless of the impact parameter χ, we find coalescence of the two drops, as illustrated in case V (Fig. 2, movie S13, and fig. S8). In this regime, the air layer between the drops is unstable, which results in direct contact and subsequent coalescence. The coalesced drop reaches a maximum spreading diameter during impact (column C in Fig. 2). During retraction, the drop elongates vertically and ultimately detaches from the surface. Occasionally, drops coalesce without subsequent bouncing (case VI, movie S14 and fig. S9). Although this outcome is rarely observed and likely caused by surface defects, we present this result for the sake of completeness to demonstrate all observed outcomes. Moreover, to consider the generality of the scenarios presented for oil-on-oil drop impact, we also tested water-on-water drop impact. Similar scenarios are observed, as illustrated in fig. S10.

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Direct numerical simulations

Although the experimental observations consistently illustrate how We and χ dictate the observed impact outcomes, they lack detailed information on the velocity fields and on how energy is transferred between both drops. To ascertain this information, we ran DNS and compared these results with our experimental data. For simulating noncoalescing droplets, we use geometric volume of fluid (VoF) (51) method with two distinct VoF tracers (see the Simulation methodology section for detailed discussions and implementation). This formulation ensures that drops cannot coalesce, reflecting the experimental situation where a finite air layer between the drops is preserved throughout the process.

We first ran four simulations choosing We and χ values within the regimes for cases I to IV, as denoted by open symbols in Fig. 3. The results are displayed in Fig. 4. The normalized times (t* = t/t_γ, where t_γ is the inertial capillary time scale, (ρR3)/γ) correspond to the stages of the process, as described by columns A to D in Fig. 2. As is evident from the top rows (orange drops), the simulations reproduce the general collision outcomes consistent with the snapshots of the impact dynamics (Fig. 2). Moreover, the DNS allow for quantifying the velocity vector fields for each of the cases (Fig. 4, bottom rows). These vector fields, combined with a calculation of the energy budget, render it possible to quantitatively explore the dynamics of the oil drop-on-drop collision process. To account for the kinetic energy (E_k), gravitational potential energy (E_p), surface energy (E_s), and dissipative losses (E_d), we numerically calculated the total energy of the system as E=Em+Es+Ed(2)

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In Eq. 2, the total mechanical energy E_m = E_k + E_p, the surface energy E_s, and the energy dissipation E_d are calculated using a method similar to the one developed by Wildeman et al. (52). E_k includes the kinetic energy of the center of mass as well as the oscillation and rotational energies obtained in the reference frame that is translating with the center of mass of the individual drops. The details of these calculations are provided in Materials and Methods.

While keeping the Weber number at We ∼ O(1), the cases appear in order from I to IV with increasing offset position from head-on alignment χ. For all cases, the energy is initially contained in the mechanical energy of the impacting drop (i.e., its kinetic and potential energy) and the surface energy of the sessile drop. To describe the system energy of the DNS results presented in Fig. 4, we plot the full energy balances for each case in Fig. 5. For comparison convenience, the energies in Fig. 5 are normalized with this initial energy of the system.

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Let us consider first a head-on collision where χ = 0 (Figs. 4A and 5A and movies S2 and S3, case I), which is defined by a symmetric configuration. First, the momentum is transferred from the impacting drop to the sessile drop, as the sessile drop deforms. This transfer results in deceleration of the impacting drop. Moreover, the kinetic energy of the impacting drop transforms into the surface energy of the system. This transfer continues until t* = 1.84 (Fig. 4A: column B) when the deformation in the two drops is maximum. Even at the moment of maximal elongation of both drops, the kinetic energy remains finite because of rotational flow within the drops (Fig. 4A, column B, velocity field) (52). The mechanical energy passes a minimum (t* = 1.84) when the surface energy is maximal. For t* > 1.84, the surface energy of the two drops is converted back into kinetic energy. Retraction of the sessile drop is hindered by the impacting one (Fig. 4A, column C), directly sitting on top of it. As a result, the sessile drop cannot lift off from the substrate, but it releases any extra energy by oscillations (movies S1 and S2). During impact, the drops lose approximately 20% of their initial energy through viscous dissipation inside the drops and the thin air layer between them (Fig. 5A). This dissipation occurs mainly during the initial stages of the process (t* < 3). It should be noted that the surface tension (γ), viscosity (μ), and impact velocity (U₀) all affect viscous dissipation (46). These properties are related to the Ohnesorge number (Oh=μ/ργR≈0.03), which compares viscous forces to inertia and surface tension forces, and the Weber number, We=ρU02R/γ∼O(1) [see Eq. 16 and (53)]. The dissipation observed in our case is lower than that reported previously for a single drop impact at comparable Oh and We on superhydrophobic (52) and superamphiphobic substrates (46). In the case of a single drop impact, the velocity of the drop goes to zero quickly as it approaches a rigid substrate (54), leading to high dissipation close to the substrate (in the thin air layer and near the contact line). In the case of drop-on-drop impact, the sessile drop is deformable, decreasing the deceleration experienced by the impacting drop. As a result, the system retains almost 80% of its initial energy in the form of mechanical and surface energy of the drops.

For slightly off-center collisions where χ = 0.08 (Figs. 4B and 5B and movies S5 and S6, case II), the initial collision is similar to case I; the drops collide, followed by vertical compression and lateral spreading. However, unlike case I, the impacting and the sessile drops lift off from the substrate. This feature results from the loss of axial symmetry of the velocity field for χ > 0. During retraction, transfer of momentum from the compressed sessile drop back to the impacting drop occurs mainly along a vector pointing normal to the apparent contact zone. Moreover, the sessile drop attempts to regain its spherical shape (minimum surface energy state). As a result, the velocity field of the sessile drop is almost parallel to the contact zone, i.e., pointing to the upper left. These opposing orientations of the velocity fields cause the impacting drop to bounce off the sessile drop and the sessile drop to lift off from the substrate (see the velocity vector fields in Fig. 4B and movie S6). Viscous dissipation increases compared with a head-on-collision but still is maximum during the initial stages of the process (t* < 3.5; Fig. 5B).

As the offset is further increased to χ = 0.25 (Figs. 4C and 5C and movies S8 and S9, case III), the impacting drop glides over the sessile drop (facilitated by the thin air layer), and sufficient energy is transferred to lift the sessile drop from the substrate. This can be understood from the interplay of the velocity field and the contact time (Fig. 4C and movie S9). The relatively large offset from head-on alignment causes the averaged velocity field of the restoring impacting drop to point both almost parallel to the surface and downward, while the velocity field of the sessile drop is pointing upward. The large deformations of both drops are reflected in the evolution of the surface energy (Fig. 5C). The large deformations of both drops also cause an increase in viscous dissipation (E_d); at the end of the process, almost 50% of the initial energy is lost. Moreover, unlike cases I and II, viscous dissipation occurs not only in the drops but also in the thin air layer as the impacting drop approaches the substrate (4).

Last, if the offset from head-on alignment is increased even more to χ = 0.625 (Figs. 4D and 5D and movies S11 and S12, case IV), the time of contact is insufficient to transfer enough energy to the sessile drop for liftoff (55). Moreover, the vector normal to the drop-drop contact area is farthest from vertical as compared with the normal vectors in other cases. That is, it points nearly horizontal. As a result, the sessile drop rolls along the substrate and the impacting drop instead rebounds from the surface, resembling typical drop-surface impact. In this case, most of the energy is retained by the impacting drop, as illustrated in Fig. 5D. Similar to case III, viscous dissipation accounts for almost 50% of the initial total energy. Although in cases I and IV the impacting drop rebounds while the sessile drop remains on the surface, we discriminate between both cases. For case I, the vector fields are symmetric around the X = Y = 0 axis, whereas for case IV, the vector fields are highly asymmetric, and the sessile drop rolls along the surface. Furthermore, in case IV, the impacting drop bounces off the substrate, as opposed to the sessile drop in case I.

These results indicate that the DNS provide a quantitative description of the impact dynamics. At this point, we investigate whether there is a one-to-one match of the experimental data and numerical simulations; this is done by comparing the drop boundaries and experimentally determined mechanical energies with the numerical predictions. Because we cannot exactly predict the impact parameter experimentally beforehand, we adjust the numerical simulations to the experimental data. We achieve a nearly quantitative agreement of the drop boundaries and experimental mechanical energies (Fig. 6). The different snapshots in Fig. 6 (i to iv) refer to the following time steps: (i) just at collision, (ii) sessile drop at maximum compression, (iii) droplet shape just before separation, and (iv) final outcome of the impact. We expect that slight deviations between the experimental and numerically determined drop boundaries result from marginal inaccuracies in the experimental determination of the offset parameter. However, the agreement is remarkably good, keeping in mind that there are no fitting parameters.

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In Fig. 6 (A-v and B-v), we compare the measured experimental mechanical energies (data points) with those calculated using simulations (dotted lines). The calculated mechanical energies exceed the experimentally determined energies. To understand the origin of this discrepancy, one needs to consider that experimentally we are only able to measure the vertical and horizontal displacements to approximate the mechanical energy of each drop. The image analysis does not offer an easy route to quantify the contribution of the rotational and oscillation energies that are included in the numerically calculated mechanical energy, E_m. Therefore, to test whether neglecting the rotational and oscillation energies in our experiments causes the discrepancy, we calculated the center of mass mechanical energies (EmCM) for the two drops numerically (Fig. 6, A-v and B-v; see Materials and Methods for a detailed discussion). The zero of the potential energy (EpCM=0) refers to the center of mass of the sessile drop at t = 0. This implies that EpCM of the sessile drop becomes negative during compression. The center of mass kinetic energy (EkCM) is added to this value to get EmCM, namely, EmCM=EkCM+EpCM. As illustrated in Fig. 6 (A-v and B-v), the numerical results (solid lines) now nearly overlay the experimental results (data points). This holds for the temporal development of the energy for both the sessile drop as well as the impacting drop. We suppose that the small discrepancies may arise from finite adhesion of the sessile drop to the substrate (which is not accounted for in the simulations). An additional source of error may arise from the selection of time t = 0. We choose t = 0 based on the time instant when the sessile drop starts to feel the presence of the velocity field of the impacting drop, i.e., when the center of mass kinetic energy of the sessile drop becomes nonzero. Nevertheless, the remarkable agreement between the experimental and numerical results for the center of mass mechanical energies illustrate that the DNS are able to describe the oil drop-on-drop impact physics; this allows for quantifying the contribution of the rotational and oscillatory energies.