Macrotextured spoked surfaces reduce the residence time of a bouncing Leidenfrost drop

Upon impact, a drop will spread outward radially from the contact point. This radial expansion occurs on both the flat and the spoked macrotextures, where it is most easily observed from a top–down view (figure 3). We observe that the distance that the drops spread from the contact point, which we refer to as the spreading radius r, depends on the underlying macrotexture. In figure 5, the results of this spreading radius r as a function of time t are plotted for all seven surfaces under the same impact conditions (R  =  1.2 mm, U  =  1.3 m s⁻¹). Here we have normalized the spreading radius r by the initial drop radius R and normalized the time t by the inertial-capillary timescale τ relative to the time of maximum spreading on the spoke t_s. Because we rely on top-view images, we are only able to reliably discern the spreading radius when it is large enough to appear from under the impacting drop, that is r(t)/R  >  1. As is apparent in figure 3(b), there is a significant difference in the spreading radius dynamics between the liquid on the spokes and that bisecting the spokes. In figure 5, the results from each surface are denoted by a different symbol, with the results along the spoke denoted with a closed symbol and the results between spokes denoted by an open symbol. On the flat surface n  =  0, there is no spoke orientation. To illustrate the degree that the dynamics are axisymmetric, the spreading radius in two orthogonal directions are plotted (blue asterisks and line in figure 5) and shown to be nearly identical.

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It is noteworthy that the 14 data sets of the spreading radius r(t,n) fall onto one of three main curves (figure 5). The spreading dynamics on the flat surface (n  =  0) is remarkably similar to the spreading dynamics between spokes (open symbols in figure 5) when there are three or less spokes ($n\leqslant 3$). In these cases, the film spreads to a maximum radius ${{r}_{b}}\approx 2.5R$ before retracting. Meanwhile, the simultaneous spreading on the spokes (closed symbols in figure 5) align on a different curve, reaching a maximum radius ${{r}_{s}}\approx 2R$. The spreading on the spokes when n  >  3 follows the same dynamics as when $n\leqslant 3$; however for these surfaces, the spreading between the spokes (open symbols) follow a third curve that continues to rise throughout the experiment.

The fact that the spreading between the spokes for $n\leqslant 3$ is similar to the spreading dynamics without any spoke suggests that the spokes have minimal influence in setting up the shape of the first of these three main curves, but a significant influence on the other two. The maximum spreading distance r_b has been previously documented for impact on various flat surfaces, and this literature can provide predictions to what might be expected for impact under the Leidenfrost condition. For drops impacting onto wettable surfaces, the total surface energy is reduced as the drop expands, and therefore the contact angle dynamics have a significant role in the extent of this expansion phase [25, 26]. However, for impact on non-wetting surfaces, the surface energy increases as the drop deforms from the initial spherical shape. It is tempting to conclude that the amount that the drop will deform can be calculated from an energy conservation in which all of the initial kinetic energy $\sim \rho {{R}^{3}}{{U}^{2}}$ is converted into surface energy $\sim r_{b}^{2}\gamma$. Such a scaling would predict that maximum spreading normalized by the initial radius would scale as ${{r}_{b}}/R\sim \text{W}{{\text{e}}^{1/2}}$. However, it has been shown that non-wetting drops impacting hydrophobic surfaces follow a different scaling ${{r}_{b}}/R\sim \text{W}{{\text{e}}^{1/4}}$ [27]. This scaling has been rationalized by recognizing that accelerations, such as that due to gravity, will flatten drops to a shape in which the hydrostatic and Laplace pressures balance. By analogy, the rapid deceleration of impact will also flatten the drop to a puddle with thickness h. Following Clanet et al [27], if it is assumed that the drop is decelerated from its initial vertical velocity to zero in time R/U, the resulting acceleration acting on the drop is U²/R. At the interface, the pressure balance between the inertial 'hydrostatic' pressure and the Laplace pressure can be described as:

$$\begin{eqnarray}&&\rho \frac{{{U}^{2}}}{R}h=\frac{\gamma}{h/2}.\end{eqnarray} \tag{ 1 }$$

Independently, conservation of mass of the incompressible liquid requires the flatted drop volume (approximated as a disk) to be equal to the volume of the initial spherical drop

$$\begin{eqnarray}&&\pi r_{b}^{2}h=\frac{4\pi}{3}{{R}^{3}}.\end{eqnarray} \tag{ 2 }$$

Combining these two conservation equations reveals that the maximum radius r_b and thickness h can be expressed as

$$\begin{eqnarray}&&\frac{{{r}_{b}}}{R}={{\left(\frac{8}{9}\text{We}\right)}^{1/4}};\;\;\;\;\frac{h}{R}={{\left(\frac{2}{\text{We}}\right)}^{1/2}}.\end{eqnarray} \tag{ 3 }$$

The expression for the maximum radius r_b is predicted to have a prefactor of $\sqrt[4]{8/9}=0.97$; however, for drops impacting on superhydrophobic surfaces, Clanet et al [27] find that their data is better fit to this scaling with a prefactor of 1.1. This minor adjustment is likely related to limitations associated with the assumptions of a uniform deceleration and a cylindrical final state.

This scaling relation is appropriate for drops impacting superhydrophobic surfaces [28]; however it is less clear whether it is appropriate for the Leidenfrost impact presented here. Specifically, in one past study, the results suggest that the maximum spreading follows the $\text{W}{{\text{e}}^{1/4}}$ scaling [21]; whereas another more recent studies have argued that scaling relations of $\text{W}{{\text{e}}^{2/5}}$ and $\text{W}{{\text{e}}^{3/10}}$ are more appropriate [24, 29].

To determine which scaling relationship best describes the dynamics in our system, we vary the impact velocity of drop onto the flat heated surface (n  =  0) and measure the maximum spreading radius r_b. Figure 6 shows the maximum deformation for impacting drops over a Weber number range between 4–40. It should be noted that the only working fluid considered is deionized water. In the case of higher viscosity flows, the viscous impact drives the system, and the maximum deformation has been shown to be dependent on the Reynolds number [27]. Additionally, there is evidence that the flow of the vapor under the drop can influence drop spreading on Leidenfrost surfaces [24, 30]. The surface that we use has been finished with a cutting tool so that it matches the finish on the other six surfaces. This finish is not optically smooth and therefore may reduce the outward flow of this trapped vapor layer.

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Figure 6 suggests that under Leidenfrost conditions the drop maximum diameter does indeed scale with $\text{W}{{\text{e}}^{1/4}}$. Similar to the results of Clanet et al [27], our data is better fit using a prefactor of 1.1. Relating this expression back to the data shown in figure 5 ($\text{We}=29$), the predicted maximum radius would be r_b/R  =  2.55 when spoke effects are negligible. This value is depicted by a dot-dashed line in figure 5 and is consistent with the maximum spreading of the drop when n3. From mass conservation, the height of the liquid film would be approximately $h=280~\mu$m.

The maximum spreading radius over the spokes r_s is noticeably less than the spreading radius on the flat interface r_b, suggesting that the presence of the spoke is responsible for this decrease. One might suspect that viscous drag could be responsible for the decreased spread over the spoke; however the vapor film under the drop mitigates the shear stress within the liquid film so that viscous effects are likely negligible. Instead, we explore the possibility that the earlier 'hydrostatic' argument could be extended to the thinner, more highly curved film on top of the spoke. Assuming that the drop spreads out to a thickness h, the film on top of the spokes would have a thickness of h  −  a, where a is the spoke thickness (figure 1(b)). The radius of curvature across the spread-out drop on the spoke becomes (h  −  a)/2 so that the balance of inertial and capillary pressures can be expressed as:

$$\begin{eqnarray}&&\frac{\rho {{U}^{2}}}{R}(h-a)=\frac{\gamma}{(h-a)/2}.\end{eqnarray} \tag{ 4 }$$

Combining this equation with the expression for r_b (equation (3)), leads to

$$\begin{eqnarray}&&\frac{{{r}_{s}}}{R}=\frac{{{r}_{b}}}{R}{{\left(1+\frac{a}{h}\right)}^{-1/2}}.\end{eqnarray} \tag{ 5 }$$

It is apparent from this expression that r_s decreases relative to r_b as the spoke amplitude a increases. Substituting the empirical expression for r_b, the spreading radius on the spoke r_s can be expressed as

$$\begin{eqnarray}&&\frac{{{r}_{s}}}{R}\approx 1.1\text{W}{{\text{e}}^{1/4}}{{\left(1+\frac{a}{1.3R}\text{W}{{\text{e}}^{1/2}}\right)}^{-1/2}}.\end{eqnarray} \tag{ 6 }$$

Returning to the data in figure 5—in which $\text{We}=29$ and $a=160~\mu$m—the predicted maximum spreading radius over the spokes is r_s  =  2.05. This value is depicted by a dashed line in figure 5 and is consistent with the experimentally observed maximum spreading over the spokes (figure 5, closed symbols).

The last set of curves in figure 5 corresponds to the spreading radius bisecting the spokes when $n\geqslant 4$ (open symbols). These curves continue to spread beyond the maximum spreading for the flat surface r_b. Reviewing the images in figure 3, it is apparent that the excess liquid is gathering along the bisection so that the thickness is locally greater than h. Therefore qualitatively, both the inertial 'hydrostatic' pressure increases and the interface curvature decreases, leading to the observed, larger spreading radius. A more quantitative analysis of this spreading is complicated by the ligament dynamics and is beyond the scope of this manuscript. The transition in film dynamics between n  =  3 and n  =  4 is explored further in the next section, retraction.

Once the drop spreads to its maximum extent on a non-wetting surface, it begins to retract. Figure 5 illustrates the dynamics of this retraction both on and between the spokes for various number of spokes n. The film on the spoke retracts from the maximum value r_s to the initial radius R over a time $0.5\tau$. Top-down images of the liquid film at these two points in time, t_s and ${{t}_{s}}+\tau /2$, are depicted in figure 7.

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The images in figure 7 give the impression that the film retracts significantly faster along the spoke than off of the spoke. Indeed this difference in speed is used to rationalize the center-assisted recoil of macrotextured surfaces in Bird et al [13]. Capillary forces are expected to retract an inviscid liquid sheet at a velocity V that depends on the initial film thickness h as $V=\sqrt{2\gamma /\rho h}$ [31, 32]. Note that the initially uniform film thickness does not thicken during the retraction, but rather the film remains stationary until engulfed by the retracting rim. This retraction velocity has been used to estimate the contact time of non-wetting drops on flat surfaces with some success [12, 33].

For the macrotextured surfaces explored in this manuscript, the film thickness between the spokes $h\approx 280~\mu$m is expected to be larger than the film thickness over the spokes $h-a\approx 120~\mu$m. Therefore the retraction velocities between the spokes V_b and over the spokes V_s would be expected to follow

$$\begin{eqnarray}&&{{V}_{b}}=\sqrt{\frac{2\gamma}{\rho h}};\;\;\;\;{{V}_{s}}=\sqrt{\frac{2\gamma}{\rho (h-a)}}.\end{eqnarray} \tag{ 7 }$$

A prominent feature visible as the film begins to retract is an angled notch in the film over each spoke. The angle of this notch or wedge α is identified on several of the images in figure 7. Under the experimental conditions, it appears that this angle is independent of the number of spokes n with a value of approximately $\alpha ={{120}^{\circ}}$. As the film over the spoke retracts, this angle persists and becomes slightly sharper. Indeed, we attribute a change in retraction dynamics between $n\leqslant 3$ and $n\geqslant 4$ to this feature. Specifically, when there are 4 or more spokes, the angle between the spokes ${{360}^{\circ}}/n$ is less than the angle α and causes neighboring notches to overlap as they retract. This overlaped region accumulates liquid and becomes thicker, a result that we attribute to the deviation in spreading curves between values of n in figure 5.

The existence of the angled notch in the film is consistent with the retraction speed over the spoke V_s being faster than the rim retraction speed everywhere else V_b. Indeed, we can make an analogy between the wedge in the retracting film and a Mach cone that surrounds a supersonic object. Retraction waves propagate radially outward from the rim of the liquid over the spoke; however this propagation speed V_b is slower than the rim moves on the spoke V_s so that there is a 'Zone of Silence', or stationary film, beyond the angled wedge. Similarly the angle α, as defined in figure 7, can be related to the inverse sine of the velocity ratio as

$$\begin{eqnarray}&&\sin \left(\frac{\alpha}{2}\right)=\frac{{{V}_{b}}}{{{V}_{s}}}\approx \sqrt{1-\frac{a}{h}}.\end{eqnarray} \tag{ 8 }$$

In our experiments, the spoke height relative to the film thickness $a/h\approx 0.57$. Therefore the retraction velocity along the spoke would be expected to be faster than along the bisection by a factor of 1.5, leading to a wedge angle of $\alpha ={{81}^{\circ}}$. This angle is smaller than what is observed in figure 7, a discrepancy that may be attributable to a spoke retraction velocity that is accelerating and not yet reached the predicted retraction speed (figure 5).

Under conditions in which neighboring wedge angles do not contact during retraction ($n\leqslant 3$ in our experiments), the film between the wedges is still intact. By contrast, under conditions in which neighboring wedge angles overlap during retraction ($n\geqslant 4$ in our experiments), the liquid in the overlapped region accumulates into a cylinder (figure 7). We denote the cylinder radius as r_c and the length as the maximum spreading radius between the spokes r_b. Rather than retract along the length, these cylinders break up into smaller droplets from the Rayleigh–Plateau instability.

An important feature of impact on superhydrophobic ridged surfaces is the fragmentation of the initial drop into smaller droplets [13, 14], a process that occurs after the film retracts over the spokes (figure 3). Figure 8 shows a top view of the droplets that leave the surface under the conditions in which the residence time reduction was measured (figure 4). As has previously been documented for superhydrophobic surfaces, the liquid recoils off of a flat surface (n  =  0) as a single drop, two droplets leave when n  =  2 and three droplets leave when n  =  3 [13, 14]. Less expected is the number of droplets produced for the other the geometries. Two smaller droplets accompany a larger drop after breakup on a single spoke (figure 8, n  =  1), and when n  >  3, the number of droplets n_d can far exceed the number of spokes n.

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The images in figure 8 show a striking uniform size distribution of daughter droplets for n  >  1. For the n  =  2 and n  =  3 geometries, the initial drop fragments into the expected one daughter droplet per spoke. These daughter droplets are of approximately equal size. In the n  =  4 and n  =  5 geometries, the daughter droplet count becomes twice the spoke count (figure 9). In the n  =  6 geometry, we observe a tripling effect with three droplets per spoke. Even when the number of droplets doubles or triples, the size of the fragmented daughter droplets are approximately equal size to one another.

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In an effort to understand and predict the doubling and tripling behavior, we develop a scaling argument based on the Rayleigh–Plateau instability. Specifically, the cylindrical geometry of the ligaments that immediately precedes the droplet breakup—as seen in figure 7—will be unstable and can breakup into a series of droplets provided that its length is longer than its circumference [8]. The breakup of finite, retracting ligaments are known to require significantly larger aspect ratios to break-up into more than one droplet [34, 35]. Yet the ligaments that form on the spoked surfaces, appear to be neither significantly expanding nor retracting when they break up (figure 5). Under these conditions, a criterion for the formation and size of droplets would be that the surface energy decreases. We model the ligament before breakup as a cylinder with hemispherical caps and after breakup as multiple equal-sized spherical droplets such that the total volume before and after breakup is conserved. By equating the surface area before and after breakup, we find that the transition between single and double drops per spoke is estimated to occur when the ligament length r_b is 7.46 times radius of the cylindrical ligaments r_c, and the transition between double and triple is estimated to occur when ${{r}_{b}}/{{r}_{c}}=12.04$ (figure 10). Note that due to end effects, these transition values are slightly different than would be estimated using the most unstable wavelengths from linear stability analysis for an infinitely long cylinder [36, 37].

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The aspect ratio $\Gamma ={{r}_{b}}/{{r}_{c}}$ of the cylindrical ligaments prior to breakup (figure 7) can be predicted based on their length r_b and the number of spokes n. From mass conservation, the volume of the initial drop is equal to the collective volume of the n cylindrical ligaments

$$\begin{eqnarray}&&\frac{4}{3}\pi {{R}^{3}}=n\pi r_{c}^{2}{{r}_{b}}.\end{eqnarray} \tag{ 9 }$$

The aspect ratio of the liquid cylinder can thus be estimated as

$$\begin{eqnarray} &&\Gamma =\;\frac{{{r}_{b}}}{{{r}_{c}}}=\sqrt{\frac{3}{4}n{{\left(\frac{{{r}_{b}}}{R}\right)}^{3}}}.\end{eqnarray} \tag{ 10 }$$

The spreading radius between the spokes r_b/R is reported in figure 5 for different values of n. When $n\geqslant 4$, the corresponding spreading radius increases to approximately r_b/R  =  3.5, and the aspect ratio is predicted to follow $\Gamma =5.7\sqrt{n}$. This curve, along with the drop doubling and tripling transition points, is plotted in figure 10. The curve lies in the doubling region when n  =  4 and the drop tripling region when n  =  5 and n  =  6. We can also calculate the aspect ratio  γ  directly from the experiments, and the averages across ligaments are also plotted in figure 10. These direct measurements have slightly lower values of aspect ratios  γ  than the prediction, which is likely due to our assumptions of the ligament shape. The directly-measured aspect ratios  γ  lie in the predicted doubling region when n  =  4 and n  =  5, and tripling region when n  =  6, consistent with the fragmentation results.